robust control: proceedings of a workshop held in tokyo, japan, june 23 â€“ 24, 1991
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Editors: M. Thoma and W. Wyner
S. Hosoe (Ed.)
Robust Control Proceedings of a Workshop held in Tokyo, Japan, June 23  24, 1991
SpringerVerlag Berlin Heidelberg NewYork London Paris Tokyo HongKong Barcelona Budapest
Advisory Board
L.D. Davisson • A.GJ. MacFarlane" H. K w a k e r n ~ J.L. ~ s e y " Ya Z. Tsypkin • AJ . Vit~rbi
Editor
ISBN 3540559612 SpringerVerlag Berlin Heidelberg NewYork ISBN 0387559612 SpringerVerlag NewYork Berlin Heidelberg
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© SpringerVerlag Berlin Heidelberg 1992 Printed in Germany
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PREFACE
The workshop on Robust Control was held in Tokyo, Japan on July 24 25, 1991. From 10 countries, more than 70 reseachers and engineers gathered together, and 33 general talks and 3 tutorial ones, all invited, were presented. The success of the workshop depended on their high level of scientific and engineering expertise.
This book collects almost all the papers devoted to the meeting. The to pics covered include: Hoo control, parametric#tructured approach to robust control, robust adaptive control, sampleddata control systems, and their ap plications.
All the arrangement for the workshop was executed by a organizing com mittee formed in the technical committee for control theory of the Society of Instrument and Control Engineers in Japan.
For financial support and continuing cooperation, we are grateful to Casio Science Promotion Foundation, and many Japanese companies.
Shigeyuki Hosoe
C O N T E N T S
Out l ine of the W o r k s h o p
P a p e r s
• H.KIMURA (J, J~)Lossless Factorization Using Conjugations of Zero and Pole Ex tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL Mixed H2/H~ Filtering by the Theory of Nash Games . . . . . . . . . . . . . . . . . . . . . 9
• M.MANSOUR The Principle of the Argument and its Application to the Stability and Robust Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
• L.H.KEEL, J.SHAW, S.P.BHATTACHARYYA Robust Control of Interval Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
• B.BAMIEH, M.A.DAHLEH, J.B.PEARSON Rejection of Persistent, Bounded Disturbances for SampledData Sys tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
• Y.OHTA, H.MAEDA, S.KODAMA Rational Approximation of LiOptimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . 40
• M.ARAKI, T.HAGIWARA Robust Stability of Sampled Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
• S.HARA, P.T.KABAMBA Robust Control System Design for SampledData Feedback Systems . . . . . . . 56
• Y.YAMAMOTO A Function State Space Approach to Robust Tracking for SampledData Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
• F.B.YEH, L.F.WEI SuperOptimal HankelNorm Approxmations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
• J.R.PARTINGTON
• R.F.CURTAIN FiniteDimensional Robust Controller Designs for Distributed Parameter Systems: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
• 3.H.XU, R.E.SKELTON Robust Covariance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
VIII
• T.FUJII, T.TSUJINO An Inverse LQ Based Approach to the Design of Robust Tracking System with Quadratic Stability ................................................. 106
• T.T.GEORGIOU, M.C.SMITH Linear Systems and Robustness: A Graph Point of View .................. 114
• M.FUJITA, F.MATSUMURA, K.UCHIDA Experimental Evaluation of H~ Control for a Flexible Beam Magnetic Suspension System ...................................................... 122
• K.OSUKA Robust Control of Nonlinear Mechanical Systems ......................... 130
• A.S.MORSE HighOrder Parameter Tuners for the Adaptive Control of Nonlinear Systems ................................................................. 138
• H.OHMORI, A.SANO Design of Robust Adaptive Control System with a Fixed Compensator .... 146
• S.HOSOE, JF.ZHANG, M.KONO Synthesis of Linear Multivariable Servomechanisms by Hc¢ Control ....... 154
• T.SUGIE, M.FUJITA, S.HARA ]/ooSuboptimal Controller Design of Robust Tracking Systems ........... 162
• M.KHAMMASH Stability and Performance Robustness of ~I Systems with Structured NormBounded Uncertainty .............................................. 170
• S.SHIN, T.KITAMORI A Study of a Variable Adaptive Law ..................................... 179
• M.FU Model Reference Robust Control ......................................... 186
* T.MITA, K.Z.LIU Parametrization of H~ FI Controller and Hoo/H2 State Feedback Control ... 194
i A.A.STOORVOGEL, H.L.TRENTELMAN The Mixed H2 and Hoo Control Problem ................................. 202
• D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL: Nash Games and Mixed H2/H~ Control ................................. 210
• J.B.PEARSON Robust g1Optimal Control .............................................. 218
( J , J ' )  L O S S L E S S F A C T O R I Z A T I O N U S I N G C O N J U G A T I O N S O F Z E R O A N D P O L E E X T R A C T I O N S
Hidenori Kimura Department of Mechanical Engineering for ComputerControlled Machinery, Osaka University
1. Introduction A matrix G(s) in L** is said to have a (J, J')losslessfactorization, if it is represented as a product
G(s)=O(s)rl(s), where O(s) is (J, J')Iossless ]11] and rl(s) is unimodular in rl**. The notion of O, J') lossless factorization was first introduced by Ball and Helton [2] in geometrical context. It is a generalization of the wellknown innerouter factorization of matrices in H"* to those in L**. It also includes the spectral factorization for positive matrices as a special case. Furthermore, it turned out that the (J, J')lossless factorization plays a central role in H"* control theory [1][4][7]. Actually, it gives a simple and unified framework of t l " control theory from the viewpoint of classical network theory. Thus, the (J, J')lossless factorization is of great importance in system theory.
The (J, J')lossless factorization is usually treated as a (J, J')spectralfactorization problem which is to find a unimodular n(s) such that G(s)JG(s) = rl~(s)jrl(s). Ball and Ran [4] first derived a statespace representation of the (J, J')spectral factorization to solve the Nehari extension problem based on the result of Bart et.al. [5]. It was generalized in [ 1] to more general modelmatching problems. Recently, Green et.al. [8] gave a simpler statespace characterization of the (J, J')spectral factorization for a stable G(s). Ball et al. 13] discussed some connections between the chainscattering formulation of H** and (J, J')lossless factorization.
In this paper, we give a simple derivation of the (J, J')lossless factorization for general unstable G(s) based on the theory o f conjugation developed in [9]. The conjugation is a simple operation of replacing some poles of a given rational matrix by their mirror images with respect to the origin by multiplication of a certain matrix. It is a statespace representation of the NevanlinnaPick interpolation theory [ 11] and the Darlington synthesis method of classical circuit theory. The (J, J')lossless factorization for unstable matrix, which is first established in this paper, enables us to solve the general I1 °° control problem directly without any recourse to Youla parameterization [12].
In Section 2, the theory of conjugation is briefly reviewed. The (J, J')lossless conjugation, a special class of conjugations by (J, J')lossless matrix, is developed in Section 3. Section 4 gives a frequency domain characterization of the (J, J')lossless factorization. Section 5 is devoted to the state space representation of the (J, J')lossless factorization. The existence condition for the (J, J ')lossless factorization is represented ill terms of two Riccati equations, one of which is degenerated in the sense that it contains no constant tema. Notations :
C(sIA)" B+D := [A~1 R(s) = RT(.s), R'(s) = RT(~), tC IDJ,
~A) ; The set of eigenvalues of a matrix A.
RU:x r ; The set of proper rational nmtrices of size mxr without poles on the imaginary axis,
Rllmxr; The set of proper stable rational matrices of size m×r.
2. Con juga t ion The notion of conjugation was first introduced in [9]. It represents a simple operation which replaces
the poles of a transfer function by their conjugates. The conjugation gives a unified framework for treating the problems associated with interpolation theory in state space [10]. In this section, the notion of conjugation is generalized to represent the relevant operations more naturally.
To define the conjugation precisely, let
G(s) =[ A~DB I R.. De Rr , A ~ ( 2 . t )
be a minimal realization of a transfer function O(s).
Definition 2.1 Let a(A) = A1UA2 be a partition of o'(A) into two disjoint sets of k complex numbers At and nk complex numbers A2. A matrix e(s) is said to be a AFconjugaror of G(s), if the poles of the product
G(s)O(s) are equal to {A l} UA2 and for any constant vector ~, ~ 0,
~(sl  A)'[email protected](s) #0 ( 2 . 2 )
for some s, where At is a set composed of the elements of At with reversed sign. As a special case, we can choose At composed of the RHP elements of o(A). If A has no eigenvalue
on the jo~  axis. ^l conjugator of G(s) stabilizes the product G(s)e(s). In that case, Alconjugator is called a stabilizing conjugator. In the same way, we can define an antistabilizing conjugator.
Remark. The condition (2.2) is imposed to role out the trivial case like e(s)  0. It is automatically satisfied for the case where e(s) is right invertiblc. T H E O R E M 2.2 e(s) is a Atconjugator of G(s) given by (2.4), if and only if O(s) is given by
t C. I I J (2.3)
where Co, De, X, are such that X is a solution of the Riccati equation
XA + ATX + XBC.,X = 0 (2.4)
such that A~ := A + BC,X satisfies a(AO = {At } ,..) A2 and ( A , , BD, ) is controllable. In that case, the product G(s)e(s) is given by
G(s) A(s)=[ A + B C¢ X~~] Dc c ~ ~ co x IE, J (2.5)
(Proof) From the definition, the Amatrix of the product G(s)O(s) has eigenvalues in cI(A)uo'(AT). Hence, it can be assumed, without loss of generality, that the Amatrix of O(s) is equal to A T. Hence, we put
 t C ~ I Do J (2.6)
for some B c C¢ and De. Then, the product rule yields
G(s)O(s) = 0 A T Be (2.7)
C DC~ DDc.
Since At c o(A T) and A2 c o(A), we have
BC_~ Ml M2 Mt M2 At 0
0 A T NI N2 Nt N2 0 a~ (2.8)
for some MI , M2, Nt and N2 such that 6(AI)= {At } u A2 • Assume that O(s) is a hi.conjugator. Since the modes associated with Alr are uncontrollable in
(2. I0), we have
for some BL. From (2.8), it follows that AMI + BCdN! = MIAI, ATNt = NtAI. (2.10)
Hence, we have (sl A) Mi  BC.~I = Mt (sl Al) (2.1 1)
3
Premultiplication of (2.11) by (sl  A) "t and postmultiplication by (sI  A)aa l yield
Idt (sI  Al) "1Bt = (st  A) "l BO(s). (2.12)
Here, we used the relation N~ (sl  At) "~ B~ = (sl + A'r) "t B,, which is due to (2.9) and the second relation of (2. I0). The condition (2.5) implies that Mt is nonsingular and (At, 130 is controllable.
Taking X=NIMI "l in (2.10) yields (2.4). Since MIIAcMI=AI, A c should be stable. It is clear that (^¢, BDc)=(MflAtMt, MIBI) is controllable because (Ab BI) is controllable.
Due to (2.9), we have Be=NIBI=NIMI'IBDc=XBDc, from which (2.3) follows. The derivation of (2.5) is straightforward.
Remark. Theorem 2.2 implies that the conjugator of G(s) in (2.4) depends only on (A, B). Therefore, we sometimes call it a conjugator of (A. B).
Remark It is clear, from (2.7), that the order of O(s) given in (2.6) is equal to the rank of X which is equal to the number of elements in Lt.
3. (J, J ' ) Loss l e s s C o n j u g a t o r s
A matrix O(s)~ RL" is said to be (J, J')unitary, if tm*r)x(p*q)
O(s) J O (s) = J' (3.1)
for each s. A (J, J')unitary matrix e (s ) is said to be (J, J')tossless, if O'(s) J O (s) g J', (3.2)
for each Re[s]>0, where
j = [ I,, 0 ] j ,=[ I~ 0 ] m>_p, r_>q (3.3) 0 I, , 0 I~ ,
A (J, J')lossless matrix is called simply a Jlossless matrix.
A conjugator of G(s) which is (L J')lossless is called a (J, J')lossless conjugator of G(s). The existence condition of a Jlossless conjugator, which is of primary importance in what follows, is now stated. T H E O R E M 3.1. Assume that (A, B) is controllable and A has no eigenvalue on the jc0axis. There exists a (J, J')lossless stabilizing (antistabilizing) conjugator of (A, B), if and only if there exists a nonnegative solution P of the Riccati equation
PA + ATP  PBJB'rp = 0 (3.4)
which stabilizes (antistabilizes) A, : A BJB'rp. In that case, the desired (J, I')lossless conjugator of G(s) and the conjugated system are given respectively by
L JB T [ I I , C  DJBTp D ( 3 . 5 )
where Do is any matrix satisfying D~J D, = J'. (ProoJ) Since (2.4) implies X(sl + A T) = (sl  Ac)X, O(s) in (2.3) is represented as
O (s) = Oo (s) De,
O 0 ( s ) = I + C c X ( s I  A c ) "j B, A c = A + B C c X . (3.6)
Assume that P is a solution of the Lyapunov equation P a , + AcT P+ X "r c,'rJ Co X =0 . (3.7)
Then, it follow that
O~ (s) J O0(s) = J + BT (sIAcV) a (XrC~ J + P B) + (J C¢ X + B'r P) ( s t  A¢) "~ B.
Since ~ is stable and (A¢, B) is controllable from the assumption, O(s) is Junitary if and only if J C, X + B r I, = 0. (3.8)
In that case, we have O~(s)JOu(s) = J (s + ~BT(sl  A~)"P(sl  Ac)4B
Hence, in order that Oo(s) is JIossless, P should be nonnegative. From (3.8), (3.7) is written as (3.4) and
O0(s) in (3.6) can be written as O0(s) = I  Jffr(sl + Ar)'IPB, from which the representation (3.5) follows. Now, we shall show a cascade decomposition of the (J, J')Iossless conjugation of (A, B) according
to the modal decomposition of the matrix A. Assumethat the pair (A, B) is of the form
As! A,.,
The solution P of (3.4) is represented in conformable with the partition (3.9) as p__. [ P,, PIa]
PITS P22 . (3.10)
L E M M A 3.2. If the pair (A, B) in (3.9) has a (J, J')lossless conjugator O(s) of the form (3.5), then O(s) is represented as the product
O(s) = O,(s) OKs) (3.11)
of a Jlossless conjugator Ol(s) of (At t, Bt) and a (J, J')lossless conjugator 02(s) of (An, 82) with Bz=Ba+Ptff'hB, , where P~a is a peudo inverse of P,, in (3.10). Actually, e~(s) and OXs) are given respectively as
[J B; ' I 4 B; ' ( 3 . 1 2 )
where ~ = PH  P ~ { ~ a and D is any constant (J, J')unitary matrix. (ProoJ) Since P ~ 0, we have Pt2(I  P~2Pza) = 0. Therefore,
UPU'r =[ A0 Paa0 ] , (U.I)'rB =[_~] . U=[ 10 "PtalP~ ]
From this relation and (3.4), it follows that
0 Pal Aal A~ 0 A T 0 P~ 0 P22 0 P22 (3.13)
where Azz = Aat + P~aPTaAn  A~zl'~ff'rz. Hence, t~AH + A~IA  AB~]B TA = 0. Since ~ ~ 0, Or(s) in (3.12) is a J lossle_ss conjugator of (A,. B~). Also, it is clear that Oz(s) given in (3.12) is a (J, J')lossless conjugator of (Aa:. B9. Since (3.11) implies that Az~ = BzlB TA, we have
e~(s)O~s) = 0 A~ Pzalh 19 = = e(s).
sa~ J~I I t  jdu ' The proof is now complete.
4. (J, J')Lossless F a c t o r i z a t i o n s
A matrix G(s) e RL~',,. o x c**.) is said to have a (J, J')losslessfactorization, if it can be represented as
G(s) = O(s) l'I(s), (4.1)
where O(s)¢ RL~'m,,r)x(v+*) is (J, J')lossless and rI(s)is unimodular in H ' , i.e., both rI(s)and lI(s) "* are in
RH~,,0,, c,,o • It is a generalization of the innerouter factorization of H" matrices to L" matrices.
Since rI(s) is unimodular, it has no unstable zeros nor poles. Therefore, O(s) should include all the unstable zeros and the poles of G(s). The (J, J')lossless conjugation introduced in the previous section gives a suitable machinery to carry out these procedures of "zero extraction" and "pole extraction."
If (4.1) holds, it follows that
rlCs)"J'e(s) J G(s) = lp~. (4.2)
This implies that G(s) should be left invertible. Hence, there exists a complement G'(s) of G(s) such that
[G(s) G'(s) l : = [G'(s) ] LoJ'(s)] (4.3)
exists. Obviously, G*(s) is a left inverse and G'(s) is a left annihilator of G(s), i.e., G'(s) G(s) = I~, s , G'(s) G(s) = 0. (4.4)
The following result gives a method of carrying out a (J, J')iossless factorization based on sequential conjugation of zeros and poles extraction.
T I I E O R E M 4.1. (J, J')lossless factorization of G(s) exists, if and only if the left inverse G*(s) of G(s) has a Jlossless stabilizing conjugator e l ( s ) such that G~(s)Je(s) has a (J, J')iossless antistabilizing
conjugator O2(s) such that GJ(s )e , ( s )e2(s ) :O (4.5)
(ProojO To prove the necessity, assume that G(s) allows a (J, J')lossless factorization (4.1). Ciearly, we have
G(s) J rr'(s)J' Therefore, e(s) is a (J, J')Iossless stabilizing conjugator of G+(s), as well as a (J, J')iossless anti stabilizing conjugator of G~(s)J. Due to Lemma 3.2, e(s) can be factored as (3.11) where e l ( s ) is stable J lossless matrix and e2(s) is an antistable (I, J')lossless matrix. From the wellknown property of (1, I') lossless matrix, O2(s) has no unstable zero. From
G(s )=e t(s)e2(s)rl(s), (4.6)
6+(s)et (s) = Ilt(s) satisfies nt(s)e2(s)n(s) = I. Since e2(s)rI(s) has no unstable zeros, rlt(s) is stable. Therefore e t (s ) is a stabilizing conjugator of G+(s). Since G~(s)Jet(s)e2(s) =rI~(s)J ' , and rl(s)J is anti stable, e2(s) is a (J, J ')dossless antistabilizing conjugator of G~(s)Jel(s). The relation (4.5) is obvious from (4.6). Now, the necessity of the theorem has been proven.
Assume that there exist Or(s) and O2(s) satisfying the condit ions of the theorem. Let lql(s)=G+(s)et(s) and lq2(s) = G~(s)JOt(s)O2(s). From the assumption, lIt(s) is stable and l12(s ) is anti stable. Since e l (s ) is Jlossless, e l (s )Je l (S)=J (see[6]). Therefore, we have lqt(s)JOl~(S)JG(s) =I. Since lIt(s) is stable, G~(s)JOI(S) has no stable zero. Since conjugator does not create zero, rlz(s) has no stable zero. Therefore, 112~(s) is unimodular.
Finally, we shall show that the relation (4.1) holds with e(s)=et(s)o2(s) and lI(s)=J'rlz(s). Due to (4.5), Ot(s)O2(s)=G(s)U(s) for some U(s). Since Fl(s) = e2(s)e](s)JG(s), we have II2~(s)U(s) = J'. The assertion follows immediately.
Based on the above theorem and its proof, we can state an algorithm of (J, J')lossless factorization.
[Algor i thm of (J, J ' )  Ioss less Fae to r i za t ion ] Step 1. Find a Jlossless stabilizing conjugator el(s) of G+(s). Step 2. Find a (J, J')lossless antistabilizing conjugator O2(s) of G~(s)Jet(s) satisfying (4.5). Step 3. Put o(s)=ot(s)e2(s) and II(s)=J'O~(s)JG(s), which give a (J, J31ossless factorization.
In the above algorithm, Ol(s absorbs all the unstable zeros of G(s). Hence, Step 1 is regarded as a procedure of zero extraction. Similarly, O2(s) in Step 2 absorbs all the unstable poles of G(s), and hence, Step 2 is regarded as pole extraction.
5. StateSpace Theory of (J, J')Lossless Factorizations In this section, we derive a statespace forms of (J, J')lossless factorizations. Let
G(s)=[ A ~ D B ] eR nx,,, Rem*r)x(p*q) A D~ (5.1) be a statespace form of G(s). The following assumptions are made :
(A0 The matrix A has no cigenvalucs on the jo)axis.
(A2) The matrix D is of full column rank.
(A3) G~(s)JG(s) admits no unstable polezero cancellation.
Due to the assumption (A2), there exists a column complement D' of D such that
[D D, ]I =[ D* D" ] (5.2)
exists. Obviously,
D'D = Imq D'D = 0. (5.3)
Now, we assume that O(s) has a (J, J')lossless factorization (4.1). Since G(s)JG(s)=rI(s)Jn(s), we have
D T J D = D 'r Z' D, (5.4)
where Dx = rl(~,) e R ~p'q)x~p'q). Since Il(s) is unimodular, D~ is nonsingular. Hence, DTjD must b~ non singular.
The left inverse G+(s) and a left annihilator Gl(s) of G(s) satisfying (4.6) are represented in the state space as
[O'(s)J D~ (5.5) D.tC
whcrc U is an arbitrary matrix which is detcrmincd latcr. Now, wc shall carry out the (J, J')losslcss factorization of G(s) based on thc algorithm dcrivcd in
Section 4.
Step l (Zero Extraction) Due to Theorem 3.1, a Jlossless stabilizing conjugator Or(s) of G+(s) in (5.5) is calculated to be
] [ JL 'r [ 1 ], (5.6)
where X is a nonnegative solution of the Riccati equation
X(A LC) + (A LC)TX  xLTjL'rX = 0 (5.7) which stabilizes
At := A L(C + jL'rX) (5.8i
Step 2 (Pole Extraction) Straightforward computation using LD = B yields
Ar 'r I I C'rj+XL] G(s)JO,(s) = [ ~ ] O,(s) = "At [ B "r I DTJ ] .B v DTj , (5.9)
Due to Theorem 3.1, a (J, J')lossless antistabilizing conjugator of (5.9) is given by
O.2(s)_[ A ]Y(C'rj + XL) ] DDk ' LcJL'rX[ I (5.10)
A T 0
r, o o l r , o xl o / /o , ° t
LOY I J LO 0 I J ,
where Dr is any matrix satisfying (5.4) and v is a nonnegative solution of the Riccati equation
VA T + AY + Y(c'r+ XLJ) J (C + jLTx)y = 0 (5. I 1) which stabilizes
Az := A + Y(cT+ XLI) J (C + JLTX), (5.12)
NOW, concatenation rule yields
O(s) = 8Ks)O~s) = 0 A y(CTJD+XB) Dn (5.I3)
JL T CJLTX) D
We use the free parameter U in (5.5) to satisfy (4.5). From (5.5), it follows that
G±(s)B(s) = [ At I (I+YX)B+yCTjD ] D't(C+JL'rX ") 0 .
Assumption (A3) implies that ( A1, (I+YX)B+yCTjD) is controllable. Hence, (4.5) holds, if and only if
Dt(C+jLTx) = 0. (5.14)
Thus, the free parameter U in (5.5) must be determined such that (5.14) holds. The relation (5.14) holds, if and only if
LTX = J (DF+ C) (5.15)
for some matrix F. Since LD=B for any choice of U, we have
F= (DTjD)'I(BTx + DTjc). (5.16)
Straightforward computation using (5.15) and (5.16) verifies that the Riccati equation (5.7) is written as
XA + ATx  (cTjD+XB)(DTjD)'I(DTJC+BTX)+cTjc = 0. (5.17)
The matrix At given in (5.8) is calculated to be
Al = A +BF. (5.18) Also, the Riccati equation (5.11) and A2 in (5.12) are represented respectively, as
YAT+AY+yF'rDTJDFY = 0 , A2 = A + yFTDTJDF (5.19)
Step 3. Using (5.7), we can write the statespace form of O(s)=Ol(s)O2(s) in (5.13) as
[ A, BF B ] ! A yFTDTJD [ Dk! O(s) = 0 _ • ( 5 . 2 0 )
L(C+DI) DF D j
where Dg is any matrix satisfying (5.4). It remains to compute the unimodular factor rl(s) = J'O(s)JG(s). The concatenation rule yields
(C+DF)Tjc .(C+DF)'rjc FTDTJC FTDTJD
A B
we obtain
Fl(s) = D" [ ~ IB'YFITDTJD 1. (5.21)
Now, we can state the main result of this paper. THEOREM 5.1. Assume that G(s) given in (5.1) satisfies Assumptions (AI)(A2)(A3). Then, G(s) has a (J, J')losslcss factorization, if and only if there exist a nonsingular matrix D= ~ R ~'e'L''e satisfying (5.4), a solution × > 0 of the Riccati equation (5.17) which stabilizizes A I given in (5.18)(5.16) and a solution Y 2_ 0 of the Riccati equation (5.19) which stabilizes A2 given in (5.19). In that case, a (J, J')losslcss factor e(s) is given in (5.20) and a unimodular factor rl(s) is given in (5.2t).
The case of stable G(s) is much simpler. The result is exactly the same as was obtained in [8]. COROLLARY 5.2. Assume that A is stable in addition to (AI)(A2) and (A3). Then, G(s) has a (J, J') lossless factorization (4.1), if and only if there exists a nonsingular matrix Dx ~ R t* '¢° '¢ satisfying (5.4) and the Riccati equation (5.17) has the solution x > 0 which stabilizes Al given in (5.18).
If the above condition hold, the factors O(s) and rl(s) of (4.1) are given respectively by
'1. C+DFI DJ , LFIIJ . (5.22)
(Prooj~ IrA is stable, we can take Y0 in solving (5.19). The assertions of the corollary follow immediately from (5.20) and (5.21) for Y=0.
6. Conclusion A necessary and sufficient condition for the existence of (J, J')lossless factorization in RL '~' has
been established in the state space based on the method of conjugation. It includes the result of [8] for RH** as a special case. The derivation is simple, exhibiting the power of conjugation method.
The result obtained in this paper is directly applied to !1 ¢~ control problems without any recourse to Youla parameterization, giving a simpler result for both standard and nonstandard plants. This will be published elsewhere.
REFERENCES [ 11 J.A. Ball and N. Cohen, "The sensitivity minimization in an H** norm : Parameterization of all optimal solutions," Int. J. Control, 46, pp.785816 (1987) [21 J.A. Ball and J.W. Hehon, A BeurlingLax Theorem for the Lie group U(m, n) which contains most classical imterpolation," J. Operator Th., 9, pp. 107142 (1983) [3] J.A. Ball, J.W. Hehon and M. Verma, "A factorization principle for stabilization of linear control systems," Reprint. [4] J.A. Ball and A.C.M. Ran, "Optimal Hankel norm model reduction and WienerHopf factorization I : The canonical case,'SIAM J. Contr.Optimiz., 25, pp.362382 (1987). [5] H. Ban, I. Gohberg, M.A. Kaashoek and P. van Dooren, "Factorizations of transfer functions," SlAM J. Contr. Optimiz. 18, pp. 675696 (1980). [61 P. Dewilde, "Lossless Chainscattering matrices and optimum linear prediction : The vector case," Circuit Th. & Appl., 9, pp.135175 (1981). [7] B.A. Francis, A Course in Ho, Control, Springer, New York (1987). [8] M. Green, K. Glover, D. Limebeer and J. Doyle, "A Jspectral factorization approach to H,o control," SlAM J. Contr. Optimiz., 28, pp.13501371 (1990). [9] H. Kimura, "Conjugation, interpolation and modelmatching in Hoo ," Int. J. Control, 49, pp.269307 (1989). [ 10] H. Kimura, "State space approach to the classical interpolation problem and its applications, "in Three Decades of Mathematical System Theory, H. Nijmeijer and J.M. Schumacher (eds.), SpringerVerlag, ,pp. 243275(1989) [11] H. Kimura, Y. Lu and R. Kawatani, "On the structure of H,o control systems and related extensions," IEEE Trans. Auto. Contr., AC36, pp. 653667 (1991). [12] H. Kimura, "Chainscattering formulation of H,,* control problems," Proc. Workshop on Robust Control, San Antonio (1991).
Mixed H2/H Filtering by the Theory of Nash Games
1D.J.N. Limebeer 2B.D.O. Anderson 1 B. Hendel
Department of Electrical Engineering, I Department of Systems Engineering, 2 Imperial College, Australian National University, Exhibition Rd., Canberra, London. Australia.
A b s t r a c t
The aim of this paper is to study an H2/Hoo terminal staSe estimation problem using the classical theory of Nash equilibria. The H2/Hoo nature of the problem comes from the fact that we seek an estimator which satis fies two Nash inequalities. The first reflects an Hoo filtering requirement in the sense alluded to in [4], while the second inequality demands that the estimator be optimal in the sense of minimising the variance of the termi nal state estimation error. The problem solution exploits a duality with the H2/Hoo control problem studied in [2, 3]. By exploiting duality in this way, one may quickly cxtablish that an estimator exists which staisfies the two Nash inequalities if and only if a certain pair of cross coupled Riceati equations has a solution on some optimisation interval. We conclude the paper by showing that the Kalman filtering, Hoo filtering and H2/Hoo fil tering problems may all be captured within a unifying Nash game theoretic framework.
1 I n t r o d u c t i o n In this paper we seek to solve a mixed H2/Hoo terminal state estimation problem by formulating it as a two player nonzero sum Nash differential game. As is well known [1, 5], two player nonzero sum games have two performance criteria, and the idea is to use one performance index to reflect an Hoo filtering criterion, while the second reflects the H2 optimality criterion usually associated with Kalman filtering. Following the precise problem statement given in section 2.1, we reformulate the filtering problem as a deterministic mixed H~/Hoo control problem in section 2.2. This approach offers two advantages in that the transformation is relatively simple, and we can then use an adaptation of the mixed H~/Hoo theory given in [2] to derive the necessary and sufficient conditions for the existence of the Nash equilibrium solution to the derived control problem, and therefore for the existence of optimal linear estimators which solve
10
the filtering problem. These conditions are presented in section 2.3 where we also derive the dynamics of an online estimator. Some properties of the resulting mixed filter are given in section 2.4. The aim of section 2.5 is to provide a reconciliation with Kalman and Hoo filtering. In particular, we show that Kalman, Hoo and Hz/Hoo filters may all be captured as special cases of a another two player Nash game. Brief conclusions appear in section 3.
Our notation and conventions are standard. For A E C "nxm, A ~ denotes the complex conjugate transpose, g{.} is the expectation operator. We denote I]RII21 as the operator norm induced by the usual 2norm on functions of time. That is :
IIRII21 = sup [IRu[[2
2 The H2/H~ filtering problem
2 . 1 P r o b l e m S t a t e m e n t
We consider a plant of the form
k(t) = A(t)z(t) + B(t)w(t) x(to) = zo ( 2.1 )
with noisy observations
zt(t) = Cl(t)z(t) ( 2 .2 ) ~ ( t ) = c2(t)x(t) + n~(t), ( 2.3 )
in which the entries of A(t), B(t), C1 (t) and C2(/) are continuous functions of time. The inputs w(t) and n2(t) are assumed to be independent zeromean white noise processes with
~{w(t)w'Cr)} = Q(t)6(t  ~') ( 2.4 ) E{n2(t)n2'Cr)} = R2Ct)6(t 7) Vt ,r e [t0,tfl. ( 2.5 )
The matrices Q :> 0 and R~ > 0 are symmetric and have entries which are continuous functions of time.
Our final assumption concerns the initial state. We let z0 be a gaussian random variable with
~{x0} = m0, c{(~0  m0)(~0  m0)'} = P0. ( 2.8 )
Finally, we assume that z0 is independent of the noise processes, i.e.
£{z0w'Ct)} = 0, ~{~0n~(t)} = 0; V t e [t0,t:].
The estimate of the terminal state z( t l ) is based on both observations zx(t) and z2(t), and the class of estimators under consideration is given by
~t~ ! ~(tj) = [MI(~, ts)~l(~) + M2(~,tf)z2(~)] aT (2.7)
where Mi (r, ty) and M2(r, if) are linear timevarying impulse responses.
11
The filtering problem is formulated as a twoplayer differential Nash game. The first player has access to the observation zi(t) and tries to maximise the variance of the estimation error by choosing an M1 (.,t/) which minimises the Hoo filtering cost function [4]
JI(M1, Mr) = CIIMdl ]  ~ {[=(tD  ~(tl)1'[=(~1)  ~(ts)]}. (2.8)
7rllMt Ill is a penalty term introduced to prevent player one from assigning an arbitrarily large value to M1 thus driving J1 to minus infinity. The second player has access to the observation z2(t) and attempts to minimise the error variance by selecting an Mr which minimises the Kalman filtering payoff function
Jr(Mr, Mr) = ~ {[x(t/)  fc(t/)]'[z(t/)  ~:(t/)]} . ( 2 . 9 )
We therefore seek two linear estimators M~" and M~ which satisfy the Nash equilibria
JI(M[,M~) < JI(Mt,M~) (2 .10 )
J2(M[,M~) <_ J2(M~,M2). ( 2.11 )
2 . 2 P r o b l e m R e f o r m u l a t i o n
In order to solve the stochastic filtering problem we transform it into a deterministic control problem. This approach has two advantages in that the reformulation is relatively simple, and we can then use an adaptation of the mixed Hr/Hoo theory [2] to derive the necessary and sufficient conditions for the existence of the Nash equilibria M~' and M~.
We begin by introducing the timevarying matrix differential equation [4]
Z(t, tl) = A'( t)Z(t , tl) + C~(t)Ml'(t, tl) + C~(t)M2'(t,tl) (2.12)
with Z(lc,t/) = I. The aim is to rewrite the cost functions (2.8) and (2.9) as quadratic performance indices involving Z(.), MI(.) and Ms(). By direct calculation
d (Z(,, t j)',(,)) dt
I I I I = [  A ' Z + C I M t ' + C r M 2 ] (t)x(t)
+ Z(t,tl)'[m(t)z(t) + B(t)w(t)] = Ml(t,lf)zl(t) + Mr(t, tl)[zr(t)  n2(t)]
+ Z(t,t+)'B(t)w(t).
Integrating over [to,if] gives
=(ty)  z ' ( to, ty)~o = { M l z l + M2[z2  n2] + Z 'Bw} (Odt.
Rearranging and using (2.7) yields
fi' • ( t l )  ~ (~I ) = z ' ( t o , t . l ) ~ o  { M ~ , ~  Z ' B ~ , } ( t ) d t .
12
In the next step we square both sides and take expectations making use of the assumed statistical properties of the noise processes and the initial state. This gives
f + z'(to,ts)poZ(to,~1). ( 2.13 )
We can now use (2,13) to rewrite J1 and ,/2 as
= ~/'s(7~MIM 1' M' Z'BQB'Z)(t)dt} J, tr  M2R2 2  k ~o
 ~r{Z'C~o,t~)PoZC~o,tj)} ( 2a4 )
k J t o
+ fr{Z'(to,tf)PoZ(to,t])}. ( 2.15 ) Notice that both payoff functions are now deterministic.
The filtering problem has thus been transformed into the mixed H2/Hoo control problem of finding equilibrium solutions MI" and M~ which satisfy
JI(M~,M~) <_ J,(MI,M~) ( 2 . 16 )
J2(M{,M{) < J2(M~,M2) ( 2.17 )
where J1 and J~ arc given by (2.14) and (2.15); the matrix Z(t) is constrained by (2.12).
R e m a r k s (i) Notice that the "state" Z(Q is matrix valued and the 'dynamics ' have a terminal condition Z(tl) = I. (it) The optimal linear estimators M~ and M~ are matrix valued too. (iii) The initial condition data in the form of Z'(to)PoZ(to) appears in the cost func tionMs.
2 . 3 F i n d i n g a n o n  l i n e e s t i m a t e
The necessary and sufficient conditions for the existence of Nash equilibrium solutions to the mixed H2/Hoo control problem, and therefore for the existence of linear estimators M~" and M~ follow directly from [2], and are presented in the following theorem. Scaling arguments have allowed us to assume without loss of generality that R2 = Q = I .
T h e o r e m 2.1 Given the system and observations described by (2.1) to (2.3), there exist transformatio~,s M~ and M~ which satisfy the Nash equilibria (2.16) and (2.17) with Jx and J2 defined in (2.14) and (2.15) if and only if the coupled Riecati differential equations
/'I(L) = AP,(t) + PI ( t )A '  BB'
 [P l ( t ) P ~ ( , ) ] l 7~C[C1 C~C2 Pl(t)
P2(t) = mP2(t) + P2(t)A'+ BB'
13
with Pt(to) = Po and P~(to) = Po, have sotu.ons Pt( t ) <_ O, e~(t) > 0 on (2.18) and (2.19) have solutions,
7 2Z' ( t , tl)P~(t)C I Z ' ( t , t l ) P z ( t ) C ~.
M~( t , t s ) =
[to,~sl  ~'/'
( 2.20 ) ( 2.21 )
Since Z ( t , t l ) = Z~(t l , t ) , we have
d~I Z(L,tj)= d~Z~(t~,,).
 z ~(t:,t) ~~, z(tj,t)z~(,1, t)
= Z t ( t l , t ) (A '  72C~01P1  C~C~P2)(tl),
and then (2.25) :=>
~ Z'(t, is) = ( A  72 ptc[ c1  P~C~C2)(Q )Z'(t, tl).
Now, from (2.7)
/,i' Differentiating with respect to t! and using (2.26) we obtain
or equivalently
= (A  72P1C[C1  PzC~C2)(tI)~(tl)
+ (v"P,C'~zl + P2C~z2)(t:) e ( to ) = ,no
 K,(t l )[C~(~l)~Ctl) zl(e/)]
 z<2Cts)[C~(t f )~( t f )  z~(tl)]
KI(f / ) = 72Pi(ll)Ci(tI) ( 2.29 )
K~(ts) = P~(t:)C~(tf) . ( 2.30 )
The matrices Pl( t l ) and P2(t/) satisfy the coupled Riccati equations (2.18) and (2.19).
(2.28)
(2.26)
( 2.27 )
= ~o~(tj) ( 2.22 ) = P~(t~). ( 2.23 )
With a little manipulation we can now find an online estimate for z( t l ) based on the observations zl(tl) and z2(tI), in the same sense that the Kalman filter provides an on line state estimate using current observations. Implementing the equilibrium strategies in (2.12) gives
Z(t, t]) = ( A ' + 7~PIC~C1 + P2C~C2)Z(t,II) Z ( t l , t l ) = I. ( 2.24 )
14
2.4 T h e m i x e d H 2 / H c ¢ f i l ter In this section we show how the H2/He¢ filter is defined in terms of the above analysis, and we can present some of its properties. Suppose M~ is replaced by a dormant player as follows : Calculate M[ and M~ as above, and then in (2.27) replace the term multiplying zl by zero, i.e. we take M~ = 0 • M~. This means that in (2.28) we replace g~(t l ) by zero to obtain
~ ( t / ) = [A(tl) K~(t/)C~(~/)l~(t D + g2(*1)C~(tl)z~(tS); ~(t0) = ITI0}
which is the defining equation of the H2/Hoo filter. It is easy to see from this equation that the tI2/Hoo filter has thc usual observer structure associated with Kalmaa and Hoo filters. The following result provides an upper bound on the variance of the terminal state estimation error, and it shows that if the white noise assumption on the disturbance inputs w and n2 is dropped, the energy gain from the disturbances to the generalised estimation error Ct(~:(tl)  z( tI) ) is bounded above by 7 for all w, n2 E £2[to, tl].
T h e o r e m 2.2 If M1 ~ O,
(i) the online state estimate is generated by
d~l~:(tf) = (A  P2C~C2)(t/)~(tf) + P2(tl)C~(tl)z2(tl) ~(to) = ( 2.31 ) mo~
(it) the covariance of the estimation error e(tl) = ~(tl)  z ( t l ) satisfies
£{e(tf)e'(t /) } <  P l ( t l ) ( 2.32 )
and (iii) when mo = 0, Ilnll~ < ~ for all w, n2 E £z[t0,tl], where the operator T"g is
described by
z = C1e(tI) (2 .34 )
2.5 R e c o n c i l i a t i o n o f K a l m a n , H ~ a n d H 2 / H ~ fil t e r i n g
Here we establish a link between Kalman, H~ and mixed H2/H~ filtering. Each of the three theories may be generated as special cases of the following nonzero sum, twoplayer Nash differential game:
Given the system described by (2.1) to (2.3) and a state estimate of the form given in (2.7), we seek linear estimators M~' and M~ which satisfy the Nash equilibria JI(M~, M~) < JI(MI, M~) and J2(M~, M~) < J2(M~, M2), where
J2 = ~: {[z(tf) ~(tl)]'[z(ts) ~(tj)]} p211Mdl ~.
15 This game can be reformulated as a deterministic control problem and is solved
by M~(t,tl) = 72Z'(t,t/)Sl(t)C~ and M~(t,~l) = Z'(t , t l)S2(t)C ~, where Z(t , t l ) is generated by (2.12) and SI(t) and $2(/) satisfy the coupled Riccati differential equations
Sl(t) = A S t ( t ) + S t ( t ) A '  B B '
 [ s ct) s (l) ] 1 72qC, C C2 St(t)
s2(t) $2(~) = AS2(t) + S=(t)A' + BB'
 [ s,(t) s=(t) ] / p2740~C * 72C',C, C~C2 ][ S2(t)] 7~C~C1 S,(t) L
with Sl(t l) = S2(tl) = O.
(i) The Kalman filtering problem is recovered by setting p = 0 and 3' = oo, in which case S l ( t ) = S2(t) = Pc(t). The matrix Pe(~) is the solution to the usual Kalman filter Riccati equation. (ii) The pure Hoo filtering problem is recovered by setting p = 3'. Again, S , ( t ) = S2(t) = P~(t). The matrix Poo(t) is the solution to the Hc¢ filtering Riccati equation [4]. (iii) The mixed H2/Hoo filtering problem comes from p = 0.
3 C o n c l u s i o n s We have shown how a mixed H2/Hoo filtering problem may be formulated as a deter ministic mixed II2/Hoo control problem. The control problem can then solved using the Nash game techniques explained in [2]. The necessary and sufficient conditions for the existence of a solution to the filtering problem are given in terms of the existence of solutions to a pair of crosscoupled Riccati differential equations. We have derived an upper bound on the covariance of the state estimation error in terms of one of the solu tions to the coupled equations and have shown that, under certain conditions, the energy gain from the disturbance inputs to the generalised estimation error Cl (x ( t / )  ~(t/) ) is bounded above by 7
R e f e r e n c e s [1] T. Basar and G. 3. Olsder,"Dynamic noncooperative game theory," Academic Press,
New York, 1982
[2] D. J. N. Limebeer, B. D. O. Anderson and B. IIendel,"A Nash game approach to mixed H2/Hoo control," submitted for publication
[3] D. :I. N. Limebeer, B. D. O. Anderson and B. Hendel,"Nash games and mixed H2/Hoo control," preprint
[4] D. J. N. Limebeer and U. Shaked,"Minimax terminal state estimation and Hoo filtering," submitted to IEEE Trans. Auto. Control
[5] A. W. Start and Y. C. Ho,"Nonzerosum differential games," J. Optimizalion Theory and Applications, Vol.3, No. 3, pp 184206, 1967
The Principle of the Argument and its Application to the Stability and Robust Stability Problems
Mohamed Mansour Automatic Control Laboratory
Swiss Federal Institute of Technology Zurich, Switzerland
Abstrac t
It is shown that the principle of the argument is the basis for the different stability criteria for linear continuous and discrete systems. From the principle of argument stability criteria in the frequency domain are derived which lead to HermiteBieler theorems for continuous and discrete systems. RouthHurwitz crite rion and its equivalent for discrete systems, SchurCohn criterion and its equivalent for continuous systems are directly obtained. Moreover, the monotony of the argu ment for different functions can be proved using HermiteBieler theorem. Using the principle of the argument and the continuity of the roots of a polynomial w.r.t, its coefficients, all known robust stability results can be obtained in a straightforward and simple way.
1 Principle of the Argument
Consider a function f(z) which is regular inside a closed contour C and is not zero at any point on the contour and let Ac{arg f(z)} denotes the variation of arg f(z) round the contour C, then
T h e o r e m 1 [1] AcTarg fCz)} = 2~N (1)
where N is the number @zeros o f f (z ) inside C.
2 Stability Criterion using Principle of the Argu ment
2.1 C o n t i n u o u s S y s t e m s
Consider the characteristic polynomial of a continuous system
f ( s ) = aos n + a , s n1 + . . . + a .  l s + a n (2)
T h e o r e m 2 f (s) has all its roots in the left half of ~he splane if and only if f( jw) has a change of argument ofn~ when w changes from 0 to ~ .
17
The proof of this criterion which is known in the literature as CremerLeonhardMichailov Criterion is based on the fact that a negative real root contributes an angle of ~ to the change of argument, and a pair of complex roots in the left half plane contributes an angle of ~r to the change of argument as to changes from 0 to oo. It is easy to show geometrically that the argument is monotonically increasing.
f(s) can be written as f(s) = h(s 2) + sg(s 2) (3)
where h(s ~) and sg(s z) are the even and odd parts of f (s) respectively.
f (jto ) = h(to 2) k jtog(to2) (4)
2.2 D i s c r e t e S y s t e m s
Consider the characteristic polynomial of a discrete system
/ ( z ) = aoz" + a l z " 1 + ... + a . _ : + a . (5)
T h e o r e m 3 f ( z ) has all its roots inside the unit circle of the zplane if and only if f ( J °) has a change of argument of nrc when 0 changes from 0 go re.
The proof of this criterion is based on the fact that a real root inside the unit circle contributes an angle of 7r to the change of argument, and a pair of complex roots inside the unit circle contributes an angle of 2~r to the change of argument as 0 changes from 0 to lr. It is easy to show geometrically that the argument is monotonically increasing
f (z) can be written as f (z) = h(z) + g(z) (6)
where 1[ h(z) = 7 f(z) q znf = C~o zn + oqz '~x q" ... I" otlz "I" O:o (7)
and 1 [ ( 1 ) ]
g(Z) = ~ f(Z) "q" znf = fro zn "~"/~I zn1 21" .. /~I z  80 (8)
are the symmetric and antisymmetric parts of f (z) respectively. Also
f(e i') = 2ein°/2[h" k J9*] (9)
where for n even, n = 2v
OtV h*(O) = £I' 0 cosvO I" 0~I COS(// 1)0 + ... + o~,_x cosO+ ~
(1o) g'(O) = 3 0 s i n v O + 3 1 s i n ( u  1 ) 0 + . . . + P .  l s i n 0
From theorem 3 and equation (9) we get
T h e o r e m 4 f ( z ) has all its roots inside ~he unit circle of the zplane if and only if f* = h* + jg* has a change of argument of n~ when 0 changes from 0 to r.
f" is a complex function w.r.t, a rotating axis. It can be shown geometrically that the argument of f* increases monotonically. Fig. 1 shows the change of coordinates.
g*
Figure 1: Change of coordinates
3 Hermi te B ie l er T h e o r e m
3.1 Continuous Systems
Consider the polynomial in (3).
T h e o r e m 5 (HermiteBieler Theorem) f(s) has all its roots in the left half of the s plane if and only if h(A) and g()~) have simple real negative alternating roots and ~o > O.
a 1
The first root next to zero is of h(a).
The proof of this theorem follows directly from theorem 1; [2]. If theorem 5 is satisfied then h(,~) and 0(,~) are called a positive pair.
3 .2 D i s c r e t e S y s t e m s
Consider polynomial (5).
T h e o r e m 6 (Discrete HermiteBieler Theorem) f(z) has all its roots inside the unit circle of the zplane if and only if h(z) and g(z) have simple alternating roots on the unit
~ ' l < 1; [3], [4]. circle and a,
The proof can be sketched as follows: Due to theorem 4, h ° and g" and hence h and g will go through 0 alternatively n times
I"1 < 1 distinguishes between f (z) as 0 changes from 0 to r . The necessary condition ao
and its inverse which has all roots outside the unit circle. I I
4 M o n o t o n y of the argument
In [5] it is shown that the argument of a Hurwitz polynomial f ( s ) = h(s 2) + sg(s 2) in the frequency domain f(j•) = h( w 2) + jwg(_~2) as well as the argument of the modified functions f~'(~) = h (  ~ ~) + j 0 (  ~ 2) and f~(~) = h (  ~ 2) + j ~ 2 g ( _ ~ ) are monotonically increasing functions of w, whereby w varies between 0 and co. Similarly it is shown that given a Schur polynomial f (z) = h(z) + g(z) where h(z) and g(z) are the
19
symmetric and the antisymmetric parts of f (z) respectively, the same monotony property can be obtained for an auxiliary function .f*(0) defined as f(e j°) = 2J"° /2f f (0) . The argument of f*(0) = h*(0) + jg*(O) as well as the argument of the modified functions
and
h'(0) .g.(0)
are monotonically increasing functions of 0 whereby 0 varies between 0 and a'. These results were proved in [5] using mathematical induction. However, a simpler proof can be obtained using networktheoretical results which is based on ttermiteBieler theorems, [6]. The results obtained can be directly applied to the robust stability problem [5].
5 H e r m i t e  B i e l e r T h e o r e m and Cri ter ia for H u r w i t z S tab i l i ty
In this section it is shown how different criteria for Hurwitz s~ability like RouthHurwitz criterion and equivalent of SchurCohn criterion for continuous systems are derived from HermiteBieler theorem. Also Kharitonov theorem [7], [8] can be similarly derived, [9].
5 .1 R o u t h  H u r w i t z C r i t e r i o n
Without loss of generality consider n even. The first and second rows of the Routh table are given by the coefficients of h and g respectively. The third row is given by the coefficients of h  a°s2g. The following theorem shows that Routh table reduces
. . a l .  .
the stablhty of a polynomial of degree n to another one of degree n  1. Repeating this operation we arrive at a polynomial of degree one whose stability is easily determined.
T h e o r e m 'i t [2], [10] f ( s ) is Hurwitz slable if and only if h  ~~s2g + sg is Hurmitz stable.
Proof : Let f ( s ) = h + sg be Hurwitz stable i.e. ao,al,...a,~ > 0 and according to theorem 5, h(~) and g(,~) have simple real negative alternating roots, h  a 2 ~ s g and h
have the same sign at the roots of g and the number of roots of h ~ 2  als g + s g is n  1.
Moreover a l a 2  aoa3 > 0. Therefore, according to theorem 5, h  aa~s2g + sg is Hurwitz stable. The proof of the if part is similar. []
5 . 2 E q u i v a l e n t o f S c h u r  C o h n  J u r y C r i t e r i o n f o r C o n t i n u o u s S y s 
t e m s
T h e o r e m 8 [11] f ( s ) is Hurtoitz stable if and only if f ( s )  l ~ f (  s ) is Hurwitz
20
 1 is a root of f(s)  t ~ f (  s ) which can be eliminated, so that the stability of a polynomial of degree n is again reduced to the stability of a polynomial of degree n  1. This theorem can be proved using Rouch~ theorem. Here we give a proof based on HermiteBieler theorem and on the following Lemma.
L e m m a 1 h + sg is gurwitz stable if and only if 7h + 6sg is gurwitz stable for 7, 5 positive numbers.
Proof : The interlacing property does not change by multiplying h and g by positive numbers. Also ~ > 0. I"3
7ao
Proof : Proof of theorem 8. Let f(s) be tlurwitz stable. If l ( s ) = h + s g then/(X) = h  s g
Therefore
Using Lemma 1 we get I(s)  l ~ I (  s ) is Hurwitz stable. The proof of the if part is similar. El
6 Hermi te B ie l er T h e o r e m and Criteria for Schur Stabi l i ty
In this section it is shown how different criteria for Schur stability like the equivalent of RouthHurwitz criterion for discrete systems and SchurCohnJury criterion are derived from HermiteBieler theorem. Also analog result of Kharitonov theorem can be obtained; [91, [121. Consider f(z) given by equation (5). Without loss of generality we assume a0 > 0 and n even. For n odd a similar treatment can be made. A necessary condition for Schur
[~~1 < 1 which results in a0 > 0 and fl0 > 0. stability of (5) is o,
6 .1 E q u i v a l e n t o f R o u t h  H u r w i t z C r i t e r i o n f o r D i s c r e t e S y s t e m s
If h(z) and g(z) given by (7) and (8) fulfill the discrete HermiteBieler theorem then their projections on the real axis h*(x) and g*(x) have distinct interlacing real roots between  1 and +1. In this case h*(x) and g*(x) form a discrete positive pair. From [9] h*(x) and g'(x) are given by
v  1
(11)
where n = 2v and Tk, Uk are Tshebyshev polynomials of the first and second kind respectively.
21
Let
g*(X) = 6oX ~1 +6zX ~2+ . . .
T h e o r e m 9 [11] f ( z ) is Schur stable if and only if g*(X ) and h ' ( x )  ~ ( X  1)g*(X) form a discrete positive pair and h*(1) < 0 for v odd and h*(1) > 0 for v even.
Proof : Assume f ( z ) be Schur stable, h*(x)  ~o(X  1)g*(x) has the same value at the roots of g*(X) as h*(x). Therefore, the interlacing property is preserved. The condition h*(1) < 0 for v odd and h*(1) > 0 for v even guarantees that the reduction of the order is not due to a root of h*(x) between  1 and  c o . Therefore g*(X) and h*(x)  ~.(X  1)g*(X) form a discrete positive pair. The if part can be proved similarly. I"1
This theorem can be implemented in table form.
6.2 SchurCohnJury Criterion
I ~  [ < 1 i s S c h u r s t a b l e i f a n d o n l y i f T h e o r e m 10 [4], [111 The polynomial (5) with a.
the polynomial 1 I f ( z )  aa~z" f (1)] is Schur stable.
This was proved in [4] using the following Lemma:
L e m m a 2 h + g is Schur stable if and only if 7h + ~g is Schur stable for 7, 6 positive numbers.
Proof : Proof of Lemma 2: The interlacing property does not change by multiplying h
and g by positive numbers. Also l ~° '6~° [ < 1 if I~t < l. The reverse is true. [] [ 7 ~ o + 6 P 0 [ ao
a n ] Proof : Proof of theorem 10: f ( z )  "a~oz f (7) = h(z) + g(z)  ~o [h(z)  g(z)] = ( a. [1 only if part ( 1  o ) h ( z ) + . + , o ) g ( z ). The of theorem l0 follows directly from
~I < 1 and Lemma 2. all I
The if part can be proved similarly, rq
7 Robust Stability in the Frequency Domain
Let f ( s ,a ) be given by (2) and a E A. A is a closed set in the coefficient space. Let D be an open subset of the complex plane. Robust Dstability means that all roots of the polynomial family lie in D. Let r(s*) be the value set of the polynomial family at s = s" and 0F(s*) its boundary. Using the principle of the argument and the continuity of the roots w.r.t, the coefficients of a polynomial, we get the following result for robust Dstability.
T h e o r e m 11 [13], [14]. f(s,a_) with a_ E A is robust Dstable if and only if the following conditions are satisfied:
22
is) r(s') # 0 for some s" e aD
iii) or(s) # 0 for all s e aD
Using theorem 11 we can solve the following problems in a simple way
i) an edge theorem for the robust stability of polytopes in the coefficient space, [13]
it) stability of edges
iii) vertex properties of edges, [5], [15]
iv) Kharitonov theorem and its analog and counterpart for discrete systems, [16], [17], [12], [xsl
v) robust stability of controlled interval plants continuous and discrete, [14], [19], [20]
A survey of all the above problems is given in [21].
8 C o n c l u s i o n s
In the above it was shown that the stability criteria for linear and discrete systems as well as the solution of the robust stability problem all can be derived from the principle of the argument.
R e f e r e n c e s
[1] E.G. Phillips, Functions of a complex variable with applications, Oliver and Boyd, 1963.
[2] H. Chapellat, M. Mansour, and S. P. Bhattacharyya, Elementary proofs of some classical stability criteria, IEEE Transactions on Education, 33(3):232239, 1990.
[3] H.W. Schiissler, A stability theorem for discrete systems, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP24:8789, 1976.
[4] M. Mansour, Simple proof of SchurCohnJury criterium using principle of argu ment, Internal Report 9008, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[5] M. Mansour and F.J. Kraus, Argument conditions for Hurwitz and Schur stable polynomials and the robust stability problem, Internal Report 9005, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[6] M. Mansour, Monotony of the argument of different functions for Routh and Hut wits stability, Internal Report 9111, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1991.
[7] V. L. Kharitonov, Asymptotic stability of an equifibrium position of a family of systems of linear differential equations, Differentsial'nye Uravneniya, 14:20862088, 1978.
23
[8] B. D. O. Anderson, E. I. Jury, and M. Mansour, On robust Hurwitz polynomials, IEEE Transactions on Automatic Control, 32:909913, 1987.
[9] M. Mansour and F.J. Kraus, On robust stability of Schur polynomials, Internal Report 8705, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1987.
[10] M. Mansour, A simple proof of the RouthHurwitz criterion, Internal Report 8804, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1988.
[11] M. Mansour, Six stability criteria & HermiteBieler theorem, Internal Report 9009, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[12] M. Mansour, F. J. Kraus, and B.D.O. Anderson, Strong Kharitonov theorem for discrete systems, in Robustness in Identification and Control, Ed. M. Milanese, R. Tempo, A. Vicino, Plenum Publishing Co~oration, 1989. Also in Recent Advances in Robust Control, Ed. P. Dorato, R.K. Yedavaili, IEEE Press, 1990.
[13] S. Dasgupta, P. J. Parker, B. D. O. Anderson, F. J. Kraus, and M. Mansour, Fre quency domain conditions for the robust stability of linear and nonlinear systems, IEEE Transactions on Circuits and Systems, 38(4):389397, 1991.
[14] F. J. Kraus and W. TruSl, Robust stability of control systems with polytopical uncertainty: a Nyquist approach, International Journal of Control, 53(4):967983, 1991.
[15] F. J. Kraus, M. Mansour, W. TruSl, and B.D.O. Anderson, Robust stability of control systems: extreme point results for the stability of edges, Internal Report 9106, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1991.
[16] R. J. Minnichelli, J. J. Anagnost, and C. A. Desoer, An elementary proof of Kharitonov's stability theorem with extensions, IEEE Transactions on Automatic Control, 34(9):995998, 1989.
[17] F. J. Kraus and M. Mansour, Robuste Stabilitiit im Frequenzgang (german), In ternal Report 8706, Automatic Control Laboratory, Swiss Federal Institute of Tech nology, Zurich, 1987.
[18] F. J. Kraus and M. Mansour, On robust stability of discrete systems, Proceedings of the 29th IEEE Conference on Decision and Control, 2:421422, 1990.
[19] F. J. Kraus and M. Mansour, Robust Schur stable control systems, Proceedings of the American Control Conference, Boston, 1991.
[20] F. J. Kraus and M. Mansour, Robust discrete control, Proceedings of the European Control Conference, Grenoble, 1991.
[21] M. Mansour, Robust stability of systems described by rational functions, to appear in Advances in Robust Systems Techniques and Applications, Leondes Ed., Academic Press, 1991.
Robust Control of Interval Systems*
L.H. Keel
Tennessee State University
Nashville, TN. U.S.A
Department of Electrical Engineering
Texas A&M University
College Station, TX. U.S.A.
ABSTRACT
In this paper we consider a feedback interconnection of two linear time invariant systems, one of which is fixed and the other is uncertain with the uncertainty being a box in the space of coefficients of the numerator and denominator polynomials. This kind of system arises in robust control analysis and design problems. We give several results for the Nyquist plots and stability margins of such families of systems based oil the extremal segments of rational functions introduced by Chapellat and Bhattadaaryya [1]. These CB segments play a fundamental characterizing role in many control design problems.
1. I N T R O D U C T I O N
The problem of designing control systems to account for parameter uncertainty has become very important over the last few years. The main reason for this is the lack of effective methods for dealing with systems containing uncertain or adjustable parameters. Significant progress on this problem occured with the introduction of Kharitonov's Theorem [2] for interval polynomials which provided a new way of deal ing with parametric uncertainties. However, despite its mathematical elegance, its applicability to cozltrol systems was very limited because the theorem assumes that
*Supported in part by NASA grant NAGI863 and NSF Grant ECS8914357
25
all coefficients of the given polynomial family perturb independently. Under this as sumption the theorem can only provide conservative results for the control problem where the uncertain parameters enter the characteristic polynomoial in a correlated fashion
A breakthrough was reported in the theorem of Chapellat and Bhattacharyya [1] which dealt with the case in which the characteristic polynomial is a linear combina tion of interval polynomials with polynomial coeffidents. The CB Theorem provided Kharitonovlike results for such control problems by constructing certain line seg ments, the number of which is independent of the polynomial degrees, that completely and nonconservatively characterizethe stability of interval control systems.
Using this novel theorem, it has been discovered that the worst case H** stability margin of an interval control system occurs on one of these CB segments. The details and constructive algorithm to compute H °° stability margin are given in [3]. Similarly the worst case parametric stability margin occurs on the CB segments [4].
The main result of this paper is that the Nyquist plot boundaries of all the transfer functions of interest in an interval control system occurs on the CB segments. This key result has farre~ching implications in the development of design methods for interval control systems. Throughout this paper, proofs are omitted and may be found in [5]. A similar result has recently been independently discovered by Tesi and Vicino [6].
2. NOTATION AND P R E L I M I N A R Y RESULTS
Figure 1. Feedback System
We consider the feedback interconnection (see Figure 1) of the two systems with respective transfer functions
2v(s) Q~(s) P(~): . . . . . (1)
Q(~) := Q~(,) D(~)
where N(s), D(s) and Q~(s), i = 1,2 are polynomials in the complex variable a. In general Q(s) is fixed and P(s) will be assumed to lie in an uncertainty set described as follows.
Write
N(s) := nvs p+np_ls pI +. . .+his+no DCs) := dq~q + d ,_~ ,~ + . . . + d~, + do (2)
26
~nd let the coefficients lie in prescribed intervals
. , e l . . , . + ] , i e {0,1,. . . ,p} :=~ die[d~,d+], ie{O, 1,...,q}:=q.
Introduce the interval polynomial sets
H(~) := {N(,) = n,s"+n~_,~'~ + . . . n , ~ + , , o : n~ e ["~',",+l, i e E} D(,) := {D(s)=d,8'+d,,sqt+...dz*+do : d /e ld~/ ,d~/ ] , ieq} (3)
and the corresponding set of interval transfer hmctions (orintervM systems)
LD(a) : (N(,) ,D(s)) e Af(a)xV(s)}. (4)
The four Kharitonov vertex polynomials associated with Af(s) are
K~(,) := o + 'q* +~+*' +  D 3 + " ~ " + ' q ' ~ + " "
K~(~) := o +",+~ +"~+~ ÷ " ~ + " ~ ' +"~+'~ + . . . . K~(~) := ,,o+ + ,q~ + , q ~ + .+,~ + ,,,%, + , q , s + . . . (5) K.~(~) := .+ + .+~ + . ~ + .~ ,~ ' + , ,+ , ' +,,~%~ + . . . .
and we write ~.,,,'C~) " {K~C.~), K~(.,), g~c.,),.,"qC.~)} (6)
Similarly the four Kharitonov polynomials associated with 29(s) are denoted K~(a), i = 1,2,3,4 and
Jc.~(.,) := {K~C.,), K,~C..,), x~',~C,,,),_rc,,/(s)} (7) The four Kharitonov segment polynomials associated with N'(s) are defined as follows:
st¢0) :=
[AK~,Cs) + (1  A)K~(s) : A E [0,1], (i,j) E {(1,2), (1,3),(2,4),(3,4)}] (8)
and the four Kharitonov segment polynomials associated with :D(s) are
S,(~) :=
[id(~(s)+ (1 #)K~(s) : /~E [0,11, Ci,j) E {(1,2),(1,3),(2,4),(3,4))] (9) Following Chapellat and Bhattacharyya [1] we introduce the following subset of
(H(~)xV(~))c8 = {(2v(~),D(~)) : N(~) e ~:~f(~), D(s) 6 Sv(s) or N(s) E Sa'(s), n(s) e K:v(*)}. (10)
Referring to the control system in Figure 1, we calculate the following transfer functions of interest in an,alysls and design problems:
yCs) = p(~) "(~) .(.,) ~ = Q(~) 01)
T°(s) := r  ~ = i + P(s)Q(s) (14)
u(s) Q(s) (15) T"(s) := r  ~ = 1 + P(s)QCs)"
The characteristic polynomial of the system is denoted as
~r(s) = Q2(s)D(s) + Ql(s)N(s). (16)
As P(s) ranges over the uncertainty set P(s) (equivalently, (N(s), D(s)) ranges over Af(s)xD(s)) the transfer functions T°(s), TY(s), T"(s), T'(s) range over correspond ing uncertainty sets T°(s), T~(s), T"(s), and T'(s), respectively. In other words,
T'(~) := { PC~)Q(~) : P(~)e ~(~) } (17)
T'(~) := { ~ : P ( ~ ) ~ ( ~ ) } (lS)
T"(~) := { ~ : P(~) ~ ~(~) } (10)
IICs) = {Q2Cs)D(s) + Q,Cs)N(s) : (N(s),D(s)) E AfCs)xD(s)}. (21)
We now introduce the CB subset of the family of interval systems P(s):
PcBCs) := { N9]DO : (NCs),DCs)) E (N'(s)xDCs))cn }. (22)
These subsets will play a central role in all the results to be developed later. We note that each element of Pea(s) is a 0he parameter of transfer functions and there are at most 32 such distinct elements.
The CB subsets of the transfer function sets (17)  (20) and the polynomial set (21) are also introduced:
T~:BCs ) := { PCs)Q(~) : PCs) E PCB(8) } (23)
T~B(S ) : { ~ : P(s) E PCB(S) } (24) l+P(o)e(,)
T~.(~) := { Q(.) : P(~) c ro.(~) } (25) l+P(s)Q(s)
TS.(~) := { ' l+P(~')O(,) : P(s) E PCB(S) } (26)
28
and
ncB(s) := {Q.( , )D(s) + 0,( ,)IV(s) : (N( , ) . D( , ) ) e (~ (~)xV( , ) )cB} (27)
In the paper we shall deal with the complex plane image of each of the above sets evaluated at s = fla. We denote ¢ar3a of these two dimensional sets in the complex plane by replacing s in the corresponding argument by w. Thus, for example,
n ( ~ ) := {n(~) : • = j~} (28)
and T~s(o~) := {T~s(~) : s = jo~} (29)
The Nyquist plot of a set of functions (or polynomials) T(s) is denoted by T:
X := Uo<~<ooX(w) (30)
Stabili ty and Stabil i ty Mar$ins
The control system of Figure 1 is stable for fixed Q(s) and P(s) if the charac teristic polynomial
(s) = Q~(s)DCs) + Q1Cs)N(s) (31)
is Hurwitz, i.e. has all its n = q+ degree [Q2(s)] roots in the open left half of the complex plane. The system is robustly stable if and only if each polynomial in l'I(s) is of degree n (degree D(s). remains invariant and equal to q as D(s) ranges over :D(s)) and every polynomial in II(s) is Hurwitz.
The following important result was provided in Chapdlat and Bhattacharyya [11.
Theorem 1. (CB Theorem) The control system of Figure I is stable for all P(s) E P(s) if and only if it is stable for all P(s) e Pcs(s).
The above Theorem gives a constructive solution to the problem of checking robust stability by reducing it to a problem of checking a set of (at most) 32 root locus problems.
If a fixed system is stable we can determine its gain margin V and phase margin ~b as follows:
7+(P(s),O(s)) := max{ f~" : Q2(s)D(s) + KQI(s)N(s)
is Hurwitz for It" E [1, f(] } (32)
~(P(s),Q(s)) := max{ K : Q~(s)D(s)+ kQ,(s)~(s) ' 1 is Hurwitz for It" E [K' ] } (33)
29
@+(P(s),Q(s)) := max{ ~ : Q,(s)D(s)+ei÷Ql(s)N(s)
is Hurwitz for ~ E [O, ~l } (34)
qb(PCs),Q(s)) := max { ~b : Q, Cs)DCs) + ei÷Ql(s)NCs)
is Hurwitz for ~ E [0,~ }. (35)
Note that 7 +, 7 , ~+, and ~b are uniquely determined when Q(s) and P(s) are fixed. The unstructured additive stability margin y. (H ¢* stability margin) can also be
uniquely determined for a fixed stable closed loop system and is given by
1 v.(P(s)Q(s)) := IIQ(s)(1 + p(s)O(s))_,ll, 0. (36)
In the following section we state some fundamental results on the above sets of transfer functions and the calculation of extremal stability margins over the uncertainty set HCs)xV(s).
3. MAIN RESULTS
We shall give the main results here without proof. The proofs are given in Keel, Shaw and Bhattacharyya [5]. Similar results have been recently reported independently by Tesi and Vicino [6].
Theorem 2. For every to > O,
OP(to) C [email protected]) 0T°(to) C T~.(w)
[email protected]) c T~.(to) OT"(to) c [email protected]) OT'(to) c Tt.(to)
(37) (38) (39) (40)
(41)
This result shows that at every to > 0 the image set of each transfer function in (17)  (20) is bounded by the corresponding image set of the CB segments.
The next result deals with the Nyquist plots of each of the transfer functions in (17)(20).
Corollary 1. The Nyquist plots of each of the transfer function sets T°(s), TU(s), T"(s), and T~(s) are bounded by their corresponding CB subsets:
cTr ° c T~. (42) 69T' C W~s (43)
~)T" C T"cB (44)
3 0
This result has many important implications in control system design and will be explored in forthcoming papers. One important consequence is given below:
Theorem 3. Suppose that the closed loop system is robustly stable, i.e. stable for all P(s) E P(s). Then
and
max 4~: = max ~b ~: (47) P(s)eP(s) P(s)ePcB (s)
max v. (48) max tta = P(s)eP(s) P(s)ePc,(s)
min 7 ~= = min 7 ~= (49) P(s)6P(s) P(s)6PcB(s)
min q,+ = min ~: (50) P(s) E P ( s ) P(s)EPcB(s}
min va = min va. (51) P(s)eP(.) P(s)ePc,(s)
In control systems the maximum margins are useful for synthesis and design (with P(s) being a compensator) and the minimum margins are useful in calculating worst case stability margins (P(s) is the plant). In each case the crucial point that makes the result constructive and useful is the fact that Pcs(s) consists of a finite and "small" number of line segments.
4. C O N C L U D I N G R E M A R K S
We have showe that the boundaries of various transfer function sets and various ex tremal gain and phase margins occur on the CB segments first introduced in [1]. Using this idea, construction of Nyquist and Bode bands can be given. These important tools allow us to reexamine all of classical control theory with the added ingredient of robustness in both stability and performance using the framework of interval systems. More interestingly, these new results provide a way of developing design and synthesis techniques in the field of "parametric robust control. These techniques also connect with H °° control theory and promise many new results in the analysis and design of control systems containing uncertain or adjustable real parameters.
R E F E R E N C E S
[1] H. Chapellat and S. P. Bhattaeharyya, "A generalization of Kharitonov's theorem: robust stability of interval plants," IEEE Transactions on Automatic Control, vol. A C  34, pp. 306  311, March 1989.
[2] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations," Differential Uravnsn, vol. 14, pp. 2086  2088, 1978.
31
[3] H. ChapeUat, M. Dahleh, and S. P. Bhattacharyya, "Robtmt stability under struc tured and unstructured perturbations," IEEE Transactions on Automatic Control, vol. AC  35, pp. 1100  1108, October 1990.
[4] H. Chapellat, M. Dahleh, and S. Bhattacharyya, "Extremal manifolds in robust stability," TCSP Report, Texas A&M University, July 1990.
[5] L. Keel, J. Shaw and S. P. Bhattacharyya, "Frequency domain design of interval control systems," Tech. Rep., Tennessee State University, July 1991. Also in TCSP Tech. Rep., Texas A&M University, July 1991.
[6] A. Tesi and A. Vicino, "Kharitonov segments suffice for frequency response anal ysis of plantcontroller families". To appear in Control of Uncertain Dynamic Systems, September 1991, CRC Press.
Rejection of Persistent, Bounded disturbances for SeLmpledData Systems*
Bassam Bamleh t, Munther A. Dahleh $, and J. Boyd Pearson t
June 4, 1991
Abst rac t
In this paper, a complete solution for the l 1 sampleddata problem is furnished for arbitrary plants. Then l 1 sampleddata problem is described a~ follows: Giv en a contlnuoustime plant, with contlnuoustime performance objectives, design a digital controller that delivers this performance. This problem differs from the s tandard discretetlme methods in that it takes into con~ideratlon the intersamp]L, lg behaviour of the closed loop system. The resulting dosed loop system dynamics con sists of both continuousthne and discretetime dynamics and thus such systems are known as "Hybrid" systems. It is shown that given any degree of accuracy, there exists a standard discretetime l I problem, which can be determ;ned apriori, such that for any controller that achieves a level of performance for the discretetlme problem, the same controller achieves the same performance within the prescribed level of accuracy if implemented as a sampleddata controller.
"The first and last authors' research is supported by NSF ECS8914467 and AFOSR910036. The second author is supported by WrightPatterson A.F.B. F3361590C3608, , C.S. Draper Laboratory DLH4186U and by the ARO DAAL0386K0171.
t Dept. of Electrical and Computer Engineering, Rice University, Houston, TX 77030 I Laboratory of Information sad Decision Systems, Massachusetts Institute of Technology, Cambridge,
MA
1 I n t r o d u c t i o n
33
This paper is concerned with designing digital controllers for continuoustime system s to optimaly achieve certain performance specifications in the presence of uncertainty. Contrary to discrete time designs, such controllers are designed taking into consideration the intersample behavior of the system. Such hybrid systems are generally known as sampleddata systems, and have recently received renewed interest by the control com munity.
The difficulty in considering the continuous time behavior of sampleddata systems, is that it is time varying, even when the plant and the controller are both continuous time and discretetime timeinvarlant respectively. We consider in this paper the standard problem with sampledda~a controllers (or the sampleddata problem, for short) shown in figure 1. The continuous time controller is constrained to be sampleddata controller, that is, it is of the form 7~GS~. The generalized plant is continuoustime timeinvariant and G is discretetime timeinvariant,Tf, is a zero order hold (with period r), and ,.~ is an ideal sampler (with period r). 7f¢ and ,S~ are assumed synchronized. Let ~'(G, 7f~CS¢) denote the mapping between the exogenous input and the regulated output. Jr(G, 7~¢G,S~) is in general time varying, in fact it is rperiodic where r is the period of the sample and hold devices.
Sampleddata systems have been studied by many researchers in the past in the context of LQO controllers (e.g. [121). Recently, [4] studied this problem in the context of 7"{ °° control, and were able to provide a solution in the case where the regulated output is in discrete time and the exogenous input is in continuous time. The exact problem was solved in [2],[3], and independently by [9] and [13]. The L•induced norm problem (the one we are concerned with in this paper) was considered in [7] for the case of stable plants.
In this paper we will use the framework developed in [2],[3], to study the t x sampled data problem. Precisely, the controller is designed to minimize the induced norm of the periodic system over the space of bounded inputs (i.e. L°°). This minimization results from posing time domain specifications and design constraints, which is quite natural for control system design. The solution provided in this paper is to solve the sampleddata problem by solving an (almost.) equivalent, discrete time ~i problem. While this was the approach followed in [7], the main contribution of this paper is that it provides bounds that can be computed apriori to deternfine the equivalent discretetime problem, given any desired degree of accuracy and thus provides a solution for the synthesis problem. The solution in this paper is presented in the context of the lifting framework of [2], [3], as an approximation procedure for certain infinite dimensional problems. This approach has the advantage of handling both the stable and the unstable case in the saane franlework, and results in techniques that are more transparent than those in [7].
34
. . . . . . pi

k
2 T h e L i f t i n g T e c h n i q u e
In this section we briefly summarize the lifting technique for continuoustime periodic systems developed in [2], [3], and apply it to the sampleddata problem.
For continuous time signals, we consider the usual L°°[0,oo) space of essentially bounded functions [6], and it 's extended version L~°[0,oo). We will also need to consider discrete time signals that take values in a function space, for this, we define Ix to be the space of all X valued sequences, where X is some Banach space. We define t ~ as the subspace of l x with bounded norm sequences, i.e. where for ( f l } fi t x , the norm II(f,}llz r := supl II/illx < Go. Given any .f fi L~°[o,oo), we define i t ' s / / / t in o / ¢ t~o.t,..i, as follows: / is an L°°to,~lvalued sequence, we denote it by ( / i} , and for each ~,
/~(t) := / ( t + , 0 o < t < ,.
The lifting can be visualized ~ taking a continuous time signal and breaking it up into a sequence of 'pieces' each corresponding to the function over an interval of length r (see figure 2). Let us denote this lifting by Wr : L~[o,~o) . , l~..lo., j. W,. is a linear isomorphism, furthermore, if restricted to L~°io,oo) , then W,. : L~[o,~)  ~ t~',[o., ! is an isometry, i.e. it preserves norms.
Using the lifting of signals, one can define a lifting on systems. Let C be a linear continuous time system on L~°[o,oo), then it 's li/ting G is the discrete time system G :=
35
W.GW~ "1, this is illustrated in the commutative diagram below:
IX,to,,! ~ ,. l£"°t..,l
+++
Thus G is a system that operates on Banach space (L°°[0,~]) vMued signah, we will call such systems infinite dimensional. Note that since W~ is an isometry, if G is stable, i.e. a bounded linear map on L °° then G is also stable, and furthermore, their respective induced norms are equal, Ildll = IIGll The correspondence between a system and it's lifting also preserves algebraic system properties such as addition, cascade decomposition and feedback (see [2] for the details).
To apply the lifting to the sampleddata problem, consider again the standaxd problem of figure 1, and denote the dosed loop operator by .T(G,?L.CS¢). Since the lifting is an isometry, we have that II~r(o, 7¢.cs~)[ I  IIW.~=(G, ~ C S . ) w ;  ' II, this is shown in figure 3(a).
We now present (from [2]) a state space realization for the new generalized plant 0 . The original plant G has the realization:
G = 6 A Dll D12 .
G'l 0 0
It is assumed that the sampler is proceeded with a presampling filter. It can be shown ([2]) that a realization for the generalized plant C. (figure 3) is given by
G21 G2~ C2 0 0
36
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " 1
! ! ! : [ , ,
I c I Figure 4: Decomposition of (~
[ cA" I eA('')B, ~(*')B, ] Gte" Gte'4("')10,)Bx + D t t 6 ( t  , ) Gt~I'(t)B= + Dn C= 0 0
where ~(t) := I~ eA'd's" The system O has the following input and output spaces
6 n : ~.r.,,,olo.,.i ~ .~.f,O*l,,fl Gn : la. ~ lLto,,.! 021 : tLto,,j '* t a , 02: : +,~ * Is,
Using this lifting, G can be decomposed as
° [ ~ 0 ~ ° l + [ I ° 0 0 o.10,] :J Lo, ' ~ ] [ ] L °] L~J ~10 o 0 I 0 0
This decomposition is illustrated in figure 4. The closed loop mapping ~'(G, U) is corre spondingly decomposed as
~(O,C) = b,t + ~(Oo, C) = D,, + [~t b~, ] J:( ¢., C) B,. (I)
We will use the notation C) := [ eL bt, ] , and call (~ the output operator and/~, the input operator. With this decomposition, G,~, is finite dimensional, and f), B1 axe finite rank operators
: R "+~ ~ L°°[o.fl, /~1 : L~10,fl o ~7.
It is evident from this decomposition that only the "discrete" part of the plant depends on the controller, while the infinite dimensional hybrid part depends only on the system's dynamics.
3
37
, l
F~}. "1 Figure 5: The system On
S o l u t i o n P r o c e d u r e
Using the lifting we are able to convert the problem of finding a controller to minimize the L °° induced norm of the hybrid system (figure 1) into the following standard problem with an infinite dimensional generalized plant G:
:= inf I P ' ( a , ~ , o s , ) l l =: inf IlS~(O,O)ll (2) 0 j tab l i z ing ~ 0 etaMixitto
The above infinite dimensional problem is solved by the followin 8 approximation proce dure through solving a standard MIMO 0 problem [5, 10]. Let 74~ and ,S~, be the following operators defined between L~[0,~l and l°°(.) (t°°(.) is R" with the maximum norm),
x,,: t**c.) ,/,"*to,.l (x,,,,)(O = ,4l~J) ; {,,(i)} E t**c.),
One can in fact show that the operators are well defined. Now to approximate the infinite dimensional problem, we use the approximate dosed loop system 8,,~'(G, U)~f,, (see figure 5), and for each ~z we define
#. := inf IIS~(0,c)X.ll, (3) O ,Jtabili~i~O
This new problem now involves the induced norm over l~***(,0, i.e. it is a standard MIMO
0 problem. Let us denote the generalized plant a~sociated with 7~,~'(0, C)S,, by ~ , such that
S.7(C.,C)~,, = Y(O. ,C) ,
where ~n and a realization for it is given by,
I o. :[~ ~,1o[~. ;] = S.Ol sob,,~o s. , , O, o °I =: bn b1~ •
0 0
i i i i i ~ ! ! ! ! ! !
0 Tin . . . . . . T
i I I I 1 2 . . . . . . n
I , i
Design Bounds
Let us begin with analysis. Note that since U~:(O,o)ll is a pe~odic,ny t~me v~ying system, its L'induced norm is not immediately computable. An alternative method of computing II~:(O, O)ll comes from the limit
IW(¢, C)ll = .u2~ IIS,,~(¢, c)~,,il =: ~m 1W(6",,, C)ll, (4)
for a fixed C. This equation (4) however is by far not sufficient to show the convergence of the synthesis procedure, since given only (4), the rate of convergence may depend on the choice of C. Our objective is to obtain explicit bounds on [].~r(~, C)[l in the following form
Main Inequal i ty : There are conatants Ko and K1 which depend only on G, such ~a~
IIJ:(0.,O)ll < II~:(0,O)H _< KLln + (1 +  ~ ) [l~'(¢,,C')[[, (5)
The significance of the bound (5) is that it is exactly what is needed for synthesis. When one performs an t t design on G,, the result is a controller that keeps [[.~(~,,, C)H small, but the objective is to keep the L'induced norm of the hybrid system (or equivalently II~'(~, C)ll) small, and the inequality (5) guarantees this. The proof of this bound utilizes heavily the decomposition discussed earlier, and the fact that the infinite dimensional operators O and B1 me finite dimensional, and a~e dependent only on the original plant. The details of these proofs can be found in [1].
39
References
[1] B. Bamieh, M.A. Daldeh and 3.B. Pearson, 'Minimization of the L ~ induced norm for sampleddata systems,' Rice University Technical Report, Houston, TX, June 1991.
[2] B. Bamleh and J.B. Pearson,'A General Framework for Linear Periodic Systems with Application to 7{ ~ SampledDe~ta Control', to appear in IEEE Trans. on Ant. Control.
[3] B. Bamleh, J.B. Pearson, B.A. Francis, A. Tannenbanm, 'A Lifting Technique for Linear Periodic Systems with Applications to SampledData Control', to appear in Systems and Controls Letters.
[4] T. Chen and B. Francis, 'On the L: induced 26 26 norm of a sampleddata system', Systems and Control Letters 1990, v.15, 211219.
[5] M.A. Dahleh and J.B. Pearson, 'l:optimal feedback control for MIMO discretetime systems', in Trans. Ant. Control, AC32, 1987.
[6] C.A. Desoer and M. Vidyasagar, Feedback Systems: InputOutput Properties, Aca demic Press, New York, 1975.
[7] G. DuUerud and B. Francis, ,ry Performance in SampledData Systems', preprint, Submitted to IEEE Trans. on Ant. Control.
[8] B.A. Francis and T.T. Georgiou, 'Stability theory for linear timeinvariant plemts with periodic digital controllers', IEEE Trans. Ant. Control, AC33, pp. 820832, 1988.
[9] P.T. Kabamba and S. Hara, 'Worst ca~e analysis and design of sampled data control systems', Preprint.
[i0] J.S. McDonald and J.B. Pearson, 'IZOptimal Control of Muitivariable Systems with Output Norm Constraints', Autoraatica, vol. 27, no. 3, 1991.
[11] N. Sivashankar and Pramod P. Khargonekar, '~o~Induced Norm of SampledData Systems', Proc. of the CDC, 1991.
[12] P.M. Thompson, G. Stein and M. Athans, 'Conic sectors for sampleddata feedback systems,' System and Control Letters, Vol 3, pp. 7782, 1983.
[13] H.T. Toivonen, 'Sampleddata control of continuoustime systems with an 7foo opti ramify criterion', ChemicaJ Engineering, Abo Akademi, Finland, Report 901, Jan uary 1990.
Rational Approximation of L1optimal controller
Yoshito Ohm?, Hajhnc Ma~ and Shinzo Kodama~
? Dcparuncnt of Mechanical Engineering for ComputerControlled Machinery Osaka Univcxsity 2I Yam~dmOka, Suil~t, Osaka 565, Japan Dcparm~nt of EIeclmnic Hnginccring Osaka University 2I YAm~daOka, Sulfa, Osaka 565, Japan
Abstract
This paper studies the L1optimal control problem for SISO systems with rational
controllers. It is shown that thc infimal achievable norm with rational controllers is as srrmll as
that with irrational controllers. Also a way to construct a rational suboptimal controller is
studied.
1. Introduction The problem of Llopdmal control was fwst formulated by Vidyasagar [7] to solve the
disturbance rejection problem with persistent input. A complete solution for general problems
was presented by Dahlch and Pearson for discrctctimc 1block MIMO systems [2] and for
continuoustime SISO systems [3]. In [3] they also pointed out the uscfuincss of Lloptimal
control in the following areas: (i) disturbance rejection of bounded input, (ii) tracking of
bounded input, and (iii) robustness.
For continuoustime systcms, it turns out that an LlOpdmal solution has a certain
disadvantage: delay terms appear both in the numerator and denominator of an optimal
controllcr [3]. This raises a question as to what limitation wc impose if controllers arc confined
to be rational. One obvious way to look at this is to approximate the irrational controller by
rational one. However, as it will be shown in a later section, an irrational transfer function
cannot be approximated by the set of stable rational functions in thc topology with which the
Llconlrol problems arc formulated. This means th
S. Hosoe (Ed.)
Robust Control Proceedings of a Workshop held in Tokyo, Japan, June 23  24, 1991
SpringerVerlag Berlin Heidelberg NewYork London Paris Tokyo HongKong Barcelona Budapest
Advisory Board
L.D. Davisson • A.GJ. MacFarlane" H. K w a k e r n ~ J.L. ~ s e y " Ya Z. Tsypkin • AJ . Vit~rbi
Editor
ISBN 3540559612 SpringerVerlag Berlin Heidelberg NewYork ISBN 0387559612 SpringerVerlag NewYork Berlin Heidelberg
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PREFACE
The workshop on Robust Control was held in Tokyo, Japan on July 24 25, 1991. From 10 countries, more than 70 reseachers and engineers gathered together, and 33 general talks and 3 tutorial ones, all invited, were presented. The success of the workshop depended on their high level of scientific and engineering expertise.
This book collects almost all the papers devoted to the meeting. The to pics covered include: Hoo control, parametric#tructured approach to robust control, robust adaptive control, sampleddata control systems, and their ap plications.
All the arrangement for the workshop was executed by a organizing com mittee formed in the technical committee for control theory of the Society of Instrument and Control Engineers in Japan.
For financial support and continuing cooperation, we are grateful to Casio Science Promotion Foundation, and many Japanese companies.
Shigeyuki Hosoe
C O N T E N T S
Out l ine of the W o r k s h o p
P a p e r s
• H.KIMURA (J, J~)Lossless Factorization Using Conjugations of Zero and Pole Ex tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL Mixed H2/H~ Filtering by the Theory of Nash Games . . . . . . . . . . . . . . . . . . . . . 9
• M.MANSOUR The Principle of the Argument and its Application to the Stability and Robust Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
• L.H.KEEL, J.SHAW, S.P.BHATTACHARYYA Robust Control of Interval Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
• B.BAMIEH, M.A.DAHLEH, J.B.PEARSON Rejection of Persistent, Bounded Disturbances for SampledData Sys tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
• Y.OHTA, H.MAEDA, S.KODAMA Rational Approximation of LiOptimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . 40
• M.ARAKI, T.HAGIWARA Robust Stability of Sampled Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
• S.HARA, P.T.KABAMBA Robust Control System Design for SampledData Feedback Systems . . . . . . . 56
• Y.YAMAMOTO A Function State Space Approach to Robust Tracking for SampledData Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
• F.B.YEH, L.F.WEI SuperOptimal HankelNorm Approxmations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
• J.R.PARTINGTON
• R.F.CURTAIN FiniteDimensional Robust Controller Designs for Distributed Parameter Systems: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
• 3.H.XU, R.E.SKELTON Robust Covariance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
VIII
• T.FUJII, T.TSUJINO An Inverse LQ Based Approach to the Design of Robust Tracking System with Quadratic Stability ................................................. 106
• T.T.GEORGIOU, M.C.SMITH Linear Systems and Robustness: A Graph Point of View .................. 114
• M.FUJITA, F.MATSUMURA, K.UCHIDA Experimental Evaluation of H~ Control for a Flexible Beam Magnetic Suspension System ...................................................... 122
• K.OSUKA Robust Control of Nonlinear Mechanical Systems ......................... 130
• A.S.MORSE HighOrder Parameter Tuners for the Adaptive Control of Nonlinear Systems ................................................................. 138
• H.OHMORI, A.SANO Design of Robust Adaptive Control System with a Fixed Compensator .... 146
• S.HOSOE, JF.ZHANG, M.KONO Synthesis of Linear Multivariable Servomechanisms by Hc¢ Control ....... 154
• T.SUGIE, M.FUJITA, S.HARA ]/ooSuboptimal Controller Design of Robust Tracking Systems ........... 162
• M.KHAMMASH Stability and Performance Robustness of ~I Systems with Structured NormBounded Uncertainty .............................................. 170
• S.SHIN, T.KITAMORI A Study of a Variable Adaptive Law ..................................... 179
• M.FU Model Reference Robust Control ......................................... 186
* T.MITA, K.Z.LIU Parametrization of H~ FI Controller and Hoo/H2 State Feedback Control ... 194
i A.A.STOORVOGEL, H.L.TRENTELMAN The Mixed H2 and Hoo Control Problem ................................. 202
• D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL: Nash Games and Mixed H2/H~ Control ................................. 210
• J.B.PEARSON Robust g1Optimal Control .............................................. 218
( J , J ' )  L O S S L E S S F A C T O R I Z A T I O N U S I N G C O N J U G A T I O N S O F Z E R O A N D P O L E E X T R A C T I O N S
Hidenori Kimura Department of Mechanical Engineering for ComputerControlled Machinery, Osaka University
1. Introduction A matrix G(s) in L** is said to have a (J, J')losslessfactorization, if it is represented as a product
G(s)=O(s)rl(s), where O(s) is (J, J')Iossless ]11] and rl(s) is unimodular in rl**. The notion of O, J') lossless factorization was first introduced by Ball and Helton [2] in geometrical context. It is a generalization of the wellknown innerouter factorization of matrices in H"* to those in L**. It also includes the spectral factorization for positive matrices as a special case. Furthermore, it turned out that the (J, J')lossless factorization plays a central role in H"* control theory [1][4][7]. Actually, it gives a simple and unified framework of t l " control theory from the viewpoint of classical network theory. Thus, the (J, J')lossless factorization is of great importance in system theory.
The (J, J')lossless factorization is usually treated as a (J, J')spectralfactorization problem which is to find a unimodular n(s) such that G(s)JG(s) = rl~(s)jrl(s). Ball and Ran [4] first derived a statespace representation of the (J, J')spectral factorization to solve the Nehari extension problem based on the result of Bart et.al. [5]. It was generalized in [ 1] to more general modelmatching problems. Recently, Green et.al. [8] gave a simpler statespace characterization of the (J, J')spectral factorization for a stable G(s). Ball et al. 13] discussed some connections between the chainscattering formulation of H** and (J, J')lossless factorization.
In this paper, we give a simple derivation of the (J, J')lossless factorization for general unstable G(s) based on the theory o f conjugation developed in [9]. The conjugation is a simple operation of replacing some poles of a given rational matrix by their mirror images with respect to the origin by multiplication of a certain matrix. It is a statespace representation of the NevanlinnaPick interpolation theory [ 11] and the Darlington synthesis method of classical circuit theory. The (J, J')lossless factorization for unstable matrix, which is first established in this paper, enables us to solve the general I1 °° control problem directly without any recourse to Youla parameterization [12].
In Section 2, the theory of conjugation is briefly reviewed. The (J, J')lossless conjugation, a special class of conjugations by (J, J')lossless matrix, is developed in Section 3. Section 4 gives a frequency domain characterization of the (J, J')lossless factorization. Section 5 is devoted to the state space representation of the (J, J')lossless factorization. The existence condition for the (J, J ')lossless factorization is represented ill terms of two Riccati equations, one of which is degenerated in the sense that it contains no constant tema. Notations :
C(sIA)" B+D := [A~1 R(s) = RT(.s), R'(s) = RT(~), tC IDJ,
~A) ; The set of eigenvalues of a matrix A.
RU:x r ; The set of proper rational nmtrices of size mxr without poles on the imaginary axis,
Rllmxr; The set of proper stable rational matrices of size m×r.
2. Con juga t ion The notion of conjugation was first introduced in [9]. It represents a simple operation which replaces
the poles of a transfer function by their conjugates. The conjugation gives a unified framework for treating the problems associated with interpolation theory in state space [10]. In this section, the notion of conjugation is generalized to represent the relevant operations more naturally.
To define the conjugation precisely, let
G(s) =[ A~DB I R.. De Rr , A ~ ( 2 . t )
be a minimal realization of a transfer function O(s).
Definition 2.1 Let a(A) = A1UA2 be a partition of o'(A) into two disjoint sets of k complex numbers At and nk complex numbers A2. A matrix e(s) is said to be a AFconjugaror of G(s), if the poles of the product
G(s)O(s) are equal to {A l} UA2 and for any constant vector ~, ~ 0,
~(sl  A)'[email protected](s) #0 ( 2 . 2 )
for some s, where At is a set composed of the elements of At with reversed sign. As a special case, we can choose At composed of the RHP elements of o(A). If A has no eigenvalue
on the jo~  axis. ^l conjugator of G(s) stabilizes the product G(s)e(s). In that case, Alconjugator is called a stabilizing conjugator. In the same way, we can define an antistabilizing conjugator.
Remark. The condition (2.2) is imposed to role out the trivial case like e(s)  0. It is automatically satisfied for the case where e(s) is right invertiblc. T H E O R E M 2.2 e(s) is a Atconjugator of G(s) given by (2.4), if and only if O(s) is given by
t C. I I J (2.3)
where Co, De, X, are such that X is a solution of the Riccati equation
XA + ATX + XBC.,X = 0 (2.4)
such that A~ := A + BC,X satisfies a(AO = {At } ,..) A2 and ( A , , BD, ) is controllable. In that case, the product G(s)e(s) is given by
G(s) A(s)=[ A + B C¢ X~~] Dc c ~ ~ co x IE, J (2.5)
(Proof) From the definition, the Amatrix of the product G(s)O(s) has eigenvalues in cI(A)uo'(AT). Hence, it can be assumed, without loss of generality, that the Amatrix of O(s) is equal to A T. Hence, we put
 t C ~ I Do J (2.6)
for some B c C¢ and De. Then, the product rule yields
G(s)O(s) = 0 A T Be (2.7)
C DC~ DDc.
Since At c o(A T) and A2 c o(A), we have
BC_~ Ml M2 Mt M2 At 0
0 A T NI N2 Nt N2 0 a~ (2.8)
for some MI , M2, Nt and N2 such that 6(AI)= {At } u A2 • Assume that O(s) is a hi.conjugator. Since the modes associated with Alr are uncontrollable in
(2. I0), we have
for some BL. From (2.8), it follows that AMI + BCdN! = MIAI, ATNt = NtAI. (2.10)
Hence, we have (sl A) Mi  BC.~I = Mt (sl Al) (2.1 1)
3
Premultiplication of (2.11) by (sl  A) "t and postmultiplication by (sI  A)aa l yield
Idt (sI  Al) "1Bt = (st  A) "l BO(s). (2.12)
Here, we used the relation N~ (sl  At) "~ B~ = (sl + A'r) "t B,, which is due to (2.9) and the second relation of (2. I0). The condition (2.5) implies that Mt is nonsingular and (At, 130 is controllable.
Taking X=NIMI "l in (2.10) yields (2.4). Since MIIAcMI=AI, A c should be stable. It is clear that (^¢, BDc)=(MflAtMt, MIBI) is controllable because (Ab BI) is controllable.
Due to (2.9), we have Be=NIBI=NIMI'IBDc=XBDc, from which (2.3) follows. The derivation of (2.5) is straightforward.
Remark. Theorem 2.2 implies that the conjugator of G(s) in (2.4) depends only on (A, B). Therefore, we sometimes call it a conjugator of (A. B).
Remark It is clear, from (2.7), that the order of O(s) given in (2.6) is equal to the rank of X which is equal to the number of elements in Lt.
3. (J, J ' ) Loss l e s s C o n j u g a t o r s
A matrix O(s)~ RL" is said to be (J, J')unitary, if tm*r)x(p*q)
O(s) J O (s) = J' (3.1)
for each s. A (J, J')unitary matrix e (s ) is said to be (J, J')tossless, if O'(s) J O (s) g J', (3.2)
for each Re[s]>0, where
j = [ I,, 0 ] j ,=[ I~ 0 ] m>_p, r_>q (3.3) 0 I, , 0 I~ ,
A (J, J')lossless matrix is called simply a Jlossless matrix.
A conjugator of G(s) which is (L J')lossless is called a (J, J')lossless conjugator of G(s). The existence condition of a Jlossless conjugator, which is of primary importance in what follows, is now stated. T H E O R E M 3.1. Assume that (A, B) is controllable and A has no eigenvalue on the jc0axis. There exists a (J, J')lossless stabilizing (antistabilizing) conjugator of (A, B), if and only if there exists a nonnegative solution P of the Riccati equation
PA + ATP  PBJB'rp = 0 (3.4)
which stabilizes (antistabilizes) A, : A BJB'rp. In that case, the desired (J, I')lossless conjugator of G(s) and the conjugated system are given respectively by
L JB T [ I I , C  DJBTp D ( 3 . 5 )
where Do is any matrix satisfying D~J D, = J'. (ProoJ) Since (2.4) implies X(sl + A T) = (sl  Ac)X, O(s) in (2.3) is represented as
O (s) = Oo (s) De,
O 0 ( s ) = I + C c X ( s I  A c ) "j B, A c = A + B C c X . (3.6)
Assume that P is a solution of the Lyapunov equation P a , + AcT P+ X "r c,'rJ Co X =0 . (3.7)
Then, it follow that
O~ (s) J O0(s) = J + BT (sIAcV) a (XrC~ J + P B) + (J C¢ X + B'r P) ( s t  A¢) "~ B.
Since ~ is stable and (A¢, B) is controllable from the assumption, O(s) is Junitary if and only if J C, X + B r I, = 0. (3.8)
In that case, we have O~(s)JOu(s) = J (s + ~BT(sl  A~)"P(sl  Ac)4B
Hence, in order that Oo(s) is JIossless, P should be nonnegative. From (3.8), (3.7) is written as (3.4) and
O0(s) in (3.6) can be written as O0(s) = I  Jffr(sl + Ar)'IPB, from which the representation (3.5) follows. Now, we shall show a cascade decomposition of the (J, J')Iossless conjugation of (A, B) according
to the modal decomposition of the matrix A. Assumethat the pair (A, B) is of the form
As! A,.,
The solution P of (3.4) is represented in conformable with the partition (3.9) as p__. [ P,, PIa]
PITS P22 . (3.10)
L E M M A 3.2. If the pair (A, B) in (3.9) has a (J, J')lossless conjugator O(s) of the form (3.5), then O(s) is represented as the product
O(s) = O,(s) OKs) (3.11)
of a Jlossless conjugator Ol(s) of (At t, Bt) and a (J, J')lossless conjugator 02(s) of (An, 82) with Bz=Ba+Ptff'hB, , where P~a is a peudo inverse of P,, in (3.10). Actually, e~(s) and OXs) are given respectively as
[J B; ' I 4 B; ' ( 3 . 1 2 )
where ~ = PH  P ~ { ~ a and D is any constant (J, J')unitary matrix. (ProoJ) Since P ~ 0, we have Pt2(I  P~2Pza) = 0. Therefore,
UPU'r =[ A0 Paa0 ] , (U.I)'rB =[_~] . U=[ 10 "PtalP~ ]
From this relation and (3.4), it follows that
0 Pal Aal A~ 0 A T 0 P~ 0 P22 0 P22 (3.13)
where Azz = Aat + P~aPTaAn  A~zl'~ff'rz. Hence, t~AH + A~IA  AB~]B TA = 0. Since ~ ~ 0, Or(s) in (3.12) is a J lossle_ss conjugator of (A,. B~). Also, it is clear that Oz(s) given in (3.12) is a (J, J')lossless conjugator of (Aa:. B9. Since (3.11) implies that Az~ = BzlB TA, we have
e~(s)O~s) = 0 A~ Pzalh 19 = = e(s).
sa~ J~I I t  jdu ' The proof is now complete.
4. (J, J')Lossless F a c t o r i z a t i o n s
A matrix G(s) e RL~',,. o x c**.) is said to have a (J, J')losslessfactorization, if it can be represented as
G(s) = O(s) l'I(s), (4.1)
where O(s)¢ RL~'m,,r)x(v+*) is (J, J')lossless and rI(s)is unimodular in H ' , i.e., both rI(s)and lI(s) "* are in
RH~,,0,, c,,o • It is a generalization of the innerouter factorization of H" matrices to L" matrices.
Since rI(s) is unimodular, it has no unstable zeros nor poles. Therefore, O(s) should include all the unstable zeros and the poles of G(s). The (J, J')lossless conjugation introduced in the previous section gives a suitable machinery to carry out these procedures of "zero extraction" and "pole extraction."
If (4.1) holds, it follows that
rlCs)"J'e(s) J G(s) = lp~. (4.2)
This implies that G(s) should be left invertible. Hence, there exists a complement G'(s) of G(s) such that
[G(s) G'(s) l : = [G'(s) ] LoJ'(s)] (4.3)
exists. Obviously, G*(s) is a left inverse and G'(s) is a left annihilator of G(s), i.e., G'(s) G(s) = I~, s , G'(s) G(s) = 0. (4.4)
The following result gives a method of carrying out a (J, J')iossless factorization based on sequential conjugation of zeros and poles extraction.
T I I E O R E M 4.1. (J, J')lossless factorization of G(s) exists, if and only if the left inverse G*(s) of G(s) has a Jlossless stabilizing conjugator e l ( s ) such that G~(s)Je(s) has a (J, J')iossless antistabilizing
conjugator O2(s) such that GJ(s )e , ( s )e2(s ) :O (4.5)
(ProojO To prove the necessity, assume that G(s) allows a (J, J')lossless factorization (4.1). Ciearly, we have
G(s) J rr'(s)J' Therefore, e(s) is a (J, J')Iossless stabilizing conjugator of G+(s), as well as a (J, J')iossless anti stabilizing conjugator of G~(s)J. Due to Lemma 3.2, e(s) can be factored as (3.11) where e l ( s ) is stable J lossless matrix and e2(s) is an antistable (I, J')lossless matrix. From the wellknown property of (1, I') lossless matrix, O2(s) has no unstable zero. From
G(s )=e t(s)e2(s)rl(s), (4.6)
6+(s)et (s) = Ilt(s) satisfies nt(s)e2(s)n(s) = I. Since e2(s)rI(s) has no unstable zeros, rlt(s) is stable. Therefore e t (s ) is a stabilizing conjugator of G+(s). Since G~(s)Jet(s)e2(s) =rI~(s)J ' , and rl(s)J is anti stable, e2(s) is a (J, J ')dossless antistabilizing conjugator of G~(s)Jel(s). The relation (4.5) is obvious from (4.6). Now, the necessity of the theorem has been proven.
Assume that there exist Or(s) and O2(s) satisfying the condit ions of the theorem. Let lql(s)=G+(s)et(s) and lq2(s) = G~(s)JOt(s)O2(s). From the assumption, lIt(s) is stable and l12(s ) is anti stable. Since e l (s ) is Jlossless, e l (s )Je l (S)=J (see[6]). Therefore, we have lqt(s)JOl~(S)JG(s) =I. Since lIt(s) is stable, G~(s)JOI(S) has no stable zero. Since conjugator does not create zero, rlz(s) has no stable zero. Therefore, 112~(s) is unimodular.
Finally, we shall show that the relation (4.1) holds with e(s)=et(s)o2(s) and lI(s)=J'rlz(s). Due to (4.5), Ot(s)O2(s)=G(s)U(s) for some U(s). Since Fl(s) = e2(s)e](s)JG(s), we have II2~(s)U(s) = J'. The assertion follows immediately.
Based on the above theorem and its proof, we can state an algorithm of (J, J')lossless factorization.
[Algor i thm of (J, J ' )  Ioss less Fae to r i za t ion ] Step 1. Find a Jlossless stabilizing conjugator el(s) of G+(s). Step 2. Find a (J, J')lossless antistabilizing conjugator O2(s) of G~(s)Jet(s) satisfying (4.5). Step 3. Put o(s)=ot(s)e2(s) and II(s)=J'O~(s)JG(s), which give a (J, J31ossless factorization.
In the above algorithm, Ol(s absorbs all the unstable zeros of G(s). Hence, Step 1 is regarded as a procedure of zero extraction. Similarly, O2(s) in Step 2 absorbs all the unstable poles of G(s), and hence, Step 2 is regarded as pole extraction.
5. StateSpace Theory of (J, J')Lossless Factorizations In this section, we derive a statespace forms of (J, J')lossless factorizations. Let
G(s)=[ A ~ D B ] eR nx,,, Rem*r)x(p*q) A D~ (5.1) be a statespace form of G(s). The following assumptions are made :
(A0 The matrix A has no cigenvalucs on the jo)axis.
(A2) The matrix D is of full column rank.
(A3) G~(s)JG(s) admits no unstable polezero cancellation.
Due to the assumption (A2), there exists a column complement D' of D such that
[D D, ]I =[ D* D" ] (5.2)
exists. Obviously,
D'D = Imq D'D = 0. (5.3)
Now, we assume that O(s) has a (J, J')lossless factorization (4.1). Since G(s)JG(s)=rI(s)Jn(s), we have
D T J D = D 'r Z' D, (5.4)
where Dx = rl(~,) e R ~p'q)x~p'q). Since Il(s) is unimodular, D~ is nonsingular. Hence, DTjD must b~ non singular.
The left inverse G+(s) and a left annihilator Gl(s) of G(s) satisfying (4.6) are represented in the state space as
[O'(s)J D~ (5.5) D.tC
whcrc U is an arbitrary matrix which is detcrmincd latcr. Now, wc shall carry out the (J, J')losslcss factorization of G(s) based on thc algorithm dcrivcd in
Section 4.
Step l (Zero Extraction) Due to Theorem 3.1, a Jlossless stabilizing conjugator Or(s) of G+(s) in (5.5) is calculated to be
] [ JL 'r [ 1 ], (5.6)
where X is a nonnegative solution of the Riccati equation
X(A LC) + (A LC)TX  xLTjL'rX = 0 (5.7) which stabilizes
At := A L(C + jL'rX) (5.8i
Step 2 (Pole Extraction) Straightforward computation using LD = B yields
Ar 'r I I C'rj+XL] G(s)JO,(s) = [ ~ ] O,(s) = "At [ B "r I DTJ ] .B v DTj , (5.9)
Due to Theorem 3.1, a (J, J')lossless antistabilizing conjugator of (5.9) is given by
O.2(s)_[ A ]Y(C'rj + XL) ] DDk ' LcJL'rX[ I (5.10)
A T 0
r, o o l r , o xl o / /o , ° t
LOY I J LO 0 I J ,
where Dr is any matrix satisfying (5.4) and v is a nonnegative solution of the Riccati equation
VA T + AY + Y(c'r+ XLJ) J (C + jLTx)y = 0 (5. I 1) which stabilizes
Az := A + Y(cT+ XLI) J (C + JLTX), (5.12)
NOW, concatenation rule yields
O(s) = 8Ks)O~s) = 0 A y(CTJD+XB) Dn (5.I3)
JL T CJLTX) D
We use the free parameter U in (5.5) to satisfy (4.5). From (5.5), it follows that
G±(s)B(s) = [ At I (I+YX)B+yCTjD ] D't(C+JL'rX ") 0 .
Assumption (A3) implies that ( A1, (I+YX)B+yCTjD) is controllable. Hence, (4.5) holds, if and only if
Dt(C+jLTx) = 0. (5.14)
Thus, the free parameter U in (5.5) must be determined such that (5.14) holds. The relation (5.14) holds, if and only if
LTX = J (DF+ C) (5.15)
for some matrix F. Since LD=B for any choice of U, we have
F= (DTjD)'I(BTx + DTjc). (5.16)
Straightforward computation using (5.15) and (5.16) verifies that the Riccati equation (5.7) is written as
XA + ATx  (cTjD+XB)(DTjD)'I(DTJC+BTX)+cTjc = 0. (5.17)
The matrix At given in (5.8) is calculated to be
Al = A +BF. (5.18) Also, the Riccati equation (5.11) and A2 in (5.12) are represented respectively, as
YAT+AY+yF'rDTJDFY = 0 , A2 = A + yFTDTJDF (5.19)
Step 3. Using (5.7), we can write the statespace form of O(s)=Ol(s)O2(s) in (5.13) as
[ A, BF B ] ! A yFTDTJD [ Dk! O(s) = 0 _ • ( 5 . 2 0 )
L(C+DI) DF D j
where Dg is any matrix satisfying (5.4). It remains to compute the unimodular factor rl(s) = J'O(s)JG(s). The concatenation rule yields
(C+DF)Tjc .(C+DF)'rjc FTDTJC FTDTJD
A B
we obtain
Fl(s) = D" [ ~ IB'YFITDTJD 1. (5.21)
Now, we can state the main result of this paper. THEOREM 5.1. Assume that G(s) given in (5.1) satisfies Assumptions (AI)(A2)(A3). Then, G(s) has a (J, J')losslcss factorization, if and only if there exist a nonsingular matrix D= ~ R ~'e'L''e satisfying (5.4), a solution × > 0 of the Riccati equation (5.17) which stabilizizes A I given in (5.18)(5.16) and a solution Y 2_ 0 of the Riccati equation (5.19) which stabilizes A2 given in (5.19). In that case, a (J, J')losslcss factor e(s) is given in (5.20) and a unimodular factor rl(s) is given in (5.2t).
The case of stable G(s) is much simpler. The result is exactly the same as was obtained in [8]. COROLLARY 5.2. Assume that A is stable in addition to (AI)(A2) and (A3). Then, G(s) has a (J, J') lossless factorization (4.1), if and only if there exists a nonsingular matrix Dx ~ R t* '¢° '¢ satisfying (5.4) and the Riccati equation (5.17) has the solution x > 0 which stabilizes Al given in (5.18).
If the above condition hold, the factors O(s) and rl(s) of (4.1) are given respectively by
'1. C+DFI DJ , LFIIJ . (5.22)
(Prooj~ IrA is stable, we can take Y0 in solving (5.19). The assertions of the corollary follow immediately from (5.20) and (5.21) for Y=0.
6. Conclusion A necessary and sufficient condition for the existence of (J, J')lossless factorization in RL '~' has
been established in the state space based on the method of conjugation. It includes the result of [8] for RH** as a special case. The derivation is simple, exhibiting the power of conjugation method.
The result obtained in this paper is directly applied to !1 ¢~ control problems without any recourse to Youla parameterization, giving a simpler result for both standard and nonstandard plants. This will be published elsewhere.
REFERENCES [ 11 J.A. Ball and N. Cohen, "The sensitivity minimization in an H** norm : Parameterization of all optimal solutions," Int. J. Control, 46, pp.785816 (1987) [21 J.A. Ball and J.W. Hehon, A BeurlingLax Theorem for the Lie group U(m, n) which contains most classical imterpolation," J. Operator Th., 9, pp. 107142 (1983) [3] J.A. Ball, J.W. Hehon and M. Verma, "A factorization principle for stabilization of linear control systems," Reprint. [4] J.A. Ball and A.C.M. Ran, "Optimal Hankel norm model reduction and WienerHopf factorization I : The canonical case,'SIAM J. Contr.Optimiz., 25, pp.362382 (1987). [5] H. Ban, I. Gohberg, M.A. Kaashoek and P. van Dooren, "Factorizations of transfer functions," SlAM J. Contr. Optimiz. 18, pp. 675696 (1980). [61 P. Dewilde, "Lossless Chainscattering matrices and optimum linear prediction : The vector case," Circuit Th. & Appl., 9, pp.135175 (1981). [7] B.A. Francis, A Course in Ho, Control, Springer, New York (1987). [8] M. Green, K. Glover, D. Limebeer and J. Doyle, "A Jspectral factorization approach to H,o control," SlAM J. Contr. Optimiz., 28, pp.13501371 (1990). [9] H. Kimura, "Conjugation, interpolation and modelmatching in Hoo ," Int. J. Control, 49, pp.269307 (1989). [ 10] H. Kimura, "State space approach to the classical interpolation problem and its applications, "in Three Decades of Mathematical System Theory, H. Nijmeijer and J.M. Schumacher (eds.), SpringerVerlag, ,pp. 243275(1989) [11] H. Kimura, Y. Lu and R. Kawatani, "On the structure of H,o control systems and related extensions," IEEE Trans. Auto. Contr., AC36, pp. 653667 (1991). [12] H. Kimura, "Chainscattering formulation of H,,* control problems," Proc. Workshop on Robust Control, San Antonio (1991).
Mixed H2/H Filtering by the Theory of Nash Games
1D.J.N. Limebeer 2B.D.O. Anderson 1 B. Hendel
Department of Electrical Engineering, I Department of Systems Engineering, 2 Imperial College, Australian National University, Exhibition Rd., Canberra, London. Australia.
A b s t r a c t
The aim of this paper is to study an H2/Hoo terminal staSe estimation problem using the classical theory of Nash equilibria. The H2/Hoo nature of the problem comes from the fact that we seek an estimator which satis fies two Nash inequalities. The first reflects an Hoo filtering requirement in the sense alluded to in [4], while the second inequality demands that the estimator be optimal in the sense of minimising the variance of the termi nal state estimation error. The problem solution exploits a duality with the H2/Hoo control problem studied in [2, 3]. By exploiting duality in this way, one may quickly cxtablish that an estimator exists which staisfies the two Nash inequalities if and only if a certain pair of cross coupled Riceati equations has a solution on some optimisation interval. We conclude the paper by showing that the Kalman filtering, Hoo filtering and H2/Hoo fil tering problems may all be captured within a unifying Nash game theoretic framework.
1 I n t r o d u c t i o n In this paper we seek to solve a mixed H2/Hoo terminal state estimation problem by formulating it as a two player nonzero sum Nash differential game. As is well known [1, 5], two player nonzero sum games have two performance criteria, and the idea is to use one performance index to reflect an Hoo filtering criterion, while the second reflects the H2 optimality criterion usually associated with Kalman filtering. Following the precise problem statement given in section 2.1, we reformulate the filtering problem as a deterministic mixed H~/Hoo control problem in section 2.2. This approach offers two advantages in that the transformation is relatively simple, and we can then use an adaptation of the mixed H~/Hoo theory given in [2] to derive the necessary and sufficient conditions for the existence of the Nash equilibrium solution to the derived control problem, and therefore for the existence of optimal linear estimators which solve
10
the filtering problem. These conditions are presented in section 2.3 where we also derive the dynamics of an online estimator. Some properties of the resulting mixed filter are given in section 2.4. The aim of section 2.5 is to provide a reconciliation with Kalman and Hoo filtering. In particular, we show that Kalman, Hoo and Hz/Hoo filters may all be captured as special cases of a another two player Nash game. Brief conclusions appear in section 3.
Our notation and conventions are standard. For A E C "nxm, A ~ denotes the complex conjugate transpose, g{.} is the expectation operator. We denote I]RII21 as the operator norm induced by the usual 2norm on functions of time. That is :
IIRII21 = sup [IRu[[2
2 The H2/H~ filtering problem
2 . 1 P r o b l e m S t a t e m e n t
We consider a plant of the form
k(t) = A(t)z(t) + B(t)w(t) x(to) = zo ( 2.1 )
with noisy observations
zt(t) = Cl(t)z(t) ( 2 .2 ) ~ ( t ) = c2(t)x(t) + n~(t), ( 2.3 )
in which the entries of A(t), B(t), C1 (t) and C2(/) are continuous functions of time. The inputs w(t) and n2(t) are assumed to be independent zeromean white noise processes with
~{w(t)w'Cr)} = Q(t)6(t  ~') ( 2.4 ) E{n2(t)n2'Cr)} = R2Ct)6(t 7) Vt ,r e [t0,tfl. ( 2.5 )
The matrices Q :> 0 and R~ > 0 are symmetric and have entries which are continuous functions of time.
Our final assumption concerns the initial state. We let z0 be a gaussian random variable with
~{x0} = m0, c{(~0  m0)(~0  m0)'} = P0. ( 2.8 )
Finally, we assume that z0 is independent of the noise processes, i.e.
£{z0w'Ct)} = 0, ~{~0n~(t)} = 0; V t e [t0,t:].
The estimate of the terminal state z( t l ) is based on both observations zx(t) and z2(t), and the class of estimators under consideration is given by
~t~ ! ~(tj) = [MI(~, ts)~l(~) + M2(~,tf)z2(~)] aT (2.7)
where Mi (r, ty) and M2(r, if) are linear timevarying impulse responses.
11
The filtering problem is formulated as a twoplayer differential Nash game. The first player has access to the observation zi(t) and tries to maximise the variance of the estimation error by choosing an M1 (.,t/) which minimises the Hoo filtering cost function [4]
JI(M1, Mr) = CIIMdl ]  ~ {[=(tD  ~(tl)1'[=(~1)  ~(ts)]}. (2.8)
7rllMt Ill is a penalty term introduced to prevent player one from assigning an arbitrarily large value to M1 thus driving J1 to minus infinity. The second player has access to the observation z2(t) and attempts to minimise the error variance by selecting an Mr which minimises the Kalman filtering payoff function
Jr(Mr, Mr) = ~ {[x(t/)  fc(t/)]'[z(t/)  ~:(t/)]} . ( 2 . 9 )
We therefore seek two linear estimators M~" and M~ which satisfy the Nash equilibria
JI(M[,M~) < JI(Mt,M~) (2 .10 )
J2(M[,M~) <_ J2(M~,M2). ( 2.11 )
2 . 2 P r o b l e m R e f o r m u l a t i o n
In order to solve the stochastic filtering problem we transform it into a deterministic control problem. This approach has two advantages in that the reformulation is relatively simple, and we can then use an adaptation of the mixed Hr/Hoo theory [2] to derive the necessary and sufficient conditions for the existence of the Nash equilibria M~' and M~.
We begin by introducing the timevarying matrix differential equation [4]
Z(t, tl) = A'( t)Z(t , tl) + C~(t)Ml'(t, tl) + C~(t)M2'(t,tl) (2.12)
with Z(lc,t/) = I. The aim is to rewrite the cost functions (2.8) and (2.9) as quadratic performance indices involving Z(.), MI(.) and Ms(). By direct calculation
d (Z(,, t j)',(,)) dt
I I I I = [  A ' Z + C I M t ' + C r M 2 ] (t)x(t)
+ Z(t,tl)'[m(t)z(t) + B(t)w(t)] = Ml(t,lf)zl(t) + Mr(t, tl)[zr(t)  n2(t)]
+ Z(t,t+)'B(t)w(t).
Integrating over [to,if] gives
=(ty)  z ' ( to, ty)~o = { M l z l + M2[z2  n2] + Z 'Bw} (Odt.
Rearranging and using (2.7) yields
fi' • ( t l )  ~ (~I ) = z ' ( t o , t . l ) ~ o  { M ~ , ~  Z ' B ~ , } ( t ) d t .
12
In the next step we square both sides and take expectations making use of the assumed statistical properties of the noise processes and the initial state. This gives
f + z'(to,ts)poZ(to,~1). ( 2.13 )
We can now use (2,13) to rewrite J1 and ,/2 as
= ~/'s(7~MIM 1' M' Z'BQB'Z)(t)dt} J, tr  M2R2 2  k ~o
 ~r{Z'C~o,t~)PoZC~o,tj)} ( 2a4 )
k J t o
+ fr{Z'(to,tf)PoZ(to,t])}. ( 2.15 ) Notice that both payoff functions are now deterministic.
The filtering problem has thus been transformed into the mixed H2/Hoo control problem of finding equilibrium solutions MI" and M~ which satisfy
JI(M~,M~) <_ J,(MI,M~) ( 2 . 16 )
J2(M{,M{) < J2(M~,M2) ( 2.17 )
where J1 and J~ arc given by (2.14) and (2.15); the matrix Z(t) is constrained by (2.12).
R e m a r k s (i) Notice that the "state" Z(Q is matrix valued and the 'dynamics ' have a terminal condition Z(tl) = I. (it) The optimal linear estimators M~ and M~ are matrix valued too. (iii) The initial condition data in the form of Z'(to)PoZ(to) appears in the cost func tionMs.
2 . 3 F i n d i n g a n o n  l i n e e s t i m a t e
The necessary and sufficient conditions for the existence of Nash equilibrium solutions to the mixed H2/Hoo control problem, and therefore for the existence of linear estimators M~" and M~ follow directly from [2], and are presented in the following theorem. Scaling arguments have allowed us to assume without loss of generality that R2 = Q = I .
T h e o r e m 2.1 Given the system and observations described by (2.1) to (2.3), there exist transformatio~,s M~ and M~ which satisfy the Nash equilibria (2.16) and (2.17) with Jx and J2 defined in (2.14) and (2.15) if and only if the coupled Riecati differential equations
/'I(L) = AP,(t) + PI ( t )A '  BB'
 [P l ( t ) P ~ ( , ) ] l 7~C[C1 C~C2 Pl(t)
P2(t) = mP2(t) + P2(t)A'+ BB'
13
with Pt(to) = Po and P~(to) = Po, have sotu.ons Pt( t ) <_ O, e~(t) > 0 on (2.18) and (2.19) have solutions,
7 2Z' ( t , tl)P~(t)C I Z ' ( t , t l ) P z ( t ) C ~.
M~( t , t s ) =
[to,~sl  ~'/'
( 2.20 ) ( 2.21 )
Since Z ( t , t l ) = Z~(t l , t ) , we have
d~I Z(L,tj)= d~Z~(t~,,).
 z ~(t:,t) ~~, z(tj,t)z~(,1, t)
= Z t ( t l , t ) (A '  72C~01P1  C~C~P2)(tl),
and then (2.25) :=>
~ Z'(t, is) = ( A  72 ptc[ c1  P~C~C2)(Q )Z'(t, tl).
Now, from (2.7)
/,i' Differentiating with respect to t! and using (2.26) we obtain
or equivalently
= (A  72P1C[C1  PzC~C2)(tI)~(tl)
+ (v"P,C'~zl + P2C~z2)(t:) e ( to ) = ,no
 K,(t l )[C~(~l)~Ctl) zl(e/)]
 z<2Cts)[C~(t f )~( t f )  z~(tl)]
KI(f / ) = 72Pi(ll)Ci(tI) ( 2.29 )
K~(ts) = P~(t:)C~(tf) . ( 2.30 )
The matrices Pl( t l ) and P2(t/) satisfy the coupled Riccati equations (2.18) and (2.19).
(2.28)
(2.26)
( 2.27 )
= ~o~(tj) ( 2.22 ) = P~(t~). ( 2.23 )
With a little manipulation we can now find an online estimate for z( t l ) based on the observations zl(tl) and z2(tI), in the same sense that the Kalman filter provides an on line state estimate using current observations. Implementing the equilibrium strategies in (2.12) gives
Z(t, t]) = ( A ' + 7~PIC~C1 + P2C~C2)Z(t,II) Z ( t l , t l ) = I. ( 2.24 )
14
2.4 T h e m i x e d H 2 / H c ¢ f i l ter In this section we show how the H2/He¢ filter is defined in terms of the above analysis, and we can present some of its properties. Suppose M~ is replaced by a dormant player as follows : Calculate M[ and M~ as above, and then in (2.27) replace the term multiplying zl by zero, i.e. we take M~ = 0 • M~. This means that in (2.28) we replace g~(t l ) by zero to obtain
~ ( t / ) = [A(tl) K~(t/)C~(~/)l~(t D + g2(*1)C~(tl)z~(tS); ~(t0) = ITI0}
which is the defining equation of the H2/Hoo filter. It is easy to see from this equation that the tI2/Hoo filter has thc usual observer structure associated with Kalmaa and Hoo filters. The following result provides an upper bound on the variance of the terminal state estimation error, and it shows that if the white noise assumption on the disturbance inputs w and n2 is dropped, the energy gain from the disturbances to the generalised estimation error Ct(~:(tl)  z( tI) ) is bounded above by 7 for all w, n2 E £2[to, tl].
T h e o r e m 2.2 If M1 ~ O,
(i) the online state estimate is generated by
d~l~:(tf) = (A  P2C~C2)(t/)~(tf) + P2(tl)C~(tl)z2(tl) ~(to) = ( 2.31 ) mo~
(it) the covariance of the estimation error e(tl) = ~(tl)  z ( t l ) satisfies
£{e(tf)e'(t /) } <  P l ( t l ) ( 2.32 )
and (iii) when mo = 0, Ilnll~ < ~ for all w, n2 E £z[t0,tl], where the operator T"g is
described by
z = C1e(tI) (2 .34 )
2.5 R e c o n c i l i a t i o n o f K a l m a n , H ~ a n d H 2 / H ~ fil t e r i n g
Here we establish a link between Kalman, H~ and mixed H2/H~ filtering. Each of the three theories may be generated as special cases of the following nonzero sum, twoplayer Nash differential game:
Given the system described by (2.1) to (2.3) and a state estimate of the form given in (2.7), we seek linear estimators M~' and M~ which satisfy the Nash equilibria JI(M~, M~) < JI(MI, M~) and J2(M~, M~) < J2(M~, M2), where
J2 = ~: {[z(tf) ~(tl)]'[z(ts) ~(tj)]} p211Mdl ~.
15 This game can be reformulated as a deterministic control problem and is solved
by M~(t,tl) = 72Z'(t,t/)Sl(t)C~ and M~(t,~l) = Z'(t , t l)S2(t)C ~, where Z(t , t l ) is generated by (2.12) and SI(t) and $2(/) satisfy the coupled Riccati differential equations
Sl(t) = A S t ( t ) + S t ( t ) A '  B B '
 [ s ct) s (l) ] 1 72qC, C C2 St(t)
s2(t) $2(~) = AS2(t) + S=(t)A' + BB'
 [ s,(t) s=(t) ] / p2740~C * 72C',C, C~C2 ][ S2(t)] 7~C~C1 S,(t) L
with Sl(t l) = S2(tl) = O.
(i) The Kalman filtering problem is recovered by setting p = 0 and 3' = oo, in which case S l ( t ) = S2(t) = Pc(t). The matrix Pe(~) is the solution to the usual Kalman filter Riccati equation. (ii) The pure Hoo filtering problem is recovered by setting p = 3'. Again, S , ( t ) = S2(t) = P~(t). The matrix Poo(t) is the solution to the Hc¢ filtering Riccati equation [4]. (iii) The mixed H2/Hoo filtering problem comes from p = 0.
3 C o n c l u s i o n s We have shown how a mixed H2/Hoo filtering problem may be formulated as a deter ministic mixed II2/Hoo control problem. The control problem can then solved using the Nash game techniques explained in [2]. The necessary and sufficient conditions for the existence of a solution to the filtering problem are given in terms of the existence of solutions to a pair of crosscoupled Riccati differential equations. We have derived an upper bound on the covariance of the state estimation error in terms of one of the solu tions to the coupled equations and have shown that, under certain conditions, the energy gain from the disturbance inputs to the generalised estimation error Cl (x ( t / )  ~(t/) ) is bounded above by 7
R e f e r e n c e s [1] T. Basar and G. 3. Olsder,"Dynamic noncooperative game theory," Academic Press,
New York, 1982
[2] D. J. N. Limebeer, B. D. O. Anderson and B. IIendel,"A Nash game approach to mixed H2/Hoo control," submitted for publication
[3] D. :I. N. Limebeer, B. D. O. Anderson and B. Hendel,"Nash games and mixed H2/Hoo control," preprint
[4] D. J. N. Limebeer and U. Shaked,"Minimax terminal state estimation and Hoo filtering," submitted to IEEE Trans. Auto. Control
[5] A. W. Start and Y. C. Ho,"Nonzerosum differential games," J. Optimizalion Theory and Applications, Vol.3, No. 3, pp 184206, 1967
The Principle of the Argument and its Application to the Stability and Robust Stability Problems
Mohamed Mansour Automatic Control Laboratory
Swiss Federal Institute of Technology Zurich, Switzerland
Abstrac t
It is shown that the principle of the argument is the basis for the different stability criteria for linear continuous and discrete systems. From the principle of argument stability criteria in the frequency domain are derived which lead to HermiteBieler theorems for continuous and discrete systems. RouthHurwitz crite rion and its equivalent for discrete systems, SchurCohn criterion and its equivalent for continuous systems are directly obtained. Moreover, the monotony of the argu ment for different functions can be proved using HermiteBieler theorem. Using the principle of the argument and the continuity of the roots of a polynomial w.r.t, its coefficients, all known robust stability results can be obtained in a straightforward and simple way.
1 Principle of the Argument
Consider a function f(z) which is regular inside a closed contour C and is not zero at any point on the contour and let Ac{arg f(z)} denotes the variation of arg f(z) round the contour C, then
T h e o r e m 1 [1] AcTarg fCz)} = 2~N (1)
where N is the number @zeros o f f (z ) inside C.
2 Stability Criterion using Principle of the Argu ment
2.1 C o n t i n u o u s S y s t e m s
Consider the characteristic polynomial of a continuous system
f ( s ) = aos n + a , s n1 + . . . + a .  l s + a n (2)
T h e o r e m 2 f (s) has all its roots in the left half of ~he splane if and only if f( jw) has a change of argument ofn~ when w changes from 0 to ~ .
17
The proof of this criterion which is known in the literature as CremerLeonhardMichailov Criterion is based on the fact that a negative real root contributes an angle of ~ to the change of argument, and a pair of complex roots in the left half plane contributes an angle of ~r to the change of argument as to changes from 0 to oo. It is easy to show geometrically that the argument is monotonically increasing.
f(s) can be written as f(s) = h(s 2) + sg(s 2) (3)
where h(s ~) and sg(s z) are the even and odd parts of f (s) respectively.
f (jto ) = h(to 2) k jtog(to2) (4)
2.2 D i s c r e t e S y s t e m s
Consider the characteristic polynomial of a discrete system
/ ( z ) = aoz" + a l z " 1 + ... + a . _ : + a . (5)
T h e o r e m 3 f ( z ) has all its roots inside the unit circle of the zplane if and only if f ( J °) has a change of argument of nrc when 0 changes from 0 go re.
The proof of this criterion is based on the fact that a real root inside the unit circle contributes an angle of 7r to the change of argument, and a pair of complex roots inside the unit circle contributes an angle of 2~r to the change of argument as 0 changes from 0 to lr. It is easy to show geometrically that the argument is monotonically increasing
f (z) can be written as f (z) = h(z) + g(z) (6)
where 1[ h(z) = 7 f(z) q znf = C~o zn + oqz '~x q" ... I" otlz "I" O:o (7)
and 1 [ ( 1 ) ]
g(Z) = ~ f(Z) "q" znf = fro zn "~"/~I zn1 21" .. /~I z  80 (8)
are the symmetric and antisymmetric parts of f (z) respectively. Also
f(e i') = 2ein°/2[h" k J9*] (9)
where for n even, n = 2v
OtV h*(O) = £I' 0 cosvO I" 0~I COS(// 1)0 + ... + o~,_x cosO+ ~
(1o) g'(O) = 3 0 s i n v O + 3 1 s i n ( u  1 ) 0 + . . . + P .  l s i n 0
From theorem 3 and equation (9) we get
T h e o r e m 4 f ( z ) has all its roots inside ~he unit circle of the zplane if and only if f* = h* + jg* has a change of argument of n~ when 0 changes from 0 to r.
f" is a complex function w.r.t, a rotating axis. It can be shown geometrically that the argument of f* increases monotonically. Fig. 1 shows the change of coordinates.
g*
Figure 1: Change of coordinates
3 Hermi te B ie l er T h e o r e m
3.1 Continuous Systems
Consider the polynomial in (3).
T h e o r e m 5 (HermiteBieler Theorem) f(s) has all its roots in the left half of the s plane if and only if h(A) and g()~) have simple real negative alternating roots and ~o > O.
a 1
The first root next to zero is of h(a).
The proof of this theorem follows directly from theorem 1; [2]. If theorem 5 is satisfied then h(,~) and 0(,~) are called a positive pair.
3 .2 D i s c r e t e S y s t e m s
Consider polynomial (5).
T h e o r e m 6 (Discrete HermiteBieler Theorem) f(z) has all its roots inside the unit circle of the zplane if and only if h(z) and g(z) have simple alternating roots on the unit
~ ' l < 1; [3], [4]. circle and a,
The proof can be sketched as follows: Due to theorem 4, h ° and g" and hence h and g will go through 0 alternatively n times
I"1 < 1 distinguishes between f (z) as 0 changes from 0 to r . The necessary condition ao
and its inverse which has all roots outside the unit circle. I I
4 M o n o t o n y of the argument
In [5] it is shown that the argument of a Hurwitz polynomial f ( s ) = h(s 2) + sg(s 2) in the frequency domain f(j•) = h( w 2) + jwg(_~2) as well as the argument of the modified functions f~'(~) = h (  ~ ~) + j 0 (  ~ 2) and f~(~) = h (  ~ 2) + j ~ 2 g ( _ ~ ) are monotonically increasing functions of w, whereby w varies between 0 and co. Similarly it is shown that given a Schur polynomial f (z) = h(z) + g(z) where h(z) and g(z) are the
19
symmetric and the antisymmetric parts of f (z) respectively, the same monotony property can be obtained for an auxiliary function .f*(0) defined as f(e j°) = 2J"° /2f f (0) . The argument of f*(0) = h*(0) + jg*(O) as well as the argument of the modified functions
and
h'(0) .g.(0)
are monotonically increasing functions of 0 whereby 0 varies between 0 and a'. These results were proved in [5] using mathematical induction. However, a simpler proof can be obtained using networktheoretical results which is based on ttermiteBieler theorems, [6]. The results obtained can be directly applied to the robust stability problem [5].
5 H e r m i t e  B i e l e r T h e o r e m and Cri ter ia for H u r w i t z S tab i l i ty
In this section it is shown how different criteria for Hurwitz s~ability like RouthHurwitz criterion and equivalent of SchurCohn criterion for continuous systems are derived from HermiteBieler theorem. Also Kharitonov theorem [7], [8] can be similarly derived, [9].
5 .1 R o u t h  H u r w i t z C r i t e r i o n
Without loss of generality consider n even. The first and second rows of the Routh table are given by the coefficients of h and g respectively. The third row is given by the coefficients of h  a°s2g. The following theorem shows that Routh table reduces
. . a l .  .
the stablhty of a polynomial of degree n to another one of degree n  1. Repeating this operation we arrive at a polynomial of degree one whose stability is easily determined.
T h e o r e m 'i t [2], [10] f ( s ) is Hurwitz slable if and only if h  ~~s2g + sg is Hurmitz stable.
Proof : Let f ( s ) = h + sg be Hurwitz stable i.e. ao,al,...a,~ > 0 and according to theorem 5, h(~) and g(,~) have simple real negative alternating roots, h  a 2 ~ s g and h
have the same sign at the roots of g and the number of roots of h ~ 2  als g + s g is n  1.
Moreover a l a 2  aoa3 > 0. Therefore, according to theorem 5, h  aa~s2g + sg is Hurwitz stable. The proof of the if part is similar. []
5 . 2 E q u i v a l e n t o f S c h u r  C o h n  J u r y C r i t e r i o n f o r C o n t i n u o u s S y s 
t e m s
T h e o r e m 8 [11] f ( s ) is Hurtoitz stable if and only if f ( s )  l ~ f (  s ) is Hurwitz
20
 1 is a root of f(s)  t ~ f (  s ) which can be eliminated, so that the stability of a polynomial of degree n is again reduced to the stability of a polynomial of degree n  1. This theorem can be proved using Rouch~ theorem. Here we give a proof based on HermiteBieler theorem and on the following Lemma.
L e m m a 1 h + sg is gurwitz stable if and only if 7h + 6sg is gurwitz stable for 7, 5 positive numbers.
Proof : The interlacing property does not change by multiplying h and g by positive numbers. Also ~ > 0. I"3
7ao
Proof : Proof of theorem 8. Let f(s) be tlurwitz stable. If l ( s ) = h + s g then/(X) = h  s g
Therefore
Using Lemma 1 we get I(s)  l ~ I (  s ) is Hurwitz stable. The proof of the if part is similar. El
6 Hermi te B ie l er T h e o r e m and Criteria for Schur Stabi l i ty
In this section it is shown how different criteria for Schur stability like the equivalent of RouthHurwitz criterion for discrete systems and SchurCohnJury criterion are derived from HermiteBieler theorem. Also analog result of Kharitonov theorem can be obtained; [91, [121. Consider f(z) given by equation (5). Without loss of generality we assume a0 > 0 and n even. For n odd a similar treatment can be made. A necessary condition for Schur
[~~1 < 1 which results in a0 > 0 and fl0 > 0. stability of (5) is o,
6 .1 E q u i v a l e n t o f R o u t h  H u r w i t z C r i t e r i o n f o r D i s c r e t e S y s t e m s
If h(z) and g(z) given by (7) and (8) fulfill the discrete HermiteBieler theorem then their projections on the real axis h*(x) and g*(x) have distinct interlacing real roots between  1 and +1. In this case h*(x) and g*(x) form a discrete positive pair. From [9] h*(x) and g'(x) are given by
v  1
(11)
where n = 2v and Tk, Uk are Tshebyshev polynomials of the first and second kind respectively.
21
Let
g*(X) = 6oX ~1 +6zX ~2+ . . .
T h e o r e m 9 [11] f ( z ) is Schur stable if and only if g*(X ) and h ' ( x )  ~ ( X  1)g*(X) form a discrete positive pair and h*(1) < 0 for v odd and h*(1) > 0 for v even.
Proof : Assume f ( z ) be Schur stable, h*(x)  ~o(X  1)g*(x) has the same value at the roots of g*(X) as h*(x). Therefore, the interlacing property is preserved. The condition h*(1) < 0 for v odd and h*(1) > 0 for v even guarantees that the reduction of the order is not due to a root of h*(x) between  1 and  c o . Therefore g*(X) and h*(x)  ~.(X  1)g*(X) form a discrete positive pair. The if part can be proved similarly. I"1
This theorem can be implemented in table form.
6.2 SchurCohnJury Criterion
I ~  [ < 1 i s S c h u r s t a b l e i f a n d o n l y i f T h e o r e m 10 [4], [111 The polynomial (5) with a.
the polynomial 1 I f ( z )  aa~z" f (1)] is Schur stable.
This was proved in [4] using the following Lemma:
L e m m a 2 h + g is Schur stable if and only if 7h + ~g is Schur stable for 7, 6 positive numbers.
Proof : Proof of Lemma 2: The interlacing property does not change by multiplying h
and g by positive numbers. Also l ~° '6~° [ < 1 if I~t < l. The reverse is true. [] [ 7 ~ o + 6 P 0 [ ao
a n ] Proof : Proof of theorem 10: f ( z )  "a~oz f (7) = h(z) + g(z)  ~o [h(z)  g(z)] = ( a. [1 only if part ( 1  o ) h ( z ) + . + , o ) g ( z ). The of theorem l0 follows directly from
~I < 1 and Lemma 2. all I
The if part can be proved similarly, rq
7 Robust Stability in the Frequency Domain
Let f ( s ,a ) be given by (2) and a E A. A is a closed set in the coefficient space. Let D be an open subset of the complex plane. Robust Dstability means that all roots of the polynomial family lie in D. Let r(s*) be the value set of the polynomial family at s = s" and 0F(s*) its boundary. Using the principle of the argument and the continuity of the roots w.r.t, the coefficients of a polynomial, we get the following result for robust Dstability.
T h e o r e m 11 [13], [14]. f(s,a_) with a_ E A is robust Dstable if and only if the following conditions are satisfied:
22
is) r(s') # 0 for some s" e aD
iii) or(s) # 0 for all s e aD
Using theorem 11 we can solve the following problems in a simple way
i) an edge theorem for the robust stability of polytopes in the coefficient space, [13]
it) stability of edges
iii) vertex properties of edges, [5], [15]
iv) Kharitonov theorem and its analog and counterpart for discrete systems, [16], [17], [12], [xsl
v) robust stability of controlled interval plants continuous and discrete, [14], [19], [20]
A survey of all the above problems is given in [21].
8 C o n c l u s i o n s
In the above it was shown that the stability criteria for linear and discrete systems as well as the solution of the robust stability problem all can be derived from the principle of the argument.
R e f e r e n c e s
[1] E.G. Phillips, Functions of a complex variable with applications, Oliver and Boyd, 1963.
[2] H. Chapellat, M. Mansour, and S. P. Bhattacharyya, Elementary proofs of some classical stability criteria, IEEE Transactions on Education, 33(3):232239, 1990.
[3] H.W. Schiissler, A stability theorem for discrete systems, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP24:8789, 1976.
[4] M. Mansour, Simple proof of SchurCohnJury criterium using principle of argu ment, Internal Report 9008, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[5] M. Mansour and F.J. Kraus, Argument conditions for Hurwitz and Schur stable polynomials and the robust stability problem, Internal Report 9005, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[6] M. Mansour, Monotony of the argument of different functions for Routh and Hut wits stability, Internal Report 9111, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1991.
[7] V. L. Kharitonov, Asymptotic stability of an equifibrium position of a family of systems of linear differential equations, Differentsial'nye Uravneniya, 14:20862088, 1978.
23
[8] B. D. O. Anderson, E. I. Jury, and M. Mansour, On robust Hurwitz polynomials, IEEE Transactions on Automatic Control, 32:909913, 1987.
[9] M. Mansour and F.J. Kraus, On robust stability of Schur polynomials, Internal Report 8705, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1987.
[10] M. Mansour, A simple proof of the RouthHurwitz criterion, Internal Report 8804, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1988.
[11] M. Mansour, Six stability criteria & HermiteBieler theorem, Internal Report 9009, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1990.
[12] M. Mansour, F. J. Kraus, and B.D.O. Anderson, Strong Kharitonov theorem for discrete systems, in Robustness in Identification and Control, Ed. M. Milanese, R. Tempo, A. Vicino, Plenum Publishing Co~oration, 1989. Also in Recent Advances in Robust Control, Ed. P. Dorato, R.K. Yedavaili, IEEE Press, 1990.
[13] S. Dasgupta, P. J. Parker, B. D. O. Anderson, F. J. Kraus, and M. Mansour, Fre quency domain conditions for the robust stability of linear and nonlinear systems, IEEE Transactions on Circuits and Systems, 38(4):389397, 1991.
[14] F. J. Kraus and W. TruSl, Robust stability of control systems with polytopical uncertainty: a Nyquist approach, International Journal of Control, 53(4):967983, 1991.
[15] F. J. Kraus, M. Mansour, W. TruSl, and B.D.O. Anderson, Robust stability of control systems: extreme point results for the stability of edges, Internal Report 9106, Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, 1991.
[16] R. J. Minnichelli, J. J. Anagnost, and C. A. Desoer, An elementary proof of Kharitonov's stability theorem with extensions, IEEE Transactions on Automatic Control, 34(9):995998, 1989.
[17] F. J. Kraus and M. Mansour, Robuste Stabilitiit im Frequenzgang (german), In ternal Report 8706, Automatic Control Laboratory, Swiss Federal Institute of Tech nology, Zurich, 1987.
[18] F. J. Kraus and M. Mansour, On robust stability of discrete systems, Proceedings of the 29th IEEE Conference on Decision and Control, 2:421422, 1990.
[19] F. J. Kraus and M. Mansour, Robust Schur stable control systems, Proceedings of the American Control Conference, Boston, 1991.
[20] F. J. Kraus and M. Mansour, Robust discrete control, Proceedings of the European Control Conference, Grenoble, 1991.
[21] M. Mansour, Robust stability of systems described by rational functions, to appear in Advances in Robust Systems Techniques and Applications, Leondes Ed., Academic Press, 1991.
Robust Control of Interval Systems*
L.H. Keel
Tennessee State University
Nashville, TN. U.S.A
Department of Electrical Engineering
Texas A&M University
College Station, TX. U.S.A.
ABSTRACT
In this paper we consider a feedback interconnection of two linear time invariant systems, one of which is fixed and the other is uncertain with the uncertainty being a box in the space of coefficients of the numerator and denominator polynomials. This kind of system arises in robust control analysis and design problems. We give several results for the Nyquist plots and stability margins of such families of systems based oil the extremal segments of rational functions introduced by Chapellat and Bhattadaaryya [1]. These CB segments play a fundamental characterizing role in many control design problems.
1. I N T R O D U C T I O N
The problem of designing control systems to account for parameter uncertainty has become very important over the last few years. The main reason for this is the lack of effective methods for dealing with systems containing uncertain or adjustable parameters. Significant progress on this problem occured with the introduction of Kharitonov's Theorem [2] for interval polynomials which provided a new way of deal ing with parametric uncertainties. However, despite its mathematical elegance, its applicability to cozltrol systems was very limited because the theorem assumes that
*Supported in part by NASA grant NAGI863 and NSF Grant ECS8914357
25
all coefficients of the given polynomial family perturb independently. Under this as sumption the theorem can only provide conservative results for the control problem where the uncertain parameters enter the characteristic polynomoial in a correlated fashion
A breakthrough was reported in the theorem of Chapellat and Bhattacharyya [1] which dealt with the case in which the characteristic polynomial is a linear combina tion of interval polynomials with polynomial coeffidents. The CB Theorem provided Kharitonovlike results for such control problems by constructing certain line seg ments, the number of which is independent of the polynomial degrees, that completely and nonconservatively characterizethe stability of interval control systems.
Using this novel theorem, it has been discovered that the worst case H** stability margin of an interval control system occurs on one of these CB segments. The details and constructive algorithm to compute H °° stability margin are given in [3]. Similarly the worst case parametric stability margin occurs on the CB segments [4].
The main result of this paper is that the Nyquist plot boundaries of all the transfer functions of interest in an interval control system occurs on the CB segments. This key result has farre~ching implications in the development of design methods for interval control systems. Throughout this paper, proofs are omitted and may be found in [5]. A similar result has recently been independently discovered by Tesi and Vicino [6].
2. NOTATION AND P R E L I M I N A R Y RESULTS
Figure 1. Feedback System
We consider the feedback interconnection (see Figure 1) of the two systems with respective transfer functions
2v(s) Q~(s) P(~): . . . . . (1)
Q(~) := Q~(,) D(~)
where N(s), D(s) and Q~(s), i = 1,2 are polynomials in the complex variable a. In general Q(s) is fixed and P(s) will be assumed to lie in an uncertainty set described as follows.
Write
N(s) := nvs p+np_ls pI +. . .+his+no DCs) := dq~q + d ,_~ ,~ + . . . + d~, + do (2)
26
~nd let the coefficients lie in prescribed intervals
. , e l . . , . + ] , i e {0,1,. . . ,p} :=~ die[d~,d+], ie{O, 1,...,q}:=q.
Introduce the interval polynomial sets
H(~) := {N(,) = n,s"+n~_,~'~ + . . . n , ~ + , , o : n~ e ["~',",+l, i e E} D(,) := {D(s)=d,8'+d,,sqt+...dz*+do : d /e ld~/ ,d~/ ] , ieq} (3)
and the corresponding set of interval transfer hmctions (orintervM systems)
LD(a) : (N(,) ,D(s)) e Af(a)xV(s)}. (4)
The four Kharitonov vertex polynomials associated with Af(s) are
K~(,) := o + 'q* +~+*' +  D 3 + " ~ " + ' q ' ~ + " "
K~(~) := o +",+~ +"~+~ ÷ " ~ + " ~ ' +"~+'~ + . . . . K~(~) := ,,o+ + ,q~ + , q ~ + .+,~ + ,,,%, + , q , s + . . . (5) K.~(~) := .+ + .+~ + . ~ + .~ ,~ ' + , ,+ , ' +,,~%~ + . . . .
and we write ~.,,,'C~) " {K~C.~), K~(.,), g~c.,),.,"qC.~)} (6)
Similarly the four Kharitonov polynomials associated with 29(s) are denoted K~(a), i = 1,2,3,4 and
Jc.~(.,) := {K~C.,), K,~C..,), x~',~C,,,),_rc,,/(s)} (7) The four Kharitonov segment polynomials associated with N'(s) are defined as follows:
st¢0) :=
[AK~,Cs) + (1  A)K~(s) : A E [0,1], (i,j) E {(1,2), (1,3),(2,4),(3,4)}] (8)
and the four Kharitonov segment polynomials associated with :D(s) are
S,(~) :=
[id(~(s)+ (1 #)K~(s) : /~E [0,11, Ci,j) E {(1,2),(1,3),(2,4),(3,4))] (9) Following Chapellat and Bhattacharyya [1] we introduce the following subset of
(H(~)xV(~))c8 = {(2v(~),D(~)) : N(~) e ~:~f(~), D(s) 6 Sv(s) or N(s) E Sa'(s), n(s) e K:v(*)}. (10)
Referring to the control system in Figure 1, we calculate the following transfer functions of interest in an,alysls and design problems:
yCs) = p(~) "(~) .(.,) ~ = Q(~) 01)
T°(s) := r  ~ = i + P(s)Q(s) (14)
u(s) Q(s) (15) T"(s) := r  ~ = 1 + P(s)QCs)"
The characteristic polynomial of the system is denoted as
~r(s) = Q2(s)D(s) + Ql(s)N(s). (16)
As P(s) ranges over the uncertainty set P(s) (equivalently, (N(s), D(s)) ranges over Af(s)xD(s)) the transfer functions T°(s), TY(s), T"(s), T'(s) range over correspond ing uncertainty sets T°(s), T~(s), T"(s), and T'(s), respectively. In other words,
T'(~) := { PC~)Q(~) : P(~)e ~(~) } (17)
T'(~) := { ~ : P ( ~ ) ~ ( ~ ) } (lS)
T"(~) := { ~ : P(~) ~ ~(~) } (10)
IICs) = {Q2Cs)D(s) + Q,Cs)N(s) : (N(s),D(s)) E AfCs)xD(s)}. (21)
We now introduce the CB subset of the family of interval systems P(s):
PcBCs) := { N9]DO : (NCs),DCs)) E (N'(s)xDCs))cn }. (22)
These subsets will play a central role in all the results to be developed later. We note that each element of Pea(s) is a 0he parameter of transfer functions and there are at most 32 such distinct elements.
The CB subsets of the transfer function sets (17)  (20) and the polynomial set (21) are also introduced:
T~:BCs ) := { PCs)Q(~) : PCs) E PCB(8) } (23)
T~B(S ) : { ~ : P(s) E PCB(S) } (24) l+P(o)e(,)
T~.(~) := { Q(.) : P(~) c ro.(~) } (25) l+P(s)Q(s)
TS.(~) := { ' l+P(~')O(,) : P(s) E PCB(S) } (26)
28
and
ncB(s) := {Q.( , )D(s) + 0,( ,)IV(s) : (N( , ) . D( , ) ) e (~ (~)xV( , ) )cB} (27)
In the paper we shall deal with the complex plane image of each of the above sets evaluated at s = fla. We denote ¢ar3a of these two dimensional sets in the complex plane by replacing s in the corresponding argument by w. Thus, for example,
n ( ~ ) := {n(~) : • = j~} (28)
and T~s(o~) := {T~s(~) : s = jo~} (29)
The Nyquist plot of a set of functions (or polynomials) T(s) is denoted by T:
X := Uo<~<ooX(w) (30)
Stabili ty and Stabil i ty Mar$ins
The control system of Figure 1 is stable for fixed Q(s) and P(s) if the charac teristic polynomial
(s) = Q~(s)DCs) + Q1Cs)N(s) (31)
is Hurwitz, i.e. has all its n = q+ degree [Q2(s)] roots in the open left half of the complex plane. The system is robustly stable if and only if each polynomial in l'I(s) is of degree n (degree D(s). remains invariant and equal to q as D(s) ranges over :D(s)) and every polynomial in II(s) is Hurwitz.
The following important result was provided in Chapdlat and Bhattacharyya [11.
Theorem 1. (CB Theorem) The control system of Figure I is stable for all P(s) E P(s) if and only if it is stable for all P(s) e Pcs(s).
The above Theorem gives a constructive solution to the problem of checking robust stability by reducing it to a problem of checking a set of (at most) 32 root locus problems.
If a fixed system is stable we can determine its gain margin V and phase margin ~b as follows:
7+(P(s),O(s)) := max{ f~" : Q2(s)D(s) + KQI(s)N(s)
is Hurwitz for It" E [1, f(] } (32)
~(P(s),Q(s)) := max{ K : Q~(s)D(s)+ kQ,(s)~(s) ' 1 is Hurwitz for It" E [K' ] } (33)
29
@+(P(s),Q(s)) := max{ ~ : Q,(s)D(s)+ei÷Ql(s)N(s)
is Hurwitz for ~ E [O, ~l } (34)
qb(PCs),Q(s)) := max { ~b : Q, Cs)DCs) + ei÷Ql(s)NCs)
is Hurwitz for ~ E [0,~ }. (35)
Note that 7 +, 7 , ~+, and ~b are uniquely determined when Q(s) and P(s) are fixed. The unstructured additive stability margin y. (H ¢* stability margin) can also be
uniquely determined for a fixed stable closed loop system and is given by
1 v.(P(s)Q(s)) := IIQ(s)(1 + p(s)O(s))_,ll, 0. (36)
In the following section we state some fundamental results on the above sets of transfer functions and the calculation of extremal stability margins over the uncertainty set HCs)xV(s).
3. MAIN RESULTS
We shall give the main results here without proof. The proofs are given in Keel, Shaw and Bhattacharyya [5]. Similar results have been recently reported independently by Tesi and Vicino [6].
Theorem 2. For every to > O,
OP(to) C [email protected]) 0T°(to) C T~.(w)
[email protected]) c T~.(to) OT"(to) c [email protected]) OT'(to) c Tt.(to)
(37) (38) (39) (40)
(41)
This result shows that at every to > 0 the image set of each transfer function in (17)  (20) is bounded by the corresponding image set of the CB segments.
The next result deals with the Nyquist plots of each of the transfer functions in (17)(20).
Corollary 1. The Nyquist plots of each of the transfer function sets T°(s), TU(s), T"(s), and T~(s) are bounded by their corresponding CB subsets:
cTr ° c T~. (42) 69T' C W~s (43)
~)T" C T"cB (44)
3 0
This result has many important implications in control system design and will be explored in forthcoming papers. One important consequence is given below:
Theorem 3. Suppose that the closed loop system is robustly stable, i.e. stable for all P(s) E P(s). Then
and
max 4~: = max ~b ~: (47) P(s)eP(s) P(s)ePcB (s)
max v. (48) max tta = P(s)eP(s) P(s)ePc,(s)
min 7 ~= = min 7 ~= (49) P(s)6P(s) P(s)6PcB(s)
min q,+ = min ~: (50) P(s) E P ( s ) P(s)EPcB(s}
min va = min va. (51) P(s)eP(.) P(s)ePc,(s)
In control systems the maximum margins are useful for synthesis and design (with P(s) being a compensator) and the minimum margins are useful in calculating worst case stability margins (P(s) is the plant). In each case the crucial point that makes the result constructive and useful is the fact that Pcs(s) consists of a finite and "small" number of line segments.
4. C O N C L U D I N G R E M A R K S
We have showe that the boundaries of various transfer function sets and various ex tremal gain and phase margins occur on the CB segments first introduced in [1]. Using this idea, construction of Nyquist and Bode bands can be given. These important tools allow us to reexamine all of classical control theory with the added ingredient of robustness in both stability and performance using the framework of interval systems. More interestingly, these new results provide a way of developing design and synthesis techniques in the field of "parametric robust control. These techniques also connect with H °° control theory and promise many new results in the analysis and design of control systems containing uncertain or adjustable real parameters.
R E F E R E N C E S
[1] H. Chapellat and S. P. Bhattaeharyya, "A generalization of Kharitonov's theorem: robust stability of interval plants," IEEE Transactions on Automatic Control, vol. A C  34, pp. 306  311, March 1989.
[2] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations," Differential Uravnsn, vol. 14, pp. 2086  2088, 1978.
31
[3] H. ChapeUat, M. Dahleh, and S. P. Bhattacharyya, "Robtmt stability under struc tured and unstructured perturbations," IEEE Transactions on Automatic Control, vol. AC  35, pp. 1100  1108, October 1990.
[4] H. Chapellat, M. Dahleh, and S. Bhattacharyya, "Extremal manifolds in robust stability," TCSP Report, Texas A&M University, July 1990.
[5] L. Keel, J. Shaw and S. P. Bhattacharyya, "Frequency domain design of interval control systems," Tech. Rep., Tennessee State University, July 1991. Also in TCSP Tech. Rep., Texas A&M University, July 1991.
[6] A. Tesi and A. Vicino, "Kharitonov segments suffice for frequency response anal ysis of plantcontroller families". To appear in Control of Uncertain Dynamic Systems, September 1991, CRC Press.
Rejection of Persistent, Bounded disturbances for SeLmpledData Systems*
Bassam Bamleh t, Munther A. Dahleh $, and J. Boyd Pearson t
June 4, 1991
Abst rac t
In this paper, a complete solution for the l 1 sampleddata problem is furnished for arbitrary plants. Then l 1 sampleddata problem is described a~ follows: Giv en a contlnuoustime plant, with contlnuoustime performance objectives, design a digital controller that delivers this performance. This problem differs from the s tandard discretetlme methods in that it takes into con~ideratlon the intersamp]L, lg behaviour of the closed loop system. The resulting dosed loop system dynamics con sists of both continuousthne and discretetime dynamics and thus such systems are known as "Hybrid" systems. It is shown that given any degree of accuracy, there exists a standard discretetime l I problem, which can be determ;ned apriori, such that for any controller that achieves a level of performance for the discretetlme problem, the same controller achieves the same performance within the prescribed level of accuracy if implemented as a sampleddata controller.
"The first and last authors' research is supported by NSF ECS8914467 and AFOSR910036. The second author is supported by WrightPatterson A.F.B. F3361590C3608, , C.S. Draper Laboratory DLH4186U and by the ARO DAAL0386K0171.
t Dept. of Electrical and Computer Engineering, Rice University, Houston, TX 77030 I Laboratory of Information sad Decision Systems, Massachusetts Institute of Technology, Cambridge,
MA
1 I n t r o d u c t i o n
33
This paper is concerned with designing digital controllers for continuoustime system s to optimaly achieve certain performance specifications in the presence of uncertainty. Contrary to discrete time designs, such controllers are designed taking into consideration the intersample behavior of the system. Such hybrid systems are generally known as sampleddata systems, and have recently received renewed interest by the control com munity.
The difficulty in considering the continuous time behavior of sampleddata systems, is that it is time varying, even when the plant and the controller are both continuous time and discretetime timeinvarlant respectively. We consider in this paper the standard problem with sampledda~a controllers (or the sampleddata problem, for short) shown in figure 1. The continuous time controller is constrained to be sampleddata controller, that is, it is of the form 7~GS~. The generalized plant is continuoustime timeinvariant and G is discretetime timeinvariant,Tf, is a zero order hold (with period r), and ,.~ is an ideal sampler (with period r). 7f¢ and ,S~ are assumed synchronized. Let ~'(G, 7f~CS¢) denote the mapping between the exogenous input and the regulated output. Jr(G, 7~¢G,S~) is in general time varying, in fact it is rperiodic where r is the period of the sample and hold devices.
Sampleddata systems have been studied by many researchers in the past in the context of LQO controllers (e.g. [121). Recently, [4] studied this problem in the context of 7"{ °° control, and were able to provide a solution in the case where the regulated output is in discrete time and the exogenous input is in continuous time. The exact problem was solved in [2],[3], and independently by [9] and [13]. The L•induced norm problem (the one we are concerned with in this paper) was considered in [7] for the case of stable plants.
In this paper we will use the framework developed in [2],[3], to study the t x sampled data problem. Precisely, the controller is designed to minimize the induced norm of the periodic system over the space of bounded inputs (i.e. L°°). This minimization results from posing time domain specifications and design constraints, which is quite natural for control system design. The solution provided in this paper is to solve the sampleddata problem by solving an (almost.) equivalent, discrete time ~i problem. While this was the approach followed in [7], the main contribution of this paper is that it provides bounds that can be computed apriori to deternfine the equivalent discretetime problem, given any desired degree of accuracy and thus provides a solution for the synthesis problem. The solution in this paper is presented in the context of the lifting framework of [2], [3], as an approximation procedure for certain infinite dimensional problems. This approach has the advantage of handling both the stable and the unstable case in the saane franlework, and results in techniques that are more transparent than those in [7].
34
. . . . . . pi

k
2 T h e L i f t i n g T e c h n i q u e
In this section we briefly summarize the lifting technique for continuoustime periodic systems developed in [2], [3], and apply it to the sampleddata problem.
For continuous time signals, we consider the usual L°°[0,oo) space of essentially bounded functions [6], and it 's extended version L~°[0,oo). We will also need to consider discrete time signals that take values in a function space, for this, we define Ix to be the space of all X valued sequences, where X is some Banach space. We define t ~ as the subspace of l x with bounded norm sequences, i.e. where for ( f l } fi t x , the norm II(f,}llz r := supl II/illx < Go. Given any .f fi L~°[o,oo), we define i t ' s / / / t in o / ¢ t~o.t,..i, as follows: / is an L°°to,~lvalued sequence, we denote it by ( / i} , and for each ~,
/~(t) := / ( t + , 0 o < t < ,.
The lifting can be visualized ~ taking a continuous time signal and breaking it up into a sequence of 'pieces' each corresponding to the function over an interval of length r (see figure 2). Let us denote this lifting by Wr : L~[o,~o) . , l~..lo., j. W,. is a linear isomorphism, furthermore, if restricted to L~°io,oo) , then W,. : L~[o,~)  ~ t~',[o., ! is an isometry, i.e. it preserves norms.
Using the lifting of signals, one can define a lifting on systems. Let C be a linear continuous time system on L~°[o,oo), then it 's li/ting G is the discrete time system G :=
35
W.GW~ "1, this is illustrated in the commutative diagram below:
IX,to,,! ~ ,. l£"°t..,l
+++
Thus G is a system that operates on Banach space (L°°[0,~]) vMued signah, we will call such systems infinite dimensional. Note that since W~ is an isometry, if G is stable, i.e. a bounded linear map on L °° then G is also stable, and furthermore, their respective induced norms are equal, Ildll = IIGll The correspondence between a system and it's lifting also preserves algebraic system properties such as addition, cascade decomposition and feedback (see [2] for the details).
To apply the lifting to the sampleddata problem, consider again the standaxd problem of figure 1, and denote the dosed loop operator by .T(G,?L.CS¢). Since the lifting is an isometry, we have that II~r(o, 7¢.cs~)[ I  IIW.~=(G, ~ C S . ) w ;  ' II, this is shown in figure 3(a).
We now present (from [2]) a state space realization for the new generalized plant 0 . The original plant G has the realization:
G = 6 A Dll D12 .
G'l 0 0
It is assumed that the sampler is proceeded with a presampling filter. It can be shown ([2]) that a realization for the generalized plant C. (figure 3) is given by
G21 G2~ C2 0 0
36
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " 1
! ! ! : [ , ,
I c I Figure 4: Decomposition of (~
[ cA" I eA('')B, ~(*')B, ] Gte" Gte'4("')10,)Bx + D t t 6 ( t  , ) Gt~I'(t)B= + Dn C= 0 0
where ~(t) := I~ eA'd's" The system O has the following input and output spaces
6 n : ~.r.,,,olo.,.i ~ .~.f,O*l,,fl Gn : la. ~ lLto,,.! 021 : tLto,,j '* t a , 02: : +,~ * Is,
Using this lifting, G can be decomposed as
° [ ~ 0 ~ ° l + [ I ° 0 0 o.10,] :J Lo, ' ~ ] [ ] L °] L~J ~10 o 0 I 0 0
This decomposition is illustrated in figure 4. The closed loop mapping ~'(G, U) is corre spondingly decomposed as
~(O,C) = b,t + ~(Oo, C) = D,, + [~t b~, ] J:( ¢., C) B,. (I)
We will use the notation C) := [ eL bt, ] , and call (~ the output operator and/~, the input operator. With this decomposition, G,~, is finite dimensional, and f), B1 axe finite rank operators
: R "+~ ~ L°°[o.fl, /~1 : L~10,fl o ~7.
It is evident from this decomposition that only the "discrete" part of the plant depends on the controller, while the infinite dimensional hybrid part depends only on the system's dynamics.
3
37
, l
F~}. "1 Figure 5: The system On
S o l u t i o n P r o c e d u r e
Using the lifting we are able to convert the problem of finding a controller to minimize the L °° induced norm of the hybrid system (figure 1) into the following standard problem with an infinite dimensional generalized plant G:
:= inf I P ' ( a , ~ , o s , ) l l =: inf IlS~(O,O)ll (2) 0 j tab l i z ing ~ 0 etaMixitto
The above infinite dimensional problem is solved by the followin 8 approximation proce dure through solving a standard MIMO 0 problem [5, 10]. Let 74~ and ,S~, be the following operators defined between L~[0,~l and l°°(.) (t°°(.) is R" with the maximum norm),
x,,: t**c.) ,/,"*to,.l (x,,,,)(O = ,4l~J) ; {,,(i)} E t**c.),
One can in fact show that the operators are well defined. Now to approximate the infinite dimensional problem, we use the approximate dosed loop system 8,,~'(G, U)~f,, (see figure 5), and for each ~z we define
#. := inf IIS~(0,c)X.ll, (3) O ,Jtabili~i~O
This new problem now involves the induced norm over l~***(,0, i.e. it is a standard MIMO
0 problem. Let us denote the generalized plant a~sociated with 7~,~'(0, C)S,, by ~ , such that
S.7(C.,C)~,, = Y(O. ,C) ,
where ~n and a realization for it is given by,
I o. :[~ ~,1o[~. ;] = S.Ol sob,,~o s. , , O, o °I =: bn b1~ •
0 0
i i i i i ~ ! ! ! ! ! !
0 Tin . . . . . . T
i I I I 1 2 . . . . . . n
I , i
Design Bounds
Let us begin with analysis. Note that since U~:(O,o)ll is a pe~odic,ny t~me v~ying system, its L'induced norm is not immediately computable. An alternative method of computing II~:(O, O)ll comes from the limit
IW(¢, C)ll = .u2~ IIS,,~(¢, c)~,,il =: ~m 1W(6",,, C)ll, (4)
for a fixed C. This equation (4) however is by far not sufficient to show the convergence of the synthesis procedure, since given only (4), the rate of convergence may depend on the choice of C. Our objective is to obtain explicit bounds on [].~r(~, C)[l in the following form
Main Inequal i ty : There are conatants Ko and K1 which depend only on G, such ~a~
IIJ:(0.,O)ll < II~:(0,O)H _< KLln + (1 +  ~ ) [l~'(¢,,C')[[, (5)
The significance of the bound (5) is that it is exactly what is needed for synthesis. When one performs an t t design on G,, the result is a controller that keeps [[.~(~,,, C)H small, but the objective is to keep the L'induced norm of the hybrid system (or equivalently II~'(~, C)ll) small, and the inequality (5) guarantees this. The proof of this bound utilizes heavily the decomposition discussed earlier, and the fact that the infinite dimensional operators O and B1 me finite dimensional, and a~e dependent only on the original plant. The details of these proofs can be found in [1].
39
References
[1] B. Bamieh, M.A. Daldeh and 3.B. Pearson, 'Minimization of the L ~ induced norm for sampleddata systems,' Rice University Technical Report, Houston, TX, June 1991.
[2] B. Bamleh and J.B. Pearson,'A General Framework for Linear Periodic Systems with Application to 7{ ~ SampledDe~ta Control', to appear in IEEE Trans. on Ant. Control.
[3] B. Bamleh, J.B. Pearson, B.A. Francis, A. Tannenbanm, 'A Lifting Technique for Linear Periodic Systems with Applications to SampledData Control', to appear in Systems and Controls Letters.
[4] T. Chen and B. Francis, 'On the L: induced 26 26 norm of a sampleddata system', Systems and Control Letters 1990, v.15, 211219.
[5] M.A. Dahleh and J.B. Pearson, 'l:optimal feedback control for MIMO discretetime systems', in Trans. Ant. Control, AC32, 1987.
[6] C.A. Desoer and M. Vidyasagar, Feedback Systems: InputOutput Properties, Aca demic Press, New York, 1975.
[7] G. DuUerud and B. Francis, ,ry Performance in SampledData Systems', preprint, Submitted to IEEE Trans. on Ant. Control.
[8] B.A. Francis and T.T. Georgiou, 'Stability theory for linear timeinvariant plemts with periodic digital controllers', IEEE Trans. Ant. Control, AC33, pp. 820832, 1988.
[9] P.T. Kabamba and S. Hara, 'Worst ca~e analysis and design of sampled data control systems', Preprint.
[i0] J.S. McDonald and J.B. Pearson, 'IZOptimal Control of Muitivariable Systems with Output Norm Constraints', Autoraatica, vol. 27, no. 3, 1991.
[11] N. Sivashankar and Pramod P. Khargonekar, '~o~Induced Norm of SampledData Systems', Proc. of the CDC, 1991.
[12] P.M. Thompson, G. Stein and M. Athans, 'Conic sectors for sampleddata feedback systems,' System and Control Letters, Vol 3, pp. 7782, 1983.
[13] H.T. Toivonen, 'Sampleddata control of continuoustime systems with an 7foo opti ramify criterion', ChemicaJ Engineering, Abo Akademi, Finland, Report 901, Jan uary 1990.
Rational Approximation of L1optimal controller
Yoshito Ohm?, Hajhnc Ma~ and Shinzo Kodama~
? Dcparuncnt of Mechanical Engineering for ComputerControlled Machinery Osaka Univcxsity 2I Yam~dmOka, Suil~t, Osaka 565, Japan Dcparm~nt of EIeclmnic Hnginccring Osaka University 2I YAm~daOka, Sulfa, Osaka 565, Japan
Abstract
This paper studies the L1optimal control problem for SISO systems with rational
controllers. It is shown that thc infimal achievable norm with rational controllers is as srrmll as
that with irrational controllers. Also a way to construct a rational suboptimal controller is
studied.
1. Introduction The problem of Llopdmal control was fwst formulated by Vidyasagar [7] to solve the
disturbance rejection problem with persistent input. A complete solution for general problems
was presented by Dahlch and Pearson for discrctctimc 1block MIMO systems [2] and for
continuoustime SISO systems [3]. In [3] they also pointed out the uscfuincss of Lloptimal
control in the following areas: (i) disturbance rejection of bounded input, (ii) tracking of
bounded input, and (iii) robustness.
For continuoustime systcms, it turns out that an LlOpdmal solution has a certain
disadvantage: delay terms appear both in the numerator and denominator of an optimal
controllcr [3]. This raises a question as to what limitation wc impose if controllers arc confined
to be rational. One obvious way to look at this is to approximate the irrational controller by
rational one. However, as it will be shown in a later section, an irrational transfer function
cannot be approximated by the set of stable rational functions in thc topology with which the
Llconlrol problems arc formulated. This means th