Optimising stability bounds of finite-precision controller structures for sampled-data systems in the &operator domain

S.Chen, J.Wu, R.H.lstepanian, J.Chu and J.F.Whidborne

Abstract: A tractable closed-loop stability-related measure for controller structures, realised using the 6 operator and digitally implemented with ‘finite word length’ (FWL), is derived. The optimal realisations of the general finite-precision controller are defined as those that maximise this measure and are shown to be the solutions of a constrained nonlinear optimisation problem. For the special case of digital PID controllers, the constrained problem can be decoupled into two simpler unconstrained optimisation problems. A global. optimisation strategy based on the adaptive simulated annealing (ASA) is adopted to provide an efficient method for solving this complex optimal realisation problem. Two numerical examples are presented to illustrate the design procedure, and the simulation results confirm that the optimal FWL realisations of the 6- operator based controller have better closed-loop stability margins than those of the usual shift- operator based controller, especially under fast sampling conditions.

1 Introduction

Although the number of controller implementations using floating-point processors is increasing due to their reduced price, for reasons of simplicity, speed, memory space and ease-of-programming, the use of fixed-point processors is more desired for many industrial and consumer applica- tions, particularly for mass market applications in the automotive and consumer electronics sectors. Furthermore, due to their reliability and well-understood properties, fixed-point processors predominate in critical-safety systems. It is well known that a designed stable closed- loop system may become unstable when the ‘infinite- precision’ controller is implemented using a fixed-point processor due to the finite word length (FWL) effect. The ‘robustness’ of closed-loop stability under controller para- meter perturbations is therefore a critical issue in fixed- point implementations.

Many studies have addressed the problem of digital controller realisations with finite-precision considerations [ 1-61. The first FWL stability measure, denoted as po, was proposed in 1994 [3], but computing explicitly this

0 IEE, 1999 IEE Proceedings online no. 19990745 DOI: 10.1049/ip-cta:l9990745 Paper first received 15th December 1998 and in revised form 12th August 1999 S. Chen is with the Department of Electronics and Computer Science, University of Southampton, Highfield, Southampton SO17 IBJ, UK J. Wu and J. Chu are with the National Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou 3 10027, P. U. China R.H. Istepanian is with the Department of Electrical and Computer Engineering, Ryerson Polytechnic University, 350 Victoria Street, Toronto, Ontario, Canada M5B 2K3 J.F. Whidborne is with the Department of Mechanical Engineering, King’s College London, Strand, London WC2R 2LS, UK

IEE Proc.-Control Theoiy Appl., Vol. 146, No. 6, November 1999

measure is still an unsolved open problem. Recently, two tractable FWL stability-related measures, connected to po, have been derived and the design procedures for searching for optimal FWL controller realisations have been devel- oped [5, 61. In the original works [5, 61, the term lower bound stability measure was used. We prefer to use the term stability related measure, as it is a lower bound of PO only under some restricted conditions. It can be shown that the measure of [6] is ‘closer’ to po than that of [SI, and the investigations on FWL controller realisations have mainly been based on this stability related measure [7-121.

In all the above-mentioned studies, digital controller structures were described and realized with the usual shift operator z. A discrete-time system can also be described and realized with a different operator, called the delta operator 6 [13]. Two major advantages are claimed for the use of 6 operator realisation: a theoretically unified formulation of continuous-time and discrete-time systems; and better numerical properties in FWL imple- mentations [2]. For signal processing applications, it is well known that the digital filter realized with the 6 operator has lower round-off noise, lower coefficient error and lower output error variance than the shift operator realisation [14, 151. The use of the 6 operator as opposed to the shift operator in control applications has also been promoted [ 16, 171. However, no study to date addresses the closed-loop stability issues of FWL controller structures using the 6 operator formulation.

This paper analyses the closed-loop stability of sampled- data control systems in the 6 domain with FWL considera- tions. We derive a new measure quantifying FWL effects on the closed-loop stability. For the computational purpose, a tractable stability related measure is given. The optimal digital controller realisation in the 6 domain can be obtained by maximising this measure. As the optimisation criterion is nonsmooth and nonconvex, an efficient nongradient based global optimisation method, known as the adaptive simulated annealing (ASA) [18-211, is employed to search for the true optimal controller

517

realisation. It turns out that the approach of analysing FWL digital controllers in the z domain [3, 5 , 61 can be extended to study &based FWL digital controllers, and the &based controller realisation has better closed-loop stability robustness to FWL effects over the z-domain approach.

2 The 6 operator

From a continuous-time transfer function G(s), as the result of discretisation procedure with a sampling period h, a discrete-time transfer function G,(z) based on the shift operator z can be obtained. Define the 6 operator as follows [13]:

(1) z - 1 6 = -

h The transfer function G,(z) can be re-expressed in 6 form: Gs(6) = G,(z). Obviously G,(z) and Gg(6) are two different but equivalent parameterisations representing thk same input-output relationship. The state-space models in the z and 6 domains are [22, 231

(2) I I

zx,(k) = A,x,(k) + B,u(k)

Y(k) = C,x,(k) + D,u(k) and

(3) 6xg(k) = A[jx,(k) + Bgu(k)

Y(k) = Csxs(k) + D s W respectively, where the various vectors and matrices are understood to have appropriate dimensions, and

G,(.) = C,(ZI - A,)-'B, + D,

= ~ ( ~ ( 6 ) = c 6 ( a - A,)-IB, + D, (4)

It follows from eqn. (4) that the relationships relate the two state-space representations are

A, = hAg + I, B, = hB6, C, = Cg, D, = D6 (5)

It is well known that the state-space realisation of an input- output transfer function is not unique. Define a generalised operator p, where p = z or 6, and

A 9, = {(A,, B,, C,, D,) :

G,(d = C,(PI - AJ'B, + D,} (6)

Then, if (Ap, B,, C,, D , ) E ~ , , (T-'A,T, T-IB,, C,T, D,) E 9, for any nonsingular T. Let {A,,,} be the eigen- values of A,. The following lemma relates to {A,,,}. Lemma I : A,, I = 1 + h Vi. The proof of lemma 1 is straightforward based on the definition of eigenvalue and the relationship A, = h AS + I . It is well known that the discrete-time system (A,, B,, C,, D,) is stable if and only if all the eigenvalues lA,,,l < 1. From lemma 1, we have the condition of the stability for the discrete-time system described with the 6 operator. Lemma 2: The discrete-time system (AS, Bg, Cg, Dg) is stable if and only if

3 FWL stability issue in the 6 domain

(7)

Consider the sampled-data system depicted in Fig. 1 , where P(s) is strictly proper. The discrete-time plant P(6) = ShP(s)Hh, has a state-space realisation (Ap, Bp,

518

continuous time plant c:J sampler

hold digital device controller (period h)

Fig. 1 Sampled-data system with digital controller realisation

C,, 0) in the 6 domain, where Ap E gm m , B, E grn x 1

and C p ~ , % q X r n . The controller C(6) has a state-space realisation (Ac, B,, C,, D,) with A, E gn n , B, E Bn ', C, E B?lx and D, E 9' q. The corresponding realisation (A, B, E , D) of the closed-loop system is:

B = [ ~ ] ; C = , c p o ] , D = o (9)

where all 0's are zero matrices of proper dimensions, In is the n x n identity matrix, and

I i LP(l+n-l)(q+n)+l ~ ( l + n - l ) ( q + n ) + 2 . . . P(/+n)(q+n) 1

(10)

is the controller matrix. Let C(6) be chosen to make the feedback system stable. Then all the eigenvalues of A(Xg), denoted by Ai, 1 5 i 5 m + n, satisfy IIzi + l/hl < 1/15, Vi.

When the realisation (A,, Be, C,, De) is implemented with a fixed-point digital control processor, Xg is perturbed into Xg + A X6 due to the FWL effects, where

AX6 =

r A P ~ AP2 " . APq+n 1

'PN i . . . 1 Ap(/+n-i)(q+n)+l AP(l+n-;)(q+n)+2 . . . and N = (I + n)(q + n). Each element of AX<s is bounded by €12, that is,

(12) A p(AX,) = I max S i S N IApiI SE /2

For a fixed-point processor of B, bits

IEE Proc-Control Theoiy Appl.. Vol. 146, No. 6, November 1999

Where 2Bx is a normalisation factor such that the absolute value of each element of 2-BAXcj is no) larger-than 1. With the perturbation AX(>, li is moved to Ai. The sampled-data system will be unstable i j and only if there exists ig { 1,. . . , m + n } such that (ILj + l /h l> l /h.

To see when the round off error will cause the closed- loop system to become unstable, let us introduce the following stability measure:

pd0(X6) A inf {p(AXg) : A(X6 + AX,) is unstable] (14)

From this definition, the following proposition is obvious: Proposition 1: A(X,)+AXg) is stable if p(AX6)

The larger pa0 (X,) is, the bigger FWL error the closed- loop stability can tolerate. Let B r f n be the smallest word length that, when used to implement Xg, can guarantee the closed-loop stability. It would be highly desirable to know Brin. However, except in simulation, it is impossible to test the closed-loop system by reducing B,y until it becomes unstable. Assuming that h is realised exactly, an estimate of BFi” can be provided by

BF = I t [ - 1og2(pbo(Xa))] - 1 + B, (15)

where Int[x] rounds x to the nearest integer and Int[x] 1 x. From exprs. (12) to (1 5), it can be seen that the closed-loop system is stable when X(j is implemented with a fixed-point processor of at least B F bits.

It is worth emphasising an often overlooked constraint on the FWL implementation of 8-based controllers. The state-space equation of the 6-based controller, 8x(k) = A, x(k) + B,u(k), is realised using: x(k+ 1) = x(k) + h(Ac x(k) + B,u(k)). The sampling period h should be realised exactly without FWL errors. Otherwise, analysis based on X, may not be valid. Specifically, assume that h can be realized exactly by Bh bits with the integer part of h requiring Bhl bits and the fractional part of h requiring BhF bits. A modified estimate of the ‘minimum bit length that can guarantee the closed-loop stability is

< pso (&>.

&$’ = max(Bhl, B,} + max{Bh,, ip - B,} (16)

Notice that the &domain stability measure pgo, defined in (14), is similar to the stability measure po for z-operator based controller realisations given in [3]. Like p0, how to compute explicitly the value of pgO for a given realisation Xa is also an unsolved open problem. Thus, the stability measure pc5o(X(j) has very limited practical value. Alter- native measure that can not only quantify the FWL effects on stability robustness but also be computed easily must be sought.

4 FWL stability related measure in the 6 domain

Roughly speaking, how easily the FWL error AX, can cause a stable control system to become unstable is strongly determined by how close 2, are to the unstable boundary and how sensitive they are to the controller parameter perturbations. We propose the following stabi- lity related measure:

Let A(X8 + AXa), respectively, and define

and j, be the lth eigenvalues of A(Xg) and

We have the following proposition: Proposition 2: A(X6 + AX*) is stable if AXg E P(X8) and

ProoJ For AXa E Y(X,), P,l(XS) ’ Y(AXS).

It follows from pg1(X6) > p(AX6) that

(20)

which means that b(Xa + AXg) is stable. Remarks: The requirement AXa E P(Xg) is not over

restricted. In practice, we are only interested in those AX3 which lie in a bounded region including AXg=O. More plainly, we are only interested for those AX6 lying in (see proposition 1):

%X,> A IAXa : P(AXS) < Puso(X,)) (21) Since a&Iapj is continuous,

(22)

where C is the oriented segment from Xa to Xa + AXs, aj and bj are some points on C, Re[x] and Img [x] are the real and imaginary parts of the complex number x , respectively, and i = F l . Hence,

Now, let us compare

with

(25)

519 IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999

Notice that all the N real-valued items

are in alignment; while the N complex-valued items

are generally out of alignment. Moreover, lApjl 5 p(AXh), Re[aAl/ap,] and Im [ail/apj] are differentiable. Thus, there exists a rather large IC such that VAXJE {AX6:p(AXd) 5 IC},

The above analysis shows that P(X6) exists and a rather large part of 9(Xd) can be covered by P(Xs), that is, the condition AX6 E P(X6) is not too restricted.

Notice that, although pa1(X6) can be used to describe the FWL stability characteristics, it is not generally true that “A(X8 + AXs) is stable if p(AX8) < p ~ ~ ( X 6 ) ” . This is in contrast to pao(Xs). For this reason, we prefer to call pdl(Xd) a stability related measure. Also, generally speak- ing, there is no rigor relationship between pd0(X6) and pdl(X~), but pdl(X6) is connected with a lower bound of pdo(X6) in some manners, as shown in the following corollary. First, define

CoroWY 1: PdI(X6) I pao(x6) if P(Y(Xd)) ’ PdO(Xd). From corollary 1, it can be seen that pd1(X6) can be considered as a lower bound of pdo(X6), provided that pdo(X~) is small enough. The assumption of small ,uaO(Xa) is not over restricted, as it does not make much sense to study the FWL effects on the closed-loop stability for those situations where the closed-loop systems have a very large stability robustness. It should be pointed out that most of digital control systems do have a small stability robustness, which is especially true when fast sampling is applied.

To compute pdI(Xs), {a;l,/ap,} are needed. The follow- ing theorem shows that these eigenvalue sensitivities can easily be calculated. Theorem I: Let A = MO + MlXM2 E 9’ ’ be diagonal- isable and denote { hi} its eigenvalues, where X E 9 l x ‘, and MO, MI and M2 are independent of X with proper dimen- sions. Let xi be a right eigenvector of A corresponding to the eigenvalue 2,. Denote M, = [xl x2 . . . xu] and M y = [yl y2 . . . y,,] =MX#, where 2 denotes the transpose and conjugate operation and yi is called the reciprocal left eigenvector corresponding to 2,. Then,

- a i , ax - -

. . . . . a i , a;l, __ - ... 8x12 axlr

where xkj is the (k,j)th element of X, and the superscript ‘*’ denotes the conjugate operation. Pro08 Let a be a variable independent of M1 and M2. It follows from yI“ x, = 1 that

Notice that Ax, = i ,x, and 2, = y? Ax,. Hence,

a y y aA ax 3 = L A X , + y?-xx, + yTA$

It follows from eqn. (29) and f l A =Ale that

aa aa aa

aa aA f l ax aa = Y?-% = Y, MI z M 2 X 1

Let a =x4. Then,

a/z, aa = (y?’1)k(M2x1)j

where ( Y T M ~ ) ~ and (MZxi), are the kth andjth elements of C M , and M2xj, respectively. This leads to eqn. (28).

Since pdI(Xa) is computationally tractable, for a given controller realisation Xa, we can estimate the smallest word length BYin based on pdl(Xd) using the following:

h p = 1 0 g ~ ( p ~ ~ ( ~ ~ ) ) ] - I + B, (33)

When the requirement for implementing h exactly is taken into account, the estimated smallest bit length should be modified to

B?; = max{Bh,, B,] + max{BhF, By - B,] (34)

It should be pointed out that although pLgl(Xd) can be used to estimate BYin, its importance lies in the fact that it can be used as the optimisation criterion to search for an optimal controller realisation, defined as

(35)

where

Y(X6) A (X, : C(6) = C,(dI - A,)-’B, + D,] (36)

is the set of all the realisations of the controller C(6). The realisation X:OPt is optimal in the sense that it has maxi- mum stability robustness to FWL effects. The digital controller implemented with an optimal realisation means that the stability of the closed-loop system is guaranteed with a minimum hardware requirement in terms of word length. The detailed design procedure for finding an optimal controller realisation will be discussed in the following Section.

5 Optimal realisation of FWL controller structures in the S domain

To begin with the optimal design procedure, assume that an initial controller realisation Xao is given to be

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520 IEE Proc.-Control Theovy AppL, Vol. 146, No. 6, November 1999

with the cost function where C(6) = C:(6I - A:)- ' B: + D,". Any realisation of C(6) can be expressed as:

where T c B n x n and det(T)#O. From eqn. (8) it can be shown that the transition matrix of the closed-loop system is

From eqn. (39), applying theorem 1 results in

Where

is readily computed using theorem 1. Let 2: be _the ith eigenvalue of ,";(X,,). Obviously,

b(X60) and A(X;iT) have the identical eigenvalues, and the optimisation problem (35) can be expressed as

A Topt =

det(T)#O

Given Topt, the optimal controller realisation Xhopt is readily computed usin eqn. (38). For the complex- valued matrix M E @'+$ (q + n ) with elements m,, define the matrix norm

i=l ;=I

Maximisation problem (4 1) is equivalent to the minimisa- tion problem

(43)

where

Because the cost function f;i(T) is nonsmooth and nonconvex, optimisation must be based on a direct search without the aid of cost function derivatives. The conventional optimisation methods for this kind of problem, such as Rosenbrock and Simplex algorithms [24-261, generally can only find a local minimum. Notice that, although the choice of initial realisation will not affect the closed-loop eigenvalues, the eigenvalue sensitivities a).i/aXd depend on the chosen initial realisation. Thus for different Xcs0 the shape of the cost function Ay(T) will change, giving rise to different degree of difficulty in the optimisation procedure. It is, therefore, important to use an efficient and preferably global optimisation method. We adopt a global optimisation strategy based on the ASA [18-211 to search for a true global optimum Topt.

6 Optimal realisation of FWL PID controllers in the 6 domain

In this Section we specifically discuss the optimal realisa- tion problem of FWL &based PID controllers. It is well- known that a constrained nonlinear optimisation problem is generally much more difficult to solve than an uncon- strained one. For the FWL z-based controller realisation problem, the previous works have shown that the constrained optimisation problem can be decoupled into two simpler unconstrained ones [l 1, 121 This result can readily be extended to the case of FWL b-based PID controller structures. A digital PID controller is an order n = 2 system. For notational simplicity, we will also restrict to the single-input and single-output controller, that is, 1=q= 1. Let an initial realisation for such a di ita1 PID controller C(6) be (A: E B2 2, BO, E g2 ', C, E 9' 2,

D: E 9). From eqn. (43), the optimal PID controller realisa- tion problem is defined as the optimisation problem:

%

The aim is to avoid handling the constraint det(T) # 0 directly in optimisation. The following theorem shows that optimisation problem (47) can be solved by solving for the two simpler unconstrained problems. First, define the two cost functions:

max - - 15i5m+2

and (44)

Thus finding an optimal controller realisation is equivalent to obtaining a similarity transformation that is a solution of the following constrained nonlinear optimisation problem

(45) Topt = arg T$:,,./i (TI det(T)#O

max 0 - - 1: I" "1 0

X

IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999 52 I

Theorem 2: Let

V a l = rgk) hl(4 Y , w)

Vh2 = <G(m&CO).h2(x>Y3 U , (51)

(50) IE(-m +m)

w t ( 0 +CO)

and

y+m +m) 4 0 +m) ue(0 +m)

then,

va = min{vgl, va21 (52) Moreover, if vg = v d l and (xoptl, yoptl, woPtl) is the optimal solution of problem (50), the optimal solution of problem (47) is given as

if vg =v62 and (xOpt2, yopt2, uOpt2, wOpt2) is the optimal solution of problem (51), the optimal solution of the problem (47) is given as

] (54) 1 Xopt2 (xoptzvopt2 - w o p t 2

Topt = - [ Wopt2 Uopt2 Yopt2

The proof of theorem 2 is given in the Appendix (Section 10). Because Ay,(x, y, w) and &(x, y, U , w) are still nonsmooth and nonconvex functions, an efficient global optimisation method is preferred and we will adopt the ASA optimizer to solve for these two unconstrained nonlinear optimisation problems.

7 Application examples

Two numerical examples were used to show how the optimisation approach presented earlier can be used effi- ciently for designing optimal FWL &based controller structures. For the comparison purpose, both the z and 6 based controllers were investigated in the simulation. The optimal realisation problem of FWL z-based controller structures with the stability related measure pzl (X,) was defined in the previous works [6-121.

Example 1: We consider the following IFAC93 benchmark PID control system [27]. The continuous-time plant model is

25(-0.4~ + 1) (s2 + 3s + 25)(5s + 1)

P(s) = (55)

and the designed PID controller is

(56) 0.431 1.048s

C(S) = 1.311 +- s + 1 + 12.92s

The sampled-data system with the infinite-precision digital controller in the z domain is stable when the sampling period h 5 23. The range of the sampling period tested in the simulation was 23 to 2-12, to cover the slow to very fast sampling conditions.

Given a sampling rate, the discrete-time plant model P(6) and the digital controller C(6) with the 6 operator were obtained using the discretising routines in MATLAB. The discretisation procedure was based on the bilinear (Tustin) transformation,

2 1 -z-1 h 1 +z-'

s = -~ (57)

with 6 = ( z - l ) / h . The initial realisation XaO was chosen to be the "controllable" canonical form. When XaO was

522

provided, the eigenvalues { ):} of the ideal closed-loop system without FWL effects and the eigenvalue sensitivity matrices {Oi} were computed. The ASA was then used to search for an optimal transform matrix Topt by solving for the minimisation problem (47) using theorem 2. This produced a corresponding optimal controller realisation Xgopt that maximises the stability related measure pg (X,). The entire process was repeated with the z operator para- meterisation to obtain the optimal z-based realisation X,,,, that maximises the stability related measure pzI (X,).

Fig. 2 shows the values of the FWL stability related measure pal given different sampling rates for the initial and optimal &based controller realisations Xgo and Xaopt, respectively. It can be seen that, for this example, optimi- sation achieved an improvement by more than an order of magnitude on the stability related measure. Fig. 3 and 4

0.100 1 1 = t i a,

E P 7) g! 0.010

0.0011 " " " " " " " I

- 3 - 2 -1 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 logZ(sampling rate)

Fig. 2 FWL stability related measure paI as a function of sampling rate for two different &based controller realisations (IFAC93 benchmark PID control system) 0-0 optimal 0-0 initial

2 12

._ - 2 4

._ C

7)

._ E 8 - a,

0 -3-2-1 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

logz(sampling rate)

Fig. 3 Estimated minimum bit length Bp based on pal only as a function of sampl'ng rate for the initial 8-band controller realisation ?Lao (IFAC93 benchmark PID control system)

log2(sampling rate)

Fig. 4 Estimated minimum bit length By based on p 6 , only as a function of sampling rate for the optimal 8-band controller realisation Xcyopt (IFAC93 benchmark PID control system)

IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999

depict the estimated minimum bit lengths, B,”;“‘, based only on the values of the stability related measure for Xfio and Xaopt, respectively. As mentioned previously, for the 6 operator parameterization, the sampling period h should be implemented exactly without FWL errors. Taking this into account, the modified estimate of the minimum bit length for the optimal realisation Xdopt is given in Fig. 5.

Fig. 6 compares the FWL stability related measure for X J ~ ~ ~ with that of the optimal z-operator controller realisa- tion Xzopt. It is seen that the optimal 6-based controller realisation has much larger FWL closed-loop stability margin than its z-based counterpart. Furthermore, as the sampling rate is increasing, the stability related measure for XJopt is improving slightly and eventually leveling out while the stability related measure for Xzopt is decreasing exponentially. This confirms with a well-known fact that the 6 parameterisation has significant advantages over the usual z parameterisation, especially under fast sampling conditions. Fig. 7 gives the estimated minimal bit length for the optimal z-operator controller realisation. Notice that it does not need to consider h separately in the z-operator parameterisation, as the effect of h has already been included in the controller realisation X,. Comparing Fig. 7 with Fig. 5, even taking into account the requirement of implementing h exactly, the optimal 6-based realisation requires a smaller bit length in FWL implementation than the optimal z-based realisation. Example 2: This example is the linearized model of a CH- 47 tandem-rotor helicopter in horizontal motion about a nominal airspeed [28]. The continuous-time plant model

- 3 - 2 - 1 0 1 0 1 2 3 4 5 6 7 8 9 1 0 logp(sampling rate)

Fig. 5 and h as a function of sampling rate for the optimal 6-based controller realisation X,op, (IFAC93 benchmark PID control system)

Estimated minimum bit length BAT based on

l o o 1

10 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

log2(sampling rate)

Fig. 6 Comparison of FWL stability related measures for the optimal z-based and 6-based controller realisations (IFAC93 benchmark PID control system) 0-0 6 based -0 shift based

IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999

2o I s

16

a - 12

E 8

- - ._ E

._ 2 4

c .-

U

- a,

0 - 3 - 2 -1 0 1 2 3 4 5 6 7 8 9 1011 12

log2(sampling rate)

Fig. 7 Estimated minimum bit length B,T;,”” as a function of sampling rate far the optimal z-based controller realisation (IFAC93 benchmark PID control system)

P(s) given by [28] is in the state-space form (A.s, B,, C,, D,) with

-0.02 0.05 2.4 -32

-0.14 0.44 -1.3 -30 ”=I : 0.:8 -:.6 l t ] ’

1

c s = [ 0 1 0 1, I).-[” ‘1.. (58) 0 0 0 0 0 57.3

A stabilising continuous-time controller C(s) was designed using the LQG method [6] and the controller C(s) is given in the state-space form (Af, B,, Ct, Dt) with

-0.0175 -0.1436 0.3852 -26.3518

0.0084 -17.6863 -4.0536 -13.9065

0.001 0.0018 -6.7274 -33.2584

0 0.003 1 1 -5.1 191 0.0158 -0.2405

9.0660 -0.1761

Bt = I 0.0091 0.2289 1’ -0.0031 0.0893

0.0033 -0.0472 -14.6421 -60.8894

-0.0171 1.0515 -0.2927 -3.2469

The range of sampling rate used in the simulation was 2*

Using the generalised operator p to represent 6 or z, depending on which operator is actually employed, the discrete-time plant model P(p) and the discrete-time controller C(p) were obtained for each given sampling rate using the discretising routines in MATLAB. Because the version of MATLAB, which we have, does not have the discretising routine that can provide the canonical state-

to 214.

523

space model for multi-input/multi-output transfer func- tions, the initial controller realisation X p o was chosen to be the noncanonical form as the result of a direct discretis- ing the state-space model of C(s) given in eqn. (59). The ASA was used to find the optimal Topt and hence the optimal XpOpt that maximises the stability related measure p L p l ( X p ) for both p = 6 and p =z.

Fig. 8 plots the FWL stability related measures as function of sampling rate for the initial and optimal 6- operator controller realisations Xa0 and Xaopt, where it can be seen that the optimisation very effectively improves the FWL closed-loop stability robustness. Fig. 9 compares the FWL stability related measure for the &based optimal realisation Xaopt with that of the z-based optimal realisation Xzopt. Again, as the sampling rate increases, the stability related measure for X,,,, decreases exponentially while the stability related measure for X60p! does not reduce, and the optimal &based controller realisation has much better FWL closed-loop stability robustness than its z-based counterpart. Figs. 10 and 1 1 depict the estimated minimum bit length, BzLn, based only on the value of pcyl for the optimal &operator realisation Xcyopt avd the modified estimate of the minimum bit length, Bz;ln, taking into account the sampling period h, respectively. The estimated minimum bit length for the z-operator realisation Xzopt is given in Fig. 12. Again, XGopt requires a smaller bit length to implement than Xzopt.

28

5 24-

- - B 5 20-

E 1 6 -

._ ._ E

-7

1 ° ' 2 ' 4 ' 6 ' ; ' 1 ; ' 1 ; ' ; 4 ' log2(sampling rate)

Fig. 8 FWL stability related measure pdl as a function of sampling rate for two different &based controller realisations (helicopter control system example) 0-0 optimal 0-0 initial

-

-

2

l o 1

.. U

B ._ E 12- 'j

8 . . . . . . . . . . . . . . 2 3 4 5 6 7 8 9 1'0 I 1 12 13 14

log2(sampling rate)

Fig. 10 Estimated minimum bit length B.:"' based on pa1 only as a function of sampling rate for the optimal &based controller realisation (helicopter control system example)

I a - 20

E 16

B

._ E

g 12

._

U

v)

8 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

log2(sampling rate)

Fig. 11 Estimated minimum bit length B,,"h" based on pa1 and h as a function of sampling rate , f ir the optimal &based controller realisation (helicopter control system example)

a, - - a - 20 ._ E

._ 2 12

._ E 1 6 U + - a,

F1 - 2 3 4 5 6 7 a 9 1'0 1'1 li 13

log 2(sampling)

Fig. 12 Estimated minimum bit length B.y as a function of sampling rate for the optimal z-based controller realisation (helicopter control system example)

8 Conclusions

-7

l o ' 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 2 1 4 logp(sampling rate)

Fig. 9 Comparison of FWL stability related measures for the optimal z-based and &based controller realisations (helicopter control system example) 0-0 6 based 0-0 shift based

524

The paper has addressed the problem of digital controller structures realised using the 6 operator and the relevant issues of closed-loop stability subject to FWL implementa- tion. A tractable stability-related measure, quantifying the robustness of closed-loop stability to the FWL effects in the 6 domain, has been derived. It has been shown that the optimal realisation problem of finite-precision 8-based digital controllers can be interpreted as a constrained nonlinear optimisation problem. In particular, for &based PID controller realisations, the optimisation can be decoupled into two unconstrained optimisation problems. An efficient global optimisation strategy based on the ASA

IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999

has been adopted to solve for this FWL optimal controller realisation problem in the 6 domain.

Two numerical examples have been used to illustrate the optimal design procedure. The results obtained also demonstrate that the digital controllers described with the 6 operator has much better FWL closed-loop stability robustness in fast sampling conditions, compared with the digital controllers described with the usual shift operator. In this work, the main emphasis has been focused on the important FWL closed-loop stability issues of sampled- data control systems. Ongoing work will explore the integration of the proposed optimisation procedure with the closed-loop controller performance and the sparseness consideration of optimal controller realisations. This will provide a multi-objective framework to develop the opti- mal finite-precision controller realisation that possesses the optimal trade off between minimum computational requirements, improved performance and stability robust- ness.

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References

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ISTEPANIAN, R.H., LI, G., WU, J., CHU, J.: ‘Analysis of sensitivity measures of finite-precision digital controller structures with closed-loop stability hounds’, IEE Proc. Control Theory Appl., 1998, 145, ( S ) , pp. 472478 ISTEPANIAN, R.H., PRATT, I., GOODALL, R., JONES, S.: ‘Effect of fixed point parameterization on the performance of active suspension control systems’, Proceedings of 13th IFAC World Congress 1996, San

689-693

Francisco, pp. 291-295 - ISTEPANIAN, R.H., WU, J., CHU, J., WHIDBORNE, J.F.: ‘Maximizing lower bound stability measure of finite precision PID controller realisa- tions by nonlineai programming’, Pioceedings of I998 American Control Conference, 1998, Philadelphia, pp. 2596-2600

DEAN, S.E.: ‘Finite-word-length stability issues of teleoperation motion-scaling control system’, Proceedings of UKACC Control’98, 1998, Swansea, UK, pp. 1676-1681 CHEN, S., ISTEPANIAN, R.H., WU, J.: ‘Optimizing stability bounds of finite-precision PID controllers using adaptive simulated annealing’, Proceedings of 1999 American Control Conference, 1999, San Diego, pp. 43164320 WU, J., ISTEPANIAN, R.H., CHEN, S.: ‘Stability issues of finite precision controller structures for sampled data systems’, Inf. .I Control, 1999, 72, (IS), pp. 1331-1342 CHEN, S., WU, J., ISTEPANIAN, R.H., CHU, J.: ‘Optimizing stability bounds of finite-precision PID controller structures’, IEEE Trans. Autom. Control, 1999, 44, (I), pp. 2149-2153 MIDDLETON, R.H., GOODWIN, G.C.: ‘Digital control and estimation: a unified approach’ (Prentice Hall, Englewood Cliffs, NJ, 1990) GOODALL, R.M., DONOGHUE, B.J.: ‘Very high sample rate digital filters using the d operator’, fEE Proc. G, Circuits Devices Syst., 1993, 140, pp. 199-206 SUNG, Y., KUNG, M.: ‘Lower finite word-length effect on state space digital filter by 6 operator realisation’, Int. J. Electron, 1993, 75, (6), pp. 1135-1 141 LI, G., GEVERS, M.: ‘Comparative study of finite wordlength effects in shift and delta operator parameterizations’, fEEE Trans. Autorn. Control,

ISTEPANIAN, R.H.: ‘Implementational issues for discrete PID algo- rithms using shift and delta operators parameterizations’, Proceedings of 4th IFAC Workshop on Algorithms and Architectures for Real-Time Control, 1997, Vilamoura, Portugal, pp. 117-122 INGBER, L.: ‘Simulated annealing: practice versus theory’, Math. Comput. Model., 1993, 18, (1 I) , pp. 29-57 INGBER, L.: ‘Adaptive simulated annealing (ASA): lessons learned’, JT Control Cybern., 1996, 25, (I), pp. 33-54 CHEN, S., LUK, B.L., LIU, Y.: ‘Application of adaptive simulated annealing to blind channel identification with HOC fitting’, Electron. Lett., 1998, 34, (3), pp. 234-235

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10 Appendix Proof of theorem 2

Define the diagonal matrix set: Udiag(n) A {diag(ul, u2,. . . , u,J: U, E { - 1, 1 j , 1 5 i 5 n j . From the definition (4% Lemma 3: V M E grn ‘, U1 E Udiag(m) and U2 E Udiag(n),

Define the sets:

Construct the optimisation problems:

and

Obviously, To = Y1 U Y2 and, therefore, vg = min{ vg 1,

v g 2 j . Define the function

1, x > o 0, x t 0

sgn(x) =

IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999 525

Consider the optimisation problem (6 1). V T E F1 and V i E { 1, . . . , m + 2}, utilising lemma 3 , we have,

0

1/"optl 0 Woptl

-Yoptl "opt1 ' 1 Define

Then,

F

and

If v a = v g ~ and (xoptl, yoptl, woptl) is the solution of the optimisation problem (67),

1 /Woptl

XQi[ 0 0

which means that

is the optimal solution of the problem (47). By considering the optimisation problem (62) in a

526 IEE Proc.-Control Theory Appl., Vol. 146, No. 6, November 1999