synthesis of adaptive robust controllers for a class of...
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Research ArticleSynthesis of Adaptive Robust Controllers for a Class ofNonlinear Systems with Input Saturations
Kenta Oba1 Hidetoshi Oya2 Tomohiro Kubo3 and Tsuyoshi Matsuki4
1Graduate School of Advanced Technology and Science Tokushima University 2-1 Minamijosanjima Tokushima 770-8506 Japan2Department of Computer Science Tokyo City University 1-28-1 Tamazutsumi Setagaya Tokyo 158-8557 Japan3Graduate School of Technology Industrial and Social Sciences Tokushima University 2-1 MinamijosanjimaTokushima 770-8506 Japan4Department of Electronics and Control Engineering National Institute of Technology Niihama College7-1 Yagumo Niihama Ehime 792-8580 Japan
Correspondence should be addressed to Hidetoshi Oya hidetcuacjp
Received 4 July 2016 Revised 29 November 2016 Accepted 2 February 2017 Published 4 October 2017
Academic Editor Mohammad D Aliyu
Copyright copy 2017 Kenta Oba et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with a design problem of an adaptive robust controller for a class of nonlinear systems with specified inputsaturations For the nonlinear system under consideration the nonlinearity means unknown perturbations and satisfies thematching condition In this paper we show that sufficient conditions for the existence of the proposed adaptive robust controllergiving consideration to input saturations are given in terms of linearmatrix inequalities (LMIs) Finally simple illustrative examplesare shown
1 Introduction
It is well known that robust control for uncertain dynamicalsystems is very important topic for the control engineeringcommunity and therefore various robust control problemshave been well studied (see [1] and references therein)Moreover many robust controllers achieving some robustperformance such as mixedH2Hinfin control [2] and guaran-teed cost control [3] have been suggested Additionally Siljakand Stipanovic [4] and Zuo et al [5] have presented resultsof robust stability and stabilization for linear continuous-time and discrete-time system under nonlinear perturbationsusing the linear matrix inequalities (LMIs) Although almostall of these controllers consist of a state feedback with afixed feedback gain the fixed feedback gain can be derivedby considering the worst case for unknown parametervariations In contrast with the above-mentioned controlstrategies adaptive robust controllers which are adjustedby parameter updating laws have also been presented (eg[6 7]) In the work of Maki and Hagino [6] an adapta-tion mechanism for improving transient behavior has been
introduced and the robust controller includes fixed gainparameters and adjustable ones which are tuned by someupdating laws Moreover Oya and Hagino [7] have proposeda robust controller with adaptive compensation inputs andthe adaptive compensation input consists of a state feedbackwith an adaptive gain and a compensation input One cansee that these adaptive robust controllers have time-varyingadjustable parameters which are tuned by adjustment laws
On the other hand in practical systems there are someconstraints such as some limit of actuators and electricsaturations for electronic circuits If the constraint conditionsare violated they cannot generate the desired response andat worst the systembecomes unstable From these viewpointsanalysis andor controller design of dynamical systems withconstraint conditions are very important issue and there area large number of the existing results such as reachable andcontrollable sets [8] Model Predictive Control (MPC) [9]and saturation-dependent Lyapunov functions [10] Gilbertand Tan [11] have proposed a general theory which pertainsto the maximal output admissible sets Oinfin and O119888infin for linearsystems with state and input constraints However so far the
HindawiMathematical Problems in EngineeringVolume 2017 Article ID 5345812 9 pageshttpsdoiorg10115520175345812
2 Mathematical Problems in Engineering
design problem of adaptive robust controllers with adjustableparameters for uncertain dynamical systems with inputsaturations has little been considered as far as we know
In this paper on the basis of the work of Oya and Hagino[7] we propose a new LMI-based design method of adaptiverobust controllers for a class of nonlinear systems with inputsaturations For the nonlinear system considered here thenonlinearity means unknown perturbations and satisfies thewell-known matching condition [12] By using the conceptof state reachable sets the proposed controller can flexiblybe adjusted by parameter updating laws [7] and satisfiesthe specified input saturations Namely the result of thispaper is a natural extension of the existing result [7] In thispaper we show that sufficient conditions for the existenceof the proposed adaptive robust controller are given interms of LMIsThus the proposed adaptive robust controllercan easily be derived by solving LMIs and this is also anadvantage for the proposed controller design approach
This paper is organized as follows In Section 2 notationand useful lemmas which are used in this paper are shownand in Section 3 we introduce the class of nonlinear sys-tems with input constraints under consideration Section 4contains the main results Finally we show simple illustrativeexamples to verify the effectiveness of the proposed robustcontroller
2 Preliminaries
In this section notations and useful and well-known lemmas(see [13 14] for details) which are used in this paper areshown
In the paper the following notations are used For amatrix A the inverse of matrix A and its transpose aredenoted by Aminus1 and A119879 respectively Additionally 119867119890Aand 119868119899 mean A + A119879 and 119899-dimensional identity matrixrespectively and the notation diag (A1 AN) representsa block diagonal matrix composed of matrices A119894 for 119894 =1 N TraceA represents trace ofA For real symmetricmatrices A and B A gt B (resp A ge B) means thatA minus B is positive (resp nonnegative) definite matrix Fora vector 120572 isin R119899 120572 denotes standard Euclidian normand for a matrix A A represents its induced norm Allthe eigenvalues of the matrix A are denoted by 120582A and120582maxA means the maximum value of the eigenvaluesAdditionally for a symmetric positive definite matrix P isinR119899times119899 E(P) represents a region E(P) = 120577 isin R119899 | 120577119879P120577 le1 The symbols ldquo≜rdquo and ldquo⋆rdquo mean equality by definition andsymmetric blocks in matrix inequalities respectively
Next we show some useful lemmas which are used in thispaper
Lemma 1 For arbitrary vectors 120582 and 120585 and the matricesG and H which have appropriate dimensions the followinginequality holds
2120582119879GΔ (119905)H120585 le 2 10038171003817100381710038171003817G11987912058210038171003817100381710038171003817 1003817100381710038171003817H1205851003817100381710038171003817 (1)
where Δ(119905) with appropriate dimensions is a time-varyingunknown matrix satisfying 1003817100381710038171003817Δ(119905)1003817100381710038171003817 le 10
Lemma 2 (S-procedure [14]) Let F(119909) G(119909) and H(119909)be three arbitrary quadratic forms over R119899 Then F(119909) lt0 forall119909 isin R119899 satisfying G(119909) le 0 and H(119909) le 0 if there existnonnegative scalars 1205911 and 1205912 such that
F (119909) minus 1205911G (119909) minus 1205912H (119909) le 0 forall119909 isin R119899 (2)
Lemma 3 (Schur complement [14]) For a given constantreal symmetric matrix Θ we consider the following inequalityconditions
(i)
Θ = (Θ11 Θ12Θ11987912 Θ22) ge 0 (3)
(ii)
Θ11 gt 0Θ22 minus Θ11987912Θminus111Θ12 ge 0 (4)
(iii)
Θ22 gt 0Θ11 minus Θ12Θminus122Θ11987912 ge 0 (5)
Then the inequality condition of (i) is equivalent to inequalitiesof (ii) and (iii)
3 Problem Formulation
Let us consider the nonlinear system represented by thefollowing state equation
119889119889119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) + 120585 (119909 119905) + 119861119903119903 (119905) 119909 (0) = 0 (6)
where 119909(119905) isin R119899 119906(119905) isin R119898 and 119903(119905) isin R119897 are the vectorsof the state the control input and the reference input res-pectively In (6) the reference input 119903(119905) is assumed to belongto the ellipsoidal set E(X119903) defined as
E (X119903) = 119903 isin R119897 | 119903119879X119903119903 le 1 (7)
Note that X119903 isin R119897times119897 is a given symmetric positive definitematrix In addition the vector 120585(119909 119905) isin R119899 in (6) meansnonlinearity and uncertainty in the controlled system and itsatisfies the relation 120585(119909 119905) = 119861119892(119909 119905) that is a well-knownmatching condition [12] is satisfied Although the nonlinearfunction 119892(119909 119905) isin R119898 is ldquounknownrdquo for the known matrixG isin R119898times119899 it is assumed to satisfy the following inequalitycondition [4 5] 1003817100381710038171003817119892 (119909 119905)1003817100381710038171003817 le G119909 (119905) (8)
In the sequel we deal with the case of 119898 = 1 for simplicitybecause the results for the case of119898 ge 2 can easily be obtained
Mathematical Problems in Engineering 3
by the following result (see Remark 5)Therefore we considerthe following constraint for the control input 119906(119905) isin R1|119906 (119905)| le 120583 (9)
where 120583 is a known positive constant and this relation isequivalent to 119906(119905) le 120583
Now for the nonlinear system of (6) we consider thecontrol input 119906(119905) isin R119898 described as119906 (119905) ≜ 119870119909 (119905) + 120595 (119909 119905) (10)
In (10)119870 isin R119898times119899 and 120595(119909 119905) isin R119898 are a fixed feedback gainand an adaptive compensation input respectively From (6)and (10) the closed-loop system can be described as119889119889119905119909 (119905) = 119860119870119909 (119905) + 119861120595 (119909 119905) + 119861119892 (119909 119905) + 119861119903119903 (119905) (11)
where 119860119870 isin R119899times119899 is a matrix given by 119860119870 = 119860 + 119861119870Next we consider a state reachable set R(119909 119905) for the
nonlinear system of (6) By using a symmetric positive def-inite matrixP isin R119899times119899 we introduce the quadratic functionV(119909 119905) ≜ 119909119879(119905)P119909(119905) and consider the time derivative of thequadratic functionV(119909 119905) along the trajectory of the closed-loop system of (11) The ellipsoid E(P) contains the statereachable set R(119909) that is R(119909) sub E(P) provided that forany 119909(119905) and 119903(119905) satisfyingV(119909 119905) ge 1 and 119903119879(119905)X119903119903(119905) le 1the following inequality condition holds [14]119889119889119905V (119909 119905) le 0 (12)
Therefore our objective in this paper is to develop anLMI-based design procedure of the proposed adaptive robustcontroller which guarantees the internal stability of theresultant closed-loop system of (11) and the input constraintof (9) In this paper by using the concept of the state reachableset we derive an LMI-based design method for the fixed gain119870 isin R119898times119899 and the adaptive compensation input 120595(119909 119905) isinR119898 such that the closed-loop system of (11) achieves notonly internal stability but also the control input constraint of119906(119905) le 1205834 Synthesis of Adaptive Robust Controllers
In this section we show an LMI-based design method ofthe fixed feedback gain 119870 isin R119898times119899 and the adaptivecompensation input 120595(119909 119905) isin R119898 which ensure internalstability of the closed-loop system of (11) and satisfy the inputconstraint 119906(119905) le 12058341 Analysis of State Reachable SetR(119909) The time derivativeof the quadratic functionV(119909 119905) can be expressed as119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)
+ 2119909119879 (119905)P119861120595 (119909 119905)+ 2119909119879 (119905)P119861119892 (119909 119905)+ 2119909119879 (119905)P119861119903119903 (119905)
(13)
Firstly we consider the case of 119861119879P119909(119905) = 0 Then bydefining the adaptive compensation input 120595(119909 119905) of (10) as
120595 (119909 119905) ≜ minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (14)
and using Lemma 1 the inequality
119889119889119909V (119909 119905)le 119909119879 (119905) [119867119890 P119860119870] 119909 (119905)
+ 2119909119879 (119905)P119861(minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905))+ 2 10038171003817100381710038171003817119861119879P119909 (119905)10038171003817100381710038171003817 G119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
= 119909119879 (119905) [119867119890 P119860119870] 119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
(15)
is derived Moreover one can see that the inequality of (15)can be represented as
119889119889119909V (119909 119905) le (119909 (119905)119903 (119905))119879(119867119890 P119860119870 P119861119903⋆ 0 )(119909 (119905)119903 (119905))
= (119909 (119905)119903 (119905))119879Ψ (119870P) ( 119909 (119905)119903 (119905) )
(16)
where Ψ(119870P) isin R(119899+119897)times(119899+119897) is given by
Ψ (119870P) ≜ (119860119879119870P +P119860119870 P119861119903⋆ 0 ) (17)
On the other hand if 119861119879P119909(119905) = 0 is satisfied then wesee from (13) that the following equation holds
119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)+ 2119909119879 (119905)P119861119903119903 (119905) (18)
Note that in this case the adaptive compensation input120595(119909 119905) isin R119898 can be defined as
120595 (119909 119905) ≜ 120595 (119909 119905120575) (19)
where 119905120575 is given by 119905120575 = lim120575rarr0(119905minus120575) [6] Note that the normof 120595(119909 119905) can be expressed as 1003817100381710038171003817120595(119909 119905)1003817100381710038171003817 = 1003817100381710038171003817G119909(119905)1003817100381710038171003817
From the above discussion in order for the state reachablesetR(119909) of the system of (6) to belong to E(P) the relation
(119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905)) le 0 (20)
4 Mathematical Problems in Engineering
should be satisfied Therefore we introduce F119896(119909 119903) (119896 =0 1 2) defined as
F0 (119909 119903) ≜ (119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905))
F1 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(minusP 00 0)(119909 (119905)119903 (119905)) + 1
F2 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(0 00 X119903
)(119909 (119905)119903 (119905)) minus 1(21)
Note that the inequality condition of (20) V(119909 119905) ge 10and 119903(119905) sub E(X119903) correspond to F119896(119909 119903) le 0 (119896 = 0 1 2)respectively Therefore we consider
F0 (119909 119903) le 0subject to F1 (119909 119903) le 0F2 (119909 119903) le 0 (22)
By using Lemma 2 (S-procedure) the condition of (22) isequivalent to
( 119909 (119905)119903 (119905) )119879(119867119890 P119860119870 + 1205911P P119861119903⋆ minus1205912X119903)(119909 (119905)119903 (119905))minus 1205911 + 1205912 le 0 (23)
Namely if there exist the positive definite symmetric matrixP isin R119899times119899 the fixed gain matrix 119870 isin R119898times119899 and the positivescalars 1205911 and 1205912 satisfying
(119867119890 P119860119870 + 1205911P P119861119903 0⋆ minus1205912X119903 0⋆ ⋆ minus1205911 + 1205912) le 0 (24)
then R(119909) sub E(P) Clearly we must have 1205911 ge 1205912 If theinequality condition of (24) is satisfied for some (12059110158401 12059110158402) thenit holds for all 12059110158401 ge 1205912 ge 12059110158402Therefore we can assume withoutloss of generality that 1205911 = 1205912 = 120591 and the condition of (24)can be rewritten as the following form [14]
(119867119890 P119860119870 + 120591P P119861119903⋆ minus120591X119903) le 0 (25)
Here by introducing the complementary matrices S ≜Pminus1 and W ≜ 119870S and pre- and postmultiplying (25) bydiag (S 119868119898) we have
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0 (26)
Consequently if there exist the symmetric positive defi-nitematrixS isin R119899times119899 thematrixW isin R119898times119899 and the positivescalar 120591 which satisfy the matrix inequality of (26) then thestate reachable set R(119909) for the closed-loop system of (11) isincluded inE(P)
42 Analysis of Input Constraints In Section 41 the con-dition for achieving the relation R(119909) sub E(P) has beenderived Next we consider the input constraints
From (10) and (14) we find that the control input 119906(119905) isinR1 can be written as
119906 (119905) = 119870119909 (119905) minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (27)
Moreover one can see from Section 41 that the state reach-able setR(119909) belongs to E(P)
Firstly we consider the 1st term of the right hand side of(27) Taking the Euclidian norm of119870119909(119905) we obtain
max119905
119870119909 (119905) = max119905
10038171003817100381710038171003817WSminus1119909 (119905)10038171003817100381710038171003817
le max120577isinE(P)
10038171003817100381710038171003817WSminus112057710038171003817100381710038171003817
= max120577isinE(P)
10038171003817100381710038171003817WSminus12
Sminus1212057710038171003817100381710038171003817
le radic120582max Sminus12W119879WSminus12(28)
Furthermore one can easily see that for the 2nd term of theright hand side of (27) the inequality
max119905
100381710038171003817100381710038171003817100381710038171003817 G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905)100381710038171003817100381710038171003817100381710038171003817 = max119905
G119909 (119905) (29)
is satisfied Additionally by using the similar way to thecalculation of (28) we have
max119905
G119909 (119905) le max120577isinE(P)
1003817100381710038171003817G1205771003817100381710038171003817 = max120577isinE(P)
10038171003817100381710038171003817GS12
Sminus1212057710038171003817100381710038171003817
le radic120582max S12G119879GS12 (30)
From (28) and (30) in order to satisfy the specifiedconstraint 119906(119905) le 120583 the following inequalities should besatisfied
radic120582max Sminus12W119879WSminus12 le 120583119870radic120582max S12G119879GS12 le 120583119871120583119870 + 120583119871 le 120583
(31)
In this paper in order to derive an LMI-based designmethodwe introduce the conditions
Sminus12
W119879WSminus12 le 1205832119870119868119899
S12
G119879GS12 le 12058321198711198681198991205832119870 + 1205832119871 le 120583
(32)
instead of the inequalities of (31) Note that the 1st and the 2ndinequalities of (32) are equivalent to them of (31) and the 3rdinequality of (32) is a sufficient condition of the 3rd one of(31) Thus if the conditions of (32) hold then the inequalities
Mathematical Problems in Engineering 5
of (31) are also satisfied Namely if there exist the positivescalars 120583119870 and 120583119871 satisfying the inequalities of (32) then theinput constraint is also satisfied
Now by introducing the complementary variables 120583⋆119870 and120583⋆119871 we consider the following inequalitiesSminus12
W119879WSminus12 le 120583⋆119870119868119899
S12
G119879GS12 le 120583⋆119871119868119899120583⋆119870 + 120583⋆119871 le 120583120583⋆119870 ge 1205832119870120583⋆119871 ge 1205832119871
(33)
One can see from Lemma 3 (Schur complement) that the 4thand 5th inequalities in (33) are equivalent to
(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0 (34)
Moreover by applying Lemma 3 (Schur complement) to the1st inequality of (33) we obtain
minus120583⋆119870119868119899 +Sminus12
W119879WSminus12 le 0 lArrrArr
minus120583⋆119870S +W119879W le 0 lArrrArr
minusS + 1120583⋆119870W119879W le 0 lArrrArr(minusS W119879
W minus120583⋆119870119868119898) le 0(35)
Additionally since the matrix G isin R119898times119899 is known one caneasily calculate the scalar 120574G satisfyingG119879G le 120574G119868119899Thus weintroduce the inequality condition
minus120583⋆119871119868119899 + 120574GS le 0 (36)
and this inequality is a sufficient condition of the 2ndinequality of (33)
From the above discussion one can see that for givenpositive constants 120591 and 120574G if the symmetric positive definitematrix S isin R119899times119899 the matrix W isin R119898times119899 and the positivescalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 which satisfy LMIs of (26) (34)(35) and (36) are obtained then the input constraints aresatisfied
Consequently our main result is summarized as thefollowing theorem
Theorem 4 Consider the nonlinear system of (6) and thecontrol input of (10)
For given positive constants 120591 and 120574G if there exist thesymmetric positive definite matrix S isin R119899times119899 the matrix
W isin R119898times119899 and the positive scalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 whichsatisfy the LMIs
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0(minusS W119879⋆ minus120583⋆119870119868119898) le 0(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0minus120583⋆119871119868119899 + 120574GS le 0
120583⋆119870 + 120583⋆119871 le 120583
(37)
the fixed gain matrix 119870 isin R119898times119899 and the adaptive compensa-tion input 120595(119909 119905) isin R119898 are determined as
119870 = WSminus1120595 (119909 119905)
= minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) if 119861119879P119909 (119905) = 0120595 (119909 119905120575) if 119861119879P119909 (119905) = 0
(38)
where 119905120575 = lim120575gt0120575rarr0(119905minus120575) [6]Then the closed-loop system of(11) is internally stable and the state reachable setR(119909) belongsto E(P) Moreover the input constraints for the control input119906(119905) are also guaranteedRemark 5 In this paper we introduce the assumption119898 = 1and consider the input constraint of (9)
By theway for the control input119906(119905) = (1199061(119905) 119906119898(119905))119879there exist the componentwise input constraints that is1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 le 120583119894 (119894 = 1 119898) (39)
where 120583119894 is a given positive constant If the positive constant120583119894 for the constraints of (39) satisfies 120583119894 = 10 one can adoptthe inequality [14]
119906119879 (119905)Z119906 (119905) le 1Trace Z = 1 (40)
and the proposed design approach can be easily extended tothis case
On the other hand for the case of 120583119894 = 10 although wecan consider the condition119906119879 (119905)R119906 (119905) le 1
R12 = diag (120583+ 120583+) (41)
instead of (40) where 120583+ means max119895120583119895 then the resultingcontroller which is obtained by adopting the condition of (41)
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
design problem of adaptive robust controllers with adjustableparameters for uncertain dynamical systems with inputsaturations has little been considered as far as we know
In this paper on the basis of the work of Oya and Hagino[7] we propose a new LMI-based design method of adaptiverobust controllers for a class of nonlinear systems with inputsaturations For the nonlinear system considered here thenonlinearity means unknown perturbations and satisfies thewell-known matching condition [12] By using the conceptof state reachable sets the proposed controller can flexiblybe adjusted by parameter updating laws [7] and satisfiesthe specified input saturations Namely the result of thispaper is a natural extension of the existing result [7] In thispaper we show that sufficient conditions for the existenceof the proposed adaptive robust controller are given interms of LMIsThus the proposed adaptive robust controllercan easily be derived by solving LMIs and this is also anadvantage for the proposed controller design approach
This paper is organized as follows In Section 2 notationand useful lemmas which are used in this paper are shownand in Section 3 we introduce the class of nonlinear sys-tems with input constraints under consideration Section 4contains the main results Finally we show simple illustrativeexamples to verify the effectiveness of the proposed robustcontroller
2 Preliminaries
In this section notations and useful and well-known lemmas(see [13 14] for details) which are used in this paper areshown
In the paper the following notations are used For amatrix A the inverse of matrix A and its transpose aredenoted by Aminus1 and A119879 respectively Additionally 119867119890Aand 119868119899 mean A + A119879 and 119899-dimensional identity matrixrespectively and the notation diag (A1 AN) representsa block diagonal matrix composed of matrices A119894 for 119894 =1 N TraceA represents trace ofA For real symmetricmatrices A and B A gt B (resp A ge B) means thatA minus B is positive (resp nonnegative) definite matrix Fora vector 120572 isin R119899 120572 denotes standard Euclidian normand for a matrix A A represents its induced norm Allthe eigenvalues of the matrix A are denoted by 120582A and120582maxA means the maximum value of the eigenvaluesAdditionally for a symmetric positive definite matrix P isinR119899times119899 E(P) represents a region E(P) = 120577 isin R119899 | 120577119879P120577 le1 The symbols ldquo≜rdquo and ldquo⋆rdquo mean equality by definition andsymmetric blocks in matrix inequalities respectively
Next we show some useful lemmas which are used in thispaper
Lemma 1 For arbitrary vectors 120582 and 120585 and the matricesG and H which have appropriate dimensions the followinginequality holds
2120582119879GΔ (119905)H120585 le 2 10038171003817100381710038171003817G11987912058210038171003817100381710038171003817 1003817100381710038171003817H1205851003817100381710038171003817 (1)
where Δ(119905) with appropriate dimensions is a time-varyingunknown matrix satisfying 1003817100381710038171003817Δ(119905)1003817100381710038171003817 le 10
Lemma 2 (S-procedure [14]) Let F(119909) G(119909) and H(119909)be three arbitrary quadratic forms over R119899 Then F(119909) lt0 forall119909 isin R119899 satisfying G(119909) le 0 and H(119909) le 0 if there existnonnegative scalars 1205911 and 1205912 such that
F (119909) minus 1205911G (119909) minus 1205912H (119909) le 0 forall119909 isin R119899 (2)
Lemma 3 (Schur complement [14]) For a given constantreal symmetric matrix Θ we consider the following inequalityconditions
(i)
Θ = (Θ11 Θ12Θ11987912 Θ22) ge 0 (3)
(ii)
Θ11 gt 0Θ22 minus Θ11987912Θminus111Θ12 ge 0 (4)
(iii)
Θ22 gt 0Θ11 minus Θ12Θminus122Θ11987912 ge 0 (5)
Then the inequality condition of (i) is equivalent to inequalitiesof (ii) and (iii)
3 Problem Formulation
Let us consider the nonlinear system represented by thefollowing state equation
119889119889119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) + 120585 (119909 119905) + 119861119903119903 (119905) 119909 (0) = 0 (6)
where 119909(119905) isin R119899 119906(119905) isin R119898 and 119903(119905) isin R119897 are the vectorsof the state the control input and the reference input res-pectively In (6) the reference input 119903(119905) is assumed to belongto the ellipsoidal set E(X119903) defined as
E (X119903) = 119903 isin R119897 | 119903119879X119903119903 le 1 (7)
Note that X119903 isin R119897times119897 is a given symmetric positive definitematrix In addition the vector 120585(119909 119905) isin R119899 in (6) meansnonlinearity and uncertainty in the controlled system and itsatisfies the relation 120585(119909 119905) = 119861119892(119909 119905) that is a well-knownmatching condition [12] is satisfied Although the nonlinearfunction 119892(119909 119905) isin R119898 is ldquounknownrdquo for the known matrixG isin R119898times119899 it is assumed to satisfy the following inequalitycondition [4 5] 1003817100381710038171003817119892 (119909 119905)1003817100381710038171003817 le G119909 (119905) (8)
In the sequel we deal with the case of 119898 = 1 for simplicitybecause the results for the case of119898 ge 2 can easily be obtained
Mathematical Problems in Engineering 3
by the following result (see Remark 5)Therefore we considerthe following constraint for the control input 119906(119905) isin R1|119906 (119905)| le 120583 (9)
where 120583 is a known positive constant and this relation isequivalent to 119906(119905) le 120583
Now for the nonlinear system of (6) we consider thecontrol input 119906(119905) isin R119898 described as119906 (119905) ≜ 119870119909 (119905) + 120595 (119909 119905) (10)
In (10)119870 isin R119898times119899 and 120595(119909 119905) isin R119898 are a fixed feedback gainand an adaptive compensation input respectively From (6)and (10) the closed-loop system can be described as119889119889119905119909 (119905) = 119860119870119909 (119905) + 119861120595 (119909 119905) + 119861119892 (119909 119905) + 119861119903119903 (119905) (11)
where 119860119870 isin R119899times119899 is a matrix given by 119860119870 = 119860 + 119861119870Next we consider a state reachable set R(119909 119905) for the
nonlinear system of (6) By using a symmetric positive def-inite matrixP isin R119899times119899 we introduce the quadratic functionV(119909 119905) ≜ 119909119879(119905)P119909(119905) and consider the time derivative of thequadratic functionV(119909 119905) along the trajectory of the closed-loop system of (11) The ellipsoid E(P) contains the statereachable set R(119909) that is R(119909) sub E(P) provided that forany 119909(119905) and 119903(119905) satisfyingV(119909 119905) ge 1 and 119903119879(119905)X119903119903(119905) le 1the following inequality condition holds [14]119889119889119905V (119909 119905) le 0 (12)
Therefore our objective in this paper is to develop anLMI-based design procedure of the proposed adaptive robustcontroller which guarantees the internal stability of theresultant closed-loop system of (11) and the input constraintof (9) In this paper by using the concept of the state reachableset we derive an LMI-based design method for the fixed gain119870 isin R119898times119899 and the adaptive compensation input 120595(119909 119905) isinR119898 such that the closed-loop system of (11) achieves notonly internal stability but also the control input constraint of119906(119905) le 1205834 Synthesis of Adaptive Robust Controllers
In this section we show an LMI-based design method ofthe fixed feedback gain 119870 isin R119898times119899 and the adaptivecompensation input 120595(119909 119905) isin R119898 which ensure internalstability of the closed-loop system of (11) and satisfy the inputconstraint 119906(119905) le 12058341 Analysis of State Reachable SetR(119909) The time derivativeof the quadratic functionV(119909 119905) can be expressed as119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)
+ 2119909119879 (119905)P119861120595 (119909 119905)+ 2119909119879 (119905)P119861119892 (119909 119905)+ 2119909119879 (119905)P119861119903119903 (119905)
(13)
Firstly we consider the case of 119861119879P119909(119905) = 0 Then bydefining the adaptive compensation input 120595(119909 119905) of (10) as
120595 (119909 119905) ≜ minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (14)
and using Lemma 1 the inequality
119889119889119909V (119909 119905)le 119909119879 (119905) [119867119890 P119860119870] 119909 (119905)
+ 2119909119879 (119905)P119861(minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905))+ 2 10038171003817100381710038171003817119861119879P119909 (119905)10038171003817100381710038171003817 G119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
= 119909119879 (119905) [119867119890 P119860119870] 119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
(15)
is derived Moreover one can see that the inequality of (15)can be represented as
119889119889119909V (119909 119905) le (119909 (119905)119903 (119905))119879(119867119890 P119860119870 P119861119903⋆ 0 )(119909 (119905)119903 (119905))
= (119909 (119905)119903 (119905))119879Ψ (119870P) ( 119909 (119905)119903 (119905) )
(16)
where Ψ(119870P) isin R(119899+119897)times(119899+119897) is given by
Ψ (119870P) ≜ (119860119879119870P +P119860119870 P119861119903⋆ 0 ) (17)
On the other hand if 119861119879P119909(119905) = 0 is satisfied then wesee from (13) that the following equation holds
119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)+ 2119909119879 (119905)P119861119903119903 (119905) (18)
Note that in this case the adaptive compensation input120595(119909 119905) isin R119898 can be defined as
120595 (119909 119905) ≜ 120595 (119909 119905120575) (19)
where 119905120575 is given by 119905120575 = lim120575rarr0(119905minus120575) [6] Note that the normof 120595(119909 119905) can be expressed as 1003817100381710038171003817120595(119909 119905)1003817100381710038171003817 = 1003817100381710038171003817G119909(119905)1003817100381710038171003817
From the above discussion in order for the state reachablesetR(119909) of the system of (6) to belong to E(P) the relation
(119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905)) le 0 (20)
4 Mathematical Problems in Engineering
should be satisfied Therefore we introduce F119896(119909 119903) (119896 =0 1 2) defined as
F0 (119909 119903) ≜ (119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905))
F1 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(minusP 00 0)(119909 (119905)119903 (119905)) + 1
F2 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(0 00 X119903
)(119909 (119905)119903 (119905)) minus 1(21)
Note that the inequality condition of (20) V(119909 119905) ge 10and 119903(119905) sub E(X119903) correspond to F119896(119909 119903) le 0 (119896 = 0 1 2)respectively Therefore we consider
F0 (119909 119903) le 0subject to F1 (119909 119903) le 0F2 (119909 119903) le 0 (22)
By using Lemma 2 (S-procedure) the condition of (22) isequivalent to
( 119909 (119905)119903 (119905) )119879(119867119890 P119860119870 + 1205911P P119861119903⋆ minus1205912X119903)(119909 (119905)119903 (119905))minus 1205911 + 1205912 le 0 (23)
Namely if there exist the positive definite symmetric matrixP isin R119899times119899 the fixed gain matrix 119870 isin R119898times119899 and the positivescalars 1205911 and 1205912 satisfying
(119867119890 P119860119870 + 1205911P P119861119903 0⋆ minus1205912X119903 0⋆ ⋆ minus1205911 + 1205912) le 0 (24)
then R(119909) sub E(P) Clearly we must have 1205911 ge 1205912 If theinequality condition of (24) is satisfied for some (12059110158401 12059110158402) thenit holds for all 12059110158401 ge 1205912 ge 12059110158402Therefore we can assume withoutloss of generality that 1205911 = 1205912 = 120591 and the condition of (24)can be rewritten as the following form [14]
(119867119890 P119860119870 + 120591P P119861119903⋆ minus120591X119903) le 0 (25)
Here by introducing the complementary matrices S ≜Pminus1 and W ≜ 119870S and pre- and postmultiplying (25) bydiag (S 119868119898) we have
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0 (26)
Consequently if there exist the symmetric positive defi-nitematrixS isin R119899times119899 thematrixW isin R119898times119899 and the positivescalar 120591 which satisfy the matrix inequality of (26) then thestate reachable set R(119909) for the closed-loop system of (11) isincluded inE(P)
42 Analysis of Input Constraints In Section 41 the con-dition for achieving the relation R(119909) sub E(P) has beenderived Next we consider the input constraints
From (10) and (14) we find that the control input 119906(119905) isinR1 can be written as
119906 (119905) = 119870119909 (119905) minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (27)
Moreover one can see from Section 41 that the state reach-able setR(119909) belongs to E(P)
Firstly we consider the 1st term of the right hand side of(27) Taking the Euclidian norm of119870119909(119905) we obtain
max119905
119870119909 (119905) = max119905
10038171003817100381710038171003817WSminus1119909 (119905)10038171003817100381710038171003817
le max120577isinE(P)
10038171003817100381710038171003817WSminus112057710038171003817100381710038171003817
= max120577isinE(P)
10038171003817100381710038171003817WSminus12
Sminus1212057710038171003817100381710038171003817
le radic120582max Sminus12W119879WSminus12(28)
Furthermore one can easily see that for the 2nd term of theright hand side of (27) the inequality
max119905
100381710038171003817100381710038171003817100381710038171003817 G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905)100381710038171003817100381710038171003817100381710038171003817 = max119905
G119909 (119905) (29)
is satisfied Additionally by using the similar way to thecalculation of (28) we have
max119905
G119909 (119905) le max120577isinE(P)
1003817100381710038171003817G1205771003817100381710038171003817 = max120577isinE(P)
10038171003817100381710038171003817GS12
Sminus1212057710038171003817100381710038171003817
le radic120582max S12G119879GS12 (30)
From (28) and (30) in order to satisfy the specifiedconstraint 119906(119905) le 120583 the following inequalities should besatisfied
radic120582max Sminus12W119879WSminus12 le 120583119870radic120582max S12G119879GS12 le 120583119871120583119870 + 120583119871 le 120583
(31)
In this paper in order to derive an LMI-based designmethodwe introduce the conditions
Sminus12
W119879WSminus12 le 1205832119870119868119899
S12
G119879GS12 le 12058321198711198681198991205832119870 + 1205832119871 le 120583
(32)
instead of the inequalities of (31) Note that the 1st and the 2ndinequalities of (32) are equivalent to them of (31) and the 3rdinequality of (32) is a sufficient condition of the 3rd one of(31) Thus if the conditions of (32) hold then the inequalities
Mathematical Problems in Engineering 5
of (31) are also satisfied Namely if there exist the positivescalars 120583119870 and 120583119871 satisfying the inequalities of (32) then theinput constraint is also satisfied
Now by introducing the complementary variables 120583⋆119870 and120583⋆119871 we consider the following inequalitiesSminus12
W119879WSminus12 le 120583⋆119870119868119899
S12
G119879GS12 le 120583⋆119871119868119899120583⋆119870 + 120583⋆119871 le 120583120583⋆119870 ge 1205832119870120583⋆119871 ge 1205832119871
(33)
One can see from Lemma 3 (Schur complement) that the 4thand 5th inequalities in (33) are equivalent to
(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0 (34)
Moreover by applying Lemma 3 (Schur complement) to the1st inequality of (33) we obtain
minus120583⋆119870119868119899 +Sminus12
W119879WSminus12 le 0 lArrrArr
minus120583⋆119870S +W119879W le 0 lArrrArr
minusS + 1120583⋆119870W119879W le 0 lArrrArr(minusS W119879
W minus120583⋆119870119868119898) le 0(35)
Additionally since the matrix G isin R119898times119899 is known one caneasily calculate the scalar 120574G satisfyingG119879G le 120574G119868119899Thus weintroduce the inequality condition
minus120583⋆119871119868119899 + 120574GS le 0 (36)
and this inequality is a sufficient condition of the 2ndinequality of (33)
From the above discussion one can see that for givenpositive constants 120591 and 120574G if the symmetric positive definitematrix S isin R119899times119899 the matrix W isin R119898times119899 and the positivescalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 which satisfy LMIs of (26) (34)(35) and (36) are obtained then the input constraints aresatisfied
Consequently our main result is summarized as thefollowing theorem
Theorem 4 Consider the nonlinear system of (6) and thecontrol input of (10)
For given positive constants 120591 and 120574G if there exist thesymmetric positive definite matrix S isin R119899times119899 the matrix
W isin R119898times119899 and the positive scalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 whichsatisfy the LMIs
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0(minusS W119879⋆ minus120583⋆119870119868119898) le 0(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0minus120583⋆119871119868119899 + 120574GS le 0
120583⋆119870 + 120583⋆119871 le 120583
(37)
the fixed gain matrix 119870 isin R119898times119899 and the adaptive compensa-tion input 120595(119909 119905) isin R119898 are determined as
119870 = WSminus1120595 (119909 119905)
= minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) if 119861119879P119909 (119905) = 0120595 (119909 119905120575) if 119861119879P119909 (119905) = 0
(38)
where 119905120575 = lim120575gt0120575rarr0(119905minus120575) [6]Then the closed-loop system of(11) is internally stable and the state reachable setR(119909) belongsto E(P) Moreover the input constraints for the control input119906(119905) are also guaranteedRemark 5 In this paper we introduce the assumption119898 = 1and consider the input constraint of (9)
By theway for the control input119906(119905) = (1199061(119905) 119906119898(119905))119879there exist the componentwise input constraints that is1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 le 120583119894 (119894 = 1 119898) (39)
where 120583119894 is a given positive constant If the positive constant120583119894 for the constraints of (39) satisfies 120583119894 = 10 one can adoptthe inequality [14]
119906119879 (119905)Z119906 (119905) le 1Trace Z = 1 (40)
and the proposed design approach can be easily extended tothis case
On the other hand for the case of 120583119894 = 10 although wecan consider the condition119906119879 (119905)R119906 (119905) le 1
R12 = diag (120583+ 120583+) (41)
instead of (40) where 120583+ means max119895120583119895 then the resultingcontroller which is obtained by adopting the condition of (41)
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
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Mathematical Problems in Engineering 3
by the following result (see Remark 5)Therefore we considerthe following constraint for the control input 119906(119905) isin R1|119906 (119905)| le 120583 (9)
where 120583 is a known positive constant and this relation isequivalent to 119906(119905) le 120583
Now for the nonlinear system of (6) we consider thecontrol input 119906(119905) isin R119898 described as119906 (119905) ≜ 119870119909 (119905) + 120595 (119909 119905) (10)
In (10)119870 isin R119898times119899 and 120595(119909 119905) isin R119898 are a fixed feedback gainand an adaptive compensation input respectively From (6)and (10) the closed-loop system can be described as119889119889119905119909 (119905) = 119860119870119909 (119905) + 119861120595 (119909 119905) + 119861119892 (119909 119905) + 119861119903119903 (119905) (11)
where 119860119870 isin R119899times119899 is a matrix given by 119860119870 = 119860 + 119861119870Next we consider a state reachable set R(119909 119905) for the
nonlinear system of (6) By using a symmetric positive def-inite matrixP isin R119899times119899 we introduce the quadratic functionV(119909 119905) ≜ 119909119879(119905)P119909(119905) and consider the time derivative of thequadratic functionV(119909 119905) along the trajectory of the closed-loop system of (11) The ellipsoid E(P) contains the statereachable set R(119909) that is R(119909) sub E(P) provided that forany 119909(119905) and 119903(119905) satisfyingV(119909 119905) ge 1 and 119903119879(119905)X119903119903(119905) le 1the following inequality condition holds [14]119889119889119905V (119909 119905) le 0 (12)
Therefore our objective in this paper is to develop anLMI-based design procedure of the proposed adaptive robustcontroller which guarantees the internal stability of theresultant closed-loop system of (11) and the input constraintof (9) In this paper by using the concept of the state reachableset we derive an LMI-based design method for the fixed gain119870 isin R119898times119899 and the adaptive compensation input 120595(119909 119905) isinR119898 such that the closed-loop system of (11) achieves notonly internal stability but also the control input constraint of119906(119905) le 1205834 Synthesis of Adaptive Robust Controllers
In this section we show an LMI-based design method ofthe fixed feedback gain 119870 isin R119898times119899 and the adaptivecompensation input 120595(119909 119905) isin R119898 which ensure internalstability of the closed-loop system of (11) and satisfy the inputconstraint 119906(119905) le 12058341 Analysis of State Reachable SetR(119909) The time derivativeof the quadratic functionV(119909 119905) can be expressed as119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)
+ 2119909119879 (119905)P119861120595 (119909 119905)+ 2119909119879 (119905)P119861119892 (119909 119905)+ 2119909119879 (119905)P119861119903119903 (119905)
(13)
Firstly we consider the case of 119861119879P119909(119905) = 0 Then bydefining the adaptive compensation input 120595(119909 119905) of (10) as
120595 (119909 119905) ≜ minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (14)
and using Lemma 1 the inequality
119889119889119909V (119909 119905)le 119909119879 (119905) [119867119890 P119860119870] 119909 (119905)
+ 2119909119879 (119905)P119861(minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905))+ 2 10038171003817100381710038171003817119861119879P119909 (119905)10038171003817100381710038171003817 G119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
= 119909119879 (119905) [119867119890 P119860119870] 119909 (119905) + 2119909119879 (119905)P119861119903119903 (119905)
(15)
is derived Moreover one can see that the inequality of (15)can be represented as
119889119889119909V (119909 119905) le (119909 (119905)119903 (119905))119879(119867119890 P119860119870 P119861119903⋆ 0 )(119909 (119905)119903 (119905))
= (119909 (119905)119903 (119905))119879Ψ (119870P) ( 119909 (119905)119903 (119905) )
(16)
where Ψ(119870P) isin R(119899+119897)times(119899+119897) is given by
Ψ (119870P) ≜ (119860119879119870P +P119860119870 P119861119903⋆ 0 ) (17)
On the other hand if 119861119879P119909(119905) = 0 is satisfied then wesee from (13) that the following equation holds
119889119889119905V (119909 119905) = 119909119879 (119905) (119860119879119870P +P119860119870) 119909 (119905)+ 2119909119879 (119905)P119861119903119903 (119905) (18)
Note that in this case the adaptive compensation input120595(119909 119905) isin R119898 can be defined as
120595 (119909 119905) ≜ 120595 (119909 119905120575) (19)
where 119905120575 is given by 119905120575 = lim120575rarr0(119905minus120575) [6] Note that the normof 120595(119909 119905) can be expressed as 1003817100381710038171003817120595(119909 119905)1003817100381710038171003817 = 1003817100381710038171003817G119909(119905)1003817100381710038171003817
From the above discussion in order for the state reachablesetR(119909) of the system of (6) to belong to E(P) the relation
(119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905)) le 0 (20)
4 Mathematical Problems in Engineering
should be satisfied Therefore we introduce F119896(119909 119903) (119896 =0 1 2) defined as
F0 (119909 119903) ≜ (119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905))
F1 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(minusP 00 0)(119909 (119905)119903 (119905)) + 1
F2 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(0 00 X119903
)(119909 (119905)119903 (119905)) minus 1(21)
Note that the inequality condition of (20) V(119909 119905) ge 10and 119903(119905) sub E(X119903) correspond to F119896(119909 119903) le 0 (119896 = 0 1 2)respectively Therefore we consider
F0 (119909 119903) le 0subject to F1 (119909 119903) le 0F2 (119909 119903) le 0 (22)
By using Lemma 2 (S-procedure) the condition of (22) isequivalent to
( 119909 (119905)119903 (119905) )119879(119867119890 P119860119870 + 1205911P P119861119903⋆ minus1205912X119903)(119909 (119905)119903 (119905))minus 1205911 + 1205912 le 0 (23)
Namely if there exist the positive definite symmetric matrixP isin R119899times119899 the fixed gain matrix 119870 isin R119898times119899 and the positivescalars 1205911 and 1205912 satisfying
(119867119890 P119860119870 + 1205911P P119861119903 0⋆ minus1205912X119903 0⋆ ⋆ minus1205911 + 1205912) le 0 (24)
then R(119909) sub E(P) Clearly we must have 1205911 ge 1205912 If theinequality condition of (24) is satisfied for some (12059110158401 12059110158402) thenit holds for all 12059110158401 ge 1205912 ge 12059110158402Therefore we can assume withoutloss of generality that 1205911 = 1205912 = 120591 and the condition of (24)can be rewritten as the following form [14]
(119867119890 P119860119870 + 120591P P119861119903⋆ minus120591X119903) le 0 (25)
Here by introducing the complementary matrices S ≜Pminus1 and W ≜ 119870S and pre- and postmultiplying (25) bydiag (S 119868119898) we have
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0 (26)
Consequently if there exist the symmetric positive defi-nitematrixS isin R119899times119899 thematrixW isin R119898times119899 and the positivescalar 120591 which satisfy the matrix inequality of (26) then thestate reachable set R(119909) for the closed-loop system of (11) isincluded inE(P)
42 Analysis of Input Constraints In Section 41 the con-dition for achieving the relation R(119909) sub E(P) has beenderived Next we consider the input constraints
From (10) and (14) we find that the control input 119906(119905) isinR1 can be written as
119906 (119905) = 119870119909 (119905) minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (27)
Moreover one can see from Section 41 that the state reach-able setR(119909) belongs to E(P)
Firstly we consider the 1st term of the right hand side of(27) Taking the Euclidian norm of119870119909(119905) we obtain
max119905
119870119909 (119905) = max119905
10038171003817100381710038171003817WSminus1119909 (119905)10038171003817100381710038171003817
le max120577isinE(P)
10038171003817100381710038171003817WSminus112057710038171003817100381710038171003817
= max120577isinE(P)
10038171003817100381710038171003817WSminus12
Sminus1212057710038171003817100381710038171003817
le radic120582max Sminus12W119879WSminus12(28)
Furthermore one can easily see that for the 2nd term of theright hand side of (27) the inequality
max119905
100381710038171003817100381710038171003817100381710038171003817 G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905)100381710038171003817100381710038171003817100381710038171003817 = max119905
G119909 (119905) (29)
is satisfied Additionally by using the similar way to thecalculation of (28) we have
max119905
G119909 (119905) le max120577isinE(P)
1003817100381710038171003817G1205771003817100381710038171003817 = max120577isinE(P)
10038171003817100381710038171003817GS12
Sminus1212057710038171003817100381710038171003817
le radic120582max S12G119879GS12 (30)
From (28) and (30) in order to satisfy the specifiedconstraint 119906(119905) le 120583 the following inequalities should besatisfied
radic120582max Sminus12W119879WSminus12 le 120583119870radic120582max S12G119879GS12 le 120583119871120583119870 + 120583119871 le 120583
(31)
In this paper in order to derive an LMI-based designmethodwe introduce the conditions
Sminus12
W119879WSminus12 le 1205832119870119868119899
S12
G119879GS12 le 12058321198711198681198991205832119870 + 1205832119871 le 120583
(32)
instead of the inequalities of (31) Note that the 1st and the 2ndinequalities of (32) are equivalent to them of (31) and the 3rdinequality of (32) is a sufficient condition of the 3rd one of(31) Thus if the conditions of (32) hold then the inequalities
Mathematical Problems in Engineering 5
of (31) are also satisfied Namely if there exist the positivescalars 120583119870 and 120583119871 satisfying the inequalities of (32) then theinput constraint is also satisfied
Now by introducing the complementary variables 120583⋆119870 and120583⋆119871 we consider the following inequalitiesSminus12
W119879WSminus12 le 120583⋆119870119868119899
S12
G119879GS12 le 120583⋆119871119868119899120583⋆119870 + 120583⋆119871 le 120583120583⋆119870 ge 1205832119870120583⋆119871 ge 1205832119871
(33)
One can see from Lemma 3 (Schur complement) that the 4thand 5th inequalities in (33) are equivalent to
(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0 (34)
Moreover by applying Lemma 3 (Schur complement) to the1st inequality of (33) we obtain
minus120583⋆119870119868119899 +Sminus12
W119879WSminus12 le 0 lArrrArr
minus120583⋆119870S +W119879W le 0 lArrrArr
minusS + 1120583⋆119870W119879W le 0 lArrrArr(minusS W119879
W minus120583⋆119870119868119898) le 0(35)
Additionally since the matrix G isin R119898times119899 is known one caneasily calculate the scalar 120574G satisfyingG119879G le 120574G119868119899Thus weintroduce the inequality condition
minus120583⋆119871119868119899 + 120574GS le 0 (36)
and this inequality is a sufficient condition of the 2ndinequality of (33)
From the above discussion one can see that for givenpositive constants 120591 and 120574G if the symmetric positive definitematrix S isin R119899times119899 the matrix W isin R119898times119899 and the positivescalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 which satisfy LMIs of (26) (34)(35) and (36) are obtained then the input constraints aresatisfied
Consequently our main result is summarized as thefollowing theorem
Theorem 4 Consider the nonlinear system of (6) and thecontrol input of (10)
For given positive constants 120591 and 120574G if there exist thesymmetric positive definite matrix S isin R119899times119899 the matrix
W isin R119898times119899 and the positive scalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 whichsatisfy the LMIs
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0(minusS W119879⋆ minus120583⋆119870119868119898) le 0(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0minus120583⋆119871119868119899 + 120574GS le 0
120583⋆119870 + 120583⋆119871 le 120583
(37)
the fixed gain matrix 119870 isin R119898times119899 and the adaptive compensa-tion input 120595(119909 119905) isin R119898 are determined as
119870 = WSminus1120595 (119909 119905)
= minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) if 119861119879P119909 (119905) = 0120595 (119909 119905120575) if 119861119879P119909 (119905) = 0
(38)
where 119905120575 = lim120575gt0120575rarr0(119905minus120575) [6]Then the closed-loop system of(11) is internally stable and the state reachable setR(119909) belongsto E(P) Moreover the input constraints for the control input119906(119905) are also guaranteedRemark 5 In this paper we introduce the assumption119898 = 1and consider the input constraint of (9)
By theway for the control input119906(119905) = (1199061(119905) 119906119898(119905))119879there exist the componentwise input constraints that is1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 le 120583119894 (119894 = 1 119898) (39)
where 120583119894 is a given positive constant If the positive constant120583119894 for the constraints of (39) satisfies 120583119894 = 10 one can adoptthe inequality [14]
119906119879 (119905)Z119906 (119905) le 1Trace Z = 1 (40)
and the proposed design approach can be easily extended tothis case
On the other hand for the case of 120583119894 = 10 although wecan consider the condition119906119879 (119905)R119906 (119905) le 1
R12 = diag (120583+ 120583+) (41)
instead of (40) where 120583+ means max119895120583119895 then the resultingcontroller which is obtained by adopting the condition of (41)
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
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4 Mathematical Problems in Engineering
should be satisfied Therefore we introduce F119896(119909 119903) (119896 =0 1 2) defined as
F0 (119909 119903) ≜ (119909 (119905)119903 (119905))119879Ψ (119870P) (119909 (119905)119903 (119905))
F1 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(minusP 00 0)(119909 (119905)119903 (119905)) + 1
F2 (119909 119903) ≜ (119909 (119905)119903 (119905))119879(0 00 X119903
)(119909 (119905)119903 (119905)) minus 1(21)
Note that the inequality condition of (20) V(119909 119905) ge 10and 119903(119905) sub E(X119903) correspond to F119896(119909 119903) le 0 (119896 = 0 1 2)respectively Therefore we consider
F0 (119909 119903) le 0subject to F1 (119909 119903) le 0F2 (119909 119903) le 0 (22)
By using Lemma 2 (S-procedure) the condition of (22) isequivalent to
( 119909 (119905)119903 (119905) )119879(119867119890 P119860119870 + 1205911P P119861119903⋆ minus1205912X119903)(119909 (119905)119903 (119905))minus 1205911 + 1205912 le 0 (23)
Namely if there exist the positive definite symmetric matrixP isin R119899times119899 the fixed gain matrix 119870 isin R119898times119899 and the positivescalars 1205911 and 1205912 satisfying
(119867119890 P119860119870 + 1205911P P119861119903 0⋆ minus1205912X119903 0⋆ ⋆ minus1205911 + 1205912) le 0 (24)
then R(119909) sub E(P) Clearly we must have 1205911 ge 1205912 If theinequality condition of (24) is satisfied for some (12059110158401 12059110158402) thenit holds for all 12059110158401 ge 1205912 ge 12059110158402Therefore we can assume withoutloss of generality that 1205911 = 1205912 = 120591 and the condition of (24)can be rewritten as the following form [14]
(119867119890 P119860119870 + 120591P P119861119903⋆ minus120591X119903) le 0 (25)
Here by introducing the complementary matrices S ≜Pminus1 and W ≜ 119870S and pre- and postmultiplying (25) bydiag (S 119868119898) we have
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0 (26)
Consequently if there exist the symmetric positive defi-nitematrixS isin R119899times119899 thematrixW isin R119898times119899 and the positivescalar 120591 which satisfy the matrix inequality of (26) then thestate reachable set R(119909) for the closed-loop system of (11) isincluded inE(P)
42 Analysis of Input Constraints In Section 41 the con-dition for achieving the relation R(119909) sub E(P) has beenderived Next we consider the input constraints
From (10) and (14) we find that the control input 119906(119905) isinR1 can be written as
119906 (119905) = 119870119909 (119905) minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) (27)
Moreover one can see from Section 41 that the state reach-able setR(119909) belongs to E(P)
Firstly we consider the 1st term of the right hand side of(27) Taking the Euclidian norm of119870119909(119905) we obtain
max119905
119870119909 (119905) = max119905
10038171003817100381710038171003817WSminus1119909 (119905)10038171003817100381710038171003817
le max120577isinE(P)
10038171003817100381710038171003817WSminus112057710038171003817100381710038171003817
= max120577isinE(P)
10038171003817100381710038171003817WSminus12
Sminus1212057710038171003817100381710038171003817
le radic120582max Sminus12W119879WSminus12(28)
Furthermore one can easily see that for the 2nd term of theright hand side of (27) the inequality
max119905
100381710038171003817100381710038171003817100381710038171003817 G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905)100381710038171003817100381710038171003817100381710038171003817 = max119905
G119909 (119905) (29)
is satisfied Additionally by using the similar way to thecalculation of (28) we have
max119905
G119909 (119905) le max120577isinE(P)
1003817100381710038171003817G1205771003817100381710038171003817 = max120577isinE(P)
10038171003817100381710038171003817GS12
Sminus1212057710038171003817100381710038171003817
le radic120582max S12G119879GS12 (30)
From (28) and (30) in order to satisfy the specifiedconstraint 119906(119905) le 120583 the following inequalities should besatisfied
radic120582max Sminus12W119879WSminus12 le 120583119870radic120582max S12G119879GS12 le 120583119871120583119870 + 120583119871 le 120583
(31)
In this paper in order to derive an LMI-based designmethodwe introduce the conditions
Sminus12
W119879WSminus12 le 1205832119870119868119899
S12
G119879GS12 le 12058321198711198681198991205832119870 + 1205832119871 le 120583
(32)
instead of the inequalities of (31) Note that the 1st and the 2ndinequalities of (32) are equivalent to them of (31) and the 3rdinequality of (32) is a sufficient condition of the 3rd one of(31) Thus if the conditions of (32) hold then the inequalities
Mathematical Problems in Engineering 5
of (31) are also satisfied Namely if there exist the positivescalars 120583119870 and 120583119871 satisfying the inequalities of (32) then theinput constraint is also satisfied
Now by introducing the complementary variables 120583⋆119870 and120583⋆119871 we consider the following inequalitiesSminus12
W119879WSminus12 le 120583⋆119870119868119899
S12
G119879GS12 le 120583⋆119871119868119899120583⋆119870 + 120583⋆119871 le 120583120583⋆119870 ge 1205832119870120583⋆119871 ge 1205832119871
(33)
One can see from Lemma 3 (Schur complement) that the 4thand 5th inequalities in (33) are equivalent to
(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0 (34)
Moreover by applying Lemma 3 (Schur complement) to the1st inequality of (33) we obtain
minus120583⋆119870119868119899 +Sminus12
W119879WSminus12 le 0 lArrrArr
minus120583⋆119870S +W119879W le 0 lArrrArr
minusS + 1120583⋆119870W119879W le 0 lArrrArr(minusS W119879
W minus120583⋆119870119868119898) le 0(35)
Additionally since the matrix G isin R119898times119899 is known one caneasily calculate the scalar 120574G satisfyingG119879G le 120574G119868119899Thus weintroduce the inequality condition
minus120583⋆119871119868119899 + 120574GS le 0 (36)
and this inequality is a sufficient condition of the 2ndinequality of (33)
From the above discussion one can see that for givenpositive constants 120591 and 120574G if the symmetric positive definitematrix S isin R119899times119899 the matrix W isin R119898times119899 and the positivescalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 which satisfy LMIs of (26) (34)(35) and (36) are obtained then the input constraints aresatisfied
Consequently our main result is summarized as thefollowing theorem
Theorem 4 Consider the nonlinear system of (6) and thecontrol input of (10)
For given positive constants 120591 and 120574G if there exist thesymmetric positive definite matrix S isin R119899times119899 the matrix
W isin R119898times119899 and the positive scalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 whichsatisfy the LMIs
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0(minusS W119879⋆ minus120583⋆119870119868119898) le 0(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0minus120583⋆119871119868119899 + 120574GS le 0
120583⋆119870 + 120583⋆119871 le 120583
(37)
the fixed gain matrix 119870 isin R119898times119899 and the adaptive compensa-tion input 120595(119909 119905) isin R119898 are determined as
119870 = WSminus1120595 (119909 119905)
= minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) if 119861119879P119909 (119905) = 0120595 (119909 119905120575) if 119861119879P119909 (119905) = 0
(38)
where 119905120575 = lim120575gt0120575rarr0(119905minus120575) [6]Then the closed-loop system of(11) is internally stable and the state reachable setR(119909) belongsto E(P) Moreover the input constraints for the control input119906(119905) are also guaranteedRemark 5 In this paper we introduce the assumption119898 = 1and consider the input constraint of (9)
By theway for the control input119906(119905) = (1199061(119905) 119906119898(119905))119879there exist the componentwise input constraints that is1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 le 120583119894 (119894 = 1 119898) (39)
where 120583119894 is a given positive constant If the positive constant120583119894 for the constraints of (39) satisfies 120583119894 = 10 one can adoptthe inequality [14]
119906119879 (119905)Z119906 (119905) le 1Trace Z = 1 (40)
and the proposed design approach can be easily extended tothis case
On the other hand for the case of 120583119894 = 10 although wecan consider the condition119906119879 (119905)R119906 (119905) le 1
R12 = diag (120583+ 120583+) (41)
instead of (40) where 120583+ means max119895120583119895 then the resultingcontroller which is obtained by adopting the condition of (41)
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
of (31) are also satisfied Namely if there exist the positivescalars 120583119870 and 120583119871 satisfying the inequalities of (32) then theinput constraint is also satisfied
Now by introducing the complementary variables 120583⋆119870 and120583⋆119871 we consider the following inequalitiesSminus12
W119879WSminus12 le 120583⋆119870119868119899
S12
G119879GS12 le 120583⋆119871119868119899120583⋆119870 + 120583⋆119871 le 120583120583⋆119870 ge 1205832119870120583⋆119871 ge 1205832119871
(33)
One can see from Lemma 3 (Schur complement) that the 4thand 5th inequalities in (33) are equivalent to
(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0 (34)
Moreover by applying Lemma 3 (Schur complement) to the1st inequality of (33) we obtain
minus120583⋆119870119868119899 +Sminus12
W119879WSminus12 le 0 lArrrArr
minus120583⋆119870S +W119879W le 0 lArrrArr
minusS + 1120583⋆119870W119879W le 0 lArrrArr(minusS W119879
W minus120583⋆119870119868119898) le 0(35)
Additionally since the matrix G isin R119898times119899 is known one caneasily calculate the scalar 120574G satisfyingG119879G le 120574G119868119899Thus weintroduce the inequality condition
minus120583⋆119871119868119899 + 120574GS le 0 (36)
and this inequality is a sufficient condition of the 2ndinequality of (33)
From the above discussion one can see that for givenpositive constants 120591 and 120574G if the symmetric positive definitematrix S isin R119899times119899 the matrix W isin R119898times119899 and the positivescalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 which satisfy LMIs of (26) (34)(35) and (36) are obtained then the input constraints aresatisfied
Consequently our main result is summarized as thefollowing theorem
Theorem 4 Consider the nonlinear system of (6) and thecontrol input of (10)
For given positive constants 120591 and 120574G if there exist thesymmetric positive definite matrix S isin R119899times119899 the matrix
W isin R119898times119899 and the positive scalars 120583119870 120583119871 120583⋆119870 and 120583⋆119871 whichsatisfy the LMIs
(119867119890 119860S + 119861W + 120591S 119861119903⋆ minus120591X119903) le 0(minusS W119879⋆ minus120583⋆119870119868119898) le 0(minus120583⋆119870 minus120583119870⋆ minus1 ) le 0(minus120583⋆119871 minus120583119871⋆ minus1 ) le 0minus120583⋆119871119868119899 + 120574GS le 0
120583⋆119870 + 120583⋆119871 le 120583
(37)
the fixed gain matrix 119870 isin R119898times119899 and the adaptive compensa-tion input 120595(119909 119905) isin R119898 are determined as
119870 = WSminus1120595 (119909 119905)
= minus G119909 (119905)1003817100381710038171003817119861119879P119909 (119905)1003817100381710038171003817119861119879P119909 (119905) if 119861119879P119909 (119905) = 0120595 (119909 119905120575) if 119861119879P119909 (119905) = 0
(38)
where 119905120575 = lim120575gt0120575rarr0(119905minus120575) [6]Then the closed-loop system of(11) is internally stable and the state reachable setR(119909) belongsto E(P) Moreover the input constraints for the control input119906(119905) are also guaranteedRemark 5 In this paper we introduce the assumption119898 = 1and consider the input constraint of (9)
By theway for the control input119906(119905) = (1199061(119905) 119906119898(119905))119879there exist the componentwise input constraints that is1003816100381610038161003816119906119894 (119905)1003816100381610038161003816 le 120583119894 (119894 = 1 119898) (39)
where 120583119894 is a given positive constant If the positive constant120583119894 for the constraints of (39) satisfies 120583119894 = 10 one can adoptthe inequality [14]
119906119879 (119905)Z119906 (119905) le 1Trace Z = 1 (40)
and the proposed design approach can be easily extended tothis case
On the other hand for the case of 120583119894 = 10 although wecan consider the condition119906119879 (119905)R119906 (119905) le 1
R12 = diag (120583+ 120583+) (41)
instead of (40) where 120583+ means max119895120583119895 then the resultingcontroller which is obtained by adopting the condition of (41)
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
becomes very conservative Thus the constraint condition of(41) is not desirable and efficient and tractable conditionscorresponding to the constraint of (39) have not been devel-oped Namely the design problem for such constraints is oneof our future research subjects
Remark 6 In this paper we deal with a stabilization problemof an adaptive robust controller for a class of nonlinearsystems with input constraints On the other hand in thework of Oya and Hagino [7] by introducing an error signalbetween the time response and the desired one the robustcontroller in [7] can achieve not only robust stability butalso satisfactory transient behavior Note that the proposedcontroller design approach can easily be extended to suchcontrol strategy
5 Numerical Examples
In order to demonstrate the efficiency of the proposed robustcontroller we have run a simple numerical example
Let us consider the nonlinear system
119889119889119905119909 (119905) = (minus10 4000 minus50) 119909 (119905) + (0020) 119906 (119905)+ (0020) 119892 (119909 119905) + (1020) 119903 (119905) (42)
with the input constraint119906 (119905) le 10 (43)
Additionally in this simulation the reference input 119903(119905) isin R1
and G isin R1times2 selected as 119903(119905) = Xminus1119903 times cos(120587119905) (X119903 = 22)andG = (20 10) respectively that is 120574G = 40
Firstly in order to derive the proposed adaptive robustcontroller we select the design parameter 120591 in (37) such as 120591 =20 Then by solving the LMIs of (37) the symmetric positivedefinite matrix S isin R2times2 the matrix W isin R1times2 and thepositive constants 120583119870 120583119871 120583⋆119870 and 120583⋆119871 can be calculated as
S = (24439 times 10minus1 minus28670 times 10minus2⋆ 82520 times 10minus2 ) W = (minus44815 times 10minus1 minus63870 times 10minus2) 120583119870 = 99729 times 10minus1120583119871 = 38150 times 10minus2120583⋆119870 = 99724 times 10minus1120583⋆119871 = 18800 times 10minus3
(44)
Namely the following inequality holds
120583⋆119870 + 120583⋆119871 = 99912 times 10minus1 (= 120583⋆) lt 10 (45)
Thus the constraint of (43) for the control input is satisfiedbecause 120583⋆ lt 120583
By using the symmetric positive definitematrixS isin R2times2
and the matrixW isin R1times2 we have119870 = (minus20072 minus14714) (46)
Moreover the symmetric positive definite matrix P isin R2times2
can be obtained as
P = (42676 14828⋆ 12633 times 10minus1) (47)
and the state reachable setR(119909) is included in the ellipsoidalset E(P) described as
E (P) = 119909 isin R119899 | 42676 times 10minus1 times 11990921 (119905) + 29656
times 1199091 (119905) times 1199092 (119905) + 12633 times 10minus1 times 11990922 (119905) le 10 (48)
In this example the initial value of the uncertain nonlin-ear system of (42) is selected as 119909(0) = (00 00)119879 Moreoverin this simulation we consider the following two cases forthe nonlinear function 119892(119909 119905) (note that these two cases inthis example mean a nonlinear function of Case 1 and alinear function of Case 2 as a special case for the nonlinearityrespectively)
Case 1119892 (119909 119905)= (radic2 cos (101205871199091 (119905)) minus sin (51205871199092 (119905))) 119909 (119905) (49)
Case 2
119892 (119909 119905) =
0 le 119905 lt 1 119892 (119909 119905) = (minusradic2 minus1) 119909 (119905)1 le 119905 lt 3 119892 (119909 119905) = (minusradic2 1) 119909 (119905)3 le 119905 lt 5 119892 (119909 119905) = (radic2 minus1) 119909 (119905)5 le 119905 119892 (119909 119905) = (radic2 1) 119909 (119905) (50)
The simulation result of this numerical example is shownin Figures 1ndash8 Figures 1ndash6 represent time histories of the stateand the control input and in these figures ldquoConventionalrdquoand ldquoProposedrdquo mean the conventional robust control basedon the existing result [7] and the proposed robust controllerrespectively Namely ldquoConventionalrdquo does not take inputconstraints into account Moreover ldquoUpper boundrdquo andldquoLower boundrdquo in Figures 5 and 6 represent the constraintfor control input and ldquobluerdquo line in Figures 7 and 8 meansthe ellipsoidal set E(P)
From these figures the proposed robust controller andthe conventional adaptive robust controller based on theexisting result [7] stabilize the dynamical system of (42) inspite of unknown nonlinearities Moreover from Figure 6we find that the proposed adaptive robust controller satisfiesthe control input saturation Additionally one can see fromFigures 7 and 8 that the state trajectory is contained in theellipsoidal set E(P) Namely the proposed controller canrobustly stabilize the uncertain system of (42) and also satisfythe control input saturation
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Case 1Case 2
minus1
minus05
0
05
1
Stat
e
1 2 3 4 5 6 7 80Time t
Figure 1 Time histories of 1199091(119905) Conventional
minus06
minus04
minus02
0
02
04
06
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 2 Time histories of 1199092(119905) ConventionalOn the other hand one can see from Figure 5 that the
conventional adaptive robust controller cannot satisfy theinput constraints
Therefore the effectiveness of the proposed adaptiverobust controller giving consideration to input constraints fora class of nonlinear systems has been shown
6 Conclusion
In this paper we have proposed an adaptive robust controllerfor a class of nonlinear systems with input constraints Theproposed design method is based on LMIs and thus theproposed adaptive robust controller can easily be obtainedby using software such as MATLAB and Scilab In additionthe effectiveness of the proposed robust controller has beenshown by simple numerical examples Note that the proposed
minus04
minus02
0
02
04
Stat
e
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 3 Time histories of 1199091(119905) Proposed
minus02
minus01
0
01
02St
ate
Case 1Case 2
1 2 3 4 5 6 7 80Time t
Figure 4 Time histories of 1199092(119905) Proposedadaptive robust controller is an extension of the existingresults [7]
In the proposed adaptive robust control strategy thecontrol input consists of a state feedback with fixed gains andan adaptive compensation input and the adaptive compen-sation input is defined as a state feedback with time-varyingadjustable parametersThe advantages of the proposed adap-tive robust control scheme are as follows the proposed robustcontroller is more flexible and adaptive than conventionalfixed gain robust controller which is designed for the worstcase of unknown parameter variations and satisfies the giveninput saturations
In our future work we will extend the proposed adaptiverobust controller synthesis to such a broad class of systemsas linear systems with mismatched uncertainties uncertainnonlinear systems with time delays and so on Furthermore
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 5 Time histories of 119906(119905) Conventional
1 2 3 4 5 6 7 80Time t
Case 1Case 2
Upper boundLower bound
minus15
minus1
minus05
0
05
1
15
Con
trol i
nput
Figure 6 Time histories of 119906(119905) Proposed
analysis of the conservativeness of the proposed adaptiverobust controller giving consideration to input saturations isalso an important issue for our future works Additionallythe extension of the proposed approach to more generaltypes for constraints such as (39) and the discussion forconservativeness of the proposed design approach are alsoimportant future research subjects
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03
x2(t)
Figure 7 Ellipsoidal set and trajectory of 119909(119905) Case 1
010 02 03minus02 minus01minus03
x1(t)
TrajectoryEllipsoidal set ℰ()
minus03
minus02
minus01
0
01
02
03x2(t)
Figure 8 Ellipsoidal set and trajectory of 119909(119905) Case 2Acknowledgments
The authors would like to thank Professor MohammadD Aliyu Associate Editor with Ecole Polytechnique deMontreal for his valuable and helpful comments that greatlycontributed to this paper
References
[1] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice Hall 1996
[2] P P Khargonekar and M A Rotea ldquoMixed 1198672119867infin controla convex optimization approachrdquo IEEE Transactions on Auto-matic Control vol 36 no 7 pp 824ndash837 1991
[3] I R Petersen and D C McFarlane ldquoOptimal guaranteedcost control and filtering for uncertain linear systemsrdquo IEEE
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Transactions on Automatic Control vol 39 no 9 pp 1971ndash19771994
[4] D D Siljak and D M Stipanovic ldquoRobust stabilization ofnonlinear systems the LMI approachrdquo Mathematical Problemsin Engineering vol 6 no 5 pp 461ndash493 2000
[5] Z Zuo J Wang and L Huang ldquoRobust stabilization for non-linear discrete-time systemsrdquo International Journal of Controlvol 77 no 4 pp 384ndash388 2004
[6] M Maki and K Hagino ldquoRobust control with adaptationmechanism for improving transient behaviourrdquo InternationalJournal of Control vol 72 no 13 pp 1218ndash1226 1999
[7] H Oya and K Hagino ldquoRobust control with adaptive compen-sation input for linear uncertain systemsrdquo IEICE Transactionson Fundamentals of Electronics Communications and ComputerSciences vol 86 no 6 pp 1517ndash1524 2003
[8] J E Gayek ldquoA survey of techniques for approximating reachableand controllable setsrdquo inProceedings of the 30th IEEEConferenceon Decision and Control pp 1724ndash1729 IEEE Brighton UKDecember 1991
[9] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000
[10] Y-Y Cao and Z Lin ldquoStability analysis of discrete-time systemswith actuator saturation by a saturation-dependent Lyapunovfunctionrdquo Automatica vol 39 no 7 pp 1235ndash1241 2003
[11] E G Gilbert and K T Tan ldquoLinear systems with state and con-trol constraints the theory and application of maximal outputadmissible setsrdquo IEEE Transactions on Automatic Control vol36 no 9 pp 1008ndash1020 1991
[12] F Lin R D Brandt and J Sun ldquoRobust control of nonlinearsystems compensating for uncertaintyrdquo International Journal ofControl vol 56 no 6 pp 1453ndash1459 1992
[13] F R Gantmacher The Theory of Matrices vol 1 ChelseaPublishing New York NY USA 1960
[14] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in System and Control Theory SIAM Studiesin Applied Mathematics 1994
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of