外文原文及译文.docx

1、外文原文及译文A Riccati Equation Approach to the Stabilization of Uncertain Linear SystemsIAN RPETERSEN and CHRISTOPHER VHOLLOAbstractThis paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems.The systems under consideration contain uncertain paramete

2、rs whose values are known only to with a given compact bounding setFurthermore，these uncertain parameters may be time-varying.The method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic Lyapunov function.Th

3、e main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic Lyapunov function,Once the Lyapunov function has been obtained,it used to construct the stabilizing feedback control law.The fundamental idea behind the alg

4、orithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system.This upper bound is a quadratic form.By using this upper bounding procedure,a suitable Lyapunov function can be found by solving a certain matrix Riccati equation.Another major c

5、ontribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic Lyapunov functionKey words: Feedback control;Uncertain linear systems;Lyapunov methods;Riccati equation1.INTRODUCTIO

6、NThis paper deals with the problem of designing a controller when no accurate model is available for the process to be controlled.Specifically,the problem or stabilizing an uncertain linear system using state feedback control is considered.In this case the uncertain linear system consists of a linea

7、r system containing parameters whole values are unknown but bounded.That is,the values of these uncertain parameters are known to be contained with given compact bounding sets.Furthermore,these uncertain parameters are assumed to vary with time.The problem of stabilizing uncertain linear systems of

8、this type has attracted a considerable amount or interest in recent years. In Leitman(1979,1981)and Gutman and Palmoor(1982),the uncertainty in the system is assumed to satisfy the so called “matching conditions.These matching conditions constitute sufficient conditions for a given uncertain system

9、to be stabilizable.In Corless and Leitmann(1981)and Barmish,Corless and Leltmann(1983),this approach is extended to uncertain nonlinear systems.However,even for uncertain linear systems the matching conditions are known to be unduly restrictive.Indeed,It has been shown in Barmish and Leitmann(1982)

10、and Hollot and Barmish(1980) that there exist many uncertain linear systems which fail to satisfy the matching conditions and yet are nevertheless stabilizable.Consequently,recent research efforts have been directed towards developing control schemes which will stabilize a larger class of system tha

11、n those which satisfy the matching conditions; e.g.Barmish and Leitmann(1982),Hollot and Barmish(1980),Thorp and barmish(1981),Barmlsh (1982,1985)and HoLLot(1984).The main aim of this paper is to enlarge the class of uncertain linear systems for which one can construct a stabilizing feedback control

12、 law.It should be noted however,that in contrast to Corless and Leitmann(1981) Barmish,Corless and Leitmann(1983)and Petersen and Barmish(1986),attention will be restricted to uncertain linear systems here.Lyapunov of law,enter to the 1990s non-linear controlled field succeed in will it be the eight

13、ies the 20th century while being excellent while being stupid It is a main design method with stupid and excellent and calm non-linear system. While utilizing this kind of method to design the stupid excellent composure system , suppose at first the uncertainty existing is unknown in the real system

14、, but belong to a certain set that describes,namely the uncertain factor can show in order to there is unknown parameter of the circle,gain unknown perturbation function to have circle and accuse of mark of target claim model construct a proper Lyapunov function, make its whole system of assurance s

15、teady to any element while assembling uncertain.Just because of this kind of generality,no matter used for analyzing the stability or using for being calm and comprehensive,lack contractility.People attempt ripe theory is it reach the non-linear system to delay more linear system. Introduced the non

16、- linear system.In recent years to the steps, in the non-linear system,the meaning in the steps lies in it has described the essence of the non-linear structure of the system.For imitating the non-linear system penetrated,can utilize the concept of relative steps to divide into linear and two non-li

17、near parts systematically,part it is non-linear can view,it is linear for part can have accused of can watch as well as, system that form like this zero subsystem not dynamic, having proved it under the one-dimensional situation, if the asymptotic stability of the overall situation of zero dynamic s

18、ubsystem, so whole system can be exactly booked nearer and nearer with the overall situation.Feedback as to steps linearization combines together, receive and well control the result such as document1.In all the references cited above dealing with uncertain linear systems,the stability of the close-

19、loop uncertain system is established using a quadratic Lyapunov function.This motivates the concept of quadratic stabilizability which is formalized in section 2;see also Barmish(1985).Furthermore ,in Barmlsh(1985)and Petersen(1983),it is show in section 2;see also Barmish(1985).Furthermore,in Barmi

20、sh(1985) and Petersen(1983),it is shown that the problem of stabilizing an uncertain linear system can be reduced to the problem or constructing a suitable quadratic Lyapunov function for the system consequently a major portion of this paper is devoted to this problem.Various aspects of the problem

21、or constructing suitable quadratic Lyapunov functions have been investigated in Hollot and Barmish(1980),Thorp and Barmish(1981) and Hollot(1984),Chang and Peng(1972),Noldus(1982) and Petcrsen(1983).One approach to finding a suitbale quadratic Lyapunov function involves solving an“augmented”matrix R

22、iccati equation which has been specially constructed to account for the uncertainty in the system;e.g. Chang and Peng(1972)and Noldus (1982).The results presented in this paper go beyond Noldus(l982) in that uncertainty is allowed in both the “A” matrix and “B” matrix.Furthermore,a number of classes

23、 of uncertain systems are identified,for which the success of this method becomes necessary and sufficient for the existence of a suitable quadratic Lyapunov function.The fundamental idea behind the approach involves constructing a quadratic form which serves as an upper bound for the Lyapunov deriv

24、ative corresponding to the closed loop uncertain system.This procedure motivates the introduction or the term quadratic bound method to describe the procedure used in this paper.The benefit of quadratic boundind stems from the fact that a candidate quadratic Lyapunov function can easily be obtained

25、by solving a matrix Riccati equation.For the special case or systems without uncertainty,this “augmented” Riccati equation reduces to the “ordinary” Ricccti equation which arises in the linear quadratic regulator problem,e.g.Anderson and Moore(1971).Hence,the procedure presented in the paper can be

26、regarded as being an extension of the linear quadratic regulator design procedure.2.SYSTEM AND DEFINITIONSA class of uncertain linear systems described by the state equationswhere is the state, is the control and and are vectors of uncertain parameters,is considered.The functions r()and s() are rest

27、ricted to be Lebessue measurable vector functions.Furthermore,the matrices and are assumed to be rank one matrices of the formand in the above description and denote the component of the vectors r(t) and s(t) respectively.Remarks:Note that an arbitrary n n matrix can always be decomposed as the sum

28、of rank one matrices;i.e.for the system(),one can write with rank one. Consequently,if is replaced by and the constraint is included for all i and j then this overbounding” of the uncertainties will result in a system which satisfies the rank-one assumption.Moreover,stabilizability of this larger sy

29、stem will imply stabiliabillty for(Z).At this point,observe that the rank one decompositions for the and are not unique.For example, can be multiplied by any scalar if is divided by the same scalar.This fact represents one of the main weaknesses or the approach.That is,the quadratic bound method des

30、cribed in the sequel may fail for one decomposition of and and yet succeed for another.At the moment,there is no systematic method for choosing the best rank-one decompositions and therefore,this would constitute an important area for future research.A final observation concerns the bounds on the un

31、certain parameters,it has been assumed that each parameter satisfies the same bound.For example,one has rather than separate bounds.This assumption can be made without loss or generality.Indeed,any variation in the uncertain bounds can be eliminated by suitable scaling of the matricesand.The weighti

32、ng matrices Q and R.Associated with the system()are the positive definite symmetric weighting matrices and.These matrices are chosen by the designer.It will be seen in Section 4 that these matrices are analogous to the weighting matrices in the classical linear quadratic regulator problem.The formal definition of quadratic stabilizability now presented.Definition 2.1.The system() is said to be quadratically stabilizable if there exists continuous feedback control with P(0)=0,an n n:positive definite symmetric matrix P and a constant 0 such that the foll