1、 Feedback control;Uncertain linear systems;Lyapunov methods;Riccati equation1.INTRODUCTIONThis 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 fe
2、edback control is considered.In this case the uncertain linear system consists of a linear 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 paramet
3、ers are assumed to vary with time.The problem of stabilizing uncertain linear systems of 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
4、.These matching conditions constitute sufficient conditions for a given uncertain system 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 a
5、re known to be unduly restrictive.Indeed,It has been shown in Barmish and Leitmann(1982) 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 dire
6、cted towards developing control schemes which will stabilize a larger class of system than 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 cla
7、ss of uncertain linear systems for which one can construct a stabilizing feedback control 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.Lyapu
8、nov of law,enter to the 1990s non-linear controlled field succeed in will it be the eighties 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 c
9、omposure system , suppose at first the uncertainty existing is unknown in the real system, 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 targ
10、et claim model construct a proper Lyapunov function, make its whole system of assurance steady 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
11、 theory is it reach the non-linear system to delay more linear system. Introduced the non- 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
12、penetrated,can utilize the concept of relative steps to divide into linear and two non-linear 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-dime
13、nsional situation, if the asymptotic stability of the overall situation of zero dynamic subsystem, 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
14、 references cited above dealing with uncertain linear systems,the stability of the close-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(19
15、85)and Petersen(1983),it is show in section 2;see also Barmish(1985).Furthermore,in Barmish(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
16、 a major portion of this paper is devoted to this problem.Various aspects of the problem 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
17、 to finding a suitbale quadratic Lyapunov function involves solving an“augmented”matrix Riccati 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 unc
18、ertainty is allowed in both the “A” matrix and “B” matrix.Furthermore,a number of classes 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 inv
19、olves constructing a quadratic form which serves as an upper bound for the Lyapunov derivative 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
20、 stems from the fact that a candidate quadratic Lyapunov function can easily be obtained 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 regulato
21、r problem,e.g.Anderson and Moore(1971).Hence,the procedure presented in the paper can be 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 an
22、d and are vectors of uncertain parameters,is considered.The functions r()and s() are restricted to be Lebessue measurable vector functions.Furthermore,the matrices and are assumed to be rank one matrices of the formand in the above description denote the component of the vectors r(t) and s(t) respec
23、tively.Remarks:Note that an arbitrary n n matrix can always be decomposed as the sum 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 syst
24、em which satisfies the rank-one assumption.Moreover,stabilizability of this larger system 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 represe
25、nts one of the main weaknesses or the approach.That is,the quadratic bound method described in the sequel may fail for one decomposition of 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 imp
26、ortant area for future research.A final observation concerns the bounds on the uncertain 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 unce
27、rtain bounds can be eliminated by suitable scaling of the matrices.The weighting matrices Q and R.Associated with the system()are the positive definite symmetric weighting matrices .These matrices are chosen by the designer.It will be seen in Section 4 that these matrices are analogous to the weight
28、ing 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
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