Session Keynote Lyapunov Function Candidates for Descriptor Systems Problems and Solutions

Session Keynote:Lyapunov Function Candidates for Descriptor Systems:

Problems and Solutions

Vladimir B.Baji′c

Centre for Engineering Research,T echnikon Natal,P.O.Box953,Durban4000,Republic of South

Africa

e-mail:bajic.v@umfolozi.ntech.ac.za

Keywords:Lyapunov’s direct method,Lyapunov functions,descriptor(singular)systems,stability, qualitative analysis

ABSTRACT

This paper presents in a systematic way problems encountered in the construction of Lyapunov func-tion candidates for descriptor systems.The solutions to some of the major difficulties in the applica-tion of Lyapunov’s direct method to descriptor systems are presented.Some new results regarding the extension of Lyapunov’s direct method tied to the construction of the Lyapunov functions are given. Also,the algebraic necessary and sufficient conditions for some specific properties of motions of descriptor systems are developed.

INTRODUCTION AND BACKGROUND

It is a tradition to consider the equations describing internal system dynamics of continuous time systems in the normal form of the so-called state equations

(%'s E|c%c 5U c(1) where('_*_|is the ordinary time derivative operator,and where|5U, 5U o and%5U c denote the time,the input vector and the state vector,respectively.Systems governed by the model (1)we usually call the state variable systems.However,there are physical systems for which the state models do not exist.It was recognized that the more natural models for these cases go beyond the classical state-variable description(1).These models are given by

s E|c%c(%c 'f5U (2) where R'?may be allowed.Systems with such models we denote as implicit.In the special case the models(2)contain the canonical form of models linear in(%,and they are of the form

E|c%c (%'s E|c%c (3) It should be pointed out that the matrix can even be a rectangular one.Systems governed by(3)are known as descriptor,as well as singular,semi-state,generalized state-space or differential-algebraic systems.

ST ABILITY ANAL YSIS OF DESCRIPTOR SYSTEMS

The surveys of some of the results concerning both continuous and discrete descriptor and general implicit systems can be found,for example,in the books[1]-[3],[6]-[7]and[10]and in the special issues of the journal Circuits,Systems and Signal Processing[8]-[9].The systematic introduction and presentation of part of the results relating to the general application of Lyapunov’s direct method (LDM)for the analysis of DS is given in[2],[3].For some other results on stability of DS,particularly with regard to mechanical DS,see[12]-[16].It has been shown in[2],[4],that specific structure of DS may lead to several problems regarding the construction of Lyapunov function candidates(LFCs) for the intended qualitative analysis.These problems do not appear in the application of the LDM to systems in the normal forms(1).This paper aims to contribute to the general methodology relating to the construction of the LFCs for descriptor systems.W e present in a systematic way problems

encountered in the construction of LFCs for continuous-time DS,as well as the solutions to some of the major difficulties in the application of the LDM.Some new results regarding the extension of the LDM tied to the construction of LFCs are given.

PROBLEM ST A TEMENT AND SIGNIFICANCE:LDM AND THE ROLE OF LFC There are essentially two problems that relate to the preliminary construction of the LFC:

the problems that appear in the process of evaluation of the total time derivative(TTD)of an LFC along the motions of(2)and(3).

the structure and properties of the matrix in(3)

The application of the LDM requires the selection of a concrete LFC.When the exact analytical form of an LFC is known and when the necessary and sufficient conditions that ensure the appropriate qualitative concept of motion are expressed directly in terms of system and LFC parameters,then we say that the construction of the LFC is given.Unfortunately,in the general case the main drawback of the LDM is the conceptual nonexistence of a systematic procedure for the practical construction of an LFC for the qualitative concept of interest(one of possible general solutions for this is proposed in this paper).There are only a few results of the general nature on the construction of the LFC for DS(see references in[2]).So far there are no global answers on how to select an LFC for DS.Thus any general hint that concerns the construction of an LFC is of particular importance.The aim of this presentation is to provide some suitable solutions of this problem in a very specific way by the evaluation of the derivative of an LFC along the solutions of DS.This poses practical problems in finding a TTD of an LFC for DS,and,also,directly restricts the classes of useful LFCs.T o make this more evident let the matrix and the functions s and in(3)be sufficiently smooth and such that the systems considered possess the continuous and differentiable in|solutions in some domain. For simplicity,let T E|c% be an arbitrary function of|and%,which is continuously differentiable, and let it be an LFC for systems(2)or(3).In any qualitative analysis of the solutions of systems considered by means of the LDM,specific properties of the LFC are required.In what follows the TTD of T along the solutions of any of the systems(1)to(3)we will denote by(P T In all cases,the calculation of the TTD(P T d|c E| o of the LFC T,along the system motion ,has to be found.In general,this is not possible in a direct way for systems(2)or(3),unless T E|c% depends on%in a specific manner.As is well known,(P T d|c E| o can be found without the knowledge of solutions if the system analyzed is in the normal form(1)[17,p.12].Using the same argument, the(P T d|c E| o of the LFC T E|c% along the motion is given by

(P T d|c E| o'YT d|c E| o

n

YT d|c E| o

e

( E| (4)

may be computed along the solutions of(2)or(3)if( is obtained from these equations and substi-tuted in(4).It is obvious that this can be done directly for systems in the normal form(1),in which case along the solutions of(1)we have

(P T E|c% 'YT E|c%

Y|

n

YT E|c%

Y%

e

s E|c%c (5)

However,systems(2)or(3)need not be solvable in(%.Thus,in the cases when(2)or(3)are inher-ently singular(non-solvable in(%),(4)cannot be used directly for the evaluation of(P T d|c E| o. This is one of the main difficulties relating to successful construction of the LFC for DS.In this exposition will provide some general solutions to this problem.

LFC FOR IMPLICIT SYSTEMS:ALGEBRAIC NECESSARY AND SUFFICIENT

CONDITION

W e consider a system P governed by implicit differential equations(2).Let W'i|5U G|5o|r c|s d j, |r c|s5U,|r |s.The solutions of(2)are functions G W W U c6E|c|f c%f $ E|c|f c%f 5

U c c where |f and %f are regarded as the initial moment and the initial value of a solution c re-spectively .The notation E |c |f c %f represents the value of the solution of P at the moment |c that at the moment |f had the value %f W e consider the case when (2)has only unique solutions that are continuous functions of their arguments and differentiable in |.Also,we allow that the E |f c %f 5W V 8E |f c |s c where V 8is the set of the consistent initial values %f at the moment |f such that from each %f 5V 8E |f c |s at least one solution that exists on W E |f 'd |f c |s d W c is generated.Further,we will consider properties of solutions of P only in the domain G 'W V 8E |r c |s 6E |c % .Obviously ,G is invariant w .r.t.solutions c i.e.if E |f c %f 5G c then E |c E |c |f c %f 5G for any | |f c |5W .W e will provide the algebraic necessary and sufficient conditions for a function T to be a L yapunov function for (2),by which the following useful qualitative concepts can be verified.The concepts considered are expressed by

S d |c %E |c |f c %f o d |c |f c S E |f c %f o c S E |f c %f ' d |f c |f c S E |f c %f o c (6)

where S and are scalar real functions.When all solutions of P in the domain G satisfy (6)we say that P has the property in G W e can give different meaning to the functions S and

Theorem 1.Let an auxiliary function T G W U c 6E |c % $T E |c % 5U c T E |c % 5 E W U c c be chosen and let C G W U $U define a scalar differential equation ( 'C E |c that ;E |f c f 5W U has unique solutions E |c |f c f that are continuously differentiable in |.Let the Eulerian derivative (P T along the solutions of the model of (2)be given by (P T E |c % '}E |c % If (2)has unique solutions c then for every solution that exists on W E |f the property given by (6)will hold if and only if }E |c % C d |c T E |c % o in W E |f V 8E |r c |s The last conditions are also necessary and sufficient that T E |c % S E |c % be the Lyapunov function for EP c

This result is based on [5]and is quite opposite to Gruji′c ’s method [11].It gives the algebraic nec-essary and sufficient conditions for the verification of the examined property as well as the direct construction of T Some partial results of this approach have been already successfully applied to the problems of nonlinear uncertain and fuzzy logic systems.

GENERAL PROCEDURE FOR ELIMINA TION OF (%FROM THE TTD OF AN LFC Consider the model (3)and assume that possibly all entries of the matrix are dependent on |and %.The construction of an LFC in such a situation is a complicated task.The method that follows is a general method for the calculation of the TTD of an LFC for DS (3)with nonlinear and possibly rectangular matrix .Let T be a scalar LFC,T G U U c 6E |c % $T E |c % 5U and let T be differentiable in the ordinary sense.If the matrix equation .'.7 c where .' YT E |c % Y%

e (7)has any solution 7,then the formula for the TTD o

f the LFC T alon

g the solutions of DS (3)is (P T E |c % 'YT E |c % Y|n YT E |c % Y%

e 7s E |c %c This procedure is general and can be used directly for stability problem analysis as given in the following results [2].

Theorem 2.Let W be a continuous open time interval,W U c and let l U .Let T G W l 6E |c % 5U n c T 5 E W l c and let T be a differentiable function of its arguments.Assume that >G W $U is an integrable function.If there is any solution 7of the matrix equation (7),and if ;E |c % 5W l YT E |c % Y|n YT E |c % Y% e 7s E |c %c >E | Tc (8)then any solution of (3)which starts in l at |'|f ,satisfies as long as it remains in l the

estimate

T d |c E | o T d |f c E |f o i T ]||f

>E r _r

(9)T o analyze systems (3)with inputs we considered the following.

Theorem 3.Let W be a continuous open time interval,W U c and let l U c Assume that the function s in (3)is of the form s E |c %c ' E |c % %n (E |c % ,and that is a positive defi-nite symmetric matrix .Let 7be any solution of the matrix equation .'.7 in W l c where .'Y E % *Y%e Let 8' e n Y *Y|n .7 and '(n .7( Let 8f be any positive definite constant matrix and let a solution of (3)passes through l Assume that >G W $U is an integrable function,and m E |c % m Rc m E |c % m v .Let K' %5U c G m %m Rvb 3 T E 8f .If the matrix ~f '8n 8f >E | is negative semi definite for all values of its arguments,the estimate (9)holds with T '%e %c as long as E | 5K _l c providing %f 5K _l ,and if K _l is not empty.If E |c % f ,(9)holds in l

NONDIFFERENTIABLE LFC

There are some classes of physical systems for which the ’’natural ’’LFCs are not very regular and may lead to nondifferentiable LFC.The extension of the LDM to the nondifferentiable LFC was provided for systems in the normal form by Y oshizawa [18,pp.3-4].Let (n P T denote the upper right-hand Dini derivative of an LFC T E |c % ,for the system (1).Calculation of (n P T requires a knowledge of the so-lutions of (1).However,the formula for the calculation of the (n P T ,which requires no knowledge of the system solution c is given in [18].This formula is valid under the mild assumptions that T E |c % should be continuous in the domain of interest and should locally satisfy the Lipschitz condition with respect to % It states that (n P T d |c %E | o G '* 4?

(2)or (3).T o overcome this,we introduce a new variable 5U r ,and consider the problem in terms of E |c instead of E |c % Let W be an open time interval of interest and let K %be an open set,K % U r Denote the LFC T as T E |c G W K %$U c T 5 E W K % c T 5Lip %E W K % c where Lip %E W K % denotes the class of functions that locally satisfy the Lipschitz condition with respect to when E |c 5W K %.Let (2)has solutions %E | on W .At an arbitrary moment |5W the values of all solutions %E | of (2)which exist on W determine the points in the system descriptor space U c which form the set V 8E | .Define also the set V %E | 'i 5U r G 'C E |c % c %5V 8E | j c ;|5W .Then the following hold [2].

Theorem 4.Let W be an open continuous time interval and let the function C G W U c $U r define the variable 'C E |c % and is such that:(a)for a given function G W U c ,the TTD ( d |c %c E | o along the solutions of the (2)is known,and is a bounded function when E |c belongs to a compact set in W U r ;(b)when E |c % continuously change in the set W V 8E | then E |c continuously change in the set W V %E | .Let K %be an open set,V %E | K % U r c ;|5W c and let and LFC T G W K %6E |c $T E |c 5U c T 5 E W K % c T 5Lip %E W K % If (2)has continuous solutions %E | on W ,such that E | 'C d |c %E | o satisfies E |c E | 5W V %E | c then the upper right-hand Dini derivative of T ,along the solutions %E | at the moment |5W c is given by (n P T d |c E | o '* 4t T E 3 E T E |n c n ( T E |c G $f n

TIME DISCONTINUOUS LFC

Consider the situation where and s in (3)are time-discontinuous for a fixed %.This is motivated,for example,by the usage of time-discontinuous components like switching capacitors and resistive switches in singular electrical circuits.Our interest will be in treating the case when and s are continuous in %for a fixed |,and piecewise continuous in |for a fixed %,while for a fixed %the matrix is differentiable for all |where both and s are continuous.W e will develop sufficient conditions for the upper bounds on the system response.From these bounds the conditions for the

asymptotic behavior of response will be derived.W e will utilize the LDM and a time-discontinuous Lyapunov function.Consider the singular model (3)of the form

d |c %E | o (%E | 's d |c %E | o c %c s E |c % 5U c c E |c % 5U c f c (10)

where |5W U .Let %5U c be fixed and denote by W 7 the set consisting of all points of W where either or s are discontinuous in |.W e will assume that the set W 'V i W 7 G %5U c j is such that W _E where E W and where E is bounded,has finite number of elements | i.e.W 'i | j .Denote by W the set W q W .Without loss of generality we will assume that the elements of W are ordered,i.e.; : ,|&:| .W e will assume that all discontinuities in |for and s are of the first kind,and that is continuously differentiable on A for a fixed %5U c .W e assume that and s are such as to ensure the existence and uniqueness of solutions of (10)which are continuous on W and continuously differentiable on W .Such solutions will satisfy (10)for all |5W .The set of such solutions we denote by 7 .Let W E |f 'W _i |5W G | |f j for |f 5W .In what follows U n 'i @5U G @ f j and P denotes the complement of the set P .W e can state the following results based on [4]

Theorem 5.Consider the model (10)for which V '>.Let W be a continuous open time interval,W U ,and let l U c .Assume that there exist a function T G W l 6E |c % $T E |c % 5U n ,such that:

(i)T is continuous in E |c % on W l ,T E | c % 'T E |n c % for all E | c % 5W l ,and T has only discontinuities in |of the first kind;

(ii)YT E |c % *Y|exists and is continuous in E |c % on W l ;

(iii)d YT E |c % *Y%o e exists and is continuous in E |c % on W l ;(iv)f q E | 4E % T E |c % k E | 4E % where k G W $U ,q G W $U ,k E | :f ,q E | :f ,and where k and q are continuous functions on A and bounded for finite |5W ,and 4G l 6%$4E % 5U n ,where 4is continuous on W and such that q E | 4E % :f ,;E |c % 5P ,where P 'i E |c % 5W l G T E |c % 'f j .Let >G W $U be a function continuous on W .If there is any solution 7of the matrix equation .'.7 ,where .'d YT E |c % *Y%o e ,such that ;E |c % 5W l equation (8)holds,then for any solution 5V of (10),which starts from %f 5l at |'|f 5A ,the estimate T d |c E |c |f c %f o S T E |f c %f i T ]||f >E r _r c S '3C 8E | \ '&E |f

k E |n q E |3 4D (11)holds as long as remains in l ,where is the subscript of the first moment |&of discontinuity of or s in W _d |f c n 4d .

Theorem 6Let l U c and in addition to the other conditions of Theorem 3let there exist :f c 5U ,such that E ;|W 5W for which E W _d |W c |W n d '>we have 3C 8E | \ '&E |W

k E |n q E |3 4D i T ]||W >E r _r c Z c ;|5W _d |W c |W n d where is the subscript of the first moment |&of discontinuity of or s in d |W c |W n d .Let ?'|f n ,where is a nonnegative integer .Then for any solution 5V of (10),which starts from %f 5l at |'|f 5A ,the estimate T d |c E |c |f c %f o Z ?S T E |f c %f i T ]| ?>E r _r c ;|5W _d ?c ?n d c S '3C 8E | \ '&E ? k E |n 3 4D holds.Moreover ,if W 'd @c n 4d c @5U c and q E | 6:f on W ,then * 4|

CONCLUSIONS

This paper gives a systematic approach to the elimination of some of the crucial problems in the construction of the LFC for time-continuous descriptor systems.The results obtained can be used as a basis for further development of stability and boundedness problems of descriptor systems.

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