Zhihong HUANG,Y.V.VENKATESH,Cheng XIANG,Tong Heng LEE

Department of Electrical&Computer Engineering,National University of Singapore,Singapore Received 18 November 2013;revised 2 December 2013;accepted 2 December 2013

Frequency-domain L2-stability conditions for time-varying linear and nonlinear MIMO systems

Zhihong HUANG,Y.V.VENKATESH,Cheng XIANG?,Tong Heng LEE

Department of Electrical&Computer Engineering,National University of Singapore,Singapore Received 18 November 2013;revised 2 December 2013;accepted 2 December 2013

The paper deals with the L2-stability analysis of multi-input-multi-output(MIMO)systems,governed by integral equations,with a matrix of periodic/aperiodic time-varying gains and a vector of monotone,non-monotone and quasi-monotone nonlinearities.For nonlinear MIMO systems that are described by differential equations,most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates.In contrast,a non-Lyapunov framework is employed here to derive new and more general L2-stability conditions in the frequency domain.These conditions have the following features:i)They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block.ii)For certain cases of the periodic time-varying gain,they contain,depending on the multiplier function chosen,no restrictions on the normalized rate of variation of the time-varying gain,but,for other periodic/aperiodic time-varying gains,they do.Overall,even when specialized to periodic-coefficient linear and nonlinear MIMO systems,the stability conditions are distinct from and less restrictive than recent results in the literature.No comparable results exist in the literature for aperiodic time-varying gains.Furthermore,some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented.Examples are given to illustrate a few of the stability theorems.

Circle criterion;K-P-Y lemma;L2-stability;Lur’e problem;Multiplier function;Nyquist’s criterion;Switched systems;Time-varying system

1 Introduction

It is known that switching, which is a subset of general time-varying phenomena(natural or deliberate),can introduce instability when switching between stable systems,and,on the contrary,stabilize by switching between unstable systems.The two types of problems of considerable interest in the analysis and synthesis ofswitched systems are:i)For arbitrary switching,what are the conditions on the component systems that guarantee the stability of the associated switched system?ii)If a switched system is not stable under arbitrary switching,what are the constraints(i.e.,dwell-time,switching frequency)on the switching signal such that the switched system is stable?By analyzing stability(and instability)of time-varying systems in a broader framework,there exists the facility to specialize the stability/instability conditions so obtained to switched systems.The contents of the present paper are to be considered from this point of view.

In order to appreciate the fact that the qualitative properties of linear and nonlinear time-periodic single-inputsingle-output(SISO)systems cannot,in general,be subsumed under those for the corresponding multi-inputmulti-output(MIMO)systems,it seems to be relevant and instructive to get an overview of stability results for both SISO(however briefly,for lack of space)and MIMO systems.The literature on the stability analysis of switched systems is vast,as evident from the recent survey papers and books[1–4].

a)SISO system stability.The stability results,applicable mostly to systems described by differential equations,are based on Lyapunov-type stability definitions,for which the tools used are:common quadratic Lyapunov functions(CQLF)[5–9];piecewise-quadratic Lyapunov functions[10];multiple Lyapunov functions[11,12];Lyapunov-like functions[13];and family of quadratic Lyapunov functions for switched linear system stabilization(based on minimal dwell-time computation)and Lyapunov-Metzler inequalities[14].

The existence of a CQLF is only a sufficient condition for the exponential stability of a switched linear system under arbitrary switching[15].Moreover,there seems to be no simple condition to determine the existence of a CQLF for a family of linear time-invariant(LTI)systems.The necessary and sufficient conditions,derived for a general family of stable LTI systems to have a CQLF,have been found to be complicated and difficult to check,even for the case of a pair of third order systems.Converse theorems in the Lyapunov framework have also been established:the stability of a switched system also implies the existence of a common (not necessarily quadratic) Lyapunov function for all subsystems.However,these non-quadratic Lyapunov functions are not easy to determine in general[2].For non-quadratic Lyapunov functions,see[16–18].On the other hand,the stability conditions obtained from Lie-algebraic methods[19,20]also imply the existence of CQLF;and the convex optimization-based linear matrix inequality(LMI)techniques[10,21,22]test the existence of a CQLF for a number of stable time invariant systems.

In order to arrive at the best possible stability conditions for specific switched systems(of,mostly,second and third-order),another commonly used approach is variational principles[23–27].Margaliot and Langholz[27]employ the(so-called)generalized first integrals(in a Hamiltonian formulation)to study the absolute stability of second-order systems.It is not clear yet how to use such integrals to determine the worst-case switching law for higher dimensional SISO systems.

b)MIMO system stability.The literature on the stability analysis of time-varying(including time-periodic)MIMO systems with(first-and-third-quadrant)nonlinearities is somewhat sparse.See[32–34]for some early work on(frequency-domain)circle criteria,which are basically a consequence of employing(generalized)quadratic-form Lyapunov functions for finitedimensional systems or Zames’s passive operator theory for infinite-dimensional systems.In[35],it is shown that the multivariable circle criterion is valid for systems described by differential inclusions which its authors verify by simulation.For time-invariant MIMO systems with the class of diagonal nonlinearities,Haddad and Bernstein[36]proposed a multivariable parabola criterion which is a combination of the circle and Popov criteria.For time-invariant decoupled nonlinear MIMO systems associated with a fuzzy logic controller,Ray and Majumder[37]showed the applicability of the multivariable circle crierion,while Cuesta et al.[38]employed the multivariable circle criterion and the describing function method to predict the existence of multiple equilibria and of limit cycles in time-invariant MIMO Takagi-Sugeno(T-S)fuzzy control systems with Lur’etype but asymmetric(i.e.,not necessarily odd)nonlinearities.In the examples,Cuesta et al.[38]also made a comparison of the approach with Lyapunov’s method.See[39]for related results as applied to MIMO systems with symmetric nonlinearities.

It is of interest to note here that certain SISO-related stability analysis techniques,like the variational and Liealgebraic methods,do not find their counterparts in the literature on MIMO systems in view of the complexity of the multi-dimensional space in many variables.Not surprisingly,researchers turn to frequency-domain techniques.Amato et al.[40]considered,in the course of trying to extend the application of the multipliers of[41]to time-invariant MIMO systems,vector nonlinearities in which each component is a function of a single variable.In effect,they derive stability conditions for systems containing repeated scalar nonlinearities.On the other hand,Safonov and Kulkarni[42],having a similar goal(of multiplier form stability criteria)in mind,restricted either the nonlinearity to be the gradient of a potential function or the multiplier to be a real-valued function of frequency.Recently,reference[43]contains some conditions for the absolute stability of a linear MIMO system with a periodically switching matrix of a special form;and in[44],for nonlinear MIMO systems,it is assumed that i)the nonlinearities are gradients of convex functions,and satisfy a MIMO analogue of the slope restriction condition;and ii)the Hermitian of the gradient matrix of the nonlinearity satisfies an additional condition.

In general,stability conditions for nonlinear and time varying MIMO systems cannot be obtained merely by replacing the scalar transfer function of the linear block(of the SISO system)by a matrix transfer function,and the scalar nonlinear time-varying feedback gain(of the SISO system)by a time-varying-matrix-cum-nonlinearvector(or,for simplicity in terminology,time-varying nonlinear matrix)gain.Conversely,and as significantly,the stability conditions of linear and nonlinear time periodic MIMO systems,when specialized to those of the corresponding SISO systems,turn out,in general,to be more conservative than what we can establish directly for SISO systems.Therefore,stability conditions for MIMO systems cannot in effect be treated as overarching those for SISO systems.

c)Main contributions.New frequency-domainL2-stability conditions are derived for MIMO systems consisting of a LTI part in feedback with a periodic or an aperiodic time-varying linear/nonlinear matrix gain(Switching of the feedback gain is treated here as special case of the time-varying matrix gain).The stability conditions entail,in part,i)a constraint on the period of the matrix gain;and ii)the positive definiteness of the product of a multiplier matrix function and matrix transfer function of the LTI part.The multiplier,which,in effect,incorporates the minimax properties of the time-varying linear/nonlinear block,is a non-positive real matrix function,in contrast with positive real functions found in the literature on results linking Lyapunov-based stability conditions with the frequency domain matrix inequalties based on the Kalman-Popov-Yakubovic lemma[45].To be able to deal with nonlinearities not having the property of path-independence of line integrals(PILI)involving the nonlinearity and without slope restrictions,as also with those that do,we introduce for the first time certain characteristic parameters(CPs)for the nonlinear block.The CPs facilitate the derivation of stronger stability conditions.Slope restrictions on the nonlinearity can be introduced implicitly in computing the CPs,and are not reflected in the frequency-domain condition.A different CP for the time-varying gain leads to stability conditions without the need for a symmetry constraint on the time-varying gain.Furthermore,for periodic gains,under certain conditions,there is no need for a restriction on the rate of variation of the time-varying gain.However,in general,additional restrictions on the normalized rate of variation of the time-varying-gain matrix are needed forL2-stability.As applied to the dwell-time problem,these latter stability conditions are novel in the sense that they do not involve bounds on dwelltime or on average dwell-time(mode-independent or mode-dependent).The new L2-stability conditions are illustrated with an example for which the existing stability conditions of the literature either are not applicable or yield weaker results.Theoretical comparisons with the relevant results of the literature are also made.

The rest of the paper is organized as follows.In Section2,weformulate the problem of L2-stability of MIMO systems with a time-varying linear/nonlinear matrix gain and described by integral equations.In Section 3,we consider both linear and nonlinear MIMO systems with a periodic matrix gain,and state and prove the main results of the paper.In Section 4,we present,also for the same systems but with piecewise-continuous, aperiodic,including switching,gains,dwell-time-independent L2-stability results.After a comparison of our results with relevant literature in Section 5,we illustrate a typical application of the main theorems with examples in Section 6.Finally,we conclude the paper in Section 7,followed by the appendix which contain proofs of principal lemmas invoked in the proofs of the main theorems.

2 Mathematical formulation

The MIMO feedback system under consideration is described by

where the vectors of reference inputerrornonlinear gainand outputhave each dimension r,and?denotes convolution.With′denoting vector/matrix transpose,|·|,the determinant of a square matrix,and I,a unit matrix(of dimension evident from the context of its usage),the other functions in(1)are defined as follows.The time-varying matrix K(t)of dimension r×r is,in general,not symmetric,and belongs to one of the following classes:i)Cp,periodic with a common fundamental period P for all the elements of K(t),i.e.,K(t? mP)=K(t)for m=...,?2,?1,0,1,2,...,and for all t;and ii)Ca,aperiodic.For simplicity and convenience in the proof of the results,we assume that the elements kmn(t)of K(t)assume values such that K(t)is positive definite–we write K(t)?0 for t∈[0,∞).Note that such an assumption is a generalization of the assumption typically made in the stability analysis of SISO time-varying feedback systems with G(jω)as the transfer function of the forward block and for which the scalar feedback time-varying gain k(t) is assumed to take values in[0,∞).When,however,there is a need to consider,in the case of linear SISO systems,the gain k(t)∈ [?,∞),for constant＞0,the SISO system can be transformed to an equivalent SISO system with the modified transfer function G(s)/(1?G(s))of the forward block,and with the modified time-varying gain(k(t)+).The same transformation holds for the nonlinear SISO system also.See below for an analogous transformation for the MIMO system under consideration when K(t)is not positive definite.

The impulse response of the linear forward block contains two components:1)the continuous-time r×r real matrix function S(c)(t)for t∈ [0,∞);and 2)the discretetime r×r real sequence of matrix functions,one typical matrix of the sequence beingwhere the argument(δ(t? {τm,η}))indicates that a matrix of the sequence has elements which are functions of delta functions,indices m and η referring,respectively,to the matrix sequence number and sub-sequence number of the delta function in the element of the matrix itself.More specifically,the(ι,?)th element ofis given byis the coefficient of delta functions,and ρ(ι,?)is the number of delta functions in the(ι,?)th element.The convolution operation involving the discrete-time matrix sequencein(1)be used to account for initial conditions–gives as the?th element

In order to state the main assumptions needed to establish stability conditions for(1),we need to consider its special case of the linear constant matrix gain MIMO system:

where the matrix K(0)has elements which can assume constant values in such a way that K(0)?0.Furthermore,as a step towards analyzing the stability of(1),we also analyze the stability of the linear periodic/aperiodic matrix-gain MIMO system:

The system described by(1)is said to be L2-stable ifL2[0,∞),and an inequality of the typeholds where C is a constant.This definition is valid for(2),(3)and their gain-transformed counterparts(see a)and b)in the appendix).

Main assumptionsA1)The solutions to(2)are in L1∩ L2,which implies that the zeros of|(I+K(0)Γ(s))|for K(0)?0 lie strictly in the left-half(Res＜?δ≤0)of the complex plane.

A2)K(t)? 0,t∈ [0,∞).

A3)The nonlinearities(for all the classes defined below)satisfy the following inequality for an arbitrary bounded constant matrix Y,

where‖Y‖is the matrix norm of Y1The matrix norm of Y could be,for instance,.The classes of nonlinearities considered are subsets of(N,K),depending on the desired form of the stability conditions,and are defined as follows:

1)Class(N0,K):If there exists,a(scalar)potential function,such that,for?V(t,)2lternatively,is conservative in the domain of definition of the vector function.This property guarantees the path-independence of line-integrals with respect toinvolvingNote that the gradient operator? is with respect to the components ofThe limits of the line-integral used to define the class M3of nonlinearities later below involve only,

2)Class(M1,K):

3)Class(M2,K):

Piecewise(first-and-third-quadrant)nonlinearities having negative slopes also(but for which?bcan be computed)are included in this class3If?b=∞,then(M2,K)becomes(N,K)for which no version of the generalized Popov-type matrix multiplier function can be employed even for a periodic K(t),without or with a restriction on the rate of variation of the elements of K(t).It has been found,however,that a generalized Popov-type matrix multiplier can be employed for∈(N0,K),with a constraint on the rate of variation of the elements of K(t).This contrasts with the more general multiplier function meant for the class(M2,K)proposed in this paper.

2its characteristic parameter?bcan be obtained as a solution to a minimization problem:

Note that,for an odd nonlinearity,V(·,is even with respect toand(M3,K)?(M1,K)?(M2,K)?(N,K).A further sub-class of monotone functions corresponding to(M3,K)can be defined by inserting the right hand side expression of(6)into the right hand side of(8),before the equality sign.

Formulation of the problemFind conditions for the L2-stability of the system described by(1)with K(·) ∈ Cpor Ca,and(·)∈(N0,K),(M1,K),(M2,K)or(M3,K).

For simplicity in the proof of stability theorems,we assume that the generalized feedback gain is allowed to assume values in[0,∞).The stability conditions which are derived in terms of Γ(jω)and K(t(·)can then be specialized to ‘finite-gain’systems by the following algebraic operations:Replace both the transfer function of the forward block and the feedback by their corresponding gain-transformed versions.See a)and b)in the appendix.

3 Main results:K(t)∈Cp

We establish frequency-domain stability conditions for(3)and(1),in that order of increasing generality,by employing a combination of the extended Parseval theorem(in Fourier transform theory)and certain integral inequalities.Motivated by the frequency-domain interpretation of the Nyquist criterion–in terms of the real part of a multiplier function and of its product with the forward block’s transfer function – for linear SISO time-invariant feedback systems[47,48],the stability conditions are expressed in terms of the positive definiteness of the real part of the product of a multiplier matrix frequency function4See(10)below for one form of the multiplier matrix function.Y(jω)and Γ(jω),and of certain constraints on K(t)as follows.For system(3),with K(·)∈ Cp,the constraints are on the period P(of K(t))and on a norm of Y,but when K(·)∈ Ca,the constraints are on both the normalized rate of variation of the elements of K(t)and norm of Y.On the other hand,for system(1)with class(N0,K),(M1,K),(M2,K)or(M3,K)of nonlinearities,the constraints are similar to those for system(3)with K(·)∈ Cpand Ca,but with a set of different restrictions on the norm of Y.

PreliminariesFor any real-valued vector functionon[0,∞)and any T ≥ 0,the truncated functionis defined as follows:=for 0≤t≤T;and=0 for t＜ 0 and t＞ T.Furthermore,let L2ebe the space of those real-valued functionson[0,∞)whose truncations(·)belong to L2[0,∞)for all T ≥ 0.Essentially,by assuming infinite escape time for the solution to the system with∈L2,the solution belongs to L2e.Then,it is shown that,under certain conditions onand on Γ(jω),the solution actually belongs to L2[0,∞).

The matrix operator Y,needed to establish stability conditions for systems(2)and(3),has a(matrix)impulse response representation with elements in L1∩L2.For the linear,constant coefficient system(2),the form of the matrix multiplier operator,Y(l),with its Fourier transform Y(l)(jω),is general,whereas for systems(3)and(1)with K(t)∈Cp,a typical multiplier operator Y(p)has the following specific form:

where αm,1,αm,2,m=1,2,...are constants;and{Ym,1},{Ym,2},m=1,2,···,are sequences of constant matrices,which,along with their sequences of coefficients{αm,1}and{αm,2},are in ?1∩ ?2.The Fourier transform of the operator in(9)is given by

Note thatY(jω)is a non-positive real matrix function.It turns out that one of the conditions in the stability theorems for system(3)is a constraint on the generalized eigenvalue of a(matrix-)weightedK(t);and that for system(1)is a constraint on a further generalized pseudo-eigenvalue involving bothK(t)andthe latter with a(matrix-)weighted argument.

Inequalities are the workhorses of stability analysis.The new stability conditions derived in the paper are based on certain matrix-algebraic and integral inequalities.For arbitrary,real scalarsaandb,the well-known algebraic inequality(AI)

We introduce its two generalizations tailored to the methodology adopted here for theL2-stability analysis of MIMO systems:the first involves the ratios of biquadratic forms,and the second,the ratios of pseudobi-quadratic forms with vector nonlinearities. In the case of the former,we consider real vectorsand boundedr×rmatricesA(t)? 0,B(t)? 0 andC(t),t≥ 0,in terms of which the bi-quadratic formsatisfies the inequalitywhere the(constant)parameter νs＞ 0 is defined by

The nonlinear generalization of the AI involves the pseudo bi-quadratic form(with the real vectorswhich satisfies the inequality where the(constant)parameters γi＞ 0,γs＞ 0 are defined by

νs,γiand γsare obtained as solutions to an optimization problem,as made explicit in(11)and(12),in terms ofgeneralized(pseudo-)eigenvalues.Fortheproof of the stability theorems below,these parameters will assume different subscripts and superscripts,depending on the matrices that replaceA(t),B(t),C(t)andin(11);andin(12).

Stability conditionsSee Table 1 for a summary view of the new results and their domains of applicability.

Before we can state and prove the stability theorems for systems(3)and(1),we need to specify the multiplier matrix operator to be used in the theorems,and define the parameters that appear in them.For both(3)and(1),we choose the multiplier matrix-operator Y(p)defined by(9).With respect to(3),1,2;andare vectors(of compatible dimension withK(t))5These are identified later,in the proof of Theorem 1,with(t)and(t?mP),respectively..The ch aracteristic parameter

Table 1 Summary of the basic properties of the new MIMO stability results.

When the multiplier(9)is chosen with αm,1= ?αm,2=which is skew-symmetric,the CPis re-labelled for clarity:

For system(1),the CPs of both(M2,K)are defined by

Finally,for class(N0,K),recall that,for(N0,K),there exists a(scalar)potential functionsuch thatwhere t∈ [0,∞)is a parameter.Since K(t)∈Cpor Cais arbitrary,we can invoke a special case of the property of(N0,K)to definewhere Q is a constant r×r matrix,and define the corresponding CP’s:

Remark 1The above parameters are needed in Theorems 1–4 as follows:in Theorems 1 and 3(Section 4);in Corollary T1-1 to Theorem 1;in Theorems 2A,Corollary T2-1 of Theorem 2A,and in Theorem 4B(Section 5);the pairin Theorem 2B;and the pair(δqi,δqs)in Theorem 4A(Section 5).

Recalling the assumptions that K(0)?0,K(t)is not(necessarily)symmetric,and K(t)? 0 for t∈ [0,∞),we now state the main stability theorems of the paper.In order to avoid confusion with parenthesized numbers meant for equations and square-bracketed numbers meant for references,we employ square-bracketed

3.1 Proofs of stability theorems

The proofs of the above stability theorems depend on i)an application of an extended Parseval theorem(for vector functions),and ii)establishing positivity conditions for two blocks in cascade:for system(3),the first block is linear having Y(jω)as its transfer function,and the second is K(t);and for system(1),the first block is the same,but the second isWe need the following auxiliary results to prove the theorems.

Extended Parseval’s theoremSuppose the elements ofare real-valued functionswhereare the Fourier transforms of

Lemma 1With the operator Y(p)defined by(9),the following inequality holds:

Corollary L-1With i)the operator Y(p)defined by(9);and ii)with αm,1= ?αm,2= αm;Ym,1= ?Ym,2=Ym,m=1,2,...,inequality(18)of Lemma 1 is satisfied under the condition of Corollary T1-1 that replaces condition[H-1]of Theorem 1.

Lemma 2AFor(·) ∈ (M1,K),with the operator Y(p)defined by(9),the following inequality holds:

Corollary L2-1Lemma 2A is valid for(·) ∈(M2,K),if condition[H-1A]of Theorem 2A is replaced by the corresponding condition of Corollary T2-1.

Lemma 2BLemma 2A is valid for(·) ∈ (M3,K),if,instead of condition[H-1A]of Theorem 2A,condition[H-1B]of Theorem 2B is satisfied.

Below we outline only the proofs of Theorems 1 and 2A which are based on Lemmas 1 and 2A,respectively.The proofs of the other theorems can be obtained by slight modifications of the proofs of Theorems 1 and 2A.For the proofs of Lemmas 1,2A and 2B(and their corollaries),see the appendix.Let G denote the matrix operator of the forward linear block,i.e.,

Proof of Theorem 1With the operator Y(p)defined by(9),consider the integral,for any T＞0,

which,on invoking(3),becomes

The first integral on the right hand side of(21)can be shown,by condition[H-2]of Theorem 1 in association with the extended Parseval theorem,to satisfy the inequality

for some δ＞0.The second integral on the right hand side of(21)can,by virtue of(9),be rewritten as

By invoking Lemma 1(see Proof of Lemma 2A in the appendix),the right hand side of(23)be shown to be non-negative,if condition[H-1]of Theorem 1 is satisfied.Therefore,the second integral on the right hand side of(21)is positive.Finally,by applying the extended Parseval theorem to(20),and combining the result with(22),we obtain

Proof of Theorem 2AWith the operator Y(p)defined by(9),consider,for any T＞0,the integral(20),which,on invoking(1),becomes

As in the proof of Theorem 1,the first integral on the right hand side of(25)can be shown,by condition[H-2A]of Theorem 2A,to satisfy the inequality

for some δ＞0.After replacing Y(p)in the second integral of the right hand side of(25)by the right hand side of(9),we proceed along the lines of the proof of Theorem 1 above.Finally,we invoke Lemma 2A along with conditions[H-1A]and[H-2A]of the theorem to complete the proof.

Remark 2The proof of Theorem 2B is similar to the proof of Theorem 2A,and is established by invoking Lemma 2B.

Remark 3In system(1),we have assumed,for clarity,transparency and simplicity in the derivations,a separable structure for the time-varying nonlinearity,viz.,For nonseparable nonlinearities(written typically asTheorems 2A and 2B need to be recast after suitable changes in the definitions of classesin the statement of lemmas(related to Theorems 2A and 2B)and in the proofs of those lemmas.

Remark 4In Theorems 2A(and its corollary)and 2B,the stability conditions for system(1)withrespectively,entail no restriction on the rate of variation of K(t).However,such an exemption does not seem to be possible for the caseOn the other hand,it is found that,for this case,i.e.,forwe can arrive at stability conditions by imposing a constraint on the rate of variation of K(t).See Section 4 below on the dwelltime problem.However,no similar results are known for the general case ofi.e.,without any simplifying assumptions on the vector nonlinearity.

Remark 5General multivariable nonlinearities of the type considered here are too general to be amenable for computation of their CP’s(i.e.,characteristic parameters)in the illustration of theorems in Section 6 below.Alternatively,the time-domain constraints(of the theorems)on the multiplier matrix are interpreted in such a way as to lead to stability bounds on the CP’s themselveswhich are defined respectively by(15),(7),(16)and(17)–once the frequency-domain(i.e.,real-part)conditions involving the Fourier transform of the multiplier and the system function Γ(jω)are satisfied.

4 Dwell-time and L2-stability:aperiodic matrix gains

We consider the following problem:For K(t)∈Ca,what are the constraints on the rate of variation of K(t)and on the switching discontinuities in a finite/infinite interval for the stability of system(1)and of(3)?To this end,we need to settle a few preliminaries to facilitate the derivation of new stability conditions that involve constraints on the normalized positive and negative intensities of

Let h(t)be a nonnegative,integrable and bounded function onsume that the integraanda bounded positive function.Note thatwhich is non-positive.Furthermore,we assume that i)each element of K(t)is a piecewise-continuous function of bounded variation with first-order(i.e.,jump-)discontinuities in ?1;and ii)K(t)is made up of the continuous part,Kc(t),and the discontinuous part Kd(t).

In what follows,let the subscript iq denote the(i,q)th element of the matrix K(t)and of its derivative.With kiq(t)denoting the(i,q)th element of K(t),the(i,q)th element of Kd(t)represents the discontinuities of kiq(t)at instantstm+corresponding to positive jumpsinstants tm?corresponding to negative jumps,The derivative of K(t)is then given by

Moreover,similarly,by the statement thatω(t)eζtK(t)for t∈ [0,∞)is non-decreasing,we mean that the quadratic form

In view of the assumptions on ω(t)and K(t),it can be shown that(28)and(29)together reduce to the following inequality:

In order to arrive at explicit conditions ontosatisfy(30),we need to find functions ?(t)and ?(t),t∈ [0,∞)

abas solutions to the following non-standard,generalized eigenvalue problem(which is of independent interest by itself):

In effect,we are looking for a solution to the following problem at each ‘frozen’instant of time,t′∈ [0,∞):

For an interpretation of(32)for Kd(t),i.e.,discontinuous part of K(t),at instants t=tm+for positive discontinuities,and t=tm?for negative discontinuities,see(27)above and the paragraph associated with it.Briefly,represent merely the matrices having their elements as positive and negative impulse intensities(or strengths)at instants t=tm+and t=tm?,respectively.Those matrices do not themselves contain delta functions.If Kc(t)≡0 and K(t)=Kd(t),t∈ [0,∞),then for a negative discontinuity at t=tm?and a positive discontinuity at t=tm+,equation(32)reads as follows:

In what follows,the terms in curly brackets(which are coefficients of the delta-functions)above will be denoted respectively by ψa,d;m?and ψb,d;m+;and when a reference is made to generalized eigenvalues in(32),it is understood that the sequence pair{ψa,d;m?,ψb,d;m+}is implied for the derivative of Kd(t)having the pair of negative and positive discontinuities at t=tm?and t=tm+,m=1,2,...,respectively.Changes need to be made to the sequence’s super-and sub-scripts as and when needed.

In this section,for K(t)∈Ca,we state three theorems and the lemmas needed to prove them.Theorem 3 concerns system(3);and Theorems 4A and 4B,system(1)with,respectively,∈ (M1,K)and∈(N0,K).For Theorem 3,the multiplier matrix is given by(9),where P is replaced by p6If P=p,then Theorem 1 comes into play,and there is no need for Theorem 3.,which is treated as an additional parameter to be chosen.For Theorem 4A,we choose the multiplier operator Y(q)defined by

where Q is an r×r constant matrix with the property that Q′K(t)? 0,t∈ [0,∞).Its Fourier transform is given by Y(d)(jω)=I+jωQ.Theorem 3 needs the CP’s defined by(13);Theorem 4A,the CPdefined by(15);and Theorem 4B,the CPs δqiand δqsdefined by(17).We now proceed to the statement of the theorems.

Theorem 3System(3)with K(t)∈Cais L2-stable if there exist a multiplier matrix-operator Y(p)defined by(9)with P replaced by parameter p;a bounded positive function ω(·)as defined above;and nonnegative constants, ξ,ζ,such that[H-1]: ω(t)e?ξtK(t)is non-increasing and ω(t)eζtK(t)is non-decreasing for all

The proof of Theorem 3,which is omitted because it is similar to the proof of Theorem 1 above in partial combination with the proof of Theorem 1 in[49,pages 572–573],depends on the following lemma.

Lemma 3-1With the operator Y(p)defined by(9)with P replaced by parameter p;and nonnegative constants ξ and ζ,the integral

Condition[H-1]of Theorem 3 is equivalent to an appropriate bound on the rate of variation of K(t),as made explicit by the following lemma.

Lemma 3-2With ?a(t)and ?b(t)defined in(31);denoting the positive lobes of ?b(t),anddenoting the negative lobes of ?a(t)for t ∈ [0,∞),if ω(t)e?ξtK(t)is non-increasing and ω(t)eζtK(t)is nondecreasing for all t∈ [0,∞),then,for some positive constants N1and N2,for all finite T＞0 and all t0≥0,

but for ξ = ζ,?a(t)and ?b(t)are unrestricted.

If K(t)is piecewise constant,then Kc(t)is identically a zero matrix.Let M denote the number of discontinuities in a finite interval T＞0.Then,by invoking(33),(37)

becomes,

The proof of Lemma 3-2 is similar to that of Lemma 2 of[49,pages 572,577–579],and is hence omitted.When K(t)∈Cpwith period P≠p,then(37)and(38)reduce respectively to7If P=p,then Theorem 1 comes into play,and there is no restriction on the rate of variation of K(t).

Theorem 4ASystem(1)with K(t)∈ Caand∈(M1,K)is L2-stable,if[H-1A]:for some positive ξ and ζ,and a bounded positive function ω(·)as defined in Section 4 above,ω(t)e?ξtK(t)is non-increasing and ω(t)eζtK(t)is non-decreasing;[H-2A]:there exist a multiplier matrix-operator Y(q)defined by(34)with αm,1＜ 0 and αm,2＜ 0,and parameterdefined by(15),such that

The proof of Theorem 4A requires the following lemma.

Lemma 4AWith i)the operator Y(q)defined by(34)with αm,1＜ 0 and αm,2＜ 0,ii)∈ (M1,K)and iii)a bounded positive function ω(·)as defined in Section 4 above,the following inequality holds:

Theorem 4BSystem(1)withK(t)∈ Caand∈(N0,K)isL2-stable,if[H-1B]:for some positive ξ and a bounded positive function ω(·)as defined in Section 4 above, ω(t)e?ξtK(t)is non-increasing;[H-2B]:there exists a multiplier matrix-operator Y(d)defined by(35),such that,with the constant matrixQchosen to satisfyQ′K(t) ? 0,t∈ [0,∞),and with δqsdefined by(17),ξ ＜ 1/δqs;and,for some ε＞ 0,[H-3B]:

The proof of Theorem 4B requires the following lemma.

Lemma 4BWith i)the operator Y(d)defined by(35)and the constant matrixQchosen such thatQ′K(t)?0,t∈ [0,∞),ii)(·)∈ (N0,K)and iii)a bounded positive function ω(·)as defined in Section 4 above,the following inequality holds:

Theorems 4A and 4B can be proved along the lines of Theorem 1(for SISO systems)in[49]after invoking Lemmas 4A and 4B,respectively.The proofs of the latter are outlined in the appendix.

Remark 6Theorems 4A and 4B are applicable whenK(t)∈Cp.In the case of the former,substitutemP for both τmandthe multiplier matrix.In this case,the stability constraints on the rate of variation of the elements ofK(t),can be converted to equivalent constraints over an interval P,as indicated in Lemma 3-2.Furthermore,Theorem 4A can be modified to apply to the cases of(·)∈(M2,K)and(M3,K).Lemma 3-1 is applicable,as a special case,to Theorem 4B also(In this case,ζ is non-existent).

Remark 7Theorems 3,4A and 4B forK(t)∈Caand Cp,as also for switching matrices, are novel with respect to the constraints that are imposed on a normalized rate of variation ofK(t).These constraints take the form of upper and lower bounds on the integral/summation of positive and negative lobes of,respectively,the largest and lowest,generalized eigenvalues of the derivative ofK(t)relative toK(t)itself.Since there is no constraint on dwell time directly or indirectly,these stability results are believed to be the first of their kind.However,it is not known how to arrive at stability results for both nonlinear periodic/aperiodic systems with more general multiplier matrix operators.If(·)is odd-monotone,then the constraint of negativity on αm,1,αm.2form=1,2,...,in Theorems 2A,2B and 4A can be removed.Due to lack of space,the(standard)details are not given here.

Remark 8From the explicit relations among the variables of system(1)and its ‘finite’gain counterpart(see a)and b)in the appendix),viz.,among the pairsthe following sequence of stability implications for the finite gain system can be established by invoking the assumptions made on system(1):Finally,note thatwhich can be obtained from inverting the relationship betweenK(t)andK1(t)presented in a)in the appendix for system(3).As far as system(1)is concerned,we adopt the following interpretation of the ‘gain’of the feedback block.SinceK(t)is allowed to have large values,we can assume,without loss of generality,thathas unit gain,i.e.,

Switching matricesSuppose we have two switching matricesK(1)(t),K(2)(t)both of which satisfy the conditions imposed onK(t)in Section 2.If an external control law facilitates the transition,at the instant τm,fromK(1)(t)toK(2)(t),then the change is denoted by(ΔK)1→2(τm).Similarly,for the transition,at the instant υm,fromK(2)(t)back toK(1)(t),the change is given by(ΔK)(2→1)(υm).Suppose the dynamics of the system are such that,starting withK(1)(t),there is only one switching at τ1toK(2)(t).If the composite system matrix is represented byK(t),its derivative is given by

When there is a further switch fromK(2)back toK(1)at instant υ1＞ τ1,then(42)becomes

With the transitionsK(1)(t)→K(2)(t)happening at τ?∈ [t0,∞),?=1,2,...,and vice versa at instants υ?∈ [t0,∞),?=1,2,...,where τ1＜ υ1＜ τ2＜ υ2＜ ...,(43)can be generalized by adding the required terms to its right hand side by noting that the operational domains D(1)and D(2)ofK(1)(t)andK(2)(t)are,respectively,{0 ≤t≤ τ1,υ1＜t≤ τ2,υ2＜t≤ τ3,...}and{τ1＜t≤ υ1,τ2＜t≤ υ2,τ3＜t≤ υ3,...},using which we can generalize(43)as follows:

Assuming,for simplicity,thatK(1)(t)andK(2)(t)are both continuous in the domains D(1)and D(2),respectively,(ΔK)(1→2)(τm),and(ΔK)(2→1)(υm)are to be interpreted along the lines of Section 4 above,in the paragraph dealing with(27).The only difference is that an instant of a discontinuity is applicable simultaneously to all the elements ofThe above expression forcan be extended to the case of multiple switching matrices.With that proviso,let

Then,the extremal eigenvalue problem of(32)can be reduced to the following form:

To be able to apply Lemma 3-2 to the case of switching matricesK(1)(t)andK(2)(t),the right hand side of(44)is to be substituted forin(32),and solved for ?aand ?b,as made explicit in(46).However,when the matricesK(1)andK(2)are constant,then,in(45)only the summations involving the delta functions remain.To understand the implications of this operation,note the interpretation of(32)in the paragraph containing that equation.Briefly,Theorems 3,4A and 4B hold for switching matrices directly.However,to express condition[H-1]of Theorem 3,[H-1A]of Theorem 4A and[H-1B]of Theorem 4B in terms of the(normalized)rate of change ofK(t),the optimization problem(46)is to be solved,after which Lemma 3-2 is to be invoked.Note that,in the case of Theorem 4B,the lower bound ?a(t′),t′∈ [0,∞)in(46)does not exist.The stability results are applicable to periodic/aperiodic switching with no constraints on i)the period ofK(t)∈Cpand ii)time instants of switching,as long as the norms of the sequences of matrix discontinuitiesandare in ?1.

5 Comparison with literature

1)To compare our Corollary T1-2 of Theorem 1,meant for the linear system(3)(havingK(t)∈Cp),with Theorem 5.2 in[43]using a different approach– ‘delay-integral-quadratic constraints’– it is found that the square-bracketed frequency function in the latter8As applied to the infinite gain problem,the expression κ?1I in Theorem 5.2 in[43]is to be removed.cannot be compared with the multiplier-form stability criterion of our Corollary 2B,i.e.,withYmΓ(jω).Therefore,Theorem 5.2 of[43]and our Corollary T1-2 are quite distinct.

2)There seem to be no results in the literature which are similar or comparable to our Theorem 1 which does not assume the symmetry ofK(t).The stability conditions based on Floquet’s theorem[50]for linear differential equations with periodic coefficients are basically of the existence-type,i.e.,they are not in the form of explicit conditions onK(t)and Γ(jω),and are not applicable,at least in their present form,to systems governed by integral equations(3).

3)There appear to be no results in the literature comparable to Theorem 2A for(1)with∈(M1,K)(or with its corollary for∈(M2,K)).More explicitly,note that the existing literature on the stability of nonlinear MIMO system(having a periodic gain or not)considers the class of monotone functions having the property of path-independence of line integrals(PILI)involving the nonlinearity and with slope restrictions.In contrast,we deal not only with such a class of nonlinearities,but also with the class of nonlinearities without the property of PILI. Concerning Theorem 2B for (1) with∈(M3,K),a comparison with a recent result in[44]is quite interesting and instructive.In[44],the assumption on the linear block is exponential stability,whereas we assume that the linear block’s continuous-time and discrete-time impulse responses belong,respectively,toL1∩L2,and ?1∩?2.Note this linear block is more general than that of[44].Furthermore,in[44],the nonlinearities satisfy a MIMO analogue of a slope restriction.In contrast,our Theorem 2B does not involve this assumption.

4)An interesting byproduct of the approach adopted here for system(1)is that Theorems 2A,2B,4A and 4B can be generalized to hold for nonlinearities with a slope restriction.To this end,in the definitions of the CPs(15)for Theorems 2A and 4A;(7)and(15)for the corollary of Theorem 2A;(16)for Theorem 2B;and(17)for Theorem 4B,we include,in the supremum/infimum operations,the desired constraint on the slope matrixthereby dispensing with an extra term in the multiplier function(the extra term being typical of the literature on SISO systems[41,51]).In such a case,the constraints on the multiplier matrix functions can be weakened.

5)In contrast with the results of, for instance,[52–54],no reference has been made in Theorems 3,4A and 4B to dwell time,either average or dynamic,mode dependent or independent.It is conjectured that there exist periodic/aperiodicK(t)’s which violate the dwell time-related constraints of the literature,but obey the conditions of Theorems 3,4A and 4B.

6 Example to illustrate theorems

The forward block of the MIMO system’s transfer function,the non-symmetric time-varying matrix gain and the nonlinearity are given by

wherek12(t)≠k21(t),t∈ [0,∞).For Theorems 1,2A and 2B(as also for the special case of periodic switching matrices to illustrate Theorem 3),the elements ofK(t)are(row-wise):where Δ(t)? ((1.01?

1cos(βt))/(3+ ∈));Δ2(t)? ((1.01+sin(βt))/(4+ ∈));and∈＞ 0,c1,c2,c3,and β are constants.Note that the period P=2π/β.It can be shown that i)K(t)? 0,t∈ [0,∞),if,for instance,|c1+c2|=2.6;and ii)K(t)has arbitrarily large gain(equivalent to ‘infinite’range)property.For Theorems 3,4A and 4B,to generateK(t)∈Cawe replace cos(βt)by f1(t),and sin(βt)by f2(t)),where f1(t)and f2(t),t≥0 are aperiodic/periodic subject to the constraints,|f1(t)|≤ 1,|f2(t)|≤ 1,t≥ 0,having,when they are periodic,only a finite number of first-order discontinuities in each period;when aperiodic,there exist only first-order discontinuities(whose magnitude sequence is assumed to be in?1).Atypical Routh-Hurwitz limit matrix for Γ1(s)is:=[(4,?12);(0,10)].The characteristic equation of the linear MIMO system with the constant matrix feedback gain is1978.01s5+7992.615s4+32398.33s3+74038.46s2+128871.36s+81802.4)=0,whose roots are given by(?33.77695±j23.43083,?0.03764±j2.97085,?1.45948±j1.81369,?1.01184).It can be verified that Γ(s) ? (I+KˉΓ1(s))is not positive real,thereby implying that the generalized version of the circle criterion of MIMO systems,of the type found in[35,38]meant for time invariant systems,cannot be invoked for the linear and nonlinear systems(3)and(1).We need to employ a multiplier matrix for establishing their stability.As explained in Remark 5(Section 3),results to illustrate Theorems 2A,2B,and 4A are expressed in terms of CP’s of the nonlinearities to guarantee stability.

Due to lack of space,we briefly illustrate Theorems 1,2A,2B,3,and 4B,using results obtained from a special form of the multiplier matrices:for Theorems 1–3 and withK(t)∈ Cp,Y(p)(jω)=I? 2jα sin(ωP)Y;and for Theorem 4A,Y(d)(jω)=(I+jωQ).In the former case,for Theorems 1 and 2B,the parameters ofK(t))are:P=0.69,c1=5,c2=?4,c3=10 and∈=0.01.Tailoring the notation used for(14),we choosem=1,Y?Y1,1=?Y1,2=[(?0.4,?0.2);(0.5,?1)],α ? α1,1= α1,2=0.4.In this case,which is a non-zero and skew-symmetric matrix,implying that the assumption of[43](and hence its stability theorem)is not applicable here.Using(14),we obtain=0.0271,from which we conclude that the condition of Corollary T1-1(which is special case of condition[H-1]of Theorem 1)is satisfied.We can also apply Theorem 2B.To this end,recall the CPs of(M3,K),i.e.,fined by(16).Using the notation,we have the following result:system(1)isL2-stable for an odd∈ (M3,K)andK(t)∈ Cp,if the CPs ofobey the inequalityWith the same multiplier,however,we cannot arrive at a non-trivial restriction on the CPs of an odd∈(M1,K)from condition[H-1A]of Theorem 2A,since ‖Y‖=1.118.For the same reason,we cannot employ Corollary T2-1 of Theorem 2A for an odd∈(M2,K).Moreover,this is left out,pro tem,as an open numerical problem.Note that the stability theorem of[44],which assumes path-independent integrals involvingcannot be applied to the case of∈(M1,K)or(M2,K),because in these classes of nonlinearities there is no such constraint.As far as class M3(which does assume path independent integrals involving the nonlinearity)is concerned,note that there is no restriction on the derivative of the nonlinearity.Therefore,the stability results of[44]are not applicable.Concerning Theorem 4B,choice ofQ=[(0.8,1);(0,0.8)]andK(t)∈Cpwith P=50,it is found thatQ′K(t)? 0,t≥ 0;and condition[H-3B]is satisfied.Therefore,the system with∈ (N0,K)isL2-stable,if,for any chosen value of ξ,the CP ofδqs＜ 1/ξ,and condition[H-1B]is satisfied.

Special case of Theorem 3 as applied to switching matricesA typical pair of switching matrices{K(1),K(2)}–generated from the structure ofK(t)introduced in the first paragraph of this section–and the corresponding coefficient matrices for a transition from the first matrix to the second and vice versa are

For switching from K(1)to K(2)at instant τ1,followed by switching from K(2)to K(1)at instant υ1,we haveτ1?)+where arguments τ1+and τ1?indicate positive and negative jumps respectively of elements at τ1;υ1+and υ1?,positive and negative jumps respectively of elements at instantt=contains only the positive elements ofcontains only the nonpositive elements ofWith K(τ?)=K(1)anda typical computation,based on(33),for estimating the lower and upper bounds foris of the following type:andwith similar corresponding expressions for ?a(τ?)and ?b(υ+).By generalized eigenvalue computation,it is found that the coefficients of δ(t? τ+),δ(t?τ?),δ(t?υ?)and δ(t? υ+)are,respectively,1.6118,?0.8959,?0.9392 and 0.8515.Suppose the switchings ofK(t)take place with period P=50.To invoke Theorem 3 for the chosen multiplier(with p=0.6),we compute the CP ofK(t)(with period P=50),defined by(13),which is found to be0.1145.From condition[H-2]of Theorem 3,we can determine non-negative ξ and ζ to satisfy the inequality(eξp+eζp)＜ 27.1121.For reference below,we label this inequality($).If we assume ξ = ζ,then,from Lemma 3-2,there is no need for a restriction on the switchings.On the other hand,suppose we set,for instance,ξ =5.0,then we find ζ ＜ 3.2495 to satisfy the inequality($).If the number of switchings from K(1)to K(2)and vice versa in one(semi-closed)period P ofK(t)are respectivelyM1andM2,then using the above computed generalized eigenvalues for switchings in(39),system(3)isL2-stable,if(1.6118M1+0.8515M2)＜250 and(0.8959M1+0.9392M2)＜162.4748.Since the switching is periodic(with period=50),eitherM1=M2+1 orM2=M1+1,implying thereby(the limits of)M1=89 andM2=88,or the other way round,for theL2-stability of the switched linear system. Note that no reference has been made to dwell time,average or otherwise.

7 Conclusions

For theL2-stability of periodic and aperiodic,linear and(certain classes of)nonlinear MIMO systems,we have derived new frequency-domain conditions in terms of the positive definiteness of the real part of matrices involving the transfer functions of the linear time-invariant block and a matrix multiplier function,along with constraints on the latter and,for certain cases,with constraints on the normalized rate of variation of the time varying matrix gain.These conditions are believed to be more general than those of the literature by their ap-plicability to both linear and nonlinear MIMO systems with periodic and aperiodic coefficients.When these systems have piecewise constant,periodic or aperiodic gains,new stability conditions are established in terms of average of the sum of their normalized first-order matrix-discontinuities,while not involving dwell-time constraints.One of the interesting,unsolved problems seems to be the generation of possible Lyapunov function candidates for the results of Sections 3–5,thereby generalizing the results of[55–57]on the KPY lemma for multivariable systems.

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[28,29]rely on Lyapunov techniques to deal with global stabilizability of time-invariant nonlinear systems by dynamic output feedback.However,the dynamics of nonlinear systems considered in stabilization are described by some special triangular forms which are distinct from the structure of the systems considered in the papers earlier referred to.Chitour and Sigalotti[30]consider the problem of global asymptotic stabilization of a system described by a linear(vector)differential equation,using a constant linear feedback,in association with scalar persistently excited(PE)signals.The PE signals satisfy an integral constraint that can reflect some approximately periodic or quasi-periodic phenomenon affecting the control action. They observe that Lyapunov based techniques do not seem well adapted to persistently excited systems,at least for what concerns the proof of their stability,and employ techniques that are strictly related to the analysis of the Lyapunov,Floquet,and Morse spectra developed by[31].Such an analysis is inapplicable to nonlinear SISO systems.

DOI10.1007/s11768-014-0182-2

?Corresponding author.E-mail:elexc@nus.edu.sg.

Appendix

When the feedback block of(1)has a finite gain,i.e.,whereˉ is the constant upper bound matrix,the stability results remain valid under a transformation of forward and feedback blocks.For clarity,we employ subscript 1 in the(dependent)variables of the equation governing the finite gain system,but retainKˉ as the upper bound matrix for the feedback block in both the linear and nonlinear cases.

a)Linear MIMO gain transformationLet the equations of the ‘finite’gain linear MIMO system with the upper bound matrixbe given byThe output of the ‘infinite’gain system(3)isand feedback gainKˉ,as a result of which the feedback output isis,by a quirk in the transformation adoptedNote that

b)Nonlinear MIMO Gain TransformationLet the equations of the finite gain nonlinear MIMO system with the upper bound matrixbe given byconcerning which we assume the existence ofa matrixsuch thatThe output of(1)isand the new feedback(notional)gain

Corollary T1-1Following the simplification procedure adopted in the proof of Lemma 1 above,the integral on the left-hand side of the inequality sign in(18),now defined for clarity ascan be expanded to the following form:

Recognizing that the last integral on the right-hand side of(a9)is E1(T)(with Ymreplaced by Ym,1),which was defined and simplified above,and recalling(4)under assumption A3)of the main assumptions,(a9)can be reduced to

On similar lines,(a7)can,with the help of the inequality related to E2(T)defined above,be shown to lead to the following inequality:

We now return to λ2(T)defined by(a5)to determine a condition for its positiveness.To this end,note that,in(a5),the scalars αm,1＜ 0 and αm,2＜ 0 for m=1,2,....We combine(a10)and(a11)to obtain the following inequality:

from which,we conclude that λ2(T),defined in(a5),is positive if condition[H-1A]of Theorem 2A is satisfied.The proof is completed.

Corollary T2-1The proof,which requires the use of the monotonicity property(6)ofis similar to the proof of Lemma 2A(based on the monotonicity property(5)).

Proof of Lemma 2BFor∈ M3,we start with(a5).In the governing inequality(8),we identifywith(t),andwithto obtain

In the second integral of(a13),we change the variable of integration t to τ =(t?mP),invoke the periodicity of V(t,·)and the properties of the integrands with truncation,and simplify to conclude that

from which,invoking(16),we conclude that the right hand side of(a14)can be replaced byMultiplying both the sides of the resultant inequality by αm,1＜ 0 for m=1,2,...,assuming the validity of an interchange of summation and integration operations,and summing up,we obtain

which we combine with the first integral on the right hand side of(a5)to conclude that

Proof of Lemma 3Let the first part of the expanded version of integral(36)be defined as Based on condition[H-1]of Theorem 3 that ω(t)e?ξtK′(t)is non-increasing,we invoke the second mean value theorem:there is a point T′∈ [0,T]such that the right hand side of(a17)can be simplified to give

and employing the condition of Theorem 3 that ω(t)eζtK′(t)is non-decreasing,we invoke the second mean value theorem,from which there is a point T′′∈ [0,T]such that the right hand side of(a19)can be simplified to give

We adopt the framework of the proof of Lemma 1 above,in the course of which we employ the parameters of(13)to reduce(a18)and(a20)to inequalities involving quadratic forms of,respectively,i)and(t?mp),and ii)(t)and(t+mp),and combine the resultants with the reasoning adopted in[49,pages 572–573],including the reference to condition[H-3](of Theorem 3),to conclude that λ3(T)of(36)is positive,if condition[H-2]of Theorem 3 is satisfied.

Outline of Proof of Lemma 4AWe have from(40)

where ξ is a nonnegative constant.Since ω(t)e?ξtK′(t)is nonincreasing by virtue of condition[H-1A]of Theorem 4A,by the second mean value theorem,there is a point T′∈ [0,T]for which(a22)can be written as

In(a23),we resort to the following operations:interchange of integration and summation operators(assuming the validity of such an interchange),and i)replacement of the variable of integration t in the second integral by τ =(t? τm),and ii)invoking the properties of a truncated functions in the integrand to change the integration limits,culminating in the following result:

Since ω(t)eζtK′(t)is non-decreasing by virtue of condition[H-1A]of Theorem 4A,by the(dual of the)second mean value theorem(for non-decreasing functions),there is a point T′′∈ [0,T]for which(a25)can be written as

which is the counterpart of(a23).The remaining steps amount to the reduction first of(a24),and then of(a26),to inequalities.With respect to the second integral in(a24),we let=and=(t+τm),and invoke the monotonicity property(5)ofand follow the lines of reasoning,manipulation and simplification of the integrals adopted in the proof of Lemma 2A;and repeat the operations with respect to(a26)(after changing the variable of integration in its integrand t involving the summation toand modifying the limits as was done for(a23))to arrive at condition[H-3A]of Theorem 4A.The only extra elements that appear while repeating the proof of Lemma 2A are the terms eξτmand eζτ′

m which multiply coefficients αm,1and αm,2,respectively.Due to lack of space these details are omitted here.

Proof of Lemma 4BWe have from(41)

where ξ is a nonnegative constant.Since ω(t)e?ξtK′(t)is nonincreasing by virtue of condition[H-2B]of Theorem 4B,there is,by the second mean value theorem,a pointT′∈ [0,T]for which(a28)can be written as

Integrating by parts,the last integral of(a29),invoking the property of N0,and simplifying the resulting expressions,we can rewrite(a29)as

which we combine with(17)to rewrite as the inequality

which is positive,if ξ ＜ 1/δqs,becauseK(t)Q? 0,t∈ [0,∞)(thereby implying that δqi≥ 0,δqs＞ 0),and the last term of(a31)can be made small by choosingappropriately small.The proof is completed.

his B.S.degree in Electrical and Electronic Engineering from Zhejiang University,China in 2002,and Ph.D.degree in Electrical and Computer Engineering from National University of Singapore(NUS)in 2011.He joined the Standard Chartered Bank,Singapore where he is a quantitative analyst with the modeling and analytical group.His research interests are switched systems and stochastic models.E-mail:mrhuangzhihong@hotmail.com.

Y.V.VENKATESH(SM-IEEE’91)receivedhis Ph.D.degree from the Indian Institute of Science(IIS),Bangalore.He was an Alexander von Humboldt fellow at the Universities of Karlsruhe,Freiburg,and Erlangen,Germany;a national research council fellow(USA)at the Goddard Space Flight Center,Greenbelt,MD;and a visiting professor at the Institut fuer Mathematik,Technische Universitat Graz,Austria,Institut fuer Signalverarbeitung,Kaiserslautern,Germany,National University of Singapore,Singapore and others.Apart from stability theory,his research interests include 3D computer and robotic vision,signal processing,pattern recognition,biometrics,hyperspectral image analysis,and neural networks.As a professor at IIS,he was also the Dean of Engineering Faculty and,earlier,the Chairman of the Electrical Sciences Division.He is a fellow of the Indian Academy of Sciences,the Indian National Science Academy,and the Indian Academy of Engineering.He is on the editorial board of the International Journal of Information Fusion.E-mail:yv.venkatesh@gmail.com.

Cheng XIANGreceived his B.S.degree in Mechanical Engineering from Fudan University,China in 1991,M.S.degree in Mechanical Engineering from the Institute of Mechanics,Chinese Academy of Sciences in 1994,and Ph.D.degree in Electrical Engineering from Yale University in 2000.From 2000 to 2001,he was a financial engineer at Fannie Mae,Washington D.C.He has been with the National University of Singapore since 2001.At present,he is an associate professor with the Department of Electrical and Computer Engineering,the National University of Singapore.His research interests include pattern recognition,intelligent control and systems biology.E-mail:elexc@nus.edu.sg.

Tong Heng LEEreceived his B.A.degree with First Class Honors in the Engineering Tripos from Cambridge University,England,in 1980,M.E.degree from NUS in 1985,and Ph.D.degree from Yale University in 1987.He is a professor in the Department of Electrical and Computer Engineering at the National University of Singapore(NUS)and also a professor in the NUS Graduate School,NUS NGS.He was a past vice-president(research)of NUS.Dr.Lee’s research interests are in the areas of adaptive systems,knowledge-based control,intelligent mechatronics and computational intelligence.He currently holds associate editor appointments in the IEEE Transactions in Systems,Man and Cybernetics;IEEE Transactions in Industrial Electronics;Control Engineering Practice(an IFAC journal);and the International Journal of Systems Science(Taylor and Francis,London).In addition,he is the deputy editor-in-chief of IFAC Mechatronics journal.Dr.Lee was a recipient of the Cambridge University Charles Baker Prize in Engineering;the 2004 ASCC(Melbourne)Best Industrial Control Application Paper Prize;the 2009 IEEE ICMA Best Paper in Automation Prize;and the 2009 ASCC Best Application Paper Prize.He has also co-authored five research monographs(books)and holds four patents(two of which are in the technology area of adaptive systems,and the other two are in the area of intelligent mechatronics).He is the recipient of the 2013 ACA Wook Hyun Kwon Education Prize.E-mail:eleleeth@nus.edu.sg.