Frequency-domain stability criteria for SISO and M IMO nonlinear feedback system s with constant and variable time-delays

Yedatore.V.VENKATESH

(Form erly)Division of Electrical Sciences,Indian Institute of Science,Bangalore 560012,India;Department of ECE,National University of Singapore,Singapore

1 Introduction

The main subject of this paper is theL2-stability analysis of certain classes of feedback system s(or system s which can be cast,equivalently for analytical purposes,as feedback system s)endowed with a linear timeinvariant part in the forward path and,in the feedback path,a constant gain in product with a linear combination of nonlinearities having instantaneous-action and delayed arguments.The integral equation govern-ing such SISO systems1The corresponding description of M IMO system s will be presented below in Section 7.(Fig.1)has the basic form

where time-delay constants,pandh,are each positive;and the time-delay functionq(t)satisfies the inequality 0 ＜q(t)≤Q＜ ∞ foris the impulse responseof the timeinvariant forward block with constant real sequences{gm},{τm},in which τm≥ 0,?m;g(·)a real-valued function in[0,∞);andi.e.,andg(·) ∈L1;constant gainK∈ [0,∞);f(·),v(·),σ(·)are,respectively,the input,“error”signal and output of the system;and the output of the nonlinear feedback gain is of the formwhere the(constant)coefficientsc1＞0 andci≥0,i=2,3,4;and φi(·),i=1,...,4,real-valued functions on(-∞,∞),are memoryless,first-and-third-quadrant nonlinearities having the following basic properties:φi(0)=0,i=1,...,4;there exist positive constantsq1andq2withq1＜q2such thaThis class of nonlinearities,,denoted by N,wasoriginally proposed by[1]in their pioneering Lyapunov method-based work on a certain nonlinear differential equation(without delay-terms).Additional nonlinearities of class N with delayed arguments of each of the three types can be added to the right-hand side of the first equation of(1)withoutanyneed to change theproposed framework forL2-stability analysis of(1).

Fig.1 Block schematic of a delayed-feedback nonlinear pseudo-time-invariant system.

The constant gainKand the linear combination of nonlinearities φi(·),i=1,...,4 in combination assume values in[0,∞).The transfer function of the forward block is given byG(jω).For convenience in some manipulations,we denote the operator representing the time-invariant forward block by G(·).The two functions(with delay arguments),on the right hand side of(1)are mathematically somewhat not straigt forward to hand le in the frequency domain,and hence call for some exp lanation which is given in Appendix O.

Main contributions&organization of the paper

Set against the above technical background(and against what can be subsequently gleaned from Section 3 dealing with a brief survey of literature),the main contributions of the present paper are the new frequency-domainL2-stability criteria for SISO and M IMO nonlinear feedback system s with time-delays,by em ploying a general causal+anticausal O’Shea-Zam es-Falb(OZF)multiplier function[2-4]along with certain scalar nonlinear inequalities,introduced in[2],for SISO system s,and their vector versions in a generalized quadratic form for M IMO system s in the present paper.

As far as the organization of the paper is concerned,thenextsection(Section 2)offersaglim pseofanalogous system s,i.e.,system s that apparently are different but can be cast into the structure of the system s under consideration.Section3 contains a brief survey of relevant literature,while Section 4 gives an idea of the motivating background for the stability results of the paper;and,Section 5 describes the main assumptions,problem formulation and mathematical preliminaries,including the definitions of the characteristic parameters(CPs)of the class N of nonlinearities needed in the proof of the stability theorem s for SISO system s.2See Section 7 for the definitions of CPs for vector nonlinearities that appear in M IMO system s.Section 6 presents the main stability result(Theorem 1)for theL2-stability of the SISO system(1)with φi(·)∈ N,i=1,...,4.Section7 deals with M IMO system stability analysis in the course of which basic assumptions and preliminaries are given before proving the main theorem s-Theorem s2 and 3 for linear M IMO system s,and Theorem 4 for nonlinear M IMO system s-in Section8.Comparison of the new results with the relevant literature is presented in the first part of Section 9,followed,in the second part,by computational results to serve as illustrations of Theorem s 1 and 4,with two exam ples for each.Section10 concludes the paper. Appendices contain proofs of a few basic lemmas used in the proof of the theorem s.

2 Analogous system s

Consider the scalar linear,time-invariant differential equation,

The linear system s considered in[5]and[6],and used as exam ples in[7]are governed by the coup led vector differential equation,

where,in the terminology of[7],is thendimensional state vector;AandBare constant matrices of appropriate dimensions;τ(t)=h1+ η1(t)is the time varying delay,withh1＞0,a nominal constant value and η1(t)is a time-varying piecewise continuous perturbation satisfying|η1(t)|≤ μ1,the last one being a known upper bound.System(2)can be re-cast as the following vector integral equation of a linear system with a delayed gain as a part of the feedback:

where then×nmatrixG(t)=inverse Lap lace transform of(sI-A)-1,withIdenoting the unitn×nmatrix,and?denotes convolution.

Now,consider the system governed by the coup led nonlinear delay-differential equations,

wherexi(t),i= 1,2 are the state-variables,y(t)is the output;are constants;are first-and-third-quadrant nonlinearities;andf1(t),f2(t)are input functions,which are m inor generalizations of the types of neural networks considered by m any authors3Note that neural networks with delays,of the type considered in[8]and[9]viz.in the latter reference’s term inology for the right-hand side,cannot be cast in the form of(1),unless some assumptions are made.who study their stability properties using Krasovskii-Lyapunov functionals or Razumikhin-Lyapunov functions[10,11].They can be reduced to the following vector integral equation of an equivalent feedback system:

whereis the two-dimensional vector input to the linear forward blockis the two-dimensional statevector;is the input vector;andare vector nonlinearities;G(t),the diagonal matrix with elements e-κ1tand e-κ2t,is the impulse response matrix of the forward block;andW,U,Vare constant 2×2 coefficient matrices,having elements{wij,uij,vij},i,j=1,2,respectively.

It is interesting to note that the generalized Lotka-Volterra predation-prey differential equation pair,

w herex1(t)andx2(t)are the prey and predator populations,respectively;α0is the rate of increase of prey;α1＞ 0 the coefficient limiting the grow th of prey,β2＞ 0 is the rateofdecrease of predator;andγ1＞ 0andγ2＞ 0 are the coefficients of the effect of predation onx1(t)andx2(t),can also be castinto the form(4).The vectordifferential equation(4)is a special case of the vector integral equation describinganM IMO system considered in Section7.On the other hand,the one-dimensional version of the same vector differential equation corresponds to a special case of the scalar integral equation(1)which describes the SISO system under consideration.Note,however,that we need to assume,to apply the results of the present paper to(4),that the nonlinearities obey the(generalized)first-and-third-quadrant condition,viz.andforwhere′denotes transpose.

In contrast with the analogous system s described above,[12]and[13]consider the exponential stability analysis of linear integral delay system s with multiple delays and having the following form(in our slightly modified notation to avoid possible confusion with our system(1)):

whereann-dimensional vector,Ai,i=1,...,Nare(constant)N×Nmatrices,and delaysτi＞ 0,i=1,...,Nare scalars.Assum ing the operation of differentiation of(in the Banach space of functions assumed by[13])to be valid,(6)can be reduced to the following differential equation:

which,as exp lained above can be cast into a vector version of the scalar integral equation(1)4See i)Section 7 below for a simplified version(37)of the corresponding M IMO vector integral equation,along with Rem ark 7 in the same section on the generalization of(37)to a more general M IMO vector integral equation;and ii)Section 11 for the com putational results corresponding to the example in[13]..Note that the initial conditions can be transformed to an equivalent(vector)forcing function.

3 Brief survey of literature

Tim e delays in dynamical system s,artificial or natural,are ubiquitous and appear in various form s.In artificial neural networks(ANN),the finite-time dynamics of switching causes time delays.In both artificial and naturalsystem s,time delayscause instability.And thisserves as a powerful motivation to analyse the stability properties of dynamical system s with delayed arguments.

For a survey of stability analysis of certain classes of neural networks which involve delays,see[9,14].The stability results m entioned in those papers are derived directly or indirectly in a Lyapunov framework.However,quite a few of the differential equations found in[9].[14]can be cast into a vector-matrix feedback system with multiple inputs and multiple outputs,and are am enable to(an extended form of)the frequencydomain framework proposed in the present paper.

The authors of[15]construct Lyapunov-Krasovskii(LK)functionals for linear time-delay system s using the sum of squares decom position of multivariate polynomials to solve the related infinite dim ensional linear matrix inequalities(LM Is).Fridm an and Niculescu[7]also deal with the stability of similar linear system s(but cast as a vector-differential equation)with norm-bounded uncertainties and uncertain but bounded time-varying delays(without a constraint on delay derivative).The framework is Lyapunov’s in the form of a synthesized Lyapunov-Krasovskii functional.In[11],sm all-gain theorem s for large-scale time-delay system s are developed using a Razumikhin-type approach,i.e.,the state variables with delays are treated as disturbances to the system,thereby converting the problem of stability analysis of delay-differential equations to the problem of stability analysis for system s without delay but with disturbances.

Some authors study nonlinear time-delay system s using linearization and the second method of Lyapunov.Even the LM I method has been employed in the stability analysis of recurrent neural networks[10].In[16],the authors study the stability of oscillations of continuoustime analog neural networks with delay,but with symmetric connections.They establish conditions on delay for the network to oscillate.In[17],the authors em ploy the Lyapunov-Krasovskii functional and invoke LM Is to deriveless conservative so lutions to the globalstability problemin neural networks with time-varying delays.For a class of recurrent neural networks with time-varying delay,the authors of[10]propose a secondarydelay partitioning methodin combination with a Lyapunov functional.On the other hand,the authors of[18]em ploy the Lyapunov-Krasovskii functional to derive stability conditions for a vector differential equation with a time-varying delay.

Reference[19]deals with linear system s having multiple pointwise and distributed delays for which necessary stability conditions are derived by connecting theLyapunov delay matrixand the fundamental matrix of the linear system s under consideration.In[20],the authors present the equivalence between quadratic Lyapunov functions for nonlinear Lur’e system s and the circle criterion.It is not clear why an M IMO Popov-type frequency-domain absolute stability result was not considered by invoking KPY-type lemm a.

Reference has already been made to[12]and[13]in the context of analogous systems. s.Both of them consider the exponential stability analysis of linear integral delay system s with multiple delays.While the author of[12]uses the Lyapunov-Krasovskii functional approach to derive sufficient delay-dependent stability conditions and exponential estim ates for the solutions,the authors of[13]em ploy a generalized version of the Jensen inequality,optimized to reduce the conservatism,along with Lyapunov functional-based approaches and linearization of nonlinear matrix inequalities to obtain sufficient stabiliity conditions for the exponential stability of(6).The authors of[13]propose,further,the positive operator theory to derive,in part,a spectral radius-based stability condition.

From among other references in the literature,[21]em ploys linear Lyapunov functionals and positive system s to establish conditions for stability(as also robustness against uncertainty)of time-varying systems with delays.In[22],homogeneity theory is employed for global delay independent stability analysis of certain nonlinear time-delay systems.However,it is not clear how such a theory can be applied to the class of system s considered in the present paper.The authors of[23]em ploy the method ofim plicitLyapunov-Krasovski functional for various types of stability analysis(including Lyapunovand asymptotic stability)of time-delaysystem s.For an abstract version of the Nyquist-like stability criterion as applied to linear time-invariant time-delay system s,see[24]where spectral properties ofm onodrom y operatorshave been invoked.

To summarize,the stability of dynamical system s described by scalar and vector delay-differential equations has been m ostly analysed in an extended Lyapunov framework.No related results seem to be available in the frequency-domain for the same system s or for system s described by scalar and vectorintegralequations.Note that the frequency-domain stability conditions for time-varying M IMO system s as found in[25]cannot be app lied directly,and without additional constraints,to establishing theL2-stability of the class of M IMO system s with time-delays that are considered in the present paper.

4 Motivating background(SISO system)

The starting point for the stability analysis of(1)is the linear pseudo-time-invariant system corresponding to it

a

Since there seems to be no algebraically explicit relationship between the Laplace transform s of σ(t)and σ(t-q(t))for arbitrayq(t),w e cannot,in general,derive the feedback transfer function to be able to invoke the Nyquist criterion for stability.However,to provide a rough foothold on a tentative use of the Nyquist criterion,we assume that the time-varying delay function σ(t-q(t))-recall,q(t)∈ (0,Q],t∈ [0,∞)-can be rep laced by σ(t-q?)w here constantq?∈ (0,Q].In this case,the feedback transfer function for the basic linear pseudo-time-invariant system is given by

The Aizerman-type of assumption we make(in the absence of an expression for the Laplace transform of σ(t-q(t))in term s of the Laplace transform of σ(t))is that the roots of the characteristic equation of(8),viz.

lie in the left half of the complex p lane,?s＜ 0 forK∈[0,∞).The multiplier-function form of the Nyquist criterion(referred to above)now has to contend with the phase-angle behaviour ofH(jω)G?(jω).Alternatively,the transcendental equation(10)is to be solved.(Thisdoes not,of course,guarantee,in general,that the system(8)is asymptotically stable.)

5 SISO system assumptions,problem form u lation&mathematical preliminaries

For system(1),the impulse response of the linear block is assumed to be inand the solutions of system(8)are inwhich implies(for the time being,as a starting point,but in need of more precise formulation)that the roots of the characteristic equation(10),lie strictly in the left half(?s＜ -δ≤ 0)of the complex p lane forK∈[0,∞).Invoking(9)and assuming thatc1＞0 andci≥0 fori=2,3,4,we can,as a first approximation,compute the maximalgainof the linear feedback block as(c1+c2+c3+hc4)K.However,as exp lained in Rem ark 4,Section 1,withc3=0,theeffective gain(in the frequency-domain)of the feedback block with the nonlinearities can be treated as

Problem formulationBased on the above assumptions,find conditions onandci,i=1,...,4,for theL2-stability of system(1)5When K in(1)is replaced by a time-varying gain k(t)∈ [0,∞),t≥ 0,we can obtain stability results by following the framework of[2]and[26]..

Mathematical preliminariesFor any real valued functionx(·)on[0,∞)and anyT≥ 0,w e define thetruncated function xT(·)byxT(t)=x(t)for 0 ≤t≤T;andxT(t)=0 fort＜ 0 andt＞T.LetL2ebe the space of those real-valued functionsx(·)on[0,∞)whose truncationsxT(·)belong toL2[0,∞)for allT≥ 0.In order to establish stability of the system under consideration,w e first assume infiniteescape timefor the solution to the system withf∈L2and the solution belongs toL2e.We then show that,under certain conditions on the φi(·)’s,ci’s andG?(jω),the solution actually belongs toL2[0,∞).

Consider the class of operatorssatisfying an equation of the type

where the(real)constantsα≥0andη＞0;his the same delay-history parameter that appears in(1)in the form of the lower limit(t-h)of the integral on right-hand side of the first equation;the real sequences{zm}andare in ?1,i.e.,sequences{τm}andare in[0,∞);z(·)is a real-valued function on(-∞,∞),and is inL1(-∞,∞),i.e.,Its Fourier transform is given by

The above multiplier function(12)contains the lastterm,which is apulse functionin the time-domain,as addition to the one used in[3,4](to deal explicitly with the class of monotone nonlinearities for no-time-delay system s).For later use,let

1)anddenote,respectively,the positive and negative coefficients from the set{zm},i.e.,zm=anddenote,respectively,the positive and negative coefficients from the seti.e.,2)andza(t)=z(t)fort＜0,so thatz

3)ifzc(t)＞ 0,t∈ [0,∞);else0,t∈ [0,∞);ifzc(t)＜ 0,t∈ [0,∞);elsei.e.t∈ [0,∞);and

4)ifza(t)＞ 0,t∈ (-∞,0];elset∈ (-∞,0].Similarly,ifelsei.e.,

For convenience in invoking some results from[2],we split the above multiplier operator into two parts:

from which w e haveTheir Fourier transform s are

Further preliminariesTheorem 1 and its proof are based on new algebraic inequalities-introduced,ina simplified form,in[2]-involving the nonlinearities φi(·),i=1,...,4.Based on these inequalities,the characteristic parameters(CPs)of φi(·) ∈ N(fori=1,...,4)are defined as follows.6By way of avoiding a clutter of subscripts at this stage,a generic nonlinearity φ(·)is em ployed,and its CPs are defined with a two-element parenthesized subscript in which the first element is either the Greek letter iota,ι,to denote infim um or the letter s to denote suprem um;and the second element is a center dot,“·”,to be replaced later by the index corresponding to the nonlinearity under consideration.However,while stating the theorem s and outlining their p roofs,the subscripts for the nonlinearities and their CPs will be com plete.Let Ψ(x,y)?{φ1(x)x+φ1(y)y}.Then i)for φ?(·)∈ N,?=1,...,4,we define-μι,(?,1)Ψ(x,y) ≤ φ?(x)y≤ μs,(?,1)Ψ(x,y),where the CPs μι,(?,1)＞ 0 and μs,(?,1)＞ 0 are defined by

See Table 1 in[2](page 215,column 1)for typical values of μι,(?,?)and μs,(?,?)for φ?(·) ∈ N.It is conjectured in[2]that,for an arbitrary φ(·) ∈ N,the maximum value of each of μι,(?,?)and μs,(?,?)is 1.(However,this conjecture may not benecessarilyvalid in general for μι,(?,1)and μs,(?,1)w hen ?=2,3,4.)For what follow s,let

for ?=1,...,4,where the subscript ? inrefers to the nonlinearity φ?(·)that appears later in the integrands.Further,we use generic notation for integrals involving the product of the nonlinearity(with an argument)and its argument itself.For compaction of later integrals,we need the following notation:Ψ[1]?As applied to a specific nonlinearity,a subscript is added as and when required.For instance,

Lemm a 1Awith φ1(·)∈ N and the multiplier functionZ1(jω)of the form(14),the integral

satisfies the inequality

See the proof of Lemma 1 in[2](pages 224 and 225).

Lemm a 1Bwith φ1(·)∈ N and the multiplier functionZ2(jω)of the form(14),and having η ＞ 0,the integral

See Proof of Lemma 1B.

We can combine Lemmas 1A and 1B to arrive at the following corollary:

Corollary1with φ1(·)∈ N and the multiplier functionsZi(jω)(wherei=1,2)of the form(14)and having η＞0,the integral

for all σT(t)in the domain of Z1and for allT≥ 0.

On similar lines,w e can establish the following corollary.

Corollary 2

the integral

obeys the inequality,

See the appendix for its proof.

Since no sim p le closed form lower bound can be derived for the integral

Lemm a 1Dwith α=0 in(14),

Recalling the system equation(1),letWe combine and simplify the above inequalities,to arrive at the followingcomprehensivelemm a.

if the following inequalities are satisfied:

where ??,?=1,...,4 are defined by(16);and1,...,4 by(15),for all σT(t)in the respective domains of Zi,i=1,2 and for allT≥0.

Specialcase of Lemm a 1?When the feedback is linear,then letThe integral(27)inequality now assumes the form

in which the integrand com ponent σT(t-q(t))does not haveacom pact Fourier transform in term sof the Fourier transform ΣT(jω)of σT(t).Therefore,in order to be able to express conditions for the nononegativity of λL?(T)in the frequency domain,we setc3=0.Using the Parseval relation,it can be shown that λL?(T) ≥ 0 if,for-∞＜ω＜∞,

Whenthe inequality constraints(28)can be specialized to hold for the linearsystem.However,there is a time-domain constraint on the multiplier function.

6 Main resu lt for SISO system s

In order to avoid confusion with parenthesized numbers meant for equations and square-bracketed numbers meant for references,we em ploy square-bracketed numbers with prefix“H-”for the conditions(orhypo theses)in the theorem s.with the preliminaries settled,we now present the main result for the SISO system(1).

Theorem 1The nonlinear system(1)with φi(·) ∈N,i=1,...,4,constant gainK∈[0,∞)isL2-stable,if there exists a multiplier functionZ(jω)of the form(14)such that H-1)for some positive constant δ,?Z(jω)G?(jω) ≥ δ ＞ 0,ω ∈ (-∞,∞);and H-2)the conditions of Lemm a 1?and its two inequalities(28)are satisfied.

The proof of Theorem 1 follow s the technique developed in[26]and is based on Lemm a 1?.

Proof of Theorem 1Consider the integral,for anyT＞0,

LetVT(jω)denote the Fourier transform ofvT(t)(Note that the subscriptTinVT(jω)refers to the fact that the original time-function is truncated.There is no truncation of the Fourier transform.)App lying the Parseval theorem to the first integral on the right hand side of(31),we obtain

Invoking the condition H-1)of Theorem 1,we obtain the following inequality:

By virtue of Lemma 1?,as required by condition H-2)of the theorem,and withthe second integral right hand side of(31)obeys the inequality,

Now,by app lying the Parseval theorem to(30),and combining the result with(33)and(34),we obtain

Using Cauchy-Schw arz inequality in the integral on the right-hand side of(35)leads to

7 M IMO system s

with a view to provide some newL2-stability results for M IMO feedback system s,but constrained by limitations of space,we consider the following nonlinear system with only constant and time-varying delays:

w here the vectors of reference inputerrornonlinear gainand outputhave each dimensionr,and?denotes convolution.with’denoting vector/matrix transpose,|·|,the determinant of a square matrix,andI,a unit matrix(of dimension evident from the context of its usage),the other functions in(37)are defined as follows.The constant-gain matrix K of dimensionr×ris,in general,notsymmetric.For simplicity and convenience in the proof of the results,we assume that the elementskmnof K assume values such that K is positive definite-we w rite K?0.This is a generalization of the assumption typically made in the stability analysis of SISO feedback system s withG(jω)as the transfer function of the forward block and for which the scalar feedback gainKis assumed to take values in[0,∞).For other cases,a suitable matrix-transformation[25](which is a generalization of the type made in the case of SISO systems)leads to the desired assumption on the feedback gain.

Due to limitations of space,we consider only the case of the impulse response of the linear forward block with the(continuous-time)r×rreal matrix functionfort∈ [0,∞).Now,assum ing that the elements ofare inL1∩L2,let the Fourier transform ofbe given by Γ(jω).(Note that the transfer function of the forward block is then Γ(s),wheresis a comp lex variable.)In order to deal with the case of the feedback nonlinear block having a non-positive definite K and/or obeying the inequalitywhereis a constant(lower bound)matrix,we transform the system to an equivalent M IMO system with the modified forward block transfer function and feedback gain given byandrespectively.

M IMO system-related mathematical preliminariesLetdenote a real-valued vector function,having elementsx1,x2,...,xr.If each elem ent of the vectoris inL2,the vector itselfissaid to be inL2.Then itsL2-norm is defined byIn order to state the main assumptions needed to establish stability conditions for(37),we need to consider its special case of thelinear constantmatrix gain M IMO system,

whereand the elements of the matrix K(0)are such that K(0)?0.The system described by(37)is said to beL2-stable ifand an inequality of the typeholds whereCis a constant,or,in the more general form,(whereCi,i=1,...,4 are constants),to account for initial conditions in the system.This definition is also valid for(38)and its gain-transformed counterpart.

M IMO system-related main assumptionsA1)The solutions to(38)are inL1∩L2,which im p lies that the zeroslie strictly in the lefthalfof the com plex plane.A2)K ?0;A3)The nonlinearities(for all the classes defined below)satisfy the following inequality for an arbitrary bounded constant matrixY,

wherei=1,...,4,and ‖Y‖is the matrix norm7The matrix norm of Y could be,for instance,ofY.The classes of nonlinearities considered in the literature(like monotone, odd-monotone and others) are subclasses of(N,K).Note that piecewise(first-and-thirdquadrant)nonlinearities having negative slopes are also

included in this class.withand ?=1,2,3,we define the characteristic parameters,and

Formulation of the M IMO stability prob lemFind conditions for theL2-stability of the system(38)and of system(37)with

Forsim p licity in the proofofstability theorem s,w eassume thatthegeneralized feed backgain isallowed to assume values in[0,∞).The stability conditions which are derived in term sofΓ(jω)and amultipliermatrix-function-see below-can then be specialized to “finite-gain”system s by the following algebraic operations:Rep lace both the transfer function of the forward block and the feedback by their corresponding gain-transformed versions.See Appendices 0.aand 0.bin[25].

8 Main results for M IMO system s

The underlying motivation for frequency-domainbased stability criteria for M IMO system s is,as is to be expected,the generalized Nyquist criterion as found,for instance,in[27-29]for linear time-invariant feedback system s with no delay term s.somewhat more interestingly,the matrix multiplier function-based stability criterion that is presented here for linear feedback system s with delay arguments can be adapted/modified to express stability conditions for both nonlinear and periodically time-varying gain feedback system s with delayed arguments.

For any real-valued vector functionon[0,∞)and anyT≥0,thetruncated functionis defined as follow s:for 0≤t≤T;andfort＜0 andt＞T.Further,letL2ebe the space of those real-valued functionson[0,∞)whose truncationsbelong toL2[0,∞)for allT≥0.Essentially,by assuming infiniteescape timefor the solution to the system withthe solution belongs toL2e.Then,it is shown that,under certain conditions on K,the CPs ofand Γ(jω),the solution actually belongs toL2[0,∞).

For the linear,constant coefficient system(38),the matrix multiplier operator,denoted by Y(l)with elements inL1∩L2and having the Fourier transformY(l)(jω),is general,whereas for system(37),a typical multiplier operator Y has the following specific form10Due to limitations of space,this is a special case of a possible matrix generalization of the multiplier function(11)chosen for SISO system s in Section 5.Stability results with a more general matrix multipler can be derived with additional notation.:

(Note thatY(jω)is a non-positive real matrix function.)For later use in the proof of theorem s and their lemmas,letanddenote,respectively,the positive and negative coefficients from the set{αm},i.e.,αm=Similarly,anddenote,respectively,the positive and negative coefficients from the set

We now state the main stability theorem s for the M IMO system s(38)and(37).For Theorem 2 below meant for system(38),we assume thatc3=0.This restriction is due to the fact that we do not know,just as in the case of SISO system s,how to compute the Fourier transform ofexplicitly in term s of the Fourier transform ofandq(t)(in some suitable form or the other)appearing as an extra factor.

The proof of Theorem 3 depends on the following lemm a.

Lemm a 3with the operator Y defined by(41),and νsby(23),the following inequality holds:

forallin the domain of Y and for allT≥0,if condition H-2)of Theorem 3 is satisfied.

The proof of Theorem 4 depends on the following lemma:

Lemm a 4with the operator Y defined by(41),the following inequality holds:

(sincefor allin the domain of Y and for allT≥0,if condition H-2)of Theorem 4 is satisfied.

Proofs of the M IMO stability theorem sThe proofs of the above stability theorem s depend on i)an application of an extended Parseval theorem(for vector functions),and ii)establishing positivity conditions for tw o blocks in cascade:for system(38),the first block is linear havingY(jω)as its transfer function,and the second is K;and for system(37),the first block is the same,but the second is a constant gain K in cascade with a linear combination of nonlinearities with delayedarguments. □

An outline of the proofs of Theorem s 2 and 4 is given below.The proof of the latter is based on Lemma 4,which is established in Appendix 2.The proof of Theorem 3 is om itted here because it is a special case of the proof of Theorem 4.Let G denote the matrix operator of the forward linear block,i.e.,

Proof of Theorem 2with the multiplier matrixoperator Y(l)having the Fourier transformY(l)(jω),consider the integral,for anyT＞0,

which,on invoking(38),becomes

The first integral on the right hand side of(45)can be shown,by condition H-1)of Theorem 2 in association with the extended Parseval theorem,to satisfy the inequality

for some δ＞0.We app ly the extended Parseval theorem to the second integral on the right hand side of(45)to obtain

which can be shown to be positive invoking condition H-2)of Theorem 2.Now we apply the extended Parseval theorem to(44)and combine the result with(46)to obtain

Proof of Theorem 4with the operator Y defined by(41),consider,for anyT＞0,the integral

which,on invoking(37),becom es

As in the proof of Theorem 2,the first integral on the right hand side of(50)can be shown,by condition H-1)of Theorem 4,to satisfy the inequality

Rem ark 2In case the first equation of(37)is rep laced by

Rem ark 3Know ledge of the constant delay parameterpcan weaken the stability constraints on the CPs of nonlinearities and their coefficients by choosing a multiplier matrix-function with a τm=p.However,due to limitations of space,the corresponding changes in the condition H-2)of Theorem 4 are not established here.

9 Comparison with literature and applications of theorem s

1)There seem to benofrequency-domain results in the literature on the stability of SISO nonlinear timeinvariant with multiple nonlinearities,φ?(·) ∈ N,?=1,...,4,the latter two of which have different types of delayed arguments.2)Theorem 1 of[2]can be considered as a special case of Theorem 1 of the present paper.3)It is known that Kalm an-Yakubovich-Popov(KYP)lemm a proves that Popov’s stability criterion for system(1)with φ1(·)∈ N and withc1=1,c2=c3=c4=0 is equivalent to the existence of a Lyapunov function com prising a quadratic form and an integral of the nonlinearity.An interesting open problem is to find possible Lyapunov-Krasovskii functionals and/or Lyapunov-Razum ikhin functions corresponding to Theorem 1 of the present paper.4)It seems that a graphical interpretation of Theorem 1 is extraordinarily difficult.It is,however,possible to convert the real-part condition and the time-domain constraints on the multiplier function of Theorem 1 to a non-convex optimization problem.5)Thereseem to benoL2-stability results in the frequencydomain for nonlinear M IMO system s with delayed arguments.The frequency-domain(nonlinear and timevarying)M IMO system stability results of[25]cannot be applied directly to the nonlinear M IMO system s under consideration,having the class(N,K)of vector nonlinearities,without additional constraints,even when the time-delay term s are removed.Moreover,Theorem 4,when specialized to the case ofc2=0 andc3=0 is more general than Theorem 4A of[25]in the special of the constant gain matrixK0(in the latter reference’s notation)because the vector nonlinearity belongs to the monotone class,whereas the corresponding vectornon linearity in the present Theorem 4 belongs to the more general non-monotone class.On the other hand,Theorem 4B of[25]deals with non-monotone class of nonlinearities,but its matrix multiplier function is a special case of the matrixmultiplier function of Theorem 4 of the present paper.6)In the SISO system governed by(1),if the constant gainKis replaced by the time-varying gaink(t)∈ [0,∞),t≥ 0,then the procedure outlined in[2]is to be followed.Due to lack of space,details are omitted.It turns out that,in essence,the inequalities(28)are to be rep laced by their weighted versions,the weights being eξtand eζt,where the constant parameters ξ and ζ correspond to global upper and lower bounds on the norm alizedrate of variation ofk(t).Evidently,the new inequalities are significantly more complicated than those found in[2]meant fora nonlinear time-varying feedback system with a single nonlinearity with no time-delay in its argument.A similar observation can be made with respect to nonlinear M IMO systems with a time-varying(feedback)matrix gain(see[25]for the no-delay system results)in cascade with vector nonlinearities having delayed arguments.7)Theproposed(continuous-time)approach is not applicable to hybrid system s(i.e.,system s defined on continuous-and discrete-time domains)with delay.On the other hand,due to limitations of space,we haven’t considered the following:i)time-varying system s with delay,even though the proposed framework can be extended to their stability analysis w hen cast in the form of a feedback system with a linear timeinvariant block and a time-varying gain in the feedback loop.ii)Neutral delay differential system s(for which the comm ent in item i)above holds.

The inequalities in the last column of the table are meant to be self-exp lanatory.However,we discuss briefly a few typical item s of Table 1.

Table 1 The SISO system(1)of Exam p le 1:Typical multiplier functions and stability limits/constraints on gain K and the CPs of nonlinearities.

The standard circle criterion (which is originally meant for a single time-varying nonlinearity in the feedback loop)cannot be app lied to the SISO system(with delayed arguments in the feedback loop)under consideration.Case no.1(c)gives results of what happens if we try to do so,the crucial im plication being the inequality in the last column of the table.

with the upper-bound=7.9104,the nonlinear time-delay system withc1=1.0;c2＞0;c3＞0;andc4=0 isL2-stable if the CPs of nonlinearitiesφ2(·)andφ3(·),their coefficientsc2andc3and the rate of variation of the time-varying delay functionq(t),characterized by νs,satisfy the inequalityNoting that we can trade off among the five parameters in the last inequality,suppose the CPs of the nonlinearities φ2(·)and φ3(·)are,respectively,μι,(2,1)=0.4 and μι,(3,1)=0.3,then,for theL2-stability of the system under consideration,0.8c2+0.3c3(1+νs)＜ 1.The coefficientsc2andc3of the nonlinearities can now be traded for νsto meet other system design requirements.

Now let us consider item s 2(a)and 2(b)which emp loy more general multiplier functions.In item 2(a),the constant time-delay parameterpin the argument of the nonlinearityφ(·)isnotspecified,whereas in item 2(b),it is specified thatp=1.5.Correspondingly,the multip lier function of item 2(b)has one exponent set to 1.5,which im p lies that we are taking into account the know ledge of the constant delay parameter.Comparison between the computational results for item s 2(a)and 2(b),with the multiplier functions so chosen as to satisfy condition H-1)of Theorem 1,are as follow s:

Tab le 1,item 2(a)Gain limit=54.8105;and the stability inequality,0.242μι,(1,1)+1.242μs,(1,1)+

Tab le 1,item 2(b)Gain limit=100.0;and the stability inequalitiesare6.5μs,(1,1)+0.4μι,(1,1)＜ 1;3.9μs,(2,1)+0.2μι,(2,1)＜ 0.7;and(0.7-3.9μs,(2,1)-0.2μι,(2,1))c2＞

To facilitate comparison between consequences of having no know ledge of the delay parameterpand having its specific value,we now assume the following CPs for the two nonlinearites φ1(·)and φ2(·):

μι,(1,1)=0.02; μs,(1,1)=0.15; μι,(2,1)=0.05;and μs,(2,1)=0.15.

From the inequalities in the last column of the table,the constrainton the coefficientc2forstability is follow s:

For item 2(a),c2＜1.5443;and for item 2(b),c2＞0.9211.

The last inequality implies thatc2can be greater than 1.5443,which is the upper bound on it in item 2(a).In other words,when the delay parameterpis known,this know ledge can be utilized in the choice of the multiplier function,leading to enlarging the bound on the coefficientc2of the nonlinearity φ2(·),beyond that possible whenpis not known.

For the case not considered,i.e.,c1=1 andci＞0,i=2,3,4,the form of the multiplier function of Case 4 may be chosen,and,after satisfying condition H-1)of Theorem 1,the corresponding time-domain constraint on it is obtained from(28)where the?i,i=1,...,4 are to be obtained from(16).

Now we generalize these two latter results to be applicable to system(1)withc1=1,c2＞0,c3＞0 andc4=0 by invoking Theorem 1 of the present paper.LetThe system(1)isL2-stable(i)for=11.4,if ?1＜5.0251;and(ii)for=16.0,if ?1＜2.1739.Further,we have,from the same Theorem 1,new results for θ0=1.0 and θ0=2.0.For theL2-stability of the system(1)withc1=1,c2＞0,c3＞0 andc4=0,with theG(s)as indicated above,the findings are:(i)θ0=1.0;=2.316;Z3(jω)=(1-j0.82sin(1.4ω));?1＜1.2195;and(ii)θ0=2.0;=1.5;Z4(jω)=(1-j0.82sin(1.56ω));?1＜ 1.2195.

The forward block of the M IMO system’s transfer function,Γ1(s)is given(row-w ise)by

A typical“frozen-time”coefficient matrix of exam p le in[25]is given(row-w ise)by[(9,17);(-13.6,12)],which is positive definite.From[25],a typical Routh-Hurwitzlimit matrix forfor which the characteristic equation of the linear M IMO system with the constant matrix feedback gain has roots with negative real parts,andis not positive real,thereby im plying that the generalized version of the circle criterion of M IMO system s cannot be invoked for the linear and nonlinear system s(38)and(37).As in the SISO case,forthere is a need to approximate the time-varying delay by a constant to be able to derive the feedback transfer function that can lead to an overall RH gain-matrix.However,this is not done here.The main reason is that the multiplier-matrix form of the stability result,though evidently conservative,obviates the need for exp licit R-H limits involving the transfer function of the delayed-argument feedback block for the linear system(38).See Table 2 for a summary of the numerical results,which,while possibly needing a check,may be refined/improved since they are not optimized for best results possible.Concerning the inequalities that appear in the last column of the table,note that the characteristic parameters(μι,(?,1),μs,(?,1)),?=1,...,3 are defined by(40);and νsby(23).

Table 2 The M IMO system(37)of Examp le 3:Typical multiplier functions and stability limits/constraints on gain K and the CPs of nonlinearities.

There are some interesting differences between the numerical results given in Table 1 for SISO system s and those in the present Table 2 for M IMO system s.As rem arked in the in the section on M IMO system s-in both the original and the present,revised papers-to conserve space,the case of M IMO system s w hen the value of the constant delay parameter is known is not considered.Therefore,no exp licit stability criteria for such an M IMO system have been derived.Table 2 therefore does not refer to this case.

From another point of view,when we exam ine item s 1(g),dealingwith an extension of the standard Popov criterion to the delayed system,the “upper”bound on the matrixgain isgiven by,and the stability inequality of the lastcolumn reads:On the other hand,for the same upper bound on the matrix gain,item 2(b)indicates the choice of a different matrix multiplier function,leading to the stability inequality,

From the inequalities in the last column of Table 2,the constraints on the coefficientsc2andc3for stability are follow s:from item 1(g),0.1c2+0.075c3(1+νs)＜ 1;and from item 2(b),0.2094c2+0.4451c3(1+νs)＜ 1.

Note that the latter inequality is more restrictive than the former with respect to the bounds on the coefficientsc2andc3.In other words,the matrix version of Popov multiplier function gives better results than the“periodic”function(in the frequency-domain)as a matrix multiplier function.It is not know n w hether other choices of the latter can give results better than those obtainable from a Popov multiplier function.

On the otherh and,item s2(a)and 2(b),while illustrating the choice of alternative matrix-multiplier functions for the sameandas in item s 1(e)and 1(g),respectively,can also be app lied to a periodic matrix gain K?(t)replacing the constant matrix-gain K,the period of K?(t)(as also,simultaneously,of the varying delay functionq(t))being,respectively,2.56 and 2.41.However,the constraints on the nonlinearitesφ?(·)∈ (N,K)for ?=2,3 are different from those in item s 1(e)and 1(g)of the table.

It can be verified that the equivalent feedack gain matrixsatisfies,for a suitable choice of the Kc,the condition(in the frequencydomain)(Kc-K(e))?0,-∞ ＜ω＜∞.Now,an application of our Theorem 3 needs a slight generalization which in effect amounts to establishing the conditions for-see Lemma 3(b)-the integral inequalityfor alland for allT≥0.It can be shown that this is indeed the case,if(2α + β - αβ)＜ 1.51.Since α and β can be chosen(under the pre-conditions 0＜α＜1 and 0＜β＜2)to satisfy the last inequality,the linear system exam p le of[31]isL2-stable.

Note thatandand are both functions of α and β.

Now,along the lines of the proof of Lemma 4,and invoking Theorem 4,it can be shown that the nonlinear feedback system under consideration isL2-stable,if

10 Conclusions

New frequency-domain criteria have been presented for theL2-stability of nonlinear SISO and M IMO feedback system s described by integral equations,with a time-invariant linear block in the forward path and,in the feedback path,time-invariant,first-and-thirdquadrant class of nonlinearities with different types of delay.The stability criteria are expressed in term s of the transfer function of the linear forward block and a causal+anticausal multiplier function,with a timedomainL1-norm constraint on it.This constraint depends on i)the coefficients of the nonlinearities;ii)the characteristic parameters(CPs)of the nonlinearities;and(iii)the rate of variation of the time-varying delay.It would be interesting to relate theorem s of the paper to the existence of Lyapunov-Krasovskii functionals.It is,however,quite likely that the frequency-domain conditions,being the first of their kind,need to be verified for refinem ents and possible improvements.

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Appendix

In what follow s,we assume that interchanges of the operations of summation and integration and of two integrals(one with respect to τ and the other with respect tot)are valid.Due to shortage of space,we prove only Lemm as 1B,1C and 4.The other lemmas can be proved along similar lines.

In(1),the term φ3(σ(t-q(t)))is unusual because we cannot consider it as the output of two system s in cascade,viz.i)a linear system with the input σ(t)and output σ(t-q(t)),and ii)a nonlinear gain φ2(·).In fact,more basically,we do not know the impulse response of a linear system which,with σ(t)as the input,produces σ(t-q(t))as the output.Synthesizing an impulse response function to generate the output σ(t-q(t))from the input σ(t)seem s to be an unresolved problem.Alternatively,assum ing the existence of the Fourier transform of σ(t),there is no closed-form expression for the Fourier transform of σ(t-q(t))in term s of the Fourier transform of σ(t)itself andq(t),im plicitly or otherw ise.

Proof of Lemm a 1BThe integral of(19)can be rewritten as

Wenow employ the CPsdefined in(15)and recallthatη＞ 0 to reduce(a1)to the following inequality:

The lemma is proved.

following the lines of proof of Lemma 1A in[2]and of Lemma 1B in proof of Lemma 1B above,the integral of(24)can be rewritten as

We now employ the CPs defined in(15),use(a5)and recall that η＞0 to reduce(a4)to inequality(25).The lemma is proved. □

Proof of Lemm a 4We analyse the integral on the righth and side of(43).To this end,let

Invoking(41),the first integral of(a6)can be written as

in which

In(a6),the integral

which,on invoking(40)and w ritingleads to the inequality

Along similar lines,

leads to the inequality

Com bining(a7)-(a10),along with the other(unnumbered)inequalities in between,we obtain

In much the same manner,mutatis mutandis,we can derive the following inequality concerning ?2(T)(recallc2＞ 0)in(a6):

As far as ?3(T)in(a6)is concerned,here are the steps.We have

w herec3＞0.Let

Invoking(40),we obtain

For the second integrand of the last inequality,changing the variable of integration fromtto τ ? (t-q(t)),and recalling νsdefined by(23),we obtain

For the second com ponent inside the curly brackets of(a13),let

following the steps em ployed for ?2(T),changing the dumm y variable of integration appropriately to sim p lify ?31(T)above,we can show that ?3(T)in(a6)obeys the following inequality:

Assembling the inequalities(a11),(a12),and(a14)together,and invoking condition H-2)of Theorem 4,we arrive at the inequality(43).The lemm a is proved.