Small Gain Theorems on Input-To-Output Stability

8/4/2019 Small Gain Theorems on Input-To-Output Stability

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Small Gain Theorems on Input-to-Output Stability

Zhong-Ping Jiang Yuan Wang .

Dept. of Electrical & Computer Engineering Dept. of Mathematical SciencesPolytechnic University Florida Atlantic University

Brooklyn, NY 11201, U.S.A. Boca Raton, FL 33431,[email protected] [email protected]

Abstract This paper presents nonlinear small gain theo-rems for both continuous time systems and discrete time systemsin a general setting of input-to-output stability. Most small gaintheorems in the past literature were stated for interconnectedsystems of which the subsystems are either input-to-state sta-ble, or are input-to-output stable and satisfy an additional con-dition of input/output-to-state stability. In this work we showthat when the input-to-output stability is strengthened with anoutput Lagrange stability property, a small gain theorem can beobtained without the input/output-to-state stability condition.

I. Introduction

In the analysis of control systems, it is very often thatstability properties are obtained as consequences of smallgain theorems. Typically, a system is composed by two sub-systems for which the output signals of one subsystem arethe input signals of the other subsystem. For such inter-connected systems, one would like to know how a stability

property of an interconnected system can be determinedby some gain conditions of the subsystems. Most of theclassical work on the small gain theory applies to the lin-ear (gain) case with a norm based approach. The work onsmall gain theorems for the nonlinear case began with thework [12], where a nonlinear generalization of the classicalsmall-gain theorem was proposed for both continuous- anddiscrete-time feedback systems.

In [14], the notion of input-to-state stability ( iss ) wasintroduced, which naturally encapsulates the nite gainconcept and the classic stability notions used in ordinarydifferential equations, enabling one to give explicit esti-

mates on transient behavior. In [6], a small gain theo-rem was given in the iss -framework, which led an extendedfollow-up literature. In conjunction with the results in [16],a much simplied proof of the small gain theorem was pro-vided in [1]. In [5], a small gain theorem was provided interms of iss -Lyapunov functions. In [2], the authors pre-sented an iss -type small gain theorem in terms of operatorsthat recovers the case of state space form, with applicationsto serveral variations of the iss property such as input-to-output stability, incremental stability and detectability forinterconnected systems. Recently, some iss -type small gain

Partially supported by NSF Grants INT9987317 and ECS

0093176. Partially supported by NSF Grant DMS0072620

results for discrete time systems was discussed in [7].

Since its formulation in [14], the iss notion has quicklybecome a foundational concept with wide applications innonlinear feedback analysis and design. It has becomean integrated part of several current textbooks and mono-graphs, including [4, 9, 10, 11, 13]. A variation of the issnotion, called input-to-output stability ( ios ), was also in-

troduced in [14]. In [17] and [18], several notions on input-to-output stability were studied. All the ios notions requirethat the outputs of a system be asymptotically bounded bya gain function on the inputs, yet the notions differ fromeach other on how uniform the transient behavors are interms of the initial values of state variables or the outputvariables. In this work, we will focus on small gain theoremsfor the so called output Lagrange ios property. In additionto requiring that the outputs of a system be asymptoti-cally bounded by a gain function on the inputs, the outputLagrange ios property also require the overshoot be propor-tional to the size of the initial values of the outputs (insteadof the size of the intial values of the states).

Our main contribution will be to provide small gaintheorems on the output Lagrange ios property for bothcontinuous- and discrete-time systems. In the special casewhen the output variables of a system represent the full setof state variables, the output Lagrange ios property be-comes the iss property. Thus, our results recover the iss -type small gain theorems for both continuous- and discrete-time systems.

In the past work of the iss -type smal gain theory, mostof the results were obtained either for the iss case, or forthe ios case but subject to an extra input/output-to-statestability condition (which roughly means that the state trajectories are bounded as long as the input and output arebounded. Clearly, this property is related to detectabil-ity). In the current work, we deal with the output La-grange ios case, and we show that for this case, small gaintheorems can be developed without the above-mentionedinput/output-to-state stability condition. Since the mainideas of the proofs, which are motivated by the approachesin [1], are the same for both continuous- and discrete-timesystems, we present our results for both of the cases.

The paper is organized as follows. In Section II, we brieydiscuss the output Lagrange ios property. In Section III,we present our main results an output Lagrange ios smallgain theorem for both the continuous- and discrete-time


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cases. In Section IV, we recover some iss -small results fordiscrete-time systems. In Section V, we provid proofs of the main results presented in Section III.

II. Preliminaries

We consider, for the continuous time case, systems as inthe following:

x (t) = f (x (t), u (t )) , y(t) = h (x (t)) , (1)

where the map f : R n R m R n is locally Lipschitzcontinuous, and the output map h : R n R p is continuous.We also assume that f (0, 0) = 0 and h (0) = 0.

The function u : [0, ) R m represents an input func-tion, which is assumed to be measurable and locally es-sentially bounded. For a given input function u () and aninitial state , we use x (, , u ) to denote the maximal so-lution (dened on some maximal interval) of (1) with theinput u such that x (0, , u ) = .

For the discrete time case, we consider systems of thefollowing form:

x (k + 1) = f (x (k), u (k)) , y(k) = h (x (k)) , (2)

where the maps f : R n R m R n and h : R n R p arecontinuous, and where f (0, 0) = 0 and h (0) = 0.

In the discrete time case, an input u : Z + R m is a mapfrom the set of nonnegative integers Z + to R m . Slightlyabsuing the notations, we again use x (, , u ) to denote thesolution of (2) with the input function u and the initialstate .

In both the continuous case and the discrete case, we usey(t , ,u ) to denote h (x (t , ,u )).

We will use the symbol || for Euclidean norms in R n ,R m , and R p , and to denote the L norms of a functiondened on either [0 , ) or Z + , taking values in R n , R m ,and R p .

Below we briey discuss some input-to-output stabilityproperties. For detailed discussions on the properties, see[17] and [18].

Denition 1 A forward-complete continuous system asin (1) is:

input to output stable (ios ) if there exist a KL-function and a K-function such that

|y(t , ,u )| ( | | , t ) + ( u ) (3)

for all t 0;

output-Lagrange input to output stable (ol-ios ) if it isios and there exist some K-functions 1 , 2 such that

|y(t , ,u )| max {1(|h () |), 2( u )} (4)

for all t 0.

A discrete time system as in (2) is:

ios if there exist a KL-function and a K-function such that (3) holds for all t Z + ;

ol-ios if it is ios and there exist some K-functions1 , 2 such that (4) holds for all t Z + .

Remark 2 It was shown in [17] that a system (1) is ol-iosif and only if there exist some KL , some K andsome K such that

|y(t , ,u )| |h ()| ,t

1 + ( | |)+ ( u )

for all t R 0 in the continuous-time case. This propertynicely encapsulates both the ios and the output-Lagrangeaspects of the ol-ios notion. It is not hard to see that theproperty also applies to discrete time systems (2) for allt Z + . 2

The ol-ios property can also be characterized in termsof output Lagrange stability property and the asymptoticgains.

Lemma 3 A forward-complete continuous time system asin (1) is ol-ios if and only if the following holds:

the output-Lagrange stability property holds, that is,for some 1 , 2 K , (4) holds for all t R 0 ; and

the asymptotic gain property holds: for some K ,

limt

|y(t , ,u ) | ( u ). (5)

A discrete time system as in (2) is ol-ios if and only if the following holds:

for some 1 , 2 K , (4) holds for all t Z + ;

for some K ,

limk

|y(k , ,u )| ( u ). (6)

Remark 4 Combining Remark 2 with a similar argumentas in the proofs of the corresponding results for iss (see [15]for the continuous case and [7] for the discrete case), onecan show that if a system (1) (or (2) respectively) admitsthe ol-ios estimates as in (3) and (4), then the asymptoticgain function in (5) (or (6) respectively) can be chosen tobe . 2

Remark 5 It is not hard to see that it results in an equiv-alent property if one strengthens (5) or (6) to:

limt

|y(t , ,u )| limt

(|u (t)|).

For the proof of Lemma 3, we refer to Theorem 1 of [3]for the continuous time case, where the proof was based ona theorem about approximations of relaxed trajectories byregular trajectories. The proof of the discrete time case israther straightforward: as the limit function of a sequenceof trajectories of (12) is also a trajectory of (12) (see theproofs of Theorem 1 of [7] and Proposition 3.2 of [8]), whilein the continuous time case, the limit of a sequence of trajec-tories of (1) is in general a trajectory of the relaxed systemof (1).


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III. Main Results

In this section, we present an ol-ios small gain theoremfor both continuous time and discrete time systems.

A. A Small Gain Theorem the Continuous Case

Consider a interconnected system as in the following:

x 1(t) = f 1(x 1(t), v1(t), u 1(t ))y1(t) = h 1(x 1(t )) ,x 2(t) = f 2(x 2(t), v2(t), u 2(t ))y2(t) = h 1(x 2(t )) ,

(7)

subject to the interconnection constraints

v1(t) = y2(t), v2(t ) = y1(t), (8)

where for i = 1 , 2, x i (t) R n i , u i (t) R m i , and yi (t) R p i ,and where f i is continuously differentiable, h i is continuous.

Theorem 1 Suppose both of the subsystems in (7) arecomplete and are ol-ios , and thus, there exist some iKL and some K-functions yi , ui , oi , ( i = 1 , 2), such that the following holds for all t R 0 :

|y1(t)| max 1( |x 1(0) | , t ),

y1 ( v1 ), u1 ( u 1 ) ,

|y2(t)| max 2( |x 2(0) | , t ),

y2 ( v2 ), u2 ( u 2 ) ;

(9)

and

|y1(t) | max {o1 (|h 1(1)|),

y1 ( v1 ),

u1 ( u 1 )},

|y2(t) | max { o1 (|h 2(2)|), y2 ( v2 ), u2 ( u 2 )},

(10)where we have used yi (t) to denote yi (t, i , vi , u i ). If thesmall gain condition

y1 y2 (s ) < s s > 0, (11)

then the interconnected system (7)-(8) is ol-ios with (y1 , y2) as the output and (u 1 , u 2) as the input.

B. A Small Gain Theorem the Discrete Case

In this section, we present a discrete analogue of the smallgain theorem for systems as in the following:

x 1(k + 1) = f 1(x 1(k), v1(k), u 1(k))y1(k) = h 1(x 1(k)) ,

x 2(k + 1) = f 2(x 2(k), v2(k), u 2(k))y2(k) = h 2(x 2(k)) ,

(12)

subject to the interconnection constraints

v1(k) = y2(k) , v2(k) = y1(k) (13)

where for i = 1 , 2 and for each k Z + , x i (k) R n i , u i (k)R m i , vi (k) R p i , and where f i and h i are continuous maps.

Theorem 2 Assume that both of the subsystems of (12)are ol-ios , and thus, there exist some KL -functions KL and some K-functions oi ,

yi and

ui , ( i = 1 , 2), such

that (9)(10) hold for all t Z + for any trajectory with yi (t) = yi (t, i , v i , u i ) ( i = 1 , 2). If the small gain condition (11) holds, then the interconnected system (12)-(13) is ol-ios with (y1 , y2) as the output and (u 1 , u 2) as the input.

IV. The Case of Discrete ISS

Consider a discrete time system as in (2). When h () = ,the ol-ios property becomes the input-to-state stability( iss ) property. The iss property for discrete time systemswas studied in detail in our previous work [7], where we havepresented two iss -small gain theorems (c.f. Theorems 2 and3 of [7]) for interconnected systems. Although Theorems 2and 3 of [7] are correct, their proofs carried in [7] wereawed with an incomplete statement of Lemma 4.1 of [7].One of the motivations of the current work is to x the aw

appeared in [7], and to show that, indeed, Theorems 2 and3 of [7] can be seen as corollaries of Theorem 2 presented inthe previous section. Observe that, when applying Theo-rem 2 to the case when h 1(1) = 1 , h 2(2) = 2 , the outputLagrange stability property (10) becomes redundant, as itis a direct consequence of (9).

Theorem 3 Consider an interconnected system

x 1(k + 1) = f 1(x 1(k), v1(k), u 1(k)) ,x 2(k + 1) = f 2(x 2(k), v2(k), u 2(k)) ,

(14)

subject to the interconnection constraintsv1(k) = x 2(k), v2(k) = x 1(k). (15)

Suppose both the subsystems in (14) are iss in the sensethat

|x 1(k , ,v 1 , u 1)| max 1(|1 | , k ), x1 ( v1 ), u1 ( u 1 ) ,

|x 2(k , ,v 2 , u 2)| max 2(|2 | , k ), x2 ( v2 ), u1 ( u 2 ) .

If x1 x2 (s ) < s (equivalently, x2 x1 (s ) < s ) for all s > 0,then the interconnected system (14)-(15) is iss with (u 1 , u 2)as input.

Invoking Theorem 2, one can also derive a Lyapunovsmall gain theorem for interconnected systems. Assumethat both of the subsystems in (14) are iss . By Theorem1 of [7], both of the subsystems admit iss -Lyapunov func-tions. That is, for each i = 1 , 2, there exists a continuousmap V : R n i R 0 such that

for i = 1 , 2, there exist i , i K ,

i (|i |) V i (i ) i (|i |) ; (16)


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there exist i K and xi , ui such that

V 1(f 1(1 , 2 , 1)) V 1(1) 1(V 1(1))

+ max x1 (V 2(2)) , u1 (|1 |) ,

V 2(f 2(2 , 1 , 2)) V 2(2) 2(V 2(2))

+ max x2 (V 1(1)) , u2 (|1 |) .

(17)

Without loss of generality, we assume that Id i K fori = 1 , 2 (c.f. Lemma B.1 of [7]).

Theorem 4 Assume that x 1- and x 2-subsystems of (14)admit iss -Lyapunov functions V 1 and V 2 respectively that satisfy (16)-(17), with Id i K for i = 1 , 2. If thereexists a K -function such that

11 x1

12

x2 < Id , (18)

then the interconnected system (14)-(15) is iss with (u 1 , u 2)

as the input.

Proof: For i = 1 , 2, Let xi = 1i xi , ui =

1i ui . By

Remark 3.7 of [7], one sees that

V 1(x 1(k)) max {V 1(x 1(0)) , x1 ( V 2(x 2) ), u1 ( u 1 )},

V 2(x 2(k)) max {V 2(x 2(0)) , x2 ( V 1(x 1) ), u2 ( u 2 )}.

To get the asymptotic gain conditions for V 1(x 1(k)) andV 2(x 2(k)), we apply Lemma 3.13 of [7] to get

limk

V 1(x 1(k)) max x1 (lim supk

V 2(x 2(k))) , u1 ( u 1 )

limk

V 2(x 1(k)) max x2 (limsupk

V 1(x 1(k))) , u2 ( u 2 ) .

Consequently,

limk

V 1(x 1(k)) max x1 x2 (limsup

k V 1(x 2(k))) ,

x1 u2 ( u 2 ),

u1 ( u 1 ) ,

limk

V 2(

x2(

k)) max

x1

x2 (limsupk

V 2(

x1(

k)))

,

x2 u1 ( u 1 ),

u1 ( u 2 ) .

With the small gain condition x1 x2 < Id, we concludethat

limk

V 1(x 1(k)) max { x1 u2 ( u 2 ),

u1 ( u 1 )} ,

limk

V 2(x 1(k)) max { x2 u1 ( u 1 ),

u2 ( u 2 )} .

By Theorem 2, the interconnected system (14)-(15) is ol-ios with ( V 1(x 1), V 2(x 2)) as output and ( u 1 , u 2) as input.By property (16), one sees that the system is iss .

V. Proofs of the Small Gain Theorems

The main idea of the proof in the continuous time caseand the main idea in the proof in the discrete time caseare the same. Both proofs are based on Lemma 3. Sincemuch work has been done for continuous time samll gaintheorems, and not so much has been done for the discretecounter part, we provide here a proof for the discrete time

case.Proof of Theorem 2. Suppose for some KL-function i andsome K-function oi ,

yi ,

ui (i = 1 , 2), (9)(10) hold for all

t Z + along any trajectory with yi (t) = yi (t, i , v i , u i ).Assume the small gain condition (11) holds.

To show that the interconnected system (12)(13) sat-ises the output Lagrange stability property, we considera trajectory x (k) = ( x 1(k), x 2(k)) and the correspondingoutput function y(k) = ( y1(k), y2(k)). By causality, for anyk 1,

|y1(t) | max o1 ( |y1(0) | , y1 y2 [k 1] ,

u1 ( u 1 ) ,

|y2(t) | max o2 ( |y2(0) | , y2 y1 [k 1] ,

u2 ( u 2 ) ,

where we have let w [l ] = max 0 j l |w( j )| for any functionw dened on Z + . Therefore,

|y1(t) | max o1 ( |y1(0) | , y1

o2 ( |y2(0) |),

y1 y2 y1 [k 1] ,

y1

u2 ( u 2 ),

u1 ( u 1 ) ,

|y2(t) | max o2 ( |y2(0) | , y2

o1 ( |y1(0) |),

y2

y1

y2[k 1]

, y2

u1

( u 1 ), u2

( u 2 )

for all t Z + . Taking the maximal values for t = 0 , 1, . . . k ,on both sides of the above inequality, we get:

y1 [k ] o1 (|y1(0) |),

y1

o2 (|y2(0) |),

u1 ( u 1 )

y1 y2 y1 [k 1] ,

y1

u2 ( u 2 ), ,

y2 [k ] max o2 ( |y2(0) | ,

y2

o1 ( |y1(0) |),

u2 ( u 2 )

y2 y1 y2 [k 1] ,

y2

u1 ( u 1 ), .

With the small condition (11), we get

|y1(k)| max o1 ( |y1(0) | , y1

o2 ( |y2(0) |),

y1 u2 ( u 2 ),

u1 ( u 1 )

|y2(k)| max o2 ( |y2(0) | , y2

o1 ( |y1(0) |),

y2 u1 ( u 1 ),

u2 ( u 2 ) .

Hence, one sees that the interconnected system (12)(13)satises the following output Lagrange stability estimates:

|y1(k) | max 1(|y(0) |), 1( u ) ,


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|y2(k)| max 2(|y(0) |), 2( u ) ,

for some K-functions 1 , 2 , 1 and 2 .Next we show that the interconnected system admits an

asymptotic gain. We have shown that for any given trajec-tory, there exists some c 0 (depending on |y(0) | and u )such that

|y1(k)| c and |y2(k)| c kZ

+ .

Hence, both limk

|h 1(x 1(k)) | and limk |h 2(x 2(k)) | are nite.Applying Lemma 3 together with Remarks 4 and 5 to getthe following:

limk

|y1(k)| max y1 limk |y2(k)| , u1 limk |u 1(k) | ,

limk

|y2(k)| max y2 limk |y1(k)| , u2 limk |u 2(k) | ,

Consequently,

limk

|y1(k) | max y1 y2 limk |y1(k)| ,

y1 u2 limk |u 2(k) | ,

u1 limk |u 1(k)| ,

limk

|y2(k) | max y2 y1 limk |y2(k)| ,

y2 u1 limk |u 1(k) | ,

u2 limk |u 2(k)| .

Again, with the small gain condition (11), we get the fol-lowing asymptotic gain estimates:

limk

|y1(k) | 1( u ),

limk |y2(k)| 2( u )

for some 1 , 2 K .We have proved that the interconnected system (12)(13)

satises both the output Lagrange stability property andthe asymptotic gain property. By Lemma 3, the system isol-ios .

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Small Gain Theorems on Input-To-Output Stability