Elsevier

Systems & Control Letters

Volume 38, Issues 4–5, 10 December 1999, Pages 235-248
Systems & Control Letters

Notions of input to output stability

https://doi.org/10.1016/S0167-6911(99)00070-5Get rights and content

Abstract

This paper deals with concepts of output stability. Inspired in part by regulator theory, several variants are considered, which differ from each other in the requirements imposed upon transient behavior. The main results provide a comparison among the various notions, all of which specialize to input to state stability (ISS) when the output equals the complete state.

Introduction

This paper addresses questions of output stability for general finite-dimensional control systemsẋ(t)=f(x(t),u(t)),y(t)=h(x(t)).(Technical assumptions on f, h, and admissible inputs, are described later.) Roughly, a system (1) is “output stable” if, for any initial state, the output y(t) converges to zero as t→∞. Inputs u may influence this stability in different ways; for instance, one may ask that y(t)→0 only for those inputs for which u(t)→0, or just that y remains bounded whenever u is bounded. Such behavior is of central interest in control theory. As an illustration, we will review below how regulation problems can be cast in these terms, letting y(t) represent a quantity such as a tracking error. Another motivation for studying output stability arises in classical differential equations: “partial” asymptotic stability (cf. [26]) is nothing but the particular case of our study in which there are no inputs u and the coordinates of y are a subset of the coordinates of x (that is to say, h is a projection on a subspace of the state space Rn). The notion of output stability is also related to that of “stability with respect to two measures”, cf. [9].

The main starting point for our work is the observation that there are many different ways of making mathematically precise what one means by “y(t)→0 for every initial state” (and, when there are inputs, “provided that u(t)→0”). These different definitions need not result in equivalent notions; one must decide how uniform is the rate of convergence of y(t) to zero, and precisely how the magnitudes of inputs and initial states affect convergence.

Indeed, our previous work on input-to-state stability (ISS, for short) was motivated in much the same way. The concept of ISS was originally introduced in [16] to address the problem when y=x. Major theoretical results were developed in [19], [21] and applications to control design can be found in, among others, [4], [6], [7], [8], [12], [15], [21], [25] as well as in the recent work [13] as a foundation for the formulation of robust tracking.

Actually, in the original paper [16] we had already introduced a notion of input/output stability (IOS), but the theoretical effort until now was almost exclusively directed towards the ISS special case. (There are two ways to formulate the property of input/output stability and its variants. One is in purely input/output terms, where one uses past inputs in order to represent initial conditions. Another is in state space terms, where the effect of past inputs is summarized by an initial state. In [16] an i/o approach was used, but here, because of our interest in initial-state dependence, we adopt the latter point of view. The relations between both approaches are explained in [16] and in more detail in [10], [5].)

It turns out that the IOS case is substantially more complicated than ISS, in the sense that there are subtle possible differences in definitions. One of the main objectives of this paper is to elaborate on these differences and to compare the various definitions; the companion paper [24] provides Lyapunov-theoretic characterizations of each of them.

A second objective is to prove a theorem on output redefinition which (a) extends one of the main steps in linear regulation theory to general nonlinear systems, and (b) provides one of the main technical tools needed for the construction of Lyapunov functions in [24].

The organization of this paper is as follows. Section 1.1 starts with the review of certain facts from regulation theory; this material is provided merely as an additional motivation for our study, and is not required in order to follow the paper. Because of the technical character of the paper, it seems appropriate to provide an intuitive overview; thus, the rest of that section describes the main results in very informal terms. After that, in Section 2, we define our notions carefully and state precisely the main results. The rest of the paper contains the proofs. A preliminary version of this paper appeared in [22].

Output regulation problems encompass the main typical control objectives, namely, the analysis of feedback systems with the following property: for each exogenous signal d(·) (which might represent a disturbance to be rejected, or a signal to be tracked), the output y(·) (respectively, a quantity being stabilized, or the difference between a certain variable in the system and its desired target value) must decay to zero as t→∞. Typically (see e.g. [18, Section 8.2], or [14, Chapter 15] for linear systems, and [3, Chapter 8], for nonlinear generalizations), the exogenous signal is unknown but is constrained to lie in a certain prescribed class (for example, the class of all constant signals). Moreover, this class can be characterized through an “exosystem” given by differential equations (for example, the constant signals are precisely the possible solutions of ḋ=0, for different initial conditions).

In order to focus on the questions of interest for this paper, we assume that we already have a closed-loop system exhibiting the desired regulation properties, ignoring the question of how an appropriate feedback system has been designed. Moreover, let us, for this introduction, restrict ourselves to linear time-invariant systems (local aspects of the theory can be generalized to certain nonlinear situations employing tools from center manifold theory, see [3]). The object of the study becomesż=Az+Pw,ẇ=Sw,y=Cz+Qw,seen as a system ẋ=f(x), y=h(x), where the extended state x consists of z and w; the z-subsystem incorporates both the state of the system being regulated (the plant) and the state of the controller, and the equation ẇ=Sw describes the exosystem that generates the disturbance or tracking signals of interest. This is a system without inputs; later we explain how inputs may be introduced into the model as well.

As an illustration, take the stabilization of the position y of a second-order system ÿ−y=u+w under the action of all possible constant disturbances w. The conventional proportional-integral-derivative (PID) controller uses a feedback law u(t)=c1q(t)+c2y(t)+c3v(t), for appropriate gains c1,c2,c3, where q=∫y and v=ẏ. Let us take c1=−1, c2=c3=−2. If we view the disturbances as produced by the “exosystem” ẇ=0, the complete system becomesq̇=y,ẏ=v,v̇=−q−y−2v+w,ẇ=0with output y. That is, the plant/controller state z is col(q,y,v), andS=Q=0,A=010001−1−1−2,C=(010),P=001.

In linear regulator theory, the routine way to verify that the regulation objective has been met is as follows. Suppose that the matrix A is Hurwitz and that there is some matrix Π such that the following two identities (“Francis’ equations”) are satisfied:ΠS=AΠ+P,0=CΠ+Q.(The existence of Π is necessary as well as sufficient for regulation, provided that the problem is appropriately posed, cf. [2], [3].) Consider the new variable ŷ≔z−Πw. The first identity for Π allows decoupling ŷ from w, leading to ŷ̇=Aŷ. Since A is a Hurwitz matrix, one concludes that ŷ(t)→0 for all initial conditions. As the second identity for Π gives that y(t)=Cŷ(t), one has the desired conclusion that y(t)→0.

Let us now express this convergence in a much more informative form. For that purpose, we introduce the map ĥ:x=(z,w)↦|z−Πw|=|ŷ|. We also denote, for ease of future reference, χ(r)≔r/|C| and β(r,t)=r|etA|/|C|, using |·| to denote Euclidean norm of vectors and also the corresponding induced matrix norm. So, y=Cŷ givesχ(|y|)=χ(|h(x)|)⩽ĥ(x)=|ŷ|for all x, and we also have |y(t)|⩽β(|ŷ(0)|,t) and in particular|y(t)|⩽β(|x(0)|,t),∀t⩾0along all solutions. This estimate quantifies the rate of decrease of y to zero, and its overshoot, in terms of the initial state of the system. For the auxilliary variable ŷ, we have in addition the following “stability” property:|ŷ(t)|⩽σ(|ŷ(0)|),∀t⩾0where σ(r)≔rsupt⩾0|etA|.

The use of ŷ (or equivalently, finding a solution Π for the above matrix identities) is a key step in the analysis of regulation problems. Note the fundamental contrast between the behaviors of ŷ and y: because of (4), a zero initial value ŷ(0) implies ŷ≡0, which in regulation problems corresponds to the fact that the initial state of the “internal model” of the exosignal matches exactly the one for the exosignal; on the other hand, for the output y, typically an error signal, it may very well happen that y(0)=0 but y(t) is not identically zero. The fact about ŷ which allows deriving (3) is that ŷ dominates the original output y, in the sense of (2). One of the main results in this paper provides an extension to very general nonlinear systems of the technique of output redefinition.

To illustrate again with the PID example: one finds that Π=col(1,0,0) is the unique solution of the required equations, and the change of variables consists of replacing q by qw, the difference between the internal model of the disturbance and the disturbance itself, and ĥ(q,y,v,w)=(|q−w|2+|y|2+|v|2)1/2. For instance, with x(0)=y(0)=v(0)=0 and w(0)=1 we obtain the output y(t)=12t2e−t. Notice that this output has y(0)=0 but is not identically zero, which is consistent with an estimate (3). On the other hand, the dominating output ŷ=(q−w,y,v)) cannot exhibit such overshoot.

The discussion of regulation problems was for systems ẋ=f(x) which are subject to no external inputs. This was done in order to simplify the presentation and because classically one does not consider external inputs. In general, however, one should study the effect on the feedback system of perturbations which were not exactly represented by the exosystem model. A special case would be, for instance, that in which the exosignals are not exactly modeled as produced by an exosystem, but have the form w+u, where w is produced by an exosystem. Then one may ask if the feedback design is robust, in the sense that “small” u implies a “small” asymptotic (steady-state) error for y, or that u(t)→0 implies y(t)→0. Experience with the notion of ISS then suggests that one should replace (3) by an estimate as follows:(IOS)|y(t)|⩽β(|x(0)|,t)+γ(∥u∥).By this we mean that for some functions γ of class K and β of class KL which depend only on the system being studied, and for each initial state and control, such an estimate holds for the ensuing output. We suppose as a standing hypothesis that the system is forward-complete, that is to say, solutions exist (and are unique) for t⩾0, for any initial condition and any locally essentially bounded input u. (Recall that a function γ:[0,∞)→[0,∞) is of class K if it is strictly increasing and continuous, and satisfies γ(0)=0, and of class K if it is also unbounded, and that KL is the class of functions [0,∞)2→[0,∞) which are of class K on the first argument and decrease to zero on the second argument.) The reason that inputs u are not usually incorporated into the regulation problem statement is probably due to the fact that for linear systems it makes no difference: it is easy to see, from the variation of parameters formula, that IOS holds for all inputs if and only if it holds for the special case u≡0.

Property (4) generalizes when there are inputs to the following “output Lagrange stability” property:(OL)|y(t)|⩽σ1(|y(0)|)+σ2(∥u∥).As mentioned earlier when discussing the classical tools of regulation theory, one of our main results is this: if a system satisfies IOS, then we can always find another output, let us call it ŷ, which dominates y, in the sense of (2), and for which the estimate OL holds in addition to IOS.

As ŷ̇=Aŷ and A is a Hurwitz matrix, in the case of linear regulator theory the redefined output ŷ satisfies a stronger decay condition, which in the input case leads naturally to an estimate as follows:(SIIOS)|y(t)|⩽β(|y(0)|,t)+γ(∥u∥)(we write y instead of ŷ because we wish to define these notions for arbitrary systems). We will study this property as well. For linear systems, the conjunction of IOS and OL is equivalent to SIIOS. (Sketch of proof: with zero input and any initial state x such that Cx=0, OL gives us that CetAx≡0, which means that the kernel of C coincides with the unobservable subspace O(A,C). Therefore, using the Kalman observability decomposition and the notations in [18, Eq. (6.8)], now y can be identified to the first r coordinates of the state, which represent a stable system.) Remarkably, this equivalence breaks down for general nonlinear systems, as we will show.

Finally, as in the corresponding ISS paper [19], there are close relationships between output stability with respect to inputs, and robustness of stability under output feedback. This suggests the study of yet another property, which is obtained by a “small gain” argument from IOS: there must exist some χ∈K so that(ROS)|y(t)|⩽β(|x(0)|,t)if|u(t)|⩽χ(|y(t)|)∀t.For linear systems, this property is equivalent to IOS, because applied when u≡0 it coincides with IOS. One of our main contributions will be the construction of a counterexample to show that this equivalence also fails to generalize to nonlinear systems.

In summary, we will show that precisely these implications hold:SIIOSOL&IOSIOSROSand show that under output redefinition the two middle properties coincide.

We caution the reader not to confuse IOS with the notion named input/output to state stability (IOSS) in [23] (also called “detectability” in [17], [20], and “strong unboundedness observability” in [5]). This other notion roughly means that “no matter what the initial conditions are, if future inputs and outputs are small, the state must be eventually small”. It is not a notion of stability; for instance, the unstable system ẋ=x, y=x is IOSS. Rather, it represents a property of zero-state detectability. There is a fairly obvious connection between the various concepts introduced, however: a system is ISS if and only if it is both IOSS and IOS. This fact generalizes the linear systems theory result “internal stability is equivalent to detectability plus external stability” and its proof follows by routine arguments [16], [10], [5].

Section snippets

Definitions, statements of results

We assume, for the systems (1) being considered, that the maps f:Rn×RmRn and h:RnRp are locally Lipschitz continuous. We also assume that f(0,0)=0 and h(0)=0. We use the symbol |·| for Euclidean norms in Rn, Rm, and Rp.

By an input we mean a measurable and locally essentially bounded function u:IRm, where I is a subinterval of R which contains the origin. Whenever the domain I of an input u is not specified, it will be understood that I=R⩾0. The Lm-norm (possibly infinite) of an input u is

Proofs

We will first prove Theorem 2.1, and then we will prove Lemma 2.4.

Example

In this section we show, by means of a counterexample, that ROS and IOS are not equivalent. We will then modify the example to get a system which is in addition bounded-input bounded-output stable (UBIBS), thus showing that even under this very strong stability assumption, ROS does not imply IOS.

Consider the following two-dimensional system:ẋ=ρ(|u|−1−|y|)x−yσ(x,y),ẏ=ρ(|u|−1−|y|)y+xσ(x,y),with output y (we write x and y instead of x1 and x2), where ρ is defined byρ(s)=−1ifs<−1,sif|s|⩽1,1ifs>1,

References (26)

  • M. Krstić, H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer, London,...
  • M. Krstić, I. Kanellakopoulos, P.V. Kokotović, Nonlinear and Adaptive Control Design, Wiley, New York,...
  • V. Lakshmikantham, S. Leela, A.A. Martyuk, Practical Stability of Nonlinear Systems, World Scientific, River Edge, NJ,...
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