Approximate reduction of dynamic systems

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Abstract

The reduction of dynamic systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an “exact” manner–as is the case with mechanical systems with symmetry–which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e. when it is behaviourally similar to a dynamic system on a lower dimensional space. These concepts are illustrated on a series of examples.

Introduction

Modelling is an essential part of many engineering disciplines and often a key ingredient for successful designs. Although it is widely recognized that models are only approximate descriptions of reality, their value lies precisely on the ability to describe, within certain bounds, the modelled phenomena. In this paper we consider modeling of closed-loop nonlinear control systems, i.e. differential equations, with the purpose of simplifying the analysis of these systems. The goal of this paper is to reduce the dimensionality of the differential equations being analysed while providing hard bounds on the introduced errors. One promising application of these techniques is to the verification of hybrid systems, which is currently constrained by the complexity of high dimensional differential equations.

Reducing differential equations–and in particular mechanical systems–is a subject with a long and rich history. The first form of reduction was discovered by Routh in the 1860’s; over the years, geometrical reduction has become an academic field in itself. One begins with a differential equation with certain symmetries, i.e. a differential equation invariant under the action of a Lie group on the phase space. Using these symmetries, one can reduce the dimensionality of the phase space (by “dividing” out by the symmetry group) and define a corresponding differential equation on this reduced phase space. The main result of geometrical reduction is that one can understand the behaviour of the full-order system in terms of the behavior of the reduced system and vice versa [11], [16], [5]. While this form of “exact” reduction is very elegant, the class of systems for which this procedure can be applied is actually quite small. This indicates the need for a form of reduction that is applicable to a wider class of systems and, while not being exact, is “close enough”.

In systems theory, reduced order modelling has also been extensively studied under the name of model reduction [4], [3]. The typical problem addressed in this literature consists in approximating a system Σ1 by a system Σ2 while minimizing the L2 norm: (0|y1(t)y2(t)|2dt)12 where y1 is the output of Σ1 and y2 is the output of Σ2. This kind of reduction is not adequate when one is interested in applications to formal verification of hybrid systems. A typical safety verification problem consists in determining if any trajectory of Σ1 starting in a given set of initial conditions S enters a given set of unsafe states B. If one solves this verification problem with the reduced order model Σ2, then one cannot conclude, based on an upper bound on (1), if trajectories of Σ1 do enter the B. This motivates us to study reduction problems in which trajectories of Σ1 and its reduced model Σ2 are instead related by the L norm: supt[0,[|y1(t)y2(t)|. More recent work considered exact reduction of control systems [17], [15] based on the notion of bisimulation which was later generalized to approximate bisimulation [7], [13], [8].

We develop our results in the framework of incremental stability and our main result is in the spirit of existing stability results for cascade systems that proliferate the Input-to-State Stability (ISS) literature. See, for example, [12] and the references therein. A preliminary version of our results appeared in the conference paper [14].

Section snippets

Preliminaries

A continuous function γ:R0+R0+, is said to belong to class K if it is strictly increasing, γ(0)=0 and γ(r) as r. A continuous function β:R0+×R0+R0+ is said to belong to class KL  if, for each fixed s, the map β(r,s) belongs to class K with respect to r and, for each fixed r, the map β(r,s) is decreasing with respect to s and β(r,s)0 as s.

A function φ:RnRm is said to be smooth if it is infinitely differentiable. We denote by Tφ the tangent map to φ and by Txφ the tangent map to φ at x

Exact reduction

For some dynamic systems described by a vector field X on Rn it is possible to replace X by a vector field Y describing the dynamics of the system on a lower dimensional space, Rm, while retaining much of the information about X. When this is the case we say that X can be reduced to Y. This idea of (exact) reduction is captured by the notion of φ-related vector fields.

Definition 3

Let φ:RnRm be a smooth map. The vector field (Rn,X) is said to be φ-related to the vector field (Rm,Y) if for every xRn we

Approximate reduction

The generalization of Definition 3 proposed in this section requires a decomposition of Rn of the form Rn=Rm×Rk. Associated with this decomposition are the canonical projections πm:RnRm and πk:RnRk taking Rnx=(y,z)Rm×Rk to πm(x)=y and πk(x)=z, respectively. Intuitively, Rn corresponds to the state space of the full model and we will be interested in the evolution of only the part of the state described by y=πm(x), for which we will be seeking a reduced model.

Definition 4

The vector field (Rn,X) is said

Examples

In this section, we consider examples that illustrate the usefulness of approximate reduction.

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This research was partially supported by the National Science Foundation, EHS award 0712502.

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