18th Mediterranean Conference on Control & Automation Congress Palace Hotel, Marrakech, Morocco June 23-25, 2010

Gramian based approach to model order-reduction for discrete-time switched linear systems Abderazik Birouche, J´erˆome Guillet, Benjamin Mourllion, Michel Basset Laboratoire Mod´elisation, Intelligence, Processus et Syst`emes MIPS, EA 2332 / Equipe MIAM–UHA ENSISA, 12 rue des Fr`eres Lumi`ere, 68093 Mulhouse Cedex, France [email protected]

Abstract— This paper describes the problem of model orderreduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. The paper investigates a novel method to model reduction. The approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. The error limit depends on the truncated singular values and balanced truncation procedure helps to keep the biggest singular values of the system. A numerical example shows the effectiveness of the proposed approach.

I. INTRODUCTION Realistic systems are often complex and characterized by models of a very high order. These models are usually difficult to study in a context of analysis. The interest of model order-reduction is also that high order models generate high order controllers/observers leading to numerous complexities and difficulties in synthesis, simulation and implementation. Therefore, it is necessary to reduce the order of these systems. The problem of model order-reduction has become the focus of different research areas (mathematics, mechanics and computer engineering). Several studies have been devoted to this issue following the class of system considered. State-of-the-art methods for reducing the order of linear models can be found in [1]. One important contribution to model reduction is the balanced truncation method [13]. In this approach, each state is equally controllable and observable and the reduced order model is obtained by truncating the least controllable and observable states. The balanced truncation is a common method to reduce the order of Linear Time Invariant (LTI) systems thanks to its simplicity of implementation; the stability of the reduced system is guaranteed and the approximation error is bounded. Another important solution to model reduction problems is the optimization approach which minimizes the H∞ (or H2 ) norm between the original full order model and the reduced order model [11], [18]. Recently, the Linear Matrix Inequalities (LMI) [3] technique has been used to solve model reduction problems for different linear systems, including state-space systems [7]. Balanced realization for Linear Parameter Varying (LPV) systems have also been considered, for example [15], [4], [14] and [10] where an

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explicit error limit for discrete-time balanced truncation is given. Switched systems are a particular class of hybrid systems which can be considered as the result of the interaction between a finite state automaton and a finite set of LTI subsystems. It consists of a set of linear subsystems with a switching rule which specifies which subsystem will be activated each time. The switching system is defined by its mixed state vector (x, q). The first one represents the classic state vector x ∈ Rn and the second one finds its values in a finite set q ∈ Q = {1, 2, · · · , s} and represents the subsystem indexes. The study of the properties of hybrid systems in general and switched systems in particular is still the subject of intense research, including the problems of stability [12], observability/controllability [2] and synthesis [9], [5], but the problem of model reduction for this class has received little attention. Recently, research has been dedicated to the problem of order reduction for switched systems; all the results are based on the analysis of performance and stability of error approximation. The problem of model orderreduction is expressed in terms of stability performance H∞ in [8], [16], [19], [17] and stability performance H2 in [6]. The previous papers are concerned with the evaluation of the quality of the reduction model only formulated as minimizing the approximation error between outputs of the original system and outputs of the reduced-order system. This paper is motivated by the use of other criteria such as controllability and observability, Hankel singular values of the system. The main contribution of this paper is a novel solution to model order-reduction problem for a class of discrete-time switched linear systems. Specifically, the focus is on the problem of model reduction and balanced truncation approach. This paper is organized as follows. The system class considered is presented in Section II. Section III, presents a novel approach to the balanced truncation for switched systems. The key idea is to use the energy matrix functions (Gramians) of the switched system. The error limit depends on the truncated singular values and balanced truncation procedure helps to keep the biggest singular values of the system. An example given in Section IV illustrates the effectiveness of the proposed approach.

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II. PROBLEM STATEMENT

A. Reachability and observability Gramians

Consider the class of discrete-time Switched Linear Systems (SLS) given by:  s s P P   αi (k)Ai xk + αi (k)Bi uk  xk+1 = i=1 i=1 ΣSLS : s s P P   = αi (k)Ci xk + αi (k)Di uk  yk i=1

Considers a finite interval time [ti , tf ], the next definitions give the notions of reachability and observability Gramians for LTI systems. Definition 1: The finite reachability Gramian at time k are defined for discrete-time LTI systems as

(1) where xk ∈ Rn is the state vector, yk ∈ Rp is the output vector and uk ∈ Rm is the input vector. k refers to the sample index. The switching signal vector . α(k) = [α1 (k), α2 (k), · · · , αs (k)]

specifies which subsystem will be activated at the discrete s P time k, when αi (k) ∈ {0, 1} and αi (k) = 1. For

Wc (k) =

III. GRAMIAN-BASED APPROACH TO MODEL REDUCTION FOR SWITCHED LINEAR SYSTEMS The reachability and observability Gramians play an important role in the traditional balanced truncation approach to model reduction. This section considers the Gramians for the switched system (1). It is well known that the reachability and observability Gramians for the LTI systems are given as a solution of the Lyapunov equalities. For switched systems, the reachability and observability Gramians are approximated by a set of Lyapunov inequalities.

(3)

The system is called reachable if and only if Wc (k) is positive-definite for all k and their columns span the reachability subspace. The associated infinite reachability Gramian Wc (with ti = −∞) satisfies the discrete-time reachability Lyapunov equation AWc AT − Wc + BB T = 0

(4)

The system is also called reachable if Wc > 0 and their columns span the reachability subspace. Definition 2: The finite observability Gramian at time k are defined for discrete-time LTI systems as tf X

Wo (k) =

(Al )T C T CAl

(5)

l=k

The system is called observable if and only if Wo (k, tf ) is positive-definite for all k and their columns span the observability subspace. The associated infinite observability Gramian Wo (with tf = +∞) satisfies the discrete-time Lyapunov equation AT Wo A − Wo + C T C = 0

i=1

(2) ˆi , Cˆi , D ˆ i ) are where dim(ˆ xk ) < dim(xk ) and (Aˆi , B matrices of appropriate dimensions to be determined such ˆ SLS approximates ΣSLS . The evaluation of the quality that Σ of the reduced model requires criteria such as eigenvalues, controllability and observability, Hankel singular values and in particular the minimizing of the error (y− yˆ). In [19], [17], the solution is expressed using the classical stability performance of the error between the original full-order model and the reduced-order model. In this paper, the balanced truncation problem for switched systems is considered. For this purpose, the next sections propose a novel approach based on the reachability and observability Gramians of the switched system.

Al BB T (Al )T

l=ti

i=1

example, when αi (t0 ) = 1, the active mode index at time t0 is qk = i which finds its values in the finite set i ∈ Q = {1, 2, · · · , s}, s is the number of subsystems. It means that the i-th subsystem (Ai , Bi , Ci , Di ) characterizes the dynamics of the system at instant t0 . The switching strategy is unknown a priori but available in real time (or estimated). Therefore, all results presented in this paper consider the system with an arbitrary switching rule and not with an average dwell time assumption. Here, the system ΣSLS in (1) is approximated with a ˆ SLS of the form reduced-order Σ  s s P P  ˆ i uk  αi (k)B αi (k)Aˆi x ˆk + ˆk+1 =  x i=1 i=1 ˆ ΣSLS : s s P P  ˆ i uk  = αi (k)Cˆi x ˆk + αi (k)D  yˆk i=1

k−1 X

i=1

(6)

The system is also called observable if Wo > 0 and their columns span the observability subspace. The energy associated with reaching/observing a state are defined respectively as kuk22 =

−1 X

k=−∞ +∞ X

kyk22 =

uTk uk = xT0 Wc−1 x0

(7a)

ykT yk = xT0 Wo x0

(7b)

k=0

The first is the input energy to reach a state x(0) = x0 from x(−∞) = 0. The second is the output energy for a given initial state x0 . These ideas will help to calculate the Gramians for the switched systems of the form (1). According to the two previous definitions, the Gramians are defined in a finite interval [ti , tf ] as:

P (k) =

k−1 X

l=ti

"

Φ[k,l]

s X

αi (l)Bi BiT

i=1

for the reachability Gramian and

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!

T

(Φ[k,l] )

#

(8)

tf

Q(k) =

X l=k

"

(Φ[l,k] )T

s X

αi (l)CiT Ci

i=1

!

Φ[l,k]

#

2) Given an initial state x0 , examine the output energy ∞ P ykT yk . From (I), ykT yk < −∆V (xk , k) there-

(9)

k=0

fore

k=0

for the observability Gramian. The discrete transition matrix over time [t1 , t2 ] is defined as ! t1 −1 s . Y X Φ[t2 ,t1 ] = αi (k)Ai k=t2

∞ P

k=0

xT0 Wo x0

So, these finite Gramians are also obtained from the following recurrence formulas: P (k + 1) =

+

Q(k) =

αi (k)ATi Q(k + 1)Ai +

s X

Wo
0 the infinite observability Gramian s X

s X

x0

Lemma 2: Given the switched linear system (1) with initial state x0 , let the semi-definite matrices Pi for i ∈ Q be the solution of this LMI:

for (i, j) ∈ Q × Q then: 1) The system is asymptotically stable. 2) The output energy is bounded : s X

αi (k)Qi

!

i=1

The infinite Gramians are obtained with ti = −∞ in (8) and tf = +∞ in (9). Given the complexity of the system, at every time step k, Q(k), P (k) must be calculated, and it seems difficult to find a method to calculate the infinite Gramians. The idea is to avoid this complex calculation by replacing these Gramians by set a of semi-definite matrices. Lemma 1: Given the switched linear system (1) with initial state x0 and without input uk = 0 and let the semidefinite matrices Qi for i ∈ Q be the solution of this LMI:

kyk22 < xT0