Realization theory of discrete-time linear switched systems

doi:10.1016/j.automatica.2013.07.022

Automatica

Volume 49, Issue 11, November 2013, Pages 3337-3344

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Abstract

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.

Introduction

In this paper we develop realization theory of discrete-time linear switched systems (abbreviated by DTLSS).

Realization theory. For DTLSSs, the goal of realization theory is to answer the following questions.

•
When does there exist a (preferably minimal) DTLSS state-space representation of an input/output behavior ?
•
How to characterize minimal DTLSSs which generate an input/output behavior ?

Motivation. Realization theory has already been addressed for continuous-time linear switched systems, Petreczky, 2007, Petreczky, 2011. The motivation for devoting a separate paper to realization theory of discrete-time linear switched systems is as follows.

1.
Realization theory for DTLSSs is sufficiently different from the continuous-time case.
2.
Formulating realization theory explicitly for DTLSSs will be useful for identification of these systems. For example, as it was pointed out by one of the anonymous referees, there are several methods for identification of a linear switched system in input–output form. The realization algorithm proposed in this paper then allows us to compute an equivalent minimal state-space representation. Subsequently, one can apply the available tools for analysis and control synthesis of state-space representations. In fact, the results of this paper were already used in Petreczky and Bako (2011) and Petreczky, Bako, and van Schuppen (2010) for analyzing identifiability and persistence of excitation of DTLSSs.

The main difference between realization theory of DTLSSs and that of continuous-time linear switched systems lies in the definition of Markov-parameters. In the continuous-time case, the Markov-parameters are defined as certain high-order derivatives of the input–output map. In contrast to the continuous-time case, in the discrete-time case the Markov-parameters are defined directly in terms of input–output data. The latter definition can directly be used for system identification. Once the Markov-parameters are defined, the discrete-time case becomes analogous to the continuous-time one.

Contribution of the paper. In this paper, input–output behavior will mean an input–output map, which maps sequences of continuous inputs and discrete modes to continuous outputs. Hence, the discrete modes are viewed as inputs. In addition, we will deal with DTLSSs whose initial state is fixed. Thus, there is one input–output map which corresponds to a DTLSS: this is the input–output map which maps sequences of inputs and discrete states to the corresponding outputs generated by DTLSS, when the system is started from that fixed initial state. Note that this is not the only possible formalization of the concept of input–output behavior: it could also be formalized as a set of input–output trajectories (see the behavioral approach Rocha et al., 2011, Tóth et al., 2011, Trenn and Willems, 2012, Willems and Polderman, 1998) or a family of input–output maps. The latter is obtained if instead of fixing one initial state, we allow several (possibly infinite) initial states. At this point it is not clear if the results of the current paper can be adapted to the behavioral setting. Note that Rocha et al. (2011) and Trenn and Willems (2012) assumes that the switching sequence is fixed, while in this paper we view it as an input. The approach of Tóth et al. (2011) is closer to our setting, but it is formulated for LPV systems. We will explain the relationship between the two when discussing related work. We conjecture that the results of the paper could be extended to families of input–output maps. For continuous-time linear switched systems, such an extension was presented in Petreczky, 2007, Petreczky, 2011.

We prove that a DTLSS is minimal if and only if it is span-reachable and observable. We show that minimal realizations of the same input–output map are isomorphic. We also show that any DTLSS can be transformed to a minimal one while preserving its input–output map. In addition, we formulate the concept of Markov-parameters and Hankel-matrix. We show that an input–output map can be realized by a DTLSS if and only if the Hankel-matrix is of finite rank. We also present a realization algorithm, for constructing a DTLSS state-space representation from the Hankel-matrix. Our main tool is the theory of rational formal power series (Berstel and Reutenauer, 1984, Fliess, 1973, Kuich and Salomaa, 1986, Sontag, 1979).

Related work. To the best of our knowledge, the results of this paper are new. The results on minimality were announced in Petreczky et al. (2010) but no detailed proof was provided. Algorithm 1 and Theorem 4 were announced in Petreczky and Bako (2011), but again no proof was provided there. A preliminary version of the paper was made available as a technical report (Petreczky, Bako, & van Schuppen, 2011).

The realization problem for hybrid systems was first formulated in Grossman and Larson (1995). In Paoletti, Roll, Garulli and Vicino (2010) and Weiland, Juloski, and Vet (2006) the relationship between input–output equations and the state-space representations was studied. In Petreczky, 2007, Petreczky, 2011 realization theory for continuous-time (bi)linear switched systems was developed. Hybrid system identification, reachability and observability of linear switched systems are all related topics with a vast literature. Without claiming completeness, we refer to Bako (2011), Lauer, Bloch, and Vidal (2011) and Paoletti, Juloski, Ferrari-Trecate, and Vidal (2007) for hybrid system identification, and to Alur, Dang, and Ivancic (2003), Babaali and Pappas (2005), Baglietto, Battistelli, and Scardovi (2007), Collins and van Schuppen, 2004, De Santis et al., 2003, Ge, Sun, and Lee (2001), Habets, Collins, and van Schuppen (2006), Lee and Khargonekar (2009), Sun and Ge (2005) and Vidal, Sastry, and Chiuso (2003) for observability, reachability, stabilizability of linear switched systems.

In this paper we reduce the realization problem of DTLSSs to that of rational formal power series (Berstel and Reutenauer, 1984, Fliess, 1973, Kuich and Salomaa, 1986, Sontag, 1979). Note that the realization problem of state-affine systems Sontag (1979) is also equivalent to that of rational formal power series. Hence, indirectly we also show that the realization problems for DTLSSs and state-affine systems are equivalent. In turn, state-affine systems include autonomous DTLSSs as a special case.

There are several ways to view linear switched systems as time-varying linear systems. One option is to view them as linear parameter-varying systems (LPVs), where sequences of discrete modes play the role of scheduling parameters. The realization problem studied in this paper is thus similar to that of LPV systems. For such systems, realization theory (using behavioral approach) was addressed in Tóth et al. (2011). Note that for LPV systems, the scheduling parameters are usually assumed to take values in a subset of $R^{d}$ . This assumption seems essential to Tóth et al. (2011), since it allows the use of meromorphic functions of scheduling parameters. The latter functions are defined outside a set of measure zero. Now, if we restrict attention to discrete-valued scheduling parameters, then the set of admissible scheduling parameters becomes a set of measure zero. Hence, it is not obvious how to adapt the approach of Tóth et al. (2011) to switched systems. In Tóth, Abbas, and Werner (2012) a realization algorithm for affine LPV systems was presented and it was used to formulate a model reduction procedure for affine LPV systems. The realization algorithm of Tóth et al. (2012) is similar to Algorithm 1; however, in Tóth et al. (2012) the Markov parameters were defined based on existing state-space representations and not directly for input–output maps. Note that it is possible to use the results of the current paper to obtain a Kalman-style realization theory for LPV systems with affine parameter dependence. A detailed discussion on this topic goes beyond the intended scope of the paper, more details can be found in Petreczky and Mercère (2012).

Note that a linear switched system can also be viewed as a collection of time-varying linear systems. In turn, realization theory of time-varying linear systems was published in Kamen and Hafez (1979) and Weiss (1972). The difference between Kamen and Hafez (1979) and Weiss (1972) and realization theory of linear switched systems is as follows: in the case of linear switched systems, state-space representations of several time-varying input–output maps have to be found simultaneously and these representations are strongly related. For this reason, Kamen and Hafez (1979) and Weiss (1972) is not directly applicable to linear switched systems.

Outline. Section 2 presents the formal definition of DTLSSs and related system-theoretic properties. Sections 3–4 state the main results of the paper. The proofs are presented in Section 5.

Notation. Denote by $N$ the set of natural numbers including 0. The notation below is standard in automata theory; see Eilenberg (1974) and Kuich and Salomaa (1986). Consider a finite set $X$ . Denote by $X^{*}$ the set of finite sequences of elements of $X$ . The elements of $X^{*}$ are also referred to as strings or words over $X$ . Each non-empty word $w$ is of the form $w = a_{1} a_{2} \dots a_{k}$ for some $a_{1}, a_{2}, \dots, a_{k} \in X$ . The element $a_{i}$ is called the ith letter of $w$ , for $i = 1, \dots, k$ and $k$ is called the length of $w$ and it is denoted by $| w |$ . We denote by $ϵ$ the empty sequence (word), the length of $ϵ$ is zero by definition. We denote by $X^{+}$ the set of non-empty words. We denote by $w v$ the concatenation of word $w \in X^{*}$ with $v \in X^{*}$ . For each $j = 1, \dots, m$ , $e_{j}$ is the $j$ th unit vector of $R^{m}$ , i.e. $e_{j} = (δ_{1, j}, \dots, δ_{m, j})$ , $δ_{i, j}$ is the Kronecker symbol. If $L$ is a linear map defined on a vector space $X$ , $L x$ denotes the value $L (x)$ of $L$ at $x \in X$ . If $L_{1}, L_{2}$ are two linear maps, then $L_{1} L_{2}$ denotes the composition $L_{1} \circ L_{2}$ .

Section snippets

Linear switched systems

Definition 1

A discrete-time linear switched system (abbreviated by DTLSS) is a discrete-time system $Σ {\begin{matrix} x_{t + 1} = A_{q_{t}} x_{t} + B_{q_{t}} u_{t} and x_{0} is fixed \\ y_{t} = C_{q_{t}} x_{t} . \end{matrix}$ Here $Q = {1, \dots, D}$ is the finite set of discrete modes, $D > 0$ is an integer. For each time instant $t \in N$ , $q_{t} \in Q$ denotes the discrete mode, $u_{t} \in R^{m}$ denotes the continuous input, $x_{t} \in R^{n}$ denotes the state, and $y_{t} \in R^{p}$ denotes the output at time $t$ . Furthermore, $A_{q} \in R^{n \times n}$ , $B_{q} \in R^{n \times m}$ , $C_{q} \in R^{p \times n}$ are the matrices of the linear system in mode $q \in Q$ , and $x_{0}$ is the initial continuous state. The

Main result on minimality

In the sequel, $Σ$ denotes a DTLSS of the form (1).

Theorem 1 Minimality

A DTLSS is minimal, if and only if it is span-reachable and observable. If two minimal DTLSSs are equivalent, then they are isomorphic.

The proof of Theorem 1 is presented in Section 5. Intuitively, the theorem follows from the observation that the states which are not a linear combination of the reachable ones, and the states which always yield zero output do not contribute to the input–output map of a DTLSS. Note that $Σ$ can be minimal, while

Main results on existence of a realization

We present the necessary and sufficient conditions for the existence of a DTLSS realization for an input–output map. To this end, we need the notion of the Hankel-matrix and Markov-parameters of an input–output map. In the sequel, $f$ denotes a map of the form (3).

Definition 11 Markov-parameters

Consider the input–output map $y_{0}^{f}$ such that for any hybrid input $w \in U^{+}$ of the form (2), $y_{0}^{f} (w) = f (w) - f ((q_{0}, 0) (q_{1}, 0) \dots (q_{t}, 0)) .$ Define the maps $S_{0}^{f} : Q^{+} \to R^{p}$ and $S_{j}^{f} : Q^{+} \to R^{p}$ , $j = 1, \dots, m$ by $S_{0}^{f} (v) = f ((q_{1}, 0) (q_{2}, 0) \dots (q_{k}, 0)),$ $S_{j}^{f} (v) = {\begin{matrix} y_{0}^{f} ((q_{1}, e_{j}) (q_{2}, 0) \dots (q_{k}, 0 \end{matrix}$

Proof of Theorem 2

The theorem follows from the following more general statement. Consider any family of $n \times n$ matrices $F_{q}$ , $q \in Q$ and any matrix $G \in R^{n \times l}$ for some $l > 0$ . Define the matrix $R_{k} = [\begin{matrix} F_{v_{1}} G & \dots & F_{v_{N (k)}} G \end{matrix}]$ for $k \in N$ . Here we applied Notation 3 to $F_{q}, q \in Q$ to obtain the matrices $F_{v}, v \in Q^{*}$ . Define the subspace $I$ as the space spanned by the column vectors of the matrices $F_{v} G$ , $v \in Q^{*}$ . We then claim that $Im R_{n - 1} = I$ .

From $Im R_{n - 1} = I$ the statement of the theorem follows easily. Indeed, it is easy to see that the linear span of all reachable

Conclusions

We presented realization theory for discrete-time linear switched systems, including necessary and sufficient conditions for existence and minimality of state-space representations. These results are expected to be useful in model reduction and identification of switched systems. In fact, they were already used to address the problems of identifiability and persistence of excitation for linear switched systems; see Petreczky et al. (2010) and Petreczky and Bako (2011).

Acknowledgments

The authors thank the anonymous referees for their valuable comments.

Mihaly Petreczky received an M.Sc. in computer science and a Ph.D. in mathematics from Vrije Universiteit in Amsterdam, The Netherlands, in 2002 and in 2006, respectively. In the past, he held appointments as a postdoc at Johns Hopkins University, USA, Eindhoven University of Technology, The Netherlands, and as an assistant professor at Maastricht University, The Netherlands. He is currently an assistant professor at Ecole des Mines de Douai, France. His research interests include control and

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Laurent Bako received a “diplôme d’ingénieur” in Electrical Engineering from Ecole Nationale Supérieure d’Ingénieurs de Poitiers and the M.Sc. degree in Automatic Control from Université de Poitiers, both in 2005. He was a visiting researcher in the Center for Imaging Sciences, at the Johns Hopkins University from the Spring through the Fall of 2007. In 2008, he obtained the Ph.D. degree in Automatic Control and Computer Sciences from Université Lille 1, Sciences et Technologies. Since December 2008, Dr. Bako is an assistant professor with Ecole des Mines de Douai, in the Department of Computer Sciences and Automatic Control. His research interests are mainly in control theory, system identification, hybrid systems, machine learning.

Jan H. van Schuppen is affiliated as professor emeritus with the Department of Mathematics of Delft University of Technology in Delft, The Netherlands, and is further active with his consulting company Van Schuppen Control Research in Amsterdam, The Netherlands, since his retirement in October 2012. His research interests include control of hybrid systems, control of discrete-event systems, control of decentralized and distributed systems, realization, system identification, control of motorway networks, and modeling and identification of biochemical reaction networks. He is Advising Editor of the journal Mathematics of Control, Signals, and Systems and recently was coordinator of the project Control for coordination of distributed systems which was financially supported by the European Commission.

^☆: The material in this paper was partially presented at the 13th ACM International Conference on Hybrid Systems: Computation and Control (HSCC’10), April 12–16, 2010, Stockholm, Sweden. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels under the direction of Editor Andrew R. Teel.

¹: Tel.: +33 327 712 238; fax: +33 327 712 917.

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Brief paperRealization theory of discrete-time linear switched systems☆

Abstract

Introduction

Section snippets

Linear switched systems

Main result on minimality

Main results on existence of a realization

Proof of Theorem 2

Conclusions

Acknowledgments

Automatica

Automatica

Theoretical Computer Science

Automatica

European Journal of Control

Systems & Control Letters

Progress on reachability analysis of hybrid systems using predicate abstraction

Observability of switched linear systems in continuous time

Rational series and their languages

Observability of piecewise-affine hybrid systems

Automata, languages and machines

Matrices de hankel

Journal de Mathématiques Pures et Appliqués

Reachability and controllability of switched linear discrete-time systems

IEEE Transactions on Automatic Control

Reachability and control synthesis for piecewise-affine hybrid systems on simplices

IEEE Transactions on Automatic Control

Algebraic theory of linear time-varying systems

SIAM Journal on Control and Optimization

Semirings, automata, languages

Detectability and stabilizability of discrete-time switched linear systems

IEEE Transactions on Automatic Control

System identification: theory for the user

Brief paper
Realization theory of discrete-time linear switched systems☆