Realization Theory For Linear Switched Systems: Formal Power Series Approach Mih´aly Petreczky

1 2

Center for Imaging Science, Johns Hopkins University 21218 MD, Baltimore, USA

Abstract The paper deals with the realization theory of linear switched systems. Necessary and sufficient conditions are formulated for a family of input-output maps to be realizable by linear switched systems. Characterization of minimal realizations is presented. The paper treats two types of linear switched systems. The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences is admissible, but within this restricted set the switching times are arbitrary. The paper uses the theory of formal power series to derive the results on realization theory. Key words: Hybrid systems, switched linear systems, realization theory, formal power series, minimal realization 1991 MSC: 93B15, 93B20, 93B25, 93C99

1

Introduction

Linear switched systems are one of the best studied subclasses of hybrid systems. Informally, a linear switched system is a dynamical system specified by the following equations   

Σ: 1



d x(t) dt

= Aq(t) x(t) + Bq(t) u(t)

(1)

y(t) = Cq(t) x(t)

The work presented in this paper was carried out during the author’s stay at Centrum voor Wiskunde en Informatica (CWI), The Netherlands. 2 The work presented in this paper was included into the author’s PhD thesis, [1] Email address: [email protected] ( Mih´aly Petreczky ).

Preprint submitted to Elsevier Science

11 November 2006

where x(t) ∈ Rn is the continuous state, y(t) ∈ Rp is the continuous output, u(t) ∈ Rm is the continuous input, and q(t) ∈ Q is the switching signal. The set Q is the set of discrete modes, and for each q ∈ Q, Aq , Bq , Cq are matrices of appropriate dimensions. In this paper we will think of the switching signal as an input and we will be looking at input-output maps which map continuous input functions and switching signals to continuous outputs. A more formal and detailed description of linear switched systems will be presented in Section 2. The paper addresses the realization problem for linear switched systems. The realization problem and realization theory play a central role in control systems theory. Realization theory provides a theoretical foundation for systems identification, model reduction and control design. The goal of this paper is to present a short summary of the realization theory of linear switched systems. A more detailed version containing the proofs is in preparation, see [1]. More specifically, the paper deals with the following problems. When can a set of input-output maps Φ be realized by a linear switched systems Σ ? What are the conditions for Σ being a minimal realization of Φ ? The paper considers both the case when the input-output maps are defined for all possible switching sequences and the case when the input-output maps are defined for a subset of switching sequences. In the latter case we assume that all switching times are allowed and the restriction is only on the possible sequences of discrete modes. The latter case is inspired by the following problem. Construct a realization of a set of input-output maps by a linear switched system, such that the discrete modes are states of an automaton. The automaton is specified in advance and a discrete-mode transition is triggered by an external discrete event from the event alphabet of the automaton. If we know the automaton, then we know the input-output map for all the switching sequences which are allowed by the automaton. Notice that the set of allowed sequences of discrete modes forms a regular language (see [13,3] for the definition) in this case. Remarkably, the case when the set of admissible sequences of discrete modes forms a regular language is exactly the situation, for which we have particularly nice results. Notice that if the set of admissible sequences of discrete modes is finite, then it also forms a regular language (see [13,3] for the proof). We will show that the problem of finding a linear switched system realization of a family of input-output maps Φ reduces to a purely algebraic problem. This algebraic algebraic problem is related to the well-known problem of finding a rational representation for a formal power series. Rational formal power series were used in systems theory earlier, see [7,5,8–10] the the references therein. There are a number of equivalent frameworks for rational formal power series, see [3–5]. In this paper we will present an extension of the classical theory of formal power series. This extension will allow us to deal with families of formal power series and it is based on the framework presented in [3]. The problem of dealing with families of formal power series already appeared in 2

[11], but the results presented there are not sufficient for our purposes. The reason why we need rational representations for a family of formal power series is the following. First, we develope realization theory for families of input-output maps, which give rise to families of formal power series. Another problem is that we have to find a collection of matrices Cq , Bq , q ∈ Q, and the framework of [3] does not cover this case. By extending the framework of [3] to families of formal power series, we can deal both with families of inputoutput maps and with the problem of finding a collection Cq , Bq , q ∈ Q. Note however, that the latter problem could be solved by using the framework of discrete-time state-affine system from [4,5]. Hence, an alternative route could be developing realization theory for families of input-output maps which are realizable by state-affine systems. Since discrete-time state-affine systems and the framework of [3] are essentially equivalent, the choice between the two routes seems to be a matter of taste. Note that the linear switched systems (1) can be viewed as continuous-time nonlinear systems. More precisely, one can view linear systems either as bilinear systems with a restricted set of admissible piecewise-continuous input functions, or as general nonlinear systems with discrete and continuous inputs. Due to the lack of space we will not present this correspondence formally. For both types of nonlinear systems realization theory exists, see [10,7] and the references therein. In fact, the connection between realization theory of bilinear systems and rational formal power series is well-known. However, realization theory for general nonlinear systems yields a general nonlinear realization instead of a linear switched system. Realization theory of bilinear systems assumes a much wider class of admissible inputs than what one gets by viewing linear switched systems as bilinear systems. None of the classical results account for the case of restricted switching. Hence, a direct application of the existing results does not seem possible. The approach taken in this paper can be thought of as an adaptation of the classical nonlinear realization theory to linear switched systems. In fact, we believe that the route taken in this paper is the most straightforward adaptation of the classical nonlinear realization theory. Due to the lack of space we will not spell out the relationship with the classical theory more formally. However, the reader is encouraged to consult the references and make a comparison himself. The contributions of the paper can be formulated as follows. The paper develops realization theory for linear switched systems. To the best of our knowledge, the only prior work on realization theory of linear switched systems is [2]. Compared to [2], the novelty of the current paper lies in (a) presenting realization theory for families of input-output maps by linear switched systems, (b) presenting realization theory for input-output maps which are defined only on a subset of all the switching sequences, (c) the use of formal power series theory for realization theory of continuous-time linear switched systems. In our opinion, the main contribution of the paper are the results on realiza3

tion theory of linear switched systems discussed above. An additional minor contribution of this paper is the extension of the classical theory of rational formal power series to families of formal power series. The outline of the paper is the following. Section 2 introduces the notation and concepts which are used in the rest of the paper. It also formally states the main results of the paper. Appendix A presents the theory of rational families of formal power series. Section 3 presents the details of the realization theory of linear switched systems with arbitrary switching. Section 4 presents the details of the realization construction for linear switched systems with constrained switching. Appendix A is a prerequisite for Section 4 and Section 3.

2

Switched Systems

The notation and notions described in this section are largely based on [2,1]. Denote by T the set [0, +∞) ⊆ R, i.e. T will denote the time axis. Denote by P C(T, Rm ) the class of piecewise-continuous maps from T to Rm . For a set Σ denote by Σ∗ the set of finite strings of elements of Σ. For w = a1 a2 · · · ak ∈ Σ∗ , a1 , a2 , . . . , ak ∈ Σ the length of w is denoted by |w|, i.e. |w| = k. The empty sequence is denoted by ². The length of ² is zero: |²| = 0. Let Σ+ = Σ∗ \ {²}, i.e. Σ+ is the set of all non-empty strings of letters from Σ. The concatenation of two strings v = v1 · · · vk , w = w1 · · · wm ∈ Σ∗ is the string vw = v1 · · · vk w1 · · · wm . We denote by wk the string w · · w}. The word w0 is | ·{z k−times

just the empty word ². Denote by F (A, B) the set of all functions from the set A to the set B. For any function f the range of f will be denoted by Imf . For any two sets J, X an indexed subset of X with the index set J is simply a map Z : J → X, denoted by Z = {aj ∈ X | j ∈ J}, where aj = Z(j), j ∈ J. Denote by Nk the set of k tuples of non-negative integers. If α = (α1 , . . . , αk ) ∈ Nk and β = (β1 , . . . , βm ) ∈ Nm , then (α, β) = (α1 , . . . , αk , β1 , . . . , βm ) ∈ Nk+m . For any smooth map φ : Rk → Rp , and k tuple α = (α1 , α2 , . . . , αk ) ∈ Nk denote by Dα φ the following high-order partial derivative of φ at zero, dα1 dα2 dαk D φ = α1 α2 · · · αk φ(t1 , t2 , . . . , tk )|t1 =t2 =···=tk =0 . dt1 dt2 dtk α

A linear switched system is a system of the form (1). Recall that Q is a finite set, called the set of discrete modes, X = Rn , n > 0 is the set of continuous states, U = Rm , m > 0 is the set of continuous inputs, and finally Y = Rp , p > 0 is the set of continuous outputs. The matrices Aq , Bq , Cq , q ∈ Q, are of the form Aq ∈ Rn×n , Bq ∈ Rn×m and Cq ∈ Rp×n . We will use the following 4

short-hand notation for linear switched systems of the form (1) Σ = (X, U, Y, Q, {(Aq , Bq , Cq ) | q ∈ Q}) A detailed description of the semantics of switched systems is available in [12]. We will define the dimension of Σ as dim Σ := dim X , i.e. as the dimension of the continuous state space. In this paper, the inputs of the switched system Σ are functions from P C(T, U) and sequences from (Q × T )+ . The elements of the set (Q × T )+ are called switching sequences. That is, the switching sequences are part of the input, they are specified externally. Hence, in our setting the system theoretic states of a switched systems (1) are the continuous states x(t) ∈ Rn . We will denote by xΣ (x0 , u, w) the continuous state of the P system Σ reached from the initial state x0 at time kj=1 ti , after feeding in the continuous input u and the switching sequence w = (q1 , t1 ) · · · (qk , tk ). It is instructive to present the explicit formula for xΣ (x0 , u, w). Aqk tk

xΣ (x0 , u, w) = e

Aqk tk Aqk−1 tk−1

··· + e

e

Aq1 t1

···e

x0 + +

Aq2 t2

···e

Z

0

t1

Z

0

tk

Aqk (tk −s)

e

k−1 X

Bqk u(

ti + s)ds + · · ·

1

eAq1 (t1 −s) Bq1 u(s)ds

Since feeding in empty switching sequences corresponds to evaluating the system at time 0, we adopt the following convention. We will identify xΣ (x0 , u, ²) with x0 , i.e. xΣ (x0 , u, ²) = x0 . The reachable set of the system Σ from a set of initial states X0 is defined by Reach(Σ, X0 ) = {xΣ (x0 , u, w) ∈ X | u ∈ P C(T, U), w ∈ (Q × T )∗ , x0 ∈ X0 } Σ is said to be reachable from X0 if Reach(Σ, X0 ) = X holds. Σ is semireachable from X0 if X is the vector space of the smallest dimension containing Reach(Σ, X0 ). For each x ∈ X define the input-output map yΣ (x, ., .) of the system Σ induced by x as the function yΣ (x, ., .) : P C(T, U) × (Q × T )+ 3 (u, w) 7→ Cqk xΣ (x, u, w) ∈ Y

(2)

where w = (q1 , t1 ) · · · (qk , tk ). By abuse of notation we will use yΣ (x, u, w) for P yΣ (x, ., .)(u, w). That is, yΣ (x, u, w) is the output y( kj=1 tj ) of the system (1), P if the continuous input equals u and the switching sequence up to time kj=1 tj is w. Two states x1 6= x2 ∈ X of the switched system Σ are indistinguishable if yΣ (x1 , ., .) = yΣ (x2 , ., .). Σ is called observable if it has no pair of indistinguishable states. See [2,6] for conditions for observability and reachability. Consider two linear switched systems Σi = (Xi , U, Y, Q, {(Aiq , Bqi , Cqi ) | q ∈ Q}) (i = 1, 2). The systems Σ1 and Σ2 are algebraically similar if there exists a vector space isomorphism S : X1 → X2 such that for all q ∈ Q the following holds A2q = SA1q S −1 , Bq2 = SBq1 , and Cq2 = Cq1 S −1 . 5

A set of input-output maps Φ ⊆ F (P C(T, U) × (Q × T )+ , Y) is said to be realized by a linear switched system Σ if there exists a map µ : Φ → X such that ∀f ∈ Φ : yΣ (µ(f ), ., .) = f Both Σ and (Σ, µ) are called a realization of Φ. Thus, Σ realizes Φ if and only if for each f ∈ Φ there exists a state x ∈ X such that yΣ (x, u, w) = f (u, w) for all u ∈ P C(T, U), w ∈ (Q × T )+ . A switched system Σ is a minimal realization of Φ if Σ is a realization of Φ and for each switched system Σ1 such that Σ1 is a realization of Φ it holds that dim Σ ≤ dim Σ1 . For any L ⊆ Q+ define the subset of admissible switching sequences T L ⊆ (Q × T )+ by T L := {(q1 , t1 )(q2 , t2 ) · · · (qk , tk ) ∈ (Q × T )+ | k ≥ 1, q1 · · · qk ∈ L} That is, T L is the set of all those switching sequences, for which the sequence of discrete modes belongs to L and the sequence of times is arbitrary. Notice that if L = Q+ then T L = (Q×T )+ , i.e T L is the set of all switching sequences. Let Φ ⊆ F (P C(T, U) × T L, Y). That is Φ is a subset of input-output maps which are defined only for admissible switching sequences. The system Σ realizes Φ with constraint L if there exists a map µ : Φ → X such that ∀f ∈ Φ, w ∈ T L, u ∈ P C(T, U) : yΣ (µ(f ), u, w) = f (u, w) That is, Σ is a realization of Φ with constraint L, if for each map f ∈ Φ the restriction of the input-output map yΣ (µ(f ), ., .) to the set of admissible switching sequences yields f . Both Σ and (Σ, µ) will be called a realization of Φ. Notice that if L = Q+ then Σ realizes Φ with constraint L if and only if Σ realizes Φ without constraints. If (Σ, µ) is a realization of Φ (with constraint L), then the map µ : Φ → X can be thought of as a map assigning to each input-output map f ∈ Φ an initial state of Σ, which induces an inputoutput map identical to f . We will call the realization (Σ, µ) observable, if Σ is observable, and we will call (Σ, µ) semi-reachable , if Σ is semi-reachable from Imµ. Below we will define the notion of generalized kernel representation. Recall from linear systems theory that an input-output map y : P C(T, Rm × T 3 (u, t) 7→ y(t) ∈ Rp has a realization by a linear system, only if y(t) = Rt p×m is an analytic map. Recall that 0 K(t − s)u(s)ds, where K : T → R k the Markov-parameters of y can be computed by Mk = dtd k K(t)|t=0 . A linear system (A, B, C) is a realization of y if and only if y is of the form Rt y(t) = 0 K(t − s)u(s)ds and Mk = CAk B, ∀k ≥ 0. The condition that a set of input-output maps has a generalized kernel representation is analogous R to the condition that y is of the form y(t) = 0t K(t − s)u(s)ds. The formal definition goes as follows. Consider a language (subset) L ⊆ Q+ . Define the 6

e = {ui1 ui2 · · · uik ∈ Q∗ | sets suffixL = {u ∈ Q∗ | ∃w ∈ Q∗ : wu ∈ L} and L 1 2 k u1 u2 · · · uk ∈ suffixL, uj ∈ Q, ij ≥ 0, j = 1, 2, . . . k, i1 , ik > 0, k > 0}.

Definition 1 (Generalized kernel-representation with constraint L) A set Φ ⊆ F (P C(T, U) × T L, Y) is said to have generalized kernel representae there exist functions tion with constraint L if for all f ∈ Φ and for all w ∈ L k p×m Kwf,Φ : Rk → Rp×1 and GΦ , where |w| = k w : R → R

such that the following holds. e ∀f ∈ Φ: K f is analytic and GΦ is analytic (1) ∀w ∈ L, w w e it holds that (2) For each f ∈ Φ and w, v ∈ Q∗ such that wqqv, wqv ∈ L, 0

0

0

0

f,Φ f,Φ Kwqqv (t1 , . . . , tk , t, t , tk+1 , . . . tk+l ) = Kwqv (t1 , . . . tk , t + t , tk+1 . . . tk+l ) Φ GΦ wqqv (t1 , . . . , tk , t, t , tk+1 , . . . tk+l ) = Gwqv (t1 , . . . tk , t + t , tk+1 . . . tk+l )

where k = |w| and l = |v|. e ∀f ∈ Φ : (3) ∀vw ∈ L, f,Φ f,Φ (t1 , t2 , . . . , tk+l ) if |w| > 0 (t1 , . . . , tl , 0, tl+1 , . . . , tk+l ) = Kvw Kvqw Φ GΦ vqw (t1 , . . . , tl , 0, tl+1 , . . . , tk+l ) = Gvw (t1 , . . . , tl+k )

if |v| > 0, |w| > 0

where k = |w| and l = |v|. (4) For each f ∈ Φ, q1 q2 · · · qk ∈ L, q1 , . . . , qk ∈ Q, k ≥ 1, t1 , . . . , tk ∈ T k : (t1 , . . . , tk )+ f (u, (q1 , t1 )(q2 , t2 ) · · · (qk , tk )) = Kqf,Φ 1 q2 ···qk +

k Z X

tj

j=1 0

where σj u(s) = u(s +

GΦ qj ···qk−1 qk (tj − s, tj+1 , . . . , tk )σj u(s)ds

Pj−1 1

ti ).

We say that Φ has a generalized kernel representation if it has a generalized kernel representation with the constraint L = Q+ . Using the notation above, define the function y0Φ : P C(T, U) × T L → Y by y0Φ (u, w) = f (u, w) − f (0, w) for some f ∈ Φ If Φ has a generalized kernel representation with constraint L, then y0Φ is well defined no matter which f ∈ Φ will be picked. Let u ∈ U be a continuous input value and identify u with the constant function g ∈ P C(T, U), g(t) = u for all t ∈ T . For each element f ∈ Φ and each sequence of discrete modes w = w1 . . . wk ∈ L, w1 , . . . , wk ∈ Q, define the map f (u, w, .) : T k 3 (t1 , . . . , tk ) 7→ f (u, (w1 , t1 )(w2 , t2 ) · · · (wk , tk )) ∈ Y. Similarly, we can define the map y0Φ (u, w, .) : T k 3 (t1 , . . . , tk ) 7→ y0Φ (u, (w1 , t1 )(w2 , t2 ) · · · (wk , tk )) ∈ Y. 7

Notice that if Φ has a generalized kernel representation with constraint L, then the maps f (u, w, .) and y0Φ (u, w, .) are analytic. Proposition 1 Let Σ be a linear switched system of the form (1) and let µ be a map µ : Φ → X . The pair (Σ, µ) is a realization of Φ with constraint L if and only if Φ has a generalized kernel representation with constraint L and for each w ∈ L, f ∈ Φ, j = 1, 2, . . . , m and α ∈ N|w| the following holds αl −1 αk αk−1 Dα y0Φ (ej , w, .) = Dβ GΦ wl ···wk ej = Cw1 Awk Awk−1 · · · Awl Bwl ej

Dα f (0, w, .) =

Dα Kwf,Φ

· · · Aαwll µ(f ) = Cwk Aαwkk Aαwk−1 k−1

(3)

where l = min{h | αh > 0}, ej is the jth unit vector of U, β = (αl −1, . . . , α|w| ) and w = w1 · · · wk , w1 , . . . , wk ∈ Q. PROOF. [Sketch] (Σ, µ) is a realization of Φ with constraint L if and only if Φ has a generalized kernel representation defined by Awk tk Awk−1 tk−1 e · · · eAw1 t1 Bw1 GΦ w1 w2 ···wk (t1 , t2 , . . . , tk ) =Cwk e

(t1 , t2 , . . . , tk ) = Cwk eAwk tk eAwk−1 tk−1 · · · eAw1 t1 µ(f ) Kwf,Φ 1 w2 ···wk

(4)

e w , . . . , w ∈ Q. Using Part 2 and Part 3 of Definition for each w1 w2 · · · wk ∈ L, 1 k 1, one can show that the functions {Kwf,Φ , GΦ w | f ∈ Φ, w ∈ suffixL} completely e Hence the collection of | f ∈ Φ, w ∈ L}. determine the functions {Kwf,Φ , GΦ w f,Φ Φ analytic functions {Kw , Gw | w ∈ suffixL, f ∈ Φ} determines Φ. Since for each w ∈ suffixL the maps Kwf,Φ , GΦ w are analytic, we get that the high-order Φ |w| derivatives {Dα Kwf,Φ , Dα G | α ∈ N } determineR Kwf,Φ and GΦ w w uniquely. By R t d d t applying the formula dt 0 f (t, τ )dτ = f (t, t) + 0 dt f (t, τ )dτ and Part 4 of Definition 1 one gets β Φ Dα Kwf,Φ = Dα f (0, w, .) and Dα GΦ wl wl+1 ···wk ez = D y0 (ez , w, .)

(5)

where w = w1 · · · wk , w1 , . . . , wk ∈ Q, l ≤ k, Nk 3 β = (0, 0, . . . , 0, α1 + 1, α2 , . . . , αl ) and ez is the zth unit vector of Rm , i.e eTz ej = δzj . Hence, if (Σ, µ) is a realization of Φ, then we get the statement of the proposition from (4) and (5). If the statement of the proposition holds, then by the discussion above (4) holds which implies that (Σ, µ) is a realization of Φ with constraint L. The main idea behind applying formal power series theory to realization theory of linear switched systems is the following. Assume that Φ is a set of inputoutput maps defined for switching sequences from T L. Assume that Φ has a generalized kernel representation with constraint L. We associate a set of formal power series ΨΦ with the set Φ . The values of the elements ΨΦ are 8

formed by the derivatives Dα K f,Φ , Dα GΦ w , or equivalently, by the derivatives α Φ α D y0 (ej , w, .) and D f (0, w, .) where f runs through the elements of Φ, and α runs through all the multi-indices. We define the Hankel matrix HΦ of Φ as the Hankel matrix (see Appendix A) of the family of formal power series ΨΦ . Notice that the entries of HΦ are formed by linear combinations of highorder derivatives of the input-output maps of f ∈ Φ. In fact, the Hankel matrix HΦ is related to the Hankel matrix of nonlinear input-output maps [10,7]. If L = Q+ , i.e. all switching sequences are allowed, then there is a one-to-one correspondence between rational representations of ΨΦ and linear switched system realizations of Φ. Moreover, minimal representations give rise to minimal realizations and vice versa. It can be shown that a representation of ΨΦ is reachable and observable if and only if the corresponding linear switched system is semi-reachable and observable. Hence from the results of Appendix A we get the following. Theorem 1 (Arbitrary Switching) Let Φ ⊆ F (P C(T, U) × (Q × T )+ , Y) be a set of input-output maps defined for all switching sequences. (i) Φ has a realization by a linear switched system, if and only if Φ has a generalized kernel representation and ΨΦ is rational, that is, rank HΦ < +∞, (ii) Let (Σ, µ) be a linear switched system realization of Φ. Then (Σ, µ) is a minimal realization of Φ if and only if (Σ, µ) is semi-reachable 3 and observable. If (Σ, µ) is a minimal realization of Φ, then dim Σ = rank HΦ . All minimal linear switched systems realizing Φ are algebraically similar. If L 6= Q+ , that is, not all switching sequences are allowed, then the situation is more complex. We can show that any representation of ΨΦ gives rise to a realization of Φ with constraint L. If L is regular, then any realization of Φ with constraint L gives rise to a representation of ΨΦ . Unfortunately minimal representations of ΨΦ do not yield minimal realizations of Φ. However, any minimal representation of ΨΦ yields an observable and semi-reachable realization of Φ. One can construct an input-output map y, a regular language L, and realizations Σ1 and Σ2 such that the following holds. Both Σ1 and Σ2 realize y from the initial state zero with constraint L, they are both reachable and observable, but dim Σ1 = 1 and dim Σ2 = 2. That is, observability and semi-reachability is not sufficient for minimality in case of constrained switching. Theorem 2 (Constrained Switching) Let Φ ⊆ F (P C(T, U) × T L, Y) and assume that L is regular. (i) Φ has a realization by a linear switched system with constraint L, if and 3

From [6] it follows Σ is semi-reachable from {0} if and only if Σ is reachable from {0}. Hence Theorem 1 implies the results of [2]

9

only if Φ has a generalized kernel representation with constraint L and ΨΦ is rational, that is, rank HΦ < +∞. (ii) If Φ has a realization by a linear switched system with constraint L, then there exists a realization (Σ, µ) of Φ with constraint L such that (Σ, µ) is semie µ) e is an arbitrary linear switched reachable and observable. Moreover, if (Σ, system realizing Φ with constraint L, then e , where M depends only on L dim Σ ≤ M dim Σ,

(6)

In the sequel we will present the arguments and constructions leading to the theorems above more formally. It follows from the proofs of the theorems above that linear switched system realizations of Φ arise from rational representations of ΨΦ . Hence, algorithms for constructing minimal representations of ΨΦ will immediately yield algorithms for constructing semi-reachable and observable linear switched system realizations of Φ. In particular, linear switched systems realizing Φ can be constructed from the columns of the Hankel-matrix of Φ.

3

Realization of input-output maps by linear switched systems with arbitrary switching

Let Φ ⊆ F (P C(T, U) × (Q × T )+ , Y) be set of input-output maps and assume that Φ has a generalized kernel representation. Define the formal power series Sq1 ,q2 ,z , Sf,q1 ∈ Rp ¿ Q∗ À, ( q1 , q2 ∈ Q, f ∈ Φ, z ∈ {1, 2, . . . , m} ) by Sq1 ,q2 ,z (w) = D(1,Iw ,0) y0Φ (ez , q2 wq1 , .) , Sf,q1 (w) = D(Iw ,0) f (0, wq1 , .) for each w ∈ Q∗ . Here Iw stands for Iw = (1, 1, . . . , 1) ∈ Nk , where k = |w| is the length of w. Assume that card(Q) = N and fix an enumeration Q = {q1 , q2 , . . . , qN } of the elements of Q. For each q ∈ Q, z = 1, 2, . . . , m, f ∈ Φ define the formal power series Sq,z , Sf ∈ Rp|Q| ¿ Q∗ À by ·

Sq,z (w) = Sq1 ,q,z (w) , Sq2 ,q,z (w) , · · · , SqN ,q,z (w) ·

T

T

T

Sf (w) = Sf,q1 (w) , Sf,q2 (w) , · · · , Sf,qN (w) T

T

T

¸T

¸T

for all w ∈ Q∗ . Define the set JΦ = Φ ∪ {(q, z) | q ∈ Q, z = 1, 2, . . . , m}. Define the indexed set of formal power series associated with Φ by ΨΦ = {Sj ∈ Rp|Q| ¿ Q∗ À| j ∈ JΦ }

(7)

Define the Hankel-matrix HΦ of Φ as the Hankel-matrix of the associated set of formal power series, i.e. HΦ := HΨΦ . Let Σ be a linear switched system 10

of form (1) and assume that µ is a map of the form µ : Φ → X . Define the representation associated with (Σ, µ) by e C) e RΣ,µ = (X , {Aq }q∈Q , B,

where Ce : X → Rp|Q| , Ce =

·

CqT1 ,

CqT2 ,

··· ,

CqTN

¸T

(8) , and Be = {Be j ∈ X |

j ∈ JΦ } is defined as follows. For each f ∈ Φ let Be f = µ(f·), and for all l = ¸ e e e 1, 2, . . . , m, q ∈ Q, let Bq,l be the lth column of Bq , i.e. Bq = Bq,1 , , . . . , Bq,m .

e C) e such that B e = Conversely, consider a representation R = (X , {Aq }q∈Q , B, {Be j ∈ X | j ∈ JΦ }. Then define (ΣR , µR ) the realization associated with R by

ΣR = (X, U, Y, Q, {(Aq , Bq , Cq ) | q ∈ Q}) , µR (f ) = Be f , for all f ∈ Φ

where µR

(9)

¸T · ¸ · e T T T e e e : Φ → X and C = Cq1 , Cq2 , · · · , CqN and Bq = Bq,1 , Bq,2 , . . . , Bq,m .

Proposition 1 yields the following theorem.

Theorem 3 If (Σ, µ) is a realization of Φ then RΣ,µ is a representation of ΨΦ . Conversely, if Φ has a generalized kernel representation and R is a representation of ΨΦ then (ΣR , µR ) is a realization of Φ . Recall from Appendix A the definition of reachability and observability of representations. Let (Σ, µ) be a realization of Φ. Let R = RΣ,µ be the representation associated with (Σ, µ). Using the results of [6,2] and (A.1) the following can be shown. Σ is observable if and only if R is observable, and (Σ, µ) is semi-reachable if and only if R is reachable. From the discussion after Theorem 5 we get that any realization of Φ can be transformed to an observable and semi-reachable realization of Φ. Moreover, Theorem 3 implies the following. If (Σ, µ) is a minimal realization of Φ, then RΣ,µ is a minimal representation of ΨΦ . Conversely, if R is a minimal representation of ΨΦ , then (ΣR , µR ) is a minimal realization of Φ. From the discussion above, Theorem 3 and Theorem 5, the statement of Theorem 1 follows easily.

4

Realization of input-output maps with constraints on the switching

Let L ⊆ Q+ be a subset of sequences of discrete modes and let Φ ⊆ F (P C(T, U)× T L, Y) be a set of input-output maps defined on the set of admissible switching sequences T L. Assume that Φ has a generalized kernel representation with constraint L. We define the formal power series Sq1 ,q2 ,j , Sq,f ∈ Rp ¿ Q∗ À, q1 , q2 , q ∈ Q, j = 1, 2, . . . , m, f ∈ Φ as follows. For each w ∈ Q∗ , let 11

  e  D (O|v| ,α+ ) y Φ (ej , vz, .) if w ∈ L q1 ,q2 and (v, (α, z)) ∈ Fq1 ,q2 (w) 0

Sq1 ,q2 ,j (w) = 

0 otherwise   e and (v, (α, z)) ∈ F (w)  D (O|v| ,α) f (0, vz, .) if w ∈ L q q

Sq,f (w) = 

0

otherwise

Here we used the following notation. Ol is the l tuple Ol = (0, 0, . . . , 0) ∈ Nl . For any α ∈ Nk , α+ denotes the tuple α+ = (α1 + 1, α2 , . . . , αk ) ∈ Nk . Define the set S = {(α, w) | α ∈ N|w| , w ∈ Q∗ }. For each w ∈ Q∗ , q1 , q2 ∈ Q define the sets Fq1 ,q2 (w), Fq1 (w) ⊆ Q∗ × S as follows. For each α ∈ Nk , z, v ∈ Q∗ , k ≥ 0, the pair (v, (α, z)) belongs to Fq1 ,q2 (w) if vz ∈ L and q2 wq1 = z1 z1α1 · · · zkαk zk , where z = z1 · · · zk , z1 , . . . , zk ∈ Q. Similarly, the pair (v, (α, z)) belongs to Fq1 (w), if vz ∈ L, and wq1 = z1α1 · · · zkαk zk , where e e α1 > 0, z = z1 · · · zk , z1 , . . . , zk ∈ Q. The languages L q1 ,q2 and Lq1 are defined ∗ ∗ e e as follows. L q1 ,q2 = {w ∈ Q | Fq1 ,q2 (w) 6= ∅} and Lq = {w ∈ Q | Fq (w) 6= e e ∅}, i.e. Lq1 ,q2 (Lq1 ) is the collection of all the strings w such that Fq1 ,q2 (w) (respectively Fq1 (w)) is not empty. It can be shown that if Φ has a generalized kernel representation with constraint L, then the series Sq1 ,q2 ,j and Sq,f are well-defined. That is, Sq1 ,q2 ,j (w) and Sq,f (w) do not depend on the choice of (v, (α, z)) ∈ Fq1 ,q2 (w) or (v, (α, z)) ∈ Fq (w) respectively. Assume that Q = {q1 , . . . , qN } and card(Q) = N , i.e. q1 , . . . , qN is an enumeration of elements of Q. Define the formal power series Sq,j , Sf ∈ Rp|Q| for each j ∈ {1, 2, . . . , m}, q ∈ Q and f ∈ Φ by ·

Sq,j (w) = Sq1 ,j (w) , Sq2 ,j (w) , . . . , SqN ,j (w) T

T

·

T

¸T

Sf (w) = Sq1 ,f (w)T , Sq2 ,f (w)T , . . . , SqN ,f (w)T

,

¸T

for all w ∈ Q∗ . Define the index set JΦ = Φ ∪ (Q × {1, 2, . . . , m}). Define the set of formal power series associated with Φ by ΨΦ = {Sj ∈ Rp|Q| ¿ Q∗ À| j ∈ JΦ } Define the Hankel-matrix of Φ HΦ as the Hankel-matrix of ΨΦ , i.e. HΦ = HΨΦ . Let Σ be a linear switched system of the form (1) and let µ be a map of the form µ : Φ → X . Let ΦΣ = {yΣ (µ(f ), ., .) | f ∈ Φ} be the set of all inputoutput maps defined for all switching sequences which are induced by the initial states µ(f ), f ∈ Φ. Recall the definition of the set of formal power series ΨΦΣ associated with ΦΣ as defined in (7), Section 3. Denote by Tq,z the element of ΨΦΣ indexed by (q, z) ∈ (Q × {1, 2, . . . , m}) and denote by Tf the element of ΨΦΣ indexed by yΣ (µ(f ), ., .). Define the the set of formal power series ΘΣ indexed by JΦ as ΘΣ = {Tj ∈ Rp|Q| ¿ Q∗ À| j ∈ JΦ }. It follows from Theorem 3 that ΨΦΣ is rational. Since ΘΣ is obtained from ΨΦΣ 12

by reindexing its elements, it follows that ΘΣ is rational. Let E = (1, 1, . . . , 1)T ∈ Rp . Define the power series Hq1 ,q2 , Zq ∈ Rp ¿ Q∗ À by     e e  E if w ∈ L  E if w ∈ L q q1 ,q2 and Zq (w) = Hq1 ,q2 (w) =    0 otherwise  0 otherwise

for all w ∈ Q∗ . Define the power series Hq , H ∈ Rp|Q| ¿ Q∗ À such that for ·

all w ∈ Q∗ , Hq (w) = Hq,q1 (w)T , Hq,q2 (w)T , · · · , Hq,qN (w)T ·

¸T

¸T

and H(w) =

Zq1 (w)T , Zq2 (w)T , . . . , ZqN (w)T . Define the family of formal power series ΩΦ as ΩΦ = {Hj ∈ Rp|Q| ¿ Q∗ À| j ∈ JΦ } where Hf = H for all f ∈ Φ and H(q,z) = Hq for all q ∈ Q, z = 1, . . . , m. Recall the definition of a representation associated with a linear switched system realization given by (8). Recall from (9) the definition of a linear switched system realization associated with a representation. With get the following theorem. Theorem 4 (Σ, µ) is a realization of Φ with constraint L if and only if Φ has a general kernel representation with constraint L and ΨΦ = ΘΣ ¯ ΩΦ . If Φ has a generalized kernel representation with constraint L and R is a representation of ΨΦ , then (ΣR , µR ) realizes Φ with constraint L. Below we will sketch the proof of Theorem 2. Assume that L is regular. It e and L e can be shown that then the languages L q q1 ,q2 are regular. Hence, using Appendix A or the well-known results in [3] it can be shown that then the family ΩΦ will be rational. If Φ has a realization (Σ, µ) with constraint L, then by Theorem 4, ΦΨ = ΘΣ ¯ ΩΦ . It is clear that ΘΣ is rational. If L is regular, then ΩΦ is rational, hence ΦΨ is rational. Hence, Theorem 4 implies that if L is regular, then Φ has a realization with constraint L if and only if ΨΦ is rational, i.e. rank HΦ < +∞. If R is a minimal representation of ΨΦ , then by Theorem 4, (Σ, µ) = (ΣR , µR ) is a realization of Φ with constraint L. From the discussion at the end of Section 3 it follows that (Σ, µ) is semi-reachable and observable. Finally, take M = rank HΩΦ and then (6) is a direct consequence of Lemma 1.

5

Conclusions

Solution to the realization problem for linear switched systems has been presented. The paper provides strong indication that the classical technique of rational formal power series might be an important tool for studying hybrid 13

systems. There is a paper in preparation on partial realization and other algorithmic issues for linear switched systems, see [1]. Topics of further research include realization theory for piecewise-affine systems. Acknowledgement The author thanks Jan H. van Schuppen for the help with the preparation of the manuscript. The author thanks Pieter Collins for the useful discussions and suggestions. The author thanks the anonymous reviewers for the constructive comments and suggestions.

References [1] M. Petreczky, Realization Theory of Hybrid Systems. Phd thesis, Vrije Universiteit, Amsterdam, 2006. [2] M. Petreczky, Realization theory for linear switched systems, in: Proceedings MTNS 2004. [3] J. Berstel, C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1984. [4] E. D. Sontag, Polynomial Response Maps, Vol. 13 of Lecture Notes in Control and Information Sciences, Springer Verlag, 1979. [5] E. D. Sontag, Realization theory of discrete-time nonlinear systems: Part I – the bounded case, IEEE Transaction on Circuits and Systems CAS-26 (4). [6] Z. Sun, S. Ge, T. Lee, Controllability and reachability criteria for switched linear systems, Automatica 38 (2002) 115 – 786. [7] A. Isidori, Nonlinear Control Systems, Springer Verlag, 1989. [8] M. Fliess, Functionnelles causales non lin´eaires et ind´etermin´ees non commutatives, Bull. Soc. Math. France (109) (1981) 2 – 40. [9] A. Isidori, P. D’Alessandro, A. Ruberti, Realization and structure theory of bilinear dynamical systems, SIAM J. Control 12 (3). [10] B. Jakubczyk, Realization theory for nonlinear systems, three approaches, in: Algebraic and Geometric Methods in Nonlinear Control Theory, D.Reidel Publishing Company, 1986, pp. 3–32. [11] Y. Wang, E. Sontag, Algebraic differential equations and rational control systems, SIAM Journal on Control and Optimization (30) (1992) 1126–1149. [12] D. Liberzon, Switching in Systems and Control, Birkh¨auser, Boston, 2003. [13] F. G´ecseg, I. Pe´ ak, Algebraic theory of automata, Akad´emiai Kiad´o, Budapest, 1972.

14

A

Families of Formal Power Series and Their Rationa Representations

Let X be a finite alphabet, that is, just a finite set of symbols. A formal power series S with coefficients in Rp is a map S : X ∗ → Rp . We denote by Rp ¿ X ∗ À the set of all formal power series with coefficients in Rp . It is easy to see that Rp ¿ X ∗ À forms a vector space with pointwise addition and multiplication by scalar, see [3]. An indexed set of formal power series Ψ = {Sj ∈ Rp ¿ X ∗ À| j ∈ J} is called rational if there exists a finite dimensional vector space X , dim X < +∞, linear maps C : X → Rp , Aσ : X → X , σ ∈ X and an indexed set B = {Bj ∈ X | j ∈ J} of elements of X such that for all j ∈ J, for all σ1 , . . . , σk ∈ X, k ≥ 0, Sj (σ1 σ2 · · · σk ) = CAσk Aσk−1 · · · Aσ1 Bj . The 4-tuple R = (X , {Ax }x∈X , B, C) is called a rational representation (short representation) of Ψ. The number dim X is called the dimension of the representation R and it is denoted by dim R. In the sequel the following short-hand notation will be used Aw := Awk Awk−1 · · · Aw1 for w = w1 · · · wk . We will identify A² with the identity map. A representation Rmin of Ψ is called minimal if for each representation R of Ψ it holds that dim Rmin ≤ dim R. Let R = (X , {Aσ }σ∈X , B, C) be a representation. Define the subspaces WR and OR of X by WR = Span{Aw Bj | w ∈ X ∗ , j ∈ J} , OR =

T

w∈X ∗

ker CAw

(A.1)

We will call the representation R reachable, if dim WR = dim R and we will call R observable, if OR = {0}. Two representations R1 = (X1 , {Az }z∈X , C, B) and ˆ B) ˆ are isomorphic, if there exists a linear isomorphism R2 = (X2 , {Aˆz }z∈X , C, ˆj = T Bj , ∀j ∈ J and T : X1 → X2 such that Aˆσ = T Aσ T −1 ∀σ ∈ X, B −1 Cˆ = CT . For each word w ∈ X ∗ define the left shift of S by the word w as the formal power series w ◦ S ∈ Rp ¿ X ∗ À, defined by ∀v ∈ X ∗ : w ◦ S(v) = S(wv). Let Ψ = {Sj ∈ Rp ¿ X ∗ À| j ∈ J} be a family of formal power series indexed by J. Define the linear space of formal power series WΨ = Span{w ◦ Sj ∈ Rp ¿ X ∗ À| j ∈ J, w ∈ X ∗ } ∗

∗

Define the Hankel-matrix HΨ of Ψ as the infinite matrix HΨ ∈ R(X ×I)×(X ×J) , as follows Let I = {1, 2, . . . , p} and (HΨ )(u,i)(v,j) = (Sj (vu))i , where (Sj (vu))i denotes the ith element of the column vector Sj (vu) ∈ Rp . Notice that the linear space WΨ and the linear space spanned by the columns of HΨ are isomorphic, hence dim WΨ = rank HΨ . The following theorem is an extension of the classical results from [3]. 15

Theorem 5 Let Ψ = {Sj ∈ Rp ¿ X ∗ À| j ∈ J}. The following holds. (i) Ψ is rational if and only if dim WΨ = rank HΨ < +∞, (ii) A representation R of Ψ is minimal if and only if R is reachable and observable. The representation R of Ψ is minimal if and only if dim R = rank HΨ . All minimal representations of Ψ are isomorphic. Consider the tuple RΨ = (WΨ , {Aσ }σ∈X , B, C), where for all σ ∈ X, Aσ : WΨ → WΨ is a linear map such that Aσ (T ) = σ ◦ T for all T ∈ WΨ , the indexed set B = {Bj ∈ WΨ | j ∈ J} is such that Bj = Sj for each j ∈ J, and the linear map C : WΨ → Rp , is defined by C(T ) = T (²), for all T ∈ WΨ . Then RΨ defines a representation of Ψ. The representation RΨ is called free. Since the linear space spanned by the column vectors of HΨ and the space WΨ are isomorphic, one can construct a representation of Ψ over the space of column vectors of HΨ in a way similar to the construction of RΨ . Using the theorem above it is easy to check that the free representation RΨ is minimal. One can also give a procedure, similar to reachability and observability reduction for linear systems, such that the procedure transforms any representation of Ψ to a minimal representation of Ψ. One can check observability and reachability of a given representation numerically. If R = (X , {Aσ }σ∈Σ , B, C) is a representation of Ψ, then by picking a vector space isomorphism T : X → Rn , n = dim R, we can replace R with the isomorphic representation tuple 0 R = (Rn , {T Aσ T −1 }σ∈Σ , T B, CT −1 ), T B = {T Bj | j ∈ J}. Hence w.l.g we can always assume that X = Rn holds for any representation considered. Consider a formal language L ⊆ X ∗ . For more on formal languages see [13]. ∗ ¯ ¯ If  L is a regular language then the power series L ∈ R ¿ X À, L(w) =   1 if w ∈ L

  0 otherwise

is a rational power series, see [3]. Consider two power series

S, T ∈ Rp ¿ X ∗ À. Define the Hadamard product S ¯ T ∈ Rp ¿ X ∗ À by (S ¯ T )j (w) = Sj (w)Tj (w)

(A.2)

for all w ∈ X ∗ , j = 1, . . . , p. Here Sj (w), Tj (w), (S ¯ T )j (w) denote the jth element of the column vector S(w), T (w) and (S ¯ T )(w) respectively. Let Ψ = {Sj ∈ Rp ¿ X ∗ À| j ∈ J} and Θ = {Tj ∈ Rp ¿ X ∗ À| j ∈ J} be indexed sets of formal power series. Define the Hadamard product of Ψ and Θ as Ψ ¯ Θ := {Sj ¯ Tj ∈ Rp ¿ X ∗ À| j ∈ J} (A.3) We can show that the following extension of the classical result from [3] holds. Lemma 1 If Ψ and Θ are both rational, then Ψ ¯ Θ is rational. Moreover, rank HΨ¯Θ ≤ rank HΨ · rank HΘ .

16