Realization Theory of Discrete-Time Linear Hybrid System Mih´ aly Petreczky ∗



Jan H. van Schuppen ∗∗

Systems Engineering Group, Department of Mechanical Engineering, Eindhoven University of Technology PO Box 513, 5600 MB Eindhoven, The Netherlands [email protected] ∗∗ Centrum voor Wiskunde en Informatica (CWI) P.O.Box 94079, 1090 GB Amsterdam, The Netherlands [email protected]

Abstract: The paper addresses realization theory of discrete-time linear hybrid systems without guards (abbreviated by DTHLS ). We present necessary and sufficient conditions for existence of a realization, and a characterization of minimality. In addition, we present basic results on partial realization theory of DTHLSs along with a realization algorithm. We also sketch the application of the obtained results to model reduction and system identification. Keywords: Realisation theory, minimization, identification, hybrid, identifiability 1. INTRODUCTION In this paper we present realization theory of discrete-time linear hybrid systems (DTHLS for short), along with a sketch of possible application of the developed theory to system identification and model reduction. The realization problem is one of the central topics of system theory. For DTHLSs , its can be stated as follows. • When is it possible to construct a (preferably minimal) DTHLS state-space representation which generates the specified input/output behavior ? • How to characterize minimal DTHLSs which generate the specified input/output behavior ? Our motivation for developing realization theory is that it is potentially useful for model reduction, systems identification and systems theory. We will elaborate in greater detail on potential applications in §5. The system class A DTHLS is a discrete-time piecewiseaffine systems without guards, i.e. systems where the switching is initiated externally. The system evolves as follows. As long as no discrete event has occurred, the discrete state remains unchanged and the continuous state evolves according to the current linear system. If a discrete event occurs, then the discrete state changes according to the discrete state-transition map, and the continuous state changes according to the linear reset map. Notice that the discrete events are assumed to be external inputs, and the system has no guards. DTHLSs can be thought of as the discrete-time counterparts of the continuous-time linear hybrid systems without guards of Petreczky (2006); Petreczky and van Schuppen (2008). More precisely, a DTHLS can be obtained from a linear hybrid systems in continuous-time by sampling the latter with a rate ∆ and assuming that the time between two discrete events is an integer multiple of ∆. Motivation The motivation for studying DTHLSs is the following. First, with respect to continuous-time linear hybrid systems, going to discrete-time might make system identification and control synthesis easier. Second,

DTHLSs are a subset piecewise-affine hybrid systems, hence any result on realization theory of the latter class has to be consistent with the corresponding results for DTHLSs. In turn, the relevance of piecewise-affine hybrid systems is widely recognized. In addition, a general discrete-time piecewise-affine hybrid systems can be regarded as a feedback interconnection of a DTHLS with an event generating device. Hence, understanding realization theory of DTHLSs should be useful for systems identification and model reduction of piecewise-affine hybrid systems. The latter statement is elaborated upon in §5. Related work To the best of our knowledge, the results of this paper are new. The realization problem for hybrid systems was first formulated in Grossman and Larson (1995), but no solution was provided. In Paoletti et al. (2007b); Weiland et al. (2006) the relationship between input-output equations and hybrid state-space representations was studied. In Petreczky (2006); Petreczky and Vidal (2007b, 2008) realization theory for various classes of hybrid systems were developed. In particular, the case of continuous-time (bi)linear hybrid systems was already addressed in Petreczky (2006). There is a vast literature on topics related to realization theory, such as system identification, observability and reachability of hybrid systems, see Paoletti et al. (2007a); Vidal et al. (2003); Collins and van Schuppen (2004); Alur et al. (2003). Outline of the paper §2 presents the definition and basic notions for DTHLSs. §3 presents the main theorems of the paper. §4 sketches the main ideas behind the proof of the results. §5 sketches the potential application of the results to systems identification and model reduction. 2. DISCRETE-TIME LINEAR HYBRID SYSTEMS We will start by fixing some notation and terminology. Notation Le N be the set of natural numbers including 0. Let Σ be a finite or infinite set which will be referred to as alphabet. Denote by Σ∗ the set of finite strings or words of elements of Σ, i.e. an element of Σ∗ is a finite sequence of the form w = a1 a2 · · · ak , where a1 , a2 , . . . , ak ∈ Σ, and

k ≥ 0; k is the length of w and it is denoted by |w|. The empty sequence (word) is denoted by , and its length is 0. The concatenation of two strings v = v1 · · · vk , and w = w1 · · · wm ∈ Σ∗ is the string vw = v1 · · · vk w1 · · · wm . For the empty sequence , w = w = w for all w ∈ Σ∗ . We denote by wk the string formed by repeating w consecutively k times. The word w0 is the empty word . 2.1 Definition of Moore-automaton Below we will give a brief introduction to the concept of Moore-automata, see Eilenberg (1974) for more details. Recall that a finite Moore-automaton is a tuple A = (Q, Γ, O, δ, λ) where (1) Q is the finite state-space, (2) Γ is the finite input alphabet, (3) O is the output alphabet, (4) δ : Q × Γ → Q is the state-transition map, (5) λ : Q → O is the readout map. 0

A bijective map φ : Q → Q is an automaton isomorphism 0 0 from A to A , denoted by φ : A → A if φ commutes with the state-transition and readout maps, i.e. φ(δ(q, γ)) = 0 0 δ (φ(q), γ), and λ(q) = λ (φ(q)), for all q ∈ Q, γ ∈ Γ. 2.2 Discrete-Time Linear Hybrid System A discrete-time linear hybrid systems, abbreviated by DTHLS , is a discrete-time system of the form  h(t + 1) = F (h(t), u(t), d(t)) H: (1) z(t) = N (h(t)) where h(t) = (q(t), x(t)) ∈ HH = ∪q∈Q {q} × Rnq is the hybrid state at time t ∈ N and z(t) = (o(t), y(t)) ∈ O × Rp is the output at time t. Here Q is the finite set of discrete states (modes), Rnq = Xq is the continuous state-space associated with a discrete state q ∈ Q, and HH = ∪q∈Q {q} × Rnq denotes the space of hybrid states. Furthermore, O is the finite set of discrete outputs and Rp is the space of continuous outputs. Furthermore, u(t) ∈ Rm e = Γ∪{e}, e ∈ is the continuous input and d(t) ∈ Γ / Γ is the e discrete input. Here Γ = Γ∪{e} is the set of discrete inputs, with Γ being the finite set of discrete events, and e being a symbol not in Γ describing the situation when no discrete event occurs. The space Rm is the set of continuous inputs. The dynamics of H can be described as follows. If d(t) = e∈ / Γ, then the discrete state of H remains unchanged and the continuous state changes according to the discretetime linear system associated with the current mode. If d(t) ∈ Γ, then a discrete event γ = d(t) occurred, and the state changes according to the discrete state-transition map and one of the reset maps. For each state h = (q, x), e u ∈ Rm q ∈ Q, x ∈ Rnq , and inputs d ∈ Γ,  (q, Aq x + Bq u) if d = e ∈ /Γ F (h, u, d) = (ˆ q , Mqˆ,γ,q x + Bqˆu) if d = γ ∈ Γ,ˆ q = δ(q, γ) N (h) = (λ(q), Cq x) where • δ : Q × Γ → Q is the discrete state-transition map • Aq ∈ Rnq ×nq , Bq ∈ Rnq ×m are the matrices of the state equation of the linear system in mode q ∈ Q. • The matrix Mδ(q,γ),γ,q ∈ Rnδ(q,γ) ×nq , q ∈ Q, γ ∈ Γ is the matrix of the linear reset map associated with the transition from q to δ(q, γ) under event γ ∈ Γ. • λ : Q → O is the discrete readout map • The matrix Cq ∈ Rp×nq determines the output of the linear system residing in the discrete mode q ∈ Q.

We fix an initial state (qinit , xinit ) of the system H. Notation 1. A DTHLS of the form (1) is denoted by H = (A, Rm , Rp , (Rnq , Mq )q∈Q , (qinit , xinit )) where, • A = (Q, Γ, O, δ, λ) is the Moore-automaton formed by the discrete-state transition and discrete readout map of the system H. • Mq = (Aq , Bq , Cq , {Mδ (q,γ),γ,q }γ∈Γ ) is the collection of the matrices of the linear system residing in q and of the reset matrices of transitions originating from q. • Rnq is the state-space of the linear system residing in the discrete state q ∈ Q. • (qinit , xinit ) with qinit ∈ Q, and xinit ∈ Rnqinit is the initial state of H. Notice that a DTHLS above may arise by sampling a continuous-time linear hybrid system Hc without guards, Petreczky and van Schuppen (2008) with a fixed sampling rate and with the following assumptions. We assume that at each sampling time, a continuous input is fed into the system. Furthermore, discrete events are fed into the system at sampling times only, but it might happen that no event is generated at a sampling time. The latter can be seen as generation of the dummy event e. 2.3 Basic system-theoretic concepts for DTHLSs Throughout the section H is a DTHLS of the form (1). e the Notation 2. (Hybrid inputs). Denote by U = (Rm × Γ) set of inputs of H. Notation 3. (State and output). Fix a state h ∈ HH of H, and a sequence w = e(0)e(1) · · · e(t − 1) ∈ U ∗ for some t ≥ 0, with e(i) = (u(i), d(i)), with u(i) ∈ Rm , and e i = 0, 1, 2, . . . , t − 1. d(i) ∈ Γ, Denote by ξH (h, w) the state of H reached from the initial state h with the input w at time t. If t = 0, i.e. w = , then ξH (h, w) = h. Denote by υH (h, w) the output of H under the input sequence w, if started in h, i.e. υH (h, w) = N (ξH (h, w)). Next, we define span-reachability and observability. Definition 1. (Span-reachability). The DTHLS H is called span-reachable, if for every state (q, x) of H there exist hybrid inputs wi ∈ U ∗ and reals αi ∈ R, i = 1, 2, . . . , k such that (q, xi ) = ξH ((qinit , xinit ), wi ), xi ∈ Rnq , i = Pk 1, 2, . . . , k, and x = i=1 αi xi , i.e. q is reachable, and x is a linear combination of the reachable continuous states. Definition 2. (Observability). Two distinct states h1 6= h2 of H are indistinguishable if ∀w ∈ U ∗ : υH (h1 , w) = υH (h2 , w), that is, the input-output maps induced by h1 and h2 are equal. H is observable if it has no pair of distinct indistinguishable states. It follows that the input-output maps of interest are maps of the form f : U ∗ → O × Rp . Definition 3. (Realization). The input-output map f : U ∗ → O × Rp is realized by H if the input-output map of H induced by the initial state equals f , i.e. ∀w ∈ U ∗ : υH ((qinit , xinit ), w) = f (w) We define the dimension of a DTHLS as follows.

Definition 4. (Dimension). The dimension dim H of H is definedPas a pair of natural numbers; dim H = (card(Q), q∈Q nq ) ∈ N × N, where card(Q) denotes the P number of discrete states, and q∈Q nq is the sum of dimensions of the continuous state-spaces. For each two pairs of natural numbers (m, n), (p, q) ∈ N×N define the partial order relation as (m, n) ≤ (p, q), if m ≤ p and n ≤ q. Notice that with respect to the above ordering, not all pairs are comparable. Definition 5. (Minimality). H is a minimal realization of the input-output map f , if H is a realization of f and 0 0 for any DTHLS H such that H is a realization of f , 0 dim H ≤ dim H . Remark 1. Notice that Definition 5 requires the dimension of a minimal DTHLS to be comparable with the dimension of any other DTHLS realization of f . Since we use a partial order on dimensions, the existence of a minimal DTHLS satisfying Definition 5 has to be shown. When it exists, a minimal DTHLS is a minimal element of the set of DTHLS realizations of f , taken with the partial order on dimensions defined above. Next, we define the notion of DTHLS isomorphisms. 0

Definition 6. (Isomorphism). Let H be the DTHLS 0

0

0

0

0

0

0

0

(2)

for some γ1 , γ2 , . . . , γk ∈ Γ, α1 , α2 , . . . , αk+1 ∈ N, and k ≥ 0. Recall that eα is the word obtained by repeating e consecutively α times. Informally, f has a hybrid convolution representation if,

0

with A = (Q , Γ, O, δ , λ ), and for all q ∈ Q , 0

s = eα1 γ1 eα2 γ2 eα3 γ3 · · · γk−1 eαk γk eαk+1

0

(A , Rm , Rp , (Rnq , Mq )q∈Q0 , (qinit , xinit )) 0

Remark 2. (Rank conditions). Similarly to the continuoustime linear hybrid systems, one can formulate algebraic characterization of observability and span-reachability of DTHLS. This characterization yields numerical algorithms for checking observability and span-reachability and for transforming a DTHLS realization of f to an observable and span-reachable, hence minimal, DTHLS which realizes f . Hence, there exists a minimization algorithm for DTHLSs. Existence of a realization We start with defining the notion of hybrid convolution representation of an inputoutput map, existence of which is a necessary condition for realizability. To this end, Notation 4. (Discrete and continuous-valued components). For each input-output map f , denote by fC the Rp -valued part, and by fD the O-valued part of the map f . That is, f (w) = (fD (w), fC (w)) ∈ O × Rp for all w ∈ U ∗ . e = Γ∪{e}. Every word Remark 3. Recall the definition of Γ ∗ e can be uniquely written as s∈Γ

0

(a) fD depends only on the discrete events, (b) fC is affine in continuous inputs.

0

Mq = (Aq , Bq , Cq , {Mδ0 (q,γ),γ,q }γ∈Γ ) 0

Let H be a DTHLS of the form (1). H and H are 0 isomorphic, if there exists a bijection SD : Q → Q , and non-singular matrices Sq ∈ Rnq ×nq , q ∈ Q such that 0

• SD is an automaton isomorphism from A to A . • For all q ∈ Q and qˆ = SD (q), the dimensions of the continuous state-spaces corresponding to the discrete states q and qˆ are equal, i.e. nq = nqˆ, and Sq is an isomorphism between the linear systems in q and qˆ, 0 0 0 i.e. Sq Aq Sq−1 = Aqˆ, Sq Bq = Bqˆ, Cq Sq−1 = Cqˆ • The maps Sq , Sqˆ, where qˆ = SD (q), commute with the reset maps, i.e. 0

∀γ ∈ Γ : Sδ(q,γ) Mδ(q,γ),γ,q = Mδ0 (ˆq,γ),γ,ˆq Sq

More formally, the definition goes as follows. Definition 7. (Hybrid conv. repr.). The input-output map f is said to have hybrid convolution representation if for each sequence of discrete events v = γ1 γ2 · · · γk ∈ Γ∗ , γ1 , γ2 , . . . , γk ∈ Γ, k ≥ 0, there exist sequences Kvf : Nk+1 → Rp and Gfv,j : Nj → Rp×m where j = 1, 2, . . . , k + 1, and a discrete output φfD (v) ∈ O, such that the following holds. Consider a sequence of hybrid inputs w = (u1 , s1 )(u2 , d2 ) · · · (ur , dr ) ∈ U ∗ e and ui ∈ Rm , i = 1, 2, . . . , r, and such that where di ∈ Γ e ∗ is of the form (2), and the the word s = d1 d2 · · · dr ∈ Γ word u = u1 u2 · · · ur formed by the continuous inputs is u = u1 u2 · · · uk+1

0

• The initial states of H and H 0 are related as 0 SD (qinit ) = qinit , and Sqinit xinit = xinit . 3. MAIN RESULTS ON REALIZATION THEORY First we state the main result on minimality of a DTHLS. Second, we formulate necessary and sufficient conditions for existence of a DTHLS realization. Finally, we discuss partial realization theory and realization algorithms. Throughout the section f : U ∗ → O × Rp is an inputoutput map for which a DTHLS realization is sought. Minimality We can state the following characterization of minimality of DTHLSs. Theorem 1. (Minimal realization). If f has a DTHLS realization, then f has a minimal DTHLS realization. The DTHLS H is a minimal realization of f , if and only if H is span-reachable and observable. All minimal DTHLS realizations of f are isomorphic.

(3) m

where u1 = u1,1 u1,2 · · · uα1 , u1,1 , u1,2 , . . . , u1,α1 ∈ R , and ui = ui,0 ui,1 ui,2 · · · ui,αi with ui,0 , ui,1 , . . . , ui,αi ∈ Rm for all i = 2, 3, . . . , k + 1. In addition, let u1,0 = 0, and let α1 , α2 , . . . , αk+1 ∈ N be the same as in (2) for s. Then, fD (w) = φfD (v) fC (w) = Kvf (α1 , α2 , . . . , αk+1 )+ αk+1 αk X f X Gv,1 (αk+1 − j)uk+1,j + Gfv,2 (αk − j, αk+1 )uk,j + j=0

j=0

··· +

α1 X

Gfv,k+1 (α1 − j, α2 , . . . , αk+1 )u1,j

j=0

Remark 4. (Abuse of notation). In the sequel, whenever an input-output map f admits a hybrid convolution representation, its discrete valued component fD will be identified with the map of the form fD : Γ∗ 3 v 7→ φfD (v) ∈ O.

The role of the sequences Kvf and Gfv,j is best understood by analogy with the linear case. Recall from Callier and Desoer (1991) that the map y : (Rm )∗ → Rp is realizable by a linear system, if there exist sequences G : N → Rp×m and K : N → Rp , such that y(u0 u1 · · · ut−1 ) = K(t) + Pt−1 j=0 G(t − j)uj . The requirement that f has a hybrid convolution representation is analogous to requiring that y is of the above convolution form, with Kvf and Gfv,j playing the roles of K and G. Next, we define the notion of the Hankel-matrix of an input-output map f admitting a hybrid convolution representation. The entries of the Hankel-matrix will be the generalized Markov-parameters of f . In turn, the latter can be obtained from Gfv,j and Kvf . Definition 8. (Markov-parameters). Assume that f has a hybrid convolution representation. Define the maps Zf : e ∗ → Rp and Zf,j : Γ e ∗ → Rp , j = 1, 2, . . . , m, as follows. Γ ∗ e of the form (2), For any word s ∈ Γ Zf (s) = Kvf (α1 , α2 , · · · , αk+1 )  Gv,k+2−l (αl − 1, αl+1 , . . . , αk+1 )ej if s ∈ / Γ∗ Zf,j (s) = 0 if s ∈ Γ∗ where v = γ1 γ2 · · · γk , and if s ∈ / Γ∗ , i.e. αi , i = 1, 2, . . . , k+ are not all zero, then l ∈ {1, 2, . . . , k + 1} is such that α1 = α2 = · · · = αl−1 = 0 and αl > 0, and ej is the jth unit vector in Rm . The collection {Zf , Zf,j }j=1,2,...,m is referred to as the generalized Markov parameters of f . The maps {Zf,j , Zf }j=1,2,...,m can be thought of as generalizations of Markov parameters for linear systems. They can be computed from the response of f to specific inputs. Remark 5. (Computing the Markov-parameters). For any e ∗ , one can find input sequences w0 , w1 , . . . , wm ∈ U ∗ s∈Γ of the same length as s, such that Zf (s) = fC (w0 ) and Zf,j (s) = fC (wj ) − fC (w0 ), j = 1, 2, . . . , m. As in the linear case, the Markov-parameters of f can be expressed via the matrices of a DTHLS realizing f . Proposition 1. Let H be a DTHLS of the form (1). Then H is a realization of f if and only if f has a hybrid convolution e ∗ of the form (2), and for representation and for all s ∈ Γ all j = 1, 2, . . . , m, the following holds 1 k+1 k Zf,j (s) = Cqk Aα Mqk ,γk ,qk−1 Aα qk qk−1 · · · l+1 l −1 · · · Aα Mql ,γl ,ql−1 Aα / Γ∗ ql ql−1 Bql−1 ej if s ∈ k k+1 Mqk ,γk ,qk−1 Aα Zf (s) = Cqk Aα qk qk−1 · · ·

(4)

α1 2 · · · Aα q1 Mq1 ,γ1 ,q0 Aq0 xinit

fD (γ1 γ2 · · · γk ) = λ(qk ) where for s ∈ / Γ∗ , l ∈ {1, 2, . . . , k + 1} is such that αl > 0, and α1 = . . . = αl−1 = 0, ej is the jth unit vector of Rm , q0 = qinit and qi+1 = δ(qi , γi+1 ) for each i = 0, 1, . . . , k −1. The Hankel-matrix for DTHLSs is defined as follows. Definition 9. (Hankel-matrix). We define the Hankel matrix of f , denoted by Hf , as the following infinite matrix. Consider the index set If = {f } ∪ {1, 2, . . . , m}. The 1

In (4), if l = k + 1, then the right-hand side of the first equality α −1 becomes Cqk Aqkk+1 Bqk ej . Likewise, if k = 0, then the right-hand 1 side of the second equality becomes Cq0 Aα q0 xinit .

e∗ × columns of the matrix Hf are indexed by pairs (s, l) ∈ Γ If . The rows of the matrix Hf are indexed by pairs of the e ∗ × {1, 2, . . . , p}. The element of Hf with form (g, i) ∈ Γ the row index (g, i) and column index (s, l) is defined as  (Zf (sg))i ∈ R if l = f (Hf )(g,i),(s,l) = (Zf,j (sg))i ∈ R if l = j ∈ {1, 2, . . . , m} where (Zf (sg))i and (Zf,l (sg))i denote the ith entry of the column vectors Zf (sg) ∈ Rp and Zf,l (sg) ∈ Rp respectively. The column vector of Hf indexed by (s, l), e ∗ × {1, 2, . . . , p}) 3 denoted by (Hf ).,(s,l) , is the map (Γ (g, i) 7→ (Hf )(g,i),(s,l) ∈ R. The set of maps from elements e ∗ × {1, 2, . . . , p}) to the real numbers, denoted by of (Γ ∗ Γ ×{1,2,...,p} Re , is a vector space with point-wise addition and multiplication by scalar, and hence we can speak of the linear vector space spanned by the column vectors of Hf . The rank of Hf (denoted by rank Hf ) is the dimension of the vector space spanned by the column vectors of Hf . We also need the following sub-matrix of Hf . Definition 10. (Table Hf,O ). Let Hf,O be the sub-matrix of Hf formed by those columns of Hf which are indexed by indices of the form (v, j) with j ∈ {1, 2, . . . , m} and v ∈ Γ∗ , i.e.v contains no symbol e. The number of distinct column vectors of Hf,O is denoted by card(Hf,O ). Here, two column vectors are considered to be equal, if they are e ∗ × {1, 2, . . . , p}) to R. equal as maps from (Γ The intuition behind the Hf,O will be explained after the main theorem. Next, we define the counterpart of the Hankel-matrix for the discrete-valued components of of f . Definition 11. (Hankel-table WfD ). Define the discrete Hankel-table WfD of f as the infinite table whose entries are elements of O, and whose columns and rows are indexed by sequences of discrete events. The entry of WfD indexed by the row index s ∈ Γ∗ and column index v ∈ Γ∗ equals fD (vs). The column vector of WfD indexed by v can be identified with a map Γ∗ 3 s 7→ fD (vs) ∈ O. We say that two column vectors of WfD are equal if they are equal if viewed as maps as described above. The number of distinct column vectors of WfD is denoted by card(WfD ). Theorem 2. (Existence). An input-output map f has a realization by a DTHLS if and only if (1) f has a hybrid convolution representation, and (2) the rank of the Hankel-matrix Hf is finite, and each of the tables Hf,O and WfD have finitely many distinct column vectors, i.e. rank Hf < +∞, card(WfD ) < +∞, card(Hf,O ) < +∞. Remark 6. (DTHLS from Hf and WfD ). We can compute a minimal DTHLS realization of f from the columns of the Hankel-matrix Hf and of the Hankel-table WfD . The intuition behind the theorem is the following. The finite rank of the Hankel-matrix Hf ensures that generalized Markov-parameters can be represented as products of matrices, as in Proposition 1. The finite number of column vectors of Hf,O is inspired by the following observation. If the DTHLS H realizes f , then the columns of Hf,O indexed by (v, j), j = 1, 2, . . . , m for some fixed v ∈ Γ∗ encode the continuous-valued component of the inputoutput map induced by the state (q, 0) of H. Here q is the discrete state reachable from the initial discrete state with the sequence of discrete-events v ∈ Γ∗ . Since the number of

such input-output maps is finite, Hf,O has to have finitely many column vectors. Finally, the condition that WfD has finitely many column vectors ensures that fD is realizable by a Moore-automaton. Partial-realization theory The results above allow us to construct a minimal DTHLS realization from the (infinite number) of Markov-parameters of the map f . The question arises what can be computed from finitely many Markovparameters. This leads to the partial realization problem. Definition 12. (N -partial realization). Assume that f has a hybrid convolution representation. Assume that H is a DTHLS of the form (1). Then H is said to be an N -partial realization of f , if (4) holds for all j = 1, 2, . . . , m and for e ∗ of length at most N , i.e. |s| ≤ N . all s ∈ Γ Notice that H is a realization of f if and only if H is an N -partial realization of f for all N ∈ N. Note that H being a N -partial realization involves only a finite number of Markov-parameters of f . For the results on partial realization, we need the following notation. Notation 5. Define Hf,K,L (resp. Hf,O,K,L ) as the finite matrix formed by the intersection of rows of Hf (resp. e ∗ ×{1, 2, . . . , p} with s of length Hf,O ), indexed by (s, i) ∈ Γ at most K, and columns of Hf , (resp. Hf,O ), indexed by e ∗ ×If (resp. (v, l) ∈ Γ∗ ×{1, 2, . . . , m} for the case (v, l) ∈ Γ of Hf,O,K,L ), such that v is of length at most L. Define the finite table WfD ,K,L , as the sub-table of WfD formed by the intersection of those rows of WfD which are indexed by words of length at most K, and those columns of WfD which are indexed by words of length at most L. Denote by card(Hf,O,N,N ) and card(WfD ,N,N ) the number of distinct column vectors of Hf,O,N,N and WfD ,N,N respectively. Theorem 3. Under conditions on N ∈ N, similar to the ones for partial realization of linear systems, it is possible to compute a 2N + 1-partial realization of f from Hf,N +1,N , WfD ,N +1,N and Hf,O,N +1,N . Moreover, if rank Hf,N,N = rank Hf , card(WfD ,N,N ) = card(WfD ), (5) card(Hf,O,N,N ) = card(Hf,O ) then the conditions for existence of a 2N + 1 partial realization hold, and a minimal (complete, not partial) DTHLS realization of f can be computed from Hf,N,N and WfD ,N,N . In addition, if dim H = (nd , nc ) and N ≥ max{nd , nc + nd m}, for some DTHLS H such that H is a realization of f , then (5) holds. 4. SKETCH OF PROOF OF THE RESULTS Similarly to continuous-time (bi)linear hybrid systems Petreczky (2006), the realization problem for DTHLSs can be solved using the theory of rational hybrid formal power series Petreczky (2006). The latter is an extension of the theory of rational formal power series Berstel and Reutenauer (1984). A hybrid formal power series is a combination of classical formal power series and input-output maps of Mooreautomata. A hybrid representation is an infinite-state Moore-automaton whose state-space has both discrete and continuous components. A hybrid formal power series is rational, if it is the input-output map of a hybrid representation, if the latter is viewed as a Moore-automaton.

That is, hybrid formal power series are external behaviors, and hybrid representations are potential state-space representations of these behaviors. With this correspondence in mind, one can formulate a realization problem for hybrid power series and hybrid representations. The solution of that realization problem and the corresponding algorithms are described in Petreczky (2006). The solution of the realization problem for DTHLSs proceeds then as follows. Find a correspondence between DTHLSs and hybrid representations. Find a correspondence between an input-output map of DTHLS and a hybrid power series. Furthermore, require that the hybrid representation corresponding to a DTHLS is a state-space representation of the hybrid power series corresponding to the input-output map of that DTHLS. For DTHLSs the correspondence required above is easy to find and the transformation from DTHLS to hybrid representation and back are computationally effective. In addition, the correspondence between DTHLSs and hybrid representations preserves dimension, systems isomorphism, spanreachability, observability and minimality, This allow us to reduce the realization problem for DTHLSs to the realization problem for hybrid formal power series. The results of this paper follow then from the general results for hybrid power series, by applying the correspondence described above. In addition, the corresponding algorithms for hybrid representations can be used for DTHLSs. 5. RELEVANCE FOR IDENTIFICATION AND FOR MODEL REDUCTION In this section we sketch the significance of the presented results for systems identification and model reduction. System identification algorithm As the subspace identification algorithms van Overschee and Moor (1996) demonstrate, realization theory can yield efficient systems identification algorithms. In fact, our results yield the following identification algorithm for DTHLS. Consider an input-output map f . Fix a number N ∈ N. The choice of N can be based on the estimate on the dimension of a potential realization, or our ability to make measurements for inputs of length 2N + 1. 1: Compute the Markov-parameters Zf,j (s), Zf (s) for all e ∗ of length at most 2N +1 and the values of fD (v) s∈Γ for all v ∈ Γ∗ with v being of length at most 2N +1. By Remark 5, this can be done using the responses of f to input sequences of length 2N +1. Construct the matrix Hf,N +1,N , and the tables WfD ,N +1,N , Hf,O,N,N . 2: Compute the 2N + 1 partial realization of f . It follows from Theorem 3, that if f can indeed be realized by a DTHLS , then for large enough N the procedure above returns a minimal DTHLS realization of f . In fact, the right value of N can be chosen based on the dimension of a potential DTHLS realization of f . The above algorithm computes a realization from outputs corresponding to several input sequences. For identification one needs another algorithm which computes a realization of some fixed dimension from the available single time series of measurements, at least for a sufficiently large time series. The authors have not yet succeeded in obtaining such an algorithm. For linear systems, obtaining such algorithms from realization theory took several decades.

Spaces of DTHLSs Realization theory is useful for studying the geometry and topology of the space of DTHLS. The latter is useful for system identification, fault detection and computer vision Vishwanathan et al. (2007). For linear systems, the topology and geometry of Hankel-matrices were investigated and used in systems identification, see Peeters (1994) and the references therein. We believe that similar results can be obtained for DTHLSs, using the relationship between DTHLSs and (hybrid) formal power series, and by further extending the results of Sontag (1987); Petreczky and Vidal (2007a). Model reduction First, the minimization algorithms for DTHLSs themselves represent a primitive model reduction method. Second, partial-realization theory can also be used for model reduction, by extending the moment matching method, see Antoulas and Sorensen (2001), to DTHLSs. The core of moment matching is to approximate a high-order system with a lower order one, such that the lower order system is a partial realization of a finite number of Markovparameters of the original system. For DTHLSs the role of Markov-parameters (or moments) is played by the generalized Markov-parameters. Identifiability for DTHLSs The results on existence and uniqueness of minimal DTHLS realizations are useful for studying identifiability of DTHLSs. I.e. for the structural identifiability of a parametrization of minimal DTHLSs, it is necessary and sufficient that no two different parameters yield isomorphic systems. One can then extend van den Hof (1998) to study structural identifiability. Piecewise-affine hybrid systems It is easy to see that a discrete-time piecewise-affine system with guards and linear (affine) reset maps (abbreviated by PAHS) can be thought of as the output feedback interconnection of a DTHLS with an event generator. Hence, a necessary condition for minimality of a PAHS is that its DTHLS component is minimal. Furthermore, we can replace the DTHLS component of a PAHS with a minimal one, without changing the input-output behavior of the PAHS. This implies that model reduction algorithms for DTHLSs can be used for model reduction of PAHSs. In addition, a necessary condition for identifiability of a parametrization of PAHSs is that no two distinct parameters yield PAHSs whose DTHLS components realize the same behavior. If the latter condition is violated, then the corresponding PAHSs will also yield the same inputoutput behavior, and hence the parametrization will not be identifiable. That is, (structural) identifiability of the DTHLS components is a necessary condition for (structural) identifiability of PAHSs. Hence, the results of the paper are relevant for identifiability of PAHSs. 6. CONCLUSIONS AND FUTURE WORK We have sketched realization theory of discrete-time linear hybrid systems along with its potential applications to systems identification and model reduction. Topics of further research include realization theory for piecewise-affine systems with guards, and application of the presented results to system identification and model reduction.

Acknowledgment. This work was supported in part by the ITEA project Twins 05004. REFERENCES Alur, R., Dang, T., and Ivancic, F. (2003). Progress on reachability analysis of hybrid systems using predicate abstraction. In Hybrid Systems: Computation and Control, LNCS 2623, 4–19. Springer-Verlag. Antoulas, A.C. and Sorensen, D.C. (2001). Approximation of large-scale dynamical systems: an overview. Int. J. Appl. Math. Comput. Sci., 11(5), 1093–1121. Berstel, J. and Reutenauer, C. (1984). Rational series and Their Languages. Springer-Verlag. Callier, F.M. and Desoer, C.A. (1991). Linear System Theory. Springer-Verlag. Collins, P. and van Schuppen, J.H. (2004). Observability of piecewise-affine hybrid systems. In Hybrid Systems: Computation and Control. Eilenberg, S. (1974). Automata, Languages and Machines. Academic Press, New York, London. Grossman, R. and Larson, R. (1995). An algebraic approach to hybrid systems. Theoretical Computer Science, 138, 101–112. Paoletti, S., Juloski, A., Ferrari-Trecate, G., and Vidal, R. (2007a). Identification of hybrid systems: A tutorial. European Journal of Control, 13(2-3), 242 – 260. Paoletti, S., Roll, J., Garulli, A., and Vicino, A. (2007b). Input/ouput realization of piecewise affine state space models. In 46th IEEE Conf. on Decision and Control. Peeters, R. (1994). System Identification Based on Riemannian Geometry: Theory and Algorithms. Ph.D. thesis, VU University Amsterdam. Petreczky, M. (2006). Realization Theory of Hybrid Systems. Ph.D. thesis, VU University Amsterdam. Available at http://www.cwi.nl/~mpetrec. Petreczky, M. and van Schuppen, J.H. (2008). Realization theory for linear hybrid systems. Submitted to a journal. Petreczky, M. and Vidal, R. (2007a). Metrics and topology for nonlinear and hybrid systems. In Hybrid Systems: Computation and Control, 459–472. Petreczky, M. and Vidal, R. (2007b). Realization theory of stochastic Jump-Markov linear systems. In 46th IEEE Conf. on Decision and Control. Petreczky, M. and Vidal, R. (2008). Realization theory for semi-algebraic hybrid systems. In Hybrid Systems: Computation and Control, 386–400. Sontag, D.E. (1987). A remark on bilinear systems and moduli spaces of instantons. Systems and Control Letters, 9(5), 361–367. van den Hof, J. (1998). Structural identifiability of linear compartmental systems. IEEE Transactions Automatic Control, 43, 800–818. van Overschee, P. and Moor, B.D. (1996). Subspace Identification for Linear Systems. Kluwer. Vidal, R., Sastry, S., and Chiuso, A. (2003). Observability of linear hybrid systems. In Hybrid Systems: Computation and Control. Vishwanathan, S.V.N., Smola, A.J., and Vidal, R. (2007). Binet-Cauchy kernels on dynamical systems and its application to the analysis of dynamic scenes. International Journal of Computer Vision, 73(1), 95–119. Weiland, S., Juloski, A.L., and Vet, B. (2006). On the equivalence of switched affine models and switched ARX models. In 45th IEEE Conf. on Decision and Control.

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