On the input-output representation of piecewise affine state space ...

On the input-output representation of piecewise affine state space models S. Paoletti† , J. Roll‡ , A. Garulli† , A. Vicino† Abstract This paper addresses the conversion of discrete-time PieceWise Affine (PWA) state space models into input-output form. Necessary and sufficient conditions for the existence of equivalent input-output representations of a given PWA state space model are derived. Connections to the observability properties of PWA models are investigated. Under a technical assumption, it is shown that every finite-time observable PWA model admits an equivalent input-output representation. When an equivalent inputoutput model exists, a constructive procedure is presented to derive its structure. Several examples illustrate the proposed results.

I. I NTRODUCTION PieceWise Affine (PWA) models are a special class of nonlinear models, obtained by partitioning the state-input domain into a finite number of polyhedral regions, and by considering linear/affine submodels in each region [1]. In other words, the state-update and output maps of a PWA model are both piecewise affine. Static and dynamical systems modelled by piecewise affine maps have been considered in several fields, such as neural networks [2], electrical networks [3], and time-series analysis [4]. PWA models have also been profitably used for analysis [5], [6] and control [1] of classes of nonlinear systems. Indeed, PWA models can be used to describe nonlinear and hybrid phenomena that are frequent in practical situations, e.g., when the system dynamics changes due to physical limits, dead-zones, switches and thresholds. In addition, since PWA maps have universal approximation properties, which means that every (sufficiently smooth) nonlinear function can be approximated arbitrarily well by a PWA function [7], [8], PWA models can also be used to approximate nonlinear systems that do not exhibit discontinuous or switching †

S. Paoletti, A. Garulli and A. Vicino are with Dipartimento di Ingegneria dell’Informazione, Universit`a di Siena, via Roma 56,

53100 Siena, Italy, and with Centro per lo Studio dei Sistemi Complessi, Universit`a di Siena, via Tommaso Pendola 37, 53100 Siena, Italy (e-mail: [email protected]; [email protected]; [email protected]). ‡

J. Roll is with Division of Automatic Control, Link¨oping University, SE-581 83 Link¨oping, Sweden (e-mail: [email protected]).

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behavior. However, despite the fact that they are just a composition of linear/affine time-invariant systems, the structural properties of PWA systems like observability, controllability and stability are complex and articulated (see, e.g., [9]-[16] and references therein), as is typical for nonlinear systems. Similarly to PWA models in state space form, PieceWise affine AutoRegressive models with eXogenous inputs (PWARX) are defined by building the regression vector with past system outputs and inputs, by partitioning the regressor domain into a finite number of polyhedral regions, and by considering ARX submodels in each region. Thanks to the universal approximation properties of PWA maps, PWARX models represent an attractive model structure for system identification [17]. In the last years this motivated the flourishing of several identification techniques providing PWARX models of nonlinear and hybrid systems from data. An overview and classification of different PWA system identification techniques can be found in [18] and the tutorial paper [19]. The equivalence between PWA models and other classes of hybrid models has been studied in the literature [9], [14], [20]. Equivalence results like those mentioned are important in order to establish the capabilities of representation of each class, and possibly transfer theoretical properties and tools between classes. Nevertheless, a complete realization theory for PWA models is still missing. Converting a PWARX model into an equivalent state space representation is straightforward, as is shown for example in [21], but the realization is typically not minimal. Recently, the problem of finding a PWA model which realizes a specified output trajectory has been investigated in [22] for autonomous systems. Input-output realization of PWA state space models has received even less attention so far. Some relationships between the state space and the input-output form of PWA models have been investigated in [23], where the authors consider the problem of finding an input-output representation of the same order as the state space model. Sufficient conditions are derived where all submodels are either in controllability or observability form, and the switching depends only on the input and output signals. It is apparent that results concerning the input-output realization of PWA state space models could open promising perspectives towards the development of a complete realization theory for this class of models. In particular, by clarifying the existing relationships between equivalent state space and input-output representations, the mentioned results could be useful for tackling the minimal state space realization problem for PWARX models. Moreover, the same results could be useful for transferring the prior knowledge possibly available in the state space to the identification of input-

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output models. Indeed, the physics of a system is typically known in the state space, while most PWA system identification methods (e.g., [24]-[28]) have been developed for PWARX models. It is worthwhile to note that the input-output realization problem has been more thoroughly studied for switched affine models (i.e. collections of affine models for which the switching signal is exogenous, and does not depend on the state and the input of the system). Preliminary results in [29] have shown that every observable switched affine model admits a representation as a switched ARX model. A necessary and sufficient condition for input-output realization of switched affine models has been recently presented in [30]. This paper provides the solution to the input-output realization problem for general PWA state space models in which the switching signal depends on both the state and the input. Necessary and sufficient conditions are presented for the existence of equivalent PWARX representations of a given PWA state space model. Differences with respect to the case of linear state space models are discussed. Connections to the observability properties of PWA models are investigated. While PWA models that are observable in infinite time do not admit equivalent input-output representations, it is shown that, under a mild technical assumption, finite-time observable PWA models can always be converted into the PWARX form. When an equivalent PWARX model exists, a constructive procedure is presented to derive its structure. It is shown that the number of modes and the number of parameters can grow considerably when converting a PWA state space model into a minimum-order equivalent input-output representation. Several examples are presented to illustrate the proposed equivalence results. The paper is structured as follows. Section II introduces the notation, recalls some basic notions and definitions on polyhedral sets and PWA models, and states the input-output realization problem to be addressed and solved in Section III. Connections to the observability properties of PWA models are also investigated in Section III. Section IV addresses the implementation of the results presented in Section III. Examples are reported in Section V. Finally, conclusions are drawn in Section VI. II. P RELIMINARIES AND PROBLEM FORMULATION A. Notation and definitions The sets {Xi }ni=1 form a complete partition of the set X if

Sn

i=1

Xi = X and Xi

T

Xj = ∅,

∀i 6= j. The m-ary Cartesian product of a set X is denoted by X m . The difference of two sets X and Y is denoted by X \ Y. The sets of real, integer and positive integer numbers are denoted

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by R, Z, and Z+ , respectively. The set of real matrices with m rows and n columns is denoted by Rm×n . An m × n matrix with 0 everywhere is denoted by 0m×n , while In denotes the n × n (j)

identity matrix. The n × mn matrix En,m is defined as (j) En,m = [ 0n×(j−1)n In 0n×(m−j)n ],

j = 1, . . . , m.

(1)

For a vector v ∈ Rn , kvk denotes any norm of v, while kvk∞ is the infinity norm of v. The set of all linear combinations of the row vectors v 1 , . . . , v p is denoted by span{v 1 , . . . , v p }. For a matrix A ∈ Rm×n , span(A) denotes the span of the rows of A. The stack of the values of the  ∞ discrete-time signal z(k) k=0 from time k1 to time k2 ≥ k1 is denoted by z kk21 , i.e. z kk21 = [ z(k2 )⊤ z(k2 − 1)⊤ . . . z(k1 )⊤ ]⊤ .

(2)

B. Affine and polyhedral sets This subsection recalls some basic notions on affine and polyhedral sets. For thorough definitions and proofs, the interested reader is referred to [31]. A set A ⊆ Rn is affine if it can be expressed as A = {x = x0 + v : v ∈ L}, where x0 ∈ Rn

and L ⊆ Rn is a linear subspace. The dimension of an affine set A, denoted by dim(A), is defined as the dimension of the corresponding linear subspace L.

A (convex) polyhedron P ⊆ Rn is defined as the solution set of a finite number of linear

inequalities, namely P = {x : Ax  b}, where A ∈ Rp×n , b ∈ Rp , and  denotes component-

wise inequality. The set of all affine combinations of points in P is called the affine hull of P, and denoted by aff(P): +

aff(P) = {x = α1 x1 + . . . + αk xk : k ∈ Z ,

k X i=1

αi = 1, xi ∈ P}.

(3)

The affine hull of P is the smallest affine set that contains P, in the sense that every affine set A that contains P, also contains aff(P). The dimension dim(P) of a polyhedron P is defined as the dimension of its affine hull. A polyhedron P ⊆ Rn is called full-dimensional if its dimension is n. The relative interior of P, denoted by relint(P), is the set relint(P) = {x ∈ P : B(x, ε) ∩ aff(P) ⊆ P for some ε > 0},

(4)

where B(x, ε) is the ball of radius ε and center x in some norm of Rn (the definition of relative interior does not depend on the choice of a particular norm). Note that relint(P) is never empty, unless P is itself empty. If P1 and P2 are polyhedra, and dim(P1 ∩ P2 ) = dim(P1 ), then x ∈ relint(P1 ∩ P2 ) implies x ∈ relint(P1 ).

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Given the affine transformation y = T (x) defined by y = T x + q, with T ∈ Rm×n and

q ∈ Rm , and the polyhedron X = {x : Ax  b} ⊆ Rn , denote as Y the image of X through

the affine transformation T , i.e. Y = {y : y = T (x), x ∈ X }. In addition, let {v 1 , . . . , v r } be a basis of the linear subspace corresponding to aff(X ). Then: 1) The set Y is a polyhedron in Rm .


2) aff(Y) = T (aff(X )), and dim(Y) = rank [ T v 1 . . . T v r ] .

3) y = T (x) ∈ relint(Y) if and only if x ∈ relint(X ). From item 2) it follows that:

4) If T has full-column rank, i.e. rank(T ) = n, then dim(Y) = dim(X ). 5) If X is full-dimensional, i.e. dim(X ) = n, then dim(Y) = rank(T ). C. PWA state space models Piecewise affine models in state space form are obtained by partitioning the state-input domain into convex polyhedra, and by associating affine state-update and output functions to each polyhedron [1]. Formally, a PWA state space model is defined by the relations: 

x(k + 1)





Ai

= y(k) Ci   x(k)   ∈ Ω, u(k) 

Bi Di

 

x(k) u(k)





+

fi gi

 



if 

x(k) u(k)



 ∈ Ωi ,

i = 1, . . . , s

(5a)

(5b)

where x(k) ∈ Rn , u(k) ∈ Rp and y(k) ∈ Rq are, respectively, the state, the input and the output of the system at time k ∈ Z, s is the number of affine submodels (or modes), and the real matrices/vectors Ai , Bi , Ci , Di , fi and gi have appropriate dimensions. The regions Ωi form a complete partition of the state-input domain Ω ⊆ Rn+p . The results presented in this paper are derived for PWA state space models (5) with no constraints on the inputs and the states. Hence, from now on the following assumption will be made. Assumption 2.1: The state-input domain Ω of model (5) is the whole space Rn+p . Each region Ωi in (5) is a polyhedron described by a set of linear inequalities, i.e. n hxi o x Ωi = [ u ] ∈ Rn+p : Hi u1 [i] 0 ,



(6)

where Hi ∈ Rµi ×(n+p+1) , i = 1, . . . , s, and µi is the number of linear inequalities defining Ωi . With a slight abuse of notation, in (6) the symbol [i] denotes a µi -dimensional vector whose elements can be the symbols ≤ and n ¯ , can be obtained stepwise through similar arguments. (Necessity) Let n ¯ = max{na , nb }. Without loss of generality, it can be assumed that the PWA state space model (5) admits an equivalent PWARX representation (17) with model orders na = nb = n ¯ . If either na < n ¯ or nb < n ¯ , it suffices to pad with zeros the parameter matrices Θj in appropriate positions, and to lift the regions Rj to a higher dimensional space. Given any feasible mode sequence (i0 , i1 , . . . , in¯ ) of model (5), let Pi0 ,i1 ,...,in¯ be the polyhedron

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defined as the image of the polyhedron Ωi0 ,i1 ,...,in¯ in (15) through the affine transformation (23). Since the PWARX model is equivalent to the PWA state space model, it is a fact that Pi0 ,i1 ,...,in¯ ⊆

¯ ∪sj=1 Rj . Moreover, since the number of regions s¯ is finite, there exists at least one region

¯ y Rj such that dim(Pi0 ,i1 ,...,in¯ ∩ Rj ) = dim(Pi0 ,i1 ,...,in¯ ). Let [ u ¯ ∩ Rj ). There ¯ ] ∈ relint(Pi0 ,i1 ,...,in

¯ y ¯ x exists [ u ¯ ] ∈ relint(Ωi0 ,i1 ,...,in ¯ ) which is mapped to [ u ¯ ] by the affine transformation (23). Let

r = dim(Ωi0 ,i1 ,...,in¯ ). Since (23) is continuous, there exists ε > 0 such that the set n o r x ¯ ¯ x x BΩ ([ u¯ ] , ε) = [ u ] = [ u¯ ] + Vi0 ,i1 ,...,in¯ α : α ∈ R , kαk < ε

(28)

is entirely contained in Ωi0 ,i1 ,...,in¯ , and is mapped by (23) to a subset of relint(Pi0 ,i1 ,...,in¯ ∩ Rj ).

x ∞ ¯ x Let [ u ] ∈ BΩ ([ u ¯ ] , ε), and consider an input-output trajectory {u(k), y(k)}k=0 of model (5)

x generated with initial state x(0) = x and input u(·) such that un0¯ = u. Since [ u ] ∈ Ωi0 ,i1 ,...,in¯ ,

by evaluating (9) for k = n ¯ and h = 0, it turns out that
y(¯ n) = Ci0 Ai1 ,...,in¯ x + [ Di0 Ci0 Bi1 ,...,in¯ ] u + gi0 + Ci0 fi1 ,...,in¯ ,

(29)

while, for k = n ¯ , (12) implies: y x ] = Γi1 ,...,in¯ [ u [u ] + γi1 ,...,in¯ ,

(30)

where y = y 0n¯ −1 . Since the PWA state space model and the PWARX model are equivalent, y ] be the {u(k), y(k)}∞ n) = [ u k=0 is also an input-output trajectory of the PWARX model. Let r(¯ x ¯ x initial regression vector of the PWARX model. Since [ u ] ∈ BΩ ([ u ¯ ] , ε) and (30) holds, it turns

out that r(¯ n) ∈ Rj , and the output at time k = n ¯ of the PWARX model is given by y(¯ n) = Θj

 r(¯n)  1

y = Ξj [ u ] + ξj ,

(31)

where Θj has been decomposed as in (21). By substituting (29) and (30) into (31), one gets: h i
x n ¯ f (32) − ξj − Ξj γi1 ,...,in¯ . + C [ Ξj −Iq ] νΓi0i,i11,...,i [ ] = g i ,...,i i i u n ¯ 1 0 0 ,...,in ¯

x ¯ x Equation (32) is satisfied by all points [ u ] ∈ BΩ ([ u ¯ ] , ε). Hence, it holds that i h n ¯ Vi0 ,i1 ,...,in¯ α = [ Ξj −Iq ] νΓi0i,i11,...,i ,...,in ¯ i h
¯ x n ¯ [u = gi0 + Ci0 fi1 ,...,in¯ − ξj − Ξj γi1 ,...,in¯ − [ Ξj −Iq ] νΓi0i,i11,...,i ¯ ]. ,...,in ¯

(33)

Since α ∈ Rr can vary in a ball of radius ε, which is a full-dimensional set, the only possibility is that both sides of (33) are identically equal to zero. In particular, this implies that (20) (and (22), even if it is not necessary for the proof) must hold for the considered mode sequence. Given the arbitrariness of the mode sequence, this means that Condition C2 holds.

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Now assume that there exist two initial states x and x′ , and an input u(·) such that y(k; x, u) = y(k; x′ , u) for all k = 0, 1, . . . , n ¯ −1. Let y(k) = y(k; x, u) = y(k; x′ , u) for k = 0, 1, . . . , n ¯ −1. Since the PWARX model is equivalent to the PWA state space model, there exists one region Rj h n¯ −1 i such that r(¯ n) = yu0 n¯ ∈ Rj , and the outputs y(¯ n; x, u) and y(¯ n; x′ , u) of the PWA model 0  n)  , which implies y(¯ n; x, u) = y(¯ n; x′ , u). This in turn means that are uniquely given by Θj r(¯ 1

Condition C1 holds.



The sufficient part of the proof of Theorem 1 is constructive, and provides a method to compute an equivalent PWARX representation of a PWA state space model (5) satisfying Conditions C1 and C2 for a certain n ¯ ∈ Z+ . The idea behind the proof establishes a one-to-one correspondence1 between the feasible mode sequences of length n ¯ + 1 of the original PWA state space model and the modes of the constructed equivalent PWARX model. Condition C2 makes it possible to compute the parameter matrices of the PWARX model, while Condition C1 guarantees that the PWARX model is not multi-valued in any region of the regressor domain. Remark 3.1: The result of Theorem 1 is still valid for autonomous PWA models, and all the formulas for the construction of an equivalent PWARX model can be considerably simplified in the autonomous case. The same occurs when the partition of the PWA state space model does not depend on the input, and all the matrices Di are zero, so that u(k) is not required in the regression vector (16).



The following corollary is a straightforward consequence of Theorem 1. Corollary 1: Let n ¯ ∈ Z+ . A PWA state space model (5) with Ω = Rn+p admits an equivalent PWARX representation (17) with model orders na , nb ≤ n ¯ if and only if Conditions C1 and C2 are satisfied for such n ¯. Corollary 1 requires to check Conditions C1 and C2 only for the candidate order n ¯ . Note that, if Condition C1 and Condition C2 are satisfied for a certain positive integer n ¯ , then they are ¯>n satisfied for all positive integers n ¯ . The reasoning is based on the fact that all the feasible mode sequences of length n ¯ + 1 are sub-sequences of the feasible mode sequences of length ¯ + 1. Thus, equivalent PWARX models of arbitrary order n ¯ greater than n n ¯ can still be obtained by the constructive method in the proof of Theorem 1. This motivates the following definition. Definition 3.1: Let the PWA state space model (5) admit equivalent PWARX representations. 1

Apart from slight amendments to the partition, as discussed in the proof of Theorem 1.

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An equivalent PWARX model is said to have minimum order if it has the smallest order among all equivalent PWARX models. Theorem 1 suggests the following comments. i) A given PWA state space model may not admit an equivalent PWARX representation (see Example 5.3 in Section V). A similar statement is found in [23, p. 547], but referring to the existence of equivalent input-output representations where each parameter is connected to a specific submodel of the state space model. The result of Theorem 1 is general. ii) It is known (see, e.g., [21]) that a given PWARX model (17) always admits an equivalent PWA state space representation (5). A simple way to construct such an equivalent representation is to define the state as x(k) = [ y(k − 1)⊤ . . . y(k − na )⊤ u(k − 1)⊤ . . . u(k − nb )⊤ ]⊤

(34)

by eliminating u(k) in the regression vector (16). Since there exist PWA state space models which do not admit an equivalent PWARX representation, one can conclude that the class of PWARX systems is strictly contained in the class of PWA systems. Note that the equivalent PWA state space representation constructed according to (34) does not satisfy Assumption 2.1 if the regressor domain R in (17) is not the whole space Rna q+(nb +1)p . iii) A given PWA state space model of order n admitting equivalent PWARX representations, does not necessarily admit an equivalent input-output representation of order n [23]. Indeed, there exist PWA state space models of order n which admit a minimum-order equivalent input-output representation of order n ˜ strictly greater than n. This occurs because such models do not satisfy both Condition C1 and Condition C2 for n ¯ 0 sufficiently small. Then, Oi1 ,...,in¯ x =

Oi1 ,...,in¯ x′ , and Ci0 Ai1 ,...,in¯ x 6= Ci0 Ai1 ,...,in¯ x′ . These relations, together with (9) and (11) evaluated

for k = n ¯ and h = 0, imply that y(k; x, u) = y(k; x′ , u) for k = 0, 1, . . . , n ¯ − 1, while

y(¯ n; x, u) 6= y(¯ n; x′ , u). Hence, Condition C1 is not satisfied.



In view of Lemma 2 and the equivalence of Conditions C2 and C3 under Assumption 3.1 (see the end of Section III-A), the following theorem is a direct consequence of Theorem 1. Theorem 2: Under Assumption 3.2, a PWA state space model (5) with Ω = Rn+p admits an equivalent PWARX representation (17) if and only if there exists n ¯ ∈ Z+ such that Condition C1 holds. It is worthwhile to note that Assumption 3.2 excludes the existence of feasible mode sequences (i0 , i1 , . . . , in¯ ) that can be generated only by choosing the initial state and the input sequence in a set (namely, Ωi0 ,i1 ,...,in¯ ) with measure zero. Under Assumption 3.2, if Condition C3 is violated for n ¯ ∈ Z+ , Lemma 2 implies that also Condition C1 is violated. Since Condition C3 can be checked more easily than Condition C1, Condition C3 can be used as a prerequisite for choosing a candidate model order n ¯. It is interesting to characterize the PWA state space models that may satisfy Condition C3. Note that Condition C3 is trivially satisfied if, for every feasible mode sequence (i0 , i1 , . . . , in¯ ),

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the observability matrix Oi1 ,...,in¯ has full-column rank3 . Proposition 3.1: Let Condition C3 be satisfied for some n ¯ ∈ Z+ . Then, the following statements hold for every feasible mode sequence (i0 , i1 , . . . , in¯ ):
a) rank Oi0 ,...,in¯ −1 ≥ rank (Oi1 ,...,in¯ ) − q.  
C b) If rank (Oi1 ,...,in¯ ) < rank Aiinn¯¯ , then rank Oi0 ,...,in¯ −1 < n. Proof. Since Condition C3 is satisfied, it holds that   C A 1 ,...,in ¯ rank iO0 i i,...,i = rank (Oi1 ,...,in¯ ) . n ¯ 1

In addition, it can be easily verified that      0 Oi0 ,...,in¯ −1 C C A   in¯  .  i0 i1 ,...,in¯  =  n¯ q×q Iq 0q×n Ain¯ Oi1 ,...,in¯

(37)

(38)

Hence, rank



Ci0 Ai1 ,...,in ¯ Oi1 ,...,in ¯



≤ rank



0n ¯ q×q Oi0 ,...,in ¯ −1 Iq 0q×n




= q + rank Oi0 ,...,in¯ −1 .

The last relation and (37) imply a). To show b), from (38) it follows that     Ci0 Ai1 ,...,in Cin ¯ ¯ ker Ain¯ ⊆ ker Oi ,...,in¯ . (39) 1   C If rank (Oi1 ,...,in¯ ) < rank Aiinn¯¯ , the rank-nullity theorem and (37) imply that the inclusion in (39) is strict, and there exists x ∈ Rn , x 6= 0, such that Ain¯ x 6= 0 and Oi0 ,...,in¯ −1 (Ain¯ x) = 0. This implies that Oi0 ,...,in¯ −1 does not have full-column rank.



Proposition 3.1 imposes constraints on model (5) for Condition C3 to be satisfied. In particular, item a) implies that, for fixed n ¯ ∈ Z+ , the rank of the observability matrices of the feasible mode sequences of length n ¯ cannot decrease more than q units across two consecutive switches. C. Connections to observability This subsection investigates the relationships between the observability properties of the PWA state space model (5) and the existence of equivalent PWARX representations. Observability of PWA systems has been studied in the literature (see, e.g., [1], [14], [15], [16] and references therein), and a number of different observability notions have been proposed. Here the following classical definition of observability is considered [1]. 3

Under Assumption 3.2, this is the case of PWA state space models that are observable in finite time. See Section III-C.

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Definition 3.2: Model (5) is said to be observable if the mapping where y(·) = y(·; x0 , u).



x0 u∞ 0



→

h

y∞ 0 u∞ 0

i

is invertible,

If a PWA model is observable, observability in finite time and in infinite time can be distinguished [15], [16]. Definition 3.3: If model (5) is observable, it is called observable in finite time N if there h Ni  x0  y exists a nonnegative integer N such that the mapping uN0 → u0N is invertible. Otherwise, it 0

is called observable in infinite time.

The following propositions show that observability in finite time and in infinite time determine different input-output properties. Proposition 3.2: A PWA state space model (5) with Ω = Rn+p that is observable in infinite time, does not admit any equivalent PWARX representation (17). Proof. Assume that model (5) admits an equivalent PWARX representation of order n ¯ ∈ Z+ . Hence, it satisfies Condition C1 in view of Theorem 1. Since the model is observable in infinite time, one can find two distinct initial states x and x′ and an input sequence {u(k)}∞ k=0 such

that y(k; x, u) = y(k; x′ , u), k = 0, 1, . . . , n ¯ − 1. Otherwise, the model was observable in time

¯≥n N =n ¯ − 1. Moreover, since the model is observable, there exists a smallest time n ¯ such ¯ ; x′ , u). This implies that x(n ¯−n ¯−n that y(n ¯ ; x, u) 6= y(n ¯ ; x, u) 6= x(n ¯ ; x′ , u), otherwise

¯ ; x′ , u). If one lets x ¯−n ˜ = x(n ˜ ′ = x(n ˜ (k) = y(n ¯ ; x, u) = y(n ¯ ; x, u), x ¯−n ¯ ; x′ , u), and u ¯−n ˜ and x ˜ ′ and the input sequence u(k + n ¯ ), k = 0, 1, . . ., the two distinct initial states x {˜ u(k)}∞ k=0 are such that Condition C1 is violated, which contradicts the assumption.



Proposition 3.3: If the PWA state space model (5) with Ω = Rn+p is observable in time N , and Assumption 3.1 holds for n ¯ = N + 1, then the following statements hold: a) For every feasible mode sequence (i0 , i1 , . . . , iN +1 ), it holds that rank(Oi1 ,...,iN +1 ) = n. b) Model (5) admits an equivalent PWARX representation (17) of order N + 1. Proof. The proof of a) can be found, e.g., in [15]. As regards b), in view of Corollary 2 and Lemma 2, it suffices to show that Condition C1 is satisfied. In fact, if the model is observable in time N < n ¯ , there do not exist any two initial states x 6= x′ , and input sequence {u(k)}∞ k=0 ,

such that y(k; x, u) = y(k; x′ , u) for k = 0, 1, . . . , n ¯ − 1.



Item a) of Proposition 3.3 implies that Condition C3 is satisfied with all the observability matrices Oi1 ,...,in¯ corresponding to the feasible mode sequences (i0 , i1 , . . . , in¯ ) having full-column rank.

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Moreover, it turns out from the proof of Proposition 3.3 that, for every PWA state space model (5) observable in time N , Condition C1 is satisfied for n ¯ = N + 1. However, if Assumption 3.1 does not hold for such n ¯ , one still needs to check Condition C2, even if the state space model is observable in finite time. Remark 3.2: The main result presented in [29] is the counterpart of statement b) in Proposition 3.3 for switched affine models. Indeed, it shows that finite time observability of a switched affine state space model is sufficient to guarantee the existence of an equivalent switched ARX representation. Note that the full-dimensionality assumption is meaningless for switched affine models, since they are not equipped with any partition of the state-input domain, being σ(k) an exogenous signal.



IV. A LGORITHM IMPLEMENTATION This section addresses a few practical issues concerning the application of Theorem 1 for computing an equivalent PWARX representation of a given PWA state space model. First, it is shown that Condition C1 can be checked via linear programming. When the necessary and sufficient conditions of Theorem 1 are satisfied, an equivalent PWARX model can be derived by following the sufficient part of the proof. However, while the parameter matrix of each mode of the PWARX model can be easily computed by solving the linear system (20), the corresponding region is defined implicitly as the image of the polyhedron (15) through the affine transformation (23). In this section it is shown that, when the observability matrices of the feasible mode sequences have full-column rank (e.g., in the case of models observable in finite time, see Proposition 3.3), explicit representations of the regions can be given. A. A linear programming technique for checking Condition C1 For fixed n ¯ ∈ Z+ , consider the optimization problem in Table II, where y(¯ n; x, u) = Ci0 Ai1 ,...,in¯ x + [ Di0 Ci0 Bi1 ,...,in¯ ] u + gi0 + Ci0 fi1 ,...,in¯





y(¯ n; x′ , u) = Ci′0 Ai′1 ,...,i′n¯ x′ + [ Di′0 Ci′0 Bi′1 ,...,i′n¯ ] u + gi′0 + Ci′0 fi′1 ,...,i′n¯ ,

(40) (41)

and (i0 , i1 , . . . , in¯ ) and (i′0 , i′1 , . . . , i′n¯ ) are two feasible mode sequences. The equality constraint imposes that y(k; x, u) = y(k; x′ , u) for k = 0, 1, . . . , n ¯ − 1, while the last two sets of   ′ x x ∈ Ωi′0 ,i′1 ,...,i′n¯ , respectively. Hence, inequality constraints impose that [ u ] ∈ Ωi0 ,i1 ,...,in¯ and u x the optimization problem in Table II is feasible if and only if there exist [ u ] ∈ Ωi0 ,i1 ,...,in¯ and  x′  ¯ −1]. ∈ Ωi′0 ,i′1 ,...,i′n¯ such that y(·; x, u) and y(·; x′ , u) are indistinguishable in the interval [0, n u

21

TABLE II

max

δ,x,x′ ,u

δ

subject to ky(¯ n; x, u) − y(¯ n; x′ , u)k∞ ≥ δ

Oi1 ,...,in¯ x + Di1 ,...,in¯ u + Gi1 ,...,in¯ = Oi′1 ,...,i′n¯ x′ + Di′1 ,...,i′n¯ u + Gi′1 ,...,i′n¯ hx i Ξi0 ,i1 ,...,in¯ u  0 h x1 ′ i Ξi′0 ,i′1 ,...,i′n¯ u  0 1

In such a case, the optimization problem looks for the solution that makes the outputs (40) and (41) as different as possible according to the infinity norm. Let δmax ≥ 0 be the value of δ at the optimum, with δmax = +∞ if the problem is unbounded. If δmax > 0, it means that there exist  x′  x ∈ Ωi′0 ,i′1 ,...,i′n¯ such that y(·; x, u) and y(·; x′ , u) are indistinguishable [u ] ∈ Ωi0 ,i1 ,...,in¯ and u

up to time n ¯ − 1, while y(¯ n; x, u) 6= y(¯ n; x′ , u). Hence, Condition C1 is violated. It can be concluded that Condition C1 is satisfied if and only if, for all the pairs of feasible mode sequences (i0 , i1 , . . . , in¯ ) and (i′0 , i′1 , . . . , i′n¯ )4 , either the optimization problem in Table II is infeasible, or δmax = 0. From the computational point of view, the choice of the infinity norm makes it possible to solve the optimization problem in Table II by suitably combining the solutions of 2q linear programs. Remark 4.1: As pointed out after (6) and (15), with abuse of notation the symbol  denotes componentwise inequalities that can be either strict or nonstrict. If strict inequalities are present in the definitions of the regions Ωi0 ,i1 ,...,in¯ and Ωi′0 ,i′1 ,...,i′n¯ , the max in Table II should be replaced by sup. In practice, the constraints of the type a⊤ z < b can be replaced by a⊤ z + ε ≤ b, where ε > 0 is the desired numerical accuracy.



B. Explicit representation of the regions of the constructed equivalent PWARX model Following the proof of Theorem 1, the polyhedral region Rj that is associated to the jth mode of the equivalent PWARX model is defined implicitly as the image of the polyhedron (15) through the affine transformation (23). As stated in [31], it is computationally demanding, in 4

Including (i′0 , i′1 , . . . , i′n¯ ) = (i0 , i1 , . . . , in¯ ).

22

general, to find a representation of Rj by a set of linear inequalities. However, the problem can be easily solved if Oi1 ,...,in¯ has full-column rank. From (23), consider the linear system y = Oi1 ,...,in¯ x + Di1 ,...,in¯ u + Gi1 ,...,in¯

(42)

in the unknown x. If Oi1 ,...,in¯ has full-column rank, let Oi#1 ,...,in¯ = Oi⊤1 ,...,in¯ Oi1 ,...,in¯ The unique solution of (42) is then given by x = Oi#1 ,...,in¯ (y − Di1 ,...,in¯ u − Gi1 ,...,in¯ ) ,


−1

Oi⊤1 ,...,in¯ .

(43)

provided that y − Di1 ,...,in¯ u − Gi1 ,...,in¯ belongs to the image of Oi1 ,...,in¯ . This in turn requires that the following equality constraints hold, obtained by substituting (43) into (42):
In¯ − Oi1 ,...,in¯ Oi#1 ,...,in¯ (y − Di1 ,...,in¯ u − Gi1 ,...,in¯ ) = 0.

(44)

Then, from (43) it follows that 

x





Oi#1 ,...,in¯

       u  =  0(¯n+1)pׯnq    1 01ׯnq

−Oi#1 ,...,in¯ Di1 ,...,in¯

−Oi#1 ,...,in¯ Gi1 ,...,in¯

I(¯n+1)p

0(¯n+1)p×1

01×(¯n+1)p

1



y



     u  ,   1

(45)

which, substituted into (15), provides 

Oi#1 ,...,in¯

  Ξi0 ,i1 ,...,in¯  0(¯n+1)pׯnq  01ׯnq

−Oi#1 ,...,in¯ Di1 ,...,in¯

−Oi#1 ,...,in¯ Gi1 ,...,in¯

I(¯n+1)p

0(¯n+1)p×1

01×(¯n+1)p

1



y



     u   0 .   1

(46)

The region Rj is thus defined by both equality constraints (44) and inequality constraints (46). Remark 4.2: In practical cases, one could be interested in obtaining a PWARX model with fulldimensional regions by applying the constructive procedure illustrated in the sufficient part of the proof of Theorem 1. Since each polyhedral region Rj is defined as the image of the polyhedron (15) through the affine transformation (23), a necessary condition for Rj being full-dimensional is that Oi1 ,...,in¯ has full-row rank (this requires n ¯ q ≤ n). In addition, if Assumption 3.1 holds for n ¯ , such a rank condition is also sufficient.



V. I LLUSTRATIVE EXAMPLES In this section, several examples are proposed to illustrate and clarify different aspects of the results presented in Section III.

23 1 mode #2

mode #1

y(k−2)

0.5

0

−0.5

mode #3 −1 −1

mode #4 −0.5

0 y(k−1)

0.5

1

F IG . 1 Example 5.1: Polyhedral partition of the PWARX model (48) in the (y(k −1), y(k −2))-domain. The crosses represent N = 100 vectors (y(k − 1), y(k − 2)) obtained by a simulation of the model.

Example 5.1: This example shows a PWA state space model for which both Condition C1 and Condition C3 are satisfied for n ¯ = n = 2. Hence, an equivalent PWARX representation of order n ¯ = 2 exists by Corollary 2. Since Condition C3 is satisfied with all the observability matrices of the feasible mode sequences having full-column rank, an explicit representation of the regions of the computed equivalent PWARX model can be obtained as described in Section IV-B. Consider the following PWA state space model [13]: x(k + 1)

=

y(k) = α(k)



0.8 

cos(α(k))

− sin(α(k))

sin(α(k))

cos(α(k))





 x(k) + 

[ 1 0 ] x(k)   π if [ 1 0 ] x(k) ≥ 0 (mode #1) 3 =  − π if [ 1 0 ] x(k) < 0 (mode #2), 3

0 1



 u(k)

(47)

with n = 2 and s = 2. If n ¯ = 2, all the s¯ = 8 mode sequences (i0 , i1 , i2 ) ∈ {1, 2}3 are feasible. The test described in Section IV-A provides that Condition C1 is satisfied, while Condition C3 is trivially satisfied since rank(Oi1 ,i2 ) = 2 for all pairs (i1 , i2 ) ∈ {1, 2}2 . Moreover, also Assumption 3.2 holds for model (47). The constructive method described in the sufficient part of the proof of Theorem 1 produces an equivalent PWARX model with s¯ = 8 modes. The number of modes can be reduced5 to s¯ = 4, so that one finally gets:  √   0.8 y(k − 1) − 0.64 y(k − 2) − 0.4 3 u(k − 2)      0.64 y(k − 2) + 0.4√3 u(k − 2) y(k) =  0.8 y(k − 1) − 0.64 y(k − 2) + 0.4√3 u(k − 2)       0.64 y(k − 2) − 0.4√3 u(k − 2) 5

if y(k − 1) ≥ 0, y(k − 2) ≥ 0 if y(k − 1) < 0, y(k − 2) ≥ 0 if y(k − 1) < 0, y(k − 2) < 0

(48)

if y(k − 1) ≥ 0, y(k − 2) < 0.

Submodels with the same parameter vector can be merged if the union of the corresponding regions is still a convex polyhedron.

24 1

3 mode #4

0.5

mode #1 2

Ω2,2,2

Ω2,2,1

0

mode #3 −0.5

x2(k)

y(k−2)

1

0

Ω2,1,1

−1 −1.5

−1 −2

mode #2 −2

−3 −3

Ω1,1,1

−2.5

−2.5

−2

−1.5 −1 y(k−1)

−0.5

0

0.5

−3 −4

−3

−2

−1

0

1

2

3

x (k) 1

F IG . 2 Example 5.2: (Left) Regions of the PWARX model (50) in the (y(k − 1), y(k − 2))-domain. Circles and triangles represent two trajectories of the vectors (y(k − 1), y(k − 2)) starting from mode #1. The two trajectories overlap when they enter mode #4. (Right) Corresponding state trajectories. Note the correspondence between the polyhedral sets (15) and the modes of the PWARX model.

Fig. 1 shows the polyhedral partition of the PWARX model (48) in the (y(k − 1), y(k − 2))domain. The crosses represent N = 100 vectors (y(k − 1), y(k − 2)) obtained by simulating the model with u(k) uniformly distributed in the interval [−1, 1]. All four modes of the PWARX model are visited. Example 5.2: This example shows a PWA state space model for which both Condition C1 and Condition C3 are satisfied for n ¯ = n = 2. Hence, an equivalent PWARX representation of order n ¯ = 2 exists by Corollary 2. However, differently from Example 5.1, not all the observability matrices of the feasible mode sequences have full-column rank. Consider the following autonomous PWA state space model:     0 1  x(k + 1) = A x(k) =   x(k) 1 if h⊤ x(k) + w < 0 mode #1 0 γ1   y(k) = C1 x(k) = [ 1 0 ] x(k)

      0 1 0  x(k + 1) = A x(k) + f =    x(k) +  2 2 mode #2 if h⊤ x(k) + w ≥ 0, γ2 hh12 γ2 − hh21 0.1  

(49)

y(k) = C2 x(k) = [ γ1 −1 ] x(k)

with n = 2, s = 2, γ1 = γ2 = 0.4, h1 = w = 1, and h2 = 2. The matrix A1 is asymptotically stable, thus all the state trajectories starting from mode #1 tend to the origin, and enter mode #2. Conversely, the structure of the matrix A2 is such that the region Ω2 is A2 -invariant, and all the state trajectories starting from mode #2, remain indefinitely in mode #2. This implies that, for n ¯ = 2, the only feasible mode sequences of length n ¯ + 1 = 3 are (i0 , i1 , i2 ) = (1, 1, 1), (2, 1, 1), (2, 2, 1), (2, 2, 2). As can be seen in Fig. 5.2 (Right), Assumption 3.2 holds

25

for model (49), since the regions Ωi0 ,i1 ,i2 corresponding to all the feasible mode sequences are full-dimensional. The test described in Section IV-A provides that Condition C1 is satisfied. Hence, Condition C3 is also satisfied in view of Lemma 2. However, while rank(O1,1 ) = 2, it holds that rank(O2,1 ) = rank(O2,2 ) = 1. It is worthwhile to note that the PWA state space model (49) is trivially not observable. The constructive method described in the sufficient part of the proof of Theorem 1 produces the following equivalent PWARX model with s¯ = 4 modes:

y(k) =

     0.4 y(k − 1)                  0

           −0.1           −0.5 y(k − 1) − 0.1

  0.72 y(k − 1) + 1 < 0 if  2 y(k − 1) + y(k − 2) + 1 < 0    0.72 y(k − 1) + 1 ≥ 0   if 1.8 y(k − 1) + 1 < 0     2 y(k − 1) + y(k − 2) + 1 < 0   y(k − 1) = 0 if  y(k − 2) − 1 ≤ 0

(50)

9

if 2 y(k − 1) + y(k − 2) + 0.2 = 0.

Although the matrices O2,1 and O2,2 do not have full-column rank, in this case it is simple to derive explicit representations for all the regions of the PWARX model. Fig. 2 (Left) shows the regions of the PWARX model (50) in the (y(k−1), y(k−2))-domain. It is stressed that the regions corresponding to mode #3 and mode #4 are not full-dimensional, because the matrices O2,1 and O2,2 do not have full-row rank. Moreover, all the trajectories of the vectors (y(k − 1), y(k − 2)) starting from either mode #1 or mode #2, visit mode #3 for only one time instant, and then overlap as they enter mode #4. The following comments can be made for this example: i) The regions corresponding to mode #3 and mode #4 of the PWARX model intersect in (y(k− 1), y(k − 2)) = (0, −0.2). It can be easily checked that both mode #3 and mode #4 produce the same output y(k) = −0.1 at that regression vector, as is implied by Condition C1. In practice, the partition can be easily amended to avoid the intersecting regions. ii) Since Condition C3 is satisfied, the implications of Proposition 3.1 hold. In particular, the rank of the observability matrix Oi1 ,i2 does not decrease more than one unit (q = 1 in this example) across two consecutive switches. For instance, rank(O2,1 ) = 1 = rank(O1,1 ) − 1 along the mode sequence (2, 1, 1). On the other hand, if γ2 ∈ (0, 1) is chosen such that γ2 6= γ1 , then item b) of Proposition 3.1 is violated because rank(O2,2 ) = 2, while

C1 rank(O2,1 ) = 1 < 2 = rank( A ). Hence, Condition C3 is not satisfied for n ¯ = 2, and 1

26 0.2

2.5

0

2

1.5

−0.4

y(k)

x2(k)

−0.2

−0.6

1

−0.8

0.5 −1

−3

−2

−1

0 x1(k)

1

2

3

0 0

2

4

6

8

10

k

F IG . 3 Example 5.3: (Left) Circles and diamonds represent two state evolutions corresponding to different initial conditions. (Right) The corresponding outputs coincide up to time k = 3.

an equivalent PWARX model of order n ¯ = 2 can not be found in view of Lemma 2 and Corollary 1. Nevertheless, equivalent PWARX representations exist by Corollary 2, since Conditions C1 and C3 are both satisfied for n ¯ = 3. Example 5.3: In this example, an equivalent PWARX representation of the given PWA state space model does not exist because Condition C1 is violated for all n ¯ ∈ Z+ , although Condition C3 (and hence Condition C2) is satisfied for all n ¯ ∈ Z+ .

Consider the following autonomous PWA state space model:        1 0.6 0  x(k + 1) = A x(k) + f =   x (k) ≥ 0  x(k) +   1 1 1 if mode #1 0 0 γ   x2 (k) + α < 0  y(k) = C1 x(k) = [ 1 0 ] x(k)

       x(k + 1) = A x(k) + f =  0.6 0  x(k) +  −1  2 2 mode #2 0 0 γ  

y(k) = C2 x(k) = [ −1 0 ] x(k)     0.7 0  x(k + 1) = A x(k) =   x(k) 3 mode #3 0 0.6

 

y(k) = C3 x(k) = [ −1 0 ] x(k)     0.5 0  x(k + 1) = A x(k) =   x(k) 4 mode #4 0 0.7

 

y(k) = C4 x(k) = [ 0.7 0 ] x(k)

  x (k) < 0 1 if  x (k) + α < 0 2

(51)

  x (k) < 0 1 if  x (k) + α ≥ 0 2

  x (k) ≥ 0 1 if  x (k) + α ≥ 0, 2

with n = 2, s = 4, γ = 0.8, and α = 0.5. It can be easily seen that model (51) can only switch from mode #1 to mode #4, and from mode #2 to mode #3. In addition, the model satisfies Condition C3 for every n ¯ ∈ Z+ . Nevertheless, an equivalent PWARX representation of model

(51) does not exist because Condition C1 is violated for every n ¯ ∈ Z+ . To show this, let x1,0 > 0

27

and, for fixed n ¯ ∈ Z+ , x2,0 ∈ [α/γ n¯ , α/γ n¯ −1 ). Then, let x = [ x1,0 x2,0 ]⊤ and x′ = [ −x1,0 x2,0 ]⊤ .

It turns out that y(k; x) = y(k; x′ ) for k = 0, 1, . . . , n ¯ − 1, but y(¯ n; x) 6= y(¯ n; x′ ), as can be seen in Fig. 3 for n ¯ = 4. Hence, Condition C1 is violated. Example 5.4: This example shows that the minimum order n ¯ for which an equivalent PWARX representation exists, can not be a priori bounded even for fixed n and s. Consider the following autonomous PWA state space model:        1 1 0  x(k + 1) = A x(k) + f =   0 < x (k) ≤ n  x(k) +   ˜−2 1 1 1 if mode #1 0 0 1   x2 (k) > 0  y(k) = C1 x(k) = [ 1 0 ] x(k)

    0 −1  x(k + 1) = A x(k) =   x(k) 2 mode #2 1 0   y(k) = C2 x(k) = [ 1 0 ] x(k)

    0.5 0  x(k + 1) = A x(k) =   x(k) 3 mode #3 0 0.5  

  x (k) > n ˜−2 1 if  x (k) > 0

(52)

2

otherwise

y(k) = C3 x(k) = [ 1 0 ] x(k)

with n = 2, s = 3, and n ˜ ∈ Z+ , n ˜ > 2. For the sake of clarity, it is pointed out that mode #3 is defined over a region that is non-convex, though it is the union of two convex polyhedra. However, this abuse does not affect the subsequent analysis. For n ¯ 0, x′2,0 > 0, x2,0 6= x′2,0 . Moreover, define

x = [ x1,0 x2,0 ]⊤ and x′ = [ x1,0 x′2,0 ]⊤ . It follows that y(k; x) = x0,1 + k = y(k; x′ ) for

k = 0, 1, . . . , n ¯ − 1, but y(¯ n; x) = x2,0 6= x′2,0 = y(¯ n; x′ ). Hence, Condition C1 is not satisfied for n ¯ 0 (mode #3)

(53)

28

with n = 1 and s = 3. It is worthwhile to note that modes #2 and #3 are identical, but it is necessary to consider them separately in order to have all convex polyhedral regions. Model (53) does not satisfy Assumption 3.2, since starting in x(0) < 0 and letting u(0) = −x(0), u(k) = 0, k = 1, 2 . . ., the corresponding mode sequence is (1, . . . , 1, 1), which is only obtained by that particular choice of u(k). Furthermore, model (53) does not satisfy Condition C3 for any n ¯ ∈ Z+ .

Indeed, for fixed n ¯ ∈ Z+ , the mode sequence (i0 , i1 , . . . , in¯ ) = (3, 1, . . . , 1) is feasible, and

 Ci0 Ai1 ,...,in¯ = 1 ∈ / span Cin¯ , Cin¯ −1 Ain¯ , . . . , Ci1 Ai2 ,...,in¯ = span{0}. i h Nevertheless, for n ¯ = 2 Condition C1 is satisfied, and the matrix νΓi0i,i11,i,i22 needed to check Condition C2 becomes



 

Γi1 ,i2 νi0 ,i1 ,i2

       =       

Ci1

0

0

Ci2

0

0

0

1

0

0

0

1

0

0

0

0 Ci0

Ci0

Ci1



  0    0  ,  0    1   Ci0

(54)

where C1 = 0, and Ci = 1 for i = 2, 3. It is easy to verify that, for all the feasible mode sequences h i except (i0 , i1 , i2 ) = (2, 1, 1) and (3, 1, 1), Ξ⊤ ∈ R5 can be chosen so that [ Ξ −1 ] νΓi0i,i11,i,i22 = 0. Concerning the mode sequence (i0 , i1 , i2 ) = (3, 1, 1), the set Ω3,1,1 is defined by the following

inequalities:     x(k) > 0        x(k − 1) ≤ 0   

x(k − 1) + u(k − 1) ≥ 0     x(k − 2) ≤ 0        x(k − 2) + u(k − 2) ≥ 0

   x(k − 2) ≤ 0    x(k − 2) + u(k − 2) = 0.     u(k − 1) > 0

⇔

(55)

The affine subspace aff(Ω3,1,1 ) can thus be described by 

aff(Ω3,1,1 ) = [ ux ] = V3,1,1 α : α ∈ R 

3



,

with

V3,1,1

1

0

   0 0 =   0 1  −1 0

0



  1  .  0   0

(56)

29

It follows that a solution Ξ of the linear system 



[ Ξ −1 ] 

Γ1,1 ν3,1,1



 V3,1,1

      = [ Ξ −1 ]       

0 0 0 0 −1 0

0 0



  0 0    0 1  =0  1 0    0 0   1 0

(57)

is given by, e.g., Ξ = [ 0 0 0 1 0 ]. A similar study can also be performed for the mode sequence (i0 , i1 , i2 ) = (2, 1, 1). Hence, Condition C2 is satisfied for n ¯ = 2, and an equivalent PWARX representation exists by Theorem 1. Note that, due to continuity of the output map at x = 0, the system considered in this example remains exactly the same if the inequalities x ≤ 0, x > 0 in the partition are replaced by x < 0, x ≥ 0, respectively. After this change, the new model satisfies Assumption 3.2, and also Condition C3 holds for every n ¯ ≥ 2. This suggests that violations of Assumption 3.2 may be sometimes circumvented by suitably modifying the representation of the state space model. VI. C ONCLUSIONS The conversion of PWA models from state space to input-output form was addressed by deriving necessary and sufficient conditions for a given PWA state space model to admit equivalent PWARX representations. It was shown that the class of PWARX systems is strictly contained in the class of PWA systems, and that the number of modes may grow considerably when converting a PWA state space model into a minimum-order equivalent PWARX representation. Formal relations between the observability properties of PWA models and the existence of equivalent input-output representations were established. Several numerical examples were presented to illustrate the variety of situations that may occur when addressing input-output realization of PWA state space models. Future research will concern the minimal realization of PWA models in input-output form. Since most available PWA system identification methods provide models in input-output form, while control is typically carried out in the state space, developments in this direction are advisable. The results of this paper provide a basis to start from, by clarifying the existing relations between equivalent state space and input-output representations. Finally, the results of this paper foreshadow interesting developments in PWA system identification. First, they give insights on how the prior knowledge available in the state space can be

30

transferred to the identification of PWARX models. Second, they can be used to design classes of inputs such that the number of modes of an equivalent PWARX model can be kept low. A PPENDIX Recursive formulas for some of the matrices and vectors introduced in Table I, are shown in Table III. Such formulas are useful to implement the results presented in this paper. Hereafter, the proof of Lemma 1 is reported. Proof of Lemma 1. The proof follows by induction. First, for h = n ¯ − 1 one has x(k − n ¯ + 1) = Ain¯ x(k − n ¯ ) + Bin¯ u(k − n ¯ ) + fin¯ k−¯ n = Ain¯ x(k − n ¯ ) + Bin¯ uk−¯ ¯. n + fin

Then, it is assumed that (8) holds for some h = 1, . . . , n ¯ − 1, and it is shown that it holds also for h − 1: x(k − h + 1)

= Aih x(k − h) + Bih u(k − h) + fih


k−h−1 = Aih Aih+1 ,...,in¯ x(k − n + fih+1 ,...,in¯ + Bih u(k − h) + fih ¯ ) + Bih+1 ,...,in¯ uk−¯ n
  h u(k−h) i ¯ ) + Bih Aih Bih+1 ,...,in¯ = Aih Aih+1 ,...,in¯ x(k − n ¯ uk−h−1 + fih + Aih fih+1 ,...,in k−n ¯

. ¯ ) + Bih ,...,in¯ uk−h = Aih ,...,in¯ x(k − n ¯ k−¯ n + fih ,...,in

From (8) it is immediate to derive (9): y(k − h) = Cih x(k − h) + Dih u(k − h) + gih


k−h−1 = Cih Aih+1 ,...,in¯ x(k − n + fih+1 ,...,in¯ + Dih u(k − h) + gih ¯ ) + Bih+1 ,...,in¯ uk−¯ n
¯ ) + [ Dih Cih Bih+1 ,...,in¯ ] uk−h = Cih Aih+1 ,...,in¯ x(k − n . ¯ k−¯ n + gih + Cih fih+1 ,...,in



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TABLE III   A if m = 1 ς1  A A if m > 1 ς1 ς2 ,...,ςm   B if m = 1 ς1  [B A B ς1 ς2 ,...,ςm ] if m > 1 ς1   f if m = 1 ς1  f +A f if m > 1 ς1 ς2 ,...,ςm ς1   Cς if m = 1 h 1 i A C ς1 ς2 ,...,ςm  if m > 1

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