realization theory for linear hybrid systems

2282 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 10, OCTOBER 2010

Realization Theory for Linear Hybrid SystemsMihály Petreczky and Jan H. van Schuppen

Abstract—The paper develops realization theory for linearhybrid systems, i.e., hybrid systems in continuous-time withoutguards whose continuous dynamics is determined by linear con-trol systems and whose discrete dynamics is determined by afinite-state automaton. We will formulate necessary and sufficientconditions for the existence of a realization. We will show thatminimality is equivalent to observability and span-reachability,and that minimal systems are isomorphic. In turn, observabilityand span-reachability can be characterized by rank conditions.

Index Terms—Linear hybrid systems (LHS).

I. INTRODUCTION

R EALIZATION theory is one of the central topics ofsystem theory, its aim is to answer the following ques-

tion: 1) Under which conditions is it possible to constructa (preferably minimal) system of a certain class generatingthe specified input/output behavior? 2) How to characterizeminimal systems of a certain class which generate the specifiedinput/output behavior? Realization theory answers fundamentalquestions, and serves as a theoretical foundation for modelreduction, system identification and filtering/observer design.

The Class of LHSs: This paper develops realization theory fora class of hybrid systems called linear hybrid systems (LHS). ALHS is a hybrid system in the sense of [2] with no guards, whosediscrete dynamics is determined by a finite-state automaton, andwhose continuous dynamics at each discrete state is governed bya time-invariant linear system. The reset maps are linear, and thediscrete events are externally generated inputs.

The class of LHSs is completely different from linear hybridautomata defined in [3]. The class of LHSs is similar to linearswitching systems [4]. The main difference is that in [4] the dis-crete events are viewed as disturbances not as inputs and thefinite state automaton is non-deterministic. Linear switched sys-tems in the sense of [5], [6] are a subclass of LHSs.

Motivation of the Realization Problem: The paper belongsto a series of papers addressing realization theory of hybrid sys-tems. The ultimate goal is to derive realization theory for generalhybrid systems with guards. In the first couple of papers [7]–[9]the realization theory of switched systems was explored. LHSsrepresent the next level of complexity. The final step is to addguards. At each step, the class of hybrid systems considered con-tains the class from the previous step. Hence, the previously ob-

Manuscript received April 18, 2008; revised January 19, 2009; accepted Feb-ruary 05, 2010 First published March 01, 2010; current version published Oc-tober 06, 2010. This work was supported in part by the Centrum voor Wiskundeen Informatica (CWI), Amsterdam, The Netherlands. Recommended by Asso-ciate Editor G. J. Pappas.

M. Petreczky is with Maastricht University, Maastricht 6200 MD, TheNetherlands (e-mail: [email protected]).

J. H. van Schuppen is a with Centrum voor Wiskunde en Informatica (CWI),Amsterdam 1090 GB, The Netherlands (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2010.2044258

tained results can help the next step. Realization theory of LHSscan also be useful for model reduction and system identification.

Model reduction of LHSs could be used for model reduc-tion of piecewise-affine hybrid systems with guards; piecewise-affine hybrid system can be viewed as a feedback interconnec-tion of a LHS with an event generator. By minimizing a realiza-tion of the LHS and interconnecting it with the same event gen-erator, we obtain a piecewise-affine hybrid system exhibiting thesame input-output behavior, but with a smaller state-space, see[10], [11] for similar ideas. For identification problems for hy-brid systems similar to LHSs, see [12], [13].

Contribution of the Paper: The main contribution of thepaper can be summarized as follows.

1) Existence of a realization We show that a family of input-output maps has a realization by a LHS , if and only ifthe input-output maps admit a hybrid kernel representa-tion, the rank of the generalized Hankel-matrix is finite andsome additional finiteness conditions hold. In addition, aLHS realization can be constructed from the generalizedHankel-matrix.

2) Observability, reachability We define observability andspan-reachability for LHSs and we present a characteriza-tion of observability and span-reachability via rank con-ditions. The proposed notions of observability and span-reachability allow for a neat characterization of minimality,and hence are likely to be useful for system identificationand model reduction.

3) Dimension and minimality We define dimension andminimality for LHS . We show that a LHS is minimal ifand only if it is observable and span-reachable, minimalLHSs are unique up to isomorphism, and any LHS can betransformed to a minimal LHS with the same input-outputbehavior.

The results of the paper imply the classical ones for linear sys-tems. Unlike in the classical case, here we study realizability ofa family of input-output maps instead of a single input-outputmap. This could be a first step towards a behavioral approach,[14] for hybrid systems. The case of a single input-output mapfollows from the results of the paper.

Related Work: The realization problem for hybrid systemswas first formulated in [15], but no solution was provided. Tothe best of our knowledge, the only results on realization theoryof hybrid systems are [1], [7], [16] and the references therein. In[17] the results of the paper were stated, but most of the proofswere omitted. A more detailed presentation of the results of thepaper can be found in [1], [18], [19], where the algorithmic as-pects are discussed as well. Although there is little prior workon realization theory of hybrid systems, there is a vast literatureon system identification of hybrid systems [20]–[25] and on ob-servability and reachability [4], [6], [26]–[31].

Approach of the Paper: The main tools of the paper are thetheory of rational formal power series [32], [33], and automatatheory [34], [35]. Our approach to realization theory, in partic-

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PETRECZKY AND VAN SCHUPPEN: REALIZATION THEORY FOR LINEAR HYBRID SYSTEMS 2283

ular, the translation of the realization problem to the problem ofrationality of formal power series, was inspired by nonlinear re-alization theory [36]–[41]. In the paper we use an extension ofthe classical theory of formal power series to families of formalpower series, see [1], [7], [9] for more on this extension. In con-trast, the classical literature considers a single formal power se-ries. Although the paper is self-contained, prior knowledge onrational formal power series, realization theory of Moore-au-tomata, and nonlinear realization theory is desired.

Outline of the Paper: In Section II we illustrate the main re-sults by a numerical example. Section III defines the class ofLHSs and fixes the notation. Section IV presents the definitionof the main system-theoretic concepts and it states the realiza-tion problem for LHSs. Section V presents the main result onexistence of a realization. Section VI presents the characteri-zation of span-reachability, observability and minimality. Ap-pendix I-A reviews the necessary results on the theory of ra-tional formal power series and Moore-automata. Appendix I-Bcontains the proof of the result on existence of a realization. Ap-pendix I-C presents the proof of the results on span-reachabilityand observability. Appendix I-D presents the proof of the resultson minimality.

II. NUMERICAL EXAMPLE

In this section we will demonstrate the theory presented in thepaper by a simple numerical example. We will use the notationof Sections III-A–III-C.

Let be the LHS

The discrete dynamics is determined by the Moore-automaton, where the set of discrete events is

, the set of states is . The state-transition map and the readout map are defined below

For each , the system matrices are as follows:

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Here, is the identity matrix, is the linearsystem in mode , and is the reset map to beapplied when an event occurs in mode .

Next, we define the family of input-output mapsrealizable by

for , for input , for each timed sequenceof discrete events ,and for any . Here , and

and, if and if

,

if

otherwise

where, a) , and b) and for ,if and if . Then ,

with below, is a realization of

Using Remark 14, we can transform to the followingminimal realization of

where is the automaton of ,, and the state-transition and readout maps

satisfy

and for each , the system matrices are


That is, is the linear system in mode ,and is the reset map for and .The map is defined as

From Theorem 2–3, it follows that is span-reach-able and observable. Hence, it is minimal. We only elabo-rate on span-reachability of , observability canbe shown analogously. To this end, notice that condition(i) of Theorem 3 holds, because the automaton of isreachable from . Condition (ii) of Theorem 3 holds,since ,

, and, where

is the continuous component of . Notice that noneof the linear systems of is minimal. The dimension of

is smaller than that of according to the ordering ofSection IV-B.

By Remark 13, Section B , we can compute a realizationof from the Hankel-matrix and the Hankel-table

of

where , , and

Here, is the linear system in mode , andis the reset map for , .

III. LINEAR HYBRID SYSTEMS

The goal of this section is to define LHSs. We fix the notationin Section III-A. In Section III-B. we define Moore-automata,and in Section III-C we define LHSs.

Notation

Let be a finite set, referred to as the alphabet. denotesthe set of finite strings (words) of elements of , i.e., an element

of is a sequence , where ,and ; is the length of and it is denoted by . If

, then is the empty sequence (word), denoted by . Theconcatenation of the words , and

is the word . The empty word isa unit element for concatenation. i.e., for all

. We denote by the string ; is the emptyword .

We identify any constant function with its value. For anyfunction , its range is denoted by . For any set ,is the cardinality of . For two sets , let andbe the functions which map any pair to its

-valued (resp. -valued) component, i.e.,and .

Denote by the set of natural numbers including 0. Let bethe real time-axis, i.e., . Denote bythe set of piecewise-continuous maps (i.e., maps whose restric-tion to any finite interval is piecewise-continuous in the senseof [42]) with values in . Let be a smooth mapand let be a -tuple of naturalnumbers. Then denotes for the following derivative ofat

For each and , is the standardunit basis vector of , i.e., , where

and for .

Definition of Moore-Automaton

Below we present a brief introduction to Moore-automata, see[34], [35] for more details.

Definition 1 (Moore-Automaton): A Moore-automaton is atuple where

• is a finite set, called the state-space;• is a finite set, called the input space;• is a (possibly infinite) set, called the output space;• is the state-transition map;• is the readout map.

We denote by the cardinality of the state-space , i.e.,.

Recall from [34], [35] that we can extend the functions andto act on sequences of input symbols as follows. Define the

function recursively as follows; let, and for each word and input symbol let

. Define the mapby for each input word and state

. By abuse of notation we will denote and simply byand respectively.Definition 2 (Reachability): A Moore-automaton

is called reachable from the set of initial states, if for all there exists a sequence of input

symbols and a state such that .Definition 3 (Observability): Two states of a

Moore-automaton are called indistinguish-able if for any input sequence , the output produced byequals the output produced by , i.e., .The Moore-automaton is called observable, if there are no


two distinct states , , such that and areindistinguishable.

A. LHS

Definition 4 (LHSs): A LHS is a hybrid system withoutguards of the form

(1)• is the discrete state at time , and is the finite

set of discrete states (modes);• is the discrete output at time , and is the finite

set of discrete outputs;• is discrete event at time , and is the finite set

of discrete events;• is the discrete state-transition map;• is the discrete readout map;• , , are the

matrices, and , , is the continuousstate-space, of the linear system residing in mode ;

• is the continuous state at time ;• , for , is the continuous output at time ,

and is the space of continuous outputs;• , , is the continuous input at time , and

is the space of continuous inputs;• the matrices , , ,

specify the linear reset maps.The state space of is . We willuse the following notation for :

• is the Moore-automaton formed bythe discrete-state transition and discrete readout map of thesystem; ; will also be denoted by ;

• is the collection ofthe matrices of the linear system residing in and of thereset matrices associated with the discrete state .

Notice that the discrete events are external inputs, and there areno guards. All the linear subsystems have the same inputs andoutputs, but their state-spaces might be different. The dynamicsof LHSs follows the classical definition [2]. That is, if no dis-crete event occurs, the evolution is governed by the linear systemof the current discrete state. As soon as a discrete event arrives, adiscrete-state transition occurs, the continuous state is reset ac-cording to the reset map, and the system resumes its evolutionaccording to the linear system of the new discrete state. For theformal description, we need the following notion.

Definition 5 (Timed Sequences): A timed sequence of dis-crete events is a sequencewhere , are discrete events, and

are time instances. We denote the set of alltimed sequences of discrete events by . If , then

is the empty sequence, and it is denoted by .The interpretation of above is the following. The event

took place after the event and is the elapsed time betweenthe arrival of and the arrival of , i.e., is the differenceof the arrival times of and . Hence, but we allow

, i.e., we allow to arrive instantly after . If ,then is the arrival time of the first event .

Definition 6 (Continuous-State Evolution): Consideran input , a timed sequence

, , and a time. For a state of , define the

continuous state reached fromwith inputs and timed sequence of events at timerecursively on as follows.

For , let be the solution of (2)

(2)

with , and set .If for the timed sequence of discrete events

, , thestate is already defined, then set

, and let be the solution of (3)with the initial condition

(3)

Define the continuous state as.

Note that in , the argument de-notes the time which has passed since the arrival of the last event

. The continuous state can be written as

(4)

where , .Next, we will define the input-output behavior of LHSs in-

duced by a state. To this end, notice that the inputs of the LHSare piecewise-continuous functions from and timedsequences of discrete events. The outputs consist of discrete out-puts from and continuous outputs from

Definition 7 (Input-Output Maps): The input-output map ofthe LHS induced by the state of isthe mapsuch that for each input , timed sequence

, , and

(5)

where . The value of atwill be denoted by .

That is, the input-output map induced by maps the input, the timed sequence of discrete events

, and the time instantto the pair of the discrete and continuous outputs at time


. Here stands for the time which has passed sincethe arrival of the last discrete event .

IV. REALIZATION PROBLEM FOR LHSS

The goal of the section is to state in formal terms the real-ization problem for LHSs. In Section IV-A we explain when aLHS realizes a family of input-output maps. In Section IV-B anumber of system theoretic concepts such as span-reachability,observability, etc. will be described. Finally, in Section IV-C therealization problem for LHSs will be stated.

A. Input-Output Maps Realized by LHSs

From (5) it follows that maps of the form:

(6)

are the natural models of the input-output behavior of a LHS.Intuitively a map of the form (6) is interpreted as the input-output map of an unknown (black box) LHS induced by someinitial state. In the sequel, unless stated otherwise, whenever wespeak of an input-output map, we will always mean a map of thetype (6).

In this paper we work with realizations of families of input-output maps. This means that we study systems with not one,but several initial states. Hence, we have to specify which input-output maps are realized from which initial state. To this end, weneed the following concept.

Definition 8 ( LHS Realization): Let be a LHS the form(1), and let be a family of input-output maps of the form (6).Consider a map , i.e., assigns each element ofa state of . We will call the pair a LHS realization (orsimply a realization for short).

The map just specifies a way to associate an initial state toeach element of . We will denote by the -valued compo-nent of , and by the continuous valued component of , thatis, for each , withand . The map can be thought of as a mapwhich assigns to each input-output map an initial state of thesystem .

Definition 9 (Realization of Input-Output Maps): Let be afamily of input-output maps of the form (6). A LHS realization

, where , is called a LHS realization of(or simply a realization of ), if for each , equals

the input-output map induced by , i.e., for all, ,

We say that has a LHS realization, if there exists a LHS real-ization such that is a LHS realization of .

B. System-Theoretic Concepts

The goal of this subsection is to present the definition of themain system-theoretic concepts of LHSs and LHS realizations.Throughout the section, denotes a LHS of the form (1). Wewill start with defining span-reachability and observability.

Definition 10 (Span-Reachability): Let be a subset ofthe state space and for each discrete state definethe set as the linear span of the states of the

form for all ,and , , such that

. The LHS is called span-reachablefrom if,

1) The automaton of is reachable fromthe discrete components

of ;2) for each discrete state , .

Intuitively, is just the linear span of all the con-tinuous states which belong to and which are reachable fromsome initial state in .

Definition 11 (Observability): Two distinct statesof the LHS are indistinguishable, if

the input-output maps induced by and are equal, i.e.,. The system is called observable, if it

has no pair of distinct indistinguishable states.Note that observability depends on inputs. This is in contrast

to linear systems where observability is independent of inputs.Let be a family of input-output maps and consider a LHS

realization with . Observability and span-reachability of the realization is defined as follows.

Definition 12 (Observ. and Span-Reach. of Realizations):The LHS realization is said to be span-reachable, ifis span-reachable from the range of , i.e., if is span-reach-able from . We say thatis observable, if is observable.

Next, we define the notion of LHS morphisms. LHS mor-phisms play a role, similar to algebraic similarity for linear sys-tems, i.e., they allow relating different realizations.

Definition 13 ( LHS Morphism): Consider the LHS

where

Let be a LHS of the form (1). Let be a set of input-outputmaps and let and be two maps (re-call that and denote the state-space of and re-spectively). A map is called a LHS morphismfrom to , denoted by ,if there exists a map and a linear map

such that1) For any discrete state , . We will

denote by the restriction of to , i.e., is thelinear map .

2) For any , .3) For any discrete state , ,

,4) For each discrete event , and discrete state ,

and .5) For each discrete event , and discrete state ,

6) For each , .We will identify with the pair of maps satisfyingconditions 1–6 above.

Using the identification of with a pair , is saidto be injective, surjective or isomorphism if both and arerespectively injective, surjective or bijective as maps. Two LHS


realizations are isomorphic if there exists a LHS isomorphismbetween them.

Intuitively, a LHS map is simply a map between the state-spaces of two LHSs , such that it maps state- and output-trajec-tories of one system to state- and output trajectories of the othersystem, and it maps initial states to initial states. If above is aLHS isomorphism, then the inverse map ex-ists, and it forms a LHS isomorphism .As the next step, we will define the notion of dimension for hy-brid systems.

Definition 14 (Dimension): For a LHS the di-mension of is defined as a pair of natural numbers

.The first component of indicates the number of dis-

crete state variables (i.e., number of discrete states), the secondcomponent indicates the number of continuous state variablesof the system (i.e., the sum of dimensions of the continuousstate-spaces). Here the continuous state variables belonging todifferent modes are counted separately. In some senserepresents the description complexity of . Another reason foradopting the definition above is that it leads to existence of anunique (up to isomorphism) minimal realization.

For each two pairs of natural numbersdefine the partial order relation as ,

if and . That is, the pair is smaller than orequal to the pair , if is not greater than and is notgreater than . Notice that the above ordering is a partial order.That is, there can be two LHSs with incomparable dimensions.

Definition 15 (Minimality): A LHS realization of iscalled a minimal LHS realization of , if for any LHS realization

of , .That is, is a minimal realization of , if for any LHS

realization of , the dimension of is comparablewith the dimension of , and the dimension of is not smallerthan the dimension of .

Remark 1: Since not all LHS realizations of have com-parable dimensions, the existence of a minimal LHS realizationdoes not follow trivially, it has to be proved.

Remark 2 (Consequence of Minimality): Minimal LHS re-alizations have in a sense the smallest description complexityamong all realizations of . Hence, the proposed notion of min-imality can be useful for model reduction. We will show thatminimal LHS realizations are unique up to isomorphism. Hence,there is a one-to-one correspondence between minimal LHS re-alizations and families of input-output maps. That is, minimalLHSs are identifiable, hence the proposed notion of minimalityis useful for system identification.

C. The Realization Problem

The realization problem for LHSs can be stated follows.(a) Let be family of input-output maps. Find necessary and

sufficient conditions for existence of a LHS realization of .Find a constructive procedure for obtaining a LHS realizationof based on the data obtainable directly from the elementsof .

(b) Let be family of input-output maps. Find necessary andsufficient conditions for a LHS realization of to be minimal.Are minimal LHS realizations of unique in any sense, for ex-ample, modulo isomorphism ? Find a procedure for converting

a LHS realization of to a minimal one and for checking min-imality of a LHS realization.

In this paper we will address both questions formulatedabove. Often, in addition to being constructive, the proceduresasked for in (a) and (b) above are also required to be effective,i.e., implementable by a (numerical) algorithm. In this paper wewill not study the algorithmic aspects. However, the proceduresabove can be made effective, see [1], [18], [19].

V. EXISTENCE OF A REALIZATION

In this section, denotes a family of input-output maps of theform (6). First, we define the notion of hybrid kernel represen-tation of input-output maps, existence of which is a necessarycondition for realizability.

Notation 1 (Discrete and Continuous-Valued Components):For each input-output map of the form (6), denote by the

-valued part, and by the -valued part of the map . Thatis, for all

, and .Informally, has a hybrid kernel representation if,a) depends only on the relative order of discrete events.b) is continuous and affine in continuous inputs, more-

over for constant continuous inputs, is analytic in thearrival time of the discrete events.

A formal theorem relating hybrid kernel representations withconditions (a) and (b) is presented in [1, Section 7.1], The-orem 30.

Definition 16 (Hybrid Kernel Representation): Aninput-output map has a hybrid kernel representation if

1) The function depends only on the discrete events,i.e., for any two timed sequences of events,

, and, , which differ only

in the arrival times of events, for any , inputs, it holds that

.2) For each sequence of events ,

, , there exist analytic functionsand where

, such that for alland

(7)

The family of maps has a hybrid kernel representation,if each element of has a hybrid kernel representation.

Remark 3 (Abuse of Notation): In the sequel, if aninput-output map admits a hybrid kernel representation, itsdiscrete component will be regarded as a map ,defined as follows. For any sequence

, of discrete events, set, for some arbi-

trary input and times .


Note that the existence of a hybrid kernel representation isnecessary for existence of a LHS realization of the family .The role of the maps and is best understood by analogywith linear systems. Consider a mapand recall from [43] that a necessary condition for existenceof a linear system realization of is that there exist analyticfunctions and

(8)

If the linear system with the initial state is a re-alization of , then and . Therequirement that has a hybrid kernel representation is anal-ogous to requiring that is of the form (8) with and

playing the roles of and .Remark 4: If is a realization of , and if of the form

(1), then has a hybrid kernel representation of the followingform; for each :

(9)

where are discrete events, ,, , and , for

each , with , and .Next, we define the Hankel-matrix of the family , if ad-

mits a hybrid kernel representation. The entries of the Hankel-matrix are the high-order derivatives of the elements of withthe respect to the arrival times of the discrete events. This isanalogous to the approach taken in realization theory of linearand nonlinear systems [36], [39], [43].

In order to encode these derivatives in a systematic way, weintroduce a new symbol . Consider the following finite set,

, where is chosen such that , i.e., is nota discrete event. Every word can uniquely be writtenas for some ,

, and . Recall that denotes the wordobtained by repeating the letter times. If , i.e.,contains no symbol , then . If containsno element of , then and .

The words over the alphabet will be used to index the high-order derivatives of the continuous-valued component of

. If is of the form ,then we use to index derivatives of the following type;derivative of order at 0 is taken with respect to the arrivaltime of the event for all , and derivative oforder at 0 is taken with respect to the time after thearrival of the last event . We represent these derivatives as amap mapping a word to the corresponding derivative.Formally:

Definition 17 (Generalized Markov-Parameters): Letbe an input-output map, and assume that has a hybridkernel representation. Define the maps and

, , as follows. For each word

of the form , forsome ,

(10)

Recall that is the standard unit basis vector in , i.e.,the entry of is 1 and all the other entries are 0, and that weidentify with the constant input function .Similarly, 0 denotes the constant function .The collection is called the generalizedMarkov parameters of . If the family has a hybrid kernelrepresentation, then each has a hybrid kernel represen-tation, and the maps , , are defined.We call the collection the generalizedMarkov-parameters of .

Since admits a hybrid kernel representation, the derivativeson the right-hand side of (10) exist. In fact, the functionsappearing on the right-hand side are entire analytic functionsin , and the maps and ,completely determine the continuous part of the input-outputmap . Notice that for all sequences of dis-crete events . The generalized Markov-parameters

can be thought of as generalizationsof Markov parameters for linear systems. Their role is similarto that of the Markov parameters of linear systems or coeffi-cients of the Fliess-series of input-output maps of input-affinesystems. Namely, the generalized Markov parameters ofform the entries of the generalized Hankel-matrix of .

The definition of as high-order derivatives of input-output maps is analogous to what was done in realization theoryof nonlinear systems [36]. Due to lack of space we do not elab-orate on the relationship between the maps , and non-linear realization theory. Instead, we try to provide some in-tuition based on the well-known results on linear realizationtheory [43], [44]. Recall that if is an input-output map of theform (8), then the Markov parameters of can be represented as

and ,, . Moreover, from the classical linear systems theory

[44] we know the linear system with the initial stateis a realization of if and only if and

holds for all . Similar relationship holds betweenthe maps , , and system matrices of a LHS real-ization of . Before presenting this relationship, additional no-tation is introduced.

Notation 2 (Products System Matrices): Consider a LHSof the form (1). Let and consider any sequence of dis-crete events , , and a sequence ofnatural numbers for some . Pickdiscrete states such that for each ,

. Consider the product of matrices

(11)


Following the widespread convention, if for some, then is interpreted as the identity matrix. If ,

then (11) is interpreted as the matrix . Notice that (11)is uniquely defined by the choice of and ,and . In the rest of the paper, unless statedotherwise, if we use an expression of the form (11), we willalways assume that , .holds. In the sequel, we will use the shorthand notation

instead of (11).Proposition 1 ([1], [18]): Let be a LHS of the form (1)

and let be a map . Then is a realiza-tion of if and only if has a hybrid kernel representationand for each input-output map , the following holds. As-sume that . For any word of the form

for some , discrete events, and indices

(12)

In addition, for any , the following holds. If theword has no symbol , i.e., ,then . Otherwise, let be thesmallest index such that , i.e., and

. Then

(13)

where is the standard basis vector of , i.e., theentry of is one, and all the other entries are zero. In (12) and(13) above the convention of Notation 2 was used and hence

for each .The proof of Proposition 1 is a straightforward calculation

and it can be found in [1], [18]. Now we are ready to define thenotion of Hankel-matrix for hybrid input-output maps.

Definition 18 (Hankel-Matrix): We define the Hankel matrixof , denoted by , as the following infinite matrix (see [45]for definition) formed by values of and . Consider theindex set formed by elements ofand pairs where is an element of and .The columns of the matrix are indexed by pairs

. The rows of the matrix are indexed by pairs ,where and . The element of with therow index and column index is defined as

ifif

where and denote the th entryof the vectors and respectively.

That is, , with the notation of[45]. The column of indexed by , denotedby , is the map

. The set of maps from

to reals, denoted by , is a vector space with point-wise addition and multiplication by scalar, and hence we can

speak of the linear span of the columns of . The rank of(denoted by ) is the dimension of the vector space

spanned by the columns of .Notice that the classical Hankel matrix of linear systems is a

special case of the Hankel matrix defined above. We will alsobe interested in the following subset of columns of .

Definition 19: Let be the set of those columns ofwhich are indexed by , for some input-output map

, and sequence of discrete events (i.e., containsno symbol ) and , i.e.

(14)

The cardinality of is denoted by . Note thattwo columns belonging to are considered equal, if theyare equal as maps from to .

The role of is the following. For each , considerthe map defined by . If

is realized by a LHS, then is completely determined by thediscrete state component of the initial state which induces . Infact, can be considered as an additional discrete output ofthe LHS. The collection is an encoding of .Hence is an encoding of .

Next, we define an analog of the Hankel-matrix for the dis-crete-valued components of .

Definition 20: For each sequence of discrete eventsand for each input-output map , define the shift by ofthe discrete-valued component of as the map

.Notice that the value of at is the value of for the

sequence where is preceded by , hence the use of the wordshift. If , then .

Definition 20 (Hankel-Table of ): Define discreteHankel-table of as the set

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That is, is the set of all the shifted maps of the form, and . It can be indeed thought of as a

table, columns of which are the maps . Now we are readyto state the main results on existence of a LHS realization.

Theorem 1 (Realization by LHSs): There exists a LHS real-ization of the family of input-output maps , if and only if

1) has a hybrid kernel representation;2) the rank of the Hankel-matrix is finite, and the

sets and are finite, i.e., ,and .

The proof of the above theorem can be found in Appendix I-B.We can compute a realization of from the columns of theHankel-matrix and the elements of the Hankel-table ,see Remark 13 in Appendix I-B or [1], [18]. In fact, this con-struction can be carried out using finite data, see [1], [18].

The intuition behind the theorem is the following. The finiterank condition on the Hankel-matrix makes sure that the ele-ments of the Hankel-matrix (i.e., the generalized Markov-pa-rameters) can be represented as products of matrices, as inProposition 1. It is analogous to the finite rank condition forthe Hankel-matrix for linear, bilinear and nonlinear systems[36], [39], [43]. The condition that the number of columns of

is finite is inspired by the following observation. Each


discrete state of the LHS gives rise to an unique input-outputmap induced by the state . The elements of encodethe continuous-valued parts of these input-output maps for thereachable discrete states. Hence, if has a realization, thenthe number of such input-output maps has to be finite, hence

has to be finite. Finally, the condition that is finitefollows from the fact that the discrete-part of the input-outputmaps have to be realizable by a finite-state Moore-automaton.

VI. MINIMALITY, SPAN-REACHABILITY, OBSERVABILITY

Below we present the main results of the paper onspan-reachability, observability and minimality. We startwith a rank condition for observability of LHS. In the restof the section, is a LHS of the form (1). For each dis-crete state define the subspace as theintersection of the kernels of all the matrices of the form

, for eachsequence of discrete events and each collec-tion , with ,

, i.e.

(16)

Here the convention of Notation 2 is used for the products ofthe system matrices of a LHS. The space is a generaliza-tion of the observability space for linear systems. In fact,is contained in the observability space of the linear system ofthe mode . However, also takes into account theoutput after first, second, etc., discrete-state transition. That iswhy products of the matrices of the linear subsystems and ofthe reset maps are considered too.

Theorem 2 (Observability): The LHS is observable if andonly if conditions (i) and (ii), described below, hold.

i) For each two discrete states , if andonly if conditions (a) and (b) below hold.

a) Equality of Discrete OutputsFor all , and for any sequences of events

, the corresponding discreteoutputs are the same, i.e.,

, andb) Equality ”Gen. Markov-parameters”

For all , for each sequence of discrete events, for each index

, and for each

where and. .

ii) For each , the zero vector is the only element of thesubspace , i.e., .

The proof of the theorem is presented in Appendix I-C. Theintuition behind the theorem is the following. Condition (ii)ensures that there are no two indistinguishable states with thesame discrete state components, i.e., there are no two indis-tinguishable states of the form and with .Condition (i) ensures that no two states of the form and

are indistinguishable. More precisely, condition (i) is

based on the observation that with each discrete statewe can associate an additional ”virtual” discrete output, namely,the input-output map induced by the initial state

. Condition (i) characterizes the observability of the dis-crete states of , where each state produces a dis-crete output . In particular, condition (a) of (i) isequivalent to the equality of the -valued (discrete) componentsof and , and condition (b) of (i) isequivalent to the equality of the -valued (continuous) compo-nents of and . It can be shown thatcondition (i) and condition (ii) of Theorem 2 can be checkedby a numerical algorithm, see [1], [19] for more details. Thisimplies that observability for LHSs whose matrices have onlyrational numbers as entries is decidable.

Remark 5 (Relationship With Existing Work): There existsa vast literature on observability of hybrid system, see for ex-ample [4], [6], [26], [27], [46]. As far as we know, none of theexisting work covers the class of systems considered. Note thatthe definition of observability presented in this paper is orientedtowards the need of realization theory and systems identifica-tion. Whether the proposed notion is suitable for observer de-sign remains a topic of future research.

Next, we state the rank condition for span-reachability. Tothis end, we need some additional notation and terminology.Let be a set of input-output maps. Let be a LHS of theform (1) and let be a map which assigns initialstates of to elements of . Recall the definition of the map

from Section IV-A, i.e., . Foreach , define the set of all the statessuch that either , or for someand for some sequence of events , i.e.

With the notation above and using Notation 2, let be thefollowing subspace of :

That is, is spanned by all the vectors, such that

, for some . The space canbe thought of as a generalization of the controllability space oflinear systems. In fact, the image of the controllability matrixof the linear system residing in the state is asubspace of . In addition, contains the linear span ofthose states which are reached by switching from some otherdiscrete state to too. Hence, the presence of product of resetmatrices and matrices of the linear components.

Theorem 3 (Span-Reachability): The LHS realizationis span-reachable if and only if

i) The automaton of is reachablefrom ;

ii) for each discrete state , .The proof of the theorem can be found in Appendix I-C. Theintuition behind the theorem is the following. First, in order forthe LHS realization to be span-reachable, we need to be able toreach every discrete state by a suitable choice of discrete events.


Second, the continuous state-space component associated witheach discrete state should contain only those states, which are re-ally necessary, i.e., should be the smallest vector space, whichcontains all time derivatives of the state-trajectories which endin . The latter derivatives are exactly the products of systemsmatrices and initial states which span . Span-reachability ofan LHS from a finite set of initial states can be checked by a nu-merical algorithm, see [1], [19]. In particular, span-reachabilityof an LHS from a finite set of initial states is decidable, if allthe entries of the system matrices and of the continuous statecomponents of the initial states are rational numbers.

Remark 6 (Related Work): Reachability of hybrid systemshas been a subject of intensive research, [28]–[31]. However, tothe best of our knowledge, no prior work has been done on span-reachability of LHSs. In addition, in the literature the systemis usually considered reachable, if all the states are reachable.In contrast, span-reachability means that the linear span of thereachable states is the whole state-space.

Theorem 4 (Minimality): Let be a family of input-outputmaps. If there exists a LHS realization of , then there also ex-ists a minimal LHS realization of . If is a LHS realiza-tion of , then the following are equivalent;

i) is a minimal LHS realization of ;ii) is span-reachable and it is observable;

iii) For any span-reachable LHS realizationof , there exists a surjective morphism

.All minimal LHS realizations of are isomorphic.

The proof of the above theorem can be found in Ap-pendix I-D. One can formulate an algorithm for transforminga LHS to a minimal one, see [1], [19], or Remark 14 inAppendix I-D of this paper. The intuition behind the char-acterization of minimality is similar to that of for linear andnonlinear systems, see [36], [38], [39], [43]. That is, a min-imal system should contain only states which are absolutelynecessary for representing the family of input-output mapsat hand. A realization satisfying condition (iii) of Theorem 4automatically satisfies this condition. In fact, in the abstractrealization theory [47] versions of condition (iii) are takenas definition of minimality. Due to the properties of LHSmorphisms, if a realization satisfies (iii) then it has a smallerdimension than any span-reachable realization. In turn, anyrealization can be transformed to a span-reachable one withoutincreasing the dimension. Hence, any realization satisfying (iii)is minimal. Moreover, the LHS morphism of condition (iii)automatically becomes an isomorphism, if is observable.Hence, span-reachable and observable systems are minimaland vice versa. The only thing which is left to do is to finda realization satisfying (iii). The free realization constructedfrom the columns of the Hankel-matrix as described in Remark13, Appendix I-B, has this property.

Remark 7: In Section II we presented a family of input-output maps and minimal LHS realization of ,such that the automaton and the individual linear subsystems of

are not all minimal. In fact, due to the uniquenessof a minimal LHS, the family above cannot have a LHS real-ization where the automaton and the linear subsystems are allminimal. That is, not all input-output maps which can be real-ized by a LHS can be realized by a LHS whose automaton andall linear subsystems are minimal.

VII. CONCLUSION

In this paper we have formulated and solved the realizationproblem for LHSs. The paper combines the theory of formalpower series with the classical automata theory to derive theresults.

Topics of further research include realization theory for piece-wise-affine systems on polytopes, non-LHSs without guards,and application of the presented results to subspace identifica-tion and model reduction.

APPENDIX

Review of Formal Power Series and Moore-Automata: InAppendix I-A-I we will review an extension of the theory of ra-tional formal power series. In Appendix I-A-XVIII an extensionof the classical realization theory of Moore-automata will be re-viewed.

Formal Power Series: The material of this section is basedon the classical theory of formal power series, see [32], [37].Unlike in the classical case, we are interested in rationality ofa set of formal power series, hence the original framework hasto be extended. We only state the main results and we omit theproofs, which can be found in [1], [9] and which are similar tothe classical ones. Let be an arbitrary set and let be a finiteset, which is referred to as the alphabet.

Definition 22 (Rational Representation): A rational repre-sentation with the index set , or simply a representation, is atuple such that

• is a finite-dimensional vector space;• is a linear map;• for each letter , is a linear map;• is a set of elements of indexed

by . Notice that the vectors are not required tobe all distinct.

The dimension of is called the dimensionof and it is denoted by .

Intuitively, a rational representation can be viewed as a gen-eralized Moore-automaton with the infinite state-space , withinput space , and with output space . The state transi-tion function is given by the linear maps

, the output map is given by ,an the set of initial states is given by . Wewill use the following notation in the sequel. If , are linearmaps, then denotes the composition ; for any inthe domain of , denotes the value .

Notation 3: For each word , where, , denote by the following linear

map. If , i.e., is the empty word, thenis defined as the identity map on . If , then is

composition of the linear maps , i.e.

With the notation of Definition 22, define the following sub-spaces of

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Definition 23 (Reachability): The representation iscalled reachable if . We will refer to thesubspace as the reachability subspace of .

Definition 24 (Observability): The representation iscalled observable if . We will refer to the subspace

as the observability kernel ofThe space is analogous to the kernel of the observability

matrix, and is analogous to the image of the reachabilitymatrix of a linear system.

Definition 25 (Morphism): Consider two representationswith the same index set , ,

Assume that , . Arepresentation morphism is a linear map

such that for any letter , ,for any index , , and . Themorphism is called surjective, injective, a representation iso-morphism if is a surjective, injective or isomorphism respec-tively, if considered as a linear map. Two representations are saidto be isomorphic, if there exists a representation isomorphismbetween them.

Intuitively, a representation morphism is a linear map be-tween the state-spaces of the representations commuting withthe linear maps of the representations involved.

Definition 26 (Formal Power Series): A formal power se-ries with coefficients in over is a map ,i.e., it is simply a function which maps finite words over tovectors in . We denote by the set of all formalpower series with coefficients in .

Notice that the set can be regarded a vectorspace with point-wise addition and multiplication by scalar [32].More precisely, if and , thendefine the linear combination as

, .Definition 27: A family of formal power series indexed by

is just a collection offormal power series whose elements are indexed by the elementsof . Notice that we do not require two formal power seriesindexed by different elements of to be distinct.

From now on denotes the family of formal power series.

Definition 28 (Rational Formal Power Series): Thefamily is called rational if there exists a representation

with the index set , such that

(18)

If (18) holds, then we say that is a (rational) representationof .

Intuitively, a family of formal power series is rational, if itselements are the input-output maps of a rational representation,when the latter is viewed as a generalized Moore-automaton.

Definition 29 (Minimality): A representation ofis minimal if for each representation of , it holds that

.Definition 30 (Hankel Matrix): Define the Hankel matrixof as follows. The columns of are indexed by pairs

where and . The rows of are indexedby pairs where and . The elementof indexed by the column index and the row index

is the entry of the vector . That is, if

denotes the entry of , thenfor all , and . The

column of indexed by is the map. Maps of the

above type form a vector space with point-wise addition andmultiplication by scalar, hence we can speak of the linear spacespanned by the columns of . The dimension of the vectorspace spanned by the columns of is the rank of , and itis denoted by .

Definition 31 (Shift of a Formal Power Series): Letbe a word over and for any formal power series

define the left shift of by asfor all .

Definition 32: The shift-invariant space generated by isthe linear subspace of spanned by formalpower series of the form with and , i.e.,

.Remark 8 (Equivalence of and ): There is a

one-to-one correspondence between the columns of in-dexed by and the series , hence the vector spacespanned by the columns of and the space are isomor-phic.

Theorem 5 ([1], [9]): With the notation above,• Rationality. is rational if and only if the Hankel-matrix

of is of finite rank, i.e., .• Minimality. If is rational, then there exists a minimal

rational representation of . A representation ofis minimal if and only if one of the following equivalentconditions holds: (i) is reachable and observable, (ii)for any reachable representation of , there exists a sur-jective representation morphism .

All minimal representations of are isomorphic.Remark 9: Let be a representation of . It can be shown

that can be transformed to a minimal representation ofand this transformation can be made effective, see [1], [19].

Remark 10: If is rational, then we can construct a min-imal representation of on , or, which is the same, onthe column space of the Hankel matrix , as follows;

, where for each , is the leftshift by , i.e., for any , ;

; and the map simply evaluates eachformal power series at the empty sequence, i.e., ,

. It can be shown that is a minimal representa-tion of . is called the free representation of .

Finite Moore-Automaton: Realization theory of Moore-au-tomata is a classical topic, see [34], [35]. However, in this paperwe need realization of families of input-output maps and this isnot covered by the classical theory. In this section we only statethe main results. The proofs of them are analogous to the clas-sical ones and they can be found in [1].

Natural candidates for input-outputs maps of a Moore-au-tomaton are maps of the form which map wordsover to elements in . Let be afamily of maps indexed by elements of some set . Notice thatwe do not require the maps to be all distinct. Below we definethe concept of a Moore-automaton realization of the family ofinput-output maps. In the sequel , denotes the family of maps

.Definition 33 (Moore-Automaton Realizations): Con-

sider a Moore-automaton and a map


. The pair will be called a Moore-automatonrealization.

Definition 34 (Realization of ): The Moore-automatonrealization , where and ,is a Moore-automaton realization of , if for each index ,sequence of inputs , .

Notice that the map is used to specify those states of ,which generate an input-output map which belongs to , i.e.,the input-output map of induced by equals .

Definition 35 (Reachability and Observability): A Moore-automaton realization is called reachable if is reach-able from the set ; and iscalled observable if is observable.

Definition 36 (Minimality): A Moore-automaton realiza-tion of is called minimal, if has the smalleststate-space cardinality among all realizations of .

Definition 37 (Morphisms): Let and betwo Moore-automaton realizations, such that the domain ofboth and is . Assume that and

. A map is said to be anautomaton morphism from to , denoted by

, if commutes with the state-transition andreadout maps, that is,, , . Themorphism is called injective, surjective, an isomorphism ifthe map is injective, surjective, bijective respectively. TwoMoore-automata realizations are called isomorphic, if thereexists an isomorphism between them.

Definition 38 (Shift of an i/o Map): For a mapand for any sequence define the left shift of by asthe map .

Definition 39: The shift-invariant set generated byis the set of all the maps of the form with , and

, i.e., .The set plays a role analogous to the Hankel matrix in

linear realization theory. Notice that if we apply the definitionof to the family , thenwe obtain the set already defined in (15).

Theorem 6 ([1], [34], [35]): With the notation above,• Existence.There exists a Moore-automaton realization of

if and only if is finite, i.e., .• Minimality. If there exists a Moore-automaton realization

of , then there also exists a minimal one. A realizationof is minimal if and only if one of the following

equivalent conditions holds; (i) is reachable and ob-servable, or (ii) for each reachable realization of

there exists a surjective morphism .Any two minimal Moore-automaton realizations of areisomorphic.Remark 11 (Minimization): If is a realization of the

family , then can be transformed to a minimal realiza-tion of , and with additional assumptions this trans-formation can be made effective, see [1], [19].

Remark 12: If is finite, then we can define a minimalrealization of with the state-space as fol-lows; , for all ,and for all , , andfor all . It can be shown that is minimal.

Existence of a LHS Realization: Proof of Theorem 1: Letbe a set of input-output maps. It follows from Proposition

1 that if has a LHS realization, then has a hybrid kernelrepresentation. Therefore, from now on we assume that hasa hybrid kernel representation. Informally, the proof consists ofthe following steps.

Step 1. We construct a certain family of formal power seriesand a family of discrete valued input-output maps from

. The family has the property that its Hankel matrix ofequals the Hankel-matrix of , i.e., . In addition,we show that there exists a Moore-automaton realization ofif and only if the sets and are finite.

Step 2. Let be a realization, where is of the form(1) and is a map assigning initial states. In The-orem 7 below we show that if is a realization of , thenwe can construct from a representation , and aMoore-automaton , such that is a representation of

and is a realization of . Here, denotes the-valued part of . That is, if is a realization of , then

is rational and has a realization by a Moore-automaton.We call the representation associated with . Wewill call the Moore-automaton realization associ-ated with .

Step 3. In Theorem 8 below we show that if is an observablerepresentation of , and is a reachable realization of

, then we can construct from and a LHS realiza-tion of , and we call the LHS realization

the LHS realization associated with and. By Theorem 5, if is rational, then it has a minimal

representation and this representation is observable. Similarly,from Theorem 6 it follows that if there exists a Moore-au-tomaton realization of , then there exists a minimal, and hencereachable, Moore-automaton realization of . Then

is a well-defined LHS realization of . Hence,if is rational and there exists Moore-automaton realizationof , then we can construct a LHS realization of .

Step 4. From Step 2 and Step 3 it follows that there exists aLHS realization of if and only if is rational and there existsa Moore-automaton realization of . From this, the statementof the theorem follows easily, by noticing that by Theorem 5,

is rational if and only if ;and by Lemma 1 (to be presented below) there exists Moore-automaton realization of if and only ifand .

Below we will carry out the steps outlined above in Step1—Step 2 more formally. Recall from Section V the definitionof set , , and of the generalized Markov-pa-rameters and , and . It is easy tosee that and are formal power series over the alphabet

with the coefficients in , i.e., .Definition 40: Define the family of formal power series

associated with as

where we identify with for and ,and we set .

That is, is the family of formal power series formed bythe formal power series , and indexed by the elementsof , where is set of elements of and pairs ,and . It is easy to see that the Hankel-matrix of

defined in Section V and the Hankel-matrix of are equal,


i.e., . Consider the LHS of the form (1) and letbe a map assigning initial states.

Construction 1 (Repr. Asscociated With LHS): Define therepresentation associated with from Step 2. as

where (19)

State-space . Assume that has elements, i.e.,. Define as the direct sum of the vector spaces ,

and , i.e., .Next, we fix the following basis in , which will be used

throughout the construction

(20)

Notice that the vector spaces , and can be viewedas subspaces of .

Linear maps. The linear maps ,and , , are defined as follows.

For all and : , ,and .

For all , : ,, and .Here, are the elements of the basis of fixed in (20),

and is the standard unit basis vector of . That is, therestriction of , and to any of the subspaces ofequals , and respectively. The application of

to each yields the column of , therestriction of to is zero, and the restriction of to

simulates the discrete-state transition map.Initial states. The family is defined

by and , for each, . That is, is the continuous component

of the initial state and is the vector , whereis the discrete component of the initial state , and

denotes the corresponding element of the basis (20).Notice that is always an element of .

The idea behind the choice of is the following. By“stacking up” the matrices and taking the “state-space” , we encoded most of the information on thediscrete-state dynamics which has an effect on the continuousinput-output behavior. But we still need to keep track of the ma-trices , and for that we need to simulate the discrete-state tran-sitions. This is done by introducing the vectors and definingthe action of on them accordingly.

Let be the set of

tuples of formal power series from . Define for each, the map as a pair, whose first component is simply the

discrete-valued part of and the second component mapseach sequence of discrete events to the tuple ofshifted formal power series , , i.e., forall

Definition 41: Define the set of discrete-valued mapsassociated with as the family formed by all the maps ,

, indexed by elements of . More formally

Now we are ready to define the notion of Moore-automaton re-alization associated with .

Construction 2 (Aut. Asscociated With LHS): Let be ahybrid system of the form (1) and let . Definethe automaton realization associated with the re-alization described in Step 2. as

, i.e., the state space and state-transition map ofare the same as those of the automaton of

, and the readout map is defined byfor all , where the formal power

series , are as follows. Forany word of the formfor some , , , isdefined as follows. If contains no letter , i.e.,

, then let . Otherwise, letbe such that and

, and define

where , , and is thestandard unit basis vector of . The map is

the -valued part of , i.e., .The intuition behind the definition of is the following.

For each discrete state , the continuous valued partof the input-output map induced by the

hybrid state contains information which cannot beencoded by continuous states only. That is why we have toconsider it as an additional discrete output associated with thediscrete state . Notice that has a hybrid kernelrepresentation and the maps can bedefined as in (10) by taking . In fact, isidentical to . Since, , the collection

completely determines the continuous-valuedcomponent of . With the above definitions we canformulate the following theorem.

Theorem 7: If is a realization of , then is arepresentation of and is a realization of . Theproof of the Theorem 7 is relatively straightforward and it canbe found in [1], [18].

Construction 3 (LHS From a Repr. and Automaton): Con-sider an observable representationof , and let be a reachable Moore-automaton realiza-tion of . Define the LHS realization as-sociated with and as follows. Require to be aLHS of the form (1), and require that the system parameters of

and the map are defined as follows.Moore-automaton. Assuming that is of the form

, define the automaton ofas , where the

discrete state space and the state-transition map of are thesame as those of , and the readout map of is definedas , i.e., for each , is the -valuedcomponent .

Continuous state space. For each , the continuousstate-space component belonging to is defined as follows.Denote by the set of all words such that if

is of the form withand , then . I.e. isreachable from by the word in the automaton

. Let be the subspace of spanned by all the elements


of the form and , , ,, such that , i.e.

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Note that in the expression , is a sequence ofdiscrete events, i.e., contains no . It is clear that is a finite-dimensional subspace of . Assume that and fixa basis in . By identifying the elements of with the vectorof their coordinates in this basis, we can identify with ,and we can identify linear maps from to or to with

, or matrices respectively.System Matrices. For each , define the matrices

, and , asfollows. We will view , , as linear maps

, and ,which are defined as restrictions of , and respectively

to . That is, for all

Notice that the subspace is invariant by construction,i.e., , and maps elements to elements of

, i.e., , for all .Define the matrix as the matrix such that for all

, the column of , viewed as an element of, equals for some and sequence of

discrete events such that , i.e.,.

Notice that is indeed well-defined for each . In-deed, since is reachable, it follows that for eachthere exists a map and a word such that

. It is left to show that the definition of is indepen-dent of the choice of and . If ,then , since is a realization of . Butthen , i.e., for all ,

. Since is a representation of we get that

for each . Hence, observ-ability of implies that .

The map . Define the map as follows. Foreach , let ,where is viewed as an element of .

It should be clear now why we needed observability of andreachability of . If is not observable, we could havemultiple choices for the matrices . If is not reachable,then for some discrete state we would be unable to definea continuous state space.

Theorem 8: If is an observable representation of andis a reachable realization of , then

is a span-reachable realization of .The proof of Theorem 8 is routine and it can be found in

[18]. Finally, the following result relates realizability of withfiniteness of the sets and .

Lemma 1: A Moore-automaton realization of exists ifand only if and .

Sketch: Follows from Theorem 6 , and from the observa-tion that finitiness of and is equivalent to the finit-ness of . For more details see [1], [18].

Remark 13 ( LHS From Hankel-Matrix): By Remark 10,we can construct a minimal representation of fromthe Hankel-matrix . By Remark 12 we can con-struct a minimal Moore-automaton realizationof from the set asscociated with . Notice that

is determined by and by the columns of .indexed by , , , .Since is observable, and is reachable, itfollows thatis a well-defined realization of . I.e. a LHS realization ofcan be constructed from the the Hankel-matrix and theHankel-table .

Proof of Theorem 2 and Theorem 3: Below we sketch theproofs of Theorem 2 and Theorem 3. For a more detailed expo-sition see [1], [19].

Sketch of the Proof of Theorem 2: The core of the proof isto show the following.

A) if and only if part (a) andpart (b) of condition (i) holds.

B) if and only if.

From (A) and (B) the proof of the theorem fol-lows easily. In order to prove (A) and (B), foreach state , input , sequence

, with , define themap , as

, i.e.,is the -valued component of the input-output map

for a fixed input and sequence of events .Proof of (A) Notice that is

equivalent to andfor all and .

The first equality is equivalent to part (a) of the theorem. Thesecond equality is equivalent to ,

. The latter is equivalent to equality of thehigh-order derivatives of , at zero for

, which in turn is equivalent to part (b).Proof of (B) is equivalent to

. By analiticity,is zero if and only if all its high-order derivatives at zero arezero. The latter is equivalent to .

Sketch of the Proof of Theorem 3: It is enough to show thatfor all .

follows by noticing that isspanned by the high-order derivatives of continuous state tra-jectories with respect to

evaluated at zero, such thatand is constant. Since the trajectories above belong to thelinear space , it then follows that all theirhigh-order derivatives belong to .

. From (4) it follows that any ele-ment of is a sum of expressions:

(22)

(23)


where , , and ,. The expression (22) belongs to , since it is

entire analytic in and all its Taylor-coffecients belongto . The expression (23 belongs to for a constant ,since then it is entire analytic in and all its Taylor-coefficients belong to . Finally, if is a general piecewise-continuous map, then the integral in (23) is a limit of sums ofintegrals of the form (23), with being constant, and hence weget that (23) belongs to .

Proof of Theorem 4: We will use the notation and conceptsfrom Appendix I-B. We need a number of lemmas. The proofsof these lemmas are not very difficult and can be found in [1],[19].

Lemma 2: Assume that is a LHS realization andthat is of the form (1). Then is span-reachable if andonly if is reachable and is reachable.

Lemma 3: If is an observable representation of andis a minimal Moore-automaton realization of , then

the associated LHS realization is observableand span-reachable.

Lemma 4: Assume that is a span-reachablerealization and is of the form (1), is an observablerepresentation and is reachable Moore-automatonrealization. If is a representation mor-phism and is an automatonmorphism, then there exists a surjective LHS morphism

such that for alland , and .

Lemma 5: Assume is a realization of , and assumethat is of the form (1). Then there exists a span-reachable LHSrealization of , such that ; and

if and only if is span-reachable. Finally,recall from [1] some properties of LHS morphisms.

Proposition 2 ([1]): Consider a LHS morphism. If is a LHS isomorphism, then

is span-reachable if and only if is span-reach-able and is observable if and only if isobservable. If is surjective, then . If

and is surjective, then is a LHS isomor-phism.

Proof of Theorem 4: First, we show that if has a LHS re-alization, then has a LHS realization satisfying condition (iii)of the theorem. Then we show that (iii), (ii), and (i) are equiva-lent. This immediately implies that if has a LHS realization,then has a minimal LHS realization.

Existence of a LHS realization satisfying (iii). Pick the freerealization de-fined in Remark 13. Recall that is a minimal representa-tion of and is a minimal realization of .By Lemma 3, is observable and span-reachable. Weshow that (iii) holds for . Let be a span-reach-able realization of and assume that is of the form (1). Thenby Lemma 2, is reachable and is reachable.By Theorem 5 and Theorem 6 there exist surjective morphisms

and . Thenby Lemma 4 there exists a surjective LHS morphism

.Next we show that (iii), (ii), and (i) are equivalent.

. Assume that satisfies (iii). Pickany LHS realization of . By Lemma 5 there ex-

ists a span-reachable realization of , such that, and hence there exists a surjective LHS

morphism . Then Proposition 2 impliesthat . Hence, is a minimalLHS realization of .

Let be any span-reachable and observablerealization of and assume that is of the form (1). We willshow that (iii) holds for . Consider the surjective LHSmorphism existence of which wasproved above. Using observability of , it can be shown (see[19], page 22) that is injective and thus it is a LHS iso-morphism. From the fact that is an isomorphism it followsthat the inverse LHS morphism is also an isomorphism.For any span-reachable realization of , there existsa surjective LHS morphism , whichimplies that is a surjective LHSmorphism. Hence, satisfies (iii).

Let be a minimal realization of. From Lemma 5 it follows that has to be

span-reachable. Hence, there exists a surjective morphism. But and are both

minimal, hence . Then Proposition 2 impliesthat is a LHS isomorphism. Since is observable,Proposition 2 implies that is observable.

Finally, we show the isomorphism of all minimal LHS real-izations. If , are two minimal realizations of

, then are both span-reachable and observable. Butfrom the proof of it follows that are bothisomorphic to , and hence to each other.

Corollary 1: If is a minimal representation of andis a minimal Moore-automaton realization of , then

is a minimal realization of .Proof: By Lemma 3, is span-reachable

and observable, and hence by Theorem 4 it is minimal.Remark 14 (Minimization): Let be a LHS real-

ization of . Construct the representation of andthe Moore-automaton realization of . Transform

to a minimal rational representation of . Transformto a minimal Moore-automaton realization

of . Construct the LHS realization . ByCorollary 1, is a minimal realization of .

ACKNOWLEDGMENT

The authors wish to thank their colleagues P. Collins and L.Habets of CWI.

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Mihaly Petreczky received the M.Sc. degree incomputer science and the Ph.D. degree in math-ematics from Vrije Universiteit, Amsterdam, TheNetherlands, in 2002 and in 2006, respectively.

He is currently affiliated with Maastricht Univer-sity, The Netherlands. His research interests includecontrol and systems theory of hybrid system and the-oretical computer science.

Jan H. van Schuppen is affiliated with the theresearch institute Centrum voor Wiskunde en Infor-matica (CWI) in Amsterdam, The Netherlands. He isEditor-in-Chief of Mathematics of Control, Signals,and Systems Journal and was Department Editor ofthe Discrete Event Dynamic Systems Journal. Hisresearch interests include control of hybrid systemsand of discrete-event systems, stochastic control,realization, and system identification. In appliedresearch his interests include engineering problemsof control of motorway traffic, of communication

networks, and control and system theory for the life sciences.Dr. van Schuppen was Associate Editor-at-Large of IEEE TRANSACTIONS

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realization theory for linear hybrid systems

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