Observability and Detectability of Linear Switching Systems: A Structural Approach
aa r X i v : . [ m a t h . D S ] F e b OBSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS:A STRUCTURAL APPROACH
ELENA DE SANTIS, MARIA DOMENICA DI BENEDETTO AND GIORDANO POLA
Abstract.
We define observability and detectability for linear switching systems as the possibility of recon-structing and respectively of asymptotically reconstructing the hybrid state of the system from the knowledgeof the output for a suitable choice of the control input. We derive a necessary and sufficient condition forobservability that can be verified computationally. A characterization of control inputs ensuring observabilityof switching systems is given. Moreover, we prove that checking detectability of a linear switching system isequivalent to checking asymptotic stability of a suitable switching system with guards extracted from it, thusproviding interesting links to Kalman decomposition and the theory of stability of hybrid systems. Introduction
Research in the area of hybrid systems addresses significant application domains with the aim of developingfurther understanding of the implications of the hybrid model on control algorithms and to evaluate whetherusing this formalism can be of substantial help in solving complex, real–life, control problems. In many appli-cation domains, hybrid controller synthesis problems are addressed by assuming full hybrid state information,although in many realistic situations state measurements are not available. Hence, to make hybrid controllersynthesis relevant, the design of hybrid state observers is of fundamental importance. A step towards a pro-cedure for the synthesis of these observers is the analysis of observability and detectability of hybrid systems.Observability has been extensively studied both in the continuous [11, 13] and in the discrete domains (see e.g.[18, 19]). In particular, Sontag in [20] defined a number of observability concepts and analyzed their relationsfor polynomial systems. More recently, various researchers investigated observability of hybrid systems. Thedefinitions of observability and the criteria to assess this property varied depending on the class of systemsunder consideration and on the knowledge that is assumed at the output. Incremental observability was in-troduced in [4] for the class of piecewise affine systems. Incremental observability implies that different initialstates always give different outputs independently of the applied input. A characterization of observabilityand the definition of a hybrid observer for the class of autonomous piecewise affine systems can be found in [6].In [10] observability of autonomous hybrid systems was analyzed by using abstraction techniques. In [2], thenotion of generic final–state determinability proposed in [20] was extended to hybrid systems and sufficientconditions were given for linear hybrid systems. The work in [22] considered autonomous switching systemsand proposed a definition of observability based on the concept of indistinguishability of continuous initialstates and discrete state evolutions from the outputs in free evolution. In [8, 1] observability of switchingsystems (with control) was investigated. Critical observability for safety critical switching systems was in-troduced in [7], where a set of “critical” states must be reconstructed immediately since they correspond tohazards that may yield catastrophic events.While observability of hybrid systems was addressed in the papers cited above, a general notion of detectabilityhas not been introduced as yet. To the best of our knowledge, the only contribution dealing with detectabilitycan be found in [16] where detectability was defined for the class of jump linear systems as equivalent to theexistence of a set of linear gains ensuring the convergence to zero of the estimation error in a stochastic setting.In this paper we address observability and detectability for the class of switching systems. General notions of
This work has been partially supported by the HYCON Network of Excellence, contract number FP6-IST-511368 and byMinistero dell’Istruzione, dell’Universita’ e della Ricerca under Projects MACSI and SCEF (PRIN05). observability and detectability are introduced for the class of linear switching systems, though our definitionsapply to more general classes of hybrid systems, since they involve only dynamical properties of the executionsthat are generated by the hybrid system. Further, we derive a computable necessary and sufficient conditionfor assessing observability. We then characterize detectability using a Kalman–like approach. In particular, weshow that checking detectability of a linear switching system is equivalent to checking asymptotic stability of asuitable linear switching system with guards associated with the original system. This result is clearly relatedto the classical detectability analysis of linear systems. It is important because it allows one to leverage awealth of existing results on the stability of switching and hybrid systems (see e.g. [17, 5, 12] and the referencestherein). A preliminary version of this paper appeared before in [8]. A characterization of observability, closeto the one of [8] and of the one presented in this paper, can be found in [1] for a subclass of the switchingsystems considered in [8]. The relation between [1], [8] and the present paper is discussed in Section 3.The paper is organized as follows. In Section 2, we introduce linear switching systems and the notions ofobservability and detectability. Section 3 is devoted to finding conditions for the reconstruction of the discretecomponent of the hybrid state. In Section 4 we give a characterization of observability and detectability. InSection 5, an example shows the applicability and the benefits of our results. Section 7 includes technicalproofs of some of the results established in Section 3. Section 6 offers some concluding remarks.2.
Preliminaries and basic definitions
In this section, we introduce the notations and some basic definitions that are used in the paper.2.1.
Notation.
The symbols N , R and R + denote the natural, real and positive real numbers, respectively.The symbol I denotes the identity matrix of appropriate dimensions. Given a vector x ∈ R n , the symbol x ′ denotes the transpose of x . The symbol k . k n denotes the Euclidean norm of a vector in the linear space R n .Given a linear subspace H of R n , the symbol dim ( H ) denotes its dimension and the symbol π H denotes theprojector on H , i.e. π H x is the Euclidean orthogonal projection of x onto H . Given a matrix M ∈ R n × m , thesymbols Im ( M ) and ker ( M ) denote respectively the range and the null space of M ; given a set H ⊆ R n thesymbol M − ( H ) denotes the inverse image of H through M , i.e. M − ( H ) = { x ∈ R m |∃ y ∈ H : y = M x } .Given a set Ω, the symbol card (Ω) denotes the cardinality of Ω.2.2. Switching systems.
We consider the class of linear switching systems and the class of linear switchingsystems with guards, which generalize the class defined in [8], following the general model of hybrid automata of[15, 21]. Switching systems are relevant in many application domains such as, among many others, mechanicalsystems, power train control, aircraft and air traffic control, switching power converters, see e.g. [12, 7, 9] andthe references therein.The hybrid state ξ of a GLSw –system H is composed of two components: the discrete state i belongingto the finite set Q = { , , ..., N } , called discrete state space, and the continuous state x belonging to thelinear space R n i , whose dimension n i depends on i ∈ Q . The hybrid state space of H is then defined byΞ = S i ∈ Q { i } × R n i . The control input of H is a function u ∈ U , where U denotes the class of piecewisecontinuous functions u : R → R m . The output function of H belongs to the set Y of piecewise continuousfunctions y : R → R l . The evolution of the continuous state x and of the output y of H is determined by thelinear control systems:(2.1) S ( i ) : (cid:26) ˙ x = A i x + B i u,y = C i x, whose dynamical matrices A i , B i , C i depend on the current discrete state i ∈ Q . The evolution of the discretestate of H is governed by a Finite State Machine (FSM), so that a transition from a state i ∈ Q to a state h ∈ Q may occur if e = ( i, h ) ∈ E , where E ⊆ Q × Q is the set of (admissible) transitions in the FSM, and ifthe continuous state x is in the set G ( e ) ⊆ R n i , called guard [14]. Whenever a transition e = ( i, h ) occurs,the continuous state x is instantly reset to a new value R ( e ) x , where R is the reset function which associates In this paper, the role of the guard G ( e ) is to enable (and not to enforce) a transition. BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 3 a matrix R ( e ) ∈ R n h × n i to each e ∈ E . We assume that R ( e ) = I , for any in-loop transition e = ( i, i ) ∈ E .A linear switching system with guards ( GLSw –system) H is then specified by means of the tuple:(2.2) (Ξ , S, E, G, R ) , with all the symbols as defined above. Given a GLSw –system H , if G ( e ) = R n i for any e ∈ E , then H iscalled linear switching system ( LSw –system) and for simplicity the symbol G is omitted in the tuple (2.2),i.e. H = (Ξ , S, E, R ). A GLSw –system H is said to be autonomous if all systems S ( i ) are autonomous, i.e. B i = 0.The evolution in time of GLSw –systems can be defined as in [15], by means of the notion of execution . Werecall that a hybrid time basis τ is an infinite or finite sequence of sets I j = [ t j , t j +1 ) , j = 0 , , ..., card ( τ ) − t j +1 > t j ; let be card ( τ ) = L . If L < ∞ , then t L = ∞ . Given a hybrid time basis τ , time instants t j are called switching times . Throughout the paper we suppose that given a hybrid time basis, the number ofswitching times within any bounded time interval is finite, thus avoiding Zeno behaviour [14] in the evolutionof the system. Let T be the set of all hybrid time bases and consider a collection:(2.3) χ = ( ξ , τ, u, ξ, y ) , where ξ ∈ Ξ is the initial hybrid state, τ ∈ T is the hybrid time basis, u ∈ U is the continuous control input, ξ : R → Ξ is the hybrid state evolution and y ∈ Y is the output evolution. The function ξ is defined as follows: ξ ( t ) = ξ , ξ ( t ) = ( q ( t ) , x ( t )) , where at time t ∈ I j , q ( t ) = q ( t j ), x ( t ) is the (unique) solution of the dynamical system S ( q ( t j )), withinitial time t j , initial state x ( t j ) and control law u . Moreover, if we set x − ( t j ) = lim t → t j − x ( t ) the followingconditions have to be satisfied for any j = 1 , ..., L − q ( t j − ) , q ( t j )) ∈ E,x − ( t j ) ∈ G ( q ( t j − ) , q ( t j )) ,x ( t j ) = R ( q ( t j − ) , q ( t j )) x − ( t j ) . The output evolution y is defined for any j = 0 , , ...,L − y ( t ) = C q ( t j ) x ( t ) , t ∈ [ t j , t j +1 ) . A tuple χ of the form (2.3), which satisfies the conditions above, is called an execution of H [14].2.3. Observability and Detectability.
In this section, we introduce the notions of observability and de-tectability for the class of
GLSw − systems.Given a GLSw –system H , we equip the hybrid state space with a metric: δ (( i, x i ) , ( h, x h )) = (cid:26) ∞ , if i = h, k x i − x h k n i , if i = h. The pair (Ξ , δ ) is a metric space.
Definition 2.1. A GLSw –system H is detectable if there exist a control input b u ∈ U and a function b ξ : Y × U →
Ξ such that:(2.4) ∀ ε > , ∀ ρ > , ∃ ˆ t > t : δ ( b ξ ( y | [ t ,t ] , b u | [ t ,t ) ) , ξ ( t )) ≤ ε, ∀ t ≥ ˆ t, t = t j , j = 0 , , ..., L, for any execution χ with control input b u and hybrid initial state ξ = ( i, x ) with k x k n i ≤ ρ . If condition(2.4) holds with ε = 0, then H is observable.By Definition 2.1, an observable GLSw –system is also detectable. By specializing Definition 2.1 to linearsystems, the classical observability and detectability notions are recovered. Note that the reconstruction ofthe current hybrid state is required at every time t ≥ ˆ t with t = t j . Time instants t j are ruled out as it is forobservable linear systems, where the current state may be reconstructed only at every time strictly greater ELENA DE SANTIS, MARIA DOMENICA DI BENEDETTO AND GIORDANO POLA than the initial time. However, observability and detectability for linear systems are defined independentlyfrom the control function, while here we assume to choose a suitable control law. The two definitions coincidefor linear systems but not for
GLSw –systems. In fact, if the observability (or detectability) property wererequired for any input function, then any
GLSw –system would never be observable (or detectable), see e.g.[8, 1]. However, we will show in Section 3 that if a switching system is observable in the sense of Definition2.1, then it is observable for “almost all” input functions.Definition 2.1 requires the reconstruction of the discrete and of the continuous state. We consider these twoissues separately, by stating conditions that ensure the reconstruction of the discrete state in Section 3 and ofthe continuous state in Section 4. 3.
Location observability
In this section, we focus on the reconstruction of the discrete component of the hybrid state only . By special-izing Definition 2.1, we have:
Definition 3.1. A GLSw –system H is location observable if there exist a control input ˆ u ∈ U and a function b q : Y × U → Q such that:(3.1) ∀ ρ > , ∃ ˆ t > t : b q ( y | [ t ,t ] , b u | [ t ,t ) ) = q ( t ) , ∀ t ≥ ˆ t, t = t j , j = 0 , , ..., L − , for any execution χ with control input b u and hybrid initial state ξ = ( i, x ) with k x k n i ≤ ρ .A GLSw –system H is said to be location observable for a control input ˆ u ∈ U if there exists a function b q : Y × U → Q such that condition (3.1) is satisfied. The definition of location observability guarantees thereconstruction of the discrete state, but not of the switching times, as the following example shows. Example 3.2.
Consider a
GLSw –system H = (Ξ , S, E, G, R ), where Ξ = { } × R , E = { e } with e = (1 , G ( e ) = R . Let the dynamical system S (1) and the reset function R ( e ) be described by the followingdynamical matrices: A = (cid:18) (cid:19) , B = (cid:18) (cid:19) , C = (cid:0) (cid:1) , R ( e ) = (cid:18) (cid:19) . The system H is trivially location observable for any control input u . However since for any x ∈ R ,( R ( e ) − I ) x belongs to the kernel of the observability matrix associated with S (1), it is not possible toreconstruct the switching times, for any choice of the control input u .For later use, given i, h ∈ Q , define the following augmented linear system S ih :(3.2) ˙ z = A ih z + B ih u , y ih = C ih z ,where: A ih = (cid:18) A i A h (cid:19) , B ih = (cid:18) B i B h (cid:19) , C ih = (cid:0) C i − C h (cid:1) .Let V ih ⊆ R n i + n h be the maximal controlled invariant subspace [3] for system S ih contained in ker( C ih ), i.e.the maximal subspace F ⊆ R n i + n h satisfying the following sets inclusions:(3.3) A ih F ⊆ F + Im ( B ih ) , F ⊆ ker ( C ih ) . Define ˆ J = { ( i, h ) ∈ Q × Q : i = h } and consider the set: U ∗ = n u ∈ U : u = e u, a.e., ∀ e u ∈ e U o , BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 5 where:(3.4) e U = S ( i,h ) ∈ ˆ J U ih , U ih = (cid:26) u ∈ U : u ( t ) = K ih z ( t ) + v ih ( t ) ,t ≥ ˆ t , for some ˆ t ∈ R (cid:27) , the gain K ih is such that ( A ih + B ih K ih ) V ih ⊆ V ih , v ih ( t ) ∈ B − ih ( V ih ) , ∀ t ≥ ˆ t and z ( t ) is the state of system S ih at time t , under control u with z (ˆ t ) ∈ V ih . The set U ∗ is composed of the control inputs u such thatafter a finite time ˆ t the output y ih of S ih with any initial state x ∈ R n i + n h and the control input u is notidentically zero for any choice of ( i, h ) ∈ ˆ J . We will show that control inputs in U ∗ ensure the reconstructionof the discrete state. The following result identifies conditions for nonemptyness of U ∗ . Lemma 3.3.
Given a
GLSw − system H , the set U ∗ is nonempty if (3.5) ∀ ( i, h ) ∈ ˆ J, ∃ k ∈ N , k < n i + n h : C i A ki B i = C h A kh B h . The proof of the above result requires some technicalities and is therefore reported in the Appendix. We nowhave all the ingredients for characterizing location observability of switching systems.
Theorem 3.4. A GLSw –system H is location observable if and only if condition (3.5) holds.Proof. (Necessity) Suppose by contradiction, that ∃ ( i, h ) ∈ ˆ J such that condition (3.5) is not satisfied andconsider any u ∈ U and any executions χ = (( i, , τ, u, ξ , y ) and χ = (( h, , τ, u, ξ , y ) with τ = { I } and I = [0 , ∞ ). It is readily seen that y = y and therefore the discrete state cannot be reconstructed.(Sufficiency) By Lemma 3.3, condition (3.5) implies that U ∗ = ∅ ; choose any u ∈ U ∗ and consider anyexecution χ = ( ξ , τ, u, ξ, y ). Consider any j < L and let ξ ( t ) = ( i, x ( t )) , t ∈ [ t j , t j +1 ). Given any h ∈ Q ,denote by y ih ( t, t j , z, u | [ t j ,t ) ) the output evolution at time t of system S ih with initial state z ∈ R n i + n h atinitial time t j and control law u | [ t j ,t ) . Since u ∈ U ∗ then for any ε >
0, for any h = i and for any w ∈ R n h there exists a time t ∈ ( t j , t j + ε ) such that y ih ( t, t j , ( x ( t j ) w ) ′ , u ) = 0. This implies that y ( t ) = y h ( t ),where y h is the output associated with the execution ( ξ h , τ, u, ξ h , y h ) with ξ h ( t ) = ( h, x h ( t )) , t ∈ [ t j , t j +1 ).Hence, the discrete state can be reconstructed for any t ∈ ( t j , t j +1 ), and the statement follows. (cid:3) It is seen from the above result that if a
GLSw –system H is location observable then it is location observablefor any input function u ∈ U ∗ . A control law that ensures location observability is derived in the proof ofLemma 3.3. Moreover, if the set of control inputs is the set C ∞ ( R m ) of smooth functions u : R → R m (insteadof the set U of piecewise continuous functions), then U ∗ contains all and nothing but the control inputs whichensure location observability. Remark . Condition (3.5) was first given in [8] as a necessary and sufficient condition for guaranteeinglocation observability of linear switching systems. A subclass of switching systems was then considered in [1]where similar observability conditions can be found. While the notion of observability of [1] and the one in thepresent paper (Definition 2.1 or equivalently the definition in [8]) are slightly different, the notions of locationobservability coincide in the two papers. This translates in a characterization of location observability in [1]which is equivalent to the one in [8] and hence to the one of the present paper (compare Theorem 3 of [1],Theorem 8 of [8] and Theorem 3.4 of this paper).4.
Characterizing Observability and Detectability
Definition 2.1 implies that a GLSw –system is observable if and only if it is location observable and S ( i ) isobservable for any i ∈ Q .The intuitive algorithm for the reconstruction of the (current) hybrid state of an observable GLSw –system H , ELENA DE SANTIS, MARIA DOMENICA DI BENEDETTO AND GIORDANO POLA processes the output y ∈ Y and the input u ∈ U ∗ . It first reconstructs the current discrete state, by lookingfor the unique i ∈ Q such that (4.1) Y ( n i ) ( t ) ∈ Im ( O i ) + F i u ( t ) , where Y ( n i ) ( t ) = ( y ( t ) ′ ˙ y ( t ) ′ . . . y ( n i − ( t ) ′ ) ′ , O i is the observability matrix associated with S ( i ) and F i = C i . . . C i A i C i B i . . . . . . . . . . . . C i A n i i C i A n i − i B . . . C i B i ;Then, on the basis of the knowledge of i , it reconstructs the current continuous state x ( t ), by computing:(4.2) { x ( t ) } = O − i (cid:16) Y ( n i ) ( t ) − F i u ( t ) (cid:17) . We now focus on
LSw –systems and derive conditions that ensure detectability. Since location observability isa necessary condition for a switching system to be observable or detectable, we assume now that this propertyholds for all systems considered in this section. Given a
LSw –system H = (Ξ , S, E, R ), define the autonomous LSw –system:(4.3) H ′ = (Ξ , S ′ , E, R ) , where S ′ ( i ) is defined as S ( i ) in (2.1) with B i = 0. We assume that H ′ is with full discrete evolutioninformation, i.e. that the discrete state and the switching times are known at any time. Clearly, detectabilityof H implies detectability of H ′ . Under some appropriate conditions, the converse implication is true: Lemma 4.1.
A location observable
LSw –system H is detectable if H ′ is detectable and H satisfies the followingproperty: (4.4) E (cid:8) = ∅ or Im ( R ( e ) − I ) ∩ ker( O i ) = { } , ∀ e ∈ E (cid:8) , where E (cid:8) = { ( i, h ) ∈ E : i = h } and O i is the observability matrix associated with S ( i ) . Under condition (4.4), if a transition ( i, i ) ∈ E (cid:8) occurs in H at time t j from a hybrid state ( i, x − ) to ahybrid state ( i, x + ) with x + = R ( i, i ) x − = x − then x + − x − / ∈ ker ( O i ). Hence the switching time t j can bereconstructed . Then, the proof of the result above just follows from the linearity of the continuous dynamicsin H and from the definition of H ′ .The result of Lemma 4.1 reduces the analysis of detectability of a linear switching system with control , to thatof an autonomous linear switching system.For analyzing detectability of H ′ it is useful to first perform a discrete state space decomposition.Given H ′ = (Ξ , S ′ , E, R ) as in (4.3) and a set ˆ Q ⊆ Q let H ′ | ˆ Q = ( Ξ | ˆ Q , S ′ | ˆ Q , E | ˆ Q , R | ˆ Q ) , be the switching sub–system of H ′ obtained by restricting the discrete state space Q of H to ˆ Q , i.e. such thatΞ | ˆ Q = S i ∈ ˆ Q { i } × R n i , S ′ | ˆ Q ( i ) = S ′ ( i ) , E | ˆ Q = { ( i, h ) ∈ E : i, h ∈ ˆ Q } and R | ˆ Q ( i, h ) = R ( i, h ). Proposition 4.2.
The
LSw –system H ′ is detectable if and only if the LSw –system H ′ | b Q with b Q = { i ∈ Q : S ( i ) is not observable } is detectable.Proof. (Necessity) Obvious. (Sufficiency) Consider any execution χ of H ′ . If q ( t ) ∈ b Q for any time t ≥ t then the detectability of H ′ | b Q implies the asymptotic reconstruction of the hybrid state evolution of χ . If q ( t ) / ∈ b Q for some finite time t , then S ′ ( q ( t )) is observable and hence it is possible to (exactly) reconstruct the If the switching system H is location observable and u ∈ U ∗ , Theorem 3.4 guarantees that such discrete state i is unique. Note that the switching system of Example 3.2 does not satisfy condition (4.4) and therefore switching times in that casecannot be reconstructed.
BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 7 continuous state of H ′ in infinitesimal time. Once the continuous state x ( t ′ ) is known at time t ′ > t , locationobservability of H ′ ensures the reconstruction of the hybrid state for any time t ′′ ≥ t ′ with t ′′ = t j . (cid:3) By Proposition 4.2 there is no loss of generality in assuming that system S ′ ( i ) is not observable for any i ∈ Q .Moreover, we assume that S ′ ( i ), i ∈ Q , are in observability canonical form, i.e. that dynamical matricesassociated with S ′ ( i ) are of the form: A i = A (11) i A (21) i A (22) i ! , C i = (cid:16) C (1) i (cid:17) , where A (22) i ∈ R d i × d i , 0 < d i ≤ n i matrices A (11) i , A (21) i are of appropriate dimensions and ( A (11) i , C (1) i ) is anobservable matrix pair, for any i ∈ Q . This assumption is made without loss of generality: suppose that, forsome i ∈ Q , the dynamical matrices A i , C i of the switching system H ′ are not in the observability canonicalform. Then, we define an invertible linear transformation T i : R n i → R n i such that T i A i T − i and C i T − i arein the observability canonical form. For all j ∈ Q such that the dynamical matrices A j , C j of the switchingsystem H ′ are in the observability canonical form, we let T j be the identity matrix. We then define the hybridstate space transformation T : Ξ → Ξ such that for any ( i, x ) ∈ Ξ, T ( i, x ) := ( i, T i x ). The reset function in thenew coordinates is given by T h R ( e ) T − i , for any e = ( i, h ) ∈ E . The continuous component x of the hybridstate ( i, x ) of H ′ can be partitioned as x = ( x ′ x ′ ) ′ , with x ∈ R n i − d i , x ∈ R d i , and the reset matrix R ( e ) can be partitioned as: R ( e ) = (cid:18) R (11) ( e ) R (12) ( e ) R (21) ( e ) R (22) ( e ) (cid:19) , where R (22) ( e ) ∈ R d h × d i and R (11) ( e ), R (12) ( e ), R (21) ( e ) are of appropriate dimensions. Given the LSw –system H ′ as in (4.3), define the GLSw –system:(4.5) H = (Ξ , S , E, G , R ) , where: • Ξ = S i ∈ Q { i } × R d i ; • S ( i ) is described by dynamics ˙ z ( t ) = A (22) i z ( t ), for any i ∈ Q ; • G ( e ) = ker( R (12) ( e )), for any e ∈ E ; • R ( e ) = R (22) ( e ), for any e ∈ E .There is a strong connection between detectability of H ′ and asymptotic stability of H . Set B := S i ∈ Q { i } × B i ,where B i = { x ∈ R n i : k x k n i ≤ } . We also define ε B := S i ∈ Q { i } × ε B i for any ε ∈ R + . An autonomous GLSw –system H is asymptotically stable if the continuous component of the hybrid state of any execution χ of H converges to the origin as time goes to infinity, or equivalently: ∀ ε > , ∀ ρ > , ∃ ˆ t > t : ξ ( t ) ∈ ε B , ∀ t ≥ ˆ t, for any execution χ with hybrid initial state ξ ∈ ρ B . The following holds: Proposition 4.3.
The
LSw –system H ′ is detectable if and only if the GLSw –system H is asymptoticallystable.Proof. (Sketch.) Let E be the set of executions of H ′ such that C q ( t ) x ( t ) = 0 , ∀ t ≥ t . The continuouscomponent x ( t ) of the hybrid state ( q ( t ) , x ( t )) of any execution in E belongs to the subspace ker ( O i ) with i = q ( t ) for any t ∈ I j and j = 0 , , ..., L . By definition of E , H ′ is detectable if and only if the continuouscomponent of the hybrid state ξ of any χ ∈ E converges to the origin, i.e. ∀ ε > , ∀ ρ > , ∃ ˆ t ≥ t such that ξ ( t ) ∈ ε B , ∀ t ≥ ˆ t , for any χ ∈ E with hybrid initial state ξ ∈ ρ B . By definition of the observability canonicalform, this is equivalent to asymptotic stability of H . (cid:3) ELENA DE SANTIS, MARIA DOMENICA DI BENEDETTO AND GIORDANO POLA
By combining Lemma 4.1 and Propositions 4.2 and 4.3 we obtain the following characterization of detectabilityof
LSw –systems.
Theorem 4.4. A LSw –system H is detectable if the following conditions are satisfied: i): H is location observable, and ii): H satisfies condition (4.4), and iii): H is asymptotically stable.Conversely, if H is detectable then conditions i) and iii) are satisfied. Since the executions associated with a
GLSw –system (Ξ , S, E, G, R ) are also executions of the
LSw –system H = (Ξ , S, E, R ), the conditions of Theorem 4.4 are also sufficient for a GLSw –system to be detectable.Detectability of switching systems has also been addressed in [8]. The above result provides a deeper analysisthan the one in [8] since it reduces detectability of
LSw –systems to asymptotic stability of
GLSw –systems(compare Theorem 9 of [8] with the above result). This allows one to leverage the rich literature on stabilityof hybrid systems (see e.g. [17, 5, 12] and the references therein) for checking detectability. While checkingconditions i) and ii) is straightforward, checking condition iii) requires the analysis of asymptotic stability ofswitching systems with guards.We now derive sufficient conditions for assessing the asymptotic stability of H , by abstracting H with linearswitching systems with no guards. Given the autonomous GLSw –system H as in (4.5) define the followingautonomous LSw –systems:(4.6) H = (Ξ , S , E, R ) , H = (Ξ , S , E, R ) , where R ( e ) = R (22) ( e ) π ker( R (12) ( e )) . The following result holds: Proposition 4.5.
The autonomous
GLSw –system H is asymptotically stable if either H or H is asymp-totically stable. Since transitions in
LSw –systems H and H are independent of the continuous state, the asymptotic stabilityanalysis of H and H is in general easier than the one of H . An application of this result is shown in thenext section. 5. An illustrative example
In this section, we present an example that shows the interest and applicability of our results. Consider thelinear switching system H = (Ξ , S, E, R ), where: • Ξ = (cid:0) { } × R (cid:1) ∪ (cid:0) { } × R (cid:1) ∪ (cid:0) { } × R (cid:1) ∪ ( { } × R ) ∪ (cid:0) { } × R (cid:1) ∪ (cid:0) { } × R (cid:1) ; • S associates to any i ∈ Q = { , , , , , } the linear control system S ( i ) of (2.1), where: BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 9 A = − − , B = , C = (cid:0) (cid:1) ,A = − − , B = , C = (cid:0) (cid:1) ,A = (cid:18) − (cid:19) , B = (cid:18) (cid:19) , C = (cid:0) (cid:1) ,A = 3 , B = 1 , C = 1 ,A = − − , B = , C = (cid:0) (cid:1) ,A = (cid:18) − (cid:19) , B = (cid:18) (cid:19) , C = (cid:0) (cid:1) ; • E = { (1 , , (2 , , (2 , , (2 , , (3 , , (3 , , (4 , , (4 , , (5 , , (5 , , (6 , } ; • R is defined by: R (1 ,
2) = − −
30 0 1 00 0 0 1 , R (2 ,
1) = − ,R (2 ,
3) = (cid:18) − (cid:19) , R (2 ,
5) = ,R (3 ,
3) = (cid:18) (cid:19) , R (3 ,
6) = (cid:18) (cid:19) ,R (4 ,
1) = ( 1 − ′ , R (4 ,
2) = ( 1 1 1 ) ′ ,R (5 ,
4) = ( 1 1 1 ) , R (5 ,
6) = (cid:18) (cid:19) ,R (6 ,
5) = . . The Finite State Machine associated with system H is depicted in Figure 1. Let us analyze observability and1 2 34 5 6 Figure 1.
Finite State Machine associated with the linear switching system H .detectability properties of the linear switching system H . The linear systems S ( i ) associated with discretestates i = 1 , , , , H is not observable. We now check detectability of H . For this purpose, we apply Theorem 4.4. We start by checking condition i).The Markov parameters associated with systems S ( i ), i ∈ Q are given for any k ∈ N by:(5.1) C A k B = 1 , C A k B = 2 k , C A k B = 0 ,C A k B = 3 k , C A k B = 4 , C A k B = 5 k . Hence, condition (3.5) is satisfied for k = 1. Thus by Theorem 3.4, the linear switching system H is location ob-servable. We now check condition ii) of Theorem 4.4. In this case E (cid:8) = { (3 , } and Im ( R (3 , − I ) ∩ ker( O ) = { } ;thus condition ii) is satisfied. Finally, we check condition iii). Since the linear system S (4) is observ-able, by Proposition 4.2 the switching system H ′ associated with H is detectable if and only if H ′ | b Q with b Q = { , , , , } , is detectable. The resulting linear switching system H ′ | b Q is characterized by the FiniteState Machine in Figure 2.We can now introduce the GLSw –system H of (4.5) associated with H ′ | b Q :1 2 35 6 Figure 2.
Finite State Machine associated with the linear switching system H ′ | b Q , with b Q = { , , , , } .(5.2) H = (Ξ , S , E, G , R ) , where: • Ξ = (cid:0) { } × R (cid:1) ∪ (cid:0) { } × R (cid:1) ∪ ( { } × R ) ∪ (cid:0) { } × R (cid:1) ∪ ( { } × R ); • S ( i ) is described for any i ∈ b Q by dynamics ˙ z ( t ) = A (22) i z ( t ), where: A (22)1 = (cid:18) − − (cid:19) ,A (22)2 = (cid:18) − − (cid:19) , A (22)3 = − ,A (22)5 = (cid:18) − − (cid:19) ,A (22)6 = − • E = { (1 , , (2 , , (2 , , (2 , , (3 , , (3 , , (5 , , (6 , } ; • G ( i, h ) = ker( R (12) ( i, h )) for any ( i, h ) ∈ E , where: R (12) (1 ,
2) = ( 2 − , R (12) (2 ,
1) = (cid:18) (cid:19) ,R (12) (2 ,
3) = ( − , R (12) (2 ,
5) = ( 0 0 ) ,R (12) (3 ,
3) = 1 , R (12) (3 ,
6) = 0 ,R (12) (5 ,
6) = ( 1 0 ) , R (12) (6 ,
5) = 0;
BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 11 • R ( i, h ) = R (22) ( i, h ) for any ( i, h ) ∈ E , where: R (22) (1 ,
2) = (cid:18) (cid:19) , R (22) (2 ,
1) = (cid:18) (cid:19) ,R (22) (2 ,
3) = ( 1 1 ) , R (22) (2 ,
5) = (cid:18) (cid:19) ,R (22) (3 ,
3) = 10 , R (22) (3 ,
6) = 1 ,R (22) (5 ,
6) = ( 10 0 ) , R (22) (6 ,
5) = (cid:18) (cid:19) . The Finite State Machine associated with H (Figure 2) is composed by three strongly connected components ,i.e. one involving the discrete states 1 , ∈ ˆ Q , the one involving the discrete state 3 ∈ ˆ Q and the other involvingthe discrete states 5 , ∈ ˆ Q . It is well–known that the asymptotic stability of a switching system can be assessedby studying this property in each strongly connected component. More precisely, H is asymptotically stable ifand only if switching system H | Q , with Q = { , } , switching system H | Q , with Q = { } , and switchingsystem H | Q , with Q = { , } are asymptotically stable.We first consider H | Q . We recall from [12] that an autonomous GLSw –system (with identity reset map) isasymptotically stable if it admits a common Lyapunov function V . By defining for any x ∈ R the function V ( x ) = x ′ P x with P = I , we obtain:( A (22)1 ) ′ P + P A (22)1 ≤ −Q , ( A (22)2 ) ′ P + P A (22)2 ≤ −Q , where: Q = (cid:20) − − (cid:21) ≥ . Hence V is a common Lyapunov function for sub–systems S (1) and S (2) of H | Q and by Theorem 2.1 in[12] we conclude that H | Q is asymptotically stable .Let us consider H | Q . The GLSw –system H | Q is characterized by dynamical matrix A (22)3 = −
1, resetmatrix R (3 ,
3) = 0 and guard G (3 ,
3) = { } and hence it is asymptotically stable.Let us now consider H | Q and let us apply Proposition 4.5 to investigate stability properties of H | Q . It isreadily seen that the abstraction H of (4.6), that corresponds to H | Q is unstable. Let us now consider theabstraction H of H | Q . The reset map R ( e ) with e = (5 ,
6) associated to H is given by: R (5 ,
6) = R (22) (5 , π ker( R (12) (5 , = ( 0 0 ) . Therefore, since dynamical matrices A (22)5 and A (22)6 are Hurwitz it is easy to see that the LSw –system H isasymptotically stable. Thus, by Proposition 4.5 also H | Q is asymptotically stable.We conclude that the switching system H is asymptotically stable and therefore condition iii) of Theorem4.4 is satisfied. Hence, by Theorem 4.4, the linear switching system H is detectable.6. Conclusions
We addressed observability and detectability of linear switching systems. We derived a computable necessaryand sufficient condition for a switching system to be observable. Further, we derived a Kalman decompositionof the switching system, which reduces detectability of linear switching systems to asymptotic stability ofsuitable linear switching systems with guards associated with the original systems. The study of detectabilityis a fundamental step towards the design of a hybrid observer. In fact, by Definition 2.1, a necessary conditionfor the existence of a hybrid observer for a
LSw –system H is that H is detectable. On the other hand,as shown in Section 3, observability of H implies the existence of an algorithm that reconstructs the currenthybrid state; in particular, the combination of (4.1) and (4.2) can be thought of as a hybrid observer. However, We recall that a strongly connected component of a FSM is a FSM, with a path between any two discrete states. Dynamical matrices A (22)1 and A (22)2 have been taken from [17]. such an observer requires an infinite precision in the computation of the vector Y ( n ) ( t ). Further work willidentify appropriate conditions on linear switching systems, for the existence and design of hybrid observers. References [1] M. Babaali and G.J. Pappas. Observability of switched linear systems in continuous time. In M. Morari, L. Thiele, andF. Rossi, editors,
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Lecture Notes in Computer Science , pages 526–539. SpringerVerlag, Berlin, 2003. Appendix: Proof of Lemma 3.3
We first need two preliminary technical lemmas.
Lemma 7.1.
If condition (3.5) is satisfied then for any ( i, h ) ∈ ˆ J , B − ih ( V ih ) = R m . BSERVABILITY AND DETECTABILITY OF LINEAR SWITCHING SYSTEMS: A STRUCTURAL APPROACH 13
Proof.
By contradiction, suppose that B − ih ( V ih ) = R m for some ( i, h ) ∈ ˆ J . Then Im ( B ih ) ⊆ V ih and by(3.3), A ih V ih ⊆ V ih + Im ( B ih ) ⊆ V ih , i.e. V ih is A ih − invariant and contains Im ( B ih ). Since the minimal A ih − invariant subspace containing Im ( B ih ) is Im ( B ih A ih B ih . . . A n − ih B ih ), with n = n i + n h , then Im ( B ih A ih B ih . . . A n − ih B ih ) ⊆ V ih ⊆ ker ( C ih ) , which implies C ih ( B ih A ih B ih . . . A n − ih B ih ) =0. Thus condition (3.5) is not satisfied and hence a contradiction holds. (cid:3) Lemma 7.2.
Let (cid:8) M i ∈ R m × nT , i ∈ Q (cid:9) be a family of nonzero matrices. There exists z ∈ R n and λ ∈ R suchthat M i z = 0 , ∀ i ∈ Q , where (7.1) z ′ =( z ′ λz ′ λ z ′ . . . λ T − z ′ ) ′ . Proof.
By setting M i = ( M i M i . . . M iT − ) and M i ( λ ) = M i + λM i + λ M i + . . . + λ T − M iT − ,with M ij ∈ R m × n , for any z ∈ R n , M i z = M i ( λ ) z . Given i ∈ Q , since M i = 0, there are a finite number ofvalues θ such that M i ( θ ) = 0. Choose λ such that M i ( λ ) = 0, ∀ i ∈ Q . Then there exists z / ∈ S i ∈ Q ker ( M i ( λ ))which implies M i z = 0, ∀ i ∈ Q . (cid:3) We now give the proof of Lemma 3.3.
Proof.
By contradiction, suppose that the set U ∗ is empty and let be n = n i + n h . Then(7.2) ∀ u ∈ U , ∃ t ′ , t ′′ ∈ R , ∃ ( i, h ) ∈ ˆ J and e u ∈ U ih s.t.u ( t ) = e u ( t ), ∀ t ∈ [ t ′ , t ′′ ] . Let V ih be the set of smooth functions v : R → B − ih ( V ih ) and let b U ⊂ U be the set of smooth, not identicallyzero functions. By definition of U ih , condition (7.2) implies:(7.3) ∀ u ∈ b U , ∃ t ′ , t ′′ ∈ R , ∃ ( i, h ) ∈ ˆ J , z ∈ V ih and v ih ∈ V ih s.t. u ( t ) = K ih z ( t ) + v ih ( t ), ∀ t ∈ [ t ′ , t ′′ ] , where ˙ z ( t ) = b A ih z ( t ) + B ih v ih ( t ) , b A ih = A ih + B ih K ih and z ( t ′ ) = z ∈ V ih . Condition (7.3) implies: ∀ u ∈ b U , ∃ t ′ ∈ R , ∃ ( i, h ) ∈ ˆ J s.t. ∀ ¯ N ≥ u ( t ′ )˙ u ( t ′ ) . . .u ( ¯ N ) ( t ′ ) ∈ M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih ) , where F ih = B − ih ( V ih ) and M ¯ Nih = K ih K ih b A ih . . .K ih b A ¯ Nih ∈ R m ( ¯ N +1 ) × ¯ N , F ¯ Nih = I . . . K ih B ih I . . . . . . . . . . . . K ih b A ¯ N − ih B ih K ih b A ¯ N − ih B ih . . . I ∈ R m ¯ N × m ¯ N . The matrix F ¯ Nih is nonsingular. By setting dim( F ih ) = ν , one obtains:dim( F ¯ Nih ( F ih × F ih × . . . × F ih )) = ν (cid:0) ¯ N + 1 (cid:1) , and since (3.5) holds, dim( M ¯ Nih V ih ) < n ; thusdim( M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih )) ≤ ν (cid:0) ¯ N + 1 (cid:1) + n. Therefore since by Lemma 7.1, ν < m , we obtain that ν (cid:0) ¯ N + 1 (cid:1) + n < m (cid:0) ¯ N + 1 (cid:1) for any ¯ N > nm − ν −
1; thus M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih ) is a proper subspace of R m ( ¯ N +1 ). Hence there exists a sufficiently large¯ N such that the set M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih ) is a proper subspace of R m ( ¯ N +1 ) for any ( i, h ) ∈ ˆ J .Given some z ∈ R m and λ ∈ R let be u ( t ) = z exp ( λt ) ∈ b U . It follows that: u ( t ) ˙u ( t ) . . . u ( ¯ N ) ( t ) = zλz ... λ T − z exp ( λt ) . Set M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih ) = ker ( G ih ), for some matrix G ih . By Lemma 7.2 there exist z and λ such that G ih z = 0, ∀ ( i, h ) ∈ ˆ J where z is as in (7.1). This implies that the vector u ( t ) ˙u ( t ) . . . u ( ¯ N ) ( t ) = z exp( λt )does not belong to M ¯ Nih V ih + F ¯ Nih ( F ih × F ih × . . . × F ih ), for all ( i, h ) ∈ ˆ J and t ∈ R , and hence condition (7.4)is false; thus the result follows. (cid:3) Department of Electrical Information Engineering,Center of Excellence DEWS, University of L’Aquila,, Poggiodi Roio, 67040 L’Aquila (Italy)
E-mail address : { desantis,dibenede,pola }}