The distinguishability of linear control systems

Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

The distinguishability of linear control systemsI Hongwei Lou ∗ , Pengna Si School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China

article

a b s t r a c t

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Article history: Received 9 January 2008 Accepted 10 October 2008

When observabilities of hybrid systems such as piecewise-affine hybrid systems are considered, the distinguishability of subsystems takes a very important role. However, properties and even definitions of such distinguishabilities have not been extensively studied in the literature. This paper will give a definition of the distinguishability of two linear control systems and present the necessary and sufficient condition of the distinguishability, giving an answer for the problem posed by Vidal et al. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Distinguishability Linear control systems Necessary and sufficient condition

1. Introduction Consider the following piecewise-affine hybrid system composed by time-invariant subsystems (i = 1, 2, . . . , N ):

( Si :

dx

= Ai x(t ) + Bi u(t ), dt y(t ) = Ci x(t ),

(1.1)

where x(t ) ∈ Rn , u(t ) ∈ Rm and y(t ) ∈ Rk . Naturally, we suppose that: Ai ∈ Rn×n ,

Bi ∈ Rn×m ,

Ci ∈ Rk×n .

(1.2)

We will study the distinguishability of subsystems. The motivation to study such a problem is that distinguishability takes a crucial role in considering the observability of the above hybrid system. The system considered may switch from one mode/subsystem to another one when some event occurs, say it jumps from Si to Sj at time t if (t , x(t )) ∈ Ωij with Ωij being a given subset of Rn+1 . Moreover, any subsystem is possible to be the beginning mode of the system. Thus observability in infinitesimal time of the system1 is equivalent to that of each two subsystems Si and Sj (i 6= j) are distinguishable (see [6] for example). Recently, the observability/distinguishability of hybrid systems has been paid much attention. However, it seems that there is still no strict definition of distinguishability for linear hybrid systems in continuous time and no mathematical study of it. In [15], observability and distinguishability problems were considered, but only for automation systems (i.e., systems without input/control). However, as pointed out by Vidal et al. [15], when the non-automation system is considered, the input plays a crucial role and the problem becomes very difficult. The aim of this paper is to give a definition of distinguishability for nonautomation systems, and yield the necessary and sufficient condition of distinguishability. We think we found an answer to the problem posed by Vidal et al. about the effect of measured inputs on the observability (c.f. [15], Section 4).

I This work was supported in part by NSFC (No. 10671040 and 10831007), FANEDD (No. 200522) and NCET (No. 06-0359).

∗

Corresponding author. Tel.: +86 21 55665006; fax: +86 21 65646073. E-mail addresses: [email protected], [email protected] (H. Lou).

1 i.e., for any ε > 0, the initial mode/subsystem and initial state can be determined by the output on [0, ε]. 1751-570X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.10.003

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H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

In the study of distinguishability/observability for hybrid systems, we would like to mention that many researchers have studied the distinguishability/observability of discrete linear hybrid systems. Among them, there are [2–4,8,9,11,12,14], for example. For continuous cases, the authors of [1] considered the controllability and observability of switched linear systems, where the switch manner is observable. For unobserved switching cases, sufficient conditions of generic final state determinability were given in [1], and discernibility/detectability of switched linear systems were considered in [5,13]. We note that the word distinguishability is widely used by engineers in different aspects with different meanings. It even appears in the study of control problems in a way similar but different to that which we want to consider (see, for examples, [7,10,16]). In our case, a rough definition of two subsystems S1 and S2 being distinguishable on time interval [0, T ] is that for any x10 , x20 ∈ Rn (x10 , x20 denote the initial state of S1 and S2 , respectively), u(·) : [0, T ] → Rm (u(·) denote the input or control), the corresponding outputs of S1 and S2 are different. However, if both the initial state and control u(·) are 0, then no matter what subsystem is running, the output will always be 0. So a strict definition of distinguishability is needed to replace the above loose one. Hereafter, we denote by yi (·) ≡ yi (·; xi0 , u(·)) the output of Si corresponding to the initial state x(0) = xi0 and control u(·). Definition 1.1 (Distinguishability). Systems S1 and S2 are said to be distinguishable on [0, T ], if for any non-zero

(x10 , x20 , u(·)) ∈ Rn × Rn × L1 (0, T ; Rm ), the outputs y1 (·) and y2 (·) cannot be identical to each other on [0, T ]. From the definition we can see that distinguishability of two subsystems means that as long as one of the two subsystems is running, we can distinguish them. In this paper, we will submit that the necessary and sufficient condition of distinguishability is that the infinite dimensional equation (4.1) gives only trivial solutions. To this end, we will give some basic properties and introduce some other auxiliary concepts of distinguishability in Section 2. Among these auxiliary concepts, there are ‘‘polynomial input distinguishability’’, ‘‘analytic input distinguishability’’, ‘‘smooth input distinguishability’’ and ‘‘L1 input distinguishability’’, etc. Section 3 — Section 4 are contributed to conditions of the polynomial input distinguishability and the analytic input distinguishability. In Section 5, equivalence between the analytic input distinguishability and the smooth input distinguishability will be proved. While in Section 6, we will prove the equivalence between the smooth input distinguishability and the distinguishability. Finally, in Section 7, we gave a short conclusion of the main results of this paper. We will also give the necessary and sufficient condition for the distinguishability of systems

( e Si :

dx

= Ai x(t ) + Bi u(t ), dt y(t ) = Ci x(t ) + Di u(t ),

i = 1, 2.

(1.3)

2. Basic properties of distinguishability and other auxiliary concepts Immediately from Definition 1.1, we have the following proposition: Proposition 2.1. Let T > 0. If S1 and S2 are distinguishable on [0, T ], then both S1 and S2 are observable. Proof. Let S1 and S2 be distinguishable on [0, T ]. Choose x20 = 0, and u(·) ≡ 0, then the output y2 (·) is 0. Thus, according to the definition of the distinguishability, the output y1 (·) of S1 equals to 0 identically on [0, T ] if and only if the initial state x10 = 0, which means S1 is observable. Similarly, system S2 is observable too.  It is easily to see the observabilities of S1 and S2 are only the necessary conditions of distinguishability. We can illustrate it by a trivial example that if two observable systems are identical to each other, then they are clearly not distinguishable. In this paper we will submit the necessary and sufficient condition of distinguishability. For this purpose, we first introduce some concepts such as ‘‘polynomial input distinguishability’’, ‘‘analytic input distinguishability’’, ‘‘smooth input distinguishability’’ and ‘‘L1 input distinguishability’’, etc. We define that Definition 2.2. Given T > 0. Let U ⊆ L1 (0, T ; Rm ) be a function space. We say that S1 and S2 are U input distinguishable on [0, T ] if for any non-zero

(x10 , x20 , u(·)) ∈ Rn × Rn × U , the outputs y1 (·) and y2 (·) cannot be identical to each other on [0, T ].

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

23

Especially, when U is the set of polynomial function class, analytic function class

( u(·)|u(t ) =

+∞ X αj j =0

j!

) t converges in an open interval including [0, T ] , j

smooth function class C ∞ ([0, T ]; Rm ) and L1 (0, T ; Rm ) respectively, then the corresponding distinguishability is called ‘‘polynomial input distinguishability’’, ‘‘analytic input distinguishability’’2 ‘‘smooth input distinguishability’’ and ‘‘L1 input distinguishability’’, respectively. Clearly, the ‘‘L1 input distinguishability’’ is just the ‘‘distinguishability’’ defined in Definition 1.1. Then the following conclusions are immediate: Proposition 2.3. Let T > S > 0, then (i) On [0, T ], distinguishability/L1 input distinguishability =⇒ smooth input distinguishability

=⇒ analytic input distinguishability =⇒ polynomial input distinguishability; (ii) U input distinguishability on [0, S ] =⇒ U input distinguishability on [0, T ]. For the convenience of the forthcoming discussions, we list some properties of infinite dimensional matrices and linear algebraic equations. For a matrix which has (countable) infinite rows and finite columns, the column rank and the row rank can be welldefined in a natural way and it is easy to prove that the column rank equates to the row rank. Therefore the rank of such matrix can be well-defined. We now consider the infinite dimensional linear homogeneous algebraic equation:

(

a11 z1 + a12 z2 + · · · + a1q1 zq1 = 0, a21 z1 + a22 z2 + · · · + a2q2 zq2 = 0,

(2.1)

.............

Let



a11  a A =  21

.. .

a12 a22

.. .

a13 a23

.. .

z1  z2   z=  z3  .



 ... ...   .. .



.. .

For convenience, an infinite dimensional matrix C which having infinite rows and infinite columns is called an F-type matrix if in every row of C , there are only finite non-zero elements. Thus, A is an F-type matrix. If B is another F-type matrix. Then the matrix product BA of A and B can be defined in a traditional way and the product matrix is F-type. Moreover, one can prove that lower matrices with nonzero elements on its main diagonal are invertible. In particular,

   P =  

1 p21 p31 p41

.. .

0 1 p32 p42

.. .

0 0 1 p43

.. .

0 0 0 1

.. .

... ... ... ... .. .

     

(2.2)

is invertible. Now rewrite (2.1) as A z = 0.

(2.3)

We have that (2.3) is equivalent to

(PA )z = 0, where P has form (2.2).

2 For convenience, the definition of the analytic input distinguishability we used does not consider all of the (vector valued) real analytic functions on

[0, T ]. Yet, we will see that this definition is equivalent to other reasonable definitions (see Theorem 4.8).

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H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

3. Necessary and sufficient condition of the polynomial input distinguishability The polynomial input distinguishability is the easiest one we shall study. For notational simplicity, hereafter, we will denote

 A=

A1 0

0 A2



 

,

B1 B2

B=

,

C = C1


−C 2 ,

(3.1)

and

 x0 =

x10 x20



,

Y (·) = y1 (·) − y2 (·).

(3.2)

Then, immediately from Definition 2.2, we have the following result: Theorem 3.1. The polynomial input distinguishability of S1 and S2 is independent of T > 0, which is equivalent to that of every sub-matrix composed of the left finite column vectors of M has full column rank, where

 C  CA 2 M =  CA .. .

0 CB CAB

.. .

0 0 CB

0 0 0

.. .

.. .

... ··· ··· .. .

  . 

(3.3)

More precisely, let



C CA CA2

    .. MN =  .   CAN +1  .. .

0 CB CAB

.. .

··· ··· ··· .. .

0 0 0

CAN B

.. .

··· .. .

CB

.. .

.. .

     .   

Then the necessary and sufficient condition for the N-th polynomial input distinguishability of S1 and S2 is that matrix MN has full column rank. While the necessary and sufficient condition for the polynomial input distinguishability of S1 and S2 is that for any N ≥ 1, MN has full column rank. Proof. Let u(·) ∈ L1 (0, T ; Rm ) be the input. Then the corresponding outputs of S1 and S2 with initial state x10 and x20 are y1 (t ) = C1 eA1 t x10 + C1

Z

t

eA1 (t −s) B1 u(s)ds,

(3.4)

eA2 (t −s) B2 u(s)ds.

(3.5)

0

y2 (t ) = C2 e

A2 t

Z

t

x20 + C2 0

Thus Y (t ) ≡ y1 (t ) − y2 (t ) = C eAt x0 + C

t

Z

eA(t −s) Bu(s)ds.

0

Now, let u(·) be an N-th Rm -valued polynomial on [0, T ]:

αN N t , t ∈ [0, T ], N! where αj ∈ Rm . Then y1 (·) and y2 (·) are both analytic. Therefore u(t ) = α0 + α1 t + · · · +

y1 (t ) ≡ y2 (t ),

on [0, T ]

holds if and only if all derivatives of Y (·) at t = 0 equates to zero: Y hji (0) = 0,

∀ j = 0, 1, 2 . . . .

(3.6)

Calculating the derivatives by (3.4)— (3.5), we get that (3.6) is equivalent to the following infinite dimensional equation3 : MN [x0 ; α0 ; . . . ; αN ] = 0.

(3.7)

3 We use ‘‘MATLAB notation’’: for matrices P , . . . , P with the same number of columns, [P ; P ; . . . ; P ] denotes the matrix obtained by writing P under 1 j 1 2 j 2 P1 , P3 under P2 , etc.

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

25

Therefore the N-th polynomial input distinguishability of S1 and S2 is equivalent to that (3.7) admits only trivial solution. That is to say MN has full column rank. Since the polynomial input distinguishability S1 and S2 is equivalent to that for any N, S1 and S2 are N-th polynomial input distinguishable. Therefore it is equivalent to that for any N ≥ 1, MN has full column rank. The above conclusion also indicates that the polynomial input distinguishability of S1 and S2 is independent of T .  The following result is an immediate consequence of the above theorem: Corollary 3.2. If S1 and S2 are polynomial input distinguishable, then 2n ≥ m. Proof. Let M be defined by (3.3). If S1 and S2 are polynomial input distinguishable, then by Theorem 3.1, the sub-matrix composed of the left finite column vectors of M must have full column rank. In particular, 0  CB   CAB   





 . .  .  CAn B  .. .

    

N =

has full column rank m. For any l ≥ 1, we have

 CB  CAB  .  . rank  .  .  . .

  C        = rank      

C

..

.

Al

C

CAl B

 I 2n  A ≤ rank   ...

 I 2n  A  .   ..

      B    

   = 2n, 

Al

where Ir denote the r × r unit matrix. Consequently, m = rank (N ) ≤ 2n.  Remark 3.1. Assume that the orders of the minimum polynomials of A1 and A2 are n1 and n2 , then by Cayley–Hamilton Theorem, we can prove that the polynomial input distinguishability implies (n1 + n2 )k ≥ m. We omit the proof since a result much better than it will be given in Theorem 4.5. The following example shows that the case m = 2n is possible for the polynomial input distinguishability. Example 3.1. Let m = 2, n = 1 and the corresponding matrices of S1 and S2 be defined by A1 , A2 ∈ R,

  C1 =

1 , 1

  C2 =

1 , 0

B1 = 1

1 ,




B2 = 1

0 .




Then we can easily verify that S1 and S2 are polynomial input distinguishable. In fact, by Theorem 7.1, we can see that S1 and S2 are distinguishable. 4. Study on the analytic input distinguishability We can see the following result similar to Theorem 3.1. Theorem 4.1. The analytic input distinguishability of S1 and S2 on [0, T ] are equivalent to the following infinite dimensional equation M [x0 ; α0 ; α1 ; . . .] = 0

(4.1)

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H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

having only trivial solution such that the corresponding series u(t ) =

∞ X αj j =0

j!

tj

converges in an open interval including [0, T ]. Now, we give an example to illustrate that the polynomial input distinguishability is weaker than the analytic input distinguishability. Example 4.1. Consider S1 , S2 with their matrices being A1 = 1,

B1 = 3,

A2 = 3,

B2 = −2,

C1 = −1, C2 = 1.

Then S1 and S2 are polynomial input distinguishable but not analytic input distinguishable. In fact at this time

 −1  −1  −1 M =  −1  .. .

−1 −3 −32 −33 .. .

0 −1 −3 + 2 · 3 −3 + 2 · 32

.. .

0 0 −1 −3 + 2 · 3

.. .

0 0 0 −1

.. .

··· ··· ··· ··· .. .

     

(4.2)

and the following processes will imply that S1 and S2 are polynomial input distinguishable but not analytic input distinguishable. For any t ∈ R, (4.1) admits a unique solution satisfying x10 = t:

 x10 = t ,    x20 = −t ,    α = −x − 3x , 0 10 20 α = − x10 − 32 x20 + (−3 + 2 · 3)α0 ,  1   3 2    α2 = −x10 − 3 x20 + (−3 + 2 · 3 )α0 + (−3 + 2 · 3)α1 , ···························

(4.3)

Then we can easily see that t > 0 ⇐⇒ ∀ j ≥ 1, αj > 0, t = 0 ⇐⇒ ∀ j ≥ 1, αj = 0, t < 0 ⇐⇒ ∀ j ≥ 1, αj < 0. This means that for any N, (3.7) has only a trivial solution. Consequently S1 and S2 are polynomial distinguishable. Now, let t = 1. Then (4.3) gives a non trivial solution of (4.1). We will prove that this solution makes the series +∞ X αj j =0

j!

tj

converging in (−∞, +∞). We have

|α0 | ≤ 4, |α1 | ≤ (32 + 1) + 2 · 3 |α0 |, |α2 | ≤ (33 + 1) + 2 · 32 |α0 | + 2 · 3 |α1 |, ············ Let

β0 = 6, β1 = 2 · 32 + 2 · 3β0 , β2 = 2 · 33 + 2 · 32 β0 + 2 · 3β1 , ············

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

27

Then

βn = (3 + 2 · 3)βn−1 = 9βn−1 ,

n = 1, 2, . . .

and consequently,

|αn | ≤ βn = 6 · 9n ,

∀ n = 0, 1, 2, . . . .

Therefore the series +∞ X αj j=0

j!

tj

converges in (−∞, +∞). Thus S1 and S2 is not analytic input distinguishable on [0, T ] for any T > 0. The above example is just a simple one. In the following theorem we will give further conclusion. Theorem 4.2. Assume that CB has full column rank m. (i) Then the analytic input distinguishability is independent of T . In fact, it is equivalent to that (4.1) admits only trivial solution. (ii) If k = m, then S1 and S2 are analytic input distinguishable if and only if C has full column rank. (iii) If k > m, then S1 and S2 are analytic input distinguishable if and only if there exists an ` ≥ 0 such that the (`+1)k×(2n+`m) sub-matrix H` of M has full column rank 2n + `m, where

 C  CA  CA2 H` =   .  .. CA`

0 CB CAB

0 0 CB

CA`−1 B

CA`−2 B

.. .

.. .

... ··· ··· .. .

0 0  0  . 

···

CB



..  .

(4.4)

Proof. (i) By Theorem 4.1, the analytic input distinguishability of S1 and S2 is equivalent to that (4.1) has only trivial solution such that ∞ X αj j =0

j!

tj

converges in an open interval including [0, T ]. Now suppose that S1 and S2 are analytic input distinguishable on [0, T ] for some T > 0, we want to prove that for any ε > 0, S1 and S2 are analytic input distinguishable on [0, ε]. It suffices to prove that (4.1) has a only trivial solution. Let [x10 ; x20 ; α0 ; α1 ; . . .] be a solution of (4.1). Since rank (CB) = m, there must exist a matrix P ∈ Rm×k such that G ≡ PCB is invertible. Thus, by (4.1), we have

α0 = −G−1 PCAx0 ; ,
α1 = −G−1 P CA2 x0 ; +CABα0 , ............ Denote M = max(kG−1 P k, kAk, kBk, kC k, |x0 |, 1). we have

|α0 | ≤ M 4 , |α1 | ≤ M 5 + M 4 |α0 |, |α2 | ≤ M 6 + M 5 |α0 | + M 4 |α1 |, ............. Let

β0 = M 4 , β1 = M 5 + M 4 β0 , β2 = M 6 + M 5 β0 + M 4 β1 , .............

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H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

Then it is easy to see that

βj+1 = (M + M 4 )βj ≤ 2M 4 βj ≤ · · · ≤ (2M 4 )j+1 β0 ≤ (2M 4 )j+2 and

|αj | ≤ βj ≤ (2M 4 )j+1 ,

∀ j = 0, 1, 2, . . . .

Therefore ∞ X αj j =0

j!

tj

converges in (−∞, +∞). By the assumptions we have x0 = 0,

α0 = α1 = · · · = 0.

We get the conclusion. (ii) Now, CB is an invertible m × m matrix. If rank C = 2n, then one can easily verify that (4.1) has only a trivial solution. If rank C < 2n, then in (4.1), the equation Cx0 = 0 admits nontrivial solutions. Thus, since CB is invertible, (4.1) admits corresponding non-trivial solutions. By (i), we know that S1 and S2 are not analytic input distinguishable. (iii) If there exists an ` > 0 such that H` (see (4.4)) has full column rank 2n + `m, then any solution of (4.1) must satisfy x0 = 0,

α0 = · · · = α`−1 = 0.

Consequently, it follows easily from rank (CB) = m that

α` = α`+1 = · · · = 0. Thus (4.1) has only a trivial solution, which means that S1 and S2 are analytic input distinguishable. On the other hand, suppose that for any ` ≥ 0, the sub-matrix H` of M does not have full column rank. Denote by X` ⊆ R2n+`m the solution space of H` z = 0. We have X` 6= {0}. For ` ≥ 0, denote V` = {x0 | [x0 ; α0 ; . . . ; αl−1 ] ∈ X` } , W` = {[x0 ; α0 ; . . .] | [x0 ; α0 ; . . . ; αl−1 ] ∈ X` } . Then it follows easily from rank (CB) = m and X` 6= 0 that V` 6= 0,

∀ ` = 0, 1, . . . .

Moreover, we have V`+1 ⊆ V` , ∀ ` = 0, 1, . . . , W`+1 ⊆ W` , ∀ ` = 0, 1, . . . ,

{z |M z = 0} =

∞ \

W` ,

j=0

{x0 |M [x0 ; α0 ; . . .] = 0} =

∞ \

V` .

j=0

Thus, to prove that (4.1) admits non-trivial solutions, it suffices to prove that ∞ \

V` 6= {0}.

(4.5)

`=0

Noting that V` ’s dimensional dim(V` ) is monotonically decreasing and each dim(V` ) is a nonnegative integer, we know that there must exist an r > 0 such that dim(V` ) = s ≡ lim dim(Vj ), j→∞

∀ ` ≥ r.

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

29

Consequently ∞ \

V` = Vr 6= {0}.

`=0

Therefore, (4.1) admits non-trivial solutions. By (i), S1 and S2 are not analytic input distinguishable.



Besides the cases of CB having full column rank, the following theorem concerns the cases of CB having full row rank: Theorem 4.3. Let k < m and CB has full row rank k. Then S1 and S2 are not analytic input distinguishable. Proof. Under the given condition we know that CB has k × k sub-matrix which is invertible. Without loss of generality, we suppose that CB = (E F ) with F ∈ Rk×k being invertible and E ∈ Rk×(m−k) . Since m > k, there exists an

α0 6= 0 such that

(E F )α0 = 0. Let x0 = 0 and all of the beginning m − k components of αj (j = 1, 2, . . .) be zero. Then by the invertibility of F , the other components can be solved from (4.1) and we get a non-trivial solution [x0 ; α0 ; α1 ; . . .] of (4.1). Similar to that in the proof of Theorem 4.2, the corresponding series u( t ) =

∞ X αj j=0

j!

tj

converges in (−∞, +∞). Thus, by Theorem 4.1, S1 and S2 cannot be analytic input distinguishable.



Corollary 4.4. If C are 1 × 2n matrices and CB 6= 0. Then S1 and S2 are not analytic input distinguishable. Proof. The conclusion follows from Theorem 4.2(ii) for m = 1 and from Theorem 4.3 for m > 1.



The conclusions mentioned above indicate that under the circumstances we have discussed, if m > k, then S1 and S2 cannot be analytic input distinguishable. Is it general? The following interesting result shows that the answer is yes, which means that if there are too many control variables, then systems are not distinguishable. Theorem 4.5. Let k < m. Then S1 and S2 is not analytic input distinguishable. To prove the above theorem, we need to analyze a special kind of F -type matrix. We call G1  G2 G ≡  G3



.. .

0 G1 G2

.. .

0 0 G1

.. .

 ... ...  ...  , .. .

a k × m G-type matrix if

 ∞ k×m {Gj }∞ , j=1 ∈ Qk,m ≡ {Qj }j=1 |Qj ∈ R

kQj k ≤ M j ,

for some M > 0 .



It is easy to see that the product of two k × m G-type matrices is still a k × m G-type matrix. We introduce three types of invertible transformations on a k × m G-type matrix G . Type I: Left-multiply G by an invertible k × k G-type matrix P . Type II: If for some l = 1, 2, . . . , k, and J ≥ 0, all (jk + l)-th row vectors of G (j = 0, 1, 2, . . . , J) are zero, but the ((J + 1)k + l)-th row vector is not zero, then replace (jk + l)-th row by ((J + j)k + l)-th row (j = 0, 1, 2, . . . , J). Type III: If for some l = 1, 2, . . . , k, all (jk + l)-th row vectors (j = 0, 1, 2, . . .) are zero, then delete these rows.

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H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

We have Lemma 4.6. Let G1  G2 G ≡  G3

0 G1 G2



.. .

.. .

 ... ...  6 0 ...  = .. .

0 0 G1

.. .

be a k × m G-type matrix. Then there exist an s ≤ min(k, m), an invertible m × m matrix Q and an invertible transform P , which is composed by finite transformations of types I–III , such that G1 Q  G2 Q P  G3 Q



.. .

0 G1 Q G2 Q

0 0 G1 Q

0 G1 Q G2 Q

0 0 G1 Q

.. .

.. .

  ... Is ...   0  ...  = 0 .. .. . .

0 Is 0

.. .

 ... ...  ...   .. .

0 0 Is

.. .

(4.6)

when s = m or G1 Q  G2 Q P  G3 Q



.. .

.. .

.. .

 0 Is  ...  G2 ...   0 e  = ...    e  0 G3 .. .. .. . . .

0

0

0

0

···



Is

0

0

0

···

0

e G2 .. .

Is

0

··· .. .

    

.. .

.. .

.. .

(4.7)

when s < m, where e Gj (j = 1, 2, . . .) are s × (m − s) matrices, {e Gj } ∈ Qs,m−s . Proof. We prove the lemma by induction on m − rank (G1 ). Note that rank (G1 ) ≤ min(k, m). Step I: Let m − rank (G1 ) = 0. Then k ≥ m and there exists an invertible m × m matrix Q and an invertible k × k matrix P such that PG1 Q = Im

(4.8)

when k = m or

  PG1 Q =

Im 0

(4.9)

when k > m. Without loss of generality, we can suppose that k > m and (4.9) holds. Let P  0 P1 =   0



.. .

0 P 0

0 0 P

.. .

.. .

 ... ...  ...  . .. .

(4.10)

Then



G1 Q

 G2 Q P1   G3 Q .. .

0 G1 Q G2 Q

.. .

0 0 G1 Q

.. .

... ... ... .. .

 I m  0    e  G2,1   = e G    2,2  e  G3,1  e  G3,2 .. .

where



e Gj,1 e Gj,2

∞ j =1

 ∞ = PGj Q j=1 ∈ Qk,m .

0 0 Im 0

e G2,1 e G2,2 .. .

0 0

.. . .. .

Im 0

.. .

··· ··· ··· ··· ··· ··· .. .

       ,     

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

31

Let

 I m  0  0   0   0 P2 = · · · · · · · · · · · ·   0   0  0  .. .  I m  0  0   0   · 0  0   0  0  .. .        ·     

0

.. .

.. .

Ik−m 0 0 0 0

.. .

0 0 Im 0 0 0 0 0

Ik−m 0 0 0 0 0 0

.. .

0 0 0 Ik−m 0 0 0 0

.. .

0 0 0

.. .

0 0 0 0 Im 0 −G21 −G2,2

Ik−m 0 0 0 0

.. .

0 0 0

.. .

0

.. .

0 0 Im 0 0 0 0 0

Ik−m 0 0 0 0 0 0

0 0 Im 0 −G21 −G2,2 −G3,1 −G3,2

Ik−m 0 0 0 0 0 0

Im 0 −G21 −G2,2 −G3,1 −G3,2 −G3,1 −G3,2

0

.. .

0 0 0 0 Im 0 0 0

.. .

0 0 0 0 Im 0 0 0

.. .

0 0 0 0 0 Ik−m 0 0

.. .

0 0 0 0 0

0 0 0 0 0 0 Im 0

Ik−m 0 0

.. .

0 0 0 0 0 0 Im 0

.. .

.. .

0 0 0 0 0

0 0 0 0

Ik−m 0 0

0 Im 0

.. .

··· ··· ··· ··· ··· ··· ··· ··· .. .

Ik−m

.. .

Ik−m

.. .

0 0 0 0 0 0 0

··· ··· ··· ··· ··· ··· ··· ··· .. .

0 0 0 0 0 0 0

           

0 0 0 0

··· ··· ··· ···

0 0

··· ··· ··· .. .

Ik−m

.. .

.. .

           



       .     

···

.. .



Then it is easy to see that P2 is an invertible k × k G-type matrix and

G1 Q  G2 Q P2 P1   G3 Q

0 G1 Q G2 Q



.. .

... ... ... .. .

0 0 G1 Q

.. .

.. .

 I m  0     0   =   0   0  0  .. .

0 0

··· ···

0 0

.. . .. .

Im 0 0 0

     .     

··· ··· ··· ··· .. .

Im 0

.. .



.. .

Then, a transformation P3 of type III makes G1 Q  G2 Q P3 P2 P1   G3 Q



.. .

0 G1 Q G2 Q

.. .

0 0 G1 Q

.. .

  Im ...  ...   0  ...  = 0  .. .. . .

0 Im 0

.. .

0

···



Im

··· ··· .. .

  .  

.. . .. .

Therefore, we see that the lemma’s results hold when m − rank (G1 ) = 0. Step II: Suppose that the lemma’s results hold when m − rank (G1 ) ≤ h for some 0 ≤ h ≤ m − 1. We want to prove that the lemma’s results hold for the cases of m − rank (G1 ) = h + 1. Now, let m − rank (G1 ) = h + 1. Then m = rank (G1 ) + h + 1 > rank (G1 ). Case 1: rank (G1 ) = k. Then there exists an invertible m × m matrix Q and an invertible k × k matrix P such that PG1 Q = Ik

0 .




(4.11)

32

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

We have G1 Q  G2 Q P1   G3 Q

0 G1 Q G2 Q



.. .

 Ik  ...  e . . .   G2,1  ...  = e  G3,1 .. .. . .

0 0 G1 Q

.. .

.. .

0

0

0

0

0

···



e G2,2

Ik

0

0

0

···

e G3,2 .. .

e G2,1 .. .

e G2,2 .. .

Ik

0

··· .. .

  ,  

.. .

.. .

(4.12)

where P1 is defined by (4.10) and



e Gj,1

e Gj,2


∞ j =1

 ∞ = PGj Q j=1 ∈ Qk,m .

Similar to Step I, we can find an invertible k × k G-type matrix P2 such that G1 Q  G2 Q P2 P1   G3 Q

0 G1 Q G2 Q



.. .

0 0 G1 Q

.. .

.. .

 Ik  ...  ...   0  ...  = 0  .. .. . .

0

0

0

0

0

···



G2,2

Ik

0

0

0

···

G3,2

0

G2,2

Ik

0

··· .. .

    

.. .

.. .

.. .

.. .

.. .

(4.13)

with

{Gj,2 }∞ j=2 ∈ Qk.m−k . Case 2: r ≡ rank (G1 ) < k. Then there exists an invertible m × m matrix Q and an invertible k × k matrix P such that

 PG1 Q =



Ir 0

0 . 0

(4.14)

We have

 G1 Q  G2 Q P1   G3 Q

0 G1 Q G2 Q



.. .

   e ...  G2,11 ...    e G2,21 ...  =  e G  3,11 ..  e .  G3,21 .. .

0 0 G1 Q

.. .

Ir 0

.. .

0 0

0 0

0 0

0 0

0 0

··· ···

e G2,12 e G2,22

Ir 0

0 0

0 0

0 0

··· ···

e G3,12 e G3,22 .. .

e G2,11 e G2,21 .. .

e G2,12 e G2,22 .. .

Ir 0

0 0

··· ··· .. .

.. .

.. .

      ,    

(4.15)

where P1 is defined by (4.10) and



e Gj,11 e Gj,21

e Gj,12 e Gj,22

∞

 ∞ = PGj Q j=1 ∈ Qk,m .

j=1

Thus, we can find an invertible k × k G-type matrix P2 such that

G1 Q  G2 Q P2 P1   G3 Q



.. .

0 G1 Q G2 Q

.. .

0 0 G1 Q

.. .

... ... ... .. .

 Ir 0     0   0 =     0   0 .. .

0 0

0 0

0 0

0 0

0 0

··· ···

G2,12 G2,22

Ir 0

0 0

0 0

0 0

··· ···

G3,12 G3,22

0 0

G2,12 G2,22

Ir 0

0 0

··· ··· .. .

.. .

.. .

.. .

.. .

.. .

      ,    

(4.16)

with



Gj,12 Gj,22

∞

∈ Qk.m−k .

j =2

If Gj,22 = 0,

∀ j ≥ 2,

(4.17)

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

33

then a transformation P3 of type III makes G1 Q  G2 Q P3 P2 P1   G3 Q

0 G1 Q G2 Q



.. .

0 0 G1 Q

.. .

.. .

 Ik  ...  ...   0  ...  = 0  .. .. . .

0

0

0

0

0

···



G2,12

Ik

0

0

0

···

G3,12

0

G2,12

Ik

0

··· .. .

  .  

.. .

.. .

.. .

.. .

.. .

Therefore, we see that the lemma’s results hold. If (4.17) fails, then there exists a J ≥ 2, such that GJ ,22 6= 0 and Gj,22 = 0,

∀ 2 ≤ j < J.

Thus a transformation P3 of type II makes

G1 Q  G2 Q P3 P2 P1   G3 Q

0 G1 Q G2 Q



.. .

0 0 G1 Q

.. .

.. .

... ... ... .. .

 Ir 0     0   0 =     0   0 .. .

GJ ,22

0 0

0 0

0 0

0 0

··· ···

G2,12 GJ +1,22

Ir 0

0 GJ ,22

0 0

0 0

··· ···

G3,12 GJ +2,22

0 0

G2,12 GJ +1,22

Ir 0

0 GJ ,22

··· ··· .. .

0

.. .

.. .

.. .

.. .

.. .

      .    

Since

 m − rank

Ir 0

0



≤ h,

GJ ,22

we get the conclusion by the induction assumption. Step III: We get the proof by induction.  Now, we can give a proof of Theorem 4.5. Proof of Theorem 4.5. By the assumption, k < m. Then it follows from Lemma 4.6 that there exists an s ≤ k < m, an invertible m × m matrix Q and an invertible transform P , which is composed by finite transformations of types I–III, such that CBQ  CABQ 2 P  CA BQ

0 CBQ CABQ



.. .

0 0 CBQ

.. .

.. .

 Is 0  ... e ...   0 G  2  ...  = 0 e G 3  .. .. .. . . .

0

0

0

0

···



Is

0

0

0

···

0

e G2 .. .

Is

0

··· .. .

  ,  

.. .

.. .

.. .

where e Gj (j = 1, 2, . . .) are s × (m − s) matrices, {e Gj } ∈ Qs,m−s . Consequently, the equation

 CBQ  CABQ  CA2 BQ  .. .

0 CBQ CABQ

.. .

... ... ... .. .

0 0 CBQ

.. .

 α0 α  1   α  = 0  2  .. . 

(4.18)

is equivalent to 0

0

0

0

0

···

 0 e G2    0 e G3  .. .. . .

Is

0

0

0

···

0

e G2 .. .

Is

0

··· .. .



Is

.. .

.. .

.. .



Q −1 α   −1 0   Q α1   Q −1 α  2  .. .

    = 0. 

(4.19)

34

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

Denote

 α˜ j,1  α˜ j,2   Q −1 αj =   ..  . 

α˜ j,m

and let



(α˜ 0,s+1 , . . . , α˜ 0,m ) = (1, 0, . . . , 0), α˜ j,` = 0, ∀ j ≥ 1, s + 1 ≤ ` ≤ m.

(4.20)

One can easily see that (4.19)/ (4.18) admits a unique solution [α0 ; α1 ; . . .] 6= 0 satisfying (4.20). Moreover,

{αj }∞ j=0 ∈ Qm,1 and consequently, 4

u(t ) =

∞ X αj j =0

j!

tj

converges in (−∞, +∞). Thus, noting that [0; α0 ; α1 ; . . .] is a solution of (4.1), by Theorem 4.1, we get that S1 and S2 is not analytic input distinguishable.  The following lemma is a corollary of Lemma 4.6. Lemma 4.7. Let l, k, m ≥ 1. Suppose that

{Gj }∞ j=1 ∈ Qk,m ,

{Dj }∞ j=1 ∈ Qk,l .

(4.21)

Then if the infinite dimensional equation D1  D2 Gz ≡   D3



.. .

G1 G2 G3

0 G1 G2

.. .

.. .

0 0 G1

.. .

   z0 ... z1  ...   z    ...  2  =0  z3  .. .. . .

admits non-trivial solutions, it must admit some non-trivial solution such that ∞ X zj

j! j =1

tj

converges in (−∞, +∞). Proof. Apply Lemma 4.6 to G1  G2 G1 =   G3



.. .

0 G1 G2

.. .

0 0 G1

.. .

 ... ...  ...  . .. .

If (4.7) holds, then, similar to that in proof of Theorem 4.5, (4.22) admits a nontrivial solution such that z0 = 0 and ∞ X zj

j! j =1

tj

converges in (−∞, +∞).

(4.22)

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

35

If (4.6) holds, then (4.22) implies4

    

e D1 e D2 e D3 .. .

Im 0 0

.. .

0 Im 0

.. .

... ... ... .. .

0 0 Im

.. .





     

Q −1 z0 Q −1 z1 Q −1 z2 Q −1 z3

.. .

     = 0,  

where

{e Dj }∞ j=1 ∈ Qm,m . Then, one can easily see that

{zj }∞ j=0 ∈ Qm,1 and consequently, ∞ X zj

j! j=1

tj

converges in (−∞, +∞).



As a corollary of the above lemma, we have the following interesting and important result. Theorem 4.8. The analytic input distinguishability of S1 and S2 on [0, T ] is equivalent saying to that (4.1) admits only a trivial solution. Consequently, it is independent of T . 5. The equivalence between the smooth input distinguishability and the analytic input distinguishability We will now show that Theorem 4.8 implies the equivalence between the smooth input distinguishability and the analytic input distinguishability. Theorem 5.1. The analytic input distinguishability of S1 and S2 is equivalent to the smooth input distinguishability of S1 and S2 . Proof. It suffices to prove that the analytic input distinguishability implies the smooth input distinguishability. To see this, let u(·) be a C ∞ (Rm valued) function, and Y (t ) = y1 (t ) − y2 (t ) ≡ 0,

on [0, T ],

where y1 (·) = y1 (·; x10 , u(·)), y2 (·) = y2 (·; x20 , u(·)). We want to prove that x0 = 0 and u(t ) = 0,

∀ t ∈ [0, T ].

By (5.1) and noting that Y (·) is smooth, we have Y hji (t ) = 0,

∀ t ∈ [0, T ]; j = 0, 1, 2, . . . .

Thus, similar to (4.1), we have M [x1 (t ); x2 (t ); u(t ); u0 (t ); u00 (t ); . . .] = 0,

where: x1 (t ) = eA1 t x10 +

t

Z

e−A1 (t −s) B1 u(s)ds,

0

x2 (t ) = e

A2 t

t

Z x20 +

e−A2 (t −s) B2 u(s)ds,

0

4 But is not necessary equivalent to.

(5.1)

36

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

and u0 , u00 denote the first and second derivatives of u. Therefore, by Theorem 4.8, the analytic input distinguishability implies

[x1 (t ); x2 (t ); u(t ); u0 (t ); u00 (t ); . . .] = 0,

∀ t ∈ [0, T ].

(5.2)

In particular, u(t ) = 0,

∀ t ∈ [0, T ]

and x10 = x1 (0) = 0, We get the proof.

x20 = x2 (0) = 0.



6. The equivalence between the distinguishability and the smooth input distinguishability We will prove the equivalence between the distinguishability and the smooth input distinguishability and consequently get the equivalence between the distinguishability and the analytic input distinguishability. Theorem 6.1. Two systems S1 and S2 are distinguishable if and only if they are smooth input distinguishable. Proof. We need only to prove that the smooth input distinguishability implies the distinguishability. Suppose that S1 and S2 are smooth input distinguishable. Let u(·) ∈ L1 (0, T ; Rm ) and x10 , x20 ∈ Rn such that the corresponding outputs y1 (·) and y2 (·) are equal on [0, T ]. We want to prove that x10 = x20 = 0 and u(·) = 0. It suffices to prove that y1 (·) ≡ y2 (·) implies the smoothness of u(·). By calculating all derivative y1 (·) − y2 (·) in the sense of distribution function in (0, T ), we have

 CB  CAB  CA2 B  .. .

0 CB CAB

.. .

0 0 CB

.. .

... ... ... .. .

u f1 0   u   f2    u00  =  f     2 







 (6.1)

.. .

.. .

with fj being continuous on [0, T ]. Since S1 and S2 are smooth input distinguishable, then by Theorem 4.1, when applying Lemma 4.6 to

 CB  CAB 2 G =  CA B .. .

0 CB CAB

0 0 CB

.. .

.. .

... ... ... .. .

  , 

the corresponding s must be m (see the proof of Lemma 4.7) and we have

    

Q −1 u Q − 1 u0 Q −1 u00

.. .



 CBQ    CABQ   2 = P  CA BQ  .. .

0 CBQ CABQ



.. .

f˜1 f1  f  2   f˜2  = P ˜  f2  =   f2





.. .



.. .

0 0 CBQ

.. .

... ... ... .. .

   Q −1 u  Q − 1 u0       −1 00    Q u   .. .

   . 

(6.2)

Thus, since we can see that all f˜j (j = 1, 2, . . .) are continuous, Q −1 u, Q −1 u0 , Q −1 u00 , . . . are continuous.5 Consequently, u, u0 , u00 , . . . are continuous. Therefore u(·) is smooth on [0, T ]. We get the proof. 

5 We say that u(·) is continuous (or smooth, etc.) means that it equates to a continuous (or smooth, etc.) function in the sense of distribution function.

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

37

7. Conclusion We summarize our results in this section. By Theorems 5.1 and 6.1, we see that the distinguishability of S1 and S2 is equivalent to the analytic input distinguishability of S1 and S2 . Thus, results about analytic input distinguishability can be transferred to that of distinguishability. By Theorem 4.8, we have Theorem 7.1. The distinguishability of S1 and S2 on [0, T ] is equivalent to the equation that

 C  CA  CA2  .. .

0 CB CAB

... ··· ··· .. .

0 0 CB

.. .

.. .



x0



  α0   α  = 0  1  .. .. . .

admits only trivial solution. Consequently, it is independent of T . Remark 7.1. If we set m = 0, then systems S1 and S2 become automation systems. At this time, the distinguishability of automation (i.e. k = 0) systems S1 and S2 on [0, T ] is equivalent to that

 C 1  C1 A1 2 M0 =   C1 A1 .. .

C2 C2 A 2  C2 A22  .



.. .

has full column rank. This is equivalent to saying that the rank of

     

C1 C1 A1 C1 A21

C2 C2 A2 C2 A22

C1 A12n−1

−1 C2 A2n 2

.. .

     

.. .

is 2n. By Theorem 4.2, we have Theorem 7.2. Let k ≥ m and CB have full column rank. Then systems S1 and S2 are distinguishable if and only if there exists an ` > 0 such that H` (see (4.4)) has full column rank. In particular, if k = m, then systems S1 and S2 are distinguishable if and only if C has full column rank; By Proposition 2.1, Corollary 3.2 and Theorem 4.5, we have Theorem 7.3. If S1 and S2 are distinguishable, then (i) S1 and S2 are are observable; (ii) m ≤ min(k, 2n). For systems e S1 and e S2 defined in (1.3), distinguishability can be defined similarly. Furthermore, results similar to Theorems 7.1–7.3 can also be established. We list them in the following: Theorem 7.10 . The distinguishability of e S1 and e S2 on [0, T ] is equivalent to saying that the infinite dimensional equation

 C  CA  CA2  .. .

D CB CAB

.. .

0 D CB

.. .

0 0 D

.. .

... ··· ··· .. .





x0



 α0   α   1  = 0   α2    .. .

admits only trivial solution, where D = D1 − D2 . Consequently, it is independent of T .

38

H. Lou, P. Si / Nonlinear Analysis: Hybrid Systems 3 (2009) 21–38

Theorem 7.20 . Let k ≥ m and D ≡ D1 − D2 have full column rank. Then systems e S1 and e S2 are distinguishable if and only if there exists an ` > 0 such that He` has full column rank, where

 C  CA  CA2 e H` =   .  .. CA`

D CB CAB

0 D CB

0 0 D

CA`−1 B

CA`−2 B

CA`−3 B

.. .

.. .

.. .

... ··· ··· .. .

0 0  0  . 

···

D



..  .

In particular, at this time, the distinguishability of e S1 and e S2 implies k > m. Theorem 7.30 . If e S1 and e S2 are distinguishable, then (i) e S1 and e S2 are observable; (ii) m ≤ k. References [1] A. Balluchi, L. Benvenuti, M.D. Di Benedetto, A.L. Sangiovanni-Vincentelli, Observability for hybrid systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, vol. 2, 2003, pp. 1159–1164. [2] M. Babaali, M. Egerstedt, Observability for switched linear systems, in: Hybrid Systems: Computation and Control, in: Lecture Notes in Computer Science, vol. 2993, Springer-Verlag, 2004, pp. 48–63. [3] M. Babaali, M. Egerstedt, On the observability of piecewise linear systems, in: Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, vol. 1, 2004, pp. 26–31. [4] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, IEEE Transactions on Automatic Control 45 (10) (2000) 1864–1876. [5] M. Babaali, G.J. Pappas, Observability of switched linear systems in continuous time, in: Hybrid Systems: Computation and Control 2005, in: Lecture Notes in Computer Science, vol. 3414, Springer-Verlag, 2005, pp. 103–117. [6] P. Collins, Jan H. van Schuppen, Observability of piecewise-affine hybrid systems, in: Hybrid Systems: Computation and Control, in: Lecture Notes in Computer Science, vol. 2993, Springer, 2004, pp. 265–279. [7] T.J. Debus, P.E. Dupont, R.D. Howe, Distinguishability and identifiability testing of contact state systems, Advanced Robotics 19 (5) (2005) 545–566. [8] J. Ezzine, A.H. Haddad, Controllability and observability of hybrid systems, International Journal of Control 49 (6) (1989) 2045–2055. [9] G. Ferrari-Trecate, M. Gati, Computation observability regions for discrete-time hybrid systems, in: Proceedings of 42nd IEEE Conference on Decision and Control, Maui, Hawaii, vol. 2, 2003 pp. 1153–1158. [10] A.A. Lomov, Distinguishability conditions for stationary linear systems, Differential Equation 39 (2) (2003) 283–288. [11] M. Oishi, I. Hwang, C. Tomlin, Immediate observability of discrete event systems with application to user-interface design, in: Proceedings of IEEE Conference on Decision and Control, Maui, Hawaii, vol. 3, 2003, pp. 2665–2672. [12] C. m Özveren, A.S. Willsky, Observability of discrete event dynamic systems, IEEE Transactions on Automatic Control 35 (7) (1990) 797–806. [13] E. De Santis, M.D. Di Benedetto, G. Pola, On observability and detectability of continuous-time linear switching systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, vol. 6, 2003, pp. 5777–5782. [14] R. Vidal, A. Chiuso, S. Soatto, Observability and identifiability of jump linear systems, in: Proceedings of IEEE Conference on Decision and Control, Las Vegas NV, vol. 4, 2002, pp. 3614–3619. [15] R. Vidal, A. Chiuso, S. Soatto, S. Sastry, Observability of linear hybrid systems, in: Hybrid Systems: Computation and Control, in: Lecture Notes in Computer Science, vol. 2623, Springer-Verlag, 2003, pp. 526–539. [16] E. Walter, L. Pronzato, On the identifiability and distinguishability of non-linear parametric systems, Mathematics and Computers in Simulation 42 (1996) 125–134.