Input distinguishability of linear dynamic control systems

ABSTRACT When observabilities of hybrid dynamic systems are considered, the distinguishability of subsystems takes a very important role. Necessary and sufficient conditions for distinguishability of linear dynamic systems are obtained. Some illustrative examples are presented.


Introduction
The switched system is an important case of a hybrid system. As a special kind of linear switched systems has been extensively investigated [1][2][3][4][5]. When we consider the observability of switched systems composed by time-invariant subsystems, distinguishability plays a crucial role (see [6]). Among the references about distinguishability of hybrid systems, we would like to refer the readers to the papers [7][8][9]. Distinguishability of switched systems is concerned with recovering the initial state as well as the switching signal from the output (and input) and has been widely studied, see e.g. [10] for continuous linear control homogenous systems, Lou et al. [4] for continuous linear control inhomogeneous systems and recently, relatively easy equivalent conditions to verify distinguishability are presented in [3]. For discrete switched case, Baglietto et al. [1,2], considered the problem of identifying a discrete-time nonlinear system, within a finite family of possible models, from data sequences of a finite length. The problem is approached by resorting to the notion of output distinguishability.
However, there are applications of switched systems where their temporal nature cannot be represented by the continuous line or a discrete uniform time domain [11,12]. Indeed, a closed-loop system consisting of a continuous-time system and an intermittent controller is one application [13,14]. The consensus problem under intermittent information due to communication obstacles and limitations of sensors is another example.
Time scale theory is very useful since it is an appropriate tool to study continuous and discrete-time systems in a uniform framework [7,[15][16][17][18]. The objective of this paper is to extend the distinguishability results of a class of continuous-time linear control switched systems in [3,4]. Necessary and sufficient conditions for input distinguishability of linear switched dynamic systems on time scales.
The rest of the paper is organized as follows. Section 2 recalls some preliminaries on time scale theory. The studied class of systems, input distinguishability concept, necessary and sufficient conditions for input distinguishability for linear control dynamic switched systems are obtained in Section 3.

Preliminaries
We recall some basics on time scale theory (for more details see [16,17]). A nonempty subset of real line R is called time scales and it is denoted by T.
If t 0 ∈ T and δ > 0, then we define the following neighbourhoods of t 0 : Let us consider some examples of time scales (see [17]). Then and For a function f : T → R n . We define f (t) ∈ R (provided it exists) with the property that for every ε > 0, there exists δ > 0 such that . We call f (t) the delta derivative (derivative for short) of f at t 0 . Moreover, we say that f is delta differentiable ( -differentiable for short) on T κ provided f (t) exists for all t ∈ T κ . A function f is called rd-continuous provided that it is continuous at right-dense points in T, and has a finite limit at left-dense points, and the set of rd-continuous functions are denoted by C rd (T, R n ). The set of functions C 1 rd (T, R n ) includes the functions f whose derivative is in C rd (T, R n ) too.
It is known [9] that for every δ > 0 there exists at least one partition P : a = t 0 < t 1 < · · · < t n = b of [a, b) T such that for each i ∈ {1, 2, . . . , n} either t i − t i−1 ≤ δ or t i − t i−1 > δ and ρ(t i ) = t i−1 . For given δ > 0 we denote by P([a, b) T , δ) the set of all partitions P : a = t 0 < t 1 < · · · < t n = b that possess the above property.
Let f : T → R be a bounded function on [a, b) T , and let P : choose an arbitrary point ξ i and form the sum We call S a Riemann -sum of f corresponding to the partition P.
We say that f is Riemann -integrable from a to b (or on [a, b) T ) if there exists a number I with the following property: for each ε > 0 there exists δ > 0 such that |S − I| < ε for every Riemann -sum S of f corresponding to a partition P ∈ P([a, b) T , δ) independent of the way in which we choose ξ i ∈ t i−1 , t i ) T , i = 1, 2, . . . , n. It is easily seen that such a number I is unique. The number I is the Riemann -integral of f from a to b, and we will denote it by Assume that a, b ∈ T, a < b and f : T → R is rdcontinuous. Then the integral has the following properties.
It is well known that each rd-continuous function has a -antiderivative [17,Theorem 1.74].
Let f : The set of regressive functions is denoted by R(T, R n×n ) (or shortly denoted by R). For simplicity, we denote by R c (T, C) the set of complex regressive constants and similarly, we define the set of R + c (T, C). For definition of the exponential function on time scales see [16]. A function p : T → R is called to be positively regressive if 1 + μ(t)p(t) > 0 for all t ∈ T. If p : T → R is a positively regressive function and t 0 ∈ T, then (see [16]) the exponential function e p (·, t 0 ) is the unique solution of the initial value problem The definition of generalized monomials on time scales (see [17, Section 1.6]) h n : T × T → R is given as for s, t ∈ T. It follows that where h 1 n denotes -derivative of the h n with respect to t.
Using induction, it is easy to see that h n (t, s) ≥ 0 holds for all n ∈ N 0 and all s, t ∈ T with t ≥ s and (−1) n h n (t, s) ≤ 0 holds for all n ∈ N and all s, t ∈ T with t ≤ s.
Throughout this paper, we assume that sup T = ∞. The next definitions and results were given in [19]. Let Now we give the definition of the Laplace transform (see [19,Definition 4.5]): Let f ∈ C rd (T, C) be a function. Then the Laplace transform L{f }(·; s) about the point s ∈ T of the function f is defined by For h > 0, let and for h ≥ 0, define By using equation (1), The following results about the Laplace transform were proved in [19].
be of exponential order α 1 , α 2 , respectively. Then for any c 1 , c 2 ∈ R, we have Recently, Zada et al. [20], proved the following spectral decomposition theorem on time scales: Let A be a regressive matrix of order n. For each w ∈ C n there exist z j ∈ ker(A − λ j I) n j (j = 1, 2, . . . k) such that Moreover, if z j (s) := e A (s, 0)z j then z j (s) ∈ ker(A − λ j ) n j for all s ∈ T and there exist C n -valued polynomials t j (s) with deg(t j ) ≤ n j − 1 such that z j (s) = e λ j (s, 0)t j (s), s ∈ T and (j = 1, 2, . . . k).
On the other hand, if we have a proper rational function, then by using partial fractions method and applying Laplace inverse it is easy to get the form (2).
For various properties of the Laplace transform on time scales, we refer to [16,17,19,21].
Before addressing the problem of distinguishability, it helps to recall the solution frame work assumptions with the following time-invariant dynamic linear system where x(·) ∈ R n , u(·) ∈ R m are the state and input vectors and A ∈ R n×n , B ∈ R n×m are constant matrices. For a regressive matrix A and rd-continuous input u, the time-invariant dynamic linear system (3) with initial condition x(t 0 ) = x 0 has a unique solution of the form

Distinguishability
Consider a switched dynamic system composed by time-invariant subsystems (i = 1, 2, . . . , q): where x(·) ∈ R n , u(·) ∈ R m and y(·) ∈ R p . Naturally, Without loss of generality, we can assume only two subsystems i.e. i = 1, 2. Denote and Recently, in [4], the authors gave a notion of distinguishability for linear non-autonomous systems and yielded a necessary and sufficient condition for distinguishability of two linear systems. We refer the reader to [8,9,16,22] for a broad introduction to -measure and integration theory. We say that S 1 and S 2 are said to be distinguishable on J if for any non-zero (x 10 , x 20 , u(·)) ∈ R n × R n × L 1 (J 0 ; R m ), the corresponding outputs y 1 (·) and y 2 (·) cannot be identical to each other on J.
To study the distinguishability of two subsystems, some auxiliary concepts of distinguishability have been stated here: Definition 3.2: (see [4]) Let U ⊆ L 1 (J 0 ; R m ) be a function space. We say that S 1 and S 2 are U input distinguishable on J if for any non-zero (x 10 , x 20 , u(·)) ∈ R n × R n × U, the outputs y 1 (·) and y 2 (·) cannot be identical to each other on J.
Especially, when U is the set of generalized polynomial function class, the set of analytic function class and the set of smooth function class C ∞ (J; R m ), then the corresponding distinguishability is called "generalized polynomial input distinguishability", "analytic input distinguishability" and "smooth input distinguishability", respectively.
The distinguishability of S 1 and S 2 on J is equivalent to that for the following system: Thus the problem relates to the notion of zero dynamics.
The closed form solution with initial condition of the system (8) is as follows Let us define the following infinite order matrices Our first result characterized generalized polynomial input distinguishability:

Theorem 3.3:
The generalized polynomial input distinguishability of S 1 and S 2 is independent of T > 0 which is equivalent to that of every sub-matrix composed of the left finite column vector ofM has full column rank. Moreover, the necessary and sufficient conditions for the N-th generalized polynomial input distinguishability of S 1 and S 2 is thatM N has full column rank. While the necessary and sufficient conditions for the generalized polynomial input distinguishability of S 1 and S 2 is that for any N ≥ 1,M N has full column rank.
Proof: For the given input u(·) ∈ R m , the corresponding output of the system (8) is as follows Let u(·) be an N-th R m -valued generalized polynomial on J: where α j ∈ R m . Therefore, we have It follows that Y(·) is analytic. Therefore Y ≡ 0 holds if and only if all -derivatives of Y(·) at t = 0 equal to zero: Then the equation (13), implies the following system of equations: We get that (14) is equivalent to the following infinite dimensional equation: Therefore, the Nth generalized polynomial input distinguishability of S 1 and S 2 is equivalent to that (15) admits only trivial solution. That is to sayM N has full column rank.
Hence it is generalized polynomial input distinguishable and also independent of T.
The following result is an immediate consequence of the above theorem and proof is similar to [4 Corollary 3.2].
having only trivial solution such that the corresponding series converges in an open interval including J.
To prove our next results, we need to recall some concepts from [4].
It is well known that a special kind of F-type matrix It is easy to see that the product of two p × m G-type matrices is still a p × m G-type matrix.

Lemma 3.6: Let
be a p × m G-type matrix. Then there exist an s ≤ min(p, m), an invertible m × m matrix Q and an invertible transform P, which is composed by finite transformations of types I-III, such that The following consequence of Lemma 3.6 is as follows. The proof is similar to the one in [4,Theorem 4.5] and therefore omitted.
Then if the infinite order linear equation admits non-trivial solutions, it must admit some nontrivial solution such that Under the light of Lemma 3.8, we have the following interesting and important result.

Theorem 3.9:
The analytic input distinguishability of S 1 and S 2 on J is equivalent to that (16) admits only trivial solution. Consequently, it is independent of T.
We will now show that Theorem 3.9 implies that the smooth input distinguishability and analytic input distinguishability are equivalent. Theorem 3.10: The analytic input distinguishability of S 1 and S 2 is equivalent to the smooth input distinguishability of S 1 and S 2 . Theorem 3.9 gives us conditions not only necessary but also sufficient to the smooth input distinguishability of the linear systems. However, the conditions are not easy enough to verify. Let us go further for an equivalent conditions which is relatively more easy to verify.
From Theorem 3.7, if S 1 and S 2 are not analytic input distinguishable, then there exists a pair (X 0 , u(·)) such that with a j j ≤ Mα j , for all j = 0, 1, . . .

Lemma 3.12:
If S 1 and S 2 are not distinguishable, then we can find a pair (X 0 ,ũ(·)) satisfying (19) with where λ ∈ C and ς ∈ C m .
By using the above results, it implies that the necessary and sufficient conditions for 0-th generalized polynomial input distinguishability and the N-th generalized polynomial input distinguishable are equivalent. Hereafter, we can see that for any N ≥ 0, the matrix ⎛ 2 CAB · · · · · · . . . . . . · · · . . .
has full column rank if and only if ⎛ . . .
has full column rank. From the classical Cayley-Hamilton theorem (22) is has a full column rank.
Since Y(t) = e λ (t, 0)Ỹ(t) = 0, by considering the real part or imaginary part of X 0 , u(·), X(·) and Y(·), it follows that S 1 and S 2 are not distinguishable. This is a contradiction. Therefore, Theorem 3.3, implies that has full column rank. Summarizing the above arguments, we obtain our main result. Theorem 3.13: Subsystems S 1 and S 2 are analytic input distinguishable if and only if for any λ ∈ C, the matrix M λ has a full column rank. Example 3.14: Consider S 1 , S 2 with their matrices being It is easy to see that Then S 1 and S 2 are generalized polynomial input distinguishable. However, does not satisfy the full column rank condition for λ = 7. Therefore, S 1 and S 2 are not analytic input distinguishable. It implies that generalized polynomial input distinguishability is weaker than the analytic input distinguishability.