Distinguishability criteria of conformable hybrid linear systems

: We relate this article to the emerging idea of distinguishability of conformable linear hybrid time - invar iant control systems. To obtain the necessary and su ﬃ cient conditions of α - distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techni -ques, a more simple criterion of α - distinguishability for the fractional linear system is developed.


Introduction
Fractional calculus (FC) is what which is seen in heavy tails in electrical engineering and is also used in the modeling of anomalous diffusion [1]. Other applications involve fractional oscillators and fractional mechanics in refs. [2][3][4]. One of the newly defined derivatives in FC is "conformable fractional derivative (CFD)" stated by Khalil et al. in ref. [5]. Also the significant characteristics of CFD are listed in refs [5,6]. Interested readers can also see refs [7,8]. Later, in ref. [9], it is verified that CFD is actually conformable derivative (CD) not a fractional derivative.
There are chain rules in FC that are not good enough since various results can be obtained from these. There is a mega study on chain rules by Jumarie, for example, in ref. [10]. Inspired from this work, a chain rule of CD of a function of two variables is developed and a Leibnitz rule for CDs is given in Section 2. In this article, we have to interpret the notion of α-distinguishability for the conformable switched hybrid linear system (SHLS).
While there are two types of behaviors in a hybrid system including the discrete and continuous dynamical behavior, that is, the system which has the potency to both jumps which is interpreted by a state machine or automaton and flow which is described in the form of the differential equation. Consider a linear switched hybrid system as follows: X z t C z t G w t y t E z t Fw t i :˙, for 1,2, There is a discussion of the distinguishability of (1) in ref. [11]. The objective to analyze such a problem is that distinguishability takes a key role in assuming the observability of this system. The assumed subsystems in (1) may change from one phase to another phase when any situation arises, say at time t it bounces from where Ω ij n 1 ⊆ + . Not a long ago, the distinguishability of hybrid linear systems (HLSs) has seized more importance. Input is not included in case of autonomous systems while it has an important role in nonautonomous systems. Distinguishability and observability problems were inspected in ref. [12] for systems without input. Since, in the case of a non-autonomous system, w( ) ⋅ plays a key role and it becomes difficult to resolve the problem. In ref. [11], the authors have defined distinguishability for non-autonomous systems and in ref. [13], the equivalent criterion for distinguishability is examined which is as follows: has full column rank (FCR) for any λ ∈ if and only if has FCR. Hence, the distinguishability of the assumed subsystems in (1) is obvious from the FCR of λ , see in ref. [13]. Many other authors also have studied distinguishability/ observability. They discussed the notion of distinguishability for discrete HLSs as in refs [14][15][16][17]. The researchers in ref. [18] interrogated the observability and controllability of SHLS in the case of continuous systems. Distinguishability is noticeable in other control problems in refs [19,20]. The idea of the distinguishability is that two subsystems to be distinguishable on T 0, [ ] is that for any z 10 and z 20 represents the initial states of the systems X 1 and X 2 , respectively, with z z , where w( ) ⋅ the input such that their outputs are not similar. It is observed that output will be necessarily zero if z 0 10 = , z 0 20 = and w 0 ( ) ⋅ = . Therefore, we restrict the definition of distinguishability with z z w , , 0 10 20 There is no study of distinguishability for conformable HLSs.
We have developed the idea of α-distinguishability for conformable HLS. Assume the switched conformable HLS, which is composed of the time-invariant subsystems as follows: The assumed subsystems in (4) may alter from one state to another when any situation occur, say at time t it bounces from I i to I j if t z t , ij ( ( )) ∈ ϝ where ij n 1 ϝ ⊆ + . We have generalized the definition of distinguishability for conformable linear systems and obtained the equivalent condition for α-distinguishability by using Leibniz rule for CD. To reduce the comparative efforts in α-distinguishability condition, we consider the conformable Laplace techniques and a simplified form of distinguishability criterion is achieved. This article is organized in the following manner: In Section 2, we give the elementary facts of CFD/integral which is actually a CD and also define the α-distinguishability for a conformable linear system. In Section 3, some results related to α-distinguishability for P T 0, of the conformable system are proved. In Section 4, more consequences of the generalized α-distinguishability are examined and a simple criterion is stated.

Preliminaries
In this section, we give some fundamental ideas and develop results which are supportive of the main consequences.
. It is obvious that the CD of the constant function is 0.
Remark 2.2. It is possible that a function has CD at a point while the function is not differentiable for example Now, the definition of conformable integral is given as follows: and g b c : , Then conformable power series expansion of g is Now, we prove the chain rule for a function of two variables for CD.
while T α denotes CD of order α.
, from CD on i x ( ) and by chain rule for classical derivative gives (6).
Next, we give the proof of CD of an integrable function, named as Leibniz rule. then provided that T α be CD.
Proof. To deduce Eq. (8), let us define and also, since then, applying comformable derivative and using Theorem 2.5, we obtain for α 1 = we obtain the classical formula. Hence, the required objective is achieved.
Next, the α-distinguishability of the conformable linear system is given.
Definition 2.7. Two systems I 1 and I 2 are said to be α-dis- Definition 2.8. Suppose T 0 < , also consider be a function space such that such that the obtained outputs are not likewise on ]. In next discussion, we will consider to be one of the following classes: Some conclusions drawn immediately are exactly as described in [11, Propositions 2.1 and 2.3]. For the next discussion, some properties of matrices of infinite dimensional and also the characteristics of linear algebraic equations, which are elaborated in ref. [11], are helpful.

α-Distinguishability for P T 0, α [ ]
From now onward, we use the notations as then instantly from definition of α-distinguishability, we obtain: ]of I 1 and I 2 is independent of T, where T 0 < . Equivalently, every submatrix consisting of left finite column vectors of has FCR, where More precisely, consider that where β j m ∈ . Then y 1 ( ) ⋅ and y 2 ( ) ⋅ are conformable differentiables. Therefore, y t y t T , on 0, First determining,

Y t NKe z
NMw t N e eKMw τ d τ, Likewise determining all the CD of Eqs. (15) and (16) for higher orders, we obtain (19), which is equivalent to ] of I 1 and I 2 is equivalent to that (4) has unique solution, that is, Q has FCR.  Theorem 3.6. Both the systems I 1 and

α-Distinguishability for
Example 3.7. Consider the hybrid system with m 2 = and k 1 = . And NM is calculated as which implies that NM has 1 as a full row rank. Hence, from Theorems 3.4 and 3.6, it is clear that I 1 and I 2 are not As a corollary of [11,Lemma 4.6], we obtain a more useful result which is mentioned as follows: We have to prove Hence, similar to (23), we have z t z t w t w t w t ; ; denote the CFD of order α and α 2 , respectively. Thus, from Theorem 3.3, the α-distinguishability for A T 0,

More consequences
Consider that with λ ∈ and b k ∈ . For g T : 0, Next, a lemma explains some more properties: Proof. It is easy to verify parts (1), (2), (3), and we give the proof of part (4), Now,  ] → , £ g α ( ) is a proper rational function (PRF) ⇔ g t e P t e P t e P t , with λ k ∈ and P k ( ) ⋅ is a polynomial where k 1, 2, = ….
Proof. Given that P k ( ) ⋅ be a polynomial. Suppose Applying fractional form of Laplace transform of (30) and by linearity of Laplace transform and determining Laplace of every term by using [22,Theorem 2.4] implies It is obvious from Eq. (31) that £ g t α [ ( )] is PRF. Now, suppose that £ g t α [ ( )] is PRF, we show that g t ( ) has the form of (28). Applying fractional form of Laplace inverse transform on (31), we obtain g t e P t e P t e P t , In the end of this article, we state our main consequence where proof is exactly as in classical case mentioned in ref. [13]. Next, we give the result which is equivalent to previous one.  For any value of λ, FCR of the matrix in Eq. (39) implies FCR ofˆλ . Hence, the considered system is α-distinguishable.

Conclusion
In this article, we have proved a formula for CD of an integral function, which is called Leibnitz rule. Moreover, distinguishability idea for conformable HLSs is discussed. Some equivalent criteria are developed, which seems complicated to verify. To resolve this issue, necessary and sufficient condition of distinguishability for conformable HLS is also established with the help of conformable Laplace techniques, which are easy to verify. In future, we can study how the distinguishability work in real-life problems. We can analyze the concept of weak distinguishability for fractional HLSs. We can elaborate the idea of weak distinguishability with distinguishability for fractional linear systems.
Funding information: The authors state no funding involved.