Stability Analysis of Fractional Switched Systems With Stable and Unstable Subsystems


 This article addresses stability of fractional switched systems (FSSs) with stable and unstable subsystems. First, several algebraic conditions are presented to guarantee asymptotic stability by applying multiple Lyapunov function (MLF) method, dwell time technique and fast-slow switching mechanism. Then, some stability conditions which have less conservation are also provided by utilizing average dwell time (ADT) technique and the property of Mittag-Leffler function. In addition, sufficient conditions on asymptotic stability of delayed FSSs are obtained by virtue of fractional Razumikhin technique. Finally, several examples are given to reveal that the conclusions obtained are valid.


Introduction
In recent years, many efforts focus on analysis and applications of switched systems because they can be used in various fields including electric cars [1], electric aircraft [2], genetic regulatory networks [3] and so on. In general, switched systems consists of a switching rule and a number of subsystems, which have some interesting and complicated dynamical behaviors, a switched system may be unstable under some switching rule even if its all subsystems are stable [4]. Most existing results pay attention to stability of switched systems which have only stable subsystems, common Lyapunov function(CLF) approach [5] and MLF approach [6,7,8] are main methods. In many actual circumstances, however, both stable and unstable subsystems may be contained in a switched system, the fast-slow switching scheme is often used to deal with stability of such systems. Based on fast-slow switching scheme and MLF and mode-dependent ADT approach, stability of linear switched systems [9], singular switched systems [10], linear and nonlinear delayed switched systems [11,12] is solved.
The above papers only concern integer-order dynamical models, however, fractional calculus has shown outstanding advantages on depicting many complicated process and phenomena [13,14] during past thirty years. The complexity of fractional calculus and switching rules bring great difficulties on the stability analysis, many efforts are paid to deal with stability of FSSs with stable subsystems. The stability under arbitrary switching is investigated in by employing CLF [15,16] and MLF [17] method. Some other approaches including state-dependent switching [18], ADT [19,20,21] and mode-dependent ADT [22,23] are also adopted. In [24], the formulas of solutions are provided to deal with finite-time stability of FSSs, [25,26] investigate stability of impulsive FSSs including unstable and stable subsystems by utilizing MLF approach and dwell time technique.
It is noteworthy that time delays are usually inevitable for many actual systems, which usually have particular impact on sytems' dynamics, Nguyen et al. [27] design a state-dependent switching law to implement finite-time stability of delayed singular FSSs. Employing integer-order Lyapunov function method, an observer-based controller is chosen to analyze stability of Riemann-Liouville delayed FSSs [28]. He et al. [29] provide fractional differential inequalities to study exponential stability of impulsive delayed FSSs. It should be noted that time delays usually take on a spatial extent and the real systems cannot be well modeled by dynamical models with discrete delays, but distributed delays may better describe such complicated lag phenomena and have some important applications [30,31]. Up to now, there are no effective time-domain ways to handle stability of FSSs with distributed delay in Caputo sense. Inspired by the above analysis, asymptotic stability of FSSs with stable and unstable subsystems is first investigated in this paper. Sufficient conditions are presented by utilizing MLF method, dwell time technique and fast-slow switching mechanism. Then, a less conservative criterion is provided by employing ADT technique and the property of Mittag-Leffler functions. Finally, a class of delayed FSSs are addressed and algebraic conditions on asymptotic stability are given. The highlights of this paper are listed as follows: 1) Two different approaches are developed to deal with stability of FSSs and several simple algebraic criteria are provided. 2) FSSs with distributed and discrete delays are addressed, an effective approach is developed to overcome the difficulties arising from time delays, fractional calculus and switching rules with the help of fractional Razumikhin technique, MLF method and dwell time.

Preliminaries
In this subsection, several necessary definitions and useful lemmas are first stated.

Remark 1.
Here we take y(t 0 ) as the initial state of the first activated subsystem and the terminal state y(t − q ) on interval [t q−1 , t q ) as the initial state on next interval [t q , t q+1 ) (see also [20,21,25,26]).

Asymptotical stability of the FSS with unstable subsystems
This subsection will give several algebraic conditions and lower and upper bounds of dwell times to ensure asymptotic stability of FSS (10) by employing MLF approach, dwell time technique and fast-slow switching mechanism. Let positive scalars τ s and τ u be the dwell times of stable and unstable subsystems, respectively. Here t q+1 − t q ≥ τ s and t q+1 − t q ≤ τ u for any two consecutive switchings t q , t q+1 , we now may provide our main results. Theorem 1. FSS (10) will be asymptotically stable if there are scalars ξ s , ξ u > 0, ν s > 1, 0 < ν u < 1 and positive definite matrices P i , P j such that where

and the dwell times satisfy
Proof. Assuming η(t) = i ∈ F for t ∈ [t q , t q+1 ), writing y(t) as y. Then constructing the MLF From Lemma 2, one has Based on Lemma 3 and relationship (11), we can obtain that Defining ξ i = −ξ s , i ∈ S and ξ i = ξ u , i ∈ U , we get by the inequalities (12) and (13) that From Lemma 1, taking the µ-order integral on both sides of inequality (20) from t q to t for any t ∈ [t q , t q+1 ) yields In the light of Lemma 4, one has Without loss of generality, we suppose that l stable and (q − l) unstable subsystems have been activated before switching instant t q , then one has from (14) and (22) It follows that Inequalities (15) and (16) mean ν s exp{− ξs Γ(µ+1) τ µ s } < 1 and ν u exp{ ξu Γ(µ+1) τ µ u } < 1, therefore, we can get lim t→+∞ V η(t) (y) = 0 from lim q→+∞ V η(tq) (y(t q )) = 0, which yields y → 0 as t → +∞. Finally, we conclude that FSS (10) is asymptotically stable.
Remark 2. The inequality P i ≤ ν s P j , ν s > 1 of relationship (14) implies that it is impossible that MLF (17) always decreases on the switching sequence {t q , q = 0, 1, · · · } because the FSS has only finite different subsystems, the MLF must increase at some switching instants. Otherwise, if the mth and lth subsystems are sequentially activated at t q−1 , t q , t q+1 , then, there is a scalar 0 < ν u < 1 satisfying V l (y(t q )) ≤ ν u V m (y(t − q )) and V m (y(t q+1 )) ≤ ν u V l (y(t − q+1 )), which means P l ≤ ν u P m and P m ≤ ν u P l , this concludes a contradictory.
On the other hand, to get a upper bound of τ u , we need that the MLF decrease at switching instants when unstable subsystems are activated. However, unstable subsystems cannot be consecutively activated according to above analysis, hence the relationship P i ≤ ν u P j for any i ∈ U, j ∈ S is necessary. It follows that the MLF will increase on some switching instants and time intervals. To make the MLF converge to 0, we need eliminate these increments, this may be completed by the decrements brought by stable subsystems. Therefore, stable subsystems must operate long time enough and unstable subsystems stay a shorter time and hereby we provide the lower bound of τ s and upper bound of τ u .
In order to derive a less conservative criterion, we introduce the following definition.
Definition 4. For any t 2 ≥ t 1 ≥ 0, denote N u (t 1 , t 2 ) or N s (t 1 , t 2 ) as the switching numbers of active unstable or stable subsystems in interval [t 1 , t 2 ], and T u (t 1 , t 2 ) or T s (t 1 , t 2 ) as the total running time of active unstable or stable subsystems. If there exist N 0p ≥ 0 and τ dp > 0, p = {s, u} such that then τ ds and τ du are called the ADTs of stable and unstable subsystems, respectively, N 0p are called the chatter bounds, here we choose N 0p = 0.
Theorem 2. FSS (10) will be asymptotically stable if there are scalars ξ s , ξ u > 0, ν > 1, k ≥ 1 and positive definite matrices P i , P j such that and where Proof. Similarly to Theorem 1, according to (27) and (28), the µ-order derivative of MLF (17) is given by where ξ i = −ξ s for i ∈ S and ξ i = ξ u for i ∈ U .
Remark 3. Be different from Theorem 1, the unstable subsystems can be consecutively activated in Theorem 2 since inequality (29) implies that the MLF will increase at these switching instants.
Remark 4. In [25], the authors obtain a more complex condition on dwell time to achieve the asymptotic stability of FSSs with unstable subsystems, which is described as an inequality of multiplication of N Mittag-Leffler functions. [26] considers stability of impulsive FSSs with unstable subsystems, however, all impulsive strengths must be less than zero, which has great conservation and the method is not suitable for FSSs without impulses. Here, two simple upper and lower bounds of dwell times are proposed to ensure the stability.

Asymptotical stability of delayed FSSs with unstable subsystems
Consider the following nonlinear delayed FSS for i ∈ F: where y(·) ∈ R n , A i , B i ∈ R n×n are constant matrices, r i > 0 are constant delays,r = max i∈F {r i }, and To obtain our main results, a useful lemma will be introduced.
Lemma 6 [38]. The zero solution of system (41) will be stable if there are continuous nondecreasing Theorem 3. FSS (39) will be asymptotically stable if there are scalars ξ s , ξ u > 0, ν s > 1, 0 < ν u < 1 and positive definite matrices P i , P j such that P i ≤ ν s P j for any i ∈ S, j ∈ F and P i ≤ ν u P j for any i ∈ U, j ∈ S, where i λ min P i P i , and the dwell times τ s and τ u of stable and unstable subsystems satisfy Proof. According to Lemma 2, the µ-order derivative of MLF (17) along system (39) is presented as follows Whenever that is, λ min (P i )y T (t + θ)y(t + θ) ≤ y T (t + θ)P i y(t + θ) ≤ y T P i y, θ ∈ [−r, 0].
We can obtain that from Lemma 3 and (40) Submitting (51) into (52) yields We can get that D µ V i (y) ≤ −ξ s V i (y) ≤ −ξ s λ min (P i ) y 2 from ∆ 2 i + ξ s P i ≤ 0 of (44), which shows the zero solution of the ith subsystem is stable by using Lemma 6 and thus i ∈ S.
It follows that from inequalities (44) and (45) The rest of the proof is similar to Theorem 1, so it's omitted here.

Remark 5.
Here we suppose that time delays are less than the dwell time τ s of stable subsystems (see relationship (47)  Theorem 4. FSS (39) will be asymptotically stable if there are scalars ξ s , ξ u > 0, ν > 1, k ≥ 1 and positive definite matrices P i , P j such that and where i λ min P i P i , α s = ξ s − lnkν τ ds and α u = ξ u + lnkν τ du , T u or T s is total running time of unstable or stable subsystems. Proof. The proof is similar to Theorem 3, so it is omitted here.
Remark 6. In [27], based on state-dependent switching, stability of FSSs with discrete delays is solved. [29] proposes a useful fractional differential inequality to analyze impulse FSSs with discrete delays. Riemann-Liouville delayed FSSs is studied in [28] by using integer-order Lyapunov function method, which can not be applicable to Caputo's sense systems since Caputo fractional operators have no composition property. Here, Caputo FSSs with discrete and distributed delays are addressed and its stability is solved by means of fractional Razumikhin technique and MLF method.

Numerical Examples
We now provide numerical examples to further demonstrate the availability of our main results.

Conclusion
The asymptotic stability of nonlinear FSSs with unstable and stable subsystems has been discussed in this paper. Under the assumption that unstable subsystems cannot be consecutively activated, employing fast-slow switching mechanism, MLF and dwell time technique, several algebraic conditions and the lower and upper bounds of dwell times have been derived to guarantee stability. Then, we obtain a less conservative criterion in terms of ADT technique and the property of Mittag-Leffler functions. Finally, an effective way is provided to analyse a class of delayed FSSs by virtue of fractional Razumikhin technique, the proposed method may handle the troubles well caused by delays and fractional derivatives. Our future work will focus on stability analysis of neutral and singular FSSs.