Stability of impulsive switched systems with sampled‐data control

Funding information National Natural Science Foundation of China, Grant/Award Number: 61773089 Abstract This paper investigates the sampled-data control of impulsive switched systems with asynchronous switching. The impulses are not required to synchronise the switching. Since it is possible that switches may occur in sampling intervals, the mismatched problem may happen between controllers and system modes. Aiming for this, a functional consisting of multiple impulse-dependent Lyapunov functions and looped functionals is constructed, which does not increase at impulsive times. By using ADT method, some sufficient conditions for exponential stability are proposed in terms of linear matrix inequalities. Furthermore, sampled-data controllers are presented to stabilise the impulsive switched systems. The efficiency of the proposed results is verified by an F-18 aircraft.


INTRODUCTION
Hybrid systems are composed of the continuous dynamic behaviour, which is governed by differential equations, and the discrete dynamic behaviour [1]. Switched systems [2][3][4][5] and impulsive systems [6,7] are two important classes of hybrid systems. Switched systems consist of some subsystems described by a collection of continuous dynamics and a switching rule deciding the mode transition among these subsystems. Impulsive systems exhibit continuous evolutions described by differential equations and instantaneous state jumps. Such systems have a lot of practical applications in a broad range of areas, for example, networked control systems [8][9][10], robot control systems [11], flight control systems [12]. As is known, stability analysis and stabilisation are the fundamental problems. Many methods such as the piecewise Lyapunov function method and the average dwell time (ADT) method are proposed for switched or impulsive systems and some marked results can be found in [13][14][15][16][17][18][19][20].
In practice, impulses and switches exist synchronously in many physical systems, which are usually called impulsive switched systems and have been intensely studied. Up to now, This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Control Theory & Applications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology many results have been obtained. In [21], a new adaptive control scheme for switched systems is proposed. Some sufficient conditions for input/output-to-state stability(IOSS) are addressed by using Lyapunov and ADT methods in [22]. [23] studies robust stabilisation for a class of uncertain impulsive switched systems. Input-to-state stability (ISS) and integral input-to-state stability (iISS) for a class of impulsive switched systems under asynchronous switching is studied in [24].
However, these results only focus on the case where switches and impulses are synchronous. In fact, some practical systems can not be simply modelled as dynamic systems mentioned above. This is because a switching rule that defines the switching signal may be unknown. Switching may be caused by unpredictable environmental factors or component failures thus the switching times are more likely not at the same time with impulses. For example, a given chemical process prone to component failures, an automobile running in a harsh environment, a robotic manipulator moving different specific loads or an aircraft flying in different flight conditions is better to be modelled as impulsive switched systems, where the switching and impulse times are different, for example, [25][26][27]. To be specific, a simplified F-18 aircraft is provided in [27] with a family of modes determined by Mach number and flight altitude. As the aircraft enters different flight altitudes and flies at different Mach numbers, the subsystem describing the dynamic of the flight changes accordingly. On top of that, since it is possible that the aircraft is subjected to shock effects, the system state including the angle of attack and the pitch rate suffers jumps. Therefore, the simplified F-18 aircraft system can be modeled as an impulsive switched system. Hence, this paper considers the scenario where switching and impulsive time instants are not necessary synchronous. This would inevitably bring the difficulty of analysis because two kinds of discontinuities in the state have to be taken into account. There are only a few results published, see [28][29][30]. In [28,29], the stability for impulsive switched timedelay systems with time and state-dependent impulses is investigated. The IOSS and integral IOSS for non-linear impulsive switched delay systems are studied in [30]. In addition, because of the developments of hard-ware technology and digital technologies, sampled-data controllers are more favorable in practical applications. Some insightful results can be seen in [31][32][33]. In [31], discontinuous Lyapunov functions are introduced for impulsive systems under the variable and bounded sampling. [32] improves the results on sampled-data systems by using new discontinuous Lyapunov functionals. Based on the discrete-time Lyapunov theorem, [33] provides stability criteria with looped functionals for the continuous-time model. Nevertheless, when applying sampled-data controllers to impulsive switched systems, the approaches proposed in [28][29][30] are difficult to handle the potential mismatched problem. The mismatched problem results from the unknown switching times. Specifically, it is because it will take a while to identify the system mode and switching time and then apply the matched controller, which means that there inevitably exists time delay between the system mode and the controller, resulting in the asynchronous switching. Still, the methods used in [31][32][33] can also not be applied to impulsive switched systems because the effect of switchings are not involved in these works and thus the mismatched problem cannot be addressed.
Based on the above discussions, in this paper, we study the stability and stabilisation problem of impulsive switched systems with sampled-data control. A functional consisting of multiple impulse-dependent Lyapunov functions and looped functionals is constructed, which does not increase at impulsive time instants. By using the ADT method, some sufficient conditions for exponential stability are given in terms of linear matrix inequalities. Furthermore, exponential stabilisation conditions are derived to design sampled-data controllers. As an application, these results are applied to ensure the stability of the simplified F-18 aircraft given in [27].
The remainder of this paper is arranged as follows: Section 2 describes the model of impulsive switched system via sampled-data control and some relevant notations. The multiple impulse-dependent Lyapunov functions and looped functionals are introduced in Section 3. Stability and stabilisation conditions are provided in Sections 4 and 5, respectively. An example is used to demonstrate the efficiency of the obtained results in Section 6. Finally, conclusions are listed in Section 7.
Notation: Throughout this paper, ℝ and ℕ are sets of real numbers and natural numbers, respectively, ℤ + is the set of positive integers. ‖ ⋅ ‖ is a vector norm defined in ℝ n . The notation M ≻ (⪰) 0 denotes a symmetric positive (semi-) definite matrix. Given two sets C 1 and C 2 , we denote by C 2 ∖C 1 the relative complement of C 1 in C 2 , that is, the set of all elements belonging to C 2 , but not to C 1 . For two integers n 1 and n 2 with n 1 < n 2 , the notation n 1 , n 2 represents the set of {n 1 , n 1 + 1 … , n 2 }. Asterisk * in a symmetric matrix denotes the entry implied by symmetry. For any square matrix A ∈ ℝ n×n , we define He{A} = A + A T . For any non-singular matrix A ∈ ℝ n×n , define A −T = (A T ) −1 .

PRELIMINARIES AND PROBLEM FORMULATION
In this paper, we consider the following impulsive switched systems:ẋ where x ∈ ℝ n is the state and u ∈ ℝ m is the control input.
. A i , B i and J i are known constant matrices with appropriate dimensions. Moreover, we assume that for each i ∈ the pair (A i , B i ) is stabilisable and J i is non-singular matrix. x(t k ) = J (t k ) x(t − k ) describes the state jump, where the matrix J (t k ) is dependent on the active subsystem, and x(t − ) = lim h→0 + x(t − h). In this paper, for given 0 and 1 with 0 < 0 ≤ 1 , ( 0 , 1 ) denotes a class of impulsive time sequences satisfying 0 ≤ t k+1 − t k ≤ 1 , k ∈ ℤ + .
Denote the sampling sequence {t c , c ∈ ℤ + }, wheret 0 = 0. The sampled-data controller is where K (t c ) is the controller gain. The sampling and switching sequences satisfy the following assumptions: Assumption 1. The lengths of sampling intervals h c =t c+1 −t c are bounded by Assumption 2. (Slow switching [3]) 1) There exists a positive number d such that any two switches are separated by at least d , that is,t s+1 −t s ≥ d .
2) There exist numbers a > d ( a called an average dwell time) and N 0 ≥ 1 such that where N (T, t ) stands for the number of switches on time interval [t, T ). For simplicity, we denote such kind of switching signal (t ) ∈ S [ a , N 0 ].
Remark 1. Observing from Assumption 1, the lengths of sampling interval are variable in [h min , h max ]. By taking h max ≤ d , one can guarantee that there is at most one switching in each sampling interval, that is, there exists at least one sampling within each switching interval. In this way, the controller is updated in each switching interval, which is helpful for the system stabilisation.
Combining (1) with (2) yields the following closed-loop system for t ∈ [t c ,t c+1 ): The objective of this paper is to exponentially stabilise system (4) through designing sampled-data controllers. The exponential stability definition is presented as follows. Definition 1. System (1) under a class of impulses ( 0 , 1 ) is said to be exponentially stable (ES) with rate of convergence > 0, if there exists > 0 such that, for any impulse t k ∈ ( 0 , 1 ) and any corresponding solution x, we have Next, we introduce the merging switching signal technique, which mainly comes from [4], to deal with mismatched switching. Rewrite (t c ) as The notations |Ξ( , t )| and |Θ( , t )| are used to denote the lengths of the interval Ξ( , t ) and Θ( , t ), respectively. For example, in Figure 1, Some lemmas needed for our main results are given in the following.

MULTIPLE IMPULSE-DEPENDENT LYAPUNOV FUNCTIONS AND LOOPED FUNCTIONALS
In this section, multiple impulse-dependent Lyapunov functions and looped functionals are introduced. The multiple impulse-dependent Lyapunov functions V (x(t )) are nonincreasing at impulsive time instants and satisfy certain growth condition at the augment switching time instants. The looped functionals v(t c+1 , t, x(t )) (t ∈ [t c ,t c+1 )), which are similar to the form in [33], are used to enhance the flexibility of Lyapunov functions.
Based on the fact that there is at most one switching in a sampling interval, two cases should be considered. One is that there is no switching happening in the sampling time interval [t c ,t c+1 ), then the total time interval [t c ,t c+1 ) is matched. The other is that there is one switchingt s occurring in the sampling time interval [t c ,t c+1 ). Thus, we are able to discuss separately for the matched interval [t c ,t s ) and mismatched interval [t s ,t c+1 ).

Matched Intervals
In this subsection, the impulse-dependent Lyapunov function and the time-dependent looped functional assigned to the matched interval Ξ(t c ,t c+1 ) (c ∈ ℕ) are introduced. Without loss of generality, for the matched interval Ξ(t c ,t c+1 ) and assuming (t c ) = i ∈ , the looped subsystem iṡ 10 (t ) as follows: , P i (t ) and i (t ) take the following form: Remark 2. 10 (t ) is used to define the matrices P i (t ) which change piecewise. i (t ) can be viewed as a weighting factor. i (t ) and P i (t ) are used to satisfy that V i (t ) are non-increasing at impulsive instants under appropriate condition.
Next the time-dependent looped functional for the matched interval (9) where > 0, R i ∈ ℝ n×n is positive definite matrix.

Mismatched intervals
In this subsection, for the mismatched interval Θ(t c ,t c+1 ) and assuming that mode i is switched to mode j att s (t c ≤t s <t c+1 ), the looped dynamics iṡ where where i j,l ≥ 1 (l ∈ 1, N ) are scalars, P i j,l ∈ ℝ n×n (l ∈ 0, N ) are positive definite matrices. Moreover, for t k ∈ Θ(t c ,t c+1 ) (k ∈ ℤ + ), Similar to analysis of subsection 3.1, set̃i j, In what follows, we construct the time-dependent looped functional for the matched interval Θ(t c ,t c+1 ), where > 0, R i j ∈ ℝ n×n is positive definite matrix.
Define functional W i (t ) := V i (x(t )) + v i (t ) that is assigned to the matched interval Ξ(t c ,t c+1 ), and W i j (t ) := V i j (x(t )) + v i j (t ) that is assigned to the mismatched interval Θ(t c ,t c+1 ). Set W (t ) = W i (t ) when t ∈ Ξ(t c ,t c+1 ), and W (t ) = W i j (t ) when t ∈ Θ(t c ,t c+1 ).
Remark 3. The functional W (t ) is characterised by the following properties. (i) W (t ) keeps the value of V (t ) at the sampling time instants, that is non-increasing at the impulsive time instants under appropriate conditions and satisfies certain growth condition at the augment switching time instants; (iii) The derivative of V (t ) is not required to be negative, while we only require the derivative of W (t ) is negative.

STABILITY
The theorem below provides conditions of exponential stability of system (4) based on the functional W (t ) discussed in Section 3.
Π i j1,lmq + hΠ i j2 ≺ 0, P i j,l ≺ P i,l , P j,l ≺ P i j,l , where h ∈ {h min , h max }, l = 1, N in (14) (15), l = 0, N in (16), m, q, = 0, 1 and Proof: Since the sampling intervals are divided into matched intervals and mismatched intervals, we shall prove that , then establish the exponential decay of the Lyapunov function on sampling instants under the ADT constraint.
Case 2. There is at least one impulse occurring in the matched interval Ξ(t c ,t c+1 ). According to (17), for k ∈ ℤ + , we have Part 2: In the mismatched interval Θ(t c ,t c+1 ) When a switching happens att s ∈ [t c ,t c+1 ), system (10) is considered. As discussed in Section 3, we choose the func- In addition, we make use of the following two slack variables Combining (28)-(31) and using (15), for t ∈ Θ(t c ,t c+1 ) ∩ [t k,l −1 , t k,l ) (k ∈ ℤ + , l ∈ 1, N ), we geṫ For the impulsive time sequence  ∈ ( 0 , 1 ), one should consider two cases as follows: Case 1. There are no impulses happening in the mismatched interval Θ(t c ,t c+1 ). From the form of W i j (t ), we can obtain that W i j (t ) varies continuously in Θ(t c ,t c+1 ); hence, it is obtained that W i j (t ) < e 2 |Θ( ,t )| W i j ( ) in the mismatched interval Θ(t c ,t c+1 ) form (32).
Case 2. There is at least one impulse occurring in the mismatched interval Θ(t c ,t c+1 ). According to (18), for k ∈ ℤ + , we have Part 3: Synthesise both parts and derive the ADT condition. Consider the mismatched interval Θ(t c ,t c+1 ) andt s ∈ (t c ,t c+1 ) with (t − s ) = i and (t s ) = j ≠ i ∈ , by (16), the relationship between W i j (t s ) and W i (t − s ) is derived as and at the next sampling timet c+1 after switchingt s , we can obtain the relationship between W i j (t − c+1 ) and W j (t c+1 ) Next consider the augment switching signal (t ), for any t > t 0 = 0, let 0 =t 0 < 1 , 2 , … , N (t 0 ,t ) be the augment switching time sequence for (t) in interval (t 0 , t ). Without loss of generality, let 0 =t 0 and N (t 0 ,t )+1 = t , from (26), (32), (34), (35) and by using of recursive calculation, one has By Lemma 1, we can get N (t 0 , t ) ≤ 2N 0 + d ∕ a + 2(t − t 0 )∕ a . Considering (19), for any ∈ ( at sampling instantt c (c ∈ ℕ), we get It is easy to derive that ||x(t c )|| ≤ √ In addition, since the impulse shall not occur infinitely in [t c ,t c+1 ), the state of system (1) is bounded in [t c ,t c+1 ) (c ∈ ℕ), there exists > 0 such that ||x(t )|| ≤ ||x(t c )||, and thus Finally, we can conclude that system (1) is exponentially stable over ( 0 , 1 ) under the ADT condition (19). □ For the proof of Theorem 1, we explore the derivative of the functional W (t ), namelyẆ i (t ) + 2 W i (t ) anḋ W i j (t ) − 2 W i j (t ). By calculating the derivatives, we can given in (26). Inequalities in (14) are required to guarantee < 0 to hold for m, q, = 0, 1, the inequalities in (14) indicate two sets of LMls.
Remark 4. It is not hard to get solutions of the LMIs in Theorem 1 for the following reasons: (1) Since i,l , i j,l are adjustable, (17) and (18) are not hard to be satisfied; (2) Inequalities in (16) are not hard to verify with > 1; (3) P i,l , P i j,l , R i , R i j are required to be positive definite while M i1 , M i2 , M i j1 , M i j2 , N i , and N i j are arbitrary matrices. Hence, (14a) and (15a) are not hard to verify; (4) In LMIs (14b) and (15b), −hR i e 2 h max and −hR i j are both negative definite, which make (14b) and (15b) easy to be satisfied.
Remark 5. It is worth noting that the impulse-dependent Lyapunov functional method proposed here allows the functional not to increase at destabilising impulses. As a tradeoff, the number of linear matrix inequalities needed to be solved is 16m 2 (N + 1) + 3m(m − 1) + m 2 , where m is the number of subsystems, and N is the number of subintervals between every two impulses. So a large partition number N and a system mode number m would increase the computational cost and complexity.
Remark 6. In [15,17,18], a linear time-varying Lyapunov function V (x(t )) = x T (t )P (t )x(t ) is used, which is, however, infeasible in this paper because the sampled state is used for feedback in our method. In addition, it is too strict to find such a Lyapunov function, which is required to be independent of the sampled state. In this paper, a new functional W (t ) consisting of Lyapunov function V (x(t )) and looped-functional v(t ) is constructed to overcome this difficulty. Furthermore, in order to handle impulses and the mismatched problem , multiple impulse-time dependent weighting factors are introduced. i,l and i j,l take value from min to max randomly; 8: Solve LMIs (14)

STABILISATION
In this section, a sampled-data controller is designed, and the exponential stabilisation conditions are established to calculate controller gains K i (i ∈ ).
Proof: Based on Theorem 1, the proof is also proceeded in three parts.
Part 1: In matched interval Ξ(t c ,t c+1 ). Without loss of generality, we analyse the matched interval Ξ(t c ,t c+1 ) and assume (t c ) = i ∈ . Let the slack matrix vari- (25), then according to (37), for t ∈ Ξ(t c ,t c+1 ) ∩ [t k,l −1 , t k,l ) (k ∈ ℤ + , l ∈ 1, N ), we can obtaiṅ In view of (41), by leveraging Schur complement, one has The rest of the derivations are the same as that of Part 1 in Theorem 1, we have W i (t ) < e −2 |Ξ( ,t )| W i ( ) in the matched interval Ξ(t c ,t c+1 ), wheret c ≤ < t <t c+1 . Part 2: In the mismatched interval Θ(t c ,t c+1 ) Without loss of generality, we analyse the mismatched interval Θ(t c ,t c+1 ) and assume that mode i is switched to mode j att s (t c ≤t s <t c+1 ). (32) can be also ensured for t ∈ Θ(t c ,t c+1 ) The rest of the analysis is the same as that of Part 2 in Theorem 1, according to (38), we have

Part 3: Synthesise both parts
The analysis is the same as that of Part 3 in Theorem 1, by using (39) and (40), system (1) with sampled-data controller (2), where K i = L i Q −1 i , is exponentially stabilisable over ( 0 , 1 ) under the ADT condition (19). □

SIMULATION EXAMPLE
In this section, we take an aircraft as an example to show the effectiveness of our proposed approach. The longitudinal linear equations of motion of the aircraft come from [27]. The simplified longitudinal linear equations are where Z , Z q , M , and M q are longitudinal stability derivatives and Z E , Z PTV , M E , and M PTV are longitudinal control derivatives, which are determined by the structure of aircraft and flight conditions.
A set of baseline aerodynamic data are obtained from wind tunnel and flight test data [27]. Considering two flight conditions of Mach 0.8, altitude 10 kft and Mach 0.8 altitude 12 kft, we obtain the following two subsystems: where A m8h10 long means the longitudinal state matrix at Mach 0.8 and altitude 10 kft. We assume that at least d is needed before the altitude changed from one interval, say (9,11) kft, to the other. Further, at some moments let t k , we suppose the system is subjected to shock effects because of which the angle of attack and the pitch rate q suffer an instantaneous increment. For simplicity, we model the impulsive time sequence as  ∈ ( 0 , 1 ) and the increment as .
Then applying sampled-data controller u(t ) = K (t c ) x(t c ) for t ∈ [t c ,t c+1 ), the dynamic equation (43) becomeṡ where x = [ T q T ] T , J (t ) = (I + D (t ) ), J (t ) = (I + D (t ) ). Thus Theorem 2 is applied to this situation. A class of impulsive time sequences (0.2, 3) is considered. Comparing with (14), an apparent distinction in (37) is ∑ 4 p=1 s ip I T p (A i Q iĨ1 − Q iĨ2 + B i L iĨ3 ) where Q i and L i are slack variables the role of which is the same as that of M i2 in (14), and s parameters provide more flexibility herein. So, it is sufficient for finding feasible solution using Algorithm 1 to select the slack variables s ip = s i jp = 0.5 for all i ≠ j ∈ {1, 2} and p ∈ 1, 4 as a starting point. Then by utilising Algorithm 1, we choose N = 2 due to the computational cost and choose a modest exponential decay rate = 0.4 and growth = 1.6. Note that affects the overshoot bound for Lyapunov stability. A large implies a large overshoot bound, which is not desirable system performance thus we let = 1.2. Notice also that we use sampled-data controllers to save the cost of calculation and avoid the frequent computation because only the states on sampling times are fed back, hence the sampling interval cannot be too small and we choose = 0.05. Initialise h min = 0.5, h max = 1, min = 1, max = 3. As a result, we figure out that when 1,1 = 1,2 = 2,1 = 2,2 = 1. ] , and the ADT should satisfy a > 1.0058s. The impulsive time sequence is illustrated in Figure 2. Choose x 0 = [2, 2] T , N 0 = 2, a = 1.01s, d = 0.11s > 0.1s, which satisfies Assumption 1. From Figure 3, one can see that the states of the switched systems converge to the origin, meaning the system (44) is stable and our method is effective. In addition, the trajectories of functionals W (t ), v(t ) and V (t ) are illustrated in Figure 4.

CONCLUSIONS
In this paper, we have investigated the stability and stabilisation of impulsive switched system with sampled-data controller. The functional consisting of multiple impulse-dependent Lyapunov functions and looped functionals has been proposed. By solving a set of LMIs, we could check the stability of impulsive switched system with sampled-data controllers. Moreover, exponential stabilisation conditions have been derived for the design of sampled-data controller. At last, the example has been given to show the effectiveness of the proposed approach.