Input-Output Finite-Time Sliding Mode Control of Discrete Time-Varying Systems Under an Adaptive Event-Triggered Mechanism

This paper is concerned with the issue of input-output finite-time stability (IO-FTS) for a class of nonlinear discrete time-varying systems. A time-varying observer-based sliding mode control method is proposed. In order to mitigate the transmission burden, an adaptive event-triggered mechanism is proposed by adjusting the threshold. Taking consideration of the effect of time-delay phenomenon and time-varying system matrices, a time-varying Lyapunov functional is designed. Based on the designed Lyapunov functional and IO-FTS theory, sufficient conditions are established for the error estimation system and closed-loop state estimation system. Moreover, the proposed observer-based sliding mode control method makes sure the reachability of the quasi-sliding mode surface in finite steps. And conditions in terms of recursive linear matrix inequalities (RLMIs) are attained to ensure IO-FTS during both reaching phase and sliding mode phase. An algorithm is provided to solve the RLMIs and obtain the time-varying observer gains. Finally, the effectiveness and superiority of the proposed method is demonstrated by an industrial continuous-stirred tank reactor system.


I. INTRODUCTION
In practice, time-varying systems represent effective tools to describe different types of dynamical systems containing some time-varying attributes, such as, periodic systems, sampled-data systems and switched systems, etc., see for instance, [1], [2], and [3]. Moreover, aperiodic sampling in networked control systems (NCSs) often results in that NCSs are transformed to discrete time-varying systems (DTVSs). Accordingly, it is of great significance to study DTVSs, and they have gained increasing attention. Observer-based finitetime H ∞ control of nonlinear DTVSs was studied in [4]. The issue of finite horizon fault detection of linear DTVSs was discussed in [5]. The consensus control problem was studied for a discrete time-varying multi-agent system in [6].
The associate editor coordinating the review of this manuscript and approving it for publication was Zheng Chen . On the other hand, one fact has been observed that, in NCSs, there are some sampled data packets that the information carried by them is less fluctuating than the last transmitted sampled data packets. This kind of transmission may introduce some unnecessary communication burden. In response to these problems, the event trigger mechanism (ETM) is proposed, for example, [7], [8], and [9]. Compared with timetriggered mechanism, ETM can reduce the network resource occupancy as well as maintaining the control performance. However, most of the proposed ETMs are designed with fixed thresholds that is not easy to give them proper values in advance, which is named as traditional ETM (TETM) in this paper. To deal with this, the adaptive event-triggered mechanisms (AETMs) with adjustable trigger conditions are proposed in [10] and [11]. It has been proved that AETM can save more network resources. Therefore, a critical issue is yet to be addressed when considering DTVSs, that is, can we VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ propose an AETM to save more resources while preserving the desired control performance? To the author's best knowledge, there are few relevant results focus on DTVSs, which motivates our present study. Since sliding mode control (SMC) was proposed in 1977, due to some appealing features of rapid transient response, robustness, and simplicity of implementation, it has become an effective method for suppressing disturbances in complex systems. Some SMC results of uncertain nonlinear systems, especially on advanced and optimization based algorithms were given in [12]. For continuous systems, there are many research results using SMC methods, see [13], [14], and [15]. Due to the emerging digital technologies, many practical systems are treated in the structure of discretized forms and then discrete-time SMC (DSMC) has become an increasingly hot topic, see, for instance [16] and [17]. In [18], the SMC design problem for discrete-time piecewise nonhomogeneous Markov jump nonlinear systems was discussed. In [19], dissipative-based DSMC of switched stochastic hybrid systems was discussed. And in [20], an optimal SMC approach was used to study the consensus of nonlinear discrete-time high-order multi-agent systems. In addition, there are many situations where the state variables can not be measured. In response to this issue, the dynamic and static output feedback SMC methods were studied in [21] and [22], respectively. Ma et al. discussed the observer-based adaptive sliding mode control problem for a kind of stochastic jump systems in [23]. Nevertheless, it is witnessed that the study of observer-based sliding mode control for DTVSs has not been fully spread, which becomes to the second motivation of this research.
It is noticed that most of the above mentioned literature about SMC method was considered during infinite time intervals. In fact, FTS proposed by Dorato in [24] and developed by Amato is a more practical concept [25]. It is applied in some cases where the required state or output variable does not exceed some a threshold during a fixed time interval, for example, flight control [26], terminal guidance system [27]. Another research frontier proposed based on FTS is IO-FTS, which defines that if a class of norm-bounded input signals is given within a specified time interval, the outputs of the system do not exceed the specified threshold during the time interval [28]. The issue of input-output finite-time generalized dissipative filter was studied for a kind of DTVSs in [29]. Based on the definitions of FTS and IO-FTS, an SMC design method was proposed in [30] and [31], respectively. In these two papers, FTS of the considered systems can be guaranteed during both reaching and sliding mode phase. Subsequently, FTS problem of uncertain neutral time-delay systems was discussed via the SMC approach in [32]. However, with regards to DTVSs, the research of IO-FTS based on observer-based sliding mode control method is still blank, which also motivates the following study.
Based on the above discussion, in this paper, the problem of IO-FTS is addressed for a nonlinear DTVS considering time-delay phenomenon and an AETM by the observer-based sliding mode control method. The main contributions are highlighted as: (1) Threshold of the proposed AETM can be adjusted. Thus, the network communication resources can be saved while sacrificing system performance as little as possible.
(2) The reachability of quasi-sliding mode is guaranteed in finite steps and the upper bound of quasi-sliding mode surface is given under the proposed event-triggered SMC law.
(3) Based on a time-varying Lyapunov functional, IO-FTS conditions during both reaching and sliding mode phase are obtained in terms of RLMIs. And the time-varying observer gains can be solved by an algorithm.
A. NOTATIONS Throughout this paper, N is the set of natural numbers. R n denotes the n-dimensional Euclidean space, and R n×m is the set of n × m real matrices. For a scalar N > 0, denote a set as T = {0, 1, 2, . . . , N }. For a symmetric matrix A, A > 0 (A ≥ 0) means that A is a symmetric positive-definite (semi-positive-definite) matrix. M + denotes the set of positive matrix-valued sequence. C + denotes the set of positive scalars. For R k ∈ M + , define W 2 (T , R k ) := {ω k ∈ L 2,T : T ,R k ≤ α}, α ∈ C + and ∥ω∥ 2 T ,R k = k∈T ω T k R k ω k . I and 0 denote the identity matrix and null matrix of the appropriate dimension, respectively. The symbol * means the symmetric terms in a symmetric matrix. Matrices without special explanation are considered to have proper dimensions.

II. PROBLEM FORMULATION
Consider a nonlinear DTVS as: where A k , D k and C k are time-varying matrix-valued sequences, B is the input matrix with rank(B)=m. x k ∈ R n is the state vector, y k ∈ R r stands for the system measured output, u k ∈ R m is the controlled input, and ω k is the disturbance input satisfying ω k ∈ W 2 (T , R k ).
is a given nonlinear vector function, which satisfies: where σ k ∈ C + is a series of known scalars.
Then, we design an observer in the following form to estimate the unmeasurable states: wherex k is the estimation of the state,ŷ k is the output of the observer,ȳ k is the plant output after the network,f k ≜ f (k,x k ) and L k is the time-varying observer gain to be designed.

A. MODELING BASED ON THE AETM
To alleviate waste of resources and avoid unnecessarily frequent transmission, an event detector is adopted between the sensor and observer as shown in Fig. 1 that describes the control process. It is assumed that sensors are time-triggered and the measured instant is denoted as l ∈ N. The measurement output is only transmitted when the following condition is satisfied: where s ∈ N, k s + l is the current measurement instant, y k s +l = y(k s + l) is the current measurement output, k s is the instant when an event happens and y k s is the transmitted state output. k ∈ M + are weighting matrices to be designed. γ k is a series of scalars determined by the following condition: where y e = y k s +l − y k s , λ 1 = 1 − 2 π atan(ς∥y e ∥ 2 ), λ 2 = 1 − 2 π atan(−ς∥y e ∥ 2 ), atan(·) is the arctangent function. It is assumed that γ k ∈ [γ m , γ M ], γ m and γ M are the lower and upper bounds of γ k , respectively, and ϵ, ς ∈ C + . Remark 1: In (5), when γ k is a constant, the adaptive event-triggered condition (4) is reduced to a TETM. When γ k = 0, it is the time-triggered case.
Remark 3: It should be noticed that introducing an ETM may sacrifice some SMC performance. However, the proposed AETM is adjusted by the transmission error, which can help improving the performance while reducing the network utilization.

B. INDUCED-TIME DELAY MODELING
Let d k s denote the transmission delay at triggered instant k s with the assumption d k s ∈ [0, d m ]. d m is a positive scalar. Considering the triggered condition (4) and during the hold- Based on [8] and [33], denoting I = [k s +d k s , k s+1 +d k s+1 ), the following two cases should be discussed.
Case I. If k s + d m + 1 ≥ k s+1 + d k s+1 , define the time delay function d k as: Since d k s ≤ d m , an integer 'q' can be found satisfying: Based on the above discussion, d k and δ k are given as: Thus, y k s can be rewritten as where Then, the error estimation system is obtained as follows:

C. SLIDING SURFACE AND SLIDING MODE CONTROLLER
We design the following sliding function as follows: where G = B T X , X > 0 thus GB is nonsingular.
To this end, an appropriate observer-based sliding mode control law is given as follows: VOLUME 11, 2023 Using SMC law (10), the closed-loop state estimation system can be obtained as follows: where G = I − B(GB) −1 G and µ k = σ k sgn(S k ).
In the following, the definition of IO-FTS proposed in [28] is extended for the overall closed-loop system composed of (8) and (11).
Definition 1: (IO-FTS) For a given positive scalar N ∈ N, a class of input signal W 2 defined on T , a matrix-valued sequence R k ∈ M + , and an output constraint scalar β > 0, the overall system (8) and (11) is IO-FTS with respect to (W 2 , R k , T ), if for all k ∈ T , it holds that:

III. MAIN RESULTS
In this section, based on the discrete-time sliding surface constructed in (9) and an event-triggered SMC law in (10), IO-FTS conditions are given in Theorem 1 and reachability for the quasi-sliding mode surface is analyzed in Theorem 2. Synthesis of the time-varying observer is given in Theorem 3. For the following proof process, we give some denotations as:
Remark 4: It is noticed that, in Lyapunov functional (14), the Lyapunov matrices P k , Q 1k , Q 2k , Q 3k are timevarying matrix-valued sequences. This design method greatly improves the solvability of obtained conditions.

B. ANALYSIS OF REACHABILITY
In this section, we focus on the reachability of the quasi-sliding mode surface of the overall closed-loop system (8) and (11)'s trajectories in finite steps.
Theorem 2: Considering the overall system (8) and (11), the state trajectories can be driven into a sliding region S around the sliding mode surface (9) within finite steps N * by SMC law (10), and IO-FTS during the reaching phase is guaranteed, if there exist matrix-valued sequences P k , Q 1k , Q 2k , Q 3k , k ∈ M + , and a scalar ν ∈ C + , such that (13) and the following conditions hold and the sliding region S is given as follows: Proof: From (9) and (10), it can be obtained that Construct a Lyapunov functional as where V (k) is defined in (14). Then, it can be derived that Considering J k defined in (22), one can obtain that Since outside the region S, it is true that Then, considering (31), (32) and (33), one can deduce that Thus, it means that state trajectories are strictly decreasing outside the region S defined in (29). Summing both sides of (34) from 0 to N * −1, denoting V s0 as the initial value of V sk , it can be seen that Combining the definition of ε and (35), one can derive N * < N . It means that, for a finite-time interval [0, N ], under the SMC law (10), the estimation state trajectory can be driven onto the quasi-sliding mode surface of S k within finite step N * . The reachability is proved.
On the other hand, by condition (28), it is easy to see that J k < 0 can be guaranteed. Based on the proof in Theorem 1, the overall closed-loop system composed of (8) and (11) is IO-FTS during the reaching phase.
Remark 5: According to Theorem 1 and 2, it can be concluded that trajectories of overall closed-loop system (8) and (11) can be driven into the quasi-sliding mode region S under the SMC law (10) and IO-FTS can be guaranteed with respect to (W 2 , R k , T ) by conditions (12), (13) and (28) during both the reaching and sliding mode phase.

C. SYNTHESIS OF THE TIME-VARYING OBSERVER
Obviously, inequality (28) ensures (12). Thus (28) and (13) can guarantee Theorems 1 and 2. In what follows, design of corresponding observer-based sliding mode controller will be discussed.
Theorem 3: Given scalars d 1 , d 2 , α, β ∈ C + , matrixvalued sequence R k ∈ M + , any scalars ι k ∈ C + , and input disturbance in W 2 , select a matrix X > 0 such that B T XB is nonsingular. The overall closed-loop system (8) and (11) is IO-FTS during both the reaching and sliding mode phase with respect to (W 2 , R k , T ), if there exist matrices P k , Q 1k , Q 2k , Q 3k , k ∈ M + , such that following conditions hold: where 22 = −2ι 2 k I + ι k P k+1 . Proof: For ι k ∈ C + and P k+1 ∈ M + , it holds that Thus, it is true that Using Schur complement lemma, conditions (36) and (37) can ensure Theorem 1 and Theorem 2, simultaneously. And matrix inequalities (36) and (37) are RLMIs. Thus, the time-varying observer gain L k and triggered weighting matrix-valued sequence k can be designed by solving (36) and (37) using Matlab Toolbox.

IV. NUMERICAL EXAMPLE
In this section, we consider an industrial continuous-stirred tank reactor system (CSTRS) in network environment [22], [34]. In the CSTRS, chemical species A reacts to form species B. Denote C ai and C a as the input and output concentration of a key reactant A, T and T c being the reaction and the cooling medium temperature, respectively. According to [22], both parameter uncertainties and system nonlinearity should be taken into consideration when model the CSTRS. Since the system is in a network environment, we attempt to use the proposed AETM in this paper to save communication resources.
Take the state variable as Considering the influence of disturbance, the discrete model of CSTRS can be represented as system (1) with the following system matrices: , ω k = 0.5e −k 0.4sin(k) , C k = I .
For simulation purposes, under the influence of timevarying parameters, disturbance and nonlinearity, our goal is to design a time-varying observer-based sliding mode controller such that the estimation of C a and T can be bounded by a given level. Give the parameters as γ m = 0.47, γ M = 0.74, d 1 = 1, d 2 = 3, α = 1, R k = I , N = 20. Let sgn(S k ) = S k ∥S k ∥+10 −3 for avoiding the chattering of signals.
Solving the RLMIs (36) and (37), some of the time-varying observer gain sequence L k can be tabulated in Table 1. The minimum value of β, denoted as β min , is obtained as β min = 4.40. Fig.2 depicts the trajectories of control signal u k . Fig.3 simulates the sliding mode surface function S k . The simulation of state estimation errors is shown in Fig.4. Fig.5 depicts the trajectories ofŷ T k R kŷk + (y k −ŷ k ) T R k (y k −ŷ k ) and β min , from which we can see thatŷ T k R kŷk + (y k −ŷ k ) T R k (y k − y k ) < β can be guaranteed. That is, IO-FTS is ensured by the proposed control method.  The triggered instants of the proposed AETM are simulated in Fig.6, from which we see that 50% communication resources are used and then the network resources are economized effectively. On the other hand, we suppose γ k = γ M such that the AETM is reduced to TETM. The release instants are simulated in Fig.7, from which we can see that 75% communication resources are used under the TETM. From Fig.6 and Fig.7, it can been seen that when the norm of state estimation error tends to zero, the proposed AETM can save more communication resources. Therefore, in terms of saving network resources, the proposed AETM is more effective than TETM.
In the end, to investigate the influence of event-triggered parameters γ k , values of γ k are given as some constants. We aim to obtain the value of β min when γ k takes different values. The number of trigger instants N t and values of β min   are given in Table 2. From Table 2, it can be seen that larger triggered parameter γ k can save more network resources but derive larger value of β min . That is to say, saving network resources may sacrifice system performance to some extent.
In conclusion, the AETM method proposed in this paper can economize the network resources meanwhile sacrifice system performance as little as possible. VOLUME 11, 2023

V. CONCLUSION
This paper has studied the issue of IO-FTS for a kind of nonlinear DTVSs by an observer-based sliding mode control method. An AETM with an adaptive law has been proposed to save network resources. The reachability of the quasisliding mode surface has been guaranteed in finite steps and the upper bound of the quasi-sliding mode surface is obtained. Moreover, IO-FTS during both reaching and sliding mode phase has been guaranteed by a series of RLMIs. And a time-varying observer has been designed by solving the RLMIs. A CSTRS has been given to verify the effectiveness and superiority of the proposed method.