Intelligent Control of Performance Constrained Switched Nonlinear Systems With Random Noises and Its Application: An Event-Driven Approach

In this paper, the adaptive fuzzy control of switched stochastic nonlinear systems with set-time prescribed performance based on event-driven mechanism is studied. The creative part of this paper is that based on the set-time performance function, a modified event-triggered strategy that considers asynchronous switching to deteriorate system performance without strict assumptions is presented, which avoids Zeno behavior and saves communication resources. Then, by using backstepping recursive design technique, It $\hat {o}$ ’s differential lemma and mode-dependent average dwell time (MDADT) method, a novel adaptive performance control scheme is proposed, which can ensure that all the variables in the system are semiglobally uniformly ultimately bounded (SGUUB) in probability and the tracking error gets into prescribed boundary no later than an arbitrarily adjusted setting time. Finally, the proposed algorithm is applied to a RLC circuit and its practicability is verified via simulation results.

I N RECENT decades, the electrical circuits and its control methods have been researched deeply in reports [1]- [4]. It is worth noting that the existence of nonlinear dynamics in electrical circuit systems can not be ignored, so the controller design of nonlinear systems has aroused great interest of scholars, and massive excellent achievements have  emerged based on neural network or fuzzy approximation approach [5]- [9]. However, the above control schemes are mainly applied for nonstochastic nonlinear systems. An enormous number practical engineering systems are subject to stochastic uncertainty, such as biological system, financial system and chemical reaction process and so on. For nontriangular multi-input and multi-output (MIMO) stochastic nonlinear systems, an adaptive tracking control scheme based on a new stochastic finite-time stability theorem was proposed in [10]. Then, Liu et al. [11] studied the control design of nonlinear stochastic systems with state constraints for the first time by constructing two different forms of barrier Lyapunov functions. For discrete-time stochastic nonlinear systems, an adaptive neural control scheme that mitigates the communication burden and improves the tracking accuracy was developed in [12]. Furthermore, for stochastic systems with unmeasurable states, some effective state observers were elegantly designed in [13]- [15] to estimate the unmeasured states. Unexceptionally, the above researches are both interesting and challenging, but their conclusions are only valid for nonswitched systems. Due to the fact that most systems are difficult to be described by one model in practice. Multimodel switching control have capacious developed foreground in practical systems. For the stability analysis and controller design of switched systems, massive outstanding achievements have been popping up (see [16]- [24]). Especially, the MDADT of milestone was proposed in [24] to analyze the stability of switched systems, which aroused the attention of many scholars. Since then, such method is extended to many kinds of switched systems to relax the restrictions of switching signals and realize the stability of system. To just name a few, Yang et al. [25] developed a transition probability-based MDADT switching mechanism for dynamic systems with mixed delays by designing a multiple Lyapunov-Krasovskii functional. In [26], the exponential stability was studied for discrete-time switched positive systems under the framework of MDADT. It was first reported in [27] that the adaptive control scheme for switched nonlinear lower triangular systems under MDADT switching. Nevertheless, up to now, the investigation of the adaptive control for switched nonlinear systems with random noises under MDADT switching is seldom. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Furthermore, the transient performance of controlled systems is not considered in the above articles. The prescribed performance control (PPC) method was first proposed in [28] and quickly applied to various nonlinear systems, such as large-scale nonlinear systems [29], MIMO nonlinear systems [30], and stochastic nonlinear systems [31], ect. Subsequently, for the convergence time of closed-loop systems, their finite-time adaptive neural networks and fuzzy PPC methods were studied in [32], [33], which effectively solved the problems of slow convergence and low accuracy of traditional adaptive PPC method. Although the above PPC schemes have satisfactory control effects, they have a common disadvantage, that is, the initial value of the performance function depends on the reference signal and the system output. But many industrial systems do not have constraints at the initial time, after the system runs for a certain time, there will be constraints on the system performance, that is, in [0, T ], there are no constraints on the system; and after t > T , the system has constraints. Therefore, how to design an effective adaptive PPC scheme to deal with this more complex constraint situation is worthy of further research.
On the other hand, event-triggered communication control (ETCC) has attracted widespread attention due to its important role in networked control system [34]- [38]. For the ETCC of the switched systems, an enormous challenge is that the asynchronous switching between the subsystem and the controller is proving elusive. Asynchronous switching is caused by the switch within two consecutive triggering instants. Most of the existing results evade this problem or make strict assumptions about the maximum asynchronous duration, resulting in a lot of restrictions on the applicability of the results in practice, e.g., [39]- [41]. Recently, some excellent reports [42], [43] have been published to solve asynchronous switching to ensure system performance. Unfortunately, the above schemes do not consider stochastic disturbances. In other words, these event-triggered controllers do not be directly applied to switched stochastic nonlinear systems.
In conclusion, we find that the event-triggered fuzzy control methods for switched stochastic nonlinear systems are numbered. Also, the existing methods do not ensure the transient performance of the controlled plant under asynchronous switching. In this paper, a fuzzy set-time PPC scheme is proposed for switched stochastic nonlinear systems. The innovations of this article can be embodied in the following points.
1) By introducing the set-time performance function into the controller design, the proposed adaptive fuzzy settime PPC scheme not only ensures that the tracking error gets into the predefined constraint region no later than a settable time T , but also eliminates the "initial condition" of the constrained variable e 1 in the traditional PPC scheme.
2) A novel mode-dependent event-triggered mechanism (MDETM) is designed for switched nonlinear systems with random noises considering the impact of asynchronous switching on system performance. The proposed control scheme achieves the expected control effect while mitigating the communication burden. 3) By using the lower bound of the control gain functions of each subsystem, the individual Lyapunov function is constructed, and a novel event-triggered fuzzy performance controller is designed so that all the variables in the system are bounded.

A. Basic Knowledge
Definition 1 [13]: Consider the stochastic system dx = f (x(t))dt + g(x(t))dw. Define the differential operator for C 2 function V (x) as: where T r(A) is the trace of A. Lemma 1 [10]: Let f (Z ) be a continuous function defined on a compact set. Then for any τ > 0, there exists a fuzzy systems ψ T S(Z ) such that Lemma 2 [8]: For ∀ ω 1 > 0 and ω 2 ∈ R, the following result hold

B. Problem Statement
Consider switched Itô stochastic nonlinear systems are unknown smooth nonlinear functions satisfying local Lipschitz. w ∈ R r denotes standard Brownian motion.
Remark 1: The above-mentioned switched stochastic nonlinear system can be applied to the RLC circuit with stochastic perturbations in the capacitor and the inductor. For example, a RLC circuit is shown in Fig. 1, where L is the inductor, R the resistor, C 1 , C 2 the two mutual switching capacitor.
Define the tracking error as e 1 = y − y d with y d being the reference signal. In this paper, the tracking error need to satisfy where T is a time parameter, ξ 1 (t) is called the set-time performance function and is defined as where ξ 0 > ξ ∞ > 0, κ 1 ≥ 0 are the design parameters. Remark 2: Whether it is the traditional PPC schemes proposed in [29]- [31] or the finite-time PPC schemes proposed in [32], [33], the performance function requires "initial condition", that is, Obviously, ξ 1 (0) introduced in this paper is independent of the initial conditions of the system output and the desired signal.
Our control objectives are as follows: 1) All signals of the controlled systems are SGUUB in probability under MDADT method; 2) The tracking error e 1 gets into a prescribed boundary no later than a settable time T ; 3) The designed MDETM is Zeno-free.
To this end, the following mapping is proposed: meanwhile, the following indirect performance function ξ 2 (t) is adopted Remark 3: According to the expression of ξ 2 , it can be seen And the proposed method removes the "initial condition" imposed on the tracking error e 1 .
Specially, the following assumptions are imposed. Assumption 1: (Slow Switching) (1) There exists a number τ * d > 0 (called a dwell time) such that any two switches are separated by at least τ * d > 0; (2) There exist numbers τ ap > τ * d (called a mode-dependent average dwell time) and N 0 p ≥ 1 such that Without losing generality, we assumes that sign(l i, p (x i )) > 0.
Remark 4: For the studied switched stochastic nonlinear systems, the MDETM that relies on switching signals is cleverly designed, which not only mitigates the communication burden, but also eliminates the impact of asynchronous switching on the system performance.
Remark 5: It can be seen that the triggering error of the designed MDETM is discontinuous at the switching moment, and switching may cause additional continuous triggers, which may lead to Zeno behavior. The introduction of variable T w effectively avoids the above problems.
From (3) and (9), one has Define the Lyapunov function candidate From (35) and (36), we have . By utilizing Young's inequality, the following inequality holds By substituting (38) into (37), it gets V n, p ≤ V n−1, p + z 3 n l n, p (x n )u + z 3 nf n, p (Z n ) − wheref n, p (Z n ) = f n, p (x n )+ 3 4 a −2 n z n φ n, p 4 −α n−1 + 3 4 z n + 1 4 l n−1, p (x n−1 )z n . Same as (19), one has where |δ p n (Z n )| ≤ τ n with τ n > 0. By applying Young's inequality, we have Next, we will divide the system dynamics into two parts for discussion based on whether the pth subsystem is synchronized with the candidate controller within the triggering interval [t k , t k+1 ).
Part 1: synchronous interval. At this time, 1]. Then, the actual controller can be expressed as Therefore, (42) can be repeated as Based on Using a process similar to Step i, it gets By means of Young's inequality, we get Part 2: asynchronous interval. 1 If . At this moment, the MDETM (33) can ensure that Similar to the derivation in Part 1, we have Take the same steps as Part 1 to get 2 This interval is nonempty only if r > 1. At this time, (33) is the same as Part 1 to ensure Then, using the similar derivation given in Part 1, we get It can be obtained by using the same procedure as in Part 1 The synchronous/asynchronous discussion between the subsystem and the candidate controller is completed. Next, by selecting the Lyapunov function candidate V p = V n, p , we have , 1, i = 1, 2, · · · , n, ∀k ∈ M}, it can ensure the following: 1) All the resulting system signals are SGUUB in probability.
2) The tracking error e 1 gets into a prescribed boundary no later than a setting time.
Proof. First of all, we prove that all signals of the control system are bounded, and the discussion is divided into two cases.
Case 1: When μ p = 1( p ∈ M), we get V p = V q , ∀ p, q ∈ M. Therefore, the common Lyapunov function V = V p for all subsystems satisfies (55), which means where η min = min{η p , p ∈ M}. Therefore, it can be concluded that all the signals in the control system are SGUUB. Case 2: When ∃μ p,q > 1( p, q ∈ M). There are , for an arbitrary T > 0, let t 0 = 0 and t 1 , t 2 , · · · , t s , t s+1 , · · · , t N σ (0,T ) are the switching times on Consider the piecewise continuous function It is shown from V p (Y (t)) ≤ μ p V q (Y (t)), one has Hence where Then get from (60) that where With the help of (31), it follows that u p is differentiable andu p is bounded. From β p (t k ) = 0 and lim t →t k+1 Case 2 (Triggering Interval With One Switch): Assume that the switch occurs at k 1 ∈ (t k , t k+1 ). Noting that, it can be seen from |u σ (t k ) (t k ) − u k 1 ( k 1 )| < λ|u σ (t k ) (t k )| + + T w that no additional trigger will be generated at k 1 . From case 1, d dt |β σ (t k ) | ≤ 1 can be obtained in (t k , k 1 ). As in case 1, it can be guaranteed that d t k+1 ), where 2 is a positive constant. It is shown from the above analysis that Case 3 (Triggering Interval With Multiple Switches): N k is the number of switches on the kth triggering interval, and obviously t k+1 − t k ≥ N k τ * d > 0. Based on the above analysis, the designed MDETM is Zeno-free. The proof is completed.
Remark 6: To prove the stability of the switched system based on the multiple Lyapunov function techniques, it is important to construct the relationship between any two Lyapunov functions. In this paper, the cumulative relationship of two Lyapunov functions is found by using uniform coordinate transformation and common adaptive law for all subsystems.

IV. SIMULATION EXAMPLES
In this section, the effectiveness of the proposed theoretical results is verified by numerical example and practical example.
Example 2: In order to verify the practicability of the proposed control method, the RLC circuit given in [16] is

V. CONCLUSION
Based on the event-triggered strategy, this paper solves the problem of fuzzy control for stochastic switched nonlinear systems with set-time predefined performance. Combined with MDADT method and Lyapunov function stability analysis, a fuzzy performance algorithm is proposed. The contribution of this study is to introduce the MDETM into the performance control design of switched stochastic nonlinear systems. The proposed control algorithm can not only ensure that the tracking error enters the predefined region no later than a setting time, but also overcome the adverse impact of asynchronous switching on the system performance. Finally, the theoretical results are verified by two simulation examples. In the future, we will study the set-time PPC design of MIMO systems, large-scale systems and multi-agent systems.