Neural Network-based Event-triggered Adaptive Asymptotic Tracking Control for Switched Nonlinear Systems

In this paper, an adaptive event-triggered asymptotic tracking control problem is addressed for switched nonlinear systems with unknown control directions. In existing control schemes, the proposed controller is not directly aimed at the original system, which affects the control performance. Different from the existing control schemes, based on the original system, an event-triggered control law is constructed in this paper. The proposed event-triggered controller guarantees that the tracking error ς1 asymptotically converges to the origin. Finally, the effectiveness of the proposed controller design scheme is proved by simulation examples.


INTRODUCTION
Since the uncertain parameters often exist in practical systems [1], it is difficult to construct an accurate mathematical model to describe it. Subsequently, an adaptive backstepping design method was proposed to deal with the control problem of uncertain nonlinear systems [2,3]. After years of development, the adaptive backstepping design has become a powerful tool to construct controller for a class of nonlinear systems with uncertain parameters, such as uncertain stochastic nonlinear systems [4], nonlinear large-scale systems [5], and switched nonlinear systems [6,7], etc. However, these conclusions may be infeasible when the nonlinear functions of uncertain nonlinear systems are completely unknown. To address these concerns, the fuzzy logic system (FLS) [8][9][10][11][12][13] and neural network (NN) [14][15][16][17][18][19] were proposed. Specifically, in view of FLS and the adaptive backstepping design approach, an adaptive finite-time control issue for nonlinear systems with actuator faults was studied in [11]. By employing command filters control technology and adaptive backstepping technology, Li et al. [17] studied the robust adaptive NN tracking control problem for uncertain nonlinear time-delay systems.
For general nonlinear systems, the signs of the control gain are usually unknown. At the same time, the unknown signs of the control gain also increase the complexity and difficulty of designing the controller. Thus, in the last decades, the aforementioned problem has re-ceived increasing attention. The Nussbaum gain technology proposed by Nussbaum [20] paved the way to handle the problem of unknown control coefficients. Subsequently, the Nussbaum gain technology has been widely used [21][22][23]. Especially, in [21], an adaptive tracking controller was presented for a class of MIMO stochastic nonlinear systems with unknown control directions. According to Nussbaum gain technology and backstepping design, the adaptive control problem of nonlinear systems with unknown control directions was considered in [22]. An adaptive neural command filtered control strategy for nonlinear MIMO systems with unknown control directions was designed in [23].
It should be pointed out that the controller designed in the aforementioned papers [21][22][23][24] can only ensure that the tracking error is bounded as t → ∞. However, the controller design scheme proposed in the above papers may be infeasible in some cases where high precision tracking errors are required. To address the aforementioned problem, an asymptotic tracking control approach was proposed. Many results about asymptotic tracking control were obtained [25][26][27]. For example, in [25], an adaptive asymptotic tracking control approach was introduced for nonlinear systems with parameter uncertainty, unknown actuator nonlinearity, and bounded external disturbance. In [26], the problem of asymptotic tracking control for switched nonlinear systems was solved by designing a novel discontinuous controller.
In the aforementioned results, the control signals were continuously transmitted to the actuator regardless of whether the system needs it or not, which increased the communication burden and transmission costs. Therefore, to reduce the communication burden and save resources, some scholars proposed event-triggered control (ETC). Thus, considerable attention has been paid to ETC [28][29][30][31][32][33][34][35][36]. To list a few, by employing adaptive backstepping design technology and the event-triggered mechanism, an adaptive even-triggered controller is designed for uncertain nonlinear systems in [31]. However, as far as we know, there are few results about the event-triggered asymptotic tracking control for switched nonlinear systems with unknown control directions.
With the above observations, in this paper, a novel controller is proposed to handle the problem of NN-based event-triggered adaptive asymptotic tracking control for switched nonlinear systems with unknown control directions. The contributions of the developed controller are highlighted as 1) For the first time, a NN-based event-triggered adaptive asymptotic tracking control strategy for switched nonlinear systems with unknown control directions is proposed. Compared with [28][29][30][31][32][33][34][35], a novel control law is proposed to address the problem of switched nonlinear systems with unknown control directions.
2) The proposed controller in [31][32][33][34] can guarantee that the tracking error converges to an any small neighborhood near the origin. Compared with the existing results in [31][32][33][34], the system output y can asymptotically track the desired signal y d under the designed controller.
3) Although researchers [37,38] proposed several control schemes for switched nonlinear systems with unknown control directions, the controllers were not designed directly for the original system, which affected the performance of the controller. Different from [37,38], a novel asymptotic tracking controller is designed for switched nonlinear systems with unknown control directions.
The remainder of this paper is arranged as follows: The adaptive event-triggered asymptotic tracking control problem is stated in Section 2. The design method of eventtriggered controller is presented in Section 3. In Section 4, simulation studies are given to prove the validity of the presented approach. The conclusion is drawn in Section 5.

PRELIMINARIES AND PROBLEM STATEMENT
Consider the following system: where y ∈ R, u ∈ R andῡ i = [υ 1 , . . ., υ i ] T denote the system output, event-triggered input and states, respectively; σ (t) : [0, +∞) → W = {1, · · · , w} denotes the switching signal which is right continuous. f σ (t),i (ῡ i ) denote unknown smooth nonlinear functions. g σ (t),i (ῡ i ) = 0 are known smooth nonlinear functions. b i = 1 or b i = −1 denote the control directions, which are unknown. Moreover, σ (t) = p means the pth subsystem is active. In general, we assume that the state of the system (1) does not jump at the switching instants, i.e., the solution is everywhere continuous, which is a standard assumption in the switched system. The switched nonlinear systems (1) can represent a class of practical plants such as mass-spring-damper system [39] etc. Our goal is to design an event-triggered controller such that 1) the system output y can asymptotically track the reference signal y d ; 2) all signals of the closed-loop system are bounded.
To construct an event-triggered asymptotic tracking controller, the following Lemmas and Assumption are introduced.
Assumption 1: For t > 0, y d and its time derivatives up to the nth order are known and bounded.
To approximate the unknown nonlinear function h(Z), the following NN is employed.
T is the center of the receptive field and ξ i is the width of the Gaussian function. From [24], it can be concluded that with sufficiently large q, the NN can approximate any continuous function h(Z) over a compact set Ω to arbitrary any accuracy ε > 0 as where δ (Z) denotes the approximation error which satisfies |δ (Z)| ≤ ε, and ψ * = arg min{sup Z∈Ω |h(Z) − ψ T S(Z)|} is the unknown ideal constant weight vector.

Virtual controller design
To develop a backstepping-based design procedure, the following coordinate transformations are defined where ς i denote the error signals, α i−1 refer to virtual control signals.
Step 1: From (1) and (2), it can be concluded thaṫ Choose the Lyapunov function candidate as where γ 1 and r 1 are positive design parameters. The time derivative of V 1 is given bẏ where where |δ 1 (Z 1 )| ≤ τ 1 denotes the approximation error. By plugging (5) into (4), one haṡ For any ϑ 1 > 0, the following inequalities hold Combining (6)-(7) with (8) results iṅ By using Lemma 1, it has It follows from (9) and (10) thaṫ Choose the virtual controller α 1 as where c 1 > 0 is a design parameter. Construct the adaptive laws aṡ Substituting (12)- (14) into (11), one can geṫ Remark 1: Since all systems share a radially unbounded common Lyapunov function, then according to [40] Theorem 2.1, it can be concluded that the asymptotic tracking of the switched system is uniform with respect to σ . Namely, the different switching signals have no effect on the asymptotic tracking of the system.
Step i: 2 ≤ i ≤ n − 1: From (1) and (2), there holdṡ Choose the Lyapunov function candidate as The time derivative of V i is computed aṡ where As is the same case of (5), the following equation can be obtained where |δ i (Z i )| ≤ τ i denotes the approximation error. By (16) and (17) one haṡ Similar to (7) and (8), it yields It follows from (18), (19) and (20) thaṫ According to Lemma 1, one has By (21) and (22), there holdṡ Now, construct the virtual controller α i as where c i > 0 is a design parameter. Design the adaptive laws aṡ Based on (23)- (26), it yieldṡ
Remark 2: This paper focuses on an event-triggered asymptotic tracking control for switched nonlinear systems with unknown control directions. Since the control directions of the system are unknown, it is difficult to design the event-triggered tracking controller. In addition, the approximation error will occur when the NN is utilized to approximate the unknown nonlinear function. However, when we study the asymptotic tracking of uncertain switched nonlinear systems, it is necessary to solve this approximation error. How to eliminate the influence of approximation error for switched nonlinear systems is also very difficult. To address the above problems, an improved control law (30) and adaptive laws are proposed.
Theorem 1: If the switched system (1) satisfies Assumption 1 and Lemmas 1-3, then the proposed controller can guarantee that all signals of the closed-loop system are bounded and the tracking error ς 1 will converge to zero as t → ∞. Meanwhile, the Zeno phenomena can be avoided.
Using (45), it can be shown that From (46), it can be concluded that ς i , i = 1, · · · , n are square integrable, namely ς i ∈ L 2 , i = 1, · · · , n. By utilizing the boundedness of other signals, it can be seen that ς i ∈ L ∞ ,ς i ∈ L ∞ , i = 1, · · · , n. Therefore, in view of Lemma 4, we get lim t→∞ ς 1 = 0. By using ϒ(t) = τ(t) − u(t), one has It can be seen from (27) and (29) that τ is differentiable andτ is a function of all the bounded signals. Thus, there exists a positive constant ρ satisfying |τ| ≤ ρ. From ϒ(t k ) = 0 and lim t→t k+1 ϒ(t) = h 1 , the Zeno phenomenon is avoided when {t k+1 − t k } ≥ h 1 /ρ. Remark 3: Since all systems share a radially unbounded common Lyapunov function and a controller, Zeno behaviors under different switching laws are not affected. Thus, in this paper, we only need to consider that event-triggered does not lead to Zeno behaviors.
Remark 4: In this paper, to approximate the completely unknown nonlinear function, the NN is introduced. In addition, some other techniques also have good approximation capabilities such as the interval type-3 fuzzy logic systems [41], type-2 fuzzy control systems [42] etc. In future research work, we will focus on the advantages and disadvantages of each technique to approximate the unknown nonlinear functions.

SIMULATION EXAMPLES
In this part, a continuous stirred tank reactor [26] and numerical example are proposed to prove the feasibility of the designed controller.
Practical Example 1: Consider the following stirred tank reactor model where D A denotes the concentration of species A in the reactor, T and T c are the reactor temperature and variable temperature of coolant stream, respectively. The physical meanings can be found in [26].

CONCLUSION
In this paper, a novel event-triggered asymptotic tracking controller is proposed for switched nonlinear systems with unknown control directions. The Nussbaum functions are exploited to deal with the issue of nonlinear systems with unknown control directions. The NN is utilized to approximate the unknown nonlinear functions. Under our presented control scheme, it is proved that the system output y can asymptotically track the desired signal y d and the closed-loop system is stable. With the progress of science and technology, the large-scale systems, multi-agent systems [43] and fractional-order chaotic systems [44] have attracted extensive attention due to their unique advantages. Therefore, how to extend the suggested methods to large-scale systems, multi-agent systems and fractionalorder chaotic systems are a problem worthy of research.