#### Abstract

This paper investigates a scheme of adaptive neural network control for a stochastic switched system with input saturation. The unknown smooth nonlinear functions are approximated directly by neural networks. A modified approach is proposed to deal with unknown functions with nonstrict feedback form in the design process. Furthermore, by combining the auxiliary design signal and the adaptive backstepping design, a valid adaptive neural tracking controller design algorithm is presented such that all the signals of the switched closed-loop system are in probability semiglobally, uniformly, and ultimately bounded, and the tracking error eventually converges to a small neighborhood of the origin in probability. In the end, the effectiveness of the proposed method is verified by a simulation example.

#### 1. Introduction

Switched systems as a class of hybrid systems is made up of a race of continuous or discrete subsystems and switching rules [1–4]. In the last decades, switched systems have attracted more and more attention due to their significance in the modeling of many engineering applications, such as chemical processes, robot manipulators, and power systems. So far, many remarkable achievements about the stability analysis and synthesis have been made in the field of switched systems. In general, there are two ways of dealing with it: one is to find a common Lyapunov function to ensure stability of the switched systems under arbitrary switching laws; the other is to use a multiple Lyapunov functional technique to stabilize the switched systems under some designed switching laws, see, e.g., [5–10] and the references therein.

Almost all feedback controls are involved in saturation, due to the limitation of maximum and minimum values, and the sensors of actuators are saturated. If the controller is designed without considering input saturation, it can degrade the performance of the control system and even lead to instability. In recent years, the problem of input saturation limitation has attracted wide attention, and many methods for designing controllers have been proposed. Among them, PID controller was an earlier antisaturation method [11]. Workman designed a time optimal controller to reduce input saturation. Later, scholars proposed a saturation compensator, for example, the CNF controller was designed by Hu et al. [12], Grimm et al. [13], and Mulder et al. Mulder et al. [14] gave antisaturation compensation controllers by using linear matrix inequalities. However, there are still many control questions about switched systems with input saturation to be studied in depth.

According to recent research, the switching strategy plays an important role in the switching process. Due to the switching effect, the originally linear characteristics will become nonlinear. In addition, the features of multiple subsystems vary greatly and interact with each other and further increase the complexity and make the whole system show complex dynamic behavior. Therefore, how to choose the switching strategy and ensure the stability of the whole system is a hot topic at present. The adaptive backstepping technology can effectively solve the stabilization and tracking control [15–33]. Such as in [26], an output feedback scheme based on discrete adaptive approximation was proposed for uncertain switched stochastic interconnected systems, and the unknown nonlinear function was modeled by combining a fuzzy logic system. For stochastic systems with output constraints, an adaptive neural controller based on approximation was constructed [27]. Niu et al. [28] investigate the global adaptive control problem for a class of switched uncertain nonlinear systems. By improving mode-dependent average dwell time (MDADT) method, a new adaptive control scheme is established which ensures the global bounded. It should be noted that an adaptive tracking controller based on neural networks was considered for the single input switching system of lower-triangular structure in [29, 34]. As for the switching systems, because it contains multiple subsystems, its motion pattern is richer and more complex than the previous single model. In all dynamic behaviors, there may be good performance, bad performance, or even unstable dynamics. Therefore, on the premise of ensuring the stability of the system, this work attempts to design an appropriate control strategy to make the system achieve better switching effect and optimized performance index.

Moreover, as it is known to all, neural networks (NNs) have strong approximation ability, which can be used to deal with the unknown nonlinear functions, and it can be discovered in many literatures [35–42]. Recently, the authors in [36] gave an adaptive scheme using multilayer neural networks. State feedback strategy was firstly investigated for nonaffine systems through neural network approximation [39]. Neural network output feedback controller has been designed in [42] for a large-scale stochastic systems. For nonstrict feedback systems, a dynamic surface method based on observer was established in [43]. However, the above control results were obtained under a conservative condition, which the considered system structures are always in strict feedback form. The drift functions may be a mixture of all state variables, which makes the systems be in the nonstrict feedback. Therefore, we aim to propose an adaptive neural networks scheme for a switched system with nonstrict feedback form. Different from the relevant literatures, we construct a new scheme of stochastic systems with input saturation and design switching strategies to resist the saturation.

Based on the above research progress, this work tries to establish a neural network framework for a stochastic switched system with input saturation. Compared with the existing literatures, the innovation points of this paper are stated as follows:(i)Applying neural networks technique, a modified backstepping algorithm for a stochastic switched system in lower-triangular is established. A novel controller is constructed to track the unknown states.(ii)It needs to adopt the It formula for stochastic system, which makes it difficult to structure Lyapunov function corresponding to gradient term and high-order Hessen term. To solve this problem, we successfully developed neural networks to stochastic switching situation, which reduces the conservativeness caused by feedback form when the input saturation happens.

The other parts of this article are organized as follows. In Section 2, we introduce some assumptions and preliminaries and describe problem statements. Furthermore, we give the neural control process and stability analysis in this section. The simulation results are shown in Section 3, and the conclusions are analyzed in Section 4.

#### 2. Preliminaries and Controller Design

In this section, the switched system and its descriptions are formulated in Section 2.1, and some primary lemmas and definitions including stability theorem and Young’s inequality are also described. In Section 2.2, we construct the adaptive neural controller and state the stability analysis.

##### 2.1. Problem Descriptions

Now, let us first present some lemmas, definitions, and assumptions, and they will be used in the subsequent developments. Consider a stochastic switched system with nonstrict feedback form:where , and and are the state and the output of the system, respectively. Let random function be smooth, and the switching signal is denoted by . For , assume that and are assumed to be unknown and locally Lipschitz functions. The unknown system functions , and are contained in the function and satisfy and , which are smooth and . The variate is an independent -dimensional standard Wiener process. It is important to note that there is a regular convention for switching systems, solutions are continuous everywhere, and the states do not jump at the switching moment. Let be the controller with input saturation, which can be written as

Lemma 1 (see [43]). *For each pair , Young’s inequality holdswhere .*

*Assumption 1. *The constants satisfy , . Consider the following stochastic system:where is the system state, is a standard Wiener process, and and are locally Lipschitz functions and satisfy .

*Definition 1 (see [44]). *For any given , which is associated with the stochastic system (4), the infinitesimal generator is defined as follows:where is the trace of a matrix .

*Definition 2. *The trajectory of system (4) is said to be semiglobally uniformly ultimately bounded in th moment if, for some compact set and any initial state , there exist constant and time constant , such that for all , . In particular, when , it is usually called semiglobally, uniformly, and ultimately bounded in mean square.

Lemma 2 (see [45]). *Consider the stochastic system (4). If there exist functions , and and constants and , such thatthen, there is a unique solution of system (4) for each , and it holds*

In this paper, approximation-based NNs will be used to approximate the unknown nonlinear functions. For any continuous unknown smooth nonlinear function over a compact set , there exists NNs , such that, for a desired level of accuracy ,where is the ideal constant weight vector and defined by is the approximation error, is the weight vector, and is the radial basis function vector with being the number of the NNs nodes and :where is the center of the receptive field and is the width of the Gaussian function.

##### 2.2. Adaptive Controller Design and Stability Analysis

A neural controller will be constructed for the stochastic switched system (1). At the same time, the adaptive laws will be given in this section.

At first, let us define some essential functions:where is the first virtual control signal and is an adaptive law with , and being positive design parameters.

Similarly, for , we design the virtual signals and adaptive laws as follows:where is the th virtual control signal and is the adaptive law with , and being positive design parameters. The actual control input and the adaptive law are defined as follows:where , , and being positive design parameters. is an auxiliary design signal, which will be defined later. The coordinate transformations are as follows:where . Combining (1) with coordinate transformations (17) and (18), the following system can be obtained:wherewith for .

Theorem 1. *Let Assumption 1 hold, the closed-loop structure be consisted of the virtual control signals in (11) and (13), the adaptive laws (12) and (14) together with (16) and the actual control signal (15), which are designed from the stochastic switched system (1). All signals are bounded with some suitable parameters and the tracking error enters inside the area for all :where ,and , , and .*

*Proof. *Step 1: choose the following Lyapunov function:where are design parameters, and with being the estimation of . In the light of the above equations in (4), (5), and (19), it yieldsEmploying Young’s inequality (3), we can obtainwhere is a positive constant. The NNs can estimate the following unknown function:According to (17), (25), and (27), formula (24) can be rewritten asTherefore, the following result can be obtained:where , , and are positive constants, andSubstituting (29)–(31) into (28), we haveSubstituting the virtual controller in (11) and the adaptive law in (12) into (33) yieldsWe can get the following result by Young’s inequality:Then, one haswhere .

Step : similar to Step 1, choose the following Lyapunov function:where are positive constants and is the estimation of . Then, we can obtain the following formula:wherewhere is a positive constant. By Young’s inequality, one hasThen, we apply NNs to replace the unknown function:Due to (17) and (39)–(42), we haveIn line with Young’s inequality (3), one can obtainwhere and are positive constants. Substituting (44)–(46) into (43), we can obtainwhere . Based on the designed virtual control signal in (13) and the adaptive law in (14), one can obtainSimilar to (23), we can derive the following conclusion:Moreover, substituting (50) into (49) implieswhere .

Step : the Lyapunov function is chosen as follows:where are positive constants and is the estimation of . From It differential formula, we can obtain the following result:We introduce the following dynamic system in order to simplify the controller design:Based on (5), one hasSimilar to (39), the following inequality can be written aswhere . Substituting (54)–(57) into (53), we obtainThe neural networks can estimate the unknown nonlinear function:Similarly, it is noted that the following formula holds:where and are positive constants, ,Based on (59)–(61), the equality of (58) can be rewritten asTaking formulas (15) and (16) into (63), it yieldswhereIn addition, substituting (65) into (64), we haveThen, (66) can be rewritten aswhere ,Based on (63), we can easily obtain the following inequality:Then, integrating inequality (69), the following formula can be derived:Thus, the inequality,holds for withThe proof is completed.

*Remark 1. *The controller in (54) contains neural networks structure, which will cause tedious calculation in practical application. Nevertheless, the inequality should also be emphasized, and it can be obtained from the theories in [46].

From Remark 1, the base functions in controller (54) can be conveniently deleted to further improve the design. Then, we give the following corollary.

Corollary 1. *For , we assume that the functions are, instead of by neural networks system, within bounded error . Consider the switching strategy among observer (11), the following are the intermediate virtual and the actual controllers:*

The adaptive laws denote as follows:

Then, the semiglobal boundedness of all signals in the closed-loop structure can be established under bounded initial values, and an arbitrary switching rate satisfies . Meanwhile, we can obtain the arbitrarily small output with some suitable parameters.

*Proof. *This proof is rather similar to Theorem 1; therefore, we omit the procedure here.

#### 3. Simulation Example

We will construct a numerical simulation example to verify the effectiveness of the proposed controllers. The following two stochastic switched second-order systems are listed as follows.

System (1) is defined aswhere the switched signal , and the nonlinear functions are

System (2) is defined aswhere the switched signal , and the nonlinear functions are

The input saturation is defined as

Furthermore, the reference signal is given as , and the designed parameters are chosen to be . The initial conditions are and .

The simulation results are illustrated in Figures 1–5, respectively. Figure 1 demonstrates control output and target. Figure 2 gives the trajectories of adaptive parameter and . Figure 3 shows the control input signal . The input signal is asymptotically stable with the change of time. Figure 4 shows the response curves of state variable , and it can be drawn that converge to zero after a few seconds. Figure 5 shows the response curves of the giving switching signal . From the above five figures, it can be seen that all signals in the closed-loop structure are bounded and the states can track the given reference signals. Based on the simulation results, we can conclude that the proposed control scheme is effective for the switched system.<