Adaptive Neural Control with Prespecified Tracking Accuracy for a Class of Switched Systems Subject to Input Delay
This paper is concerned with the adaptive tracking control design for a class of uncertain switched systems subject to input delay. Unlike the existing results on uncertain switched systems, the new proposed control scheme ensures that the tracking error converges to the accuracy given a priori according to the requirement. To achieve this aim, some nonnegative switching functions are introduced to replace the conventional Lyapunov function. In addition, neural networks are used to approximate the unknown simultaneous domination functions. By combining the backstepping technique and some common nonnegative switching functions, a stable adaptive neural controller is established. It can be shown that the closed-loop system is semiglobally uniformly ultimately bounded (SGUUB) and the tracking error satisfies the predefined accuracy. The effectiveness of the proposed control scheme is verified by a simulation example.
As we all know, switched systems are a special class of hybrid systems, which have been paid much attention due to their practical applications. Many practical engineering systems are well described by switched systems such as networked system, robotic system, and traffic surveillance and control system [1, 2]. It provides a strong motivation for investigating the switched systems. Over the past two decades, a lot of scholars focused on dealing with the control issues of the switched systems, and a number of interesting and meaningful results have been presented in [3–10]. In [3, 4], based on the multiple linear copositive Lyapunov function method, the stability problem for switched positive linear systems with the average dwell time switching is studied. By using the technique of adding a power integrator, Fu et al.  discuss the finite-time control problem, and an adaptive controller is constructed to achieve the globally finite-time stabilization of the switched nonlinear systems with the powers of positive odd rational numbers. It is worth pointing out that the mentioned references focus on dealing with the control problem for the switched systems without unknown system functions.
It is noted that in many practical industrial processes, not all the system functions can be accurately modeled. Thus, an important problem is to overcome the foregoing uncertainties. In recent years, some approaches have been proposed to overcome the restriction and a large amount of encouraging results have been reported [11–24]. When the considered systems are unknown, neural networks (NNs)/fuzzy logic systems are often used to approximate the uncertain function to construct a controller. Most works on adaptive NN control are based on backstepping technique in [15–24]. For example, a NN-based adaptive controller has been designed with backstepping method, and NN-based design methods warrant the semiglobal stability for the nonlinear systems in a strict-feedback form in . In , a backstepping controller containing a recurrent-neural-network-based uncertainty observer and a robust controller has been developed to address the position control problem for induction servomotors. By utilizing backstepping technique and NN appropriate method in [17, 18], the stability problem of a class of stochastic nonlinear strict-feedback systems has been explored. It is worth noting that the aforementioned results on adaptive NN control design only consider the uncertain nonswitched nonlinear systems. Recently, several works have reported on uncertain switched nonlinear systems [25–27]. For uncertain nonlinear multiagent systems, distributed finite-time control scheme and cooperative adaptive neural partial tracking errors constrained control have been addressed in  and , respectively.
In addition, input delay plays an increasing important role in the real engineering systems. It has been attracted much attention, and numerous results have been presented in [30–38]. For example, in , the augmented complete Lyapunov-Krasovskii functional method has been adopted to solve the stability issue for linear systems with input delay. Based on the stability conditions proposed in , a state transformation technique has been presented to tackle the stability problem for linear systems with input delays. In , an improved reciprocally convex approach has been proposed to achieve the asymptotic stabilization of the continuous-time systems with time-varying input delays. With respect to the switched systems with input delay, a few results have been reported in [37, 38].
Inspired by the above discussions, this paper addresses the adaptive neural tracking control problem for a class of uncertain switched systems subject to input delay. The main work of this paper can be summarized as follows. (i) By introducing some nonnegative switching functions, the desired adaptive neural controller is developed, and it is guaranteed that the tracking error can achieve the accuracy which can be adjusted based on the actual demands. (ii) The Pade approximation approach is borrowed to deal with the problem of input delay in the considered systems. (iii) In each backstepping design, NN is employed to approximate the simultaneous domination function rather than the switched system function. Thus, the number of adaptive learning parameters can be reduced.
This section introduces some preliminaries on the system stability and NN approximation theory.
Definition 1 ((SGUUB) ). For a general nonlinear system where is the system state, is a continuous vector-valued function, and and denote the initial time and the initial state vector, respectively. If there exists a compact set such that for all , there exists a and a number such that for all ; we say that the solution of this system is semiglobally uniformly ultimately bounded (SGUUB).
Lemma 2 ((Young’s inequality) ). For , the following inequality holds: where , , , and
The following switching functions are introduced and they will be used to design the desired adaptive neural controller. andwhere is the given accuracy and is a positive integer.
The main characteristics of functions and are given by the following lemma.
Lemma 3 (see ). Functions and have the following properties:
(1) and , where is the set of -th order continuously differentiable functions.
(2) For , the -th order derivatives of are (3) is a nonnegative function, and when and only when , .
(4) For ,
In addition, in this paper, some unknown continuous functions , , which will be defined later, are adopted to design the desired controller, and some RBF NNs are used to approximate these functions on a compact set , i.e.,where the input vectors , weight vectors , the NN node number , are the NN inherent approximation errors which are bounded over the compact sets, i.e., with unknown constants , and are known smooth vector functions with being chosen as the commonly used Gaussian functions, which have the form where are the centers of the receptive field and are the spreads of the Gaussian functions. The optimal weight vector is defined as where is the estimate of .
3. Problem Formulation, Control Design, and Stability Analysis
3.1. Problem Description
Consider a class of switched nonlinear systems described bywhere is the state vector. and denote the system output and the control input, respectively. represents a switching signal, and are unknown continuous functions for , . represents the delayed time.
To deal with the problem of input delay in system (55), the Pade approximation approach used in  is introduced. Subsequently, we can have where is the Laplace transform of and is Laplace variable. For further analysis, a new variable is proposed and it conforms to the following relation Then the following equation is obtained That is, we have the following equationwhere .
Remark 4. In this paper, Pade approximation method is introduced to deal with small delay. Since Pade approximation has some limitations in handling delay, the proposed scheme cannot work in large-delay case. Relaxed control design for systems with long delay and actuators saturation deserves further investigation.
Based on the above transformation, system (6) can be rewritten as followsThe control objective of this paper is addressed as follows: for a given reference signal , to develop an adaptive neural controller such that (i)all the closed-loop signals remain SGUUB;(ii)the tracking error as .
Remark 5. In fact, some control schemes have been proposed for uncertain systems to achieve the objective of ensuring that the tracking error converges to the accuracy given a priori according to the requirement; e.g., see [6, 40]. However, this paper studies the control problem for a class of uncertain switched systems with input delay. For such more general system, the existing control methods cannot be directly employed to solve this control issue. As far as the authors know, this is the first work to address the practical control for uncertain switched systems with input delay.
To design the desired adaptive neural controller, the following assumption is given.
Assumption 6. The reference signal and its derivative are continuous and bounded for .
3.2. Adaptive Neural Controller Design and Stability Analysis
To develop the desired control scheme for system (14), the following error variables are definedwhere , denote the virtual control variables and will be designed later.
Following the adaptive backstepping control method, we present the design process of the desired controller and the adaptation laws.
Step 1. Consider the error variable in (15) and the first subsystem in (14). Then, we haveSelect the following nonnegative functionand then along the trajectory of (16), by using Lemma 3, the time derivative of is given aswhere is a simultaneous domination function which is continuous and unknown. For example, here it can be selected as Since is unknown, it is obvious that is unknown and it cannot be used to design the controller directly. As shown in (6), an RBF NN is employed to approximate online, and then one can havewhere we define with
The Young’s inequality is used for the following analysis:where and is a design parameter, and
The first virtual controller is designed from the above information.where is a positive constant and and denote the estimates of and .
Obviously, (22) can be changed intowith estimate errors and .
For further study, we establish the nonnegative function as belowwhere and are design parameters. We choose the adaptation laws as follows:
Then we can obtain
Step . The method of recursion is used in each step when the value of increases from to . Similar to Step , the time derivative of iswhere with + + .
Consider the following nonnegative functionswhere and are design parameters, estimate errors and , and and denote the estimates of and which will be defined later.
Then we can obtain
Meanwhile the following inequality holds
Substituting (31) into (30), we gainwhere and are simultaneous domination functions which are continuous and unknown. For example, here they can be chosen as , . Define with , and an RBF NN (6) is used to approximate online. Based on this information, (32) can be changed as below
where , is a design parameter, and
We construct the -th virtual controller as follows:
Obviously, based on (37), the undermentioned inequality is true:
It is obvious that when . When , using the Young’s inequality, we obtainHence, holds all the time.
Based on the above information, we choose
Thus, we can have
Choose , , where and are design parameters, and design the actual controller as belowwhere and are design parameters, estimate errors and , and and denote the estimates of and which will be defined later.
Similar to the design process of Step , let . Then we can have and where is the upper bound of the RBF NN approximation error. The following adaptive laws are chosenwhere we define , , ∈
According to the above information, we have
Based on inequality (46), the main result of this paper is summarized by the following theorem.
Theorem 7. Under Assumption 6, consider the closed-loop system that comprises the plant (9), virtual control variables (23) and (37), and actual control law (44) with adaptive laws (26), (41), and (45). For the bounded initial condition on a compact set , assume there exist sufficiently large compact sets , , such that for all ; the following statements hold. (i)The tracking error as .(ii)All the closed-loop signals remain semiglobal bounded.
Proof. (i) From (46), by employing Barbalat’s Lemma , we can easily conclude that the tracking error satisfies as .
(ii) Considering inequality (46), we can see that is nonincreasing. Therefore, it can be concluded that, for all , error signals including and are bounded. To prove the boundedness of all the closed-loop signals, it can be shown that and are bounded according to the boundedness of and . Next, it follows from the boundedness of and that is also bounded. Then, the boundedness of given in (23) can be easily obtained. The boundedness of and further implies that is bounded. Continuing in the same fashion, we conclude that all the closed-loop signals are bounded. In addition, similar to the existing works on adaptive neural network control, the proposed control scheme just guarantees that the closed-loop system is semiglobally stable since the initial condition of the controlled system must remain on a compact set .
Remark 8. To show the semiglobal stability of the controlled system, the following analysis is presented. Firstly, from , we have , where is the upper bound of reference signal . Thus, the boundedness of is obtained as . Furthermore, from the above analysis, it can be seen that and are bounded. Assume that and with positive constants and . Since , then we can obtain the following inequality for ,where , when , and basis function satisfies with a positive constant . From the above inequality, it can be seen that we may increase the size of the attraction region with increasing the gain .
By the same way, we can show that the signals are semiglobally bounded.
Remark 9. For system (9) without input delay, some adaptive NN/fuzzy control schemes have been reported; e.g., see [25–27, 42–44]. For the control problem considered in this paper, two differences are summarized as follows. First, for the controlled switching system, the input delay phenomenon is considered. In addition, all the existing control methods for the uncertain switched system just can guarantee the tracking error converging to a small compact set, and the accurate size cannot be determined. This paper addresses the adaptive tracking control issue with the known tracking accuracy.
4. Simulation Examples
Example 1. Consider the following uncertain switched systemwhere , , , , The reference signal is given as The tracking accuracy is assigned as , i.e., as
According to the design process of Section 3, the virtual controller and the actual controller are presented asandThe adaptive laws are designed as
The design parameters of the adaptive neural controllers and learning laws are chosen as , , , , and The input delay is In the simulation, two RBF NNs are employed to approximate uncertain functions in the control scheme. The first RBF vector contains 125 nodes with the centers evenly placed on and the width . The second RBF vector contains nodes with the centers evenly placed on and the width .
The initial conditions are given as , , and , . The simulation results are shown in Figures 1–4, and Figure 2 shows that the tracking error achieves the given accuracy after 20 seconds, which implies that the control objective is achieved by using the proposed control scheme.
To further show the superiority of the proposed control method, a comparison simulation is presented. Considering the control scheme developed in , we have the following controllers and parameters learning laws andwhere the definitions and values of , and can be found in . For the simulation, the parameters are chosen as and . The rest of the conditions remain the same as before. The simulation results are shown in Figures 5 and 6, from which it can be seen that by using the method proposed in , the boundedness of the tracking error can be ensured, but the accurate size cannot be determined. This drawback can be overcome by employing the control scheme established in this paper.