#### Abstract

This paper investigates a class of nonlinear time-delayed systems with output prescribed performance constraint. The neural network and DOB (disturbance observer) are designed to tackle the uncertainties and external disturbance, and prescribed performance function is constructed for the output prescribed performance constrained problem. Then the robust controller is designed by using adaptive backstepping method, and the stability analysis is considered by using Lyapunov-Krasovskii. Furthermore, the proposed method is employed into the unmanned helicopter system with time-delay aerodynamic uncertainty. Finally, the simulation results illustrate that the proposed robust prescribed performance control system achieved a good control performance.

#### 1. Introduction

Time-delay systems have drawn considerable attention in the past decade [1, 2]. The adaptive backstepping technology was employed into the uncertain nonlinear time-delay system in [3]. The dynamic surface method was presented for the nonlinear time-delay system in [4]. In [5], the nonlinear stochastic system with time delay was studied. The finite-time control method was proposed for a class of time-delay systems in [6, 7]. In the previous studies on time-delay system, the uncertain nonlinear systems consisting of both constraint and external disturbances were not considered. In this paper, we will study a class of uncertain nonlinear time-delay systems subject to constraint.

It is well known that the uncertainty and external disturbance have an effect on the tracking performance of closed systems. Neural network is popular for its ability to cope with uncertainty [8]. In [9], the neural network was introduced into a class of nonlinear systems with unknown coefficient matrices. Combining RBFNN (radial basis function neural network) and disturbance observer, fault tolerant control method was presented to deal with input saturated system with actuator faults in [10]. Moreover, the disturbance observer is a valid method to deal with external disturbance [11]. In [12], the disturbance observer was proposed for permanent-magnet synchronous motor drivers. In [13], the sliding mode disturbance observer was presented to deal with mismatched disturbance. In [14], the disturbance observer was employed into a transport aircraft control system subject to continuous heavy cargo airdrop. In this paper, the neural network and disturbance observer will be utilized to tackle uncertainties, time delay, and external disturbance.

Another challenging problem in controller design lies in the constrained condition of the nonlinear systems [15]. The existence of constraint condition may degrade the performance or cause the instability of the closed control systems [16]. Using the Barrier Lyapunov function and adaptive backstepping technology, a robust constrained controller for a class of nonlinear strict systems was presented in [17]. In [18], the Barrier Lyapunov function was employed into the switched systems subject to output constraints. In [19], the Barrier Lyapunov function and high-gain observer were introduced to deal with the constrained trajectory tracking problem of the marine surface vessel. Additionally, prescribed performance is another method to cope with output constraints, by defining the appropriate prescribed performance. In [20], the prescribed performance-based feedback linearization method was proposed to deal with output tracking error constraints for the MIMO (multiple-input multiple-output) nonlinear systems. In [21], the prescribed performance and adaptive fuzzy logic were employed into the nonlinear adaptive controller design. To the best of the authors’ knowledge, there is still no research about uncertain nonlinear time-delay system considering uncertainties, external disturbance, and output constraints. Thus, in this paper, we will present a prescribed performance-based adaptive constrained control method for the time-delay nonlinear systems.

Nowadays, the unmanned helicopter system has received an increasing attention, and there is an amount of studies about the flight control approaches [22, 23]. In [24], chattering-free sliding mode was proposed for the miniature helicopter system. To solve the tracking problem with nonlinearity, the model predictive control method for unmanned helicopters was presented in [25]. In [26], a trajectory tracking control method was proposed for unmanned helicopter system with constraint conditions. However, with the increasing demands for real time and accuracy, the aerodynamic disturbance caused by transmission delay for unmanned helicopter control system cannot be ignored. In this paper, we will apply the prescribed performance-based robust adaptive control approach for the uncertain unmanned helicopter systems with external disturbance, time delay, and output constraints.

This paper is organized as follows. In Section 2, problem statement and preliminaries of time-delay system and prescribed performance are introduced. Section 3 presents the entire adaptive controller design and stability analysis. In Section 4, the prescribed performance-based control method is employed into the unmanned helicopter system. Finally, simulation and conclusion are given in Sections 5 and 6, respectively.

#### 2. Problem Statement and Preliminaries

In this subsection, we will review some preliminary knowledge about nonlinear time-delay system, prescribed performance, and neural network, which are necessary in the following controller design. Firstly, consider a class of uncertain MIMO nonlinear time-delay systems in the form of where , , are the system state vectors which are assumed to be measurable, , is the output vector, and is the control input vector. , , are the known smooth nonlinear function, , , represent the known control coefficient matrices, and , , indicate the known time-delay functions. , , denote the unknown nonlinear functions which contain both parametric and nonparametric uncertainties. , , stand for constant unknown time delays. , , mean the unknown external disturbance. In this paper, we impose the following assumptions and lemmas.

*Assumption 1 (see [5]). *The ideal tracking signals and their derivatives are known and continuous.

*Assumption 2 (see [11]). *The external disturbance means the slow varying signal, while it is restricted in the bound of , .

*Assumption 3 (see [10]). *For , the known continuous function satisfies .

Lemma 4 (see [9]). *Consider a class of nonlinear systems . For any initial conditions , if there exists a continuous and positive definite Lyapunov function satisfying , , , mean the K class functions, such that , , , then one can conclude that the solution is uniformly bounded.*

Lemma 5 (see [3]). *For , if there exists one set defined by , then one can conclude that, for any , the inequality is satisfied.*

Lemma 6 (see [10]). *For any continuous function , there exists a valid linear neural network to approximate it by choosing enough nodes on the compact set . The basic function can be selected as Gaussian function , where is the width of function and represent the center of function. The neural network approximator consists of the weight estimate vector and Gaussian function , which can be written as . Assume that means the optimal approximating weight, such that In addition, the optimal approximator of continuous function can be written as where indicates the optimal approximate error.*

#### 3. Controller Design and Stability Analysis

In this section, the objective is to propose a robust prescribed performance control law for uncertain nonlinear systems such that the closed-loop errors converge to a small neighborhood of the origin.

##### 3.1. Prescribed Performance Controller Design

*Step 1. *Define the tracking errors and as follows: where is the ideal tracking signal and is the immediate control. The output error transformation can be defined in the form of [18] where and are the positive constants and indicates the performance function, which can be chosen as [26] The constant is the maximum amplitude of the tracking error at the steady state. The decreasing rate of represents the desired convergence speed of the tracking error. Therefore, the appropriate choice of the performance function and the design constant imposes bounds on the system output trajectory.

Define and . Thus, the time derivative of becomes In order to simplify the analysis, we define and as follows: Furthermore, we can define and , then we have Since is unknown, using the RBFNN to approximate it, we obtain where , and Substituting (10) into (1) results inInvoking (4), (9), and (10), the time derivative of can be rewritten as Construct updating law of the RBFNN where is a design parameter. Furthermore, the DOB can be chosen as According to (15), we obtain According to the neural network updating law (14) and DOB (15), the immediate control is chosen where . is the time derivative of reference trajectory, and is the constant positive definite matrices. is the estimated values of , represents the estimated error, and . Substituting (17) into (13), the time derivative of becomes Choose the Lyapunov-Krasovskii functional candidate as where the positive function can be designed as follows: Then the time derivative of is Substituting (14) into (21), we obtain Substituting (16) into (22) yields

*Step i*. Define the error variables and where is a virtual control law. Combining (1) and (24) and differentiating with respect to time, we have Since is unknown, we use RBFNN to approximate uncertain function. indicates the estimate of , and is the approximate error. represents the estimated error, which is defined as . Define the variable . where , and Substituting (26) into (1) yields Moreover, substituting (26) into (25), we have Construct updating law of the RBFNN where is a design parameter. Furthermore, the DOB can be chosen as According to (31), we obtain Hence, the virtual control law is proposed as Choose the Lyapunov-Krasovskii functional candidate as where the positive function is given in the form of Similar to Step , the time derivative of can be rewritten as

*Step n*. Define the error variable : Invoking (1) and (37), differentiating with respect to time yields Since is unknown, the RBFNN is used to approximate . indicates the estimate of , and is the approximate error. represents the estimated error, which is defined as . Define the variable . where , and Substituting (39) into (1) yields Invoking (38) and (39), one obtains Construct updating law of the RBFNN where , , is a design parameter. Furthermore, the DOB can be chosen as According to (44), we obtain Therefore, design the control law as where , is design matrix. Choose the Lyapunov-Krasovskii functional candidate as and the positive function can be designed as follows: According to the derivatives in Step and Step i, we have From the above inductive design procedure, moreover, we can conclude the following theorem.

##### 3.2. Stability Analysis

Theorem 7. *Considering the system dynamics described by (1), under the adaptive neural updated laws (14), (30), and (43), the disturbance observers (15), (31), and (45), and the prescribed performance-based control laws (17), (33), and (46), we can conclude that the trajectories of the closed-loop system are semiglobally uniformly bounded, while the tracking error converges to a compact set asymptotically, where and are defined in (50).where can be made as small as desired by appropriately choosing design parameters. represents the minimum eigenvalue of the matrix. represents the minimum eigenvalue of the matrix.*

*Proof. *For analytical purposes, define the total Lyapunov function where definition of can be referred to (19), (34), and (47). According to (23), (36), and (49), we obtain Considering the fact that and , we have Then we have the following results: In addition, it is clear that there exists , ; we have the following facts: where and are the constants, and then we have Invoking Lemma 5, we obtain are defined in (50). Therefore, according to Lemma 4, we can conclude that the solution of the closed-loop system remains within a compact subset.

#### 4. Application to Unmanned Helicopter System

In this section, we will apply the proposed robust adaptive control strategy to solve the problem of attitude tracking for a class of uncertain unmanned helicopter systems.

The unmanned helicopters rigid-body dynamics consist of two parts, attitude angular dynamics and flapping dynamics. The unsteady aerodynamics bring out the time-delay nonlinear uncertainty; thus in this section we consider the attitude control for attitude control subject to time delay. Firstly, the attitude angular and angular velocity dynamics can be described as [26] where indicates the attitude angle and represents the attitude angle velocity. stands for the disturbance, and stands for the uncertainties; means the unknown nonlinear block of time delay. , denotes and denote In the unmanned helicopter attitude control loop, the main rotor thrusts are fixed. indicates the location of the main rotor relative to the center of gravity; indicates the location of the tail rotor relative to the center of gravity. means the stiffness of the rotor hub. and are constants, which are associated with the antitorque. The control design procedure can be presented as follows.

*Step 1. *Define the attitude angle tracking errors in the form of , , , and , where , , and are the reference trajectories and . Then the time derivative of is achieved as According to the analysis in the previous section, the prescribed performance function is chosen as and the definition of can be referred to in (6). Choose the virtual control signal as Choose the Lyapunov function:

*Step 2. *Define the tracking errors , , , and . Using the neural network to compensate the uncertainties and time-delay terms, such that , assume that is the optimal approximate error, and define . Design the neural updated law as The DOB can be constructed in the form of Based on the above design, the adaptive control law is achieved as Choose the Lyapunov function: Similar to Theorem 7, we can conclude the following theorem.

Theorem 8. *For the unmanned helicopter system (59), combining the DOB (67) and neural network update law (66), the control procedure can be achieved as (64) and (68); thus the closed-loop signals are bounded, and the output trajectories are satisfying the prescribed performance conditions.**For Theorem 8, the detailed proof can be referred to in Theorem 7 in the previous section.*

#### 5. Simulation Results

In this section, the numerical simulation about unmanned helicopter system is utilized for illustrating the effectiveness of the previous control method in this paper. Consider the unmanned helicopter system with the following parameters (Table 1) [24, 26].

Now let us look at the design of adaptive controller design, the DOB design parameters are chosen as . The neural update law has been given by 70, and its parameters are chosen as , ,