Adaptive Constrained Control for Uncertain Nonlinear Time-Delay System with Application to Unmanned Helicopter

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.


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 finitetime 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 timedelay 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 2 Mathematical Problems in Engineering [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 (multipleinput 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 performancebased 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.
Lemma 4 (see [9]).Consider a class of nonlinear systems ẋ = ().For any initial conditions (0) ∈ In addition, the optimal approximator of continuous function () can be written as where  * indicates the optimal approximate error.

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.

Prescribed Performance Controller Design
Step 1. Define the tracking errors z1 () ∈   and  2 () ∈   as follows: where   () ∈   is the ideal tracking signal and  * 2 () ∈   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  ∞ > 0 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 In order to simplify the analysis, we define  1 and  1 as follows: Furthermore, we can define  1 () = diag{ 11 (), . . .,  1 ()} and  1 () = [ 11 (), . . .,  1 ()]  , then we have Since Δ 1 ( 1 ()) is unknown, using the RBFNN to approximate it, we obtain where  1 () = [ 1,1 (), . . .,  1, ()]  , and Mathematical Problems in Engineering Substituting (10) into (1) results in Invoking ( 4), (9), and (10), the time derivative of  1 () can be rewritten as Construct updating law of the RBFNN where  1 > 0 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  1 () = diag{ 11 (), . . .,  1 ()}.ẏ  () is the time derivative of reference trajectory, and  1 ∈  × is the constant positive definite matrices.Ŵ1 is the estimated values of  * 1 , W1 represents the estimated error, and W1 =  * 1 − Ŵ1 .Substituting ( 17) into (13), the time derivative of  1 () becomes Choose the Lyapunov-Krasovskii functional candidate as where the positive function  1 ( 1 ()) can be designed as follows: Then the time derivative of  1 is Substituting ( 14) into ( 21), we obtain Substituting ( 16) into ( 22) yields Step i. Define the error variables   () ∈   and  +1 () ∈ where  *  () is a virtual control law.Combining (1) and ( 24) and differentiating   () with respect to time, we have where   () = [ ,1 (), . . .,  , ()]  , and Substituting ( 26) into (1) yields Moreover, substituting ( 26) into (25), we have Construct updating law of the RBFNN where   > 0 is a design parameter.Furthermore, the DOB can be chosen as According to (31), we obtain Hence, the virtual control law  * +1 () is proposed as Choose the Lyapunov-Krasovskii functional candidate as (36) Step n.Define the error variable   () ∈   : Invoking ( 1) and (37), differentiating   () with respect to time yields Construct updating law of the RBFNN where   ∈ ,   > 0, is a design parameter.Furthermore, the DOB can be chosen as According to (44), we obtain Therefore, design the control law () as where   ∈  × ,   > 0 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 1 and Step i, we have From the above inductive design procedure, moreover, we can conclude the following theorem.
where   can be made as small as desired by appropriately choosing design parameters. min represents the minimum eigenvalue of the matrix. min 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 Mathematical Problems in Engineering and Then we have the following results: In addition, it is clear that there exists ‖  ‖ ≤   ,   > 0; we have the following facts: where   > 0 and   > 0 are the constants, and then we have Invoking Lemma 5, we obtain > 0,  > 0 are defined in (50).Therefore, according to Lemma 4, we can conclude that the solution of the closedloop system remains within a compact subset.

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] 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 Ω * () = [ *  (),  *  (),  *  ()]  .Using the neural network to compensate the uncertainties and time-delay terms, such that Δ Ω (Θ(), Ω()) +  Ω () =  −1 Ω  *  Ω  Ω +  * Ω , assume that  * Ω is the optimal approximate error, and define  Ω () =  * Ω + Ω ().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.
From the curves in Figures 1-3, it can be seen that the tracking trajectories soon reach the target trajectories, and   the tracking errors are bounded in the appointed region.Figures 4-6 show the attitude angular velocities.Figures 7-9 show the control input signals.From the numerical simulation, we can conclude that the proposed control approach is valid for a class of uncertain unmanned helicopter systems with unsteady aerodynamics.Furthermore, it can be illustrated that the closed-loop system output signals are asymptotically tracking the ideal trajectories, and they are restricted in the region of output constraints.

Conclusion
In this paper, an adaptive prescribed performance control procedure has been proposed for a class of nonlinear timedelay systems with uncertainties, external disturbances, and 10 Mathematical Problems in Engineering    output constraints.Additionally, the robust controller has been applied to unmanned helicopter systems with unsteady aerodynamics.At last, the simulation illustrates that the proposed control approach is valid for the uncertain constrained time-delay system.