Robust Adaptive Control for a Class of Uncertain Nonlinear Systems with Time-Varying Delay

We present adaptive neural control design for a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form. Based on a neural controller, it is obtained by constructing a quadratic-type Lyapunov-Krasovskii functional, which efficiently avoids the controller singularity. The proposed control guarantees that all closed-loop signals remain bounded, while the output tracking error dynamics converge to a neighborhood of the desired trajectories. The simulation results demonstrate the effectiveness of the proposed control scheme.


Introduction
In the practical control process, control system is usually required to meet the stability and the corresponding performance index, which affects the system stability factors mainly including the uncertainties and time delays. On the study of the uncertain time-delay many scholars have achieved valuable fruits [1,2]. Paper [3] has analyzed and designed the optimal ∞ feedback controller by the LMI method. In recent decades, the delay nonlinear systems with neural network research have received extensive attention [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Paper [4] has solved the problem of chaotic synchronization phenomenon by the neural network method. In [5][6][7][8][9][10][11], the study of the nonlinear continuous system and discrete nonlinear system is based on adaptive neural network control. The tracking and stabilization problem of nonlinear systems has been studied by neural network backstepping method [12,13]. In [14], neural network control has been applied to a piece of triangle structure of multiple-input multipleoutput nonlinear time-delay system, in which a dynamic system neural network is used mainly for unknown function approximation and separation. In multiple input and multiple output nonlinear system, [15] presents a new adaptive neural network controller design method but does not consider with external disturbance and time-varying delay. In [18], the problem of the adaptive neural networks control for a class of nonlinear state-delay systems with unknown virtual control coefficients is considered. In [19], a control scheme combined with backstepping, radial basis function (RBF) neural networks and adaptive control are proposed for the stabilization of nonlinear system with input and state delay. This paper mainly aims at studying the simultaneous presence of uncertainties and time-varying delay MIMO nonlinear system. By defining the new quadratic Lyapunov-Krasovskii functionals, it has analyzed and designed the adaptive neural network controller by neural network approximation method in [15,16].

Description of the Problem
Let us consider the following block-triangular structure with the disturbance of nonlinear MIMO systems with timevarying delays: We make the following assumptions for the system (1).
In this paper, the following radial basis function neural network is used to approximate unknown continuous function (in [13] once had been put forward): where the input vector ∈ Ω ⊂ ; = [ 1 , 2 , . .

Adaptive Neural Network Controller Design
In this section, we will introduce a novel adaptive NN control design procedure. There are design steps in the design procedure for the th subsystem. In each step, the unknown nonlinear function , ( , ) will be approximated by a radial neural network approximation function. Define an unknown constant as where the constant 0 is defined as in Assumption 2; function , and vector , will be specified in each step. Furthermore, for = 1, 2, . . . , and = 1, 2, . . . , −1 , choose the virtual control laws as follows: where , > 0 and , > 0 are design parameters,̂represent the estimation of the unknown constant , and (⋅) is the basis function vector, and define the variables , as follows: for = 1, . . . , , = 2, . . . , . Choose the adaptive lawsȧ s follows:̇= where > 0 and > 0 are design parameters.
The control law design is thus completed.

Stability Analysis
Now, the main result in this paper can be presented as follows.
In a similar way, we can get Now, choose the Lyapunov function as = , . Combining (42)-(46) gives thaṫ is a constant. Thus, by (47) the boundedness follows immediately from the same line used in the proof in [9][10][11].

Conclusion
For a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form, an adaptive neural control design is presented. Although there are some fluctuations of the systems and control output under the influence of interference, the required performance can be achieved in a short period of time by using the controller designed in this paper and guarantees the boundedness of all the signals in the closed-loop system. It is further extended on the bases in [14,15], which makes it suitable for wider range of applications. The effectiveness of the proposed approach is provided by a simulation example.