Dynamic Output Feedback Control for Nonlinear Uncertain Systems with Multiple Time-Varying Delays

This paper addresses the adaptive dynamic output-feedback control problem for a class of nonlinear discrete-time systems with multiple time-varying delays. First, the guaranteed cost function is introduced for the nonlinear system to reduce the effect of the time-varying delays. Secondly, in order to deal with the multiple time-varying delays, the nonlinear system is decomposed into two subsystems. Then the compensator is designed for the first subsystem, and the adaptive dynamic output-feedback controller is constructed based on the subsystems. By introducing the new discrete Lyapunov-Krasovskii functional, it can be seen that the solutions of the resultant closed-loop system converge to an adjustable bounded region. Finally, the simulations are performed to show the effectiveness of the proposed methods.


Introduction
Many practical systems are the nonlinear systems and consist of time-delay, such as the urban traffic networks system, digital communication system, and the power systems [1][2][3]. As universal approximators, fuzzy systems or neural networks have been successfully applied to solve the control design problem for various kinds of such nonlinear systems, and many interesting results have been obtained see [4][5][6] and the references therein. It is noted that with those control schemes, the stability of the closed-loop systems can be guaranteed, and the tracking errors can be confined to a small residual set, while the size of the residual sets is often unknown, and the transient and/or steady state performance cannot be prescribed.
The Lyapunov-Krasovskii functional method and Lyapunov-Razumikhin method are always employed for the system design. In recent years, many adaptive fuzzy/neural output-feedback control approaches have been proposed for uncertain SISO/MIMO nonlinear systems with unmeasured states in [7,8]. Note that all the aforementioned adaptive fuzzy/neural output-feedback control schemes are for the nonlinear strict-feedback uncertain systems [9], instead of the nonlinear nonstrict-feedback ones [10,11]. In [12], the robust-control problem for the time-delay systems was considered, and the nonlinear uncertainties are bounded by the high-order polynomials. A tracking control system has a more general form than a general nonlinear system, and yet the system functions contain the whole state variables [13,14]. But how to apply this method into the nonlinear time-delay systems is a challenging subject [15]. By employing the neural network technique, the state feedback controller is designed for the time-delay nonlinear systems. Compared with the previous work, in the effort to develop new adaptive control strategies, adaptive neural back-stepping state feedback control methodologies for nonlinear systems were proposed in [18]. There are few results on dynamic output feedback control for nonlinear system with time delays It has been early recognized that the multiple time-varying characteristic deserve further research. In practice, time delay is one of the most important problems which usually appears in many industrial control systems [19]. For single nonlinear system, the time delay had been investigated in many references [20]. Recently, some static output-feedback control methods were proposed for the robotic system.
Based on the neural networks, a robust adaptive control method for a class of uncertain nonlinear systems in the presence of input saturation and external disturbance was proposed [21]. And an adaptive tracking control was designed for a class of uncertain multi-input and multi-output nonlinear systems with non-symmetric input constraints by using an auxiliary system [22]. Then, the input saturation problem was investigated for stochastic nonlinear systems [23]. In [24], the output feedback control problem for identical linear dynamic systems with input saturation was addressed. To our best knowledge, there are no results on output feedback control results for feedback nonlinear systems with time-delays and multiple subsystems.
In this paper, the dynamic output feedback control approach is proposed for the mobile robot system with multiple time-varying delays. The contributions of this paper can be summarized as follows: (1) The cost function is introduces to deal with the time-delays.
(2) The nonlinear system is decomposed into two subsystems based on the input matrix and output matrix. The dynamic compensator is developed for the first subsystem. And the dynamic output-feedback controller is designed based on the second subsystem.
(3) By introducing the new Lyapunov-Krasovskii functional, it can be seen that the solutions of the resultant closed-loop system converge to an adjustable bounded region. This paper is organized as follows. The preliminary knowledge for the nonlinear multiple time-varying delays system with parametric uncertainties are described in Section 2. The dynamic output-feedback controller for the nonlinear system is designed in Section 3. The results are further extended to the general nonlinear discrete-time case in Section 4. The simulation results are performed for a mobile robot case in Section 5. Finally, Section 6 concludes with a summary of the obtained results.
The rest of paper is organized as follows. Section 2 presents some preliminary knowledge for the dynamicsystem with. In Section 3, the dynamic output-feedback controller is presented. The proving process is shown in Section 4. The simulation results are presented in Section 5. Finally, Section 6 concludes with a summary of the obtained results

System Formulation and Preliminaries
Consider a nonlinear discrete system with multiple time-varying delays and parametric uncertainties as follows.  x x x α α ≥ . Remark 1. In this paper, the adaptive control theory was presented to estimate the system parameters or uncertain bound parameters. In [25], the adaptive control theory was discussed with some unknown interconnections. Since the research contains the nonlinear functions, it can be seen that the schemes are not enforceable to system (1). In this paper, there are three challenging problems as follows: how to design a dynamic compensator with the output feedback signal; how to reduce the influence of uncertain parameters; and how to design the adaptive output feedback controller for the multiple time-delays system. The aim of this paper is to solve the above issues, and then the controller will be easy to implement in practical systems.
With the above analysis, the system (1) is further rewritten as follows:

Controller Design
First, a cost function is introduced for the nonlinear system to deal with the uncertain parameters j A ∆ . Secondly, a dynamic compensator is designed for the subsystem- 1 x , and the adaptive output-feedback controller is constructed base on the second subsystem-2 y and the compensator.

Cost Function Design
In this study, the uncertainties are norm-bounded, described as follows: [ ] 11 12 Consider the cost function for the system (1) as follows 1 2 0 Now, in order to improve the stability of system (3), one has where ˆ( ) x k is the state vector. Then applying (6) to the system (3), the new closed-loop system is obtained as follows 2 0 For the closed-loop system (7), design the cost function as follows With the cost function (5), the Assumption 2 is imposed on system (3): Assumption 2. Consider the nonlinear system (3) and cost function (5), the initial states of system (3) are given. If there exist the equation (6) and a positive scalar J * such that J J * ≤ , then the closed-loop system is stable.

Compensator Design
For subsystem-1 ( 1) x k + , the augmented dynamic system is designed as follows: For equation (10), choose a discrete Lyapunov-Krasovskii functional as follows: where , j P R , and j Z are the positive matrices, c and σ are the positive scalars. Now taking the forward difference of (11), one has: Note that: where T ij is a weight matrix. With equations (10), (12) and (13), one has: where a and ε are the positive scalars, . G is defined as follows: ( ) ( )( ) ( )( ) With the above analysis, the new results arise: Theorem 1. ∀ positive scalar ε , there exist the positive matrices , , , T  (14), taking the forward difference of 1 V along (10), one has Remark 2. The matrix T il is employed to derive the conservative conditions. The use details of T il is shown in [28,29]. The compensator ( n m − order) is designed for subsystem-1 x . The matrices , , , By direct verification, one can obtain ( ) With the above analysis and the compensator (9), the adaptive dynamic output-feedback controller will be designed in next section.

Adaptive Dynamic Output-Feedback Controller Design
3 ) With (4), one has: where ( ( )) k δ Φ is defined as follows: ( ) Proof. Choose the discrete Lyapunov-Krasovskii functional (29) for the system (4) as follows: where ( ) i W is an increasing positive function, such that: ∫ is employed to deal with the nonlinear function ( ( )) k δ Φ . Then, taking the forward difference of V yields With (30), one has: , one has: , combining (11), (19), (29) and (33), one has:  With the above analysis, the system state ( ) . Then the proof is completed.

Case Expansion
Consider a nonlinear discrete system with multiple time-varying delays as follows: x z y such that: ( 1) ( , , , , , , , For the system (37), design the dynamic compensator as follows:  The adaptive law is designed as then the solutions of the close-loop system converge to a bounded region. Proof. Consider the Lyapunov-Krasovskii functional as follows: in which c is a scalar,

Simulation Example
Consider the nonlinear discrete multiple time-delays mobile robot system as follows:

C R × ∈
are the known matrices.
The state responses are shown in Figures 1 and 2. The control input is shown in Figure 3. From the three figures, it can be seen that the proposed method is effective and can stabilize the mobile robot system quickly.

Conclusions
This paper addressed the dynamic output feedback control problem for a class of nonlinear system with multiple time-varying delay and parametric uncertainties. The nonlinear uncertainties are in the nonlinear form and bounded by nonlinear functions with gains unknown. The dynamic compensator is designed and the control design condition is relaxed. The dynamic output feedback controller is constructed such that the solutions of the closed-loop system converge to an adjustable bounded region. The result is further extended to the general nonlinear case. Finally, the simulations for the mobile robot are performed and the results demonstrate the effectiveness of the proposed method.