Observer-Based Bounded Control for Discrete Time-Delay Uncertain Nonlinear Systems

Abstract

A bounded controller is proposed for a class of uncertain discrete time-delay systems with nonlinearity and disturbance based on state estimator and disturbance observer technique. A state estimator is developed to estimate the unmeasured system state vector. Suppose that the disturbance is generated by an exogenous system; a disturbance observer is designed to estimate the unknown disturbance. The parameters of the state estimator and the disturbance observer are calculated by solving linear matrix inequalities (LMIs). By applying the outputs of the state estimator and the disturbance observer, the sufficient condition for the existence of the bounded controller is derived based on an appropriate Lyapunov function candidate. Under the developed bounded controller, the stability of the closed-loop system can be guaranteed. Simulation examples are provided to show the effectiveness of the proposed bounded control scheme.

Bei Wu 1, 2 and Mou Chen 1, 3 and Xiaoming Chen 1, 3

Academic Editor:Chenguang Yang

1, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China
2, Nanhang Jincheng College, Nanjing, Jiangsu 211156, China
3, Jiangsu Key Laboratory of Internet of Things and Control Technologies, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China

Received 1 September 2015; Accepted 21 October 2015; 16 November 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Time-delay phenomenon widely exists in most of industrial systems, such as biological systems, hydraulic systems, transmission systems, electrical networks, and chemical systems [1-5]. The existence of time-delay leads to performance degradation or even system instability and causes the complication to the analysis and design of the controller. Up to now, many control schemes have been studied for time-delay systems, such as guaranteed cost control [1], [figure omitted; refer to PDF] control [2, 4], adaptive neural network control [5, 6], sliding mode control [7, 8], and disturbance-observer-based control [9-14]. However, most of the above-mentioned research results are dealing with continuous-time systems. Due to the fast development of computers and digital signal processor (DSP) chips, considerable attention has been paid to the study of discrete time-delay systems [15, 16]. The stability control problem of discrete time-delay systems can be divided into delay-dependent stability control [17-19] and delay-independent stability control [20]. Since delay-dependent conditions are less conservative than delay-independent conditions, more concentrations have been given on the delay dependent stability analysis.

Usually, the model errors, the measurement errors, and the external disturbances inevitably exist in dynamic systems, which can also further cause the degradation of system performance and even instability. The problem of robust stability analysis and robust controller design has been extensively studied for discrete-time systems with uncertainty and/or disturbance. A novel adaptive-critic-based neural network (NN) controller was proposed for nonlinear single-input-single-output (SISO) discrete-time systems in [21]. The novel adaptive control technique was proposed for discrete-time multi-input-multi-output (MIMO) systems in [22]. The improved dynamic surface control design was studied for constrained hypersonic flight models in [23]. In [24], based on the key ideas of "future outputs prediction" and "nearest-neighbour compensation," the adaptive predictive control laws were developed for nonlinear autoregressive moving average systems. In [25], the problem of robust stabilization was investigated for discrete-time singular large-scale systems with time-invariant norm-bounded uncertainties. Based on the dynamic surface control technique, both indirect and direct global neural controllers were developed for the strict-feedback systems in [26]. In [27], the output feedback adaptive neural network controllers were developed for nonlinear discrete-time systems. The output feedback adaptive control technique was investigated for discrete-time systems with unknown control directions in [28]. In [29], a robust adaptive sliding mode controller was constructed for discrete time-delay systems with mismatched uncertainties and matched external disturbances. Since the disturbances widely exist in practical systems and most of them are difficult to be measured, many advanced control approaches, such as adaptive control and sliding mode control, have been proposed. However, the control approaches mentioned above are rejecting disturbances via feedback control based on the tracking error, which can not deal with strong disturbances directly and fast [30]. Then, the disturbance observer was proposed to estimate unmeasured disturbances for engineering application systems such as direct-current servomotor systems [31], robotic manipulators [32], and data storage systems [33]. A reduced order disturbance observer was studied for discrete-time linear systems in [33]. Based on the disturbance observer, an antidisturbance controller was investigated for discrete time-varying delay systems with multiple disturbances under actuator failures in [34]. In [35], disturbance-observer-based control and [figure omitted; refer to PDF] control were discussed for A4D aircraft at a flight condition of 15000 ft altitude and 0.9 Mach, the states were made up with the forward velocity, the angle of attack, the pitching velocity, and the pitching angle, and the state delay was considered. A composite fuzzy control was investigated for uncertain nonlinear systems based on the disturbance observer technique in [36]. However, the input time-delay was seldom considered in the research results mentioned above.

Time delay in control inputs brings new challenges in the controller design. The problem of [figure omitted; refer to PDF] control was investigated for discrete time-varying input-delay systems via Riccati difference equation in [37]. In [38], the state feedback prediction-based control law was developed for a discrete-time system with input time-varying delay. Applying Artstein's reduction method and the scaled bounded real lemma, a sufficient condition was presented to stabilize linear discrete-time systems with time-varying input delay and model uncertainties in [39]. In [40, 41], the guaranteed cost control problem was studied for a class of discrete time-delay systems via state feedback. The problem of multiple input delays was discussed in [42]. A robust [figure omitted; refer to PDF] state feedback controller was designed for discrete time systems with interval time-varying delays in both states and control inputs in [43]. The state variables mentioned above were required as mensurable. However, not all of the state variables can be measured directly in practical applications. Thus, it is necessary to estimate the unmeasured state variables.

State observer, the product of the development of modern control engineering with the characteristics of good stability and strong robustness, has been widely used in various fields, such as diesel engines [44], stirred tank reactor [45], and permanent magnet synchronous motors [46]. The problem of the state-observer-based finite-time [figure omitted; refer to PDF] control was studied for singular stochastic systems using the descriptor system approach in [47]. Based on the future states prediction, adaptive control technique was investigated for a class of discrete-time MIMO nonlinear systems in [48]. In [49], a robust delay-dependent state estimator was investigated for a class of discrete-time Bidirectional Associative Memory (BAM) neural networks. A reduced-order Kalman-type observer was designed for nonlinear discrete-time systems in [50]. In [51], a fuzzy observer was proposed for discrete time-delay systems with nonlinearity. An observer-based finite-time [figure omitted; refer to PDF] controller was developed for discrete-time Markovian jump nonlinear systems with time delays in [52]. In [53], a state observer was designed for discrete time-delay impulsive switched systems with nonlinearity. A nonlinear observer was proposed for discrete-time uncertain nonlinear systems in [20]. In [54], the problem of reliable observer-based [figure omitted; refer to PDF] control was studied for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time-varying delays and stochastic actuator faults. To the best of author's knowledge, the discrete-time system subject to input saturation, parameter uncertainties, nonlinearity, disturbance, and time delays in both the states and the control inputs has not been investigated.

Motivated by the above discussion, we develop a bounded control scheme for uncertain discrete-time nonlinear system with disturbance and time delays in both the states and the control inputs in this paper. A state estimator and a disturbance observer are developed separately to estimate the unknown states and the disturbances generated by an exogenous system. Using the outputs of the state estimator and the disturbance observer, a bounded controller is designed. The Lyapunov function candidate is constructed so that the closed-loop system is asymptotically stable via LMI method. The paper is organized as follows: the problem formulation and preliminaries are given in Section 2. In Section 3, a state estimator and a disturbance observer are presented, respectively, and the bounded controller is designed for the discrete time-delay uncertain systems. Simulation results are given in Section 4, followed by concluding remarks in Section 5.

2. Problem Formulation and Preliminaries

Consider the following uncertain discrete-time nonlinear system with disturbance and time delays: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are unmeasured state vector, control input, output, and disturbance, respectively. [figure omitted; refer to PDF] is the nonlinear function. [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are known constant matrices with appropriate dimensions. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are parameter uncertainties. [figure omitted; refer to PDF] is a corresponding weighting matrix with appropriate dimensions. [figure omitted; refer to PDF] is a known constant delay. The system parameter uncertainties are defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are known constant matrices and [figure omitted; refer to PDF] is an unknown matrix satisfying [figure omitted; refer to PDF] Here, [figure omitted; refer to PDF] denotes the identity matrix with appropriate dimension.

Assume that [figure omitted; refer to PDF] is a saturated nonlinear function. Each element [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the saturation level of [figure omitted; refer to PDF] and is assumed as known, [figure omitted; refer to PDF] .

To design the bounded controller for the uncertain discrete time-delay system with nonlinearity and disturbance, the following assumptions and lemmas are needed.

Assumption 1.

Assume that state vector [figure omitted; refer to PDF] of the discrete time-delay system (1) is unmeasured.

Assumption 2 (see Guo and Chen [55]).

Assume that the disturbance [figure omitted; refer to PDF] is generated by the following exogenous system: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are matrices with corresponding dimensions. As shown in [9, 55, 56], a wide class of disturbances, such as harmonic disturbances, can be described by this model. This model has been extensively used in the disturbance observer design.

Assumption 3 (see Wei et al. [35]).

Assume that there exists a constant matrix [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Assumption 4 (see Wei et al. [35]).

[figure omitted; refer to PDF] is controllable; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are observable.

Lemma 5 (see Shi et al. [57]).

Given symmetric constant matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and constant matrix [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] hold if and only if [figure omitted; refer to PDF]

Lemma 6 (see Zhu et al. [58]).

Assume that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are real matrices with appropriate dimensions, [figure omitted; refer to PDF] is a symmetric matrix, and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] holds if and only if there exists a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Lemma 7 (see Song and Wang [59]).

Assume that [figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] diagonal matrices with diagonal elements being either 0 or 1. If [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Lemma 8 (see Zheng and Wu [60]).

Assume that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , the saturated input [figure omitted; refer to PDF] can be represented as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

The control objective of this paper is to find a sufficient condition in order to guarantee the stability of the uncertain discrete time-delay system described by (1) under a bounded controller.

3. Design of Bounded Controller Based on the State Estimator and the Disturbance Observer

In this section, the state estimator and the disturbance observer are designed to estimate the unmeasured state vector and disturbance of the discrete time-delay uncertain system. The output signals of the disturbance observer and the state estimator are delivered to the bounded controller as its input signals. The stability condition of the closed-loop system is derived by the chosen Lyapunov function as well as LMIs method. The control diagram of the closed-loop system is shown in Figure 1, where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the estimations of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Figure 1: The control diagram of system.

[figure omitted; refer to PDF]

3.1. Design of State Estimator

For the unmeasured state vector in the time-delay uncertain nonlinear system (1), a state estimator is needed to estimate [figure omitted; refer to PDF] . The state estimator is designed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an observer gain matrix which is needed to be designed by the comprehensive consideration between response speed and the sensitivity to disturbance and noise.

Define [figure omitted; refer to PDF] . Considering (1) and (10) yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

3.2. Design of Disturbance Observer

In order to monitor the disturbance of the discrete time-delay uncertain system (1), a disturbance observer is proposed as follows [35]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimation of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the auxiliary vector as the state of the disturbance observer.

Define [figure omitted; refer to PDF]

Considering (5), (13) and invoking (10), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Substituting (5), (12), and (13) into (11) gives [figure omitted; refer to PDF]

3.3. Design of Bounded Controller

The state estimator and the disturbance observer have been designed to estimate the unmeasured state vector and the unknown disturbance. Based on the output signals of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the bounded controller is designed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are feedback gain matrices.

The saturated linear region of [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Using Lemma 8, (16) can be rewritten as [figure omitted; refer to PDF]

Substituting (12) and (18) into (1), the closed-loop system is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Combining (14), (15), and (19), the whole closed-loop system is expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

According to Assumption 3, we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Disturbance gain matrix [figure omitted; refer to PDF] , state estimator gain matrix [figure omitted; refer to PDF] , and feedback gain matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] play an important role in closed-loop system stability and estimation error stability. The values of these matrices can be derived by solving LMIs. The stability condition of the closed-loop system (20) is given in the following theorem.

Theorem 9.

For given scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the closed-loop system (20) is asymptotically stable if there exist symmetric positive-defined matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that the following matrix inequality holds: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes the transposed elements in the symmetric matrix.

Proof.

Choose the Lyapunov function as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Define [figure omitted; refer to PDF] . Then, we have [figure omitted; refer to PDF]

Denote that [figure omitted; refer to PDF]

Taking the forward difference of [figure omitted; refer to PDF] along the trajectories of system (20), we have [figure omitted; refer to PDF]

Then, (29) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] can be proved to be less than 0, we have [figure omitted; refer to PDF] . Then, the closed-loop system (20) is asymptotically stable. Applying Schur complement theorem, [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Substituting (21) and (26) into (32) yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Using Lemmas 5 and 6, [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Left and right multiplying both sides of (36) by [figure omitted; refer to PDF] and using Schur complement theorem, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Left and right multiplying both sides of (38) by [figure omitted; refer to PDF] , we obtain the matrix [figure omitted; refer to PDF] of (23). Because [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . Namely, the closed-loop system (20) is asymptotically stable. This concludes the proof.

4. Simulation

In order to verify the correctness of the proposed method, simulation examples are given in the following.

4.1. The Case for Two-Dimension System

Consider the uncertain nonlinear discrete time system (1) with disturbance and time delays in both the states and the control inputs with [figure omitted; refer to PDF]

The parameter matrices of exogenous system (5) are chosen as [figure omitted; refer to PDF]

Suppose that the initial values of state variable are [figure omitted; refer to PDF] , the initial values of state estimator are [figure omitted; refer to PDF] , the initial values of exogenous inputs are [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , by solving (23), we obtain [figure omitted; refer to PDF]

Under the observer-based bounded controller, simulation results are obtained for the two-dimension uncertain discrete time-delay system with nonlinearity and disturbance, as shown in Figures 2-7. It is shown in Figures 2 and 3 that the stability of the closed-loop system can be obtained under the proposed bounded controller. Figure 4 shows that the state estimation errors of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are small, and the outputs of the state estimator can estimate the system state effectively. The simulation results shown in Figures 5 and 6 indicate that the outputs of the disturbance observer can effectively and quickly approximate the unknown external disturbance. Figure 7 shows that the control input is bounded under the designed controller. From these simulation results of the example, we can know that the state estimator and the disturbance observer can well approximate the system state and the disturbance, and the designed bounded control scheme based on the state estimator and the disturbance observer is valid for the discrete-time system subject to input saturation, parameter uncertainties, nonlinearity, disturbance, and state and input time delays.

Figure 2: The system state [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 3: The system state [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 4: The estimation error outputs of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 5: The disturbance [figure omitted; refer to PDF] and the estimation output of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 6: The disturbance [figure omitted; refer to PDF] and the estimation output of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 7: The control inputs [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

4.2. The Case for Three-Dimension System

Considering the following discrete-time uncertain time-delay system with disturbance and nonlinearity, originally given by [41], the parameter matrices of system (1) are [figure omitted; refer to PDF]

The parameter matrices of exogenous system (5) are chosen as [figure omitted; refer to PDF]

Suppose that the initial values of state variable are [figure omitted; refer to PDF] , the initial values of state estimator are [figure omitted; refer to PDF] , the initial values of exogenous inputs are [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , by solving (23), we have [figure omitted; refer to PDF]

Using the developed bounded controller, simulation results for the three-dimension uncertain discrete time-delay system with nonlinearity and disturbance are shown in Figures 8-14. From Figures 8-10, we can observe that the stability of the closed-loop system can be guaranteed under the observer-based bounded controller. According to Figure 11, we can see that the outputs of the state estimator can estimate the system state effectively. The disturbance estimate outputs [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and the disturbance [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are presented in Figures 12 and 13 which illustrate the effectiveness of the proposed disturbance observer. Figure 14 indicates that the control input is bounded under the designed controller. Based on these simulation results, we can obtain that the state estimator and the disturbance observer can effectively approximate the system state and the disturbance, and the proposed observer-based bounded control is valid for the discrete-time uncertain nonlinear system with time delays and disturbance.

Figure 8: The system state [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 9: The system state [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 10: The system state [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 11: The estimation error outputs of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 12: The disturbance [figure omitted; refer to PDF] and the estimation output of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 13: The disturbance [figure omitted; refer to PDF] and the estimation output of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 14: The control inputs [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

5. Conclusion

Stability of the uncertain discrete time-delay systems with disturbance and nonlinearity has been studied in this paper. The state estimator and the disturbance observer have been designed to estimate the unmeasured state vector and the unknown disturbance generated by the exogenous system. A bounded controller has been developed by the outputs of the state estimator and the disturbance observer. The stability of the closed-loop system has been proved via LMIs. Simulation examples have been provided to show the effectiveness of the proposed scheme. In the future, the time-varying state delay and input delay can be further considered for the discrete-time uncertain nonlinear systems with disturbance and input saturation.

Acknowledgments

This research is supported by National Nature Science Foundation of China (no. 61573184), Jiangsu Natural Science Foundation of China (no. SBK20130033), Program for New Century Excellent Talents in University of China (no. NCET-11-0830), and Specialized Research Fund for the Doctoral Program of Higher Education (no. 20133218110013).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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