Observer-Based Sliding Mode Control for Stabilization of a Dynamic System with Delayed Output Feedback

1 School of Applied Mathematics, University Electronic Science and Technology of China, Chengdu 610054, China 2 School of Electrical and Information Engineering, Xihua University, Chengdu 610096, China 3 School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia 4College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia 5 Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway


Introduction
For system control, state feedback control is a powerful tool when the full information of the system states is assumed to be accessible.However, in real engineering, not all of them can be available.Hence, the research on observer-based control system is a meaningful topic.Up to now, many relevant investigations [1][2][3][4][5][6][7][8][9][10] have been carried out.For instance, in [1], by imposing some restrictions on an open-loop system, two classes of the observer-based output feedback controllers, one finite dimensional and the other one infinite dimensional, are constructed; in [2], a static gain observer for linear continuous plants with intrinsic pulse-modulated feedback is designed to asymptotically drive the state estimation error to zero; in [3], by using the orthogonality-preserving numerical algorithm, a nonlinear discrete-time partial state observer is designed to realize the attitude determination for a kind of spacecraft system; in [4], with the two-step observation algorithms, an observer is constructed to force an underactuated aircraft to asymptotically track a given reference trajectory; in [5], through employing an adaptive backstepping approach, a modified high-gain observer is introduced to realize the global asymptotic tracking for a class of nonlinear systems.
In real engineering, time delays exist objectively, which makes the observer-based system control issue more complicated.Many existing control methods that have been well developed based on the conventional feedback observer, such as ones in [1][2][3][4][5], are not directly applicable.In addition, the LMI technique is known as a powerful tool for system control.However, it is not easy to use any more when time delays appear in the feedback output of system.Hence, the corresponding research is meaningful.Sliding mode control (SMC) has been proven to be an effective robust control strategy and has been successfully applied to a wide range of engineering systems such as spacecrafts, robot manipulators, aircraft, underwater vehicles, electrical motors, power systems, and automotive engines [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].In [11], a novel integral sliding surface is introduced to achieve the control for a kind of system based on the LMI technique, with the more available stability condition compared to the conventional linear sliding surface.However, the system with delay is not included in its research yet.Hence, all of the above motivate our work.
In this paper, the problem on the SMC for a kind of delay dynamic system will be discussed.By utilizing Lyapunov stability theory and a linear matrix inequality technique, a new observer-based on delayed output feedback will be Mathematical Problems in Engineering constructed.An integral sliding surface will be presented to realize the SMC of the system with the more available stability condition.Finally, some numerical examples will be included to demonstrate the validity of the proposed control method.
Notation used in this paper is fairly standard.Let   be the -dimensional Euclidean space,  × represents the set of  ×  real matrix, * denotes the elements below the main diagonal of a symmetric block matrix, diag{} represents the diagonal matrix, the notation  > 0 means that  is the real symmetric and positive definite,  denotes the identity matrix with appropriate dimensions, and ‖ ⋅ ‖ refers to the Euclidean vector norm.

Problem Statement and Preliminaries
In this paper, the following dynamic system is considered: where () ∈   is the system state, () ∈   is the system input, () ∈   is the system output,  is discrete delay, () is the measurable feedback output, and , , ,  are the system matrices with appropriate dimension.
Then, an observer of system ( 1) is built as follows: where x() ∈   is the state vector,  is the feedback control matrix, and  is the measurable feedback delay.Define the observation error: The following theoretical result will be used throughout the paper.
Lemma 1.For observer (2), the following equation holds: Proof.The derivative of ( 4) is as follows: The proof of Lemma 1 is thus completed.

Design of Observer
Theorem 2. If there exists a matrix , and positive-definite symmetric matrices  and  satisfy then, observer (2) can realize the observation of system (1).
Then, the following equation holds: where Mathematical Problems in Engineering 3 Let  = , and, with LMI (7), we can conclude Hence, lim From system (1), the following equation can be derived: Considering ( 4), we have Finally, combined with ( 16), we can get lim  → ∞ () → 0, which means the realization of the observation.The proof of Theorem 2 is thus completed.

Design of Sliding Motion Control Law
First, we introduce the following integral sliding surface: For system (1), we have Hence, the following equation holds: By ṡ () = 0, we get the equivalent control law as When () = (), we can obtain Then, based on Lyapunov theory and LMI techniques, the following theoretical result is derived.
Theorem 3. If there exists a matrix , and positive-definite symmetric matrices  and  satisfying then, dynamic system (23) is stable.
Proof.Choose the Lyapunov functional candidate as Then, the time derivative of  1 () is as Hence, we have where Let  =  −1 1 ; pre-and postmultiply Ξ by diag{, }; we have Let  =  2 ,  = , and consider LMI (24); we can get Hence, dynamic system ( 23) is stable.The proof of Theorem 3 is thus completed.
Next based on the above theoretical results, a sliding mode controller will be constructed to realize the stabilization of dynamic system (1).Theorem 4. For dynamic system (1), the state trajectory will be driven onto the sliding surface () in a finite time by the following SMC law: where  and  are positive scalars.
Proof.Choose the Lyapunov functional candidate as With ( 21) and (32), we have Hence, the system state trajectory can converge to the sliding surface () and stay on the surface.This concludes the proof of Theorem 4.

Numerical Example
In this section, we will verify the proposed theoretical theorems by giving some illustrative examples.First consider dynamic system (1) with the following parameters: Corresponding simulation results are shown in Figure 1.
Then, the presented observer-based SMC method in this paper is applied to system (1).Based on Theorem 3, we can get the control parameters as follows: In addition, to avoid the chattering phenomenon of SMC, sign(()) is recommended to be replaced by ()/(|()| + 0.05), and the corresponding simulation results are shown in Figures 2 and 3.

Remark 5.
From Figure 1, it can be seen that the state variables of the system diverge to zero as time passes by.This means that the considered system is not stable.Figure 2 shows the time response of system state () and system state estimate x() based on the given observer-based SMC method.Figure 3 shows the time response of sliding mode control input ().It can be seen that x() converges to () during 4.0 seconds, and the original unstable dynamic system becomes stable for the utilization of our controller, which demonstrates the effectiveness of the presented theoretical results.

Conclusion
In this paper, the problem on the stabilization for a kind of dynamic system has been discussed.Through utilizing Lyapunov stability theory and linear matrix inequality techniques, a delayed output feedback observer has been designed, and a sliding mode controller has been constructed to realize the stabilization of the system.Finally, some typical numerical examples have been included to verify the validity of the proposed SMC method.

Figure 3 :
Figure 3: The time response of control input () of system (1).