H∞ state-feedback control of time-delay systems using reciprocally convex approach
Introduction
The phenomena of time-delays are often encountered in physical systems like communication systems, aircraft dynamics, nuclear reactors, process control systems and so on [1]. These delays are usually time-varying in nature, and have an adverse impact not only on the system performance but also on its stability; therefore, neglecting the effects of delay in the analysis may lead to instability and disastrous consequences. Various methods have been proposed in the recent years for stability analysis and synthesis of the controller for time-delay systems [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Most of the results assume that the lower bound of the delay range is zero [2], [4], [6], [7], [11], [12], [13], [20], [22]. However, in practice, the delay-range may have a non-zero lower bound, and such systems are referred to interval time-varying delay systems [3], [5], [8], [19], [21], [23]. Typical examples for interval time-delay systems are networked control systems [6], which are basically feedback control systems wherein the feedback loop is closed through a real-time communication network. A great number of delay-dependent stability results have been reported, and many effective results have been provided to reduce the conservatism of stability results for further improving the quality of delay-dependent stability criteria [7], [8], [10], [12], [14], [17], [20], [21], [23].
H∞ control theory considers the worst case of external disturbances to design an optimal controller to achieve the desired performance. More recently, H∞ control theory has also been applied to an actual building in Tokyo, Japan using a pair of mass dampers to reduce the bending-torsion motion due to earthquakes [24]. Further, a liquid monopropellant rocket motor with a pressure feeding system has been considered as a numerical design example in [25], [26], [27]. In [11], [28], the problem of controlling the yaw angles of a satellite system with delays has been discussed. This satellite system consisting of two rigid bodies joined by a flexible link were assumed to have the state-space representations. Recently, H∞ optimal control techniques have been found to be an effective solution to treat robust stabilization and tracking problems, in the presence of external disturbances and system uncertainties [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. Authors in the paper [29] have discussed an H∞ state feedback control algorithm to reduce seismic responses of a structure equipped with an active tendon bracing system. A non-convex optimization problem has been formulated and solved using an iterative solution method to obtain the desired control gains. The problem of robust H∞ control for a half-vehicle active suspension system with input delay have been considered in [39], [42]. In the H∞ control technique, the main design goal is to force the gain from unmodeled dynamics, external disturbances, and approximation errors to be equal or less than a prescribed disturbance attenuation level (H∞ attenuation constraint). This goal is generally represented as an LMI problem.
In the time-domain approach, the direct Lyapunov method is a powerful tool for studying the problem of stability and H∞ control for systems with delay. There are two different ideas of how one can employ this method. They are the Lyapunov–Krasovskii Functional (LKF) approach and the Lyapunov–Razumikhin Function approach and both approaches can be used to handle systems with time-varying delay. The former usually requires both the upper bound of the time-varying delay and additional information on the derivative of the time-varying delay while the latter has no restriction on the derivative of the time-varying delay, which allows a fast time-varying delay. The obtained results using LKF approach are usually less conservative than those using the Lyapunov–Razumikhin approach since the former takes advantage of the additional information of the delay. A great number of delay-dependent stability and stabilization results for H∞ controller design have been reported, and many effective results have been provided to reduce the conservatism of stability results for further reducing the disturbance attenuation level [35], [36], [37], [38], [39], [40], [41], [42].
In this paper, a lower bound lemma for such a linear combination of positive functions with inverses of convex parameters proposed in [17], [22], [26] have been taken to derive this novel result. Based on the lemma, a stability criterion have been derived that directly handles the inversely weighted convex combination of quadratic terms of integral quantities, which not only achieves performance behavior identical for approaches based on the integral inequality lemma, but also with significantly fewer decision variables, comparable to those based on the Jensen inequality lemma [43]. By virtue of the developed novel criteria based on the new lemma introduced in [17], the proposed H∞ controller is not only able to reduce the disturbance attenuation level with less conservatism but also it does not require any restrictions on the free matrix variables hence reduces computational complexity. More importantly, the design guarantees asymptotic stability of the closed-loop system with disturbances. For the illustrative examples the physical model of liquid monopropellant rocket motor with a pressure feeding system and the problem of controlling the yaw angles of a satellite system with delays are considered along with its numerical simulations to show effectiveness of derived results.
The paper is organized as follows. Section 2 gives the problem description and preliminaries used in this paper. In Section 3, H∞ control is designed via LMI techniques and the reciprocally convex approach. In Section 4, the problem of controlling the yaw angles of a satellite system with delays are considered along with their numerical simulations to show the effectiveness of the criterion. Furthermore some other numerical examples are also given to illustrate the advantages of the derived results. In Section 5, the conclusion is drawn.
Notations: Throughout this paper, and denote the n-dimensional Euclidean space and the set of all n × n real matrices respectively. The superscript T denotes the transposition and the notation X ≥ Y (similarly, X > Y), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (similarly, positive definite). ∥· ∥ is the Euclidean norm in . The notation * always denotes the symmetric block in one symmetric matrix. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.
Section snippets
Problem description and preliminaries
Consider the following time-delay system
where is the state at time t; is the input; is the disturbance input; is the signal to be estimated; τ(t) is a non-negative time-varying delay satisfying 0 ≤ τm ≤ τ(t) ≤ τM; is the initial value function and , , , , , , and are constant matrices of appropriate dimension.
Main results
For a given control gain , the H∞ – norm bound of system (2) under assumption (A1) can be solved by applying the following Theorem 3.1.
Theorem 3.1 For some given scalars 0≤ τm < τM, μ < ∞ and γ, system (2) is asymptotically stable with a prescribed H∞ performance γ if there exist symmetric positive definite matrices P, Q1, Q2, Q3, R1, R2, further for any matrices S with compatible dimensions, and , such that the following matrix inequality holds: where
Numerical examples
In this section, four examples are given to illustrate the effectiveness and advantages of the results derived in this paper. In examples, MATLAB is used to solve LMI problems.
Example 4.1 Consider the time-delay system (2) with parameters defined as followsand τ(t) = 0.75 + 0.25sin(t). Applying Theorem 3.3 with γ = 0.3 and ρ = 1.0, the controller gain is computed as K = [−0.0019, − 5.8305] which means that under the controller u(t) = Kx(t), the considered
Conclusion
In this paper, the problem of asymptotic H∞ control of linear systems with interval time-varying delay based on LKF approach has been illustrated. The convex optimization approach has been developed for determining delay-dependent conditions for H∞ control and hence sufficient conditions for the solvability of this problem in terms of LMIs have been established. No restrictions on the derivative of the interval time-varying delay have been imposed, which has allowed a fast time-varying delay.
Acknowledgments
The authors would like to express their sincere gratitude to the Editor-in-Chief, Associate Editor, and Anonymous Reviewers for their valuable comments and suggestions to improve the quality of the manuscript. The research work of Mr. V. Vembarasan is supported by DST INSPIRE Fellowship, Ministry of Science and Technology, Government of India under the grant numbers DST/INSPIRE Fellowship/2011/278 dated 21.12.2011 and DST/AORC-IF/UPGRD/2013/327 dated 18.06.2013.
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