Stabilization of Linear Time-Delay Systems by Higher-Order TPF
This chapter complements the first-order TPF we proposed in the previous chapters by establishing a class of higher-order TPF. Differently from the traditional predictor-based feedback, the proposed TPF guarantees that the closed-loop system is a retarded
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Stabilization of Linear Time-Delay Systems by Higher-Order TPF
In Chap. 2, we have proposed a TPF approach for stabilization of linear system with input delay by safely dropping out the distributed terms in the traditional predictorbased feedback to result in a static stabilizing feedback law. By realizing that the TPF proposed in Chap. 2 only retains the first-order term of the nominal feedback (thus, we refer to it as the first-order TPF), we aim in this chapter to develop general TPF that contains higher-order terms of the nominal feedback gains. We will show that, similar to the first-order TPF, the proposed higher-order TPF can also solve the global and semi-global stabilization problems for the concerned time-delay systems. We further show that, although the higher-order TPF contains distributed terms as in the traditional predictor feedback approach, it can be safely implemented via numerical approximation. As a second objective of this chapter, we will carry out a comparison between the first-order TPF and the higher-order TPF in terms of the closed-loop performance. In particular, we will show by numerical study that, in spite of the fact that the higherorder TPF utilizes more information of the state than the first-order TPF does, the first-order TPF actually outperforms the higher-order TPF. The presentation of this chapter is based on our work [197]. The remainder of this chapter is organized as follows. After formulating the problems to be considered in Sect. 7.1, the higher-order TPF is designed in Sect. 7.2. In Sect. 7.3, we prove the stability of the closed-loop system by the higher-order TPF. The higher-order TPF will be compared with the first-order TPF via numerical examples in Sect. 7.5. Finally, Sect. 7.6 concludes this chapter. For easy presentation, the system matrices and delays considered in this chapter are constant. However, all the obtained results can be extended to the time-varying setting without any difficulties.
B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, DOI 10.1007/978-3-642-54206-0__7, © Springer-Verlag Berlin Heidelberg 2014
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7 Stabilization of Linear Time-Delay Systems by Higher-Order TPF
7.1 Problem Formulation and Preliminaries Consider the following linear system with input delay xP .t / D Ax .t / C Bu .t /;
(7.1)
where x .t / 2 Rn and u .t / 2 Rm are, respectively, the state and input vectors, and > 0 represents the delay in the control input. The problems we consider in this chapter are as follows. Problem 7.1 (Global Stabilization). For any given, arbitrarily large, bounded delay , find a control u .t / D u .xt / such that the closed-loop system is asymptotically stable. Problem 7.2 (L1 and L2 Semi-global Stabilization). For any given, arbitrarily large, bounded delay and any given, arbitrarily large, bounded set ˝ Cn; , find a control u .t / D u .xt / such that the closed-loop system is asymptotically stable at m U . the origin and x0 2 ˝ H) u 2 Um 1 2 It is well known that, even in the case D 0, Problem 7.2 is solvable
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