Global Stabilization of Planar Systems with Input Delay and Saturation
This chapter investigates the problem of global stabilization of planar linear systems with both actuator saturation and delay. For a double integrator system, two families of TPF-based linear feedback solutions to the problem are proposed. Both of these
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Global Stabilization of Planar Systems with Input Delay and Saturation
In the previous chapters, we have shown that any ANCBC linear system with arbitrarily large input delay can be semi-globally stabilized by TPF-based controllers in the presence of input saturation. In this chapter, we will go a further step to investigate the global stabilization problem of linear systems with both input delay and saturation. However, due to the complexity of the global stabilization problem, we restrict our attention to ANCBC planar system. Particularly, we will study global stabilization of the double integrator system and an oscillator system that are subject to simultaneously input delay and saturation. For the double integrator system, we will propose two TPF-based solutions to the global stabilization problem. The first solution is delay-dependent in the sense that the delay information is directly used in the feedback design. The second solution is delay independent as the delay information is not directly used in the feedback. For the oscillator system, we will also provide a delay-dependent TPF-based controller. All of these controllers are parameterized by a single scalar , which in fact ensure a family of solutions. In all of the designs, we provide explicit ranges of the value of the parameter within which the resulting closed-loop system is globally asymptotically stable. Our presentation of this chapter draws on materials from our work [206] and [209]. This chapter is organized as follows. The delay-dependent and delay-independent TPF-based controllers for the double integrator system are, respectively, presented in Sects. 6.1 and 6.2. The oscillator system is then investigated in Sect. 6.3. Section 6.4 presents an example to show the effectiveness of the proposed results. Finally, Sect. 6.5 concludes this chapter.
6.1 The Double Integrator System: Delay-Dependent TPF Consider the following double integrator system subject to both input delay and input saturation: xP .t / D Ax .t / C B .u .t // ; B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, DOI 10.1007/978-3-642-54206-0__6, © Springer-Verlag Berlin Heidelberg 2014
(6.1) 147
148
6 Global Stabilization of Planar Systems with Input Delay and Saturation
where x .t / 2 R2 and u .t / 2 R are, respectively, the state and input, A and B are, respectively, given by
01 AD ; 00
0 BD ; 1
(6.2)
and the scalar < 1 is a given positive number denoting the input delay. We have assumed, without loss of generality, the unity saturation level in the input. The nonunity saturation level can be absorbed by the matrix B and the feedback gain. We are interested in the design of control u .t / such that the closed-loop system is globally asymptotically stable.
6.1.1 The Main Theorem Our solutions to the global stabilization problem are based on the following parametric ARE: A| P C PA PBB | P D P;
(6.3)
which has a unique positive definite solution 3 2 : P . / D 2 2
(6.4)
We first present in this section a delay-dependent resu
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