Parametric Lyapunov Approach To The Design Of Control Systems With Saturation Nonlinearity And Its Applications

Linear system theory has been well developed during the past several decades. How-ever, controller designed by linear system theory and applied on physical plant generallycannot lead to satisfactory system performance, and sometimes cannot even guaranteestability. One possible reason is the difference between the mathematical model and thephysical model. But another important reason is that the actuator saturation nonlinear-ity on the physical actuator (e.g., the circumrotation rate of a wheel should not exceedsome prescribed limitation) is not fully taken into consideration during the controller de-sign. As a result, the control signal computed by theory cannot be applied on the plantcompletely. Therefore, motivated by the fact that any physical actuator is subject to sat-uration nonlinearity constraint, control of such kind of systems has been received muchattention during the past 30 years, and a plenty of researchers has proposed different ap-proaches to deal with different control problems. However, these approaches are eitherdifficult to be implemented due to their complexity or able to guarantee satisfactory sys-tem performance, which have significantly restricted the applications of these approachesin engineering. This thesis, based on in-depth study on previous works of others, system-atically and deeply proposes a new parametric Lyapunov approach to deal with a seriesof control problems for systems subject to actuator saturation nonlinearity in a unifiedframework. The effectiveness of the proposed work is validated by the controller designfor spacecraft rendezvous problem. The proposed approach·can deal with not only marginally unstable plant but also unstable plant;·can design not only linear controller but also nonlinear controller;·can deal with not only saturation nonlinearity but also time-delay in the actuator;·can deal with not only time-invariant systems but also time-varying periodic sys-tems.Based upon the introduction to the background and existing results on the topics inChapter 1, Chapter 2 develops the basic theory on the parametric Lyapunov approach bystudying a class of Lyapunov equations containing a single parameter. As the first appli-cation of this new approach, the problem of robust global stabilization of control systemssubject to actuator saturation and input additive uncertainties is solved. This parametricLyapunov approach is further used in Chapter 3 to design global stabilization and re-stricted tracking controllers for general marginally unstable linear plant. Different from the nonlinear Lyapunov function based approach existing in the literature, the approachin Chapter 3 is based on quadratic Lyapunov function which not only leads to simpercontroller but can also provide explicit conditions to guarantee stability. Especially, com-pared with the existing approaches, the new controller utilizes the full state and can thussignificantly improve the dynamic performances of the closed-loop system. The resultsin this chapter indicate that the parametric Lyapunov approach is not only effective indesigning linear controller but also effective in designing nonlinear controller.Chapter 4 investigates the approximated computation of the maximal invariant el-lipsoid for control systems subject to actuator saturation nonlinearity. Especially, if thecontroller is designed by the parametric Lyapunov approach, explicit formulation for themaximal invariant ellipsoid is provided, based on which it is further proved that the max-imal invariant ellipsoid is monotonically increasing with respect to the scalarγrepresent-ing the convergence rate of the closed-loop system. This result indicates that controllerdesign for system subject to actuator saturation nonlinearity should be a tradeoff betweenmaximizing the domain of attraction and maximizing the convergence rate. This resultperforms an important theoretic guide for controller design for this class of systems.Chapter 5 systematically studies the stabilization problem of systems subject to ac-tuator saturation and time-delay by using the parametric Lyapunov approach. First, con-troller for systems containing only a single constant or varying delay is designed andexplicit conditions are provided to guarantee the stability. However, the technique usedfor single delay case cannot be used for multiple delays case. As a result, Chapter 5 con-structs artfully a new controller based upon the parametric Lyapunov approach. It is fur-ther proved that the proposed controllers semi-globally stabilize the system if it is simul-taneously subject to actuator saturation. Finally, the global stabilization problem of thedouble integrator system subject to both actuator saturation nonlinearity and time-delayby using the parametric Lyapunov approach is solved in the last section of this chapter.Two classes of approaches, namely, delay-independent approach and delay-dependent ap-proach, are proposed. These results further imply that the parametric Lyapunov approachis effective in designing nonlinear controller.Chapter 6 establishes the parametric Lyapunov approach for periodic linear systems.The foundation of parametric Lyapunov approach for periodic linear systems is built bystudying properties of two classes of periodic Lyapunov (differential) equations. As anapplication of this new approach, semi-global stabilization of the periodic linear systemssubject to actuator saturation nonlinearity is solved for the first time. Especially, as a special case of periodic system, control problems for multi-rate sample system subjectto actuator saturation nonlinearity is considered, which enlarges the applications of theparametric Lyapunov approach.Chapter 7 utilizes part of the results in Chapter 2, Chapter 5 and Chapter 6 to de-sign controllers for spacecraft rendezvous with actuator saturation. Based on a deep studyin the model for spacecraft rendezvous, an explicit periodic Lyapunov transformationis constructed to transform the model to its Jordan canonical form, which gives a pro-found understanding of its stability. Based on this analysis, when the target spacecraftare respectively in elliptical and circular orbit, controllers are designed by the parametricLyapunov approach developed in Chapter 6 and Chapter 2, and some parameters in thecontrollers can be turned to guarantee that the required thrust acceleration satisfies theactuator saturation constraint. When the actuator is simultaneously subject to time-delay,controller designed according to the results in Chanter 5 can not only accomplish the ren-dezvous mission but also fulfil the constraint on the control signals if some parameters arewell adjusted. These results constitute an attempt of applying the parametric Lyapunovapproach to practical engineering problems, which not only enriches the parametric Lya-punov approach but also provides a design example and theoretic guarantee for practicalengineering.