Robust Control Of Generalized Hamiltonian Systems And Its Applications

Generalized Hamiltonian systems are a kind of important nonlinear systems them-selves and an important tool for stability studies and controller design for nonlinear con-trol systems. This dissertation mainly focuses on robust controller parameterization androbust fault-tolerant control (FTC) against actuator faults for generalized Hamiltoniansystems and their applications to stability control of power systems. The main results arelisted as follows.1. A new method, namely Successive Normal Substitution(SNS), for checking non-negativity of polynomials, is proposed based on simplex subdivisions, which will bewidely used in the remained of this dissertation. Unlike existing research, we providea geometric perspective to study variable substitutions, and establish a one-to-one corre-spondence between subdivisions of a simplex and normal substitutions. Then, we give asufficient and necessary condition for the self-similar subdivision sequence of a simplexto be convergent, and prove that the positivity of a form which is indeed positive de?nitecan be checked by SNS. Thus, we obtain various effective substitutions for checking non-negativity of forms, which are beyond weighted difference substitutions characterized by“difference”. Examples show the great effectiveness of SNS.2. Parameterization methods of robust controllers for dissipative Hamiltonian sys-tems are proposed under the full-information and partial-information cases, respectively.First, a family of robust controllers with full information is provided by interconnectinga robust controller with a generalized ZEG detectable, free generalized Hamiltonian sys-tem. These controllers thus obtained have strong robustness against external disturbances.Then, a family of robust controllers with partial information is presented in terms of thesolution to an inequality only in 2n independent variables (twice as many as the one usedto characterize the state feedback) and without imposing an additional cascade condition.These proposed controllers also have strong robustness against external disturbances. Fi-nally, when the systems involve parameter perturbations, a family of adaptive robust con-trollers with full information and the one with partial information are proposed. Both ofthem have strong robustness against external disturbances and parametric perturbations.Due to the fact that the Hamiltonian function can be used to build a Lyapunov function for the corresponding closed-loop systems, the proposed parameterization methods avoidsolving Hamilton-Jacobi-Issacs equations (or inequalities), a dif?cult and necessary pro-cess required in many available methods, and the controllers thus obtained are relativelysimple in form and easy in operation, of which the parameters provide a degree of free-dom to achieve various desired control performance and objectives.3. Using structure properties of dissipative Hamiltonian systems, the problem ofactuators’fault-tolerant control (FTC) for dissipative Hamiltonian systems is dealt with.First, a robust FT controller is proposed for the systems subject to additive actuator faults,which has strong disturbance attenuation and desired fault-tolerance for the faults. How-ever, the proposed FT controller has the drawback of being discontinuous, which may leadto the chattering effect sometimes. To reduce this effect, we use the saturated function toapproximate the symbolic function in the FT controller, and the controller thus obtainedensures that the state is uniformly bounded and uniformly ultimately bounded. Then, a ro-bust FT controller is given for the systems subject to loss-of-effectiveness actuator faults,which also shows disturbance attenuation and no sensitiveness to the faults. Especially, itis continuous and avoids the chattering effect of the discontinuity. Finally, when the sys-tems involve parameter perturbations, an adaptive robust FT controller is proposed for thesystems subject to each of two types of actuator faults, additive and loss-of-effectivenessfaults, which has strong robustness against external disturbances and parametric perturba-tions and ensures local uniform asymptotic stability of the equilibrium point of the faultysystems. In comparison with existing FT controllers, the FT controllers given in this dis-sertation avoid solving the Lyapunov function of the stable nominal system. As a result,these controllers have simple structure, good operability and high practicality.The work of this dissertation expands and enriches the current research on general-ized Hamiltonian systems, and gives some new methods and thoughts for stability studiesand controller design for nonlinear control systems.