Finite-time Stability And Control For Generalized Hamiltonian Systems

As an important kind of nonlinear systems, generalized Hamiltonian systems are a kind of open systems which can have energy exchange with external envi-ronment, energy dissipation and energy generation. Because of explicit physical significance and clear structure, Hamiltonian function is a good Lyapunov candi-date in many cases. This class of systems has special superiority in many control problems, such as stability analysis, stabilization control, H∞control, adaptive control, simultaneous stabilization, and so on. Hamiltonian function methods are widely used in many practical control problems, such as electrical power system control problems and mechanical system control problems.As a special asymptotic stability, finite-time stability describes that system states can come back and remain in the equilibrium point. Finite-time stable systems have properties of non-Lipschitz, fast convergence, good robustness and great capability of resisting disturbance. Finite-time control methods have well application in many branches of control theory and many practical control prob-lems.First, finite-time stability of a class of time-invariant Hamiltonian systems is investigated. Via proving that 'globally finite-time stability equals globally asymptotic stability and locally finite-time stability', a new criterion of finite-time stability is obtained for a class of generalized Hamiltonian systems, that is, for generalized Hamiltonian systems,'the sufficient condition of finite-time stability is Hamiltonian systems with special energy function are strictly dissipative'. By choosing another energy function, it shows that the energy function is not unique for time-invariant Hamiltonian systems of finite-time stability. Meanwhile, uti-lizing'damping injection'technique and stability criteria obtained, continuous finite-time controllers are proposed for a class of time-invariant affine nonlinear systems. And several examples and simulations are given to support corresponding results.Second, the results of finite-time stability and control can be expanded to time-varying Hamiltonian systems from time-invariant Hamiltonian systems, and a criterion of uniform finite-time stability and continuous finite-time controllers are given for time-varying Hamiltonian systems and time-varying affine nonlinear systems, respectively. Meanwhile, example and simulation are presented to prove corresponding results.