Passive system theory was applied to propose a new passive control method with nonlinear observer of the Permanent Magnet Synchronous Motor chaotic system. Through constructing a Lyapunov function, the subsystem of th...Passive system theory was applied to propose a new passive control method with nonlinear observer of the Permanent Magnet Synchronous Motor chaotic system. Through constructing a Lyapunov function, the subsystem of the Permanent Magnet Synchronous Motor chaotic system could be proved to be globally stable at the equilibrium point. Then a controller with smooth state feedback is designed so that the Permanent Magnet Synchronous Motor chaotic system can be equivalent to a passive system. To get the state variables of the controller, the nonlinear observer is also studied. It is found that the outputs of the nonlinear observer can approximate the state variables of the Permanent Magnet Synchronous Motor chaotic system if the system’s nonlinear function is a globally Lipschitz function. Simulation results showed that the equivalent passive system of Permanent Magnet Synchronous Motor chaotic system could be globally asymptotically stabilized by smooth state feedback in the observed parameter convergence condition area.展开更多
A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a...A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.展开更多
基金Project supported by the Natural Science Foundation of Zhejiang Province (No. Y104414) and the Science and Technology Plan of Zhejiang Province (No. 2005C21084), China
文摘Passive system theory was applied to propose a new passive control method with nonlinear observer of the Permanent Magnet Synchronous Motor chaotic system. Through constructing a Lyapunov function, the subsystem of the Permanent Magnet Synchronous Motor chaotic system could be proved to be globally stable at the equilibrium point. Then a controller with smooth state feedback is designed so that the Permanent Magnet Synchronous Motor chaotic system can be equivalent to a passive system. To get the state variables of the controller, the nonlinear observer is also studied. It is found that the outputs of the nonlinear observer can approximate the state variables of the Permanent Magnet Synchronous Motor chaotic system if the system’s nonlinear function is a globally Lipschitz function. Simulation results showed that the equivalent passive system of Permanent Magnet Synchronous Motor chaotic system could be globally asymptotically stabilized by smooth state feedback in the observed parameter convergence condition area.
文摘A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.
