Pseudo-Hamiltonian realization and its application

Abstract

Abstract. In this paper, the problem of pseudo-Hamiltonian realization of a control system is studied. Several sufficient conditions are obtained. The stability of a dynamic system is investigated via dissipative pseudo-Hamiltonian realization, and the stabilization of a control system is also investigated via feedback dissipative pseudo-Hamiltonian realization. Some relations between the stability (asymptotical stability) with the dissipative (strict dissipative) realization are revealed. Relations between the affine dissipative control systems and the dissipative pseudo-Hamiltonian realization are also investigated. These results show that the set of pseudo-Hamiltonian systems represent a very large class of interesting dynamic systems, and this approach is a powerful tool. Particularly, a generalization of the Krasovskii's Theorem is obtained. Then the problem of L 2 disturbance attenuation of a nonlinear system via pseudo-Hamiltonian realization is investigated. It is shown that for a class of pseudo-Hamiltonian systems the disturbance attenuation problem is solvable and an estimation of the boundary of the L 2 gain is obtained. Finally the results are applied to the excitation control of power systems. The stabilization and the H∞ control problems are investigated for the singlemachine infinite bus power systems.

Local dissipative Hamiltonian description of reversible reaction networks

“…A more general system class is the set of pseudo-Hamiltonian systems [3], where the algebraic system structure is the same as in the case of Hamiltonian systems, but the invariance or the decreasing nature of the Hamiltonian function cannot be proved globally from the properties of the structure matrix of the system. In [17], an effective general algorithm is presented for obtaining the approximate dissipative Hamiltonian realization and constructing local Lyapunov functions for continuous time nonlinear systems.…”

“…It is shown in [3] and [11] that Lyapunov-stable nonlinear systems can be algorithmically transformed to a special Hamiltonian form, but in this case, the structure matrix is generally nonsmooth at the equilibrium point, and not much physical insight is gained about the system structure.…”

Passivity and Passive Feedback Stabilization for a Class of Mixed Potential Systems

“…studied extensively in the literature, for example in (van der Schaft, 2000;Ortega et al, 2002;Cheng et al, 2002), if one pick H(x) to be the free energy at unit temperature, H(x) = E(x) − S(x) (Favache et al, 2010). However, the problem proposed above differs by the nature of the output, which is usually taken as…”

A family of H∞ controllers for dissipative Hamiltonian systems

“…An elegant and efficient way towards solving these more complex control problems is to search over the set of controllers solving the underlying H ∞ control problems for a controller satisfying the additional design objectives. Therefore, controller parameterization is a very fundamental problem in the control theory and has drawn considerable attention over the recent decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

“…Owing to their good structure, clear physical expression and the fact that the Hamiltonian function in such a system is a good candidate of Lyapunov functions, they have many advantages over others in control designs. By now, the generalized Hamiltonian systems have been successfully applied to the control of many practical systems and obtained a lot of achievement [11][12][13][14][15][16]. However, to our best knowledge, no discussion has yet been given on how to obtain a family of H ∞ controllers for generalized Hamiltonian systems.…”