Simultaneous stabilization of a class of nonlinear descriptor systems via Hamiltonian function method

Abstract

This paper studies simultaneous stabilization of a class of nonlinear descriptor systems via the Hamiltonian function method. Firstly, based on the Hamiltonian realization of the nonlinear descriptor systems and a suitable output feedback, two nonlinear descriptor systems are equivalently transformed into two nonlinear Hamiltonian differential-algebraic systems by a nonsingular transformation, and a sufficient condition for two closed-loop systems to be impulse-free is given. The two systems are then combined to generate an augmented dissipative Hamiltonian differential-algebraic system by using the system-augmentation technique, based on which a simultaneous stabilization controller and a robust simultaneous stabilization controller are designed for the two systems. Secondly, the case of more than two nonlinear descriptor systems is investigated, and two new results are proposed for the simultaneous stabilization and robust simultaneous stabilization, respectively. Finally, an illustrative example is studied by using the results proposed in this paper, and simulations show that the simultaneous stabilization controllers obtained in this paper work very well.nonlinear descriptor systems, simultaneous stabilization, robust simultaneous stabilization, Hamiltonian function method

H ∞ control of a class of nonlinear Hamiltonian descriptor systems

“…Based on the equivalent transformation in section 2, we can obtain a dissipative Hamiltonian differential algebraic form for system (12), which leads to the following lemma.…”

“…where J 1 (x) and R 1 (x) are the same as in (6) (1), we can obtain equivalent form (14) of system (12). Now, we first design an H ∞ controller for equivalent system (14), and then use the obtained controller to derive the desired H ∞ control law for the original system (12).…”

“…Now, we first design an H ∞ controller for equivalent system (14), and then use the obtained controller to derive the desired H ∞ control law for the original system (12). Theorem 3.6.…”

“…Using the linear matrix inequality approach, Masubuchi et al [3] and Xu et al [4] investigated the H ∞ control of both linear continuous descriptor systems and linear discrete descriptor systems, respectively, while Lin et al [5] discussed the robust stabilization problem for uncertain linear descriptor systems by using output feedback technique. In general, it is difficult to design a controller for a nonlinear descriptor system (NDS), and accordingly, there are fewer studies on NDS except several special cases [6][7][8][9][10][11][12][13]. When the singular matrix E = Diag{I r , 0} [6][7][8][9], Liu et al [6,7] studied the stabilization problem and robust stabilization problem of NDSs by the feedback linearization approach, while Liu et al considered the same problem via the Hamiltonian function method in [8].…”

“…In [10], Yang et al studied the generalized absolute stability for linear descriptor systems with feedback-connected nonlinearities by the linear matrix inequality method. Sun et al investigated the stabilization and H ∞ control problem for a class of nonlinear Hamiltonian descriptor systems of constant structure matrix in [11], and discussed the simultaneous stabilization problem for a class of NDSs via the Hamiltonian function method in [12].…”