Bifurcation stabilization of nonlinear systems by dynamic output feedback with application to rotating stall control

Abstract

The paper deals with the problem of stabilization of stationary bifurcation solutions of nonlinear systems via dynamic output feedback. It is emphasized that the parameter of the system is not directly available. We introduce the concepts of uniform observability of the inverse of a function of state and input and N -orderinput-to-state bifurcation stability. Based on the concepts, we propose a new method for designing dynamic compensators that guarantee bifurcation stability for the closed-loop system. As an example, we apply the general theory to active control of rotating stall in axial flow compressors by designing a stabilizing dynamic compensator for the three-state Moore-Greitzer model with a class of cubic compressor characteristics. CitationChen P N, Qin H S, Wang Y, et al. Bifurcation stabilization of nonlinear systems by dynamic output feedback with application to rotating stall control. 201via a dynamic output feedback. Washout filters and high-pass filters were used in [16] for bifurcation stabilization of nonlinear systems via dynamic output feedback. The methods presented in [5,10,11,16] are applicable to dynamic output feedback bifurcation stabilization only in the case where the dimension of the dynamic compensator is known. However, in general, the dimension of the dynamic compensator cannot be determined a priori. Thus, the theory on bifurcation stabilization via dynamic output feedback is far from complete. Moreover, even in the rotating stall control, the output feedback design for the compressor system with pressure rise as output measurement is still a challenging problem [10,11]. In the cases where some state variables are not measurable, output feedback control design of nonlinear systems becomes necessary and has been studied in different ways [17,18]. Center manifold and normal form theory provides an effective method for both nonlinear control and bifurcation analysis [19,20]. In the output feedback control design related to center manifold, two concepts of the approximation stability [18,21] and uniform observability [17,18] have often been used.In this paper, we propose a new method for dynamic output feedback stabilization of stationary bifurcation. Here the bifurcation parameters of the system are not available, and it is difficult to make conventional uniform observability of an input function and approximation stability applicable in this case. To deal with the difficulties, two generalized concepts are proposed. One is uniform observability of the inverse of a function of state and input as a generalization of uniform observability of a function of state, and the other is N -order-input-to-state bifurcation stability as an extension of approximation stability. Based on these two concepts, we provide an explicit procedure of constructing an dynamic compensator for bifurcation stabilization. As an application, we construct a dynamic compensator for the Moore-Greitzer model in active control of rotating stall for general cubic compressor characteristic curves with pressure-rise as ...

Static output feedback stabilization for systems with time-varying delay based on a matrix transformation method

“…Remark 2. Theorem 1 applies the non-uniform delay-dividing approach to obtain the less conservative results of closed-loop system (5). Because the sub-intervals with different sizes are obtained by non-uniformly dividing delay interval, less conservative results can be obtained based on the different combinations formed by these sub-intervals, which is certain to be better than the uniform delay-dividing methods, see [28].…”

“…In the following paragraphs, the Theorem 2 of [26] will be rewritten to deal with the static output feedback stabilization of system (5). It is given as follows.…”

“…Static output feedback stabilization and control problems of the delayed system are still the difficult topics, even if only a linear system with one delay is considered [1]. However, because it can transform the original delayed systems to simple closed-loop ones, the static output feedback problem can be considered as an elementary scheme for the stabilization and control of systems and is more easily realized in both theory and practice see [2][3][4][5][6][7][8][9][10][11]. Thus, it has attracted much attention of researchers, especially because of the development of the delayed systems such as networked control systems [12][13][14][15][16].…”