Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

Intl J Robust & Nonlinear

2022

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We consider the n-mode truncation of the Moore-Greitzer partial differential equation model for axial compressors with bleed valve as actuator. It is well known that all the rotating stall modes are linearly unstabilizable past the peak of the compressor characteristic. By using the center manifold and normal form reduction, we obtain a model describing the nonlinear interactions of rotating stall and surge modes when the compressor operates in a neighborhood of the peak pressure rise. The normal form is a special class of the the famous Lotka-Volterra system. Typically, there are 2 n equilibria as the throttle coefficient varies. We analyze the stability of these equilibria for the normal form, and interpret their relevance to the robustness of the nominal equilibrium near the peak pressure rise. We also derive the necessary and sufficient conditions for stabilizing the peak of the characteristic using a quadratic feedback control law.

“…When only one stall mode is considered, our result is consistent with the previous result in Reference 24.…”

Section: Feedback Controlsupporting

confidence: 93%

Bifurcation based quadratic feedback control for Moore‐Greitzer model with surge and multiple stall modes

Intl J Robust & Nonlinear

2022

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“…σiφi(x, μ,ηi) +φ l (x, μ, η), (A20) and (A21) are just (15) and (19), respectively. The derivation is completed.…”

Section: (A15)mentioning

confidence: 99%

“…Most existing results on bifurcation stabilization are based on control synthesis via state feedback or static output feedback [5,10,14,15], and few authors have addressed the issue of stabilizing a bifurcation…”

Section: Introductionmentioning

confidence: 99%

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

Chen

Qin

et al. 2011

Sci. China Inf. Sci.

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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 ...

“…In order to provide tools for these types of engineering problems, in this paper we answer the general question of when an equilibrium bifurcation with one critical linearly unstabilizable mode can be stabilized at the bifurcation point. A second paper [32] considers the case of Hopf bifurcations.…”

Section: Motivation: Active Control Of Rotating Stall In Axial Comprementioning

confidence: 99%

“…If q 11 =0; then the bifurcation is a transcritical bifurcation (see [34]) and is unstabilizable via output feedback (32). If q 11 ¼ 0; then the bifurcation is a pitchfork bifurcation and it is stabilizable via output feedback (32) if there exists K 1 2 R mÂðnÀ1Þ ; such that the following inequality holds:…”

Section: Corollary 34mentioning

confidence: 99%

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part I: equilibrium bifurcations

Intl J Robust & Nonlinear

Murray

2006

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SUMMARYUnder certain non-degeneracy conditions, necessary and sufficient conditions for stabilizability are obtained for multi-input nonlinear systems possessing a simple equilibrium bifurcation with the critical mode being linearly uncontrollable. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the bifurcation of the closed-loop system is a supercritical pitchfork bifurcation, which is equivalent to local asymptotic stability of the system at the bifurcation point. Stabilizability is reduced to the existence of solutions to a third order algebraic inequality, coupled with a quadratic algebraic equation, with the unknowns being the feedback gains. Explicit conditions for the existence of solutions to the algebraic equation and the inequality are derived. Part of the sufficient conditions are equivalent to the rank conditions of augmented matrices. Conditions under which there exists a stabilizing linear feedback law are also derived. The theoretical results are applied to active control of rotating stall in axial compressors using bleed valve as actuator.