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

Wang, Yong; Murray, Richard M.

doi:10.1002/rnc.1134

Cited by 6 publications

(5 citation statements)

References 35 publications

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“…The Moore-Greitzer model (see [8][9][10]) for an axial compression system is given by ' F¼ mðÀC þ c c ðFÞ þ c 00 c ðFÞjzj 2 Þ þ h:o:t:…”

Section: An Example In Active Control Of Rotating Stallmentioning

confidence: 99%

“…where the meanings of all the variables and parameters are given in [10]. It is clear that the system has a Hopf bifurcation at the peak of the compressor characteristic c c ðFÞ: By defining the nominal equilibria as the steady and axisymmetric flow condition: ðF; C; zÞ ¼ ðF 0 ðgÞ; C 0 ðgÞ; 0Þ; satisfying C 0 ðgÞ ¼ c c ðF 0 ðgÞÞ and F 0 ðgÞ ¼ g ffiffiffiffiffiffiffiffiffiffiffi ffi C 0 ðgÞ p…”

Section: An Example In Active Control Of Rotating Stallmentioning

confidence: 99%

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

Wang

Murray

2006

Intl J Robust & Nonlinear

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SUMMARYIn this paper we derive necessary and sufficient conditions of stabilizability for multi-input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non-degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed-loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non-degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov-Belevitch-Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators.

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“…The Moore-Greitzer model (see [8][9][10]) for an axial compression system is given by ' F¼ mðÀC þ c c ðFÞ þ c 00 c ðFÞjzj 2 Þ þ h:o:t:…”

Section: An Example In Active Control Of Rotating Stallmentioning

confidence: 99%

Section: An Example In Active Control Of Rotating Stallmentioning

confidence: 99%

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

Wang

Murray

2006

Intl J Robust & Nonlinear

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“…Clearly, since the critical modes are neither controllable nor observable. Let (13) where The Lyapunov function is in the form:…”

Section: Local Gain Analysismentioning

confidence: 99%

“…For the past two decades, stabilizing control of bifurcated systems has drawn a lot of attention from the control community [1], [6], [8], [13]. Regarding these progresses, an interesting question raised further is: to what extent, can these stabilizing control designs be deemed as robust?…”

Section: Introductionmentioning

confidence: 98%

Local Robustness of Hopf Bifurcation Stabilization

Yang

Chen

2009

IEEE Trans. Circuits Syst. II

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Local robust analysis via 2 gain method is presented for a class of Hopf bifurcation stabilizing controllers. In particular, we first construct a family of Lyapunov functions for the corresponding critical system, then derive a sufficient condition to compute the 2 gain by solving the Hamilton-Jacobi-Bellman (HJB) inequalities. Local robust analysis can be conducted through computing the local 2 gain achieved by the stabilizing controllers at the critical situation. The theoretical results obtained in this brief provide useful guidance for selecting a robust controller from a given class of stabilizing controllers under Hopf bifurcation. As an example, application to a modified Van der Pol oscillator is discussed in details.

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“…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

Wang

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 ...

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

Cited by 6 publications

References 35 publications

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

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

Local Robustness of Hopf Bifurcation Stabilization

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

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