Effects of actuator limits in bifurcation control with applications to active control of fluid instabilities in turbomachinery

Citation

Wang, Yong (2000) Effects of actuator limits in bifurcation control with applications to active control of fluid instabilities in turbomachinery. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/229f-f376. https://resolver.caltech.edu/CaltechETD:etd-06282005-103901

Abstract

Feedback stabilization is one of the most dominant issues in modern control theory. The validity of linear control design is based on the assumption that the system is stabilizable. With rapid broadening of control applications to complex systems during the past two decades, the attainability of linear stabilizability sometimes has to compromise with system constraints and affordability of distributed actuation. The goal of this thesis is to tackle some of the problems in control of nonequilibrium behavior and to apply the theory to active control of fluid instabilities in gas turbine engines. We consider two of the simplest nontrivial scenarios in local smooth feedback stabilization: the steady-state case, when the linearly unstabilizable eigenvalue is zero; and the Hopf case, when the unstabilizable eigenvalues are a pair of pure imaginary numbers. Under certain nondegeneracy conditions, we give explicit algebraic conditions for stabilizability. And when the system is stabilizable, the stabilizing feedback can be explicitly constructed. The problem of local smooth feedback stabilization for systems with critical unstabilizable modes is closely related to bifurcation control. Under certain nondegeneracy conditions, a steady-state/Hopf bifurcation can be turned into a supercritical pitchfork/Hopf bifurcation if and only if the system is locally stabilizable at the bifurcation point. Algebraic necessary and sufficient conditions are derived under which the criticality of a simple steady-state or Hopf bifurcation can be changed to supercritical by a smooth feedback. The effects of magnitude saturation, bandwidth, and rate limits are important issues in control engineering. We give qualitative estimates of the region of attraction to the stabilized bifurcating equilibrium/periodic orbits under these constraints. We apply the above theoretical results to the Moore-Greitzer model in active control of rotating stall and surge in gas turbine engines. Though linear stabilizability can be achieved using distributed actuation, it limits the practical usefulness due to considerations of affordability and reliability. On the other hand, simple but practically promising actuation schemes such as outlet bleed valves, a couple of air injectors, and magnetic bearings will make the system loss of linear stabilizability, thus the control design becomes a challenging task. The above mentioned results in bifurcation stabilization can be applied to these cases. We analyze the effects of magnitude and rate saturations in active stall and surge control using bleed valves and magnetic bearings using the Moore-Greitzer model. The analytical formulas for bleed valve actuation give good qualitative predictions when compared with experiments. Our conclusion is that these constraints are serious limiting factors in stall control and must be addressed in practical implementation to the aircraft engines.

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