# Effects of Actuator Limits in Bifurcation Control with Applications to Active Control of Fluid Instabilities in Turbomachinery

### Yong Wang

PhD Dissertation, California Institute of Technology, June 2000

Bifurcations are ubiquitous in engineering applications.

Subcritical bifurcations are typically associated with hysteresis and catastrophic instability inception, while supercritical bifurcations are usually associated with gradual and more benign instability inception. With the assumption that the bifurcating modes are linearly unstabilizable, we give a constructive procedure of designing feedback laws to change the criticality of bifurcations from subcritical to supercritical. Algebraic necessary and sufficient conditions are obtained 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 theory in bifurcation stabilization can be applied to these cases. We analyze the effects of magnitude and rate saturations in active control of rotating stall using bleed valves. Analytic formulas are obtained for the operability enhancement as a function of system parameters, noise level, and actuator magnitude and rate limits. The formulas give good qualitative predictions when compared with experiments. Our conclusion is that actuator magnitude and rate limits are serious limiting factors in stall control and must be addressed in practical implementation to the aircraft engines. <p>

- PhD Dissertation: http://www.cds.caltech.edu/~murray/preprints/yw00-phd.pdf
- Project(s): Template:HTDB funding::AFOSR