Hsin-Hsiung Wang

Optimizing bioreactors by extremum seeking

The optimization of the operation of biological reactors is an interesting non-linear problem whose solution offers potential economic benefit. We apply a peak seeking method to approach the maximum biomass production rate in a continuous stirred tank bioreactor. Two models, Monod and Haldane, are investigated and it is shown by simulation that the peak seeking scheme achieves optimization for both cases. A stabilizing feedback controller with a washout filter is designed to extend the operating range for the Haldane model. Copyright

Experimental application of extremum seeking on an axial-flow compressor

We show an application of the method of extremum seeking to the problem of maximizing the pressure rise in an axial-flow compressor. First we apply extremum seeking to the Moore-Greitzer model and design a feedback scheme actuated through a bleed valve which simultaneously stabilizes rotating stall and surge and steers the system towards the equilibrium with maximal pressure. Then we implement the scheme on a compressor rig in Murrays laboratory at the California Institute of Technology. We perform stabilization of rotating stall via air injection and implement extremum seeking through a slow bleed valve. The experiment demonstrates that extremum seeking ensures the maximization of the pressure rise starting on either side of the stall inception point. The experiment also resolves a concern that extremum seeking requires the use of periodic probing-the amplitude of probing needed to achieve convergence is far below the noise level of the compressor system (even outside rotating stall).

Design and stability analysis of extremum seeking feedback for general nonlinear systems

In this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis. Our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block-we allow the plant to be a general nonlinear dynamic system (possibly non-affine in control and open-loop unstable) whose reference-to-output equilibrium map has a maximum, and whose equilibria are locally exponentially stabilizable.

Stabilization of period doubling bifurcation and implications for control of chaos

The stabilization of period doubling bifurcations for discrete-time nonlinear systems is investigated. It is shown that generically such bifurcations can be stabilized using smooth feedback, even if the linearized system is uncontrollable at criticality. In the course of the analysis, expressions are derived for bifurcation stability coefficients of general n-dimensional systems undergoing period doubling bifurcation. A connection is determined between control of the amplitude of a period doubled orbit and the elimination of a period doubling cascade to chaos. For illustration, the results are applied to the Henon attractor.<<ETX>>

Control of Deep-Hysteresis Aeroengine Compressors

Frequencies of higher-order modes of fluid dynamic phenomena participating in aeroengine compressor instabilities far exceed the bandwidth of available (affordable) actuators. For this reason, most of the heretofore experimentally validated control designs for aeroengine compressors have been via low-order models-specifically, via the famous Moore-Greitzer cubic model (MG3). While MG3 provides a good qualitative description of open-loop dynamic behavior, it does not capture the main difficulties for control design. In particular, it fails to exhibit the so-called right-skew property which distinguishes the deep hysteresis observed on high-performance axial compressors from a small hysteresis present in the MG3 model. In this paper we study fundamental feedback control problems associated with deep-hysteresis compressors. We first derive a parametrization of the MG3 model which exhibits the right skew property. Our approach is based on representing the compressor characteristic as a convex combination of a usual cubic polynomial and a nonpolynomial term carefully chosen so that an entire family of right-skew compressors can be spanned using a single parameter ∈. Then we develop a family of controllers which are applicable not only to the particular parametrization, but to general Moore-Greitzer type models with arbitrary compressor characteristics. For each of our controllers we show that it achieves a supercritical (soft) bifurcation, that is, instead of an abrupt drop into rotating stall, it guarantees a gentle descent with a small stall amplitude. Two of the controllers have novel, simple, sensing requirements: one employs only the measurement of pressure rise and rotating stall amplitude, while the other uses only pressure rise and the mass flow rate (1D sensing). Some of the controllers which show excellent results for the MG3 model fail on the deep-hysteresis compressor model, thus justifying our focus on deep-hysteresis compressors. Our results also confirm experimentally observed difficulties for control of compressors that have a high value of Greitzers B parameter. We address another key issue for control of rotating stall and surge-the limited actuator bandwidth-which is critical because even the fastest control valves are often too slow compared to the rates of compressor instabilities. Our conditions show an interesting trade-off: as the actuator bandwidth decreases, the sensing requirements become more demanding. Finally, we go on to disprove a general conjecture in the compressor control community that the feedback of mass flow rate, known to be beneficial for shallow-hysteresis compressors, is also beneficial for deep-hysteresis compressors.

Extremum seeking for limit cycle minimization

In many physical problems equilibrium stabilization is not possible and the controlled system is in a limit cycle (of a possibly reduced size). If the size of the limit cycle depends on some of the control parameters, then a reasonable objective would be to tune this parameter to minimize the size of the limit cycle. In this paper we propose a method for achieving this. This method is an extension of our earlier result (CDC 1997) on extremum seeking for equilibria. We illustrate the method with a Van der Pol oscillator example and present analysis using averaging and singular perturbations.

Control of deep-hysteresis aeroengine compressors. II. Design of control laws

In this paper we continue the development of a methodology for control of deep-hysteresis compressors initiated in a companion paper. We develop a family of controllers which are applicable not only to the particular model presented in Part I of this paper, but also to general Moore-Greitzer type models with arbitrary compressor characteristics. For each of our controllers we show that it achieves a supercritical (soft) bifurcation. We also address another key issue for control of rotating stall and surge - the limited actuator bandwidth. Our results show an interesting trade-off: as the actuator bandwidth decreases, the sensing requirements become more demanding.

Voltage Collapse Dynamics and Control in a Sample Power System

Abstract Nonlinear phenomena, including bifurcations, chaos and crises, have been determined to be crucial factors in the inception of voltage collapse in power system models. The issue of controlling voltage collapse in the presence of these nonlinear phenomena is addressed in this paper. The work employs a example power system model akin to one studied in several recent papers. The bifurcation control approach is employed to modify the bifurcations and to suppress chaos and crises. The control law is shown to result in improved performance of the system for a greater range of parameter values

Optimum-Seeking by Feedback For Bioreactor Models

Abstract The optimization of the operation of biological reactors is an interesting nonlinear problem whose solution offers potential economic benefit. We apply a peak seeking method to approach the maximum biomass production rate in a continuous stirred tank bioreactor. Two models, Monod and Haldane, are investigated and it is shown by simulation that the peak seeking scheme achieves optimization for both cases. A stabilizing feedback controller with a washout filter is designed to extend the operating range for the Haldane model.