Several control engineering problems require the determination of parameters that optimize a system level cost function in real time. Real-time optimization is necessary to seek optimal control parameters in many application areas such as vibration and noise attenuation, flow separation, combustion control, and control of flying formations. In these problems the control architectures that improve system operation (e.g., minimize a noise or vibration figure, minimize flow separation in an airfoil, minimize the unsteady pressure fluctuation in a combustion chamber) are known in advance. On the other hand, the optimal control parameters are not known in advance and must be determined.
Off-line calculation of the optimal parameters is impractical when no reliable model is available to predict the variation of the cost function with time, the optimization parameters, or the system's operating conditions. It is generally possible, however, to make real-time measurements of the cost function through the addition of sensors and data processing. This additional hardware and software opens up the possibility of calculating optimal parameters by 'experimenting with the system' in order to determine the parameter setting that leads to a cost function improvement. The practical implementation of this idea requires an iterative algorithm that seeks the optimal parameters in real time.
The main objective of this project is the development of new iterative algorithms for the real-time optimization of a measurable cost function. Iterative 'extremum-seeking algorithms' that make use of function evaluations only, to estimate optimal parameters, will be considered. The challenge is to derive algorithms that (I) track fast variations in the optimal parameters, (II) are insensitive to the noise present in the measurements of the cost function, and (III) exhibit monotonic improvement of the cost function during the course of the optimization.
The focus shall be on 'gradient-based iterative algorithms' for which the necessary gradient information is not available and it must be estimated from the measurements of the cost function. Analysis of the algorithm performance, and sensitivity to modeling assumptions, will be carried out using methods and tools from the theory of nonlinear uncertain dynamical systems. Performance and sensitivity (or robustness) bounds will be obtained from the combination of traditional differential equations methods (averaging techniques) with recent tools from uncertain dynamical systems analysis. The bounds will later be used to synthesize algorithms that optimize criteria (I)-(III) above from available prior information.
If successful, the proposed research will advance the state-of-the-art of real-time gradient-based optimization algorithms. A novel framework for analysis that takes into account fast variations in optimal parameters and measurement noise will be developed. New systematic procedures to design extremum-seeking algorithms that work in uncertain noisy environments will be created. These design procedures would allow engineers to implement extremum-seeking algorithms with minimal development effort. ***
