A simple optimization scheme for the case of multidimensional parameter

Optimal control problems are one of the most challenging research problems with the main objective to locate the minimum or maximum of the cost function for some physical, social, economic or any other processes. The work presented in this paper is motivated by the perturbation-based extremum seeking algorithm or the so called “model-free” real time optimization where no information about the system dynamics is available. The method uses singular perturbation and averaging theories to extract the information about the gradient. We use this approach to develop a simple stabilizing algorithm for the case of vector input signal assuming that the measurements are available all the time. The output is averaged and the gradient estimated with the help of a periodic signal added to the input signal. The structure of input-output correlation is used to find the necessary conditions that would force the estimated gradient towards zero, and the output to the optimal set-point. Finally, we prove that the system operating in a small vicinity of the optimal point is stable.