Self-tuning extremum control with constraints

The objective of extremum control is to find the (optimal) value of the control variable(s) that yields the maximum (minimum) value of a process output (or performance measure) in the presence of noise. The static response curve relating input to output in extremum control systems is inherently nonlinear and we are therefore faced with a hill-climbing problem in which the detailed contours of the hill may be unknown and other sources of uncertainty may be present due to measurement noise and/or coarseness of the hill surface. One example is the adjustment of the spark ignition angle of an automotive engine to maintain optimal torque. Model-based extremum control methods generate the control action by making use of a model obtained by some kind of identification. The authors generalise previous work. First, it is of interest to investigate as wide a class of parametric models as possible (subject to reasonable smoothness conditions). This work was driven by the (reasonable) intuition that all smooth summits look alike, and clarifies the conditions under which the estimation burden can be reduced by fixing a subset of model parameters. Secondly, the problem of input constraints has not yet been previously addressed within this framework and, therefore, as a step towards developing a self-tuning extremum control algorithm that encompasses both equality and inequality constraints, the case of a single equality constraint on the inputs is investigated.