Adapting the manipulated variables of industrial processes on line in order to seek optimal operation forms the focus of this work. For optimizing the steady-state of a nonlinear dynamic system, one approach therein, termed the extremum seeking, transforms the optimization problem into a problem of controlling the gradient to zero. In the first part of the talk, several algorithms available for this class of problems will be reviewed and a simple algorithm using a local static approximation of the system to achieve an extremum seeking control will be proposed.
In the second part, we examine in more details the perturbation-based extremum seeking methods. For these methods the crucial problem of gradient estimation is addressed by using input perturbations. The gradient is computed from the correlation between the input and output variations and then pushed to zero. In other words, the perturbation method seeks an operating point (A) that provides zero correlation between the input and output variations at the given frequency. However, what is sought is the steady-state optimum (B) of a nonlinear dynamical system, where the static gain is zero.
Though this approach is widely used, it is shown in the current work that these two operating points (A and B) do not necessarily match. The above methodology can lead to the optimum on the average (A = B) only when the linear transfer function obtained from the nonlinear system at the static optimum is identically zero. Otherwise, the algorithm converges, on the average, to a false solution and the error is proportional to the frequency of excitation. Simulation results on an isothermal continuous stirred tank reactor and a non-isothermal plug flow reactor are used to illustrate the concepts presented in this work.