Real-time optimization using a jamming-free switching logic for gradient projection on active constraints

This paper addresses the problem of extremum-seeking (real-time steady state optimization of dynamic systems via control of the gradient) under inequality constraints in continuous time. A gradient projection framework is adopted in this paper, where the main challenge lies in the identification of the set of active constraints. It is first shown that if the discrete-time switching logic widely used in the literature is directly used in continuous-time, it could cause jamming, i.e., getting stuck in a non-optimal solution. So, an appropriate normalization is proposed to get a jamming-free continuous-time switching logic. The absence of jamming is rigourously proved and is also illustrated on a numerical example. In addition, real-time optimization of the desired product of a reaction taking place in an isothermal continuous stirred-tank reactor is studied, where the gradient of the objective and the constraints are calculated simultaneously using a multi-unit framework.