Optimal Control of a MIMO Bioreactor System Using Direct Approach

In this paper, the optimal control of a continuous type bioreactor with multi-input-multi-output signals is presented for the two active phases: growth and stationary. The underlying criterion to be minimized generalizes the classic quadratic forms to address some crucial objectives in controlling the bioreactor. In particular, the protection of actuators against fast switching in the controller output is considered by including a weighting term of the control signal derivatives. The direct optimal control approach is used to carry out the optimization in the presence of various limiting constraints. Direct methods are based on transcribing the infinite-dimensional problem to a finite-dimensional one. In this manuscript, direct single shooting and trapezoidal collocation methods are used for transcription, and the successive quadratic programming method is employed to solve the resulting nonlinear programming problem. It is shown that the trapezoidal method is an effective method for controlling the bioreactor in all the active phases, whereas the single shooting fails in dealing with the unstable one (i.e., growth). To analyze solutions in a more accurate manner, an auxiliary criterion is defined, and then the cheap control analysis is studied. The convergence to the lowest value of the auxiliary cost function and the effects on the optimal state and control trajectories are then examined by varying cheap parameters. Several numerical simulations support the presented theoretical formulation.