Prescribing Closed-Loop Behavior Using Nonlinear Model Predictive Control

Abstract

In this work, we address the problem of control of nonlinear systems to deliver a prescribed closed-loop behavior. In particular, the framework allows for the practitioner to first specify the nature and specifics of the desired closed-loop behavior (e.g., first order with smallest time constant, second order with no more than a certain percentage overshoot, etc.). An optimization based formulation then computes the control action to deliver the best attainable closed loop behavior. To decouple the problems of determining the best attainable behavior and tracking it as closely as possible, the optimization problem is posed and solved in two tiers. In the first tier, the focus is on determining the best closed-loop behavior attainable, subject to stability and tracking constraints. In the second tier, the inputs are tweaked to possibly improve the tracking of the optimal output trajectories given by the first tier. The efficacy of the proposed method and the various specific formulations needed are illustrated through implementation on a linear system subject to output feedback, a nonlinear CSTR subject to uncertainty and rate of change of input constraints, and a reactor separator system. The simulation results demonstrate significantly improved adherence to the prescribed performance criteria over a predictive controller representative of existing approaches.

Responsive Economic Model Predictive Control for Next-Generation Manufacturing

“…To demonstrate this challenge, consider the LEMPC of Equations (17)- (23). The theoretical results for LEMPC which guarantee closed-loop stability and recursive feasibility under sufficient conditions when no changes occur in the underlying process dynamics rely on the constraints of Equations (22) and (23) being present in the control design [24]. Therefore, ad hoc constraint development in an attempt to optimize end-user "satisfaction" with the process response would not be a means for providing closed-loop stability and recursive feasibility guarantees.…”

“…The oscillatory behavior of the states before 0.5 h is caused by the fact that the profit is maximized for this process at the boundary of Ωρ e . Without plant-model mismatch, the LEMPC is able to maintain the closed-loop state exactly on the boundary of Ωρ e and therefore always operates the process using the constraint of Equation (22); however, when the plant-model mismatch occurs (induced by the use of different integration steps to simulate the process dynamic model within the LEMPC and for the simulation of the process under the computed control actions), the closed-loop state then exits Ωρ e when the LEMPC predicts it will stay inside of it under the control actions computed by the controller. The result is that the constraint of Equation (23) is then activated until the closed-loop state reenters Ωρ e .…”

“…The subset of EMPC formulations which track a steady-state [21] possess a form of interpretability in that the reference behavior is understood by engineers and operators. Reference [22] developed an EMPC formulation in which the desired closed-loop process response specified or restricted by an operator or engineer is tracked by the controller. However, developing the best means for ensuring interpretability for EMPC to appropriately trade off end user understanding with economic optimality remains a largely open question.…”

Adaptive Model Predictive Batch Process Monitoring and Control

“…Q y ∈ R n×n and R du are positive definite and positive semidefinite matrices, respectively, and they are chosen so the nominal closed-loop system is stable. 26 x̃and x̂are predicted and the estimate of the subspace state, obtained using an appropriate state estimator.…”

“…A representative model predictive control formulation for trajectory control of batch processes is as follows: where n t denotes the end of the batch, y ̃ k SP is the desired output (desired trajectory), and, y ̃ k and u ̃ k are the predicted output trajectory and input at the time k Δ t . Q y ∈ R n × n and R du are positive definite and positive semidefinite matrices, respectively, and they are chosen so the nominal closed-loop system is stable …”