(570f) Homotopy Continuation Solution Method with Advanced Step Implementation for Nonlinear Model Predictive Control

Nonlinear model predictive control (NMPC) is a powerful control algorithm that enables the satisfaction of complex control objectives, and the systematic consideration of all process dynamic interactions for multivariable control systems. The implementation of model predictive control in highly nonlinear dynamic systems that exhibit fast dynamics may become prohibitive if the solution time of the associated dynamic optimization problem is comparable to the duration of the control interval suitable for adequate control performance. Therefore, either a delayed control action is applied to the system or a simplified model usually a linear realization of the nonlinear dynamic model is used hence compromising the achieved control performance.

Over the years a large number of efficient real-time strategies have been proposed such as explicit NMPC, Newton-type controllers and NLP sensitivity-based controllers. In the present work, the NMPC problem is formulated as a homotopy between the calculated known optimal solution at the previous control interval and the seeking optimal solution at the current control interval. Specifically, the nonlinear dynamic optimization problem is discretized using orthogonal collocation on finite elements and transformed into a large-scale nonlinear program. The Karush-Kuhn-Tucker optimality conditions parameterized with respect to the initial conditions are then solved within a homotopy continuation framework. Starting from a known optimal control sequence that corresponds to the initial states of the previous control interval the optimal solution is pathfollowed using a sparse arc-length continuation solution technique to the new set of initial conditions that reflect the current plant state after the latest measurement information has been taken into consideration. Solution of the sparse Newton step is performed using UMFPACK /1, 2/ whereas the implementation of the continuation solution method is performed using PITCON /3/. Pathfollowing the optimal solution is appropriately modified in order to identify active set changes imposed by either the violation of model constraints or the complementarity slackness condition along the solution trajectory. Furthermore, other singular points due to linear independence constraint qualification or second-order optimality condition violations are effectively treated along the continuation path.

The implementation of the homotopy NMPC exploits the advanced-step algorithm /4/ by utilizing the control interval for the pathfollowing procedure to target the predicted state for the initial state vector of the system during the current control interval for further reduction in the computational time. The correction to the state values imposed after the incorporation of the new measurement set starts from the predicted initial state values. The method appears to provide acceptable speed of solution with enhanced robustness in terms of convergence for a highly nonlinear system with considerable fast dynamics (control interval equal to 50 ms) consisted of a cart with a double pendulum. A second example considers the control of the molecular weight distribution for a bulk polymerization reactor.

1. A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, T. A. Davis, ACM Transactions on Mathematical Software, vol 30, no. 2, June 2004, pp. 165-195.
2. Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, T. A. Davis, ACM Transactions on Mathematical Software, vol 30, no. 2, June 2004, pp. 196-199.
3. Rheinboldt W.C., and J.V. Burkardt (1983). A locally parameterized continuation process. ACM Trans on Mathematical Software, 9, 215-235.
4. Zavala V. M., Biegler L. T., (2009). The advanced-step NMPC controller: Optimality, stability and robustness, Automatica, 45, 86-93.