Advances in Decision-making Under Uncertainty with Nonlinear Model Predictive Control

Abstract

Model predictive control (MPC) has been a very successful advanced process control technique for many applications especially in process industries because of its ability to handle hard constraints and multiple inputs and outputs. However, the presence of uncertainty deteriorates the performance of nominal MPC, and a robust model predictive control becomes necessary. The existing development of robust (nonlinear) MPC has yet to be widely applied in industry, mainly due to conservatism of the algorithm and also the impracticality of implementation; thereby robust NMPC has been mostly conceptual until a scenariobasedrobust NMPC recently emerged. A scenario tree is generated to represent the evolution of states with respect to uncertain parameters, and a multistage stochastic programming formulation has been employed to continuously solve for the optimal control actionin a moving horizon fashion. This is a good place to start, however, many challenges still remain and need to be addressed. This thesis develops easily implementable robust NMPC strategies which provide performance guarantees and computational efficiency. One of the major issues of multistage NMPC approaches is computational complexity. Due to the construction of the scenario tree, multistage NMPC models are inevitably larger than their nominal counterparts, and their size grows exponentially with respect to the number of uncertain parameters and the length of robust horizons. To solve this issue, we present an efficient parallelizable advanced-step multistage NMPC (as-msNMPC) approach, which explicitly deals with two types of uncertainty: model parameters and unmeasured noise. The first type is attended to by incorporating multistage scenario trees and the second by applying nonlinear programming (NLP) sensitivity. The framework of as-msNMPC has been demonstrated on two examples with robust performance andsignificantly faster online computation compared to benchmark methods. We also conduct a stability analysis for the newly constructed robust NMPC schemes. Based on Lyapunov stability theory, we show that the origin is asymptotically stable under standard NMPC if the model is perfect. And when additive disturbance is present, thesystem is robustly stable to a neighborhood around the origin. When the system allows both types of uncertainty, we first show that recursive feasibility can be ensured for fully expanded multistage NMPC under mild assumptions. With both advanced-step and ideal multistage NMPC, we then show that robust stability can be achieved with input-to-statepractical stability (ISpS). Last but not least, a sensitivity-assisted multistage NMPC (samNMPC) algorithm hasbeen proposed to deal with the size of the multistage formulation on a linear algebra level. A block-bordered-diagonal structure of the KKT matrix naturally arises with themultistage NLP problem, and Schur complement decomposition can be performed to decouple scenarios. In this case, we can approximate many scenarios with the solution to the nominal scenario. In addition, we apply a scenario generation technique to determine which scenario is most likely to violate constraints. We then formulate an approximate multistage formulation that has a much smaller problem size. The tracking performance of samNMPC and exact multistage NMPC have been compared and similar performance has been reached with only a fraction of the computational effort of the exact multistage formulation.