Efficient Methods for Control of Dynamical Systems

Abstract

The thesis addresses several critical challenges in the implementation of Model Predictive Control (MPC) for online settings, with a focus on the numerical strategies employed in solving the inherent optimisation problem at the centre of MPC.

First, an MPC-specific early termination condition is considered for the family of interior-point solvers. The proposed condition allows the computational efforts associated with solving a class of MPC problems to be reduced without compromising the stability properties of the closed-loop system.

Second, it is assumed that an optimisation algorithm has already been selected, and the design of a suboptimal MPC algorithm without terminal conditions is required. The proposed design approach considers the MPC problem horizon length and an acceptable suboptimality degree to minimise the algorithmic complexity associated with finding a solution. To this end, the stabilising properties of the feasible suboptimal solutions (with an appropriately defined measure of suboptimality with direct links with the closed-loop performance of the system) are utilised, along with the ability to estimate the algorithmic complexity of the process of obtaining such solutions. Through numerical simulations, it is shown that the smallest stabilising prediction horizon is not necessarily the optimal choice, and the complexity can be further reduced using a larger horizon length. This is shown to be consistent with the predictions obtained from the developed framework.

Third, the case where the constraint-respecting stabilising control law is to be constructed using a set of precomputed (sub)optimal control laws. A framework for approximating the optimal control law with a special family of barycentric functions and the corresponding stability certification method is proposed. The proposed stability certificate is less conservative than the state-of-the-art approaches, which results in the method to require fewer precomputed control laws. The proposed methodology demonstrates sub-exponential growth of the number of approximation sub-regions, and potentially allows for Approximate Explicit MPC to be applied to a broader range of systems.

Finally, a novel family of algorithms for solving finite-time optimal control problems with state and input constraints is proposed. The aforementioned family, termed interior-point DDP algorithms (IPDDP), are a product of combining the interior-point and differential dynamic programming (DDP) ideas. The interior-point DDP algorithms are of linear complexity in the problem's size and can either handle infeasible solution guesses or preserve the feasibility at all times. The IPDDP method is shown to have a local quadratic convergence without appealing to any convexity properties of the associated problem.

Once these three main contributions of the thesis are completed, further potential research directions and extensions are outlined as avenues for future work.