In Model Predictive Control (MPC), optimization problems are solved recurrently to produce control actions. When MPC is used in real time to control safety-critical systems, it is important to solve these optimization problems with guarantees on the worst-case execution time. In this thesis, we take aim at such worst-case guarantees through two complementary approaches:

(i) By developing methods that determine exact worst-case bounds on the computational complexity and execution time for deployed optimization solvers.

(ii) By developing efficient optimization solvers that are tailored for the given application and hardware at hand.

We focus on linear MPC, which means that the optimization problems in question are quadratic programs (QPs) that depend on parameters such as system states and reference signals. For solving such QPs, we consider active-set methods: a popular class of optimization algorithms used in real-time applications.

The first part of the thesis concerns complexity certification of well-established active-set methods. First, we propose a certification framework that determines the sequence of subproblems that a class of active-set algorithms needs to solve, for every possible QP instance that might arise from a given linear MPC problem (i.e., for every possible state and reference signal). By knowing these sequences, one can exactly bound the number of iterations and/or floating-point operations that are required to compute a solution. In a second contribution, we use this framework to determine the exact worst-case execution time (WCET) for linear MPC. This requires factors such as hardware and software implementation/compilation to be accounted for in the analysis. The framework is further extended in a third contribution by accounting for internal numerical errors in the solver that is certified. In a similar vein, a fourth contribution extends the framework to handle proximal-point iterations, which can be used to improve the numerical stability of QP solvers, furthering their reliability.

The second part of the thesis concerns efficient solvers for real-time MPC. We propose an efficient active-set solver that is contained in the above-mentioned complexity-certification framework. In addition to being real-time certifiable, we show that the solver is efficient, simple to implement, can easily be warm-started, and is numerically stable, all of which are important properties for a solver that is used in real-time MPC applications. As a final contribution, we use this solver to exemplify how the proposed complexity-certification framework developed in the first part can be used to tailor active-set solvers for a given linear MPC application. Specifically, we do this by constructing and certifying parameter-varying initializations of the solver.