Pablo Krupa’s research while affiliated with Gran Sasso Science Institute and other places

Model predictive control (MPC) typically includes a terminal constraint to guarantee stability of the closed-loop system under nominal conditions. In linear MPC, this constraint is generally taken on a polyhedral set, leading to a quadratic optimization problem. However, the use of an ellipsoidal terminal constraint may be desirable, leading to an optimization problem with a quadratic constraint. In this case, the optimization problem can be solved using second-order cone (SOC) programming solvers, since the quadratic constraint can be posed as a SOC constraint, at the expense of adding additional slack variables and possibly compromising the simple structure of the solver ingredients. In this brief, we present a sparse solver for linear MPC subject to a terminal ellipsoidal constraint based on the alternating direction method of multipliers (ADMM) algorithm in which we directly deal with the quadratic constraints without having to resort to the use of a SOC constraint nor the inclusion of additional decision variables. The solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint, and requires no external libraries. We compare its performance against other approaches from the literature.

The main benefit of model predictive control (MPC) is its ability to steer the system to a given reference without violating the constraints while minimizing some objective. Furthermore, a suitably designed MPC controller guarantees asymptotic stability of the closed-loop system to the given reference as long as its optimization problem is feasible at the initial state of the system. Therefore, one of the limitations of classical MPC is that changing the reference may lead to an unfeasible MPC problem. Furthermore, due to a lack of deep knowledge of the system, it is possible for the user to provide a desired reference that is unfeasible or non-attainable for the MPC controller, leading to the same problem. This chapter summarizes MPC formulations recently proposed that have been designed to address these issues. In particular, thanks to the addition of an artificial reference as decision variable, the formulations achieve asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints. We show a recent formulation which extends this idea, achieving better performance and larger domains of attraction when working with small prediction horizons. Additional benefits of these formulations, when compared to classical MPC, are also discussed and highlighted with illustrative examples.

This paper provides a comprehensive tutorial on a family of Model Predictive Control (MPC) formulations, known as MPC for tracking, which are characterized by including an artificial reference as part of the decision variables in the optimization problem. These formulations have several benefits with respect to the classical MPC formulations, including guaranteed recursive feasibility under online reference changes, as well as asymptotic stability and an increased domain of attraction. This tutorial paper introduces the concept of using an artificial reference in MPC, presenting the benefits and theoretical guarantees obtained by its use. We then provide a survey of the main advances and extensions of the original linear MPC for tracking, including its non-linear extension. Additionally, we discuss its application to learning-based MPC, and discuss optimization aspects related to its implementation.

Model Predictive Control (MPC) is a popular control approach due to its ability to consider constraints, including input and state restrictions, while minimizing a cost function. However, in practice, these constraints can result in feasibility issues, either because the system model is not accurate or due to the existence of external disturbances. To mitigate this problem, a solution adopted by the MPC community is the use of soft constraints. In this article, we consider a not-so-typical methodology to encode soft constraints in a particular MPC formulation known as MPC for Tracking (MPCT), which has several advantages when compared to standard MPC formulations. The motivation behind the proposed encoding is to maintain the semi-banded structure of the ingredients of a recently proposed solver for the considered MPCT formulation, thus providing an efficient and fast solver when compared to alternative approaches from the literature. We show numerical results highlighting the benefits of the formulation and the computational efficiency of the solver.

Harmonic model predictive control (HMPC) is a model predictive control (MPC) formulation that displays several benefits over other MPC formulations, especially when using a small prediction horizon. These benefits, however, come at the expense of an optimization problem that is no longer the typical quadratic programming problem derived from most linear MPC formulations due to the inclusion of a particular class of second-order cone constraints. This article presents a method for efficiently dealing with these constraints in operator splitting methods, leading to a computation time for solving HMPC in line with state-of-the-art solvers for linear MPC. We show how to apply this result to the alternating direction method of the multipliers algorithm, presenting a solver that we compare against other solvers from the literature, including solvers for other linear MPC formulations. The results show that the proposed solver, and by extension the HMPC formulation, is suitable for its implementation in embedded systems.

In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of optimization algorithms. However, a typical underlying assumption is that the operations performed by the algorithm are exact, i.e., that there is no numerical error during the mathematical operations, which is hardly a valid assumption in a real hardware implementation. This is particularly true in the case of fixed-point hardware, where computational inaccuracies are not uncommon. This article presents a certification procedure for the proximal gradient method for box-constrained QP problems implemented in fixed-point arithmetic. The procedure provides a method to select the minimal fractional precision required to obtain a certain suboptimality bound, indicating the maximum number of iterations of the optimization method required to obtain it. The procedure makes use of formal verification methods to provide arbitrarily tight bounds on the suboptimality guarantee. We apply the proposed certification procedure on the implementation of a non-trivial model predictive controller on 32-bit fixed-point hardware.

Model Predictive Control (MPC) is typically characterized for being computationally demanding, as it requires solving optimization problems online; a particularly relevant point when considering its implementation in embedded systems. To reduce the computational burden of the optimization algorithm, most solvers perform as many offline operations as possible, typically performing the computation and factorization of its expensive matrices offline and then storing them in the embedded system. This improves the efficiency of the solver, with the disadvantage that online changes on some of the ingredients of the MPC formulation require performing these expensive computations online. This article presents an efficient algorithm for the factorization of the key matrix used in several first-order optimization methods applied to linear MPC formulations, allowing its prediction model and cost function matrices to be updated online at the expense of a small computational cost. We show results comparing the proposed approach with other solvers from the literature applied to a linear time-varying system.

Many popular approaches in the field of robust model predictive control (MPC) are based on nominal predictions. This paper presents a novel formulation of this class of controller with proven input-to-state stability and robust constraint satisfaction. Its advantages are: (i) the design of its main ingredients are tractable for medium to large-sized systems, (ii) the terminal set does not need to be robust with respect to all the possible system uncertainties, but only for a reduced set that can be made arbitrarily small, thus facilitating its design and implementation, (iii) under certain conditions the terminal set can be taken as a positive invariant set of the nominal system, allowing us to use a terminal equality constraint, which facilitates its application to large-scale systems, and (iv) the complexity of its optimization problem is comparable to the non-robust MPC variant. We show numerical closed-loop results of its application to a multivariable chemical plant and compare it against other robust MPC formulations.