Real-time Suboptimal Model Predictive Control Using a Combination of Explicit MPC and Online Optimization

Limits on the storage space or the computation time restrict the applicability of model predictive controllers (MPC) in many real problems. Currently available methods either compute the optimal controller online or derive an explicit control law. In this paper we introduce a new approach combining the two paradigms of explicit and online MPC to overcome their individual limitations. The algorithm computes a piecewise affine approximation of the optimal solution that is used to warm-start an active set linear programming procedure. A preprocessing method is introduced that provides hard real-time execution, stability and performance guarantees for the proposed controller. By choosing a combination of the quality of the approximation and the number of online active set iterations the presented procedure offers a tradeoff between the warm-start and online computational effort. We show how the problem of identifying the optimal combination for a given set of requirements on online computation time, storage and performance can be solved. Finally, we demonstrate the potential of the proposed warm-start procedure on numerical examples

Quantization Design for Unconstrained Distributed Optimization

We consider an unconstrained distributed optimization problem and assume that the bit rate of the communication in the network is limited. We propose a distributed optimization algorithm with an iteratively refining quantization design, which bounds the quantization errors and ensures convergence to the global optimum. We present conditions on the bit rate and the initial quantization intervals for convergence, and show that as the bit rate increases, the corresponding minimum initial quantization intervals decrease. We prove that after imposing the quantization scheme, the algorithm still provides a linear convergence rate, and furthermore derive an upper bound on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithm and the theoretical findings for solving a randomly generated example of a distributed least squares problem

Complexity Certification of the Fast Alternating Minimization Algorithm for Linear Model Predictive Control

In this paper, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. We extend previous theoretical results of FAMA to a more general case, where convex constraints are allowed to be imposed on the strongly convex objective and all convergence properties of FAMA are still preserved. Two splitting strategies for MPC problems are presented. Both of them satisfy the assumptions of FAMA and result in efficient implementations by reducing each iteration of FAMA to simple operations. We derive computational complexity certificates for both splitting strategies, by providing bounds on the number of iterations for both primal and dual variables, which are of particular relevance in the context of real-time MPC to bound the required online computation time. For MPC problems with polyhedral and ellipsoidal constraints, an off-line preconditioning method is presented to further improve the convergence speed of FAMA by reducing the complexity bound and enlarging the step-size of the algorithm. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example

Robust stability properties of soft constrained MPC

In Model Predictive Control, the enforcement of hard state constraints can be overly conservative or even infeasible, especially in the presence of disturbances. This work presents a soft constrained MPC approach that provides closed- loop stability even for unstable systems. Two types of soft constraints are employed: state constraints along the horizon are relaxed by the introduction of two different types of slack variables and the terminal constraint is softened by moving the target from the origin to a feasible steady-state. The proposed method significantly enlarges the region of attraction and preserves the optimal behavior whenever all state constraints can be enforced. Asymptotic stability of the nominal system under the proposed control law is shown, as well as input-to- state stability of the system under additive disturbances and the robust stability properties are analyzed

Input-to-state stabilization of feasible model predictive controllers for linear systems

Research on sub-optimal Model Predictive Control (MPC) has led to a variety of optimization methods based on explicit or online approaches, or combinations thereof. Its foremost aim is to guarantee essential controller properties, i.e. recursive feasibility, stability, and robustness, at reduced and predictable computational cost, i.e. computation time and storage space. This paper shows how the input sequence of any (not necessarily stabilizing) sub-optimal controller and the shifted input sequence from the previous time step can be used in an optimal convex combination, which is easy to determine online, in order to guarantee input-to-state stability for the closed-loop system. The presented method is thus able to stabilize a wide range of existing sub-optimal MPC schemes that lack a formal stability guarantee, if they can be considered as a continuous map from the state space to the space of feasible input sequences

Learning a feasible and stabilizing explicit model predictive control law by robust optimization

Fast model predictive control on embedded sys- tems has been successfully applied to plants with microsecond sampling times employing a precomputed state-to-input map. However, the complexity of this so-called explicit MPC can be prohibitive even for low-dimensional systems. In this pa- per, we introduce a new synthesis method for low-complexity suboptimal MPC controllers based on function approximation from randomly chosen point-wise sample values. In addition to standard machine learning algorithms formulated as convex programs, we provide sufficient conditions on the learning algo- rithm in the form of tractable convex constraints that guarantee input and state constraint satisfaction, recursive feasibility and stability of the closed loop system. The resulting control law can be fully parallelized, which renders the approach particularly suitable for highly concurrent embedded platforms such as FPGAs. A numerical example shows the effectiveness of the proposed method

Real-time MPC - Stability through robust MPC design

Recent results have suggested that online Model Predictive Control (MPC) can be computed quickly enough to control fast sampled systems. High-speed applications impose a hard real-time constraint on the solution of the MPC problem, which generally prevents the computation of the optimal controller. In current approaches guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC scheme based on robust MPC design that recovers these guarantees while allowing for extremely fast computation. We show that a simple warm-start optimization procedure providing an enhanced feasible solution guarantees feasibility and stability for arbitrary time constraints. The proposed method can be practically implemented and efficiently solved for dynamic systems of significant problem size. Implementation details for a real-time robust MPC method are provided that achieves computation times equal to those reported for methods without guarantees. A 12-dimensional problem with 3 control inputs and a prediction horizon of 10 time steps is solved in 2msec with a performance deterioration less than 1% and thereby allows for sampling rates of 500Hz

Controller complexity reduction for piecewise affine systems through safe region elimination

We consider the class of piecewise affine optimal state feedback control laws applied to discrete-time piecewise affine systems, motivated by recent work on the computation of closed-form MPC controllers. The storage demand and complexity of these optimal closed-form solutions limit their applicability in most real-life situations. In this paper we present a novel algorithm to a posteriori reduce the storage demand and complexity of the closed-form controller without losing closed-loop stability or all time feasibility while guaranteeing a bounded performance decay compared to the optimal solution. The algorithm combines simple polyhedral manipulations with (multi-parametric) linear programming and the effectiveness of the algorithm is demonstrated on a large numerical example

Constrained Spectrum Control

A novel Nonlinear Model Predictive Control (NMPC) scheme is proposed in order to shape the harmonic response of constrained nonlinear systems. The salient ingredient is the short-time Fourier transform (STFT) of the system's output signal, which is constrained in an NMPC problem, leading to the novel formulation of so-called spectrum constraints. Recursive feasibility and asymptotic stability of the proposed NMPC scheme with such spectrum constraints are guaranteed by means of an appropriate ellipsoidal terminal invariant set. The efficacy of the proposed approach is demonstrated on a nonlinear vibration damping problem