Suboptimal Model Predictive Control Research Articles

A Suboptimal Economic Model Predictive Control Algorithm for Large and Infrequent Disturbances

We present a suboptimal economic model predictive control (MPC) algorithm that combines the strengths of two common suboptimal MPC approaches, one based on a warm start and one based on an optimality gap. This algorithm is specifically designed to address a class of large and infrequent disturbances that are relevant when considering discrete actuators and production scheduling in control problems. We establish that this algorithm provides the same nominal performance guarantee as optimal economic MPC and is inherently robust, in an economic context, to large and infrequent disturbances. This inherent robustness guarantee is not attained by either individual algorithm. We conclude with a small production scheduling example to demonstrate the benefits of the proposed algorithm for a practical application.

Parallel MPC for Linear Systems With State and Input Constraints

This letter proposes a parallelizable algorithm for linear-quadratic model predictive control (MPC) problems with state and input constraints. The algorithm itself is based on a parallel MPC scheme that has originally been designed for systems with input constraints. In this context, one contribution of this letter is the construction of time-varying yet separable constraint margins ensuring recursive feasibility and asymptotic stability of sub-optimal parallel MPC in a general setting, which also includes state constraints. Moreover, it is shown how to tradeoff online run-time guarantees versus the conservatism that is introduced by the tightened state constraints. The corresponding performance of the proposed method as well as the cost of the recursive feasibility guarantees is analyzed in the context of controlling a large-scale mechatronic system. This is illustrated by numerical experiments for a large-scale control system with more than 100 states and 60 control inputs leading to run-times in the millisecond range.

Stability and Robustness of Distributed Suboptimal Model Predictive Control

In distributed model predictive control (MPC), the control input at each sampling time is computed by solving a large-scale optimal control problem (OCP) over a finite horizon using distributed algorithms. Typically, such algorithms require several (virtually, infinite) communication rounds between the subsystems to converge, which is a major drawback both computationally and from an energetic perspective (for wireless systems). Motivated by these challenges, we propose a suboptimal distributed MPC scheme in which the total communication burden is distributed also in time, by maintaining a running solution estimate for the large-scale OCP and updating it at each sampling time. We demonstrate that, under some regularity conditions, the resulting time-distributed suboptimal MPC control law recovers the qualitative robust stability properties of optimal MPC, if the communication budget at each sampling time is large enough.

Autonomous Satellite Rendezvous and Proximity Operations with Time-Constrained Sub-Optimal Model Predictive Control

This paper presents a time-constrained model predictive control strategy for the 6 degree-of-freedom (6DOF) autonomous rendezvous and docking problem between a controllable “deputy” spacecraft and an uncontrollable “chief” spacecraft. The control strategy accounts for computational time constraints due to limited onboard processing speed. The translational dynamics model is derived from the Clohessy-Wiltshire equations and the angular dynamics are modeled on gas jet actuation about the deputy's center of mass. Simulation results are shown to achieve the docking configuration under computational time constraints by limiting the number of allowed algorithm iterations when computing each input. Specifically, we show that upwards of 90% of computations can be eliminated from a model predictive control implementation without significantly harming control performance.

Semi-Explicit Model Predictive Control of Quasi Linear Parameter Varying Systems

Explicit linear model predictive control (MPC) is a proven method for constrained optimal control of fast, linear time invariant (LTI) dynamic systems. However, an extension to quasi-linear parameter varying (Q-LPV) systems requires overcoming significant challenges, since neither the explicit MPC law is piecewise linear in state nor are the critical regions polytopic. In this work, we propose a semi-explicit solution to a sub-optimal MPC solution for a class of Q-LPV systems, which are obtained as multiple linear model approximations to nonlinear systems. The sub-optimal MPC for Q-LPV systems, which has been shown to be stabilizing in an earlier work, requires an online solution of a quadratic program (QP). Casting the QP into a multiparametric form leads to a new class of multiparametric QP (mpQP), termed herein as generalized mpQP (GmpQP). Information about the critical regions and the MPC law are obtained offline using a semi-explicit solution of the GmpQP based on an implicit enumeration of the active sets. Three algorithms are compared for the online evaluation of the control law. Offline and online complexity of the proposed method are presented for an example, including comparison of computational speeds with well-known implicit QP solvers.

Infusing Model Predictive Control Into Meta-Reinforcement Learning for Mobile Robots in Dynamic Environments

The successful operation of mobile robots requires them to adapt rapidly to environmental changes. To develop an adaptive decision-making tool for mobile robots, we propose a novel algorithm that combines meta-reinforcement learning (meta-RL) with model predictive control (MPC). Our method employs an off-policy meta-RL algorithm as a baseline to train a policy using transition samples generated by MPC when the robot detects certain events that can be effectively handled by MPC, with its explicit use of robot dynamics. The key idea of our method is to switch between the meta-learned policy and the MPC controller in a randomized and event-triggered fashion to make up for suboptimal MPC actions caused by the limited prediction horizon. During meta-testing, the MPC module is deactivated to significantly reduce computation time in motion control. We further propose an online adaptation scheme that enables the robot to infer and adapt to a new task within a single trajectory. The performance of our method has been demonstrated through simulations using a nonlinear car-like vehicle model with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(i)$</tex-math></inline-formula> synthetic movements of obstacles, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(ii)$</tex-math></inline-formula> real-world pedestrian motion data. The simulation results indicate that our method outperforms other algorithms in terms of learning efficiency and navigation quality.

An efficient MPC algorithm for switched systems with minimum dwell time constraints

This paper presents an efficient suboptimal model predictive control (MPC) algorithm for nonlinear switched systems subject to minimum dwell time constraints (MTC). While MTC are required for most physical systems due to stability, power and mechanical restrictions, MPC optimization problems with MTC are challenging to solve. To efficiently solve such problems, the on-line MPC optimization problem is decomposed into a sequence of simpler problems, which include two nonlinear programs (NLP) and a rounding step, as typically done in mixed-integer optimal control (MIOC). Unlike the classical approach that embeds MTC in a mixed-integer linear program (MILP) with combinatorial constraints in the rounding step, our proposal is to embed the MTC in one of the NLPs using move blocking. Such a formulation can speedup on-line computations by employing recent move blocking algorithms for NLP problems and by using a simple sum-up-rounding (SUR) method for the rounding step. An explicit upper bound of the integer approximation error for the rounding step is given. In addition, a combined shrinking and receding horizon strategy is developed to satisfy closed-loop MTC. Recursive feasibility is proven using a l-step control invariant (l-CI) set, where l is the minimum dwell time step length. An algorithm to compute l-CI sets for switched linear systems off-line is also presented. Numerical studies show significant speed-up and comparable control performance of the proposed MPC algorithm against the classical approach, though at the cost of significant solution quality degradation especially when the MTC are large.

Suboptimal nonlinear model predictive control with input move-blocking

This paper deals with the integration of input move-blocking into the framework of suboptimal model predictive control. The blocked input parameterisation is explicitly considered as a source of suboptimality. A straightforward integration approach is to hold back a manually generated stabilising fallback solution in some buffer for the case that the optimiser does not find a better input move-blocked solution. An extended approach superimposes the manually generated stabilising warm-start by the move-blocked control sequence and enables a stepwise improvement of the control performance. In addition, this contribution provides a detailed review of the literature on input move-blocked model predictive control and combines important results with the findings of suboptimal model predictive control. A numerical example supports the theoretical results and shows the effectiveness of the proposed approach.

Parallelized POD-based suboptimal economic model predictive control of a state-constrained Boussinesq approximation

Motivated by an energy efficient building application, we want to optimize a quadratic cost functional subject to the Boussinesq approximation of the Navier-Stokes equations and to bilateral state and control constraints. Since the computation of such an optimal solution is numerically costly, we design an efficient strategy to compute a sub-optimal (but applicationally acceptable) solution with significantly reduced computational effort. We employ an economic Model Predictive Control (MPC) strategy to obtain a feedback control. The MPC sub-problems are based on a linear-quadratic optimal control problem subjected to mixed control and state constraints and a convection-diffusion equation, reduced with proper orthogonal decomposition. To solve each sub-problem, we apply a primal-dual active set strategy. The method can be fully parallelized, which enables the solution of large problems with real-world parameters.

A Lyapunov function for the combined system-optimizer dynamics in inexact model predictive control

In this paper, an asymptotic stability proof for a class of methods for inexact nonlinear model predictive control is presented. General Q-linearly convergent online optimization methods are considered and an asymptotic stability result is derived for the setting where a limited number of iterations of the optimizer are carried out per sampling time. Under the assumption of Lipschitz continuity of the solution, we explicitly construct a Lyapunov function for the combined system-optimizer dynamics, which shows that asymptotic stability can be obtained if the sampling time is sufficiently short. The results constitute an extension to existing attractivity results which hold in the simplified setting where inequality constraints are either not present or inactive in the region of attraction considered. Moreover, with respect to the established results on robust asymptotic stability of suboptimal model predictive control, we develop a framework that takes into account the optimizer’s dynamics and does not require decrease of the objective function across iterates. By extending these results, the gap between theory and practice of the standard real-time iteration strategy is bridged and asymptotic stability for a broader class of methods is guaranteed.

A Stabilizing Sub-Optimal Model Predictive Control for Quasi-Linear Parameter Varying Systems

Quasi-Linear Parameter Varying (Q-LPV) systems are often obtained as convex combinations of LTI models and have been widely applied for the control of nonlinear systems. An attractive feature is that the model can be adapted online via state or input dependent scheduling parameters to reflect the nonlinear system dynamics while retaining the overall linear structure for design purposes. In the context of Model Predictive Control (MPC), it is desirable that the online optimization problem is a Quadratic Program (QP), which can be easily solved. However, an impediment occurs when using Q-LPV models with MPC: the variation of the scheduling parameters over the prediction horizon casts the online optimization problem into a Nonlinear Program (NLP), which is computationally difficult. Thus the benefits of using the Q-LPV model predictions in MPC are lost. A QP based sub-optimal MPC is obtained if the Q-LPV parameters are treated as constant or frozen over the prediction horizon and updated upon availability of a new measurement at each sampling instant. However, the stability of such a sub-optimal Q-LPV MPC is not clear. In this letter, we consider a class of Q-LPV systems along with an affine term (Q-LPV-A), represented as a polytope with vertices, which corresponds to a piecewise-affine model and examine stability of the quadratic, sub-optimal MPC when the Q-LPV scheduling parameters are maintained constant over the prediction horizon. We derive a design condition for obtaining a stabilizing MPC law for Q-LPV systems. These aspects are illustrated using the Van-de-Vusse benchmark example.

On the inherent robustness of optimal and suboptimal nonlinear MPC

Suboptimal model predictive control (MPC) is a control algorithm that uses suboptimal solutions to optimal control problems to provide control actions quickly. In MPC, terminal control laws and terminal region constraints are frequently used to ensure recursive feasibility and stability. Suboptimal MPC was proven to be inherently robust for systems with soft terminal region constraints in Pannocchia et al. (2011). We extend that work to systems with hard terminal region constraints. If these hard constraints are defined as sublevel sets of appropriate terminal cost functions, a well-chosen initial guess (warm start) for the optimization algorithm is robustly feasible. As a result, the system controlled by suboptimal MPC admits an ISS-Lyapunov function and is therefore inherently robust.The authors of Yu et al. (2014) noted that the result in Pannocchia et al. (2011) applied only to systems with continuous optimal cost functions. However, discontinuous optimal cost functions may be present in systems with hard terminal region constraints. We include a simple example of a continuous dynamical system with a provably discontinuous optimal value function that, as a consequence of the main result of this work, is inherently robust. This example is to our knowledge the first such system reported in the literature.

Towards Parallelizable Sampling–based Nonlinear Model Predictive Control

This paper proposes a new sampling–based nonlinear model predictive control (MPC) algorithm, with a bound on complexity quadratic in the prediction horizon N and linear in the number of samples. The idea of the proposed algorithm is to use the sequence of predicted inputs from the previous time step as a warm start, and to iteratively update this sequence by changing its elements one by one, starting from the last predicted input and ending with the first predicted input. This strategy, which resembles the dynamic programming principle, allows for parallelization up to a certain level and yields a suboptimal nonlinear MPC algorithm with guaranteed recursive feasibility, stability and improved cost function at every iteration, which is suitable for real–time implementation. The complexity of the algorithm per each time step in the prediction horizon depends only on the horizon, the number of samples and parallel threads, and it is independent of the measured system state. Comparisons with the fmincon nonlinear optimization solver on a benchmark example indicates that, as the simulation time progresses, the proposed algorithm converges rapidly to the “optimal” solution, even when using a small number of samples.

Model predictive control with discrete actuators: Theory and application

Despite the presence of discrete actuators in many industrial processes, model predictive control (MPC) theory typically considers only continuous actuators, which requires discrete decisions to be removed from the MPC layer. However, if discrete inputs are chosen optimally, process performance may be greatly improved, and thus, discrete decisions should be treated directly in MPC theory. In this paper, we develop the idea that discrete actuators can be added to MPC theory without major modification, i.e., results established with sufficient generality for standard MPC with continuous actuators hold also for MPC with discrete actuators. First, we show that standard exponential stability for suboptimal MPC can be extended without modification to cover discrete actuators by avoiding restrictive assumptions about the geometry of the control set. Then, we prove stability results for tracking MPC applied to a time-varying periodic system. Finally, we demonstrate these results with two example systems.

Input-to-state stability of efficient robust H∞ MPC scheme for nonlinear systems

In this work, we present an efficient H∞ model predictive control (MPC) scheme with guaranteed input-to-state stability (ISS) for constrained nonlinear systems subject to unknown but bounded disturbances. The nonlinear system is represented by using some given inclusion polytope. The notion of ISS is introduced to derive the sufficient condition in terms of linear matrix inequalities (LMIs), which ensures the ISS of the closed-loop system. In order to reduce the online computational demand of MPC, the non-convex min–max optimization problem is converted to a minimization problem in form of LMIs. A suboptimal MPC algorithm is then addressed to further decrease the computational demand. The applicability of the proposed results is illustrated by two numerical examples of a mass-cart and a vehicular platoon.

Cooperative economic model predictive control for linear systems with convex objectives

In this paper we propose a cooperative distributed economic model predictive control strategy for linear systems which consist of a finite number of coupled subsystems. The suggested feedback strategy is generating control input which converges to a set of Nash equilibria of the corresponding game provided infinite iterations are allowed at each sampling time. Moreover, the control for each subsystem is computed in itself without coordination layer except for a synchronization requirement between subsystems.We first introduce distributed linear systems with two subsystems and economic model predictive control, then show the convergence and stability properties of a suboptimal model predictive control strategy for the system. The optimization problem for the implementation of MPC is stated with a terminal equality constraint and a terminal cost.

Multistage Suboptimal Model Predictive Control With Improved Computational Efficiency

This paper proposes a multistage suboptimal model predictive control (MPC) strategy which can reduce the prediction horizon without compromising the stability property. The proposed multistage MPC requires a precomputed sequence of j-step admissible sets, where the j-step admissible set is the set of system states that can be steered to the maximum positively invariant set in j control steps. Given the precomputed admissible sets, multistage MPC first determines the minimum number of steps M required to drive the state to the terminal set. Then, it steers the state to the (M – N)-step admissible set if M &gt; N, or to the terminal set otherwise. The paper presents the offline computation of the admissible sets, and shows the feasibility and stability of multistage MPC for systems with and without disturbances. A numerical example illustrates that multistage MPC with N = 5 can be used to stabilize a system which requires MPC with N ≥ 14 in the absence of disturbances, and requires MPC with N ≥ 22 when affected by disturbances.

On real-time robust model predictive control

High-speed applications impose a hard real-time constraint on the solution of a model predictive control (MPC) problem, which generally prevents the computation of the optimal control input. As a result, in most MPC implementations guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC approach for linear systems that provides these guarantees for arbitrary time constraints, allowing one to trade off computation time vs. performance. Stability is guaranteed by means of a constraint, enforcing that the resulting suboptimal MPC cost is a Lyapunov function. The key is then to guarantee feasibility in real-time, which is achieved by the proposed algorithm through a warm-starting technique in combination with robust MPC design. We address both regulation and tracking of piecewise constant references. As a main contribution of this paper, a new warm-start procedure together with a Lyapunov function for real-time tracking is presented. In addition to providing strong theoretical guarantees, the proposed method can be implemented at high sampling rates. Simulation examples demonstrate the effectiveness of the real-time scheme and show that computation times in the millisecond range can be achieved.

A Reduced Dantzig-Wolfe Decomposition for a Suboptimal Linear MPC

Linear Model Predictive Control (MPC) is an efficient control technique that repeatedly solves online constrained linear programs. In this work we propose an economic linear MPC strategy for operation of energy systems consisting of multiple and independent power units. These systems cooperate to meet the supply of power demand by minimizing production costs. The control problem can be formulated as a linear program with block-angular structure. To speed-up the solution of the optimization control problem, we propose a reduced Dantzig-Wolfe decomposition. This decomposition algorithm computes a suboptimal solution to the economic linear MPC control problem and guarantees feasibility and stability. Finally, six scenarios are performed to show the decrease in computation time in comparison with the classic Dantzig-Wolfe algorithm.

A New Nonlinear Suboptimal Control Design Approach for Milling Circuits

A combined nonlinear model predictive control and nonlinear dynamic inversion design approach is proposed as a new effective control design philosophy for complex milling circuits. Nonlinear interaction between various states in the mill and the sump as well as significant time delays make the milling circuit difficult to control. The proposed innovative approach, which uses a suboptimal nonlinear model predictive control for the overall circuit with a fast acting dynamic inversion control for controlling the sump level, addresses the ambitious objectives of high product quality, high throughput, manageable loads in the mill as well as minimum power consumption.