Linear programming and model predictive control

The practicality of model predictive control (MPC) is partially limited by the ability to solve optimization problems in real time. This requirement limits the viability of MPC as a control strategy for large scale processes. One strategy for improving the computational performance is to formulate MPC using a linear program. While the linear programming formulation seems appealing from a numerical standpoint, the controller does not necessarily yield good closed-loop performance. In this work, we explore MPC with an l₁ performance criterion. We demonstrate how the non-smoothness of the objective function may yield either dead-beat or idle control performance.     

Optimal control of grinding mill circuit using model predictive static programming: A new nonlinear MPC paradigm

The recently developed reference-command tracking version of model predictive static programming (MPSP) is successfully applied to a single-stage closed grinding mill circuit. MPSP is an innovative optimal control technique that combines the philosophies of model predictive control (MPC) and approximate dynamic programming. The performance of the proposed MPSP control technique, which can be viewed as a ‘new paradigm’ under the nonlinear MPC philosophy, is compared to the performance of a standard nonlinear MPC technique applied to the same plant for the same conditions. Results show that the MPSP control technique is more than capable of tracking the desired set-point in the presence of model-plant mismatch, disturbances and measurement noise. The performance of MPSP and nonlinear MPC compare very well, with definite advantages offered by MPSP. The computational speed of MPSP is increased through a sequence of innovations such as the conversion of the dynamic optimization problem to a low-dimensional static optimization problem, the recursive computation of sensitivity matrices and using a closed form expression to update the control. To alleviate the burden on the optimization procedure in standard MPC, the control horizon is normally restricted. However, in the MPSP technique the control horizon is extended to the prediction horizon with a minor increase in the computational time. Furthermore, the MPSP technique generally takes only a couple of iterations to converge, even when input constraints are applied. Therefore, MPSP can be regarded as a potential candidate for online applications of the nonlinear MPC philosophy to real-world industrial process plants.     

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.     

Multiobjective model predictive control

This paper proposes a novel model predictive control (MPC) scheme based on multiobjective optimization. At each sampling time, the MPC control action is chosen among the set of Pareto optimal solutions based on a time-varying, state-dependent decision criterion. Compared to standard single-objective MPC formulations, such a criterion allows one to take into account several, often irreconcilable, control specifications, such as high bandwidth (closed-loop promptness) when the state vector is far away from the equilibrium and low bandwidth (good noise rejection properties) near the equilibrium. After recasting the optimization problem associated with the multiobjective MPC controller as a multiparametric multiobjective linear or quadratic program, we show that it is possible to compute each Pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular Pareto optimal solution. Furthermore, we provide conditions for selecting Pareto optimal solutions so that the MPC control loop is asymptotically stable, and show the effectiveness of the approach in simulation examples.     

Generalized terminal state constraint for model predictive control

A terminal state equality constraint for Model Predictive Control (MPC) laws is investigated, where the terminal state/input pair is not fixed a priori but it is a free variable in the optimization. The approach, named “generalized” terminal state constraint, can be used for both tracking MPC (i.e. when the objective is to track a given steady state) and economic MPC (i.e. when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state). It is shown that the proposed technique provides, in general, a larger feasibility set with respect to the existing approaches, given the same prediction horizon. Moreover, a new receding horizon strategy is introduced, exploiting the generalized terminal state constraint. Under mild assumptions, the new strategy is guaranteed to converge in finite time, with arbitrarily good accuracy, to an MPC law with an optimally-chosen terminal state constraint, while still enjoying a larger feasibility set. The features of the new technique are illustrated by an inverted pendulum example in both the tracking and the economic contexts.     

Robust model predictive control for optimal continuous drug administration

In this paper the model predictive control (MPC) technology is used for tackling the optimal drug administration problem. The important advantage of MPC compared to other control technologies is that it explicitly takes into account the constraints of the system. In particular, for drug treatments of living organisms, MPC can guarantee satisfaction of the minimum toxic concentration (MTC) constraints. A whole-body physiologically-based pharmacokinetic (PBPK) model serves as the dynamic prediction model of the system after it is formulated as a discrete-time state-space model. Only plasma measurements are assumed to be measured on-line. The rest of the states (drug concentrations in other organs and tissues) are estimated in real time by designing an artificial observer. The complete system (observer and MPC controller) is able to drive the drug concentration to the desired levels at the organs of interest, while satisfying the imposed constraints, even in the presence of modelling errors, disturbances and noise.     

An improved approach for constrained robust model predictive control

In this paper, we present a new technique to address constrained robust model predictive control. The main advantage of this new approach with respect to other well-known techniques is the reduced conservativeness. Specifically, the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several Lyapunov functions each one corresponding to a different vertex of the uncertainty's polytope.     

Robust and optimal predictive control of the COVID-19 outbreak

We investigate adaptive strategies to robustly and optimally control the COVID-19 pandemic via social distancing measures based on the example of Germany. Our goal is to minimize the number of fatalities over the course of two years without inducing excessive social costs. We consider a tailored model of the German COVID-19 outbreak with different parameter sets to design and validate our approach. Our analysis reveals that an open-loop optimal control policy can significantly decrease the number of fatalities when compared to simpler policies under the assumption of exact model knowledge. In a more realistic scenario with uncertain data and model mismatch, a feedback strategy that updates the policy weekly using model predictive control (MPC) leads to a reliable performance, even when applied to a validation model with deviant parameters. On top of that, we propose a robust MPC-based feedback policy using interval arithmetic that adapts the social distancing measures cautiously and safely, thus leading to a minimum number of fatalities even if measurements are inaccurate and the infection rates cannot be precisely specified by social distancing. Our theoretical findings support various recent studies by showing that (1) adaptive feedback strategies are required to reliably contain the COVID-19 outbreak, (2) well-designed policies can significantly reduce the number of fatalities compared to simpler ones while keeping the amount of social distancing measures on the same level, and (3) imposing stronger social distancing measures early on is more effective and cheaper in the long run than opening up too soon and restoring stricter measures at a later time.     

Quadratic programming algorithms for large-scale model predictive control

Quadratic programming (QP) methods are an important element in the application of model predictive control (MPC). As larger and more challenging MPC applications are considered, more attention needs to be focused on the construction and tailoring of efficient QP algorithms. In this study, we tailor and apply a new QP method, called QPSchur, to large MPC applications, such as cross directional control problems in paper machines. Written in C++, QPSchur is an object oriented implementation of a novel dual space, Schur complement algorithm. We compare this approach to three widely applied QP algorithms and show that QPSchur is significantly more efficient (up to two orders of magnitude) than the other algorithms. In addition, detailed simulations are considered that demonstrate the importance of the flexible, object oriented construction of QPSchur, along with additional features for constraint handling, warm starts and partial solution.     

Feedforward model predictive control

This paper examines the role played by feedforward in model predictive control (MPC). We contrast feedforward with preview action. The latter is standard in model predictive control, whereas feedforward has been rarely, if ever, used in contemporary formulations of MPC. We argue that feedforward can significantly improve performance in the presence of measurement noise and certain types of model uncertainty.     

Robust model predictive control using tubes

A form of feedback model predictive control (MPC) that overcomes disadvantages of conventional MPC but which has manageable computational complexity is presented. The optimal control problem, solved on-line, yields a ‘tube’ and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty; computational complexity is linear (rather than exponential) in horizon length. Asymptotic stability of the controlled system is established.     

An improved structure for model predictive control using non-minimal state space realisation

This paper describes a new method for the design of model predictive control (MPC) using non-minimal state space models, in which the state variables are chosen as the set of measured input and output variables and their past values. It shows that the proposed design approach avoids the use of an observer to access the state information and, as a result, the disturbance rejection, particularly the system input disturbance rejection, is significantly improved when constraints become activated. In addition, when there is no model/plant mismatch, the paper shows that the system output constraints can be realised in the proposed approach. Furthermore, closed-form transfer function representation of the model predictive control system enables the application of frequency response analysis tools to the nominal performance of the system.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

Optimal impedance via model predictive control for robot-aided rehabilitation

This paper proposes an optimal impedance controller for robot-aided rehabilitation of walking, aiming to increase the patient’s activity during the therapy. In an online procedure, the joint torques produced by the patient during the gait is estimated using the generalized momenta-based disturbance observer and the Extended Kalman filter algorithm. At the same time, a model predictive control is performed to obtain the instantaneous optimal stiffness parameters of the robot’s impedance controller, trying to maximize the patient’s active participation by increasing his/her joint torques. In this feasibility study, experiments with a healthy subject, considering a modular lower limb exoskeleton and a set of user’s behaviors, are performed to evaluate the proposed controller. The results show the robot stiffness converges to a value which increases the user’s active participation.     

Serial advanced-multi-step nonlinear model predictive control using an extended sensitivity method

Nonlinear model predictive control (NMPC) can directly handle multi-input multi-output nonlinear systems and explicitly consider input and state constraints. However, the computational load for nonlinear programming (NLP) of large-scale systems limits the range of possible applications and degrades NMPC performance. An NLP sensitivity based approach, advanced-step NMPC, has been developed to address the online computational load. In addition, for cases where the NLP solving time exceeds one sampling time, two types of advanced-multi-step NMPC (amsNMPC), parallel and serial, have been proposed. However, in previous studies, a serial amsNMPC could not be applied to large-scale problems because of the size of extended Karush–Kuhn–Tucker matrix and its Schur complement decomposition, and the robustness was analyzed under a conservative assumption for memory effects. In this paper, we propose a serial amsNMPC using an extended sensitivity method to increase the online computation speed further. We successfully apply it to a large-scale air separation unit using the sparse matrix handling packages of Python, Pyomo, and k_aug tools. Furthermore, an auxiliary NLP formulation is defined to analyze the robustness. Using this with the key properties of an extended sensitivity matrix, we can prove robustness while avoiding the memory effects term.     

Cooperative vehicle path generation during merging using model predictive control with real-time optimization

This paper proposed a cooperative merging path generation method for vehicles to merge smoothly on the motorway using a Model Predictive Control (MPC) scheme which optimizes the motions of the relevant vehicles simultaneously. The cooperative merging is a merging in where the most relevant vehicle in the main lane would accelerate or decelerate slightly to let the merging vehicle merge in easily. The proposed path generation algorithm can generate the merging path ensuring the merging vehicle can access the whole acceleration area, and do not exceed it. We have introduced a state variable to the optimization problem by which the merging point for the merging vehicle is optimized. The simulation results showed that the cooperative merging path can be successfully generated under some typical traffic situations without re-adjustment of the optimization parameters.     

Reduced order optimization for model predictive control using principal control moves

In order to reduce the computational complexity of model predictive control (MPC) a proper input signal parametrization is proposed in this paper which significantly reduces the number of decision variables. This parametrization can be based on either measured data from closed-loop operation or simulation data. The snapshots of representative time domain data for all manipulated variables are projected on an orthonormal basis by a Karhunen-Loeve transformation. These significant features (termed principal control moves, PCM) can be reduced utilizing an analytic criterion for performance degradation. Furthermore, a stability analysis of the proposed method is given. Considerations on the identification of the PCM are made and another criterion is given for a sufficient selection of PCM. It is shown by an example of an industrial drying process that a strong reduction in the order of the optimization is possible while retaining a high performance level.     

Periodic optimal control of nonlinear constrained systems using economic model predictive control

In this paper, we consider the problem of periodic optimal control of nonlinear systems subject to online changing and periodically time-varying economic performance measures using model predictive control (MPC). The proposed economic MPC scheme uses an online optimized artificial periodic orbit to ensure recursive feasibility and constraint satisfaction despite unpredictable changes in the economic performance index. We demonstrate that the direct extension of existing methods to periodic orbits does not necessarily yield the desirable closed-loop economic performance. Instead, we carefully revise the constraints on the artificial trajectory, which ensures that the closed-loop average performance is no worse than a locally optimal periodic orbit. In the special case that the prediction horizon is set to zero, the proposed scheme is a modified version of recent publications using periodicity constraints, with the important difference that the resulting closed loop has more degrees of freedom which are vital to ensure convergence to an optimal periodic orbit. In addition, we detail a tailored offline computation of suitable terminal ingredients, which are both theoretically and practically beneficial for closed-loop performance improvement. Finally, we demonstrate the practicality and performance improvements of the proposed approach on benchmark examples.     

Gain-scheduling model predictive control of a Fresnel collector field

Model predictive control strategies have been applied successfully when controlling solar plants. If the control algorithm uses a linear model associated only to an operating point, when the plant is working far from the design conditions, the performance of the controller may deteriorate.In this paper, a gain scheduling model predictive control strategy is designed for the Fresnel collector field located at the Escuela Superior de Ingenieros de Sevilla. Simulation results are provided comparing the proposed strategy with another linear MPC controller showing a better performance. Furthermore, two real tests are presented showing the effectiveness of the proposed strategy.     

LMI-based model predictive control for switched nonlinear systems

This paper proposes an LMI approach to model predictive control of nonlinear systems with switching between multiple modes. In this approach, at each mode, the nonlinear system is divided to a linearized model in addition to a nonlinear term. A sum of squares (SOS) optimization problem is presented to find a quadratic bound for the nonlinear part. The stability condition of the switching system is obtained by using a discrete Lyapunov function and then the sufficient state feedback control law is achieved so that guarantees the stability of the system and also minimizes an infinite prediction horizon performance index. Moreover, two other LMI optimization problems are solved at each mode in order to find the maximum area region of convergence of the nonlinear system inscribed in the region of stability. The performance and effectiveness of the proposed MPC approach are illustrated by two case studies.