Gabriele Pannocchia

Cooperative distributed model predictive control

Abstract In this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition. The control strategy has the following features: hard input constraints are satisfied; terminating the iteration of the distributed controllers prior to convergence retains closed-loop stability; in the limit of iterating to convergence, the control feedback is plantwide Pareto optimal and equivalent to the centralized control solution; no coordination layer is employed. We provide guidance in how to partition the subsystems within the plant. We first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class. We also establish that under perturbation from a stable state estimator, the origin remains exponentially stable. For plants with sparsely coupled input constraints, we provide an extension in which the decision variable space of each suboptimization is augmented to achieve Pareto optimality. We conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies.

Brief paper: Fast, large-scale model predictive control by partial enumeration

Partial enumeration (PE) is presented as a method for treating large, linear model predictive control applications that are out of reach with available MPC methods. PE uses both a table storage method and online optimization to achieve this goal. Versions of PE are shown to be closed-loop stable. PE is applied to an industrial example with more than 250 states, 32 inputs, and a 25-sample control horizon. The performance is less than 0.01% suboptimal, with average speedup factors in the range of 80-220, and worst-case speedups in the range of 4.9-39.2, compared to an existing MPC method. Small tables with only 25-200 entries were used to obtain this performance, while full enumeration is intractable for this example.

Combined Design of Disturbance Model and Observer for Offset-Free Model Predictive Control

This note presents a method for the combined design of an integrating disturbance model and of the observer (for the augmented system) to be used in offset-free model predictive controllers. A dynamic observer is designed for the original (nonaugmented) system by solving an Hprop control problem aimed at minimizing the effect of unmeasured disturbances and plant/model mismatch on the output prediction error. It is shown that, when offset-free control is sought, the dynamic observer is equivalent to choosing an integrating disturbance model and an observer for the augmented system. An example of a chemical reactor shows the main features and benefits of the proposed method.

Robust Disturbance Modeling for Model Predictive Control with Application to Multivariable Ill-conditioned Processes

Abstract In this paper the disturbance model, used by MPC algorithms to achieve offset-free control, is optimally designed to enhance the robustness of single-model predictive controllers. The proposed methodology requires the off-line solution of a min-max optimization problem in which the disturbance model is chosen to guarantee the best closed-loop performance in the worst case of plant in a given uncertainty region. Application to a well-known ill-conditioned distillation column is presented to show that, for ill-conditioned processes, the use of a disturbance model that adds the correction term to the process inputs guarantees a robust performance, while the disturbance model that adds the correction term to the process outputs (used by industrial MPC algorithms) does not.

Conditions under which suboptimal nonlinear MPC is inherently robust

Abstract We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.

A Model Predictive Control Strategy Toward Optimal Structured Treatment Interruptions in Anti-HIV Therapy

In this paper, model predictive control (MPC) strategies are applied to the control of human immunodeficiency virus infection, with the final goal of implementing an optimal structured treatment interruptions protocol. The MPC algorithms proposed in this paper use a dynamic model recently developed in order to mimic both transient responses and ultimate behavior, and to describe accordingly the different effect of commonly used drugs in highly active antiretroviral therapy (HAART). Simulation studies show that the proposed methods achieve the goal of reducing the drug consumption (thus minimizing the severe side effects of HAART drugs) while respecting the desired constraints on CD4+ cells and free virions concentration. Such promising results are obtained with realistic assumptions of infrequent (possibly noisy) measurements of a subset of model state variables. Furthermore, the control objectives are achieved even in the presence of mismatch between the dynamics of true patients and that of the MPC model.

Existence and computation of infinite horizon model predictive control with active steady-state input constraints

This note addresses the existence and implementation of the infinite-horizon controller for the case of active steady-state input constraints. This case is important because, in many practical applications, controllers are required to operate at the boundary of the feasible region (for instance, in order to maximize global economic objectives). For this case, the usual finite horizon parameterizations with terminal cost cannot be applied, and optimal solutions are not generally available. We propose here an iterative algorithm that generates two finite-horizon approximations to the true infinite-horizon problem, where the solution to one of the approximations yields an upper bound on the true optimum, while the other approximation yields a lower bound. We show convergence of both bounding approximations to the optimal solution, as the horizon length in the approximations is increased. We outline a procedure, based on this result, to provide a solution to the infinite-horizon problem that is exact to within any user-specified tolerance. Finally, we present an example that includes a comparison between optimal and suboptimal controllers.

On Computing Solutions to the Continuous Time Constrained Linear Quadratic Regulator

We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the decision-time points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency. Two examples are presented to show the main characteristics of the proposed method.

Partial enumeration MPC: Robust stability results and application to an unstable CSTR

Abstract We describe a partial enumeration (PE) method for fast computation of a suboptimal solution to linear MPC problems [1] with robust stability properties. Given that the suboptimal PE-based control law is non-unique (that is, a set-valued map) and (possibly) discontinuous, we treat the closed-loop system, appropriately augmented, as a difference inclusion. We derive novel robust exponential stability results for difference inclusions of this type. In particular we show that Strong Robust Exponential Stability (SRES) holds, for any sufficiently small but otherwise arbitrary perturbation. Such approach allows us to show SRES of the closed-loop system under PE-based MPC. Application to a simulated open-loop unstable CSTR with separation unit and recycle is presented to show performance and timing results for PE-based MPC, as well as to highlight its robustness to process/model mismatch, disturbances and measurement noise.

Offset-free control of constrained linear discrete-time systems subject to persistent unmeasured disturbances

This paper addresses the design of a dynamic, nonlinear, time-invariant, state feedback controller that guarantees constraint satisfaction and offset-free control in the presence of unmeasured, persistent, non-stationary, additive disturbances. First, this objective is obtained by designing a dynamic, linear, time-invariant, offset-free controller, and an appropriate domain of attraction for this linear controller is defined. Following this, the linear (unconstrained) control input is modified by adding a perturbation term that is computed by a robust receding horizon controller. It is shown that the domain of attraction of the receding horizon controller contains that of the linear controller, and an efficient implementation of the receding horizon controller is proposed.