Improvements in the efficiency of linear MPC

A model predictive control (MPC) strategy based on augmented autonomous predictions enables a highly efficient online optimization by imposing a terminal constraint at the current time. Near-optimal performance is obtained by delaying the imposition of the terminal constraint by one sampling period. However, under certain conditions the degree of optimality can be affected. An extension is proposed to remove this difficulty, yielding significant improvements in the degree of optimality, and achieving this at modest computational cost.     

Explicit/multi-parametric model predictive control (MPC) of linear discrete-time systems by dynamic and multi-parametric programming

This work presents a new algorithm for solving the explicit/multi-parametric model predictive control (or mp-MPC) problem for linear, time-invariant discrete-time systems, based on dynamic programming and multi-parametric programming techniques. The algorithm features two key steps: (i) a dynamic programming step, in which the mp-MPC problem is decomposed into a set of smaller subproblems in which only the current control, state variables, and constraints are considered, and (ii) a multi-parametric programming step, in which each subproblem is solved as a convex multi-parametric programming problem, to derive the control variables as an explicit function of the states. The key feature of the proposed method is that it overcomes potential limitations of previous methods for solving multi-parametric programming problems with dynamic programming, such as the need for global optimization for each subproblem of the dynamic programming step.     

Stochastic MPC with inequality stability constraints

Stochastic uncertainty is a common feature of many control engineering problems, and is also present in a wider class of applications, e.g. finance and sustainable development. Recent work proposed a constrained MPC approach that took explicit account of the distributions of uncertain model parameters but used terminal equality constraints to ensure stability. The present paper reformulates the problem in order to relax the stability constraints by invoking appropriate terminal inequalities. The application of the proposed strategy and its advantages over earlier work are illustrated by means of a numerical example.     

Distributed MPC of constrained linear systems with time-varying terminal sets

This paper proposes a decoupling strategy for the Distributed Model Predictive Control (DMPC) for a network of dynamically-coupled linear systems. Like most DMPC approaches, the proposed approach has a terminal set and uses a Lyapunov matrix for the terminal cost in the online optimization problem for each system. Unlike them, the terminal set changes at every time step and the Lyapunov matrix is not block-diagonal. These features result in a less conservative DMPC formulation. The proposed method is easy to implement when the network is strongly connected (or when a central collector is used). Otherwise, the computations of the terminal set require the online solutions of a series of linear programming problems but can be speeded up significantly by preprocessing. Numerical examples showing these results are provided.     

On the theory of economic MPC: ELOC and approximate infinite horizon EMPC

In recent years, the notion of Economic Model Predictive Control (EMPC) has gained significant interest. Despite a marked improvement in economic performance, it has been shown that this performance will degrade substantially if implemented with a horizon that is not sufficiently large. In the current effort, it is shown that if applied to a particular reaction process, EMPC performance will abruptly collapse at a critical horizon size. To alleviate this issue, we develop an Infinite Horizon EMPC (IH-EMPC) formulation. While this IH-EMPC problem is computationally intractable, it does lead to an approximation of the optimal policy. The resulting Approximate IH-EMPC (AIH-EMPC) is identical to the original finite horizon EMPC, but includes a final cost term that represents the objective function from the finite horizon to infinity. With two example systems, a chemical reactor and a power system with energy storage, it is shown that the AIH-EMPC policy is virtually insensitive to its computational horizon size.     

胜利炼厂第一催化裂化装置的先进控制

本文介绍了Honeywell公司的RMPCT先进控制软件在胜利炼厂第一催化裂化装置上的应用，阐述了催化裂化的工艺流程、先进控制器的设计方案和应用效果。     

Robust MPC of constrained discrete-time nonlinear systems based on approximated reachable sets

A robust MPC for constrained nonlinear systems with uncertainties is presented. Outer bounds of the reachable sets of the system are used to predict the evolution of the system under uncertainty. A method that uses zonotopes to represent the approximated reachable sets is proposed. The closed-loop system is ultimately bounded thanks to a contractive constraint that drives the system to a robust invariant set.     

Controlling the level of sparsity in MPC

In optimization algorithms used for on-line Model Predictive Control (MPC), linear systems of equations are often solved in each iteration. This is true both for Active Set methods as well as for Interior Point methods, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of greatest interest to solve them efficiently. Classically, the optimization problem has been formulated in either of two ways. One leading to a sparse linear system of equations involving relatively many variables to compute in each iteration and another one leading to a dense linear system of equations involving relatively few variables. In this work, it is shown that it is possible not only to consider these two distinct choices of formulations. Instead it is shown that it is possible to create an entire family of formulations with different levels of sparsity and number of variables, and that this extra degree of freedom can be exploited to obtain even better performance with the software and hardware at hand. This result also provides a better answer to a recurring question in MPC; should the sparse or dense formulation be used.     

A new optimization algorithm with application to nonlinear MPC

This paper investigates application of SQP optimization algorithms to nonlinear model predictive control. It considers feasible vs. infeasible path methods, sequential vs. simultaneous methods and reduced vs. full space methods. A new optimization algorithm coined rFOPT which remains feasibile with respect to inequality constraints is introduced. The suitable choices between these various strategies are assessed informally through a small CSTR case study. The case study also considers the effect various discretization methods have on the optimization problem.     

On the solution of the tracking problem for non-linear systems with MPC

This paper presents a Model Predictive Control (MPC) algorithm for non-linear systems which solves the tracking problem for asymptotically constant references. Closed-loop stability of the equilibrium and asymptotic zero-error regulation are guaranteed. The performance of the method is discussed with the classical Continuous Stirred Tank Reactor (CSTR) control application.     

Who needs QP for linear MPC anyway?

Conventional MPC uses quadratic programming (QP) to minimise, on-line, a cost over n linearly constrained control moves. However, stability constraints often require the use of large n thereby increasing the on-line computation, rendering the approach impracticable in the case of fast sampling. Here, we explore an alternative that requires a fraction of the computational cost (which increases only linearly with n), and propose an extension which, in all but a small class of models, matches to within a fraction of a percent point the performance of the optimal solution obtained through QP. The provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to QP-based MPC.     

Robust output-feedback model predictive control for systems with unstructured uncertainty

In this paper, we present novel results that parameterize a broad class of robust output-feedback model predictive control (MPC) policies for discrete-time systems with constraints and unstructured model uncertainty. The MPC policies we consider employ: (i) a linear state estimator, (ii) a pre-determined feedback gain (iii) a set of “tighter constraints” and (iv) a quadratic cost function in the degrees of freedom and the estimated state. Contained within the class, we find both well-known control policies and policies with novel features. The unifying aspect is that all MPC policies within the class satisfy a robust stability test. The robust stability test is suited to synthesis and incorporates a novel linear matrix inequality (LMI) condition which involves the parameters of the cost function. The LMI is shown to always be feasible under an appropriate small-gain condition on the pre-determined feedback gain and the state estimator. Moreover, we show, by means of both theoretical and numerical results, that choosing the cost function parameters subject to the proposed condition often leads to good nominal performance whilst at the same time guaranteeing robust stability.     

Controlled invariant feasibility — A general approach to enforcing strong feasibility in MPC applied to move-blocking

Strong feasibility of MPC problems is usually enforced by constraining the state at the final prediction step to a controlled invariant set. However, such terminal constraints fail to enforce strong feasibility in a rich class of MPC problems, for example when employing move-blocking. In this paper a generalized, least restrictive approach for enforcing strong feasibility of MPC problems is proposed and applied to move-blocking MPC. The approach hinges on the novel concept of controlled invariant feasibility. Instead of a terminal constraint, the state of an earlier prediction step is constrained to a controlled invariant feasible set. Controlled invariant feasibility is a generalization of controlled invariance. The convergence of well-known approaches for determining maximum controlled invariant sets, and j-step admissible sets, is formally proved. Thus an algorithm for rigorously approximating maximum controlled invariant feasible sets is developed for situations where the exact maximum cannot be determined.     

Min-Max MPC based on a computationally efficient upper bound of the worst case cost

Min-Max MPC (MMMPC) controllers [P.J. Campo, M. Morari, Robust model predictive control, in: Proc. American Control Conference, June 10–12, 1987, pp. 1021–1026] suffer from a great computational burden which limits their applicability in the industry. Sometimes upper bounds of the worst possible case of a performance index have been used to reduce the computational burden. This paper proposes a computationally efficient MMMPC control strategy in which the worst case cost is approximated by an upper bound based on a diagonalization scheme. The upper bound can be computed with O(n³) operations and using only simple matrix operations. This implies that the algorithm can be coded easily even in non-mathematical oriented programming languages such as those found in industrial embedded control hardware. A simulation example is given in the paper.     

Enlarging the domain of attraction of MPC controllers

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.     

Feasible parallel-update distributed MPC for uncertain linear systems sharing convex constraints

A distributed MPC approach for linear uncertain systems sharing convex constraints is presented. The systems, which are dynamically decoupled but share constraints on state and/or inputs, optimize once, in parallel, at each time step and exchange plans with neighbours thereafter. Coupled constraint satisfaction is guaranteed, despite the simultaneous decision making, by extra constraint tightening in each local problem. Necessary and sufficient conditions are given on the margins for coupled constraint satisfaction, and a simple on-line scheme for selecting margins is proposed that satisfies the conditions. Robust feasibility and stability of the overall system are guaranteed by use of the tube MPC concept in conjunction with the extra coupled constraint tightening.     

Model Predictive Control oriented experiment design for system identification: A graph theoretical approach

We present a new approach to Model Predictive Control (MPC) oriented experiment design for the identification of systems operating in closed-loop. The method considers the design of an experiment by minimizing the experimental cost, subject to probabilistic bounds on the input and output signals due to physical limitations of actuators, and quality constraints on the identified model. The excitation is done by intentionally adding a disturbance to the loop. We then design the external excitation to achieve the minimum experimental effort while we are also taking care of the tracking performance of MPC. The stability of the closed-loop system is guaranteed by employing robust MPC during the experiment. The problem is then defined as an optimization problem. However, the aforementioned constraints result in a non-convex optimization which is relaxed by using results from graph theory. The proposed technique is evaluated through a numerical example showing that it is an attractive alternative for closed-loop experiment design.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Enlargement of polytopic terminal region in NMPC by interpolation and partial invariance

This paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize offline the volume of an invariant polytopic set with an arbitrary predefined number of vertices subject to a bound on closed-loop performance. Interpolation techniques are used to determine a nonlinear control law which is optimal with respect to a closed-loop cost bound through the on-line solution of a linear program. The invariant polytope is also used to define a receding horizon control law through an appropriate terminal constraint and cost.