Large-scale Model Predictive Control Research Articles

Robust Switching Time Optimization for Networked Switched Systems via Model Predictive Control.

This article presents a model predictive control (MPC) strategy to find the optimal switching time sequences of networked switched systems with uncertainties. First, based on predicted trajectories under exact discretization, a large-scale MPC problem is formulated; second, a two-level hierarchical optimization structure coupled with a local compensation mechanism is established to solve the formulated MPC problem, where the proposed hierarchical optimization structure is actually a recurrent neural network consisting of a coordination unit (CU) at the upper level and a series of local optimization units (LOUs) related to each subsystem at the lower level. Finally, a real-time switching time optimization algorithm is designed to calculate the optimal switching time sequences.

Large scale model predictive control with neural networks and primal active sets

This work presents an explicit–implicit procedure to compute a model predictive control (MPC) law with guarantees on recursive feasibility and asymptotic stability. The approach combines an offline-trained fully-connected neural network with an online primal active set solver. The neural network provides a control input initialization while the primal active set method ensures recursive feasibility and asymptotic stability. The neural network is trained with a primal–dual loss function, aiming to generate control sequences that are primal feasible and meet a desired level of suboptimality. Since the neural network alone does not guarantee constraint satisfaction, its output is used to warm start the primal active set method online. We demonstrate that this approach scales to large problems with thousands of optimization variables, which are challenging for current approaches. Our method achieves a 2× reduction in online inference time compared to the best method in a benchmark suite of different solver and initialization strategies.

Industrial, large-scale model predictive control with structured neural networks

The design of neural networks (NNs) is presented for treating large, linear model predictive control (MPC) applications that are out of reach with available quadratic programming (QP) solvers. First, we introduce a new feedforward network architecture that enables practitioners to obtain offset-free closed-loop performance with NNs. Second, we discuss the data generation procedure to sample the state space relevant to training the NNs based on anticipated online setpoint changes and plant disturbances. Third, we use the input-to-state stability results available in the MPC literature and establish robustness properties of NN controllers. Finally, we present illustrative simulation studies on process control examples. We apply the NN design approach and compare the performance with online QP based MPC on an industrial crude distillation unit model with 252 states, 32 control inputs, and a control-sample horizon length of 140. Parallel computing is used for data generation and graphical processing units are used for network training. Anticipated plant operational scenarios with setpoints and disturbances that may change during operation must be sampled for NN training. After the offline design phase, NNs execute MPC three to five orders of magnitude faster than an available QP solver with less than 1% loss in the closed-loop performance.

A partial multiparametric optimization strategy to improve the computational performance of model predictive control

Determining the optimal manipulated action for large scale model predictive control formulations requires significant computational overhead. It has been demonstrated that the offline, explicit solution provided by multiparametric programming has the capacity to greatly improve the online computational performance of MPC strategies. For large scale problems, developing and deploying the full multiparametric solution remains an open challenge. In this work, a partial multiparametric solution is utilized to improve the initialization procedure for a hot start strategy. The hot start strategy provides an improved technique for determining the optimal solution of large scale MPC formulations, and the partial multiparametric solution ensures the initialization is suitable under varying conditions. The efficacy of the proposed strategy is verified on randomly generated large scale MPC problems.

Symmetric Linear Model Predictive Control

This paper studies symmetry in linear model predictive control (MPC). We define symmetry for model predictive control laws and for model predictive control problems. Properties of both MPC symmetries are studied by using a group theory formalism. We show how to efficiently compute MPC symmetries by transforming the search of MPC symmetry generators into a graph automorphism problem. MPC symmetries are then used to design model predictive control algorithms with reduced complexity. The effectiveness of the proposed approach is shown through a simple large-scale MPC problem whose explicit solution can only be found with the method presented in this paper.

Large-Scale MPC with Continuous/Discrete-Valued Inputs: Compensation of Quantization Errors, Stabilization, and Its Application

This paper investigates model predictive control (MPC) of large-scale systems with both continuous-valued and discrete-valued control inputs. In the authors' previously proposed method, the no...

A global piecewise smooth Newton method for fast large-scale model predictive control

In this paper, the strictly convex quadratic program (QP) arising in model predictive control (MPC) for constrained linear systems is reformulated as a system of piecewise affine equations. A regularized piecewise smooth Newton method with exact line search on a convex, differentiable, piecewise-quadratic merit function is proposed for the solution of the reformulated problem. The algorithm has considerable merits when applied to MPC over standard active set or interior point algorithms. Its performance is tested and compared against state-of-the-art QP solvers on a series of benchmark problems. The proposed algorithm is orders of magnitudes faster, especially for large-scale problems and long horizons. For example, for the challenging crude distillation unit model of Pannocchia, Rawlings, and Wright (2007) with 252 states, 32 inputs, and 90 outputs, the average running time of the proposed approach is 1.57 ms.

Model predictive control monitoring using multivariate statistics

Control loop monitoring has become an important research field over the past decade. Research has primarily targeted single-input single-output (SISO) feedback control systems with limited progress being made on the monitoring of multi-input multi-output (MIMO) control systems and large scale model predictive control (MPC) systems in particular. The size and complexity of MPC systems means that identifying and diagnosing problems with their operation can be challenging. This paper presents an MPC condition monitoring tool based on multivariate statistical process control (MSPC) techniques. The proposed tool uses intuitive charts to enable casual users of MPC technology to detect abnormal controller operation and to identify possible causes for this behaviour. Through its application to data collected from a large scale MPC system, the proposed technique is shown to be able to identify and diagnose poor control performance resulting from various issues including inappropriate interaction by process operators.

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.

Price-driven coordination method for solving plant-wide MPC problems

In large-scale model predictive control (MPC) applications, such as plant-wide control, two possible approaches to MPC implementation are centralized and decentralized MPC schemes. These represent the two extremes in the “trade-off” among the desired characteristics of an industrial MPC system, namely accuracy, reliability and maintainability. To achieve optimal plant operations, coordination of decentralized MPC controllers has been identified as both an opportunity and a challenge. Typically, plant-wide MPC problem can be formulated as a large-scale quadratic program (QP) with linking equality constraints. Such problems can be decomposed and solved with the price-driven coordination method and on-line solutions of these structured large-scale optimization problems require an efficient price-adjustment strategy to find an “equilibrium price”. This work develops an efficient price-adjustment algorithm based on Newton’s method, in which sensitivity analysis and active set change identification techniques are employed. With the off-diagonal element abstraction technique and the enhanced priced driven coordination algorithm, a coordinated, decentralized MPC framework is proposed. Several case studies show that the proposed coordination-based decentralized MPC scheme is an effective approach to plant-wide MPC applications, which provides a high degree of reliability and accuracy at a reasonable computational load.

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the object-oriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.

COORDINATED DECENTRALIZED MPC FOR PLANT-WIDE CONTROL OF A PULP MILL BENCHMARK PROBLEM

In large-scale model predictive control (MPC) applications, such as plant-wide control, the coordination of unit-based MPC controllers has been identified as both an opportunity and a challenge in enhancing the plant-wide control performance. This work discusses an efficient strategy for the coordination of decentralized MPC systems and illustrates the approach with an application to the pulp mill benchmark problem proposed by Castro and Doyle III (2001a). The decentralized unit-based MPC controllers are coordinated at the MPC steady-state target calculation stage by employing decentralized optimization techniques. The off-diagonal element abstraction technique and the price-driven coordination algorithm are used in the development of a coordination mechanism. The pulp mill case study shows that this coordinated, decentralized MPC framework is an effective approach to plant-wide MPC applications, which has high reliability, accuracy and efficiency.

PRICE-DRIVEN COORDINATION FOR SOLVING PLANT-WIDE MPC PROBLEMS

Two-level model predictive control (MPC) is the dominant multi-variable control technology in the process industries. In large-scale MPC applications, such as plant-wide control, two common approaches are centralized and decentralized MPC schemes, which represent the two extremes in the “trade-off” among the desired characteristics of an implemented MPC system. Alternatively the coordination of decentralized MPC systems may offer the best attributes of each of the extremes. The price-driven coordination scheme requires the existence of “equilibrium prices” and has extensive large-scale applications in economic planning. On-line solutions to large-scale optimization problems require an efficient price-adjustment method. As the coordination problem for decentralized MPC falls into the category of limited resource case, this work develops an efficient price-adjustment algorithm by using Newton's method, in which sensitivity analysis and active set change identification techniques are employed. The proposed price-adjustment strategy is incorporated into a coordinated, decentralized MPC scheme that shows a high degree of accuracy, while retaining the reliability of original decentralized scheme at a reasonable computational load.

Quadratic programming algorithms for large-scale model predictive control

Abstract 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.

Customization strategies for the solution of linear programming problems arising from large scale model predictive control of a paper machine

Abstract This paper presents an application of a customized linear programming (LP) based model predictive control strategy to the paper machine cross direction (CD) control problem. The objective of CD control is to maintain flat profiles of variables of interest by minimizing worst case deviations from setpoints (defects). These control problems can have as many as 200 actuators (inputs) and 400 sensor measurements (outputs). This large size coupled with the stringent real-time requirement of computing a control move in a few seconds poses a very challenging control problem. Computational results that demonstrate the effectiveness of this strategy will be presented. For typical disturbances this algorithm can compute provably optimal control moves for a 400 input ×400 output control problem in approximately 5 s versus approximately 90 s for a generic LP algorithm on a HP 9000/770 workstation.

Perspectives on model based integration of process operations

The chemical process industry has increasingly been pursuing the use of computing technology to gather, organize, disseminate and exploit enterprise information and to closely coordinate the decisions made at the various levels of the process operational hierarchy so as to optimize overall corporate objectives. The thesis advanced in this paper is that mathematical programming models and solution methods offer the most effective tools for integration of the tactical and strategic levels of the operational hierarchy. To that end, existing strategies for implementing model-based integrated applications are reviewed. Four classes of examples of integration are presented: scheduling of multiproduct plants, large-scale model predictive control, integration of planning and scheduling across single and multiple plant sites and design of multipurpose batch plants under uncertainty. The methodology used to address these model-based integration instances successfully accomodates the key features of process operations: diverse time scale, multiple reference frameworks, spatial and organizational aggregation/disaggregation and uncertainty in the enterprise information. The applications further demonstrate that for the foreseeable future no single model or reference framework will be sufficient and efficient for treating all aspects of the process operations hierarchy.