Steady-state Target Calculation Research Articles

Closed-loop robust steady-state target calculation for model predictive control

Industrial model predictive control (MPC) typically includes a steady-state target calculation (SSTC) layer that provides targets for the dynamic MPC controller to track. This paper develops the first closed-loop robust SSTC method, which explicitly addresses the closed-loop uncertainty, for stable linear systems with time-invariant parametric uncertainty. The problem is initially formulated as a stochastic bi-level optimization problem, where the lower-level optimization problem describes the nominal MPC law. The problem is then transformed into a deterministic mixed-integer second-order cone program for ellipsoidal uncertainty. Case study results show that the proposed method is more robust than the classical nominal SSTC in that it helps reduce cycling of the steady-state targets. In addition, the proposed method is less conservative than a typical open-loop robust SSTC method, in that it drives the system to the steady state more quickly and can push the steady state closer to the operating boundary when desired.

Double-layer model predictive control integrated with zone control

This paper proposed a double-layer model predictive control (DLMPC) strategy integrated with zone control. In the steady-state target calculation (SSTC) layer, the controlled output variables are pretreated based on the analysis of process principle and production data. Subsequently, the optimal input–output targets and corresponding zone parameters are obtained by solving the steady-state optimization problem and transmitted to the dynamic control layer for tracking. Meanwhile, the weighted and priority soft constraints relaxation methods are discussed when the SSTC problem is infeasible. Compared with conventional DLMPC and zone model predictive control (ZMPC), the proposed algorithm can not only achieve smooth control, but also have better robustness and economic performance. Finally, a simulation example is taken to show the effectiveness of the proposed method.

Steady-state target calculation integrating economic optimization for constrained model predictive control

Real-time optimization (RTO) in cascade model predictive control (MPC) is a classic two-layer architecture for economic optimization and dynamic control. The two-layer architecture is always integrated with steady-state target calculation (SSTC) to guarantee the feasibility of the control target. In this paper, we present a modified SSTC algorithm to recalculate the steady-state point for MPC when the optimal point calculated by RTO is no longer attainable. The proposed algorithm takes economic performance into account and retains the feasibility of solutions. Based on a classic SSTC formulation, ours has a new objective function with two terms, one for optimizing economic performance and the other for ensuring the feasibility. Moreover, the objective function of the proposed SSTC is rearranged, and the SSTC problem is formulated by quadratic programming, which can be solved in an instant for real applications. Furthermore, we provide a detailed 4-in-4-out model of the Fischer esterification reaction, which is similar to the benchmark continuous stirred tank reactor (CSTR) model, to demonstrate the effectiveness of our SSTC algorithm. The proposed algorithm is able to improve economic performances, whether a convex or non-convex economic cost function is concerned.

Output Feedback Model Predictive Control With Steady-State Target Calculation For Fuzzy Systems

In this article, an output feedback model predictive control for fuzzy systems to track a time-varying steady-state target is presented. The steady-state target is calculated through a steady-state target calculation (SSTC) module, which is combined with the controller in the lower layer designing based on the linear time invariant model. The presence of SSTC makes the recursive feasibility as well as the stability difficult to guarantee. This article presents a promising approach by specifying the solution of SSTC as a function of estimated artificial disturbance, designing the robust invariant set offline and including the invariance condition in SSTC. Hence, the proposed approach in this article guarantees that if SSTC optimization problem is feasible, then the dynamic controller steers the system to the time-varying steady-state target. Furthermore, in order to reduce the conservativeness of the proposed approach, a heuristic online approach with N free control moves is proposed where the invariant set specified offline is acting as a terminal region for the dynamic controller. A numerical example is given to show the effectiveness of the proposed approach.

Heuristic open-loop output feedback model predictive control for control of intermittent transonic wind tunnel

For intermittent transonic wind tunnel (ITWT), Mach number and total pressure are crucial features of the flow field in the test section. However, they are typically difficult to control due to system nonlinearities, uncertainties, coupling and delays. This paper proposes an output feedback model predictive control (MPC) strategy to ensure the tracking of Mach number and total pressure of the ITWT to a specified reference trajectory. The proposed approach includes open-loop prediction, steady state target calculation (SSTC) and dynamic control, where the lower layer dynamic control tracks the steady state target obtained from the upper SSTC layer. By adding an artificial disturbance in the identified linear parameter varying (LPV) model for ITWT, and utilizing the resulting LPV model in both SSTC and dynamic control layers, a heuristic open-loop output feedback MPC approach is proposed. The experimental results for the control of Mach number and total pressure for ITWT demonstrate the effectiveness of the proposed approach.

Two‐layer model predictive control for chemical process model with integrating controlled variables

AbstractThis paper proposes an overall solution to the two‐layer model predictive control (MPC) for the integrating controlled variables in the process model. The scheme includes three modules, that is, the open‐loop prediction module, the steady‐state target calculation (SSTC) module, and the dynamic control module. Based on the real‐time output measurements and past inputs, the open‐loop prediction module predicts the future outputs in the presence of disturbances. The economic optimization of SSTC is comprised of the feasibility stage and the economics stage, considering constraints of multi‐priority ranks. The dynamic control module receives the steady‐state targets from SSTC and calculates the control signals. The optimization problems of SSTC and dynamic control operate with the same frequency. This overall method guarantees the consistency of three modules with respect to the model, the constraints, and the targets. The simulation example illustrates that steady‐state targets are adjusted dynamically after the occurrence of disturbances, and offset‐free control is achieved.

Multivariable offset-free MPC with steady-state target calculation and its application to a wind tunnel system

This paper addresses a computationally efficient MPC approach for set-point optimization and its application to an intermittent transonic wind tunnel (ITWT). In the presented method, the open-loop prediction, which adopts the Kalman filter to obtain the open-loop dynamic/steady state predictions of manipulated/controlled variables (MVs/CVs), is presented. Based on the open-loop prediction, a linearization steady-state model featured by the total pressure and the Mach number is used for set-points optimization in steady-state target calculation (SSTC), being formulated as a linear programming (LP) problem. Based on these set-points of MVs/CVs, the dynamic control computes the optimal control moves by solving a quadratic programming (QP) problem. The effectiveness of the proposed method is illustrated on ITWT, and satisfactory performances are obtained.

Double-layered nonlinear model predictive control based on Hammerstein–Wiener model with disturbance rejection

This paper presents the double-layered nonlinear model predictive control method for a continuously stirred tank reactor and a pH neutralization process that are subject to input disturbances and output disturbances at the same time. The nonlinear systems can be described as a Hammerstein -Wiener model. Furthermore, two nonlinear parts of the Hammerstein -Wiener model should be transformed into linear combination of known input and unknown disturbances, respectively. By taking advantage of Kalman filter, disturbances and states can be estimated. The estimated disturbances and states can be considered to calculate steady-state target in steady-state target calculation layer. Moreover, the state feedback control law can be obtained in dynamic control layer. A simple proof for offset-free control is given in the proposed method. The simulation results show that the controlled variable can achieve the offset-free control. It can be seen that the proposed method has better disturbance rejection performance, strong robustness and practical value.

Economic Optimization and Control Based on Multi Priority Rank RTO and Double Layered MPC

AbstractA prevailing hierarchical structure in industrial process optimization and control includes three levels, i.e., a real time optimization (RTO) level, a double layered model predictive control (MPC) level (which is composed of a steady‐state target calculation (SSTC) layer and a dynamic control layer), and a distributed control level. In this paper, a multi priority rank RTO algorithm, in which a new variable is introduced to uniformly express the set points, is presented to get optimal set points according to their importance levels. In order to guarantee the feasibility of the dynamic control layer during tracking the steady‐state targets calculated in SSTC layer, the region of attraction is added into the SSTC layer as additional constraints, hence, the steady‐state targets can be calculated online and transmitted to the dynamic control layer at each instant to guide the state to achieve the steady‐state state gradually. The effect of the above methods are illustrated through an example.

An Offline Optimization and Online Table Look-Up Strategy of Two-Layer Model Predictive Control

Two-layer model predictive control is restricted in the field of high real-time control and poor computational controller, due to the high computational complexity. In this paper, we propose an off-line optimization and online table look-up strategy of two-layer model predictive control to address the problem. In the upper layer of steady-state target calculation, offline optimization, and online table look-up is developed to calculate steady-state targets. In the lower layer of dynamic optimization, an unconstrained model predictive control is adopted to track the steady-state targets from the upper layer. For the case that the online table look-up method is infeasible, multi-parametric LP and linear weighted sum methods are presented. An exhaustive simulation of a fat system is made to explore the performance of the strategy of the two-layer model predictive control. Furthermore, the properties of computation complexity, steady-state, and robustness are examined from the point of implementation. Simulated studies demonstrate the effectiveness of the proposed strategy.

An on-line constraint softening strategy to guarantee the feasibility of dynamic controller in double-layered MPC

Abstract A double-layered model predictive control (MPC), which is composed of a steady-state target calculation (SSTC) layer and a dynamic control layer, is a prevailing hierarchical structure in industrial process control. Based on the reason analysis of the dynamic controller infeasibility, an on-line constraints softening strategy is given. At first, a series of regions of attraction (ROA) of the dynamic controller is calculated according to the softened constraints; then a minimal ROA containing the current state is chosen and the corresponding softened constraint is adopted by the dynamic controller. Note that, the above measures are performed on-line because the centers of the above ROA are the steady-state targets calculated at each instant. The effectiveness of the presented strategy is illustrated through two examples.

A steady-state target calculation method based on “point” model for integrating processes

Aiming to eliminate the influences of model uncertainty on the steady-state target calculation for integrating processes, this paper presented an optimization method based on “point” model and a method determining whether or not there is a feasible solution of steady-state target. The optimization method resolves the steady-state optimization problem of integrating processes under the framework of two-stage structure, which builds a simple “point” model for the steady-state prediction, and compensates the error between “point” model and real process in each sampling interval. Simulation results illustrate that the outputs of integrating variables can be restricted within the constraints, and the calculation errors between actual outputs and optimal set-points are small, which indicate that the steady-state prediction model can predict the future outputs of integrating variables accurately.

Predictive Control Using an FPGA With Application to Aircraft Control

Alternative and more efficient computational methods can extend the applicability of model predictive control (MPC) to systems with tight real-time requirements. This paper presents a system-on-a-chip MPC system, implemented on a field-programmable gate array (FPGA), consisting of a sparse structure-exploiting primal dual interior point (PDIP) quadratic program (QP) solver for MPC reference tracking and a fast gradient QP solver for steady-state target calculation. A parallel reduced precision iterative solver is used to accelerate the solution of the set of linear equations forming the computational bottleneck of the PDIP algorithm. A numerical study of the effect of reducing the number of iterations highlights the effectiveness of the approach. The system is demonstrated with an FPGA-in-the-loop testbench controlling a nonlinear simulation of a large airliner. This paper considers many more manipulated inputs than any previous FPGA-based MPC implementation to date, yet the implementation comfortably fits into a midrange FPGA, and the controller compares well in terms of solution quality and latency to state-of-the-art QP solvers running on a standard PC.

Post-Optimality Approach to Prevent Cycling in Linear MPC Target Calculation

Model predictive control (MPC) is an advanced process control strategy which is widely applied in many industries and it is often implemented in two levels; steady-state target calculation and dynamic optimization. Target calculation level computes the optimal steady-state targets of the process and sends them to dynamic optimization for implementation. The existence of uncertainty or variation in the parameters of the target calculation layer can lead to unstable or cyclic targets which can significatively affect the overall performance of the controller. Many methods have been proposed to deal with model uncertainty using robust optimization. In this study, a new approach using post-optimality analysis is proposed to study the effect of uncertainty or variation in problem parameters on the optimal solution of linear target calculation. This approach can compute the stability limits, for simultaneous variations in objective function coefficients or process limitations, before the optimal target or basis are changed. It aims to improve the closed-loop robustness of MPC by assessing the required accuracy of parameter estimates and variation limits before the calculated target start cycling. The proposed approach offers a design-phase test for targets stability.

Offset-Free Strategy by Double-Layered Linear Model Predictive Control

In the real applications, the model predictive control (MPC) technology is separated into two layers, that is, a layer of conventional dynamic controller, based on which is an added layer of steady-state target calculation. In the literature, conditions for offset-free linear model predictive control are given for combined estimator (for both the artificial disturbance and system state), steady-state target calculation, and dynamic controller. Usually, the offset-free property of the double-layered MPC is obtained under the assumption that the system is asymptotically stable. This paper considers the dynamic stability property of the double-layered MPC.

Final Comment by the Authors

Model predictive control (MPC) is an advanced process control strategy which is widely applied in many industries and it is often implemented in two levels; steady-state target calculation and dynamic optimization. Target calculation level computes the optimal steady-state targets of the process and sends them to dynamic optimization for implementation. The existence of uncertainty or variation in the parameters of the target calculation layer can lead to unstable or cyclic targets which can significatively affect the overall performance of the controller. Many methods have been proposed to deal with model uncertainty using robust optimization. In this study, a new approach using post-optimality analysis is proposed to study the effect of uncertainty or variation in problem parameters on the optimal solution of linear target calculation. This approach can compute the stability limits, for simultaneous variations in objective function coefficients or process limitations, before the optimal target or basis are changed. It aims to improve the closed-loop robustness of MPC by assessing the required accuracy of parameter estimates and variation limits before the calculated target start cycling. The proposed approach offers a design-phase test for targets stability.

POST-OPTIMALITY ANALYSIS OF STEADY-STATE LINEAR TARGET CALCULATION IN MODEL PREDICTIVE CONTROL

Model predictive control (MPC) is an advanced process control strategy that is usually separated into two levels; steady-state target calculation and dynamic optimization. The existence of uncertainty in model parameters of target calculation can significantally affect the overall performance of the controller. Methods have been proposed to deal with model uncertainty using robust optimization. In this study, a new approach using post-optimality analysis is proposed to study the effect of uncertainty or variation in model parameters on the optimal solution of linear target calculation. This approach can compute the stability limits, for simultaneous variation in objective function coefficients or process limitations, before the optimal target or basis are changed.

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.

Dantzig-Wolfe Decomposition and Large-Scale Constrained MPC Problems

Model Predictive Control (MPC) strategies are typically implemented in two levels: a steady-state target calculation and a control calculation. The steady-state target calculation consumes excess degrees of freedom within the control problem to provide optimal steady-state performance with respect to some specified objective. In some MPC approaches, the target calculation is formulated as a Linear Program (LP) with a pre-specified objective function and a linear or Iinearized steady-state model derived from that used in the control calculation. In large-scale problems, centralized MPC schemes find the optimal solution for the plant-wide optimization problem, but may not provide sufficient redundancy or reliability and can require substantial computation. On the other hand, in a decentralized MPC scheme, the target calculations are performed independently by ignoring interactions among units, and as a result will not usually find the optimal operation. In contrast to the centralized MPC approach, a decentralized MPC provides a high degree of redundancy with respect to the failure of an individual MPC. For largescale process control problems, the desired characteristics for an MPC implementation include: accurate and quick tracking of the changing optimal steady-state operation, a high degree of reliability with respect to failure within the MPC application (i.e., failure of a portion of the control system), and low computational requirements. Fully centralized or monolithic MPC and independent block-wise decentralized MPC represent the two extremes in the “trade-off” among the desired characteristics of an implemented MPC system. In this paper, we propose a coordinated, decentralized approach to the steady-state target calculation problem. Our approach is based on the Dantzig-Wolfe decomposition principle and has been found to be effective at finding the optimal plant operation while providing a high degree of reliability at a reasonable computational load.

Disturbance modeling for offset-free linear model predictive control

Abstract An offset-free controller is one that drives controlled outputs to their desired targets at steady state. In the linear model predictive control (MPC) framework, offset-free control is usually achieved by adding step disturbances to the process model. The most widely-used industrial MPC implementations assume a constant output disturbance that can lead to sluggish rejection of disturbances that enter the process elsewhere. This paper presents a general disturbance model that accommodates unmeasured disturbances entering through the process input, state, or output. Conditions that guarantee detectability of the augmented system model are provided, and a steady-state target calculation is constructed to remove the effects of estimated disturbances. Conditions for which offset-free control is possible are stated for the combined estimator, steady-state target calculation, and dynamic controller. Simulation examples are provided to illustrate trade-offs in disturbance model design.