A robust economic MPC for changing economic criterion

This note presents a robust economic model predictive control controller suitable for changing economic criterion. The proposal ensures feasibility under any change of the economic criterion, thanks to the use of artificial variables and a relaxed terminal constraint, and robustness in presence of additive bounded disturbances. The resulting robust formulation considers a nominal prediction model and restricted constraints (in order to account for the effect of additive disturbances). The controlled system under the proposed controller is shown to be input‐to‐state stable in the sense that it is asymptotically steered to an invariant region around the best admissible steady state. An illustrative example shows the benefits and the properties of the proposed controller.     

An alternative structure for next generation regulatory controllers: Part I: Basic theory for design, development and implementation

Even though employed widely in industrial practice, the popular PID controller has weaknesses that limit its achievable performance, and an intrinsic structure that makes tuning not only more complex than necessary, but also less transparent with respect to the key attributes of the overall controller performance, namely: robustness, set-point tracking, and disturbance rejection. In this paper, we propose an alternative control scheme that combines the simplicity of the PID controller with the versatility of model predictive control (MPC) while avoiding the tuning problems associated with both. The tuning parameters of the proposed control scheme are related directly to the controller performance attributes; they are normalized to lie between 0 and 1; and they arise naturally from the formulation in a manner that makes it possible to tune the controller directly for each performance attribute independently. The result is a controller that can be designed and implemented much more directly and transparently, and one that outperforms the classical PID controller both in set-point tracking and disturbance rejection while using precisely the same process reaction curve information required to tune PID controllers. The design, implementation and performance of the controller are demonstrated via simulation on a nonlinear polymerization process.     

Multi-objective nonlinear predictive control of process systems: A dual-mode tracking control approach

There typically exist different and often conflicting control objectives, e.g., reference tracking, robustness and economic performance, in many chemical processes. The current work considers the multi-objective control problems of continuous-time nonlinear systems subject to state and input constraints and multiple conflicting objectives. We propose a new multi-objective nonlinear model predictive control (NMPC) design within the dual-mode paradigm, which guarantees stability and constraint satisfaction. The notions of utopia point and compromise solution are used to reconcile the confliction of the multiple objectives. The designed controller minimizes the distance of its cost vector to a vector of independently minimized objectives, i.e., the steady-state utopia point. Recursive feasibility is established via a particular terminal region formulation while stabilizing the closed-loop system to the compromise solution via the dual-mode control principle. In order to derive the terminal region as large as possible, a terminal control law with free-parameters is constructed by using the control Lyapunov functions (CLFs) technique. Two examples of multi-objective control of a CSTR and a free-radical polymerization process are used to illustrate the effectiveness of the new multi-objective NMPC and to compare their performance.     

指令跟踪自适应广义预测控制及其应用

现有的广义预测控制系统其闭环性能受可调参数影响较大,它的目标函数无法直接规定 闭环性能.该文提出一种具有独立跟踪和调节目标的新型自适应广义预测控制算法,并将其 应用于快速时变的导弹控制系统设计中.这种算法利用参考模型规定对指令信号的跟踪性 能,减少了可调参数对闭环性能的影响.仿真结果证实了该算法的有效性.     

Economic optimization using model predictive control with a terminal cost

In the standard model predictive control implementation, first a steady-state optimization yields the equilibrium point with minimal economic cost. Then, the deviation from the computed best steady state is chosen as the stage cost for the dynamic regulation problem. The computed best equilibrium point may not be the global minimum of the economic cost, and hence, choosing the economic cost as the stage cost for the dynamic regulation problem, rather than the deviation from the best steady state, offers potential for improving the economic performance of the system. It has been previously shown that the existing framework for MPC stability analysis, which addresses to the standard class of problems with a regulation objective, does not extend to economic MPC. Previous work on economic MPC developed new tools for stability analysis and identified sufficient conditions for asymptotic stability. These tools were developed for the terminal constraint MPC formulation, in which the system is stabilized by forcing the state to the best equilibrium point at the end of the horizon. In this work, we relax this constraint by imposing a region constraint on the terminal state instead of a point constraint, and adding a penalty on the terminal state to the regulator cost. We extend the stability analysis tools, developed for terminal constraint economic MPC, to the proposed formulation and establish that strict dissipativity is sufficient for guaranteeing asymptotic stability of the closed-loop system. We also show that the average closed-loop performance outperforms the best steady-state performance. For implementing the proposed formulation, a rigorous analysis for computing the appropriate terminal penalty and the terminal region is presented. A further extension, in which the terminal constraint is completely removed by modifying the regulator cost function, is also presented along with its stability analysis. Finally, an illustrative example is presented to demonstrate the differences between the terminal constraint and the proposed terminal penalty formulation.     

Robust hybrid predictive control of nonlinear systems

In this work, we consider nonlinear systems with input constraints and uncertain variables, and develop a robust hybrid predictive control structure that provides a safety net for the implementation of any model predictive control (MPC) formulation, designed with or without taking uncertainty into account. The key idea is to use a Lyapunov-based bounded robust controller, for which an explicit characterization of the region of robust closed-loop stability can be obtained, to provide a stability region within which any available MPC formulation can be implemented. This is achieved by devising a set of switching laws that orchestrate switching between MPC and the bounded robust controller in a way that exploits the performance of MPC whenever possible, while using the bounded controller as a fall-back controller that can be switched in at any time to maintain robust closed-loop stability in the event that the predictive controller fails to yield a control move (due, e.g., to computational difficulties in the optimization or infeasibility) or leads to instability (due, e.g., to inappropriate penalties and/or horizon length in the objective function). The implementation and efficacy of the robust hybrid predictive control structure are demonstrated through simulations using a chemical process example.     

Handling uncertainty in economic nonlinear model predictive control: A comparative case study

In the last years, the use of an economic cost function for model predictive control (MPC) has been widely discussed in the literature. The main motivation for this choice is that often the real goal of control is to maximize the profit or the efficiency of a certain system, rather than tracking a predefined set-point as done in the typical MPC approaches, which can be even counter-productive. Since the economic optimal operation of a system resulting from the application of an economic model predictive control approach drives the system to the constraints, the explicit consideration of the uncertainties becomes crucial in order to avoid constraint violations. Although robust MPC has been studied during the past years, little attention has yet been devoted to this topic in the context of economic nonlinear model predictive control, especially when analyzing the performance of the different MPC approaches. In this work, we present the use of multi-stage scenario-based nonlinear model predictive control as a promising strategy to deal with uncertainties in the context of economic NMPC. We make a comparison based on simulations of the advantages of the proposed approach with an open-loop NMPC controller in which no feedback is introduced in the prediction and with an NMPC controller which optimizes over affine control policies. The approach is efficiently implemented using CasADi, which makes it possible to achieve real-time computations for an industrial batch polymerization reactor model provided by BASF SE. Finally, a novel algorithm inspired by tube-based MPC is proposed in order to achieve a trade-off between the variability of the controlled system and the economic performance under uncertainty. Simulations results show that a closed-loop approach for robust NMPC increases the performance and that enforcing low variability under uncertainty of the controlled system might result in a big performance loss.     

Adaptive horizon economic nonlinear model predictive control

In this paper, we present a computationally efficient economic NMPC formulation, where we propose to adaptively update the length of the prediction horizon in order to reduce the problem size. This is based on approximating an infinite horizon economic NMPC problem with a finite horizon optimal control problem with terminal region of attraction to the optimal equilibrium point. Using the nonlinear programming (NLP) sensitivity calculations, the minimum length of the prediction horizon required to reach this terminal region is determined. We show that the proposed adaptive horizon economic NMPC (AH-ENMPC) has comparable performance to standard economic NMPC (ENMPC). We also show that the proposed adaptive horizon economic NMPC framework is nominally stable. Two benchmark examples demonstrate that the proposed adaptive horizon economic NMPC provides similar performance as the standard economic NMPC with significantly less computation time.     

Block factorization of step response model predictive control problems

By introducing a stage-wise prediction formulation that enables the use of highly efficient quadratic programming (QP) solution methods, this paper expands the computational toolbox for solving step response MPC problems. We propose a novel MPC scheme that is able to incorporate step response data in a traditional manner and use the computationally efficient block factorization facilities in QP solution methods. In order to solve the MPC problem efficiently, both tailored Riccati recursion and condensing algorithms are proposed and embedded into an interior-point method. The proposed algorithms were implemented in the HPMPC framework, and the performance is evaluated through simulation studies. The results confirm that a computationally fast controller is achieved, compared to the traditional step response MPC scheme that relies on an explicit prediction formulation. Moreover, the tailored condensing algorithm exhibits superior performance and produces solution times comparable to that achieved when using a condensing scheme for an equivalent (but much smaller) state-space model derived from first-principles. Implementation aspects necessary for high performance on embedded platforms are discussed, and results using a programmable logic controller are presented.     

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.     

ILC strategy for progress improvement of economic performance in industrial model predictive control systems

A novel approach to progress improvement of the economic performance in model predictive control (MPC) systems is developed. The conventional LQG based economic performance design provides an estimation which cannot be done by the controller while the proposed approach can develop the design performance achievable by the controller. Its optimal performance is achieved by solving economic performance design (EPD) problem and optimizing the MPC performance iteratively in contrast to the original EPD which has nonlinear LQG curve relationship. Based on the current operating data from MPC, EPD is transformed into a linear programming problem. With the iterative learning control (ILC) strategy, EPD is solved at each trial to update the tuning parameter and the designed condition; then MPC is conducted in the condition guided by EPD. The ILC strategy is proposed to adjust the tuning parameter based on the sensitivity analysis. The convergence of EPD by the proposed ILC has also been proved. The strategy can be applied to industry processes to keep enhancing the performance and to obtain the achievable optimal EPD. The performance of the proposed method is illustrated via an SISO numerical system as well as an MIMO industry process.     

Coherent quantum LQG control

Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as “coherent feedback control”. It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints, our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as an initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.     

Economic model predictive control of nonlinear singularly perturbed systems

We focus on the development of a Lyapunov-based economic model predictive control (LEMPC) method for nonlinear singularly perturbed systems in standard form arising naturally in the modeling of two-time-scale chemical processes. A composite control structure is proposed in which, a “fast” Lyapunov-based model predictive controller (LMPC) using a quadratic cost function which penalizes the deviation of the fast states from their equilibrium slow manifold and the corresponding manipulated inputs, is used to stabilize the fast dynamics while a two-mode “slow” LEMPC design is used on the slow subsystem that addresses economic considerations as well as desired closed-loop stability properties by utilizing an economic (typically non-quadratic) cost function in its formulation and possibly dictating a time-varying process operation. Through a multirate measurement sampling scheme, fast sampling of the fast state variables is used in the fast LMPC while slow-sampling of the slow state variables is used in the slow LEMPC. Appropriate stabilizability assumptions are made and suitable constraints are imposed on the proposed control scheme to guarantee the closed-loop stability and singular perturbation theory is used to analyze the closed-loop system. The proposed control method is demonstrated through a nonlinear chemical process example.     

Profitable and dynamically feasible operating point selection for constrained processes

The operating point of a typical chemical process is determined by solving a non-linear optimization problem where the objective is to minimize an economic cost subject to constraints. Often, some or all of the constraints at the optimal solution are active, i.e., the solution is constrained. Though it is profitable to operate at the constrained optimal point, it might lead to infeasible operation due to uncertainties. Hence, industries try to operate the plant close to the optimal point by “backing-off” to achieve the desired economic benefits. Therefore, the primary focus of this paper is to present an optimization formulation for solving the dynamic back-off problem based on an economic cost function. In this regard, we work within a stochastic framework that ensures feasible dynamic operating region within the prescribed confidence limit. In this work, we aim to reduce the economic loss due to the back-off by simultaneously solving for the operating point and a compatible controller that ensures feasibility. Since the resulting formulation is non-linear and non-convex, we propose a novel two-stage iterative solution procedure such that a convex problem is solved at each step in the iteration. Finally, the proposed approach is demonstrated using case studies.     

Linear LMI-based external anti-windup augmentation for stable linear systems

We study linear anti-windup augmentation for linear control systems with saturated linear plants in the special case when the anti-windup compensator can only modify the input and the output of the windup-prone linear controller. We also measure the arising performance in terms of the finite L₂ gain from exogenous inputs to selected performance outputs. Our main results are a system theoretic feasibility characterization for fixed order anti-windup design and a linear matrix inequality (LMI) formulation for optimal static and plant-order anti-windup design. Interpretations of lower bounds on the achievable performance are also given. The effectiveness of the design procedure is demonstrated on a simulation example.     

Consistent hierarchical economic NMPC for a class of hybrid systems using neighboring-extremal updates

A hierarchical two-layer control algorithm is developed for a class of hybrid (discrete-continuous dynamic) systems to support economically optimal operation of batch or continuous processes with a predefined production schedule. For this class of hybrid systems, the optimal control moves as well as the controlled switching times between two adjacent modes are determined online. In contrast to closely related schemes for integrated scheduling and control, the sequence of modes is not optimized. On the upper layer, the economic optimal control problem is solved rigorously by a slow hybrid economic model predictive controller at a low sampling rate. On the lower layer, a fast hybrid neighboring-extremal controller is based on the same economic optimal control problem as the slow controller to ensure consistency between both layers. The fast neighboring-extremal controller updates rather than tracks the optimal trajectories from the upper layer to account for disturbances. Consequently, the fast controller steers the process to its operational bounds under disturbances and the economic potential of the process is exploited anytime. The suggested two-layer control algorithm provides fully consistent control action on the fast and slow time-scale and thus avoids performance degradation and even infeasibilities which are commonly encountered if inconsistent optimal control problems are formulated and solved.     

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.     

Linear active disturbance rejection control for a raceway photobioreactor

In this paper, the problem of pH regulation in photobioreactors has been addressed by using a robust Proportional–Integral controller in combination with a linear active disturbance rejection approach. This formulation is based on the Generalized Proportional–Integral (GPI) observers design to estimate online external disturbances and non-modeled dynamics. The methodology uses a model reference optimization procedure with tracking and regulatory target closed-loop transfer functions for first-order plus dead-time models. The proposed controller is validated on a full-scale Raceway photobioreactor, whose results show a significant improvement in the accuracy of pH regulation and, consequently, a positive influence on biomass production. Moreover, a classical feedforward approach has been used for comparison purposes. The performance of the robust technique is evaluated with different indexes, whose results confirm the good performance of the proposed active disturbance rejection control scheme.     

Robustness studies for H∞ feedback control in a structural acoustic model with periodic excitation

Results from an extensive study on the robustness of an H^∞ compensator for a 2-D structural acoustic model are presented. The effects of frequency uncertainties in an exogenous signal are studied for both the case where the signal is contained in the controller formulation and the case where it is excluded. Delays are inserted in the input and/or output signals and their effect on the controller performance is recorded. A comparison between the standard LQG/Kalman filter and the H^∞/Min-Max compensator reveals no significant differences in the overall controller performance. Modifications in the controller structure are studied to see whether loss of information (the tracking variable) that must be calculated a priori would result in performance degradation. This study provides valuable insight into the computational and implementational issues that arise when dealing with the control of large and complex systems that are governed by partial differential equations.