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.     

Real-time energy management for electric arc furnace operation

Electric arc furnaces are used extensively in the steel industry for steel production. Development of energy savings strategies for the highly energy-intensive batch process is extremely challenging due to the complexity of the process and lack of measurements due to the harsh operating conditions. Here we introduce a new energy management approach that effectively curtails the energy cost in real-time through the implementation of economically optimal operating decisions. An economics- oriented shrinking horizon nonlinear model predictive control (NMPC) algorithm that exploits time-varying electricity prices is coupled with a multi-rate moving horizon estimator (MHE) to form an integrated decision- making framework. With a detailed first-principles dynamic model functioning at the core, the multi-variable interactions and plant variations are successfully incorporated into the control strategy to achieve reliable performance. We also present a novel initialization scheme for obtaining fast on-line solutions of the economic NMPC and multi-rate MHE dynamic optimization problems. Using this initialization algorithm, we show that the optimal input decisions are obtained with sufficient computational speed for real-time implementation. The energy usage optimization results indicate a significant reduction in the operating cost and peak electricity demand compared to the case where the electricity price profile is not updated.     

A COMPARATIVE STUDY ON COMPUTATIONAL SCHEMES FOR NONLINEAR MODEL PREDICTIVE CONTROL

This paper briefly reviews development of nonlinear model predictive control (NMPC) schemes for finite horizon prediction and basic computational algorithms that can solve the stable real‐time implementation of NMPC in space state form with state and input constraints. In order to ensure stability within a finite prediction horizon, most NMPC schemes use a terminal region constraint at the end of the prediction horizon — a particular NMPC scheme using a terminal region constraint, namely quasi‐infinite horizon, that guarantees asymptotic closed‐loop stability with input constraints is presented. However, when nonlinear processes have both input and state constraints, difficulty arises from failure to satisfy the state constraints due to constraints on input. Therefore, a new NMPC scheme without a terminal region constraint is developed using soften state constraints. A brief comparative simulation study of two NMPC schemes: quasi‐infinite horizon and soften state constraints is done via simple nonlinear examples to demonstrate the ability of the soften state constraints scheme. Finally, some features of future research from this study are discussed.     

Fast economic model predictive control based on NLP-sensitivities

This study adapts the advanced step NMPC framework to Economic NMPC. Here, sufficient conditions for nominal stability are derived for NMPC controllers that incorporate economic stage costs with appropriate regularization. To guarantee these conditions, we derive a constructive strategy to calculate the regularization term directly. Moreover, we extend the sensitivity components in the advanced step NMPC framework to consider a rigorous path-following algorithm. This approach accounts for active set changes and allows much weaker constraint qualifications. Moreover, using an ℓ₁ formulation of the NMPC problem satisfies these constraint qualifications and allows more reliable solution of the moving horizon optimization problem, even in the presence of noise. Finally, all of these concepts are demonstrated on a detailed case study with a continuously stirred tank reactor.     

Nonlinear model predictive control of an internal combustion engine exposed to measured disturbances

This work presents the design procedure of a speed controller for a large, lean burn, natural gas engine in island mode operation. This is a disturbance rejection problem with a measured, large disturbance. The core element is a nonlinear model predictive control (NMPC) algorithm that serves as outer loop controller in a cascaded control structure and generates set-points for low level control loops. The NMPC relies on a control oriented model that includes the physics based equations, assumptions on underlying control loops and constraints given by the control requirements. It is shown how to design the running cost such that the stability of the NMPC without terminal cost and constraints can be guaranteed for the nominal system and for the perturbed system exposed to parametric uncertainties and un-modeled dynamics. The functionality of the control strategy is demonstrated in simulation and by experimental results derived at the engine-testbed.     

On the stability of receding horizon control with a general terminal cost

We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for input-constrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.     

Robust stability of nonlinear model predictive control based on extended Kalman filter

This work deals with state estimation and process control for nonlinear systems, especially when nonlinear model predictive control (NMPC) is integrated with extended Kalman filter (EKF) as the state estimator. In particular, we focus on the robust stability of NMPC and EKF in the presence of plant-model mismatch. The convergence property of the estimation error from the EKF in the presence of non-vanishing perturbations is established based on our previous work [1]. In addition, a so-called one way interaction is shown that the EKF error is not influenced by control action from the NMPC. Hence, the EKF analysis is still valid in the output-feedback NMPC framework, even though there is no separation principle for general nonlinear systems. With this result, we study the robust stability of the output-feedback NMPC under the impact of the estimation error. It turns out the output-feedback NMPC with EKF is Input-to-State practical Stable (ISpS). Finally, two offset-free strategies of output-feedback NMPC are presented and illustrated through a simulation example.     

Enlarging the terminal region of quasi-infinite horizon NMPC based on T-S fuzzy model

The paper presents a method for enlarging the terminal region of quasi-infinity horizon nonlinear model predictive control (NMPC) for nonlinear systems with constraints. The main technique builds on the fact that terminal controllers are fictitious and never applied to the system in the quasi-infinite horizon NMPC [1]. Based on T-S fuzzy models of nonlinear systems, we show that a parameter-dependent state feedback law exists such that the corresponding value function and its level set can be served as terminal cost and terminal region. The problem of maximizing the terminal region is formulated as a convex optimization problem based on linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed method.     

IQC‐based robustness analysis of discrete‐time linear time‐varying systems

The paper investigates the robust stability and performance of uncertain linear time‐varying (LTV) systems using an integral quadratic constraint (IQC) based analysis approach. Specifically, previous theoretical work on IQC‐based robustness analysis of linear time‐invariant (LTI) systems is extended to discrete‐time LTV systems. In the case of a general LTV nominal system, the analysis solution is provided in terms of an infinite‐dimensional convex optimization problem. This optimization problem reduces into a finite‐dimensional semidefinite program when the nominal system in question is finite horizon, periodic, or, more generally, eventually periodic. Finally, the results are applied to an unmanned aircraft control system executing an aggressive maneuver, where the developed techniques are used to find the region in which the aircraft is guaranteed to reside at the end of its planned trajectory. Copyright © 2017 John Wiley & Sons, Ltd.     

Predictive optimal control for thermo-mechanical pulping processes with multi-stage low consistency refining

In this paper, we present a nonlinear model based dynamic optimization approach for optimal control for thermo-mechanical pulping (TMP) processes. The method is based on nonlinear model predictive control (NMPC) technique with an economic objective function. In our previous work [1], an economically oriented NMPC (econNMPC) was presented for a two-stage TMP process with high consistency (HC) refiners. The two-stage TMP process was simulated up to the secondary refiner. In this study, we extend the previous study to multiple stages that include low consistency (LC) refining. Here we can exploit the complexity of interactions of multiple stage TMP for both performance and optimal operations. The first process is a conventional refining process with two stages of HC refining with a primary and a secondary refiner, followed by a latency chest and a third-stage LC refiner. The second TMP process has only a HC primary refiner, and the secondary stage HC refiner is replaced by multiple stages of LC refiners after the latency chest. Both TMP schemes are dynamically optimized against disturbances to produce the same target final pulp qualities while minimizing specific energy consumption of the processes. The simulation results show the potential economic benefits of the proposed method, through a reduction in total specific energy, about 24%, for the second TMP process.     

A convex approach for NMPC based on second order Volterra series models

In model predictive control (MPC), the input sequence is computed, minimizing a usually quadratic cost function based on the predicted evolution of the system output. In the case of nonlinear MPC (NMPC), the use of nonlinear prediction models frequently leads to non‐convex optimization problems with several minimums. This paper proposes a new NMPC strategy based on second order Volterra series models where the original performance index is approximated by quadratic functions, which represent a lower bound of the original performance index. Convexity of the approximating quadratic cost functions can be achieved easily by a suitable choice of the weighting of the control increments in the performance index. The approximating cost functions can be globally minimized by convex optimization techniques in order to compute the input sequence. The minimization of the performance index is carried out by an iterative optimization procedure, which guarantees convergence to the solution. Furthermore, for a nominal prediction model, asymptotic stability for the proposed NMPC strategy can be shown. In the case of considering an estimation error in the prediction model, input‐to‐state practical stability is assured. The control performance of the NMPC strategy is illustrated by experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

Decentralized tube‐based model predictive control of uncertain nonlinear multiagent systems

This paper addresses the problem of decentralized tube‐based nonlinear model predictive control (NMPC) for a general class of uncertain nonlinear continuous‐time multiagent systems with additive and bounded disturbance. In particular, the problem of robust navigation of a multiagent system to predefined states of the workspace while using only local information is addressed under certain distance and control input constraints. We propose a decentralized feedback control protocol that consists of two terms: a nominal control input, which is computed online and is the outcome of a decentralized finite horizon optimal control problem that each agent solves at every sampling time, for its nominal system dynamics; and an additive state‐feedback law which is computed offline and guarantees that the real trajectories of each agent will belong to a hypertube centered along the nominal trajectory, for all times. The volume of the hypertube depends on the upper bound of the disturbances as well as the bounds of the derivatives of the dynamics. In addition, by introducing certain distance constraints, the proposed scheme guarantees that the initially connected agents remain connected for all times. Under standard assumptions that arise in nominal NMPC schemes, controllability assumptions, communication capabilities between the agents, it is guaranteed that the multiagent system is input‐to‐state stable with respect to the disturbances, for all initial conditions satisfying the state constraints. Simulation results verify the correctness of the proposed framework.     

A primal-dual semi-definite programming approach to linearquadratic control

We study a deterministic linear-quadratic (LQ) control problem over an infinite horizon, without the restriction that the control cost matrix R or the state cost matrix Q be positive-definite. We develop a general approach to the problem based on semi-definite programming (SDP) and related duality analysis. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control under a certain stability condition (which is satisfied automatically when Q is positive-definite). When the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP     

Multirate sliding mode disturbance compensation for model predictive control

In this paper, a novel hierarchical multirate control scheme for nonlinear discrete‐time systems is presented, consisting of a robust nonlinear model predictive controller (NMPC) and a multirate sliding mode disturbance compensator (MSMDC). The proposed MSMDC acts at a faster rate than the NMPC in order to keep the system as close as possible to the nominal trajectory predicted by NMPC despite model uncertainties and external disturbances. The a priori disturbance compensation turns out to be very useful in order to improve the robustness of the NMPC controller. A dynamic input allocation between MSMDC and NMPC allows to maximize the benefits of the proposed scheme that unites the advantages of sliding mode control (strong reduction of matched disturbances, low computational burden) to those of NMPC (optimality, constraints handling). Sufficient conditions required to guarantee input‐to‐state stability and constraints satisfaction by the overall scheme are also provided. Copyright © 2014 John Wiley & Sons, Ltd.     

A reduced regularization strategy for economic NMPC

Nonlinear Model Predictive Control (NMPC) enables the incorporation of detailed dynamic process models for nonlinear, multivariable control with constraints. This optimization-based framework also leads to on-line dynamic optimization with performance-based and so-called economic objectives. Nevertheless, economic NMPC (eNMPC) still requires careful formulation of the nonlinear programming (NLP) subproblem to guarantee stability. In this study, we derive a novel reduced regularization eNMPC approach with stability guarantees. Compared with full state regularization, the proposed strategy is less conservative and easier to implement. The resulting eNMPC framework is firstly demonstrated on a nonlinear continuous stirred-tank reactor (CSTR) example and a large-scale double distillation system example. Then the proposed strategy is applied to a challenging nonlinear CO₂ capture model, where bubbling fluidized bed models comprise a solid-sorbent post-combustion carbon capture system. Our results indicate the benefits of this improved eNMPC approach over tracking to the setpoint, and better stability over eNMPC without regularization.     

Real-time economic nonlinear model predictive control for wind turbine control

Nonlinear model predictive control (NMPC) is a strong candidate to handle the control challenges emerging in the modern wind energy industry. Recent research suggested that wind turbine (WT) control based on economic NMPC (ENMPC) can improve the closed-loop performance and simplify the task of controller design when compared to a classical NMPC approach. This paper establishes a formal relationship between the ENMPC controller and the classic NMPC approach, and compares empirically their closed-loop nominal behaviour and performance. The robustness of the performance is assessed for an inaccurate modelling of the tower fore-aft main frequency. Additionally, though a perfect wind preview is assumed here, the effect of having a limited horizon of preview of the wind speed via the LIght Detection And Ranging (LIDAR) sensor is investigated. Finally, this paper provides new algorithmic solutions for deploying ENMPC for WT control, and report improved computational times.     

Stability analysis of a multi-model predictive control algorithm with application to control of chemical reactors

We study a stabilizing multi-model predictive control strategy for controlling nonlinear process at different operating conditions. The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function that has finite and infinite horizon cost components. The finite horizon cost consists of free input variables that direct the system towards a terminal region which contains the desired operating point. The infinite horizon cost has an upper bound and steers the system to the desired operating point. The system is represented by a sequence of piecewise linear models. Based on the condition of the system states, the sequence of piecewise linear models is updated and the controller’s objective function switches form quasi-infinite to infinite horizon objective function. This results in a hybrid control structure. A recent approach in the analysis of hybrid systems that uses multiple Lyapunov functions is employed in the stability analysis of the closed-loop system. The stabilizing hybrid control strategy is illustrated on two examples and their closed-loop stability properties are studied.     

Intelligent state estimation for fault tolerant nonlinear predictive control

There is growing realization that on-line model maintenance is the key to realizing long term benefits of a predictive control scheme. In this work, a novel intelligent nonlinear state estimation strategy is proposed, which keeps diagnosing the root cause(s) of the plant model mismatch by isolating the subset of active faults (abrupt changes in parameters/disturbances, biases in sensors/actuators, actuator/sensor failures) and auto-corrects the model on-line so as to accommodate the isolated faults/failures. To carry out the task of fault diagnosis in multivariate nonlinear time varying systems, we propose a nonlinear version of the generalized likelihood ratio (GLR) based fault diagnosis and identification (FDI) scheme (NL-GLR). An active fault tolerant NMPC (FTNMPC) scheme is developed that makes use of the fault/failure location and magnitude estimates generated by NL-GLR to correct the state estimator and prediction model used in NMPC formulation. This facilitates application of the fault tolerant scheme to nonlinear and time varying processes including batch and semi-batch processes. The advantages of the proposed intelligent state estimation and FTNMPC schemes are demonstrated by conducting simulation studies on a benchmark CSTR system, which exhibits input multiplicity and change in the sign of steady state gain, and a fed batch bioreactor, which exhibits strongly nonlinear dynamics. By simulating a regulatory control problem associated with an unstable nonlinear system given by Chen and Allgower [H. Chen, F. Allgower, A quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica 34(10) (1998) 1205–1217], we also demonstrate that the proposed intelligent state estimation strategy can be used to maintain asymptotic closed loop stability in the face of abrupt changes in model parameters. Analysis of the simulation results reveals that the proposed approach provides a comprehensive method for treating both faults (biases/drifts in sensors/actuators/model parameters) and failures (sensor/ actuator failures) under the unified framework of fault tolerant nonlinear predictive control.     

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.     

The advanced-step NMPC controller: Optimality, stability and robustness

Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear programming (NLP) sensitivity have reduced on-line computational costs and can lead to significantly improved performance. In this study, we extend this concept through a simple reformulation of the NMPC problem and propose the advanced-step NMPC controller. The main result of this extension is that the proposed controller enjoys the same nominal stability properties of the conventional NMPC controller without computational delay. In addition, we establish further robustness properties in a straightforward manner through input-to-state stability concepts. A case study example is presented to demonstrate the concepts.