Multi-stage Nonlinear Model Predictive Control Research Articles

Multi-scenario and multi-stage robust NMPC with state estimation application on the Tennessee-Eastman process

This study presents the implementation of two discrete robust approaches to Non-linear Model Predictive Control (NMPC), multi-scenario NMPC (MSc-NMPC) and multi-stage NMPC (MS-NMPC), to the benchmark Tennessee-Eastman (TE) challenge, with Extended Kalman Filter (EKF) and Moving Horizon Estimation (MHE) as state estimators. The robust NMPC formulation results in closed-loop responses that prevent constraint violation and closely track the process set-point under parameter uncertainty, even in scenarios where traditional NMPC results in an unstable response for this process. Additionally, unconstrained state estimators such as EKF are unsuitable because the parameter uncertainty may cause estimates to fall outside the feasible region of the process, which ultimately destabilizes the process. MHE was able to overcome this challenge because it considers process constraints in its formulation. The additional computational time required to solve the robust NMPC formulations and MHE does not cause significant delays for the sampling time considered, demonstrating their applicability to challenging large-scale industrial chemical processes.

Multi-Stage NMPC for a MAV based Collision Free Navigation Under Varying Communication Delays

Time delays in communication networks are one of the main concerns in deploying robots with computation boards on the edge. This article proposes a multi-stage Nonlinear Model Predictive Control (NMPC) that is capable of handling varying network-induced time delays for establishing a control framework being able to guarantee collision-free Micro Aerial Vehicles (MAVs) navigation. This study introduces a novel approach that considers different sampling times by a tree of discretization scenarios contrary to the existing typical multi-stage NMPC where system uncertainties are modeled by a tree of scenarios. Additionally, the proposed method considers adaptive weights for the multi-stage NMPC scenarios based on the probability of time delays in the communication link. As a result of the multi-stage NMPC, the obtained optimal control action is valid for multiple sampling times. Finally, the overall effectiveness of the proposed novel control framework is demonstrated in various tests and different simulation environments.

Robust multistage nonlinear model predictive control on an autonomous marine surface vehicle

Robust multistage nonlinear model predictive control on an autonomous marine surface vehicle

Multi-stage Nonlinear Model Predictive Control with Online Scenario Update for Semi-batch Polymerization Processes

Multi-stage Nonlinear Model Predictive Control with Online Scenario Update for Semi-batch Polymerization Processes

Robust nonlinear model predictive control of cascade of fermenters with recycle for efficient bioethanol production

In this paper, dynamical behavior and control of the continuous ethanol fermentation process are studied. The process productivity is improved by using two fermenters connected in series with recycle of cells. First, dynamical behavior and productivity of the cascade system are studied by using numerical bifurcation analysis. Then, multi-stage nonlinear model predictive control (MS-NMPC), classical and offset-free NMPCs algorithms are designed to eliminate the undesired oscillations. To ensure acceptable efficiency of the cascade process, a minimal ethanol concentration is defined and included as additional constraint in the controller design. The unmeasured state variables are estimated by using the asymptotic observer. It is shown that for smaller sampling times the predictive controllers are very effective, even in presence of parameter uncertainties. It is also observed that offset-free NMPC algorithm is more sensitive to measurement noise, but MS-NMPC and classical NMPC are more sensitive to estimation errors for longer sampling times.

On the Practical Design of Tube-enhanced Multi-stage Nonlinear Model Predictive Control

Tube-enhanced multi-stage nonlinear model predictive control is a robust control scheme that can handle a wide range of uncertainties with reduced conservatism and manageable computational complexity. In this paper, we elaborate on the flexibility of the approach from an application point of view. We discuss the path to making design decisions to implement the novel scheme systematically. We illustrate the critical steps in the design and implementation of the scheme for an industrial example.

Multistage nonlinear model predictive control for pumping treatment in hydraulic fracturing

AbstractHydraulic fracturing has gained increasing attention as it allows the constrained natural gas and crude oil to flow out of low‐permeability shale formations and significantly increase production. Perilous operating states of extremely high pressure also raise some safety concerns, requiring us to formulate an appropriate dynamic model, and provide a careful engineering control to ensure safe operating conditions. Moreover, uncertainties due to spatially varying rock properties increase the difficulties in control of the fracturing process. In this work, we formulate a first‐principles model by considering the fracture evolution, mass transport of substances in the slurry, changing fluid properties, and the monitored operating pressure on the ground level. Next, we implement nonlinear model predictive control (NMPC) to control the process under a set of final requirements and process constraints. Our results show that the performance of standard NMPC degrades when the rock uncertainty causes the parameter mismatch between the process and the predictive model in the controller. With standard NMPC, designed with a nominal model, the process fails to meet the terminal requirements of fracture geometry, and pressure is violated in one of the parameter mismatch cases. Therefore, we resort to multistage NMPC, which considers uncertainty evolution in a scenario tree with separate control sequences to address constraint violations. We demonstrate that multistage NMPC presents good performance by showing constraint satisfaction whether the uncertain rock parameter realization is time‐invariant or time‐variant. We also simulate the process with multistage NMPC including different numbers of scenarios and compare their control performance. Our investigation demonstrates that multistage NMPC effectively manages parametric uncertainties attributed to non‐homogeneous rock formation, and provides a promising control strategy for the hydraulic fracturing process.

Sensitivity-Assisted multistage nonlinear model predictive control: Robustness, stability and computational efficiency

Key requirements for robust nonlinear model predictive control (NMPC) are stability, efficient performance under uncertainty, constraint satisfaction and computational efficiency. Multistage NMPC, based on a scenario tree formulation for the uncertainty, has been shown to satisfy the first three objectives under plant-model mismatch. However, a limiting factor in multistage NMPC, is the exponential scaling of the scenarios with respect to uncertain parameters and the length of the robust horizon. To address this issue, we present an approximate sensitivity-assisted multistage NMPC (samNMPC) scheme that reduces the problem size by dividing the scenario set into critical and noncritical scenarios, with the former composed of the worst-case realizations of the uncertain parameters. In this approach, the optimization is sought explicitly over the critical scenarios, while noncritical scenarios are included implicitly through nonlinear programming (NLP) sensitivity-based approximations in the objective function. A key advantage of the proposed approach is that the problem size is independent of the number of constraints and scales only linearly with the robust horizon. This allows faster computations with longer robust horizons that rigorously account for future uncertainty. In this paper, we explore the samNMPC approach and discuss its robust stability properties in context of the robust horizon. We demonstrate the applicability of the approach for the continuous stirred tank reactor (CSTR) and the quadtank case studies for tracking setpoints, and show that samNMPC compares favorably in performance and robustness to ideal multistage NMPC, but with a significant reduction in computational cost.

Fast approximate learning-based multistage nonlinear model predictive control using Gaussian processes and deep neural networks

Scenario-based model predictive control (MPC) methods introduce recourse into optimal control and can thus reduce the conservativeness inherent to open-loop robust MPC. However, the uncertainty scenarios are often generated offline using worst-case uncertainty bounds quantified a priori, limiting the potential gains in control performance. This paper presents a learning-based multistage MPC (msMPC) for systems with hard-to-model dynamics and time-varying plant-model mismatch. Gaussian Processes (GP) are used to learn state- and input-dependent plant-model mismatch in real-time and accordingly adapt the scenario tree online. Due to the increased computational complexity associated with incorporating the GP predictions into the optimal control problem, the learning-based msMPC (LB-msMPC) law is approximated by a deep neural network (DNN) that is cheap-to-evaluate online and has a small memory footprint, which makes it suitable for embedded applications. In addition, we present a novel algorithm for training the DNN-based controller that uses a GP description of the plant-model mismatch to generate closed-loop simulation data, which ensures the LB-msMPC law is evaluated in regions of the state space most relevant to closed-loop operation. The proposed LB-msMPC strategy is demonstrated on a cold atmospheric plasma jet with applications in (bio)materials processing. The simulation results indicate the promise of the approximate LB-msMPC strategy for control of hard-to-model systems with fast dynamics on millisecond timescales.

Stability properties of multi-stage nonlinear model predictive control

This paper discusses the stability properties of a robust nonlinear model predictive control (NMPC) scheme that is based on a multi-stage optimization formulation. The use of a scenario tree to represent the uncertainty makes it possible to formulate a closed-loop robust approach with recourse which improves the open-loop approach in terms of performance and domain of attraction. We show that a straightforward formulation of a multi-stage NMPC scheme does not guarantee Input-to-State stability (ISS) in a deterministic setting, in contrast to the results that one gets using stochastic stability concepts. Since for many applications deterministic stability guarantees are desired, we provide an alternative formulation to achieve deterministic ISS and recursive feasibility guarantees for the case of discrete values of the uncertainty. The design and the performance of the proposed schemes are illustrated by simulations for a highly nonlinear example.

A Multi-stage Economic NMPC for the Tennessee Eastman Challenge Process

This paper addresses the design and implementation of a robust nonlinear model predictive control (NMPC) scheme for a benchmark plant-wide control problem. The focus of our research is on the performance of direct optimizing control for a complex large-scale process which is subject to plant-model mismatch and external disturbances. As a benchmark case for control and monitoring applications, the Tennessee Eastman Challenge (TEC) process has been widely employed in many publications. We present a first NMPC implementation for this where only economics criteria are used for the control of the process. The results obtained demonstrate the viability of plant-wide economics optimizing NMPC. We also address the issue of robustness against model uncertainties and employ multi-stage NMPC to tackle these. Different possible multi-stage NMPC implementations are discussed and the trade-offs between economic performance and robustness are highlighted.

Sensitivity-assisted Robust Nonlinear Model Predictive Control with Scenario Generation

We propose a sensitivity-assisted multistage Nonlinear Model Predictive Control strategy, called samNMPC, to address multistage stochastic programs for robust NMPC. Our approach divides the scenario sets in the stochastic programming formulation into critical and noncritical sets. Critical scenarios are selected by scenario generation based on worst-case constraint determination, while stage costs for noncritical scenarios are determined by sensitivity-based approximations. The resulting multi-stage NMPC problem leads to a first order accurate control profile that satisfies all constraints under uncertainty. Moreover, computational costs of this formulation scale independently of the number of disturbance variables, and only linearly with the robust horizon and number of constraints. Our proposed approach is illustrated on a CSTR (continuous stirred tank reactor) case study with two uncertain parameters. Compared to competing approaches, samNMPC delivers robust performance of multi-stage NMPC with significantly less computational cost.

Application of Model-based Online Monitoring and Robust Optimizing Control to Fed-Batch Bioprocesses

The aim of the quality by design initiative is to assure a continuous and high-quality production of pharmaceuticals despite the presence of process variations and disturbances. This need for optimal process operation necessitates the use of accurate prediction and fault detection methods in combination with advanced control strategies. However, the critical component for the success of such an approach is a mathematical model providing an adequate representation of the bioprocess under study. This work presents a framework for bioprocess online optimization that utilizes rigorous modelling and control methods tailored for fed-batch and perfusion cultures. The basis of the methodology is a hybrid process modelling approach which enables both monitoring and optimization of cell culture processes. To account for inherent process variability of biological organisms, an adaptive state estimation approach is utilized which employs multiple models in parallel thus providing improved robustness to a possible occurrence of model-plant mismatch. Furthermore, optimal process trajectories for online optimization are calculated using a robust multistage nonlinear model predictive control approach which considers different scenarios based on the employed process models. Recent promising results from experimental fed-batch CHO fermentations are presented which show significant productivity increases.

Advanced-step multistage nonlinear model predictive control: Robustness and stability

Nonlinear model predictive control (NMPC) has been popular in many applications, especially when constraint satisfaction is critical. However, due to plant-model mismatch and disturbances, robust NMPC generally faces three challenges: robust performance, real-time implementation, and stability. In this paper, we propose a parallelizable advanced-step multistage NMPC (as-msNMPC), which provides a non-conservative robust control solution that explicitly addresses two types of uncertainty: model parameters and unmeasured noise. The first type is attended to by incorporating scenario trees and the second by applying nonlinear programming (NLP) sensitivity. In addition, robust stability concepts have been discussed for both ideal multistage NMPC (ideal-msNMPC) and as-msNMPC. Under suitable assumptions, as-msNMPC has demonstrated input-to-state practical stability properties with the presence of two types of uncertainty. Lastly, the as-msNMPC framework has been applied to continuous stirred tank reactor (CSTR) and quad-tank case studies for tracking setpoints to demonstrate its performance in robustness and efficiency.

Advanced-step multistage nonlinear model predictive control: Robustness and stability

Nonlinear model predictive control (NMPC) has been popular in many applications, especially when constraint satisfaction is critical. However, due to plant-model mismatch and disturbances, robust NMPC generally faces three challenges: robust performance, real-time implementation, and stability. In this paper, we propose a parallelizable advanced-step multistage NMPC (as-msNMPC), which provides a non-conservative robust control solution that explicitly addresses two types of uncertainty: model parameters and unmeasured noise. The first type is attended to by incorporating scenario trees and the second by applying nonlinear programming (NLP) sensitivity. In addition, robust stability concepts have been explored for both ideal multistage NMPC (ideal-msNMPC) and as-msNMPC. Under suitable assumptions, as-msNMPC has demonstrated input-to-state practical stability properties with the presence of two types of uncertainty. Lastly, the as-msNMPC framework has been applied to continuous stirred tank reactor (CSTR) and quad-tank case studies for tracking setpoints to demonstrate its performance in robustness and efficiency.

Robust NMPC using a model-error model with additive bounds to handle structural plant-model mismatch

Abstract We address the problem of robust nonlinear model predictive control (NMPC) under structural plant model mismatch in a multi-stage framework using a model-error model (MEM). Multi-stage NMPC models the presence of future feedback information in the predictions, hence it is less conservative than the other existing robust NMPC approaches. MEM consists of a stable linear model, an unknown nonlinear operator with bounded gain and a bounded additive mismatch. The computational burden of the proposed scheme is reduced by using two different approximations; 1. Constraint tightened multi-stage NMPC 2. Tube-enhanced multistage NMPC. Both the schemes are real-time implementable with a small loss in performance when compared to the standard multi-stage NMPC. The advantages of the proposed schemes are demonstrated for a benchmark continuous stirred tank reactor (CSTR) example.

Adaptive Scenario Generation for Multistage NMPC with Shrinking Horizons

Abstract We present a multistage scheme for shrinking horizon nonlinear model predictive control (NMPC). This approach generates scenario trees on-line in an adaptive manner, and assembles them from predictions of worst-case uncertainty realizations through first-order approximations of the process model. A key result of this approach is that both size and complexity of the resultant optimal control problems do not scale directly with the number of uncertain model parameters. Moreover, we apply and demonstrate this approach on a challenging industrially-relevant semi-batch polymerization process under parametric model uncertainty. The results show that adaptive scenario generation leads to improved performance, while maintaining the attained level of robustness for the considered process.

Robust Dual Multi-stage NMPC using Guaranteed Parameter Estimation

In this paper we present an implicit dual robust nonlinear model predictive control (NMPC) in the framework of (bounded-error) guaranteed parameter estimation and multi-stage NMPC, which uses a scenario tree to represent propagation of parametric model uncertainty through a dynamic system. The proposed implicit dual control scheme excites the system if the excitation signals improve the overall performance of the controller. A box (over-)approximation of the solution set of guaranteed parameter estimation which encloses the true parameter values is obtained by solving an optimization problem. The proposed approach uses approximations of the future solution sets and updates the scenario tree of the multi-stage NMPC along the prediction horizon accordingly. This gives rise to a bi-level optimization problem, which is solved as a single-level problem by implicitly solving the lower-level problem using its KKT conditions. The advantages of the proposed approach over the standard multi-stage NMPC are demonstrated for a linear and nonlinear (semi-batch reactor) simulation case studies.

A Synergistic Approach to Robust Output Feedback Control: Tube-based Multi-stage NMPC

Plant-model mismatch and estimation errors are critical issues in the practical implementation of Nonlinear Model Predictive Control (NMPC). To address these challenges, we formulate a robust output feedback NMPC scheme that is real-time implementable and provides robust constraint satisfaction in the presence of parametric and additive uncertainties, and estimation errors. The robustness is achieved by combining the tube-based and the multi-stage NMPC approaches. Two controllers are used in the proposed framework: a primary controller with tightened constraints that optimizes a given objective and an ancillary controller that tracks the trajectories provided by the primary controller. Unlike standard tube-based NMPC, the primary controller predicts different state trajectories for different realizations of the most important uncertainties using the multi-stage NMPC framework. The synergy between the two approaches leads to a better trade-off between optimality and complexity. The advantages of the proposed approach are demonstrated for an industrial-scale fed-batch polymerization reactor example.

A deep learning-based approach to robust nonlinear model predictive control

Dealing with uncertainties is one of the most challenging issues that prevent nonlinear model predictive control (NMPC) from being a widespread reality. Many different robust schemes have been presented recently, such as multi-stage NMPC, in which the uncertainty is represented as a scenario tree. While multi-stage NMPC achieves promising performance in practice, it suffers from an exponential increase in complexity with the number of uncertainties considered making its real-time application difficult for large case studies. We suggest in this work to use multi-stage NMPC as a generator of data pairs that are used to learn the robust NMPC policy by means of deep neural networks. This choice is motivated by recent practical successes of deep learning and theoretical results that explain the improved representation capabilities of deep networks with respect to shallow networks. We present empirical evidence which shows that the use of deep neural networks with many hidden layers as opposed to shallow networks with only one significantly improves the learning process of a robust NMPC control law. These findings are illustrated with simulation studies of an industrial polymerization reactor.