Structural Plant-model Mismatch Research Articles

Data-Driven Adaptive Optimal Control Under Model Uncertainty: An Application to Cold Atmospheric Plasmas

Cold atmospheric plasmas (CAPs) are increasingly used for applications requiring the processing of heat- and pressure-sensitive (bio)materials. A key challenge in model-based control of CAPs arises from the high-computational requirements of theoretical plasma models as well as lack of mechanistic understanding of plasma-surface interactions. Thus, control strategies that rely on simple, physics-based models that can be adapted to mitigate plant-model mismatch will be particularly advantageous for CAP applications. This article presents an optimal control approach for controlling the nonlinear and cumulative effects of CAPs delivered to a target surface using a simple system model. Through parsimonious input parameterization, the solution to the optimal control problem (OCP) is given by an arc sequence that does not include any singular arcs. A data-driven adaptive algorithm based on modifier adaptation is proposed to deal with the structural plant-model mismatch by estimating the mismatch in the cost and constraints of the OCP. The adaptive approach is shown to converge to a Karush–Kuhn–Tucker (KKT) point of the OCP for the true system. Moreover, a control strategy based on feedback linearization and derivative estimation is proposed for online tracking of path constraints in the presence of disturbances and model uncertainty. The proposed approach is demonstrated by simulations and real-time control experiments on a kilohertz-excited atmospheric pressure plasma jet in helium, in which the plasma treatment time is minimized while delivering a desired amount of nonlinear thermal effects to the target surface.

Online Feedback Optimization of Compressor Stations with Model Adaptation using Gaussian Process Regression

Online Feedback Optimization is a method used to steer the operation of a process plant to its optimal operating point without explicitly solving a nonlinear constrained optimization problem. This is achieved by leveraging a linear plant model and feedback from measurements. However the presence of plant-model mismatch leads to suboptimal results when using this approach. Learning the plant-model mismatch enables Online Feedback Optimization to overcome this shortcoming. In this work we present a novel application of Online Feedback Optimization with online model adaptation using Gaussian process regression. We demonstrate our approach with a realistic load sharing problem in a compressor station with parametric and structural plant-model mismatch. We assume imperfect knowledge of the compressor maps and design an Online Feedback Optimization controller that minimizes the compressor station power consumption. In the evaluated scenario, imperfect knowledge of the plant leads to a 5% increase in power consumption compared to the case with perfect knowledge. We demonstrate that Online Feedback Optimization with model adaptation reduces this increase to only 0.8%, closely approximating the case of perfect knowledge of the plant, regardless of the type of mismatch.

Generalized Parameter Estimation Method for Model-Based Real‑Time Optimization

• Generalized parameter estimation is proposed to handle structural mismatches. • Synergy between model parameters and gradient modifiers is visualized. • Convergence of the proposed algorithm is discussed. • Simulations show the proposed method is relatively noise-insensitive. Real-time optimization is the technology to find the optimal setpoint of a plant under uncertainty. Iterative two-step approach with alternating parameter estimation and optimization is popular and intuitive in practice. However, this approach does not guarantee convergence if structural plant-model mismatches exist. Using the modifier adaptation principle, we improve the iterative two-step approach by the proposed generalized parameter estimation method. The flexibility and estimation capability of the generalized parameters enable optimality in the presence of structural plant-model mismatches. The synergy between model parameters and modifiers is visualized and demonstrated by three examples. The convergence of the proposed algorithm and its relation with existing work are analyzed theoretically. Simulation shows that the proposed method converges to a plant optimum in the presence of structural plant-model mismatch in a relatively noise-insensitive way, and the model parameters are properly adapted during the iteration.

An efficient model-error model update strategy for multi-stage NMPC with model-error model

Multi-stage NMPC with model-error model (MS-MEM) handles structural plant-model mismatch present in the nominal model of the plant in a non-conservative fashion. A model-error model (MEM) that consists of a stable linear time-invariant dynamics and a static time-variant nonlinear mapping is built using the past data such that it captures the unmodeled dynamics of the plant. The scenario tree is built for the nominal and for the extreme realizations of the plant obtained using the nominal model and the model-error model, and a multi-stage decision problem is formulated. In this paper, we propose an efficient strategy to update the model-error model present in the MS-MEM approach if new measurements invalidate the model-error model. The advantages of the proposed scheme over the previous approach where only the gain of the linear model is updated are demonstrated for a continuous stirred tank reactor (CSTR) benchmark example.

110th Anniversary: A Feature-Based Analysis of Static Real-Time Optimization Schemes

This paper addresses the steady-state optimization of continuous processes in the presence of uncertainty in the form of unknown or time-varying model parameters, structural plant-model mismatch, a...

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.

Using a neural network for estimating plant gradients in real-time optimization with modifier adaptation

In the presence of structural plant-model mismatch, standard real-time optimization (RTO) schemes are prone to compute an operation point that does not coincide with the plant optimum. Modifier Adaptation (MA) methods are RTO variants that have the ability to reach plant optimality even in the case of structural plant-model mismatch. However, MA implementations require plant gradient information, which is challenging to obtain. This work proposes a method for estimating plant gradients based on neural networks (radial basis function network - RBFN). Our method is applied for obtaining the gradients of a gas lifted oil well network, which is then optimized using MA. The results show that, even with measurement noise, the gradients are estimated within an adequate precision and the MA method is able to increase production of the well network, reaching the plant optimum without any constraint violations despite the presence of plant-model mismatch.

Analysis of output modifier adaptation for real-time optimization

In the context of real-time optimization, modifier-adaptation schemes update the model-based optimization problem by adding first-order correction terms to the cost and constraint functions of the optimization problem. This guarantees meeting the plant first-order optimality conditions upon convergence, despite the presence of parametric and structural plant-model mismatch. An alternative modifier-adaptation strategy has been proposed, wherein the first-order corrections are applied to the output functions rather than to the cost and constraint functions. This paper analyzes this alternative adaptation scheme in detail and provides arguments for its use in cases where the cost and constraints are nonlinear functions of the inputs and outputs. The approach is illustrated in simulation on the Williams–Otto continuous stirred-tank reactor.

Handling Structural Plant-model Mismatch using a Model-error Model in the Multi-stage NMPC framework

Nonlinear model predictive control (NMPC) is one of the most promising advanced control approaches for multi-dimensional systems with state and input constraints. Among the several NMPC schemes that were proposed to deal with the uncertainties present in the model, the multi-stage NMPC approach achieves a low degree of conservatism because it incorporates the existence of recourse in the predictions. The future evolution of the state trajectories for different realizations of the uncertainty is modeled as a scenario tree. In this paper, we extend the multi-stage NMPC approach to deal with structural plant-model mismatch in a non-conservative fashion. An uncertainty region around the nominal model is built using a model-error model that consists of stable linear time-invariant dynamics and a static time-variant nonlinear mapping. The scenario tree is built for the nominal and for the extreme realizations of the plant obtained using the nominal model and the model-error model, and a multi-stage decision problem is formulated. We show the advantages of using a model-error model over a previously proposed approach to handle the structural plant model mismatch using multi-stage NMPC for a continuous stirred tank reactor (CSTR) example.

Real-Time Optimization of Gold Cyanidation Leaching Process in a Two-Layer Control Architecture Integrating Self-Optimizing Control and Modifier Adaptation

The gold cyanidation leaching process (GCLP) has been prevalent in the hydrometallurgical industry. To better control and optimize the plant operation, mechanistic models have been established which capture the physical behaviors of the GCLP. However, due to various disturbances and uncertainties, an optimized operation based on the nominal model is practically suboptimal. To perform real-time optimization (RTO) of the GCLP, this paper proposes a two-layer control architecture integrating the self-optimizing control (SOC) and modifier adaptation (MA), both of which are useful RTO approaches with distinct advantages. In the lower layer of the proposed control system, the SOC is implemented with measurement combinations as the controlled variables that are tracked at optimally insensitive setpoints to account for parametric disturbances. In the upper layer, the setpoints of self-optimizing controlled variables are further optimized in a tailored MA framework, such that the structural plant–model mismatch is...

Modifier Adaptation for Real-Time Optimization—Methods and Applications

This paper presents an overview of the recent developments of modifier-adaptation schemes for real-time optimization of uncertain processes. These schemes have the ability to reach plant optimality upon convergence despite the presence of structural plant-model mismatch. Modifier Adaptation has its origins in the technique of Integrated System Optimization and Parameter Estimation, but differs in the definition of the modifiers and in the fact that no parameter estimation is required. This paper reviews the fundamentals of Modifier Adaptation and provides an overview of several variants and extensions. Furthermore, the paper discusses different methods for estimating the required gradients (or modifiers) from noisy measurements. We also give an overview of the application studies available in the literature. Finally, the paper briefly discusses open issues so as to promote future research in this area.

Modifier-Adaptation approach to deal with structural and parametric uncertainty

Real-Time Optimization (RTO) is not always able to achieve optimal process operation due to the presence of significant uncertainty about the plant models that are used to make decisions and also due to the differences between control architecture layers which operate on different time-scales and use different kind of models. To overcome these issues the economic optimization problem is modified following the Modifier Adaptation methodology to bring the process to the real optimum despite the presence of uncertainty by using plant measurements.To deal with parametric and structural plant-model mismatch, a new approach is presented that combines the estimation of process gradients from transient and steady-state information. It speeds up the convergence of Modifier Adaptation methodology to the process optimum. The approach is illustrated through the simulated example of a depropanizer distillation column.

Control of particle size distribution in emulsion polymerization using mid-course correction under structural plant-model mismatch

This paper suggests a method to control the particle size distribution in semi-batch emulsion homopolymerizations under structural plant-model mismatch. In this approach, firstly a nominal monomer feed input trajectory is applied to the plant up to a predefined time instant after the start of the batch (mid-course). By means of a calorimetric observer, all states of the system except the particle size distribution are estimated using the available measurements. The estimated states and the measured PSD at the mid-course of the process are used as the initial condition for an optimization which is done to compute the trajectory of the monomer feed from the mid-course up to the end of the batch. In this optimization, considering a structural plant-model mismatch, a hybrid model which comprises the nominal model of emulsion polymerization and an empirical component that corrects the predictions of this nominal model is used.

Use of Convex Model Approximations for Real-Time Optimization via Modifier Adaptation

Real-time optimization (RTO) via modifier adaptation is a class of methods for which measurements are used to iteratively adapt the model via input-affine additive terms. The modifier terms correspond to the deviations between the measured and predicted constraints on the one hand, and the measured and predicted cost and constraint gradients on the other. If the iterative scheme converges, these modifier terms guarantee that the converged point satisfies the Karush–Kuhn–Tucker (KKT) conditions for the plant. Furthermore, if upon convergence the plant model predicts the correct curvature of the cost function, convergence to a (local) plant optimum is guaranteed. The main advantage of modifier adaptation lies in the fact that these properties do not rely on specific assumptions regarding the nature of the uncertainty. In other words, in addition to rejecting the effect of parametric uncertainty like most RTO methods, modifier adaptation can also handle process disturbances and structural plant–model mismatch. This paper shows that the use of a convex model approximation in the modifier-adaptation framework implicitly enforces model adequacy. The approach is illustrated through both a simple numerical example and a simulated continuous stirred-tank reactor.

Nonlinear MPC of a multi-zone multi-feed LDPE autoclave reactor

This paper discusses the design of a model-based multivariable nonlinear model predictive control (NMPC) formulation for the control of a multi-zone multi-feed LDPE autoclave reactor. Unscented Kalman filtering (UKF) state estimation was employed for online estimation of the plant states and disturbances. In addition to structural plant-model mismatch and noisy process measurements, the closed-loop performance of the UKF based NMPC scheme was investigated under parametric uncertainty in the kinetic rate constants. The closed-loop NMPC performance was also assessed for polymer grade changes, and for disturbance rejection at both open-loop stable and unstable operating regions.

Unscented Kalman filter based nonlinear model predictive control of a LDPE autoclave reactor

This paper presents a multivariable nonlinear model predictive control (NMPC) scheme for the regulation of a low-density polyethylene (LDPE) autoclave reactor. A detailed mechanistic process model developed previously was used to describe the dynamics of the LDPE reactor and the properties of the polymer product. Closed-loop simulations are used to demonstrate the disturbance rejection and tracking performance of the NMPC algorithm for control of reactor temperature and weight-averaged molecular weight (WAMW). In addition, the effect of parametric uncertainty in the kinetic rate constants of the LDPE reactor model on closed-loop performance is discussed. The unscented Kalman filtering (UKF) algorithm is employed to estimate plant states and disturbances. All control simulations were performed under conditions of noisy process measurements and structural plant–model mismatch. Where appropriate, the performance of the NMPC algorithm is contrasted with that of linear model predictive control (LMPC). It is shown that for this application the closed-loop performance of the UKF based NMPC scheme is very good and is superior to that of the linear predictive controller.

Practical issues in the application of the particle filter for estimation of chemical processes

AbstractIn this work, we explore practical aspects of the application of particle filters to chemical processes, with special emphasis on polymerization processes. Particle filters are potentially better suited for estimation of these nonlinear processes than Kalman filter-based estimators, since they do not make the assumption that the process and measurement noise are Gaussian. We demonstrate the improved performance of particle filters over the extended Kalman filter and the unscented Kalman filter when the innovation sequences are non-Gaussian. Another potential advantage of the particle filter is that information on full distribution of the state is obtained, and not just the expectation of the state estimate. We provide results based on k-means clustering that indicate the best method of extracting a point estimate from the full state distribution. Finally, we show that in cases where there is high plant-model mismatch, e.g., when a reduced order model is used for estimation, the basic sequential importance resampling particle filter (SIR-PF) does not provide accurate estimates, while an unscented particle filter provides excellent estimates even under high structural plant-model mismatch.

Modifier-Adaptation Methodology for Real-Time Optimization

The process industries are characterized by a large number of continuously operating plants, for which optimal operation is of economic importance. However, optimal operation is particularly difficult to achieve when the process model used in the optimization is inaccurate or in the presence of process disturbances. In highly automated plants, optimal operation is typically addressed by a decision hierarchy involving several levels that include plant scheduling, real-time optimization (RTO), and process control. At the RTO level, medium-term decisions are made by considering economic objectives explicitly. This step typically relies on an optimizer that determines the optimal steady-state operating point under slowly changing conditions such as catalyst decay or changes in raw material quality. This optimal operating point is characterized by setpoints that are passed to lower-level controllers. Model-based RTO typically involves nonlinear first-principles models that describe the steady-state behavior of the plant. Since accurate models are rarely available in industrial applications, RTO typically proceeds using an iterative two-step approach, namely a parameter estimation step followed by an optimization step. The idea is to repeatedly estimate selected uncertain model parameters and use the updated model to generate new inputs via optimization. This way, the model is expected to yield a better description of the plant at its current operating point. The classical two-step approach works well provided that (i) there is little structural plant-model mismatch, and (ii) the changing operating conditions provide sufficient excitation for estimating the uncertain model parameters. Unfortunately, such conditions are rarely met in practice and, in the presence of plant-model mismatch, the algorithm might not converge to the plant optimum, or worse, to a feasible operating point. As far as feasibility is concerned, the updated model should be able to match the plant constraints. Alternatively, feasibility can be enforced without requiring the solution of a parameter estimation problem by adding plant-model bias terms to the model outputs. These biases are obtained by subtracting the model outputs from the measured plant outputs. A bias-update scheme, where the bias terms are used to modify the constraints in the steady-state optimization problem, has been used in industry. However, the analysis of this scheme has received little attention in the research community. In the context of this thesis, such an RTO scheme is referred to as constraint adaptation. The constraint-adaptation scheme is studied, and its local convergence properties are analyzed. Constraint adaptation guarantees reaching a feasible operating point upon convergence. However, the constraints might be violated during the iterations of the algorithm, even when starting the adaptation from within the feasible region. Constraint violations can be avoided by controlling the constraints in the optimization problem, which is done at the process control level by means of model predictive control (MPC). The approach for integrating constraint adaptation with MPC described in this thesis places high emphasis on how constraints are handled. An alternative constraint-adaptation scheme is proposed, which permits one to move the constraint setpoints gradually in the constraint controller. The constraint-adaptation scheme, with and without the constraint controller, is illustrated in simulation through the real-time optimization of a fuel-cell system. It is desirable for a RTO scheme to achieve both feasibility and optimality. Optimality can be achieved if the underlying process model is able to predict not only the constraint values of the plant, but also the gradients of the cost and constraint functions. In the presence of structural plant-model mismatch, this typically requires the use of experimental plant gradient information. Methods integrating parameter estimation with a modified optimization problem that uses plant gradient information have been studied in the literature. The approach studied in this thesis, denoted modifier adaptation, does not require parameter estimation. In addition to the modifiers used in constraint adaptation, gradient-modifier terms based on the difference between the estimated and predicted gradient values are added to the cost and constraint functions in the optimization problem. With this, a point that satisfies the first-order necessary conditions of optimality for the plant is obtained upon convergence. The modifier-adaptation scheme is analyzed in terms of model adequacy and local convergence conditions. Different filtering strategies are discussed. The constraint-adaptation and modifier-adaptation RTO approaches are illustrated experimentally on a three-tank system. Finite-difference techniques can be used to estimate experimental gradients. The dual modifier-adaptation approach studied in this thesis drives the process towards optimality, while paying attention to the accuracy of the estimated gradients. The gradients are estimated from the successive operating points generated by the optimization algorithm. A novel upper bound on the gradient estimation error is developed, which is used as a constraint for locating the next operating point.

COMPARISON OF ITERATIVE SET-POINT OPTIMISATION STRATEGIES UNDER STRUCTURAL PLANT-MODEL MISMATCH

This paper considers the set-point optimisation problem under plant-model mismatch. Three iterative optimisation strategies, the two-step method, the integrated system optimisation and parameter estimation (ISOPE) method and the gradient-modification optimisation method are compared for a nonlinear chromatographic separation process using different structurally mismatched models as nominal process models. The gradient-modification optimisation method is proven to be very attractive when a large structural plant-model mismatch exists. Its ability to deal with disturbances is also shown.