Radoslav Paulen

Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies

This paper is concerned with guaranteed parameter estimation of nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding—or approximating as closely as possible—the set of all possible parameter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a tradeoff by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time.

Robust Nonlinear Model Predictive Control with Reduction of Uncertainty Via Robust Optimal Experiment Design

Abstract This paper studies the reduction of the conservativeness of robust nonlinear model predictive control (NMPC) via the reduction of the uncertainty range using guaranteed parameter estimation. Optimal dynamic experiment design is formulated in the framework of robust NMPC in order to obtain probing inputs that maximize the information content of the feedback and simultaneously to guarantee the satisfaction of the process constraints. We propose a criterion for optimal experiment design which provides a minimization of parameter uncertainty in the direction of improved performance of the process under robust economic NMPC. A case study from the chemical engineering domain is studied to show the benefits of the proposed approach.

Multi-stage Nonlinear Model Predictive Control with verified robust constraint satisfaction

This paper presents an approach to verify robust constraint satisfaction using dynamic state bounding techniques in the framework of multi-stage Nonlinear Model Predictive Control (NMPC). In multi-stage NMPC, the uncertainty is described by a tree of discrete scenarios, and the future control inputs depend on the previous realizations of the uncertainty, constituting a closed-loop approach which has been shown to provide significantly better performance than an open-loop approach. While the approach has demonstrated very promising results in practice, one of the problems of multi-stage NMPC is the fact that no guarantees can be given for the uncertainty values that are not explicitly considered in the scenario tree. In this work, we address this problem by updating the resulting optimization problem in an iterative fashion such that the constraints satisfaction is guaranteed based on the rigorous bounding of the state variables over the set of possible uncertainty realizations. We illustrate that the approach can deal in real time with challenging problems by presenting simulation results of an industrial batch polymerization reactor.

Optimal resource allocation in industrial complexes by distributed optimization and dynamic pricing

Abstract We address the distributed hierarchical optimization of industrial production complexes where the individual plants exchange resources via networks. Due to the site-wide couplings a centralized or a distributed hierarchical optimization is needed to achieve the best overall performance of the site and to balance the networks of the shared resources. We discuss market-like algorithms that set prices of the shared resources in order to influence the individual optimizers so that the overall operation converges to the site-wide optimum. A novel algorithm for price adjustment based on the quadratic approximation of the responses of the individual optimizers is presented. It shows convergence to the site-wide optimum with significantly less iterations in comparison to the standard subgradient-based method for a set of case studies, including a petrochemical complex.

Model-based design of optimal experiments for nonlinear systems in the context of guaranteed parameter estimation

Abstract An approach to the design of experiments is presented in the framework of bounded-error (guaranteed) parameter estimation for nonlinear static and dynamic systems. The guaranteed parameter estimation determines non-asymptotic confidence limits on the unknown parameters of a mathematical model. An essential part of the solution procedure is the approximation of the joint-confidence region. In this contribution, we develop and analyze the procedure and different ways of achieving a tight over-approximation of the solution set of guaranteed parameter estimation based on the expected values of parameters. Finally we propose to solve the problem of the design of experiments as a bilevel program. We demonstrate our approach and analyze the nature of the problem in the static and dynamic case studies. The proposed approach is also compared to the experiment design in the context of least-squares estimation and to the linearization-based techniques for optimal experiment design proposed in the literature earlier.

Towards dual robust nonlinear model predictive control: A multi-stage approach

This paper presents an approach to dual robust nonlinear model predictive control (NMPC). Dual control is traditionally formulated as a technique that seeks to solve the trade-off between probing actions, which result in a better estimation of the unknown parameters, and the optimal operation of the uncertain dynamic system. We propose a dual robust NMPC method based on the multi-stage approach that represents the uncertainty as a scenario tree of its possible realizations. We achieve a dual control formulation by taking explicitly into account the reduction of the uncertainty that future probing actions would provide over the prediction horizon. Using such formulation, our approach does not require any a priori decision on the relative importance of the probing actions with respect to the optimal operation of the system, as proposed in some recent approaches. The results obtained for a chemical engineering example show the advantages of using a dual NMPC approach based on multi-stage NMPC over the sequential use of parameter estimation with a posteriori approximation of the covariance of the parameter estimates.

Improving scenario decomposition algorithms for robust nonlinear model predictive control

Abstract This paper deals with the efficient computation of solutions of robust nonlinear model predictive control problems that are formulated using multi-stage stochastic programming via the generation of a scenario tree. Such a formulation makes it possible to consider explicitly the concept of recourse, which is inherent to any receding horizon approach, but it results in large-scale optimization problems. One possibility to solve these problems in an efficient manner is to decompose the large-scale optimization problem into several subproblems that are iteratively modified and repeatedly solved until a solution to the original problem is achieved. In this paper we review the most common methods used for such decomposition and apply them to solve robust nonlinear model predictive control problems in a distributed fashion. We also propose a novel method to reduce the number of iterations of the coordination algorithm needed for the decomposition methods to converge. The performance of the different approaches is evaluated in extensive simulation studies of two nonlinear case studies.

Time-optimal control of diafiltration processes in the presence of membrane fouling

Abstract This paper deals with time-optimal operation of batch diafiltration processes in the presence of membrane fouling. Fouling causes a decrease in the effective membrane area and, hence, an increase in processing time. In this work we study a time-optimal operation with several fouling models available in literature. Pontryagins minimum principle is applied to characterize the structure of the optimal operation which voids any further on-line optimization. Due to the specific structure of the problem, it is possible, in several problem setups, to derive and verify an explicit analytic solution. Obtained results are applied in case studies where we provide a comparison between traditional operation and the proposed time-optimal operation. In cases, where the optimal operation cannot be identified analytically, we analyze the performance of sub-optimal control derived from neighboring analytical solution and compare it to optimal operation found via numerical optimization techniques.

Time-optimal operation of multi-component batch diafiltration

Abstract We study the minimum-time operation of batch multi-component diafiltration processes. We employ the technique of Pontryagins minimum principle to derive the candidates for an optimal operation. The optimal operation is defined as a state-feedback strategy. Simulations and numerical optimizations are applied to confirm the optimality of the proposed time-optimal operation. Obtained results are evaluated on two case studies, a typical multi-component diafiltration processes. Standard operational approaches are compared to the optimal operation and the resulting improvements show attractivity of the proposed approach.

Time-optimal Operation of Diafiltration Processes in the Presence of Fouling

Abstract This paper deals with time-optimal operation of a batch diafiltration process in the presence of membrane fouling. Fouling causes a decrease in membrane area and hence, a decrease in permeate flux. Pontryagin’s minimum principle is applied to characterize the structure of the optimal operation. Due to the specific structure of the problem, it is possible to derive and verify an explicit analytic solution. Obtained results are used in a case study where we provide a comparison between traditional operation and the herein developed time-optimal operation.