On the efficient computation of disturbance rejection measures

doi:10.1016/S0098-1354(02)00155-2

Computers & Chemical Engineering

Volume 27, Issue 1, 15 January 2003, Pages 95-99

https://doi.org/10.1016/S0098-1354(02)00155-2 Get rights and content

Abstract

A number of open loop indicators for the determination of the disturbance rejection capability of a system have been proposed in the literature in the last decade. These tools are useful in screening regulatory process control structures in an early design stage since they are based on minimum modelling requirements. This paper presents mixed-integer linear programming formulations for the efficient determination of the disturbance rejection measures proposed by Skogestad and Wolff (IFAC Workshop on Interactions between Process Design and Process Control (1992) Oxford: Pergamon). Furthermore, an extension of these disturbance rejection measures to nonlinear systems is presented. A close relation between the ideas of steady state flexibility test and index problems (AICHE J 24 (1978) 1021; Comput. Chem. Eng. 11 (1987) 319) and the disturbance rejection measures proposed by Skogestad and Wolff (IFAC Workshop on Interactions between Process Design and Process Control (1992) Oxford: Pergamon) is shown to exist.

Introduction

A number of open loop indicators (OLI) for the determination of the disturbance rejection capability of a system have been proposed in the literature in the last decade. These tools are useful in screening regulatory process control structures (RCSS) in an early design stage since they are based on minimum modelling requirements such as steady state gains. Skogestad & Wolff, 1992, Wolff, 1994 have proposed a set of interesting indices for evaluating the disturbance rejection of plants. However, the calculation of these measures for nonsquare plants is not trivial and only recently Hovd and Braatz (2000) have proposed a method for their calculation. Their approach is based on the solution of a non-convex optimisation problem that necessitates the availability of efficient global optimisation software to achieve solution to global optimality for arbitrarily large-scale problems.

This paper presents mixed-integer linear programming (MILP) formulations for the efficient calculation of the disturbance rejection measures proposed by Skogestad and Wolff (1992). MILP problems can be solved easily to global optimality using currently available algorithms and software even for very large-scale problems (Johnson, Nemhauser & Savelsbergh, 2000). The number of integer variables in the proposed formulation is (in the worst case) of the order of 10² and is at least 4 orders of magnitude less than what is considered currently to be a large scale MILP (Barnhart, Johnson, Nemhauser, Savelsbergh & Vance, 1998). As a result, the computational time, even for the Tennessee Eastman case study, is less than a second. This, however, is not the case for the solution of general non-convex NLPs whose solution to global optimality depends strongly on the structure of the problem while the size of the problems that can be handled is significantly smaller (Floudas, 2000).

Finally, a generalisation of the disturbance rejection measures proposed by Skogestad and Wolff (1992) is presented, that extends their applicability to arbitrary non-linear systems. A close relation between the idea of steady state flexibility (Grossmann & Sargent, 1978, Grossmann & Floudas, 1987) and the disturbance rejection measures proposed by Skogestad and Wolff (1992) is shown to exist.

Section snippets

Efficient computation of disturbance rejection measures

In this section we assume that the following steady-state model of a process is available $y = Pu + P^{d} d$ where y is the m-dimensional vector of controlled variables, u is the n-dimensional vector of manipulated variables and d is the q-dimensional vector of disturbances. In order to simplify the notation we assume that model (1) has been scaled so as all variables are less or equal to one. In order to answer the question on whether small offset of the controlled variables can be achieved for the worst

An equivalent nonlinear disturbance rejection measure

For the case where the system under study is nonlinear, then Eq. (1) is replaced by $y = f (u, d)$ and formulation (P1) becomes $max d ∈ Δ min u ∈U max i∈M f_{i} (u, d)$

This problem is closely related to the flexibility test problem. Furthermore, the nonlinear analogue of formulation (P3) becomes the flexibility index problem (Swaney & Grossmann, 1985). The efficient solution of these problems has been presented by Grossmann and Floudas (1987). Recent advances in the area of global optimisation (Floudas, Gümü &

FCC process

This case study considers the fluid catalytic cracking process model presented by Wolff (1994). The process involves three measurements (y₁, riser outlet temperature; y₂, regenerator cyclone temperature; y₃, regenerator dense bed temperature), three manipulated variables (u₁, air flowrate; u₂, catalyst circulating rate; u₃, feed composition) and three disturbances (d₁, feed temperature; d₂, air temperature; d₃, feed flowrate). The scaled steady state matrices are $P = 10.16 5.59 1.43 15.52 −8.36 −0.71$

Conclusions

This paper presents mixed-integer linear programming formulations for the efficient calculation of the disturbance rejection measures proposed by Skogestad and Wolff (1992). These formulations can be solved to global optimality using currently available algorithms and software. A generalisation of these disturbance rejection measures is presented extending their applicability to cover the case of non-linear systems. A close relation between the idea of steady state flexibility and the

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A number of indicators for the determination of the disturbance rejection capability of a system have been proposed in the literature in the last two decades. These tools are useful for evaluating early stage process designs and screening regulatory control structures, since they are based on minimal modelling requirements. In previous publications (Hovd and Braatz, 2000; Hovd et al, 2003; Kookos and Perkins, 2003), it has been shown how to evaluate at steady state the disturbance rejection measures proposed by Skogestad and Wolff (1992). If acceptable values for these disturbance rejection measures are not achieved, no controller can achieve satisfactory control. In this paper, it is shown how to obtain upper and lower bounds for these disturbance rejection measures also as a function of frequency. The upper and lower bounds can be made arbitrarily accurate at the expense of increasing the size of the optimization problems involved.
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¹: Present address: UMIST, Department of Chemical Engineering, P.O. Box 88, Manchester, M60 1QD, UK.

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On the efficient computation of disturbance rejection measures

Abstract

Introduction

Section snippets

Efficient computation of disturbance rejection measures

An equivalent nonlinear disturbance rejection measure

FCC process

Conclusions

Computers and Chemical Engineering

Systematic Methods of Chemical Process Design

Branch-and-price: column generation for solving huge integer programs

Operations Research

Deterministic Global Optimization: Theory, Methods, and Applications

Global optimization in design under uncertainty: feasibility test and flexibility index problems

Industrial and Engineering Chemistry Research

A representation and economic interpretation of a two-level programming problem

Journal of Operational Research Society