Economic performance of variable structure control: a case study

doi:10.1016/S0098-1354(00)00501-9

Computers & Chemical Engineering

Volume 24, Issue 8, 1 September 2000, Pages 1821-1827

https://doi.org/10.1016/S0098-1354(00)00501-9 Get rights and content

Abstract

The operating point of a chemical process is usually computed by optimizing a steady-state objective function, e.g. the profit, subject to the steady-state characteristics of the plant. However, the resulting point typically lies in the boundary of the operating region. The presence of disturbances can easily cause constraint violations in the transient. Thus, it is necessary to move the operating point away from the active constraints into the feasible region. The magnitude of this ‘back-off’ has a direct influence on the economic side. The purpose of this paper is to study the effect of the combination of a state-observer and controller designed using structure variable techniques at the economical level of the process control.

Section snippets

Problem formulation

Consider the following system $x ̇ =f(x)+g^{r} (x)r+g^{u} (x)u+g^{w} (x)w$ $y=h(x)+l^{r} (x)r+l^{u} (x)u+l^{w} (x)w$ where $x∈ R^{n_{x}}$ is the system state, $y∈ R^{n_{y}}$ is the system output, $r∈ R^{l}$ is the optimization variable that will be considered constant all the time, $u∈ R^{k}$ is the control input and $w∈ R^{m}$ is the disturbance input belonging to the set $W= w: w ̂,∀t≥0 0,∀t<0 with w ̂_{∞} ≤1$ .

In order to complete the description of our system, consider now a set of inequalities, that should be satisfied at any time, $z_{c} (x,r,u,w)=p(x)+q^{r} (x)r+q^{u} (x)u+q^{w} (x)w≤0$

Variable structure control

The variable structure control (VSC) and their related sliding modes have been widely studied in the last 30 years (see e.g. Emelyarov, 1985, De Carlo, Zak & Mattheews, 1988, Stolite & Sastry, 1988).

Let us assume that, to apply the VSC control, the state vector x is fully measurable and the system has a strong relative vectorial degree in an open set $D∈ R^{n_{x}}$ . Let us define an auxiliary function $s: R^{n_{x}} →S⊂ R^{n}$ with entries s_i(x) for i=1, …, n. This function s(x), called commutation function, divides

Variable structure observer

In the previous section a VSC structure had been proposed under the assumption that the state vector x was fully measurable. As a result of this assumption, the control action Δ_u in Eq. (9) depends on the states and on the disturbances. However, some of these variables are typically unmeasured. In order to solve this problem, we propose the use of a second order sliding mode technique in a nonlinear estimation framework (Chiacchiarini, Desages, Romagnoli & Palazoglu, 1995, Colantonio, Desages,

Example

The case study considered in this section consists of two continuous stirred tank reactors (CSTR) in series, with an intermediate mixer introducing a second feed (de Hennin & Perkins, 1993), as shown in Fig. 1. A single irreversible, exothermic, first order reaction A→B takes place in both reactors. The dynamic model of these reactions is $V^{1} d (C^{1}) d t =−k_{o} e^{−E/RT^{1}} C^{1} V^{1} +Q_{F}^{1} (C_{F}^{1} −C^{1})$ $V^{1} d (T^{1}) d t =D_{h} k_{o} e^{−E/RT^{1}} C^{1} V^{1} +Q_{F}^{1} (T_{F}^{1} −T^{1})+ Cool^{1}$ $V^{2} d (C^{2}) d t =−k_{o} e^{−E/RT^{2}} C^{2} V^{2} +Q_{F}^{2} (C_{F}^{2} −C^{2})$ $V^{2} d (T^{2}) d t =D_{h} k_{o} e^{−E/RT^{2}} C^{2} V^{2} +Q_{F}^{2} (T_{F}^{2} −T^{2})+ Cool^{2}$

Conclusions

In this paper, the capabilities of the VSC schemes in the economical context of the process control have been studied. The controller structure is used to bring the actual operating point as close as it is possible to the optimum (in terms of dollars) in the presence of disturbances. The use of estimators to compute the non measurable disturbances does not produce any additional loss in the economic performance of the process and their implementation only implies software modifications.

Nomenclature

x	system state vector
r	optimization variables vector
u	control input vector
w	disturbance input vector
y	system output vector
W	set of possible disturbances
z_c	vector of constraints
s	vector of commutation functions
z_o	objective function for the optimization problem
x̂	observer state vector
y_m	observer output vector
v	observer disturbance vector
n_x	dimension of the state vector
l	number of optimization variables
k	number of control inputs
m	number of disturbance inputs
n_y	number of system outputs
n_c	number of constraints
l’

References (11)

J.L. Figueroa et al.
Economic impact of disturbances and uncertain parameters in chemical processes — a dynamical back-offs analysis
Computers & Chemical Engineering
(1996)
A. Maarleveld et al.
Constraint control on distillation columns
Automatica
(1970)
J.A. Bandoni et al.
On optimising control and the effect of disturbances: calculation of the open-loop back-offs. Escape-3
Supplement to Computers & Chemical Engineering
(1994)
H.G. Chiacchiarini et al.
Variable structure control with a second order sliding condition: theory and application to a steam generation unit
Automatica
(1995)
M.C. Colantonio et al.
Nonlinear control of a CSTR: disturbance rejection using sliding model control
Industrial Engineering & Chemical Research
(1995)

There are more references available in the full text version of this article.

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View full text

Economic performance of variable structure control: a case study

Abstract

Section snippets

Problem formulation

Variable structure control

Variable structure observer

Example

Conclusions

Nomenclature

Computers & Chemical Engineering

Automatica

On optimising control and the effect of disturbances: calculation of the open-loop back-offs. Escape-3

Supplement to Computers & Chemical Engineering

Variable structure control with a second order sliding condition: theory and application to a steam generation unit

Automatica

Nonlinear control of a CSTR: disturbance rejection using sliding model control

Industrial Engineering & Chemical Research