INTROMISSION

A Multivariable process admits inputs & outputs. In general,

the no. of inputs should be larger than or equal to the no. of outputs so that the process is controllable.

Thus, we will assume ≥

MULTIVARIABLE

Multivariable analysis is a tool for determining the relative

contributions of different causes to a single event.

As an example, in clinical research we need multivariable analysis,

like infectious diseases that are known to be caused by a single

pathogen, a no. of factors whether an exposed individual

becomes ill, including the characteristics of the pathogen, the root

of exposure, the intensity & the host response.

The system is supposed to have been identified in continuous time by

transfer functions.

In general, this identification is performed by

sequentially imposing signals such as steps on

each input Ui( i=1,….,Nu) & recording

the corresponding vector of the responses

Yij(j=1,…….,Ny).

From each input-output couple

(Ui,Yi), a transfer function is deduced by a least-squares

procedure.

In open loop, the Nyo/p’s Yi are linked to the Nu inputs Uj &

to the Nddisturbances Dk by the following set of Ny linear equations

• +…….+ + +……..+• = +…….+ + +……………+• Which will be written in open loop under condensed matrix

form as• Y = U+ D• Where y is the O/P vector, u is the input vector & d is the

distribution vector( the modeled disturbances), is the rectangular matrix , the elements of which are the input-output transfer functions, & is the rectangular matrix , the elements of matrix represent the principal effects b/w the I/P’s & the O/P’s, while the non-diagonal elements represent the couplings.

+,

INTERACTION

An Interaction occurs when the impact of a risk factor on outcome is changed by the value of a third variable.

RELATIVE GAIN ARRAY

The interaction b/w loops can be evaluated by a method based on the

study of the relative gain array introduced by Bristol.

The loops influence themselves in a more or less important manner & a possible effect is that some loops

destabilize the closed-loop system.

The RGA Model provides a simple way to decide how a set of input signals should be utilized to

control a given set of output signals.

The steady-state RGA matrices for the linearizedmodel in the three operating points are

Λ(Gu¯3(0)) = 0.0055 0.9945

• 0.9945 0.0055

Λ(Gu¯2(0)) = 0.0051 0.9949

• 0.9949 0.0051

Λ(Gu¯1(0)) = 0.0041 0.9959

• 0.9959 0.0041.

The RGA method is relatively easy to

implement, & for this reason is frequently used in chemical engineering.

It is limited in its original form, as it uses only

steady-state information.

It can be extended by using frequency

representations(Hovd & Skogestad)

DECOUPLERS

A Multivariable system presents the

particularly that the inputs are coupled to

the outputs.

Different methods exist, allowing us

to ensure at least a partial decoupling for a multivariable

system.

This is particularly important in the treatment by a

transfer function matrix.

DECOUPLING FOR A 2×2 SYSTEM

APPLICATION TO WOOD & BERRY DISTILLATION COLUMN

REFERENCESContents form-

Jean-Pierre Corriou,

Book, of Process

Control.

http://catdir.loc.gov/catdir/samples/cam032/98039350.pdf

http://user.it.uu.se/~bc/WWT/BHalvarsson_Avh.pdf