This article is presented by:
Dr M.J. Willis
Department of Chemical and Process Engineering
University of Newcastle upon Tyne
Multivariable Control: An introduction
Aims and Objectives
To introduce the basic concepts of multivariable control (a continuous stirred tank
reactor will be used as the motivating example). To highlight the phenomenon of
loop interactions. To learn how to model multivariable systems using input-output
descriptions. To introduce the relative gain array (RGA) - a tool for the selection of
At the end of this section of the course you should be able to select appropriate
manipulated variable - controlled variable pairings to minimise the effect of loop
interactions in multivariable systems. You should know how to formulate and
interpret the RGA.
• Define the terms SISO and MIMO.
• Introduce MIMO control and loop interaction using a CSTR as the motivating
• discuss systems modelling for MIMO systems (the transfer function approach).
• clarify discussions using a worked example (modelling and control of a mixing
• introduce the RGA and discuss how it may be used to select input-output
Processes with only one output being controlled by a single manipulated variable are
classified as single-input single output (SISO) systems. It should be noted however,
that most unit operations in chemical engineering have more than one control loop.
In fact, each unit typically requires the control of at least two variables, e.g. product
rate and product quality. There are therefore usually at least two control loops.
Systems with more than one control loop are known as multi-input multi-output
(MIMO) or multivariable systems.
A Continuous Stirred Tank Reactor (CSTR)
A continuous stirred tank reactor (CSTR) is used to convert a reactant (A) to a
product (B). The reaction is liquid phase, first order and exothermic. Perfect mixing is
assumed. A cooling jacket surrounds the reactor to remove the heat of reaction.
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