Multivariable Control Systems

Multivariable Control Multivariable Control SystemsSystems

Ali KarimpourAssistant Professor

Ferdowsi University of Mashhad

2 Ali Karimpour Sep 2009

Chapter 6Chapter 6Introduction to Decoupling Control

Topics to be covered include:

• Decoupling

• Pre and post compensators and the SVD controller

• Decoupling by State Feedback

• Diagonal controller (decentralized control)


Chapter 6

Introduction

CxyBuAxx

BAsICsG 1)()(

)()(.....)()()()()(........................................................................................................................................................................

)()(.....)()()()()(

)()(.....)()()()()(

2211

22221212

12121111

susgsusgsusgsy

susgsusgsusgsy

susgsusgsusgsy

pppppp

pp

pp

We see that every input controls more than one output and that every output is controlled by more than one input.

Because of this phenomenon, which is called interaction, it is generally very difficult to control a multivariable system.


Chapter 6

Definition 6-1

A multivariable system is said to be decoupled if its transfer-function matrix is diagonal

and nonsingular.

A conceptually simple approach to multivariable control is given by a two-steps

procedure in which

1. We first design a compensator to deal with the interactions in G(s) and

2. Then design a diagonal controller using methods similar to those for SISO systems.

)()()( sWsGsG ss

)()()( sKsWsK ss)(sK s

Decoupling

Decoupling


Chapter 6

Decoupling

• Dynamic decoupling

• Steady-state decoupling

• Approximate decoupling at frequency ω0

s.frequencie allat diagonal is )(sGs

1. We first design a compensator to deal with the interactions in G(s) and )()()( sWsGsG ss Decoupling

)()( choosecan we with exampleFor 1 sGsWIG ss

(s)l(s)GK(s)IslsK -s

1 have we)()(by Then It usually refers to an inverse-based controller.

diagonal. is )0(sG

This may be obtained by selecting a constant pre compensator )0(1GWs

possible. as diagonal as is )( 0jGs

This is usually obtained by choosing a constant pre compensator 10GWs

)( ofion approximat real a is 00 jGG s for selection good a is frequency 0BW


Chapter 6

Decoupling

The idea of using a decoupling controller is appealing, but there are several difficulties.

a. We cannot in general choose Gs freely. For example, Ws(s) must not cancel any

RHP-zeros and RHP poles in G(s)

b. As we might expect, decoupling may be very sensitive to modeling errors and

uncertainties.

c. The requirement of decoupling may not be desirable for disturbance rejection.

One popular design method, which essentially yields a decoupling controller, is the internal model control (IMC) approach (Morari and Zafiriou).

Another common strategy, which avoids most of the problems just mentioned, is to use partial (one-way) decoupling where Gs(s) is upper or lower triangular.


Chapter 6

Pre and post compensators and the SVD controller

The pre compensator approach may be extended by introducing a post compensator

)()()()( sWsGsWsG ssps

The overall controller is then

)()()()( sWsKsWsK spss


Chapter 6

Decoupling by State Feedback

In this section we consider the decoupling of a control system in state space representation.

CxyBuAxx

Let

)()()( Suppose tHrtKxtu Cxy

BHrxBKAx

)(have Thgen we

The transfer function matrix isBHBKAsICsG 1)()(

We shall derive in the following the condition on G(s) under which the system can be

decoupled by state feedback.


Chapter 6


Theorem 6-1 A system represented by

with the transfer function matrix G(s) can be decoupled by state feedback of the form

CxyBuAxx

)()()( tHrtKxtu

if and only if the constant matrix E is nonsingular

pE

EE

E..

2

1

Furthermore 11 , EHFEK

pdp

d

d

AC

AC

AC

F..

2

1

2

1

and

Proof: See “Linear system theory and design” Chi-Tsong Chen


Chapter 6


Example 6-2 Use state feedback to decouple the following system.

xyuxx

110001

001001

6116100010

Solution: Transfer function of the system is

656

656

61166

6116116

)()(

22

2323

2

1

sss

ss

ssss

sssss

BAsICsG

The differences in degree of the first row of G(s) are 1 and 2, hence d1=1 and

]01[6116

66116

116lim 2323

2

1

ssss

ssssssE

s

The differences in degree of the second row of G(s) are 2 and 1, hence d2=1 and

]10[65

665

6lim 222

ss

sss

sEs


Chapter 6


Now E is unitary matrix and clearly nonsingular so decoupling by state feedback is possible and

]01[1 E ]10[2 ESolution (continue):

5116010

2

1

2

1

2

1

CAACAC

AC

ACF

d

d

1001

,5116010 11 EHFEK

The decoupled system is

xCxy

rxBHrxBKAx

110001

001001

61166116000

)(

Exercise 1: Derive the corresponding decoupled transfer function matrix.


Chapter 6

Diagonal controller (decentralized control)

Another simple approach to multivariable controller design is to use a diagonal or

block diagonal controller K(s). This is often referred to as decentralized control.

Clearly, this works well if G(s) is close to diagonal, because then the plant to be

controlled is essentially a collection of independent sub plants, and each element in

K(s) may be designed independently.

However, if off diagonal elements in G(s) are large, then the performance with

decentralized diagonal control may be poor because no attempt is made to counteract

the interactions.


Chapter 6


The design of decentralized control systems involves two steps:1_ The choice of pairings (control configuration selection)

2_ The design (tuning) of each controller ki(s)

In this section we provide two useful rules for pairing inputs and outputs.

1_ To avoid instability caused by interactions in the crossover region one should prefer pairings for which the RGA matrix in this frequency range is close to identity.

2_ To avoid instability caused by interactions at low frequencies one should avoid pairings with negative steady state RGA elements.


Chapter 6


Example 6-3 Select suitable pairing for the following plant

8.14.01.187.04.85.154.16.52.10

)0(G

Solution: RGA of the system is

98.107.09.043.037.094.041.145.196.0

)0(

Multivariable Control Systems

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