Multivariable Control Systemsprofsite.um.ac.ir/~karimpor/multi/Multivariable_lec7.pdf · Example 7-1 Continue ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ − + − + = = s s s s s Q s G s Ka 1 3 3 (

Multivariable Control Multivariable Control SystemsSystemsSystemsSystems

Ali KarimpourpAssistant Professor

Ferdowsi University of MashhadFerdowsi University of Mashhad

Chapter 7Chapter 7Multivariable Control System Design: Nyquist-Like Techniquesu v b e Co o Sys e es g : Nyqu s e ec ques

Topics to be covered include:

• Sequential loop closing

• The characteristic-locus method

• Reversed-frame normalization

• Nyquist-array methods

• PI control for MIMO systems

Ali Karimpour July 2012

2

Nyquist array methods

Chapter 7

Sequential loop closing

The simplest approach to multivariable design is to ignore its multivariable nature.

• A SISO controller is designed for one pair of input and output variables.g p p p

• When this design has been successfully completed another SISO controller is

designed for a second pair of variables and so on.

H i t d t t i bl h ld b i d?


3

How input and output variables should be paired?

loop-assignment problem or input-output pairing

Chapter 7


• The transfer function matrix ofBenefits:

The transfer function matrix of

such a controller is diagonal.

• It also has the advantage that itIt also has the advantage that it

can be implemented by closing

one loop at a time.

Drawbacks

• It allows only a very limited class of controllers to be designed, and the design must

proceed in a very ad hoc manner.• Interactions are very important.• The only means available for the reduction of interaction (if this is a requirement) is


4

The only means available for the reduction of interaction (if this is a requirement) is

to use high loop gains

• Control difficulty in the case of element zeros.

Chapter 7


A more sophisticated version of sequential loop closing is calledsequential return-difference method

A cross-coupling stage of compensations should be introduced

This stage should consist of either a constant gain matrix


5

This stage should consist of either a constant-gain matrix

or a sequence of elementary operations. )()(

1)( sUsd

sKa =

Chapter 7


Example 7-1

⎥⎤

⎢⎡ −− ssG

111)(⎥⎥

⎦⎢⎢

⎣ −−+=

ss

sss

sG123

1)1(

)( 2

If we try to design a SISO controller for either the first or second loop here, we have difficulties, if the required bandwidth is close to unity, or greater, because the transferfunction ‘seen’ for the design, namely the (1,1) or (2,2) element of G(s), has a zero at 1.However, G(s) itself has a transmission zero at - 1 only, so there should be no inherent difficulty of this kind.

If we choose ⎥⎦

⎤⎢⎣

⎡−

=1201

aK⎦⎣ 12

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−++

==ss

sss

KsGsQ a131

31

)1(1)()( 2


6

We see that no right half-plane zero ‘appears’ when a SISO compensator is being designed for the first loop.

Chapter 7


⎤⎡ 111

Example 7-1 Continue

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−++

==ss

sss

KsGsQ a131

31

)1(1)()( 2

Once the first loop has beenOnce the first loop has been closed by

)(1 sk

The transfer function ‘seen’ in the second loop is

)(1

31

1 − sss 1)()( where 1 sksh

⎤⎡ ⎞⎛−

=


7222

122 )1(

3)()1()1(

1)(++

++−

=s

shs

ssssq ( ) )(1/

311

)(

12 skss ⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ ++

Chapter 7


1− s )()(where 1 sksh −

=

Example 7-1 Continue

222122 )1(

3)()1()1(

1)(++

++−

=s

ssh

ss

sssq ( ) )(1/

311

)( where

12 skss

sh

⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ ++

=

Now, if we assume high gain in the first loop

)1( 2 s + )1( 2

s

ssk+

+>>

31

)1()(2

1 s

ssh+

+−≈

31

)1()(

and hence

)13)(1(1)(1

22 ++≈

sssq


8

so that no right half-plane zero is seen when the compensator for the second loopis designed.

Chapter 7


• The main idea that of using a first stage of compensation to make subsequent

loop compensation easierloop compensation easier

• The main weakness of the method is that little help is available for choosing that

first stage of compensationfirst stage of compensation.

The available analysis relies on the assumption that there are high gains in the loops

which have already been closed, and such an assumption can rarely be justified, y , p y j ,

except at low frequencies.

One rather special case, in which the assumption of high gains is justified, arises when

different bandwidth is required for each loop and all the bandwidths are well separated


9

different bandwidth is required for each loop, and all the bandwidths are well separated

from each other.

Chapter 7


Mayne (1979) suggests that, if the plant has a state-space realization (A, B, C), the

product CB being non-singular (and the matrix D being zero) then the first stage ofproduct CB being non-singular (and the matrix D being zero), then the first stage of

compensation can be chosen to be

( ) 1( ) 1−= CBKa

∞→→ sasCBsG )( Since ∞→→ sasIKsG a)( sos

)(sa)(

so that each loop looks like a first-order SISO system at high frequencies.

However, an alternative choice, such as


10)(1

ba jGK ω−≈ )( or 1bba jGjK ωω −≈

Chapter 7




• Reversed frame normalization• Reversed-frame normalization




11

Chapter 7

The characteristic-locus method

Commutative compensators)()()()( sGsKsKsG =

Approximate commutative compensators)()()()( sGsKsKsG ≈

• How this can be done. • Why it should be useful

Suppose we have a square transfer-function matrix G(s), with m input and outputs

iondecomposit spectral is )()()()( 1 sWssWsG −Λ=

W(s) is a matrix whose columns are the eigenvectors or characteristic directions of G(s)W(s) is a matrix whose columns are the eigenvectors, or characteristic directions, of G(s)

{ })(,.....,)(,)()( 21 sssdiags mλλλ=Λ


12where the λi(s) are the eigenvalues, or characteristic functions, of G(s).

Chapter 7


iondecomposit spectral is )()()()( 1 sWssWsG −Λ=Let following structure as controller

)()()()( 1 sWsMsWsK −=where

{ })(,...,)(,)()( 21 sssdiagsM mµµµ=

then the return ration (loop transfer function) is

)()()()()()()()()()()( 11 sGsKsWsNsWsWsMssWsKsG ==Λ= −− )()()()()()()()()()()(

{ })(,.....,)(,)()( 21 svsvsvdiagsN m=where

)()()( λ )()()( sssv iii µλ=

A compensator having this structure is called a commutative compensator.


13• How this can be done. • Why it should be useful

Chapter 7


The strategy which suggests itself is to obtain graphical displays of the characteristic loci of the plant,

{ }ij 21)(λ{ }miji ,.....,2,1:)( =ωλThen design a compensator

)( ωµ ji for each )( ωλ ji

One would then obtain the compensator as the series connection of three systems

using the well-established single-loop techniques.

)(µ ji

p ycorresponding to

W(s)sMsW and )(,)(1−

Th i d b k i d i i t ti t llThere are some main drawback in designing commutative controller

1- Robust stability may not be derived.


142- It is not realizable in general case.

Chapter 7The characteristic-locus method1- Robust stability may not be derived.

Example 7-2 The transfer-function matrix

⎤⎡ 564721 ss

1 Robust stability may not be derived.

( )( ) ⎥⎦

⎤⎢⎣

⎡+−

−++

=25042

5647221

1)(ss

ssss

sG

has the following spectral decompositiong p p

⎥⎤

⎢⎡ −⎥⎥⎤

⎢⎢⎡+

⎥⎤

⎢⎡

=Λ= − 8701

187

)()()()( 1 ssWssWsG ⎥⎦

⎢⎣−⎥⎥

⎦⎢⎢

⎣ +

⎥⎦

⎢⎣

Λ76

22076

)()()()(

s

sWssWsG

and so W(s) and Λ(s) is

⎥⎥⎤

⎢⎢⎡+=Λ⎥

⎤⎢⎡

= 2

01

1

)(87

)( sssW


15⎥⎥

⎦⎢⎢

⎣ +

⎥⎦

⎢⎣

220

)(76

)(

s


Choose controller as


⎤⎡⎤⎡⎤⎡ 87087 k ⎥⎤

⎢⎡ 01

1

⎥⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡== −

7687

00

7687

)()()()(2

11

kk

sWsMsWsK⎥⎥⎥

⎦⎢⎢⎢

⎣ +

+=Λ

220

1)(

s

ss

)k(-1 constant aby stabilized is 1

1 clear that isIt 11 ∞<<+

ks

)k(-1constant aby stabilizedis2 clear that also isIt 22 ∞<<k )(y2 22+s

Now we can choose k 1 =k 2 =1 we have

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

−⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

1001

7687

1001

7687

)(sK


16

so we have good stability margin in each channel but what about the stability margin in multivariable system?

Chapter 7The characteristic-locus method1- Robust stability may not be derived.1 Robust stability may not be derived.


⎤⎡

( )( ) ⎥⎦

⎤⎢⎣

⎡+−

−++

=25042

5647221

1)(ss

ssss

sG

⎥⎦

⎤⎢⎣

⎡=

1001

)(sK

Suppose that we use above controller in our system but there is some uncertainty in inputs so let the value of controller as:

⎤⎡a 0⎥⎦

⎤⎢⎣

⎡=

ba

sK0

0)(

To check the stability we need to check the stability of


171)( −+= GKIKTu



1)( −+= GKIKTu

We need to check the stability of

Since K is a constant matrix we need to check the stability of1)( −+GKI

( )( )

1

1

250)2)(1(4256472)2)(1(

)2)(1(1)(

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡++++−

−+++++

=+bsssas

bsasssss

GKI

Closed loop characteristic equation is:

( ) 0)2222()47503()2)(1( 2 =++++−++++ abbaabssss


18

For stability we need:02222&047503 >+++>−+ abbaab



For stability we need:

02222&047503 >+++>−+ abbaab

For a=1 we need b>44/50=0.88

For b=1 we need a<53/47=1.128

For a=50.5/47=1.07 and b=0.95 we have lost stability.y

So robust stability may not be derived.


19

Is it possible to know this weak stability margin by some measure?

Chapter 7structure−∆MThe characteristic-locus method1- Robust stability may not be derived.

Suitable for robust stability analysis

y y

Perturbed PlantUncertainty M in M∆-structure

Additive uncertainty 21

1 )( wGKIKwM −+=12 wwGGp ∆+=

Multiplicative input uncertainty2

11 )( GwGKIKwM −+=)( 12 wwIGGp ∆+=

1Multiplicative output uncertainty2

11 )( wGKIGKwM −+=GwwIGp )( 12∆+=

Inverse additive uncertainty 21

1 )( wKGIGwM −+=( ) 112

−∆−= GwwIGGy 21 )( wKGIGwM +( )12∆ GwwIGGp

Inverse multiplicative input uncertainty2

11 )( wKGIwM −+=( ) 1

12−∆−= wwIGGp


2020Inverse multiplicative output uncertainty

21

1 )( wGKIwM −+=( ) GwwIGp1

12−∆−=

Chapter 7structure−∆MThe characteristic-locus method1- Robust stability may not be derived.

⎤⎡ 01⎥⎤

⎢⎡ − 564721)(

ssG


Multiplicative input uncertainty

⎥⎦

⎤⎢⎣

⎡=

10)(sK( )( ) ⎥

⎦⎢⎣ +−++

=2504221

)(ssss

sG

21

1 )( GwGKIKwM −+=)( 12 wwIGGp ∆+=

p p y

1Let 21 ==ww )2.24(2.16 dbM =∞

06.02.16/1)( ==∆∞

ωjSo we can tolerate:∞

There is small stability margin. Is there a method to improve the condition?


2121


( )( ) ⎥⎦

⎤⎢⎣

⎡+−

−++

=25042

5647221

1)(ss

ssss

sG


( )( ) ⎦⎣

⎥⎦

⎤⎢⎣

⎡ −=

02160706807068.00216.0

pK ⎥⎦

⎢⎣ 0216.07068.0p

=68242103143322

12345K

⎥⎦

⎤⎢⎣

⎡

++++−−−−−+−−−−++++

++++

25.25511.52778.22435.1734.936.24674.47927.20099.1544.936.20804.46278.19969.15424.20361.421017904.138

6.824.2103.1433.22

234234

234234

2345

ssssssssssssssss

sssss


2222Step response of this controller is quite better than pervious controller.

Chapter 7

The characteristic-locus method2- It is not realizable in general case

Example 7-3 The transfer-function matrix⎥⎥⎤

⎢⎢⎡

++ 12

11

)( sssG


⎥⎥

⎦⎢⎢

⎣ ++

++=

21

22

11)(

ss

sssG

( )[ ]1

has characteristic functions

( )[ ]33481632)2)(1(2

1)(,)( 221 ++±+

++= sss

ssss λλ

⎤⎡ 84)(The characteristic directions are given by

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

++±−

+=

334816184

)()(

22

1

sss

swsw

i

ig y

and so W(s) is

( ) ( )⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

++−−

+

+++−

+

=3348161

843348161

84

)(

22 sss

sss

sW


23⎥⎥⎥

⎦⎢⎢⎢

⎣

11

Chapter 7

The characteristic-locus method2- It is not realizable in general case

A practical alternative is to give the compensator the structure

)()()()( BMAK)()()()( 1WMWK


)()()()( sBsMsAsK =)()()()( 1 sWsMsWsK −=

where )()( sWsA ≈ )()( 1 sWsB −≈

are chosen to be realizable, and such that

Whichever approximation technique is chosen, we obtain an

approximate commutative compensatorapproximate commutative compensator


24

Chapter 7

The characteristic-locus method)()( sWsA ≈ )()( 1 sWsB −≈

One way of choosing constant matrices A and B is as approximations to

)()( sWsA )()( sWsB ≈

)(and)( 1 sWsW − i l i)( and )( 00 sWsW at some particular point s0

A successful algorithm for computing constant real matrices for use as the matrices,

A and B is the ALIGN algorithmA and B is the ALIGN algorithm.

Use of the approximate commutative compensator, with constant matrices A and B,

clearly relies on the eigenvectors of the plant not changing too quickly with frequencyclearly relies on the eigenvectors of the plant not changing too quickly with frequency.

If the eigenvectors do change too quickly, one can try to choose A and B so as to

i t t l f i th thapproximate at several frequencies rather than one. .

This refinement of the ALIGN algorithm frequently gives no better results


25

The reason seems to be that the price paid in not obtaining a good approximation to the eigenframe at any frequency often outweighs the advantage gained.

Chapter 7

The characteristic-locus method)()( sWsA ≈ )()( 1 sWsB −≈)()( sWsA )()( sWsB ≈

If all the characteristic loci require shaping over approximately the same range of

frequencies, then the choice of the frequency at which the ALIGN algorithm should

be used does not present a major problem.

But it often happens that different loci require compensation at different frequencies.

One can then either choose one of these frequencies, and hope that approximate

compensation is achieved on all loci satisfactorily, or compensate each locus with a

t i t t ti t

)()...()()( 21 sKsKsKsK m=

separate approximate commutative compensator:

iiii BsMAsK )()( =

The second alternative leads to a very complicated compensator and suffers from the


26

The second alternative leads to a very complicated compensator, and suffers from the

difficulty that each factor of the compensator interferes with all the others.

Chapter 7

The characteristic-locus methodDesign procedureDesign procedure

• Manipulating the characteristic loci• Stable design by satisfying the generalized Nyquist stability theorem• Large magnitudes at low frequencies

• Stay outside the 2=M

All the loci look satisfactory-when judged against classical SISO criteria.

Do one ensure satisfactory performance of the multivariable feedback system? NO.

{ } ∑=

− ==m

i

Tiiii ssvswsWsvdiagsWsKsG

1

1 )()()()()()()()(Let υ

)()(1

1)()()(1

1)()(1

1 ssv

swsWsv

diagsWsS Ti

i

m

ii

i

υ+

=⎭⎬⎫

⎩⎨⎧+

= ∑=

−

⎫⎧


27)(

)(1)(

)()()(1

)()()(

1

1 ssv

svswsW

svsv

diagsWsT Ti

i

im

ii

i

i υ+

=⎭⎬⎫

⎩⎨⎧+

= ∑=

−

Chapter 7The characteristic-locus methodDesign procedureg p

)()(1

)()()(

)(1)(

)()(1

1 ssv

svswsW

svsv

diagsWsT Ti

i

im

ii

i

i υ+

=⎭⎬⎫

⎩⎨⎧+

= ∑=

−

⎭⎩


28


)()()()( 11 swsrssr T →→υ

)()()()( 22 swsrssr T →→υ)(sy


29

.......

)()()()( swsrssr mT

m →→υ

)(y


The performance of a feedback system depends essentially on the singular values(principal gains) of functions such as S(s) and T(s).(p p g ) ( ) ( )

1 )(svi

These transfer functions have eigenvalue functions

)(1 svi+ )(1 svi+

We know that, for any matrix X,

)()()( XXX σλσ ≤≤−

If the characteristic loci of GK The sensitivity will certainly be largeIf the characteristic loci of GKhave low magnitudes

The sensitivity will certainly be large, in some signal directions

If some characteristic locus At least one principal gain of T(s)


30

If some characteristic locus penetrates the circle2=M will exhibit a resonance peak, of

magnitude greater than 2


If the eigenvectors are nearly th l t h th

Magnitudes of the smallest and largesteigenvalues are close to the smallest orthogonal to each other g

and largest singular values

Good system performance)( sviShaping y p

This situation happens if the return ratio has low skewness

Note that the discrepancies between characteristic loci and singular values (principal gains) do not make the shaping of characteristic loci a fruitless activity.

On the contrary, manipulating the characteristic loci is often the most straightforwardand productive way of designing a multivariable feedback system, or at least of

initiating the design. Of course, the properties of the resulting design must be checked


31

initiating the design. Of course, the properties of the resulting design must be checkedby methods more revealing than examination of the characteristic loci.


MIMO systems should exhibit little interaction.

So a change in one of the reference signals should cause only the correspondingoutput to change, without excessive transients occurring on the other outputs.

Alternatively, the appearance of a sudden disturbance on one of the outputs should not disturb the other outputs excessively.

isvi eachfor,1)( If >> 1)(1

)(≈

+ svsv

i

i IsT ≈)(

High loop gain demonstrates that lack of interaction.

Both for physical reasons and because theieachfor1)(If >>


32

Both for physical reasons and because thegeneralized Nyquist theorem needs to be satisfied

isvi eachfor,1)( If >>


isvi eachfor,1)( If >> 1)(1

)(≈

+ svsv

i

i IsT ≈)(

High loop gain demonstrates that lack of interaction.

Both for ph sical reasons and beca se theBoth for physical reasons and because thegeneralized Nyquist theorem needs to be satisfied

isvi eachfor,1)( If >>

Th hi h f i f db k b li d f l i iThus, at higher frequencies feedback cannot be relied upon to enforce low interaction.

The only thing that can be done is to insert a series compensator whose main purpose isy g p p pto obtain a ‘decoupled’ (that is, diagonal) return ratio of -G(s)K(s) at (some of) these

frequencies.

O f tt ti t d l th t ti t l f i i t


33

One way of attempting to decouple the return ratio at one or several frequencies is to attempt to invert G(s) at these frequencies by ALIGN algorithm. .


ieachforjv

jv bi ,2

1)(1

)(=

+ ωωjv bi 2)(1+ ω

)( Let 1bh jGK ω−−≈

( ) 1)( seigenvalue −→hb KjG ω

It may be advantageous to adjust theIt may be advantageous to adjust thesigns of the columns of Kh at thisstage.

In which case one of the characteristic loci will be very far from -1


34

In such a case changing the sign of the corresponding column of Kh may bring this characteristic locus into a region in which subsequent compensation becomes easier.


It can be shown that if the compensator has the structure

k )()( sCsksK =

andproperis)()( sCsG proper is )()( sCsG

then it is always possible to find a C(s) such that the closed-loop system is stable for a range of gains

*0 kk <<

If all the eigenvalues of G(0)C(0) have positive real parts.

Furthermore, it is impossible to find such a C(s) if any eigenvalue of G(0)C(0) has a negative real part.


35

This strongly suggests that the signs of the columns of Kh should be adjusted to bring all the eigenvalues of G(0)Kh into the right half-plane, if possible.

Chapter 7


Design procedure of characteristic-locus method

1 Compute a real )(1 jGK ω−≈1- Compute a real )( bh jGK ω−≈

2- Design an approximate commutative controller Km(s) at some frequency bm ωω <

for the compensated plant G(s)Kh, such that ∞→→ ωω asIjK m )(

3- If the low-frequency behavior is unsatisfactory (typically because there are excessive

t d t t ) d i i t t ti t ll K ( ) t lsteady-state errors), design an approximate commutative controller Kl(s) at low

frequency, for the compensated plant G(s)KhKm(s), such that ∞→→ ωω asIjKl )(

(th d bl k tt t t th t th d li ff t d b K i t(the red blocks are some attempt to ensure that the decoupling effected by Kh is notdisturbed too much).

4- Realize the complete compensator as )()()( sKsKKsK lmh=


36high, medium and low frequency

Chapter 7


Step 2, the medium-frequency compensation, is concerned with shaping the

characteristic loci in the vicinity of -1y

This is where phase lead or lag, or more complicated shaping, is applied to the loci.

To reduce the interaction between Step 1 and step 2 it is often necessary to obtain

Kh at a frequency rather higher than ωs.

The characteristic-locus design technique is best suited to plants which need a similar

bandwidth for each loop with reasonable control signals.

If a plant needs a different bandwidth for different loops, then it may be possible to

perform two (or more) characteristic-locus designs:

The first one would usually be for the fast loops, and the second for the slower loops.


37In other words, a kind of sequential loop-closing approach may be appropriate, the

characteristic locus method being used inside the sequential loop-closing approach.

Chapter 7

The characteristic-locus methodDesign exampleDesign example

Consider the aircraft model AIRC described in the following state-space model.

BuAxx +&

⎤⎡⎤⎡ 0000000101320100

DuCxyBuAxx

+=+=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡−

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

= 00000000.11200.0000

,00000.10000705.001712.00538.000000.101320.100

BA

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−

0732.005750.16650.104190.4

6859.00532.102909.000130.18556.000485.00

⎤⎡⎤⎡ 00000001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000000000

,001000001000001

DC


38the model has three inputs, three outputs and five states.

Chapter 7


THE SPECIFICATION

• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.

Littl i t ti b t t t• Little interaction between outputs.

G d d i f d d i h f f• Good damping of step responses and zero steady-state error in the face of step

demands or disturbances.

• We assume a one-degree-of-freedom control structure.


39

Chapter 7


PROPERTIES OF THE PLANT

• The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very

severe interaction between outputs.

• The poles of the plant (eigenvalues of A) are

jj 1826.00176.0,03.178.0,0 ±−±− jj 1826.00176.0,03.178.0,0 ±±

so the system is stable (but not asymptotically stable).

Th hi l h fi i d d li i i• Thus this plant has no finite zeros, and we do not expect any limitations on

performance to be imposed by zeros. since

the n mber of finite eros of the plant can be at most


40

the number of finite zeros of the plant can be at most

)(2 CBrankmn +− 013.25 =+−=

Chapter 7


2=M

111 −−− =− NPZ

111 =−−Z

21 =−Z

Negative feedback were Closed loop would be unstable.Characteristic loci of G , with logarithmic calibration


41

gapplied around the plant (Two RHP poles)

02.173.069.159.090.0:arevaluesEigen 53,41,2 −=±−=±= λλλ jj

Chapter 7


ALIGNMEIVT AT 10 rad/sec⎤⎡ −−− 669.30036.0535.71

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−=−=

378.690065.044.1895376.09984.95375.8669.30036.0535.71

))10(( jGalignKh

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−++−++−−−−

=jjjjjjjjj

KjG h

086.09962.00005.00000.0100.0009.00003.00008.0010.1005.00004.0063.0

009.00005.0003.00001.0068.0983.0)10(

⎦⎣ jjj

From input 1 to output 2 and 3, and that these interactions have been reduced to 10% or less (at that frequency).

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−−

=378690065044189

5376.09984.95375.8669.30036.0535.71

hK


42

⎥⎦⎢⎣ −−− 378.690065.044.189

Chapter 7

The characteristic-locus method)1031( <<<≤≤− θθj )10,

22:( <<<≤≤= επθπε θjes

111 −−− =− NPZ

111 −=−−Z

Characteristic loci of GKh , with logarithmic calibration 01 =−ZNegative feedback were Closed loop would be stable.


43

Negative feedback were applied around the plant

p(No RHP poles)

05.1099.953.096.924.0:arevaluesEigen 53,41,2 −=±−=±−= λλλ jj

Chapter 7


APPROXIMATE COMMUTATIVE COMPENSATOR120

A i ht b t d f

80

100

As might be expected from

hKjG )10(however, two of the loci pass

40

60

ain

(dB

)

, pextremely close to -1

1)10( −Λ= WWKjG h

0

20

Ga

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−−== −

0009.09526.08829.09999.00076.00124.00015.02049.03483.0

)( 1WalignA

-180 -160 -140 -120 -100 -80 -60 -40 -20 0-40

-20

⎦⎣

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−==

0024.00001.10391.06633.00026.07310.13728.00031.08165.1

)(WalignB


44

180 160 140 120 100 80 60 40 20 0Phase (deg)

Characteristic loci of GKh , on Nichols chart

⎦⎣

Chapter 7

The characteristic-locus methodAPPROXIMATE COMMUTATIVE COMPENSATOR

In this case it is the (1,1) and (2,2) elements of Λ that require compensation.

⎥⎥⎥⎤

⎢⎢⎢⎡

++

+

02175.00933.00

0010933.0

2175.00933.0

)(s

ss

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+=

100

010933.0

0)(s

sM BsAMsK m )()( =

To evaluate the interactions at this frequency it is appropriate to examineTo evaluate the interactions at this frequency it is appropriate to examine

( )⎥⎥⎥⎤

⎢⎢⎢⎡

= 0025.09999.00265.00071.00023.06811.0

)10()10( jKKjGabs mh


45

⎥⎦⎢⎣ 6731.00012.00763.0

This shows that decoupling has not been destroyed at l0 rad/sec.

Chapter 7


Characteristic loci of GKhKm , on Nichols chart

LOW FREQUENCY PROBLEM


46

Characteristic loci of GKhKm , on Nichols chart

Chapter 7

The characteristic-locus methodLOW FREQUENCY COMPENSATIONQ

)()()( XXX σλσ ≤≤−

10))0()0(( ≤−

mh KKGσ

Whi h ill i l hi h bj i f d i h f fWhich will certainly not achieve the objective of zero steady-state error in the face of step disturbances.


47IsT

sTsKl+

=1)(

Chapter 7


150

100

)

50

Gai

n (d

B)

0

Characteristic loci of GKhKmon Nichols chart

-240 -220 -200 -180 -160 -140 -120 -100 -80 -60-50

Phase (deg)

Characteristic loci of GKhKmKlNi h l h t


48

on Nichols chart

Chapter 7


150

100

50

Gai

n (d

B)

0

10-2 10-1 100 101 102-50

Frequency ( rad/sec )


49

Frequency ( rad/sec )

Largest and smallest open-loop singular values (principal gains).

Chapter 7


0

5

-10

-5

-25

-20

-15

Gai

n (d

B)Bandwidth is ok

-35

-30

25

10-1 100 101 102-45

-40

Frequency (rad/sec)


50

Frequency (rad/sec)

Largest and smallest closed-loop singular values (principal gains).

Chapter 7

The characteristic-locus methodDesign example

Step Response

1.2

Step Response

0.8

1

0.6

Ampl

itude

0.2

0.4

0 0 5 1 1 5 2 2 5 3-0.2

0


51

0 0.5 1 1.5 2 2.5 3

Closed-loop step responses to step demand on output 1 (solid curves), output 2 (dashed curves) and output 3 (dotted curves).

Chapter 7

The characteristic-locus methodDesign example

REALIZATION OF THE COMPENSATOR

h ll b li d fi li ( f hThe controller can be realized as a five-state linear system (two states for the phase-lead transfer functions in Km, and three for the integrators in Kl).

⎤⎡⎤⎡ 4911012602667491101260266707210

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−−

=0500005.0653.20106.0924.6491.10126.0266.7

,0000000000653.20106.0924.672.100491.10126.0266.7072.10

KK BA

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ 5.00005.00

0000000000

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−−−−−

=5265120409183

4372.0996.9369.8091.30129.045.70

,52651204091832578266

4372.0996.9369.8752.2489.7091.30129.045.704.3303.59

KK DC


52

⎥⎦⎢⎣ −−−⎥⎦⎢⎣ −−− 52.651204.09.18352.651204.09.1832.578.266

Chapter 7








53

Chapter 7

Reversed-frame normalization

The characteristic locus design technique is based on

Reversed-frame normalization The characteristic-locus design technique is based on

)()()()( 1 sWssWsG −Λ=

L t f i t d th i l l d itiLet us now focus instead on the singular-value decomposition

)()()()( sUssYsG H∑=

The singular values (principal gains) of G(s) are real valued but we can associate themwith ‘phase’ information in the following way.

( ){ }[ ]iH

i jdiagsYsUsi

ψσθψ

exp)()(minarg)( −=

( ) ( )( ){ }sjdiags iθexp=Θ


54

( ) ( )( ){ }jg ip

( )( ) ( ) ( ) ( )[ ]ssYsUsGm H Θ−= σ

Let

Chapter 7


( ){ }[ ]iH

i jdiagsYsUsi

ψσθψ

exp)()(minarg)( −= ( ) ( )( ){ }sjdiags iθexp=Θ

( )( ) ( ) ( ) ( )[ ]ssYsUsGm H Θ−= σ

( ) 0)( If =sGm ⇒ aligned. are )( and )( sYsU

In the sense that ( ) ( ) ( )( )sjsusy iii θexp=

We shall say that G(s) is aligned if m(G(s))=0. y ( ) g ( ( ))

If we now define

( ) ( ) ( )∑Θ=Γ sss ( ) ( ) ( )ssYsZ HΘ=( ) ( ) ( )∑ ( ) ( ) ( )

( ) ( ) ( )sUssZsG H)(Γ=

( ) ( ) ( ){ }ssdiags γγΓ

)()()()( sUssYsG H∑= )()()( sUssY HH ∑ΘΘ= ⇒


55

( ) ( ) ( ){ }ssdiags mγγ ,...,1=Γ

This is called the quasi-Nyquist decomposition.

Chapter 7


( ) ( ) ( )sUssZsG H)(Γ=

This is called the quasi-Nyquist decomposition.

The loci of are called quasi-Nyquist loci, and it should be noted that( )ωγ ji

Now, suppose that the compensator K(s) has the structure

( ) )( ωσωγ jj ii =

q yq

pp p ( )

( ) ( ) ( ) ( )sZsMsUsK H= ( ) ( ) ( ){ }ssdiagsM mµµ ,...,1=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sYsNsYsZsNsZsZsMssZsKsG HHH ==Γ=

( ) ( ) ( ){ }d ( ) ( ) ( )


56

( ) ( ) ( ){ }svsvdiagsN m,...,1= ( ) ( ) ( )sss iii µγν =

Chapter 7


In this case we see that the quasi-Nyquist loci are precisely the characteristic loci, and that their magnitudes are the singular values (principal gains) of both return ratiosthat their magnitudes are the singular values (principal gains) of both return ratios.

We therefore approach more nearly the classical situation of being able to predict stability, performance and robustness from the same set of loci.

A compensator with this structure is called a reversed-frame normalizing (RFN)compensator.

Of course, such a compensator is no more realizable than an exact commutative controller, but, as might be expected, one can obtain useful results by settling for an approximate RFN controller.pp

Approximate RFN controller?


57

Approximate RFN controller?

Chapter 7


Hung and MacFarlane (1982) have developed the above theory and its applications in great detail, and outlined an optimization-based technique for computing approximategreat detail, and outlined an optimization based technique for computing approximateRFN controllers.

( )( )

( ) ( ) ( )sKsGsQsKsK

−=∈κ

minarg( )sK ∈κ

Hung and MacFarlane performed least-squares optimization, and specified as the set of transfer functions having a matrix-fraction description with a particular canonicalform (Hermite form) and McMillan degree

κ

form (Hermite form) and McMillan degree.

Clearly many alternatives to these particular choices are possible.

One could also attempt approximate RFN at one specific frequency, just as we did with the approximate commutative compensator.


58

Chapter 7


( ) ( ) ( ) ( ){ } ( )0,...,0 1H

m YssdiagUsK µµ=

⎥⎤

⎢⎡⎥⎤

⎢⎡⎥⎤

⎢⎡

=Σ=643.0766.005342.0942.0

)0()0()0()0( HUYQ


59

⎥⎦

⎢⎣−⎥⎦

⎢⎣⎥⎦

⎢⎣−

=Σ=766.0643.010942.0342.0

)0()0()0()0( UYQ

Chapter 7

Nyquist-array methods

⎤⎡⎤⎡⎤⎡ 643076600534209420⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=Σ=766.0643.0643.0766.0

1005

942.0342.0342.0942.0

)0()0()0()0( HUYQ

A compensator of the form0ks ⎤⎡ +

)0(50

0)0()( HY

sks

sks

UsK⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+

=

⎦⎣

would have the desired effect since, at low frequencies, we would have

005k

⎥⎤

⎢⎡⎤⎡


60

)0(501005

)0()()(0

HY

jk

jYjKjQ

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

⎥⎦

⎤⎢⎣

⎡→→

ω

ωωωω

Chapter 7







y


61

Chapter 7

PI control for MIMO systems

P: Proportional, output of controller is proportional to error.

I: Integral action, remove steady state error.

D: Derivative action, anticipate the performance.

Ai f PID i MIMO t D i f ll i ifi ti ith i iAim of PID in MIMO systems: Derive following specification with minimuminformation of system.

* Stability of closed loop system. * Minimum value of interaction.

* Reference tracking. * Disturbance rejection.


62

Chapter 7


Design procedurea) Design based on step response.

* System is stable.

* System model is unknown. * No transmission zero on origin.

* Suitable for industrial plants.

y

* System is regular1 or irregular.

b) Design based on model.

* System model is known * No transmission zero on origin System model is known.

* System is regular or irregular.

No transmission zero on origin.


63-----------------------------------------------1 A system is regular if CB has full rank otherwise its irregular.

Chapter 7


Consider following system.Design based on step response.

)()()()()(

tCxytDdtButAxtx

=++=&

221

1)()(−−

−

++=

−=

BsCACABsCBBAsICsG

Suppose:pp

* System is stable.* A, B, C and D are unknown.

* It is possible to apply individual input and derive output.p pp y p p

Two important matrices:

CB: Output sleep at 0+ 1]][[ −= uuuyyyCB &&&CB: Output sleep at 0 .

G(0)=-CA-1B: Steady state gain of system.

2121 ]...][...[= mm uuuyyyCB

1]][[)0( −= uuuyyyG


64

2121 ]...][...[)0( = mmssssss uuuyyyGIf [u1 u2 … um]=Im

]...[)0(],...[ 2121 mssssssm yyyGyyyCB == &&&

Chapter 7


D i b d t ( l t ) kf llhCB

Design based on step response. Design based on step response(regular systems). rankfullhas CB

1

000

⎥⎥⎤

⎢⎢⎡

kk

M

L

12

21 )(),0(,

000

0)(K −+++ ==

⎥⎥⎥

⎦⎢⎢⎢

⎣

= TT

m

XXXXGK

k

kCB ε

L

OMM

ML

⎦⎣

Design based on step response(irregular systems). rank fullnot has CB

⎤⎡⎤⎡ kk LL 0000

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

= ++ kk

GKk

k

GOMM

ML

L

OMM

ML

L

00

00

)0(,0

000

)0(K 2

1

22

1

1 εεα


65

⎥⎦

⎢⎣

⎥⎦

⎢⎣ mm kk LL 0000

Chapter 7


Design based on model. D i b d th d l ( l t ) kf llhCBDesign based on the model (regular systems). rankfullhas CB

Exercise: Design a PI controller for following system.

a) Derive the eigenvalues of the open loop system.⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

−−−−

0679500

6750029458140676.5715.62077.038.1

) g p p y

b) Derive the eigenvalues of the closed loop system.

c) Draw the step response of the closed loop systemxy

ux

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥⎥

⎦⎢⎢⎢

⎣

−+

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−

=

00101101

0136.1146.3136.10679.5

104.2343.1273.4048.0893.5654.6273.4067.1675.0029.45814.0

x


66

c) Draw the step response of the closed loop system.

d) If we have input multiplicative uncertainty check the robust margin.

Chapter 7


Design based on model. D i b d th d l (i l t ) kf llthCBDesign based on the model (irregular systems). rankfullnot has CB


) D i h i l f h l ⎤⎡⎤⎡ 00100a) Derive the eigenvalues of the open loop system.

b) Derive the eigenvalues of the closed loop system.xy

ux

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

011001

284.742.11143.0717.0

00

964.039.90121.10100

x


67

c) Draw the step response of the closed loop system.

d) If we have input multiplicative uncertainty check the robust margin.

⎦⎣ − 011

Chapter 7



ux ⎥⎥⎤

⎢⎢⎡

+⎥⎥⎤

⎢⎢⎡−= 11

00016113

x

⎥⎤

⎢⎡ −

⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣

111

10300

a) Derive a PI controller for system for a=0.

xa

y ⎥⎦

⎢⎣ −

=11

a) e ve a co o e o sys e o a 0.

b) Derive a PI controller for system for a=1.


68

c) Compare the result of part a and part b and find the transmission zeros and analysis the results.

Chapter 7








69

Chapter 7


Nyquist-array methods ( Compensator structure )

We shall again assume that the plant's transfer function is square.

Suppose that a compensator is rational, invertible, and has all its poles and zeros in theleft half-plane (including the origin).p ( g g )Theorem

Let K(s) be square, rational and invertible, and have all its poles and zeros in the openl f h lf l Thleft half-plane. Then

abccba KsKsKsKsKsKKsK ′′′== )()()( and )()()(

KK ′and are perm tation matrices in other ords reorder the o tp ts or inp tsaa KK and are permutation matrices, in other words, reorder the outputs or inputs,

)( and )( sKsK bb ′ are products of elementary matrices

di l i i h i l d i h l d


70

)( and )( sKsK cc ′ are diagonal matrices, with rational, non-zero, and with poles and zeros in the open left half-plane only.

Chapter 7


This theorem implies that compensator design can be split into two stages.

• In the first stage are used to make the return ratio diagonally dominant.)( and sKK ba

• The second stage begins when dominance has been achieved and consists of d i i f SISO f h ldesigning a set of separate SISO compensators one for each loop.

At this stage no attention is paid to the remaining interactions in the system,At this stage no attention is paid to the remaining interactions in the system,

Except that Gershgorin bands of the return ratio replace SISO Nyquist loci.


71

Chapter 7


Design can be pursued using either)(GK)(sK)(sK

• Direct Nyquist arrays (DNA)

)()()()(0 sKsKKsGsQ cba=

)(sGaK)(sKb)(sKc

Since Kc(s) post-multiplies the other transfer functions, and is diagonal, hencecolumn dominance of the G(s)KaKb(s) is not destroyed by Kc(s).

Therefore when working with DNA it is usual to try to achieve column dominance

• Inverse Nyquist arrays (INA)

)()()()( 11111 GKKKQ −−−−−

Therefore, when working with DNA it is usual to try to achieve column dominance.

)(1 sG− )(1 sKb− )(1 sKc

−1−aK

)()()()( 111110 sGKsKsKsQ abc

−−−− =

Since pre-multiplies the other transfer functions, and is diagonal,hence row dominance of the is not destroyed by

)(1 sKc−

)()( 111 sGKsK −−− )(1 sK −


72

hence row dominance of the is not destroyed by .)()( sGKsK ab )(sKc

Therefore, when working with INA it is usual to try to achieve row dominance.

Chapter 7Nyquist-array methods The inverse Nyquist-array (INA) method

FsKsK dc )()( = )()()()( sKsKKsGsQ dba= FsQsQ )()(0 =

( ) FsHFsQFsQIsT )()()()( 1 =+= − [ ] (I) )()()( 1 sQFsQIsH −+=

)()(ˆ)()(ˆ 11 sQsQsHsH −−

FsQsH += )(ˆ)(ˆ

)()()()( sQsQsHsH ==

The simplicity of this relationship, as compared (I), underlies the use of INA

There is the Ostrowski bands, which allows the designer to take into account the ff f i i b l

The motivation for using inverse rather than direct arrays is that, in i i l l h ll h l d l b h i b d i d

effects of interactions between loops.


73

principle at least, they allow the closed-loop behavior to be determined more precisely from open-loop information.

Chapter 7Nyquist-array methods The direct Nyquist-array (DNA) method

The DNA has several advantages.

1- The designed compensator does not need to be inverted, and one consequence of thisis that any structure imposed by the designer, such as setting certain elements to zero,is retained.

It also gives the designer more freedom in the choice of compensator, since its inverse does not need to be realizable

2- The plant need not be square, since it need not have an inverse.

3- The inverse compensator designed by the INA method has right half-plane zeros, particularly if it is designed semi-automatically by one of the methods described in the next section.

However there is no tool such as the Ostrowski bands which allows the designer


74

However, there is no tool, such as the Ostrowski bands, which allows the designer to take into account the effects of interactions between loops.

Chapter 7


Achieving diagonal dominance

• Cut and try

• Perron-Frobenius theory

It is based on some straightforward transformation.

Perron-Frobenius theory allows us to check whether a plant can be made diagonally dominant by input and output scaling

• Pseudo-diagonalization

A way of automatically generating compensators with a more general


75

A way of automatically generating compensators with a more general structure.

Chapter 7


Achieving diagonal dominance ( Cut and try )It is sometimes possible to examine the display of a Nyquist array and observe thatIt is sometimes possible to examine the display of a Nyquist array and observe that some straightforward transformation will achieve diagonal dominance.

⎥⎤

⎢⎡ +− 1ssLet

⎥⎥⎥

⎦⎢⎢⎢

⎣ +−

++

++=

23232

2323)(

ss

ss

sssG

Its inverse is

⎥⎦

⎤⎢⎣

⎡+

+=

ssss

sG2

1)(ˆ

This is clearly neither row dominant nor column dominant anywhere on the Nyquist y y yqcontour.

⎥⎦

⎤⎢⎣

⎡+

+⎥⎦

⎤⎢⎣

⎡=

ssss

sGKa 21

0110

)(ˆˆ⎥⎦

⎤⎢⎣

⎡=

0110ˆ

aK ⎥⎦

⎤⎢⎣

⎡+

+=

12

ssss


76

⎦⎣ +⎦⎣ ss 201⎦⎣ 01 ⎦⎣ +1ss

This is clearly row dominant and column dominant anywhere on the Nyquist contour.

Chapter 7


Achieving diagonal dominance ( Cut and try )⎤⎡ +⎤⎡ ss 110⎤⎡ 10 ⎤⎡ + 2 ss⎥⎦

⎤⎢⎣

⎡+

+⎥⎦

⎤⎢⎣

⎡=

ssss

sGKa 21

0110

)(ˆˆ⎥⎦

⎤⎢⎣

⎡=

0110ˆ

aK ⎥⎦

⎤⎢⎣

⎡+

+=

12

ssss

So that the compensated plant is represented by⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

+−

+−

++

=

232

23

23231

)(

ss

ss

ss

ss

KsG a⎥⎦

⎤⎢⎣

⎡=

0110

aK

⎦⎣ ++ 2323 ssPhysically, this corresponds to nothing more than a re-ordering of the inputs (or a re-assignment of inputs to outputs.

In this artificial example both the direct and the inverse array have been made equally dominant, and the compensation required in each case is the same.

Thi i ll I i i ibl f i l k h


77

This is not usually true. It is quite possible for a particular compensator to make the direct array dominant, but not the inverse array, and vice versa.

Chapter 7


Achieving diagonal dominance ( Cut and try )The ‘elementary matrices’ are supposed to represent simple transformations devisedy pp p pby the designer.

⎥⎦

⎤⎢⎣

⎡ −=

1021

)(s

sKb

In practice, it is rarely possible to make much progress by relying on being able to findsuch transformations by ad hoc means.

An alternative strategy is to try to diagonalize a system at one frequency and hopeAn alternative strategy is to try to diagonalize a system at one frequency, and hope that the effect will be sufficiently beneficial over a wide range of frequencies.

If the system has no poles at the origin, then K=G-1(s) is a realizable (because constant)compensator.

Fortunately, we already have an algorithm (ALIGN algorithm) for performing the required approximation on other frequencies


78

required approximation on other frequencies.

Chapter 7


Achieving diagonal dominance ( Cut and try )

A l t h th i t f f ti ⎤⎡ ++ 22 ssA plant has the inverse transfer function

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+++

+

++++

=

2115

3101

2

)(ˆ2

ssss

ss

ss

sG

⎥⎦

⎤⎢⎣

⎡=

5.0132

)0(G

It is not diagonally dominant at low frequencies, since

The compensator⎥⎦

⎤⎢⎣

⎡−

−==

15.05.125.0

)0(ˆ GKb

The compensator

IGK =)0(ˆˆgives IGKb =)0(

⎤⎡ 29

which is obviously diagonally dominant, and it gives column, but not row, dominance at high frequencies, since


79∞→

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−= sasssGKb

44

18

1429

)(ˆˆ

Chapter 7


Achieving diagonal dominance ( Perron-Frobenius theory )

Perron Frobenius theory allows us to check whether a plant can be made diagonallyPerron-Frobenius theory allows us to check whether a plant can be made diagonally dominant by input and output scaling.

Theorem (Mees, 1981):

If G is square and primitive, then there exist a diagonal matrix S such that

is diagonally dominate if and only if

1~ −= SGSG )(sG1−S S

is diagonally dominate, if and only if

( ) (I) 21 <−diagp GGλ

If (I) satisfied then X which achieve diagonal dominance is

{ }nsssdiagS ,,..., 21=


80

Where λp Perron-Frobenius eigenvalue and (s1, s2, … ,sn)T is left Perron-Frobenius

eigenvector of .1−diagGG

Chapter 7


Achieving diagonal dominance ( Perron-Frobenius theory )Output scaling is physically impossible since the meaningful plant outputsOutput scaling is physically impossible, since the meaningful plant outputs (which are variables such as velocity, or thickness of steel strip) cannot be affected by mathematical operations.

But we can useBut we can use

But we must be wary of falling into the trap of believing that this return ratio tells usBut we must be wary of falling into the trap of believing that this return ratio tells us anything about interaction at the plant output.

The output variables may be interacting with each other to a considerable extent, and


81

this interaction may be being hidden by the measurement scaling S.

Chapter 7


Achieving diagonal dominance ( Perron-Frobenius theory )Fortunately the Perron-Frobenius theory gives useful results even if only pre-Fortunately, the Perron Frobenius theory gives useful results, even if only precompensation (input scaling) is allowed.

Theorem :

If G is square and primitive, then there exist a diagonal matrix K(s) such that

i di ll d i t if d l if

)()(~ sKsGG = )(sG)(sK

is diagonally row dominate, if and only if

( )( ) (I) 21 <− GGabs diagpλ

If (I) satisfied then K(s) which achieve diagonal dominance is

{ })(,,...)(,)()( 21 skskskdiagsK n=


82

Where λp Perron-Frobenius eigenvalue and (k1(s), k2(s), … ,kn(s))T is right Perron-

Frobenius eigenvector of ( ).1 GGabs diag−

Chapter 7Nyquist-array methods Achieving diagonal dominance ( Perron-Frobenius theory )Achieving diagonal dominance ( Perron Frobenius theory )

Everything else remains the same, except that if a dynamic compensator is used then the elements of compensator must be chosen to have realizable inverses.

A drawback of using diagonal compensatorsg g p

• Row dominance when using the DNA method

• Column dominance when using the INA method


83

Co u do a ce w e us g t e N et od

This is exactly the opposite of what we would like, since further diagonal compensation,for the purpose of ‘loop shaping’, may destroy the dominance which has been achieved.

Chapter 7Nyquist-array methods Achieving diagonal dominance ( Perron-Frobenius theory )

Example 7-5 Consider the transfer function

⎤⎡ + 14s

Achieving diagonal dominance ( Perron Frobenius theory )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

++++

=

122

100101

151

)5)(1(4

)(

2 ssss

ssss

sG

⎦⎣

3

3.5

4The Perron-Frobenius eigenvalue of

( ))(ˆ)(ˆ 1 ωω jGjGabs −

2

2.5

3

ain

(dB

)

( ))()( ωω jGjGabs diag

is:

1

1.5

Ga

Its value is smaller than 4 dB ( i.e. λp<2 )

at all frequencies, so it is possible to obtain

l d i b i di l


8410-2 10-1 100 101 1020

0.5

Frequency (rad/sec)

column dominance by using a diagonal

compensator.

Chapter 7Nyquist-array methods Achieving diagonal dominance ( Perron-Frobenius theory )

The second element of the Perron-Frobenius left eigenvector, when the first element is fixed at 1 is:

g g ( y )

30 31.0898.4438.0)(ˆ

2 ++

=s

ssk

20

25This matches the variation of the eigenvector very well.

5

10

15

Gai

n (d

B) The inverse compensator

{ })(ˆ1)(ˆ2 skdiagsK =

-5

0

{ })(,1)( 2 skdiagsK

Leads to column dominant.


8510-2 10-1 100 101 102

-10

Frequency (rad/sec)

Chapter 7Nyquist-array methods Achieving diagonal dominance ( Perron-Frobenius theory )g g ( y )

GK ˆˆNvquist array of

0

10

20

m 0

10

20

m

20 10 0 10 20-20

-10

0Im

20 10 0 10 20-20

-10

0Im

-20 -10 0 10 20Re

-20 -10 0 10 20Re

20 20

-10

0

10

Im

-10

0

10

Im


86-20 -10 0 10 20-20

-10

Re-20 -10 0 10 20

-20

-10

Re

Chapter 7

Nyquist-array methods Achieving diagonal dominance ( Pseudo-diagonalization )Achieving diagonal dominance ( Pseudo diagonalization )

This can be done by choosing some measure of diagonal dominance, some compensator structure, and then optimizing the measure of dominance over this structure.

We shall use the term pseudo-diagonalization for any such scheme, although the term is often reserved for the particular scheme proposed by Hawkins (1972).

H ki d th t i t b d b t hi th d b li dHawkins assumed that inverse arrays are to be used, but his method can be appliedequally well to direct arrays.

If we have a plant G(s) and a constant compensator K, with Q(s)=G(s)K, then individualelements of Q are given by

jTiij kjgjq )()( ωω =

jth column of K

h

∑ ∑∑ ∑= ≠= ≠ ⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

=N

k jijk

Tik

N

k jikijkj kjgpjqpJ

1

2

1

2)()( ωω

Hawkins proposed to minimize ith row of G


87

= ≠= ≠ ⎭⎩⎭⎩ k jik ji 11

subject to the constraint1=jk Otherwise it leads to kj=0

Chapter 7


∑ ∑∑ ∑⎫⎧⎫⎧ N

TN 22

Hawkins proposed to minimize

∑ ∑∑ ∑= ≠= ≠ ⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

=k ji

jkTik

k jikijkj kjgpjqpJ

1

2

1

2)()( ωω

subject to the constraint1k 1=jk

Hawkins method may not prevent from being made small, as well as the )( kjj jq ω

off-diagonal elements, so that diagonal dominance may not be obtained.

Suppose, however

∑ ∑⎫⎧N 2 2

∑ ∑⎫⎧ T

∑

∑ ∑= ≠ ⎭

⎬⎫

⎩⎨⎧

= N

kjjk

k jikijk

j

jqp

jqpJ

2

1

2

)(

)(

ω

ω

2

2

)(

)(

∑

∑ ∑⎭⎬⎫

⎩⎨⎧

=≠

kjk

Tjk

k jijk

Tik

jkjgp

kjgpJ

ω

ω


88

=k 1 k

The solution is given by the multi frequency ALIGN algorithm. (Maciejowski (1989)).

Chapter 7


Ford and Daly (1979) have extended this approach to dynamic compensators:

[ ])(),...,(),...,()( 1 sksksksK mj=

βskksk ++=)( ⎥⎥⎤

⎢⎢⎡ jk0

Vectorββ skksk jjj ++= ...)( 0

⎥⎥⎥

⎦⎢⎢⎢

⎣

=

j

j

kβ

η...

( ) ( )[ ])(....)()()( ωωωωωωγ β jgjjgjjgj Ti

Ti

Ti

Ti =

ith row of G

∑ ∑⎭⎬⎫

⎩⎨⎧

≠k jijk

Tik jp

J

2)( ηωγijth element of G(s)K(s) is :

ith row of G


89j

Tiij jjq ηωγω )()( = ∑

⎭⎩=

kjk

Tjk

jjp

J 2)( ηωγ

Chapter 7


A realizable compensator is therefore obtained by dividing Kj(s) by any polynomial

of degree β (or greater)of degree β (or greater).

Pseudo-diagonalization can be applied to either direct or inverse Nyquist arrays.

B t ti l diffi lt i if d i t i f d f iBut a practical difficulty arises if a dynamic compensator is found for an inverse array:

its inverse needs to be realizable.

P d di li i h d ib d i h f N i h dPseudo-diagonalization have described in the context of Nyquist array methods,

it can clearly be applied whenever approximate inverses of frequency responses are

required

In particular, it can be viewed as an extension of the ALIGN algorithm, and can

therefore be applied in the context of the characteristic locus method

required.


90

therefore be applied in the context of the characteristic-locus method.

Chapter 7

Design example

Nyquist-array methods Design example

Consider the aircraft model AIRC described in the following state-space model.

BuAxx +&

⎤⎡⎤⎡ 0000000101320100

DuCxyBuAxx

+=+=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡−

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

= 00000000.11200.0000

,00000.10000705.001712.00538.000000.101320.100

BA

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−

0732.005750.16650.104190.4

6859.00532.102909.000130.18556.000485.00

⎤⎡⎤⎡ 00000001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000000000

,001000001000001

DC


91the model has three inputs, three outputs and five states.

Chapter 7

Design example


THE SPECIFICATION

• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.

Littl i t ti b t t t• Little interaction between outputs.

G d d i f d d i h f f• Good damping of step responses and zero steady-state error in the face of step

demands or disturbances.

• We assume a one-degree-of-freedom control structure.


92

Chapter 7

Design example


PROPERTIES OF THE PLANT

• The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very

severe interaction between outputs.

• The poles of the plant (eigenvalues of A) are

jj 1826.00176.0,03.178.0,0 ±−±− jj 1826.00176.0,03.178.0,0 ±±

so the system is stable (but not asymptotically stable).

Th hi l h fi i d d li i i• Thus this plant has no finite zeros, and we do not expect any limitations on

performance to be imposed by zeros. since

the n mber of finite eros of the plant can be at most


93

the number of finite zeros of the plant can be at most

)(2 CBrankmn +− 013.25 =+−=

Chapter 7

Nyquist-array methods Design example Obtaining column dominanceg p g

)(

)( ω

j

jgji

ij∑≠column dominance measure

100

150

)( ωjg jj

column 1

50

100

ain (dB)

-50

0

Ga

10-3 10-2 10-1 100 101 102-100

Frequency (rad/sec)


94

Column dominances of the plant: column 1 (solid curve), column 2 (dashed curve)and column 3 (dotted curve).

Chapter 7

Nyquist-array methods Design example Obtaining column dominance

We use pseudo-diagonalization (the algorithm of Ford and Daly (1979)) to obtain column dominance, and apply it to one column at a time.

g p g

For this purpose, and in the rest of this design example, all frequency responses are evaluated at a set of 50 frequency points, equally spaced on a logarithmic scale between 0 001 and 100 rad/sec except for a greater density of points in the regionbetween 0.001 and 100 rad/sec, except for a greater density of points in the region of 0.18 rad/sec .

Applying pseudo-diagonahzation to the first column and optimizing over constant pp y g p g p gcompensator elements only, with uniform weighting on all frequencies, was not successful.

Di l d i i d t l f i h it t d d b tDiagonal dominance was improved at low frequencies, where it was not needed, butremained almost unchanged at high frequencies.

Frequencies above 0.1rad/sec were therefore weighted 10 times as much as lower


95

q gfrequencies, but dominance was still not achieved above l rad/sec.

Chapter 7


This produced virtually perfect diagonal dominance, with a dominance measure less than 10-6 at all frequencies.

g p g

q

⎥⎥⎥⎤

⎢⎢⎢⎡

×+×−×−×−

= −−

−−

33

54

43

1

10311100541011.31083.11078.91052.1

)( ss

sk⎥⎦⎢⎣ ×−×− −− 33 1031.11005.4 s

The lower degree of dominance may lead a simpler compensator structure. Only one element of the column may need to be dynamic, for example.element of the column may need to be dynamic, for example.

For the second column, it was again necessary to optimize over a first order dynamicstructure,

⎥⎤

⎢⎡ ×− −21022.10s

Again, almost perfect dominance was obtained everywhere.


96⎥⎥⎥

⎦⎢⎢⎢

⎣ ×−×−×−=

−

−−

2

322

1040.301050.41074.5)(

sssk

Chapter 7


The third column proved to be the most difficult to compensate.

g p g

Optimizing over a first-order structure with high weighting on low frequencies lead to:


97Reducing the weighting on the low frequencies gave no benefit at higher frequencies.

Chapter 7


So optimization over a second-order structure was attempted, with uniform weighting at all frequencies. 2)( skskksk ++

g p g

2313033 )( skskksk ++=

This achieved dominance, exceptin a narrow range of frequenciesg qnear of frequencies near 0.2 rad/s.

The weighting on frequenciesThe weighting on frequencies

between 0.1 and 1 rad/sec was

therefore increased to 10 times

as much as on other frequencies.


98

Chapter 7


⎤⎡ 1222

The design obtained for the third column of the compensator is

g p g

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

×−×−××−×−××−×−×

=−−−

−−−

−−−

1221

2223

1222

3

1052.11008.31086.21034.31058.21007.51014.11078.91021.4

)(ssssss

sk

⎦⎣

[ ])(),(),()( 321 sksksksKb =0

20

[ ])(),(),()( 321b

-60

-40

-20

ain

(dB

)

120

-100

-80

Ga


9910-3 10-2 10-1 100 101 102-140

-120

Frequency (rad/sec)

Chapter 7

Nyquist-array methods Design example Loop compensationDesign example Loop compensation

The first two columns are so dominant that the Gershgorin circles, superimposed on the(1,1) and (2,2) loci, cannot be distinguished from the loci themselves, and the C i f h fi l i diff f i f SISOCompensation of the first two loops is no different from compensation of a SISO system.

80

20

40

60

72)log(20 =k

⎥⎤

⎢⎡ 003980

-40

-20

0

Gai

n (d

B)

⎥⎥⎥

⎦⎢⎢⎢

⎣

=?000?0)(sKc

-100

-80

-60


100

10-3 10-2 10-1 100 101 102-100

Frequency (rad/sec)

Response of (1, l) element with and without compensation.

Chapter 7


The frequency response of the (2,2) element, shown as a Nyquist plot in Figure, is essentially constant at -25dB (=0.0562) at all frequencies.

⎥⎥⎥⎤

⎢⎢⎢⎡

−= 01740

003980

)(sK

⎥⎥

⎦⎢⎢

⎣

=

?00

00)(s

sKc


101Response of (2,2) element before compensation.

Chapter 7


The response of the (3,3) element is shown as a Nyquist plot with its Gershgorin band (computed column-wise), and its magnitude is shown in Bode form in Figures

-8

-7

-10

-9

-8

B)

-12

-11Gai

n (d

B

10-3

10-2

10-1

100

101

102

-14

-13


102Response of (3, 3) element before compensation.

10 10 10 10 10 10Frequency (rad/sec)

Gain of (3, 3) element before compensation.

Chapter 7


Suitable compensation of this element is obtained by first changing its sign, so that the locus starts and ends on the positive real axis, and then inserting an integrator with enough gain to add about l0 dB at l0rad/sec.

80

⎥⎥⎥⎤

⎢⎢⎢⎡

K 01740

003980

)(40

60

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣−

−=

s

ssKc

6.3100

00)(

0

20G

ain

(dB

)

10-3 10-2 10-1 100 101 102-40

-20

Frequency (rad/sec)


103

Frequency (rad/sec)

Gain of (3, 3) element after compensation.

Chapter 7


For this element the thickness of the Gershgorin band is significant, so we should checkthat the band does not overlap 1 in order to be sure that our inference of closed-loop stability is correctstability is correct.


104Response of (3,3) element after compensation,

with Gershgorin band, on a Nichols chart.

Chapter 7


The compensation is not yet finished, because the product Kb(s)Kc(s) is not realizable: each element in the first and third columns has one more zero than poles.p

REATTZATTON OF THE COMPENSATOR


105

Chapter 7

Nyquist-array methods Design example Analysis of the designDesign example Analysis of the design

0

-1 5

-1

-0.5

-2.5

-2

1.5

Im

M=3 dB-circle

-4

-3.5

-3

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-5

-4.5

Re


106Characteristic loci, with 3 dB M-circle.

Chapter 7


5

-5

0

5

-15

-10

in (d

B)

-30

-25

-20Ga

10-3 10-2 10-1 100 101 102-40

-35

F ( d/ )


107

Frequency (rad/sec)

Largest and smallest closed-loop singular values (principal gains).

Chapter 7


0.8

1

0.4

0.6

Ampl

itude

0.2

0 0.5 1 1.5 2 2.5 3-0.2

0

Time (sec)


108

Time (sec)

Closed-loop step responses to step demand on output 1 (solidcurves), output 2 (dashed curves) and output 3 (dash-dotted curves).

Chapter 7


0 5Impulse Response

0.4

0.5

0.2

0.3

plitu

de

0

0.1Am

0 1 2 3 4 5 6 7 8 9 10-0.2

-0.1

Ti ( )


109

Time (sec)

Response to impulse disturbance on plant input 1.

Chapter 7

References

� Skogestad Sigurd and Postlethwaite Ian. (2005) Multivariable Feedback Control: England, John Wiley & Sons, Ltd.

� Maciejowski J.M. (1989). Multivariable Feedback Design: Adison-lWesley.

� Mayne D.Q.(1973). Sequential design of linear multivariable systems: P di f th I tit t f El t i l E i l26 568 572Proceeding of the Institute of Electrical Engineers.l26, 568-572.

� Hung S. and MacFarlane A.G.J. (1982). Multivariable Feedback: A Quasi classical Approach Lecture Notes in Control and InformationQuasi-classical Approach, Lecture Notes in Control and Information Sciences, Vol.40. Berlin: Springer-Verlag.

� Ali Khaki Sedigh (2011) Analysis and Design of Multivariable


110

� Ali Khaki Sedigh (2011). Analysis and Design of Multivariable Control Systems

Documents

Multivariable Control Systemsprofsite.um.ac.ir/~karimpor/multi/Multivariable_lec7.pdf · Example 7-1 Continue ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ − + − + = = s s s s s Q s G s Ka 1 3 3 (