Multivariable Control Systems - Personal Datakarimpor.profcms.um.ac.ir/imagesm/354/stories/mul_con/... · References are appeared in the last slide. ... Characteristic-locus method

Dr. Ali Karimpour May 2017

Lecture 8

Multivariable Control

Systems

Ali Karimpour

Associate Professor

Ferdowsi University of Mashhad

Lecture 8

References are appeared in the last slide.


Lecture 8

2

Multivariable Control System Design(Sequential loop closing, Characteristic-locus method and PI controller)

Topics to be covered include:

• Sequential loop closing

• The characteristic-locus method

• PI control for MIMO systems


Remark: Evaluation of this lecture can be done before end of class under special circumstances.


Lecture 8

3

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.

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

designed for a second pair of variables and so on.

How input and output variables should be paired?

loop-assignment problem or input-output pairing


Lecture 8

4

• The transfer function matrix of

such a controller is diagonal.

• It also has the advantage that it

can be implemented by closing

one loop at a time.

Benefits:

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

to use high loop gains

• Control difficulty in the case of element zeros.



Lecture 8

5

A more sophisticated version of sequential loop closing is called

sequential return-difference method

A cross-coupling stage of compensations should be introduced

This stage should consist of either a constant-gain matrix

or a sequence of elementary operations. )()(

1)( sU

sdsCp



Lecture 8

6

Example 8-1:

ss

ss

ssG

123

11

)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 transfer

function ‘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

12

01pC

ss

ss

sCsGsQ p

13

1

3

1

)1(

1)()(

2

We see that no right half-plane zero ‘appears’ when a SISO compensator is being

designed for the first loop.



Lecture 8

7

ss

ss

sKsGsQ a

13

1

3

1

)1(

1)()(

2

Once the first loop has been

closed by

The transfer function ‘seen’ in

the second loop is

)(1 sk

222

1

22)1(

3

1

)()1()1(

1)(

s

s

shs

s

s

ssq )(1/

3

11

)()( where

1

2

1

skss

sksh

Example 8-1 Continue:



Lecture 8

8

222

1

22)1(

3

1

)()1()1(

1)(

s

s

shs

s

s

ssq )(1/

3

11

)()( where

1

2

1

skss

sksh

Now, if we assume high gain in the first loop

s

ssk

3

1

)1()(

2

1s

ssh

3

1

)1()(

2

and hence

)13)(1(

1)(1

22

ss

sq

so that no right half-plane zero is seen when the compensator for the second loop

is designed.

Example 8-1 Continue:



Lecture 8

9

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

loop compensation easier

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

first 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,

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

from each other.



Lecture 8

10

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 of

compensation can be chosen to be

1 CBCp

sass

CBsG )( Since sas

s

ICsG p)( so

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

However, an alternative choice, such as

)(1

bp jGC )(C or 1

bbp jGj



Lecture 8

11







Lecture 8

12

The characteristic-locus method

Approximate commutative compensators

• 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

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

)(,.....,)(,)()( 21 sssdiags m

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

Commutative compensators

)()()()( sGsKsKsG

)()()()( sGsKsKsG


Lecture 8

13

iondecomposit spectral is )()()()( 1 sWssWsG

Let following structure as controller

)()()()( 1 sWsMsWsK where

)(,...,)(,)()( 21 sssdiagsM m

then the return ration (loop transfer function) is

)()()()()()()()()()()( 11 sGsKsWsNsWsWsMssWsKsG

)(,.....,)(,)()( 21 svsvsvdiagsN mwhere

)()()( sssv iii

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

A compensator having this structure is called a commutative compensator.



Lecture 8

14

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

loci of the plant,

miji ,.....,2,1:)(

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

corresponding toW(s)sMsW and )(,)(1

Then design a compensator

using the well-established single-loop techniques.

)( ji for each )( ji

There are some main drawback in designing commutative controller

1- Robust stability may not be derived.

2- It is not realizable in general case.



Lecture 8

15


Example 8-2: The transfer-function matrix

25042

56472

21

1)(

ss

ss

sssG


has the following spectral decomposition

76

87

2

20

01

1

76

87)()()()( 1

s

ssWssWsG

and so W(s) and Λ(s) is

2

20

01

1

)(76

87)(

s

sssW


Lecture 8

16


Choose controller as


)k(-1 constant aby stabilized is 1

1 clear that isIt 11

ks

)k(-1 constant aby stabilized is 2

2 clear that also isIt 22

ks

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

in multivariable system?

76

87

0

0

76

87)()()()(

2

11

k

ksWsMsWsK

2

20

01

1

)(

s

ss

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

10

01

76

87

10

01

76

87)(sK


Lecture 8

17



1)( GKIKTu

Suppose that we use above controller in our system but there is some

uncertainty in inputs so let the value of controller as:

10

01)(sK



25042

56472

21

1)(

ss

ss

sssG

b

asK

0

0)(

To check the stability we need to check the stability of


Lecture 8

18



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(42

56472)2)(1(

)2)(1(

1)(

bsssas

bsasss

ssGKI

Closed loop characteristic equation is:

0)2222()47503()2)(1( 2 abbaabssss

For stability we need:

02222&047503 abbaab


Lecture 8

19




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.

So robust stability may not be derived.

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



Lecture 8

Perturbed PlantUncertainty M in MΔ-structure

20

structureM

Suitable for robust stability analysis

Additive uncertainty 2

1

1 )( wGKIKwM 12 wwGGp

Multiplicative input uncertainty2

1

1 )( GwGKIKwM )( 12 wwIGGp

Multiplicative output uncertainty2

1

1 )( wGKIGKwM GwwIGp )( 12

Inverse additive uncertainty 2

1

1 )( wKGIGwM 1

12

GwwIGGp

Inverse multiplicative input uncertainty2

1

1 )( wKGIwM 1

12

wwIGGp

Inverse multiplicative output uncertainty2

1

1 )( wGKIwM GwwIGp

1

12




Lecture 8

21

Example 8-3: The transfer-function matrix

2

1

2

21

2

1

1

)(

ss

sssG

33481632)2)(1(2

1)(,)( 2

21

sssss

ss

has characteristic functions

3348161

84

)(

)(

22

1

ss

s

sw

sw

i

i

The characteristic directions are given by

and so W(s) is

11

3348161

84

3348161

84

)(

22 ss

s

ss

s

sW




Lecture 8

22

A practical alternative is to give the compensator the structure

)()()()( sBsMsAsK )()()()( 1 sWsMsWsK

)()( sWsA )()( 1 sWsB where

are chosen to be realizable, and such that

Whichever approximation technique is chosen, we obtain an

approximate commutative compensator




Lecture 8

23

Design procedure

• Manipulating the characteristic loci

• Stable design by satisfying the generalized Nyquist stability theorem

• Large magnitudes at low frequencies

• Stay outside the 2M

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

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

m

i

T

iiii ssvswsWsvdiagsWsKsG1

1 )()()()()()()()(Let

)()(1

1)()(

)(1

1)()(

1

1 ssv

swsWsv

diagsWsS T

i

i

m

i

i

i

)()(1

)()()(

)(1

)()()(

1

1 ssv

svswsW

sv

svdiagsWsT T

i

i

im

i

i

i

i



Lecture 8

24

Design procedure

)()(1

)()()(

)(1

)()()(

1

1 ssv

svswsW

sv

svdiagsWsT T

i

i

im

i

i

i

i



Lecture 8

25

Design procedure

)()()()( 11 swsrssrT

)()()()( 22 swsrssrT

.......

)()()()( swsrssr m

T

m

)(sy



Lecture 8

26

Design procedure

The performance of a feedback system depends essentially on the singular values

(principal gains) of functions such as S(s) and T(s).

)(1

1

svi )(1

)(

sv

sv

i

i

These transfer functions have eigenvalue functions

We know that, for any matrix X,

)()()( XXX

If the characteristic loci of GK

have low magnitudes

The sensitivity may be large,

in some signal directions

If some characteristic locus

penetrates the circle2M

At least one principal gain of T(s)

will exhibit a resonance peak, of

magnitude greater than 2



Lecture 8

27

Design procedure

If the eigenvectors are nearly

orthogonal to each other

Magnitudes of the smallest and largest

eigenvalues are close to the smallest

and largest singular values

Good system performance

This situation happens if the return ratio has low skewness

)( sviShaping

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 straightforward

and 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

by methods more revealing than examination of the characteristic loci.



Lecture 8

28

Design procedure

The only thing that can be done is to insert a series compensator whose main purpose is

to obtain a ‘decoupled’ (that is, diagonal) return ratio of -G(s)K(s) at (some of) these

frequencies.

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. .


)( Let 1

bh jGK

1)( seigenvalue hb KjG

It may be advantageous to adjust the signs of the columns of Kh at this stage.

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.

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


Lecture 8

29

Design procedure of characteristic-locus method

1- Compute a real )(1

bh jGK

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

for the compensated plant G(s)Kh, such that asIjKm )(

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

steady-state errors), design an approximate commutative controller Kl(s) at low

frequency, for the compensated plant G(s)KhKm(s), such that asIjK l )(

(the red blocks are some attempt to ensure that the decoupling effected by Kh is not

disturbed too much).

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

high, medium and low frequency



Lecture 8

30

Design example

0732.005750.1

6650.104190.4

000

00000.11200.0

000

,

6859.00532.102909.00

0130.18556.000485.00

00000.1000

0705.001712.00538.00

0000.101320.100

BA

000

000

000

,

00100

00010

00001

DC

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

Example 8-4: Consider the aircraft model AIRC described in the following

state-space model.

DuCxy

BuAxx



Lecture 8

31

Design example

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

THE SPECIFICATION

• Little interaction between outputs.

• 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.



Lecture 8

32

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

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

• Thus this plant has no finite zeros, and we do not expect any limitations on

performance to be imposed by zeros. since

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

)(2 CBrankmn 013.25



Lecture 8

33

Negative feedback were

applied around the plant

111 NPZ

111 Z

21 Z

Closed loop would be unstable.

(Two RHP poles)

2M

Characteristic loci of G , with logarithmic calibration

02.173.069.159.090.0:arevaluesEigen 53,41,2 jj



Lecture 8

34

Design example

ALIGNMEIVT AT 10 rad/sec

378.690065.044.189

5376.09984.95375.8

669.30036.0535.71

))10(( jGalignK h

jjj

jjj

jjj

KjG h

086.09962.00005.00000.0100.0009.0

0003.00008.0010.1005.00004.0063.0

009.00005.0003.00001.0068.0983.0

)10(

From input 1 to output 2 and 3, and that these interactions have been reduced to 10% or

less (at that frequency).

378.690065.044.189

5376.09984.95375.8

669.30036.0535.71

hK



Lecture 8

35

Characteristic loci of GKh , with logarithmic calibration

)10,2

3

2

1:( jes

111 NPZ

111 Z

01 Z

Negative feedback were

applied around the plantClosed loop would be stable.

(No RHP poles)

05.1099.953.096.924.0:arevaluesEigen 53,41,2 jj



Lecture 8

36

Design example

APPROXIMATE COMMUTATIVE COMPENSATOR

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

-20

0

20

40

60

80

100

120

Phase (deg)

Gain

(dB

)

Characteristic loci of GKh , on Nichols chart

As might be expected from

hKjG )10(

however, two of the loci pass

extremely close to -1

1)10( WWKjG h

0009.09526.08829.0

9999.00076.00124.0

0015.02049.03483.0

)( 1WalignA

0024.00001.10391.0

6633.00026.07310.1

3728.00031.08165.1

)(WalignB



Lecture 8

37


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

100

010933.0

2175.00933.00

0010933.0

2175.00933.0

)(s

ss

s

sM BsAMsKm )()(

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

6731.00012.00763.0

0025.09999.00265.0

0071.00023.06811.0

)10()10( jKKjGabs mh

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



Lecture 8

38


Characteristic loci of GKhKm , on Nichols chart

LOW FREQUENCY PROBLEM



Lecture 8

39

LOW FREQUENCY COMPENSATION

)()()( XXX

10))0()0((

mh KKG

Which will certainly not achieve the objective of zero steady-state error in the face of

step disturbances.

IsT

sTsK l

1)(



Lecture 8

40


Characteristic loci of GKhKm

on Nichols chart

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

0

50

100

150

Phase (deg)

Gain

(dB

)

Characteristic loci of GKhKmKl

on Nichols chart



Lecture 8

41

Design example

10-2

10-1

100

101

102

-50

0

50

100

150

Frequency ( rad/sec )

Gain

(dB

)

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



Lecture 8

42

Design example

10-1

100

101

102

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Frequency (rad/sec)

Gain

(dB

)

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

Bandwidth is ok



Lecture 8

43

Design example

0 0.5 1 1.5 2 2.5 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Step Response

Time (sec)

Am

plit

ude

Closed-loop step responses to step demand on output 1 (solid curves),

output 2 (dashed curves) and output 3 (dotted curves).



Lecture 8

44

Design example

REALIZATION OF THE COMPENSATOR

The 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).

5.000

05.00

005.0

653.20106.0924.6

491.10126.0266.7

,

00000

00000

00000

653.20106.0924.672.100

491.10126.0266.7072.10

KK BA

52.651204.09.183

4372.0996.9369.8

091.30129.045.70

,

52.651204.09.1832.578.266

4372.0996.9369.8752.2489.7

091.30129.045.704.3303.59

KK DC



Lecture 8

45






Lecture 8

46

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

I: Integral action, Remove steady state error.

D: Derivative action, Anticipate the performance.

Aim of PID in MIMO systems: Derive following specification with minimum

information of system.

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

* Reference tracking. * Disturbance rejection.

PI control for MIMO systems

* PID Controller :

r(t) u(t)


Lecture 8

47

Design procedure

a) Design based on step response.

* System is stable.

* System model is unknown.

* System is regular1 or irregular.

-----------------------------------------------

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

b) Design based on model.

* No transmission zero on origin.

* Suitable for industrial plants.

* System model is known.

* System is regular or irregular.

* No transmission zero on origin.



Lecture 8

48

v P-controller : The idea of the proportional feedback controller is to speed

up the transient. In the multivariable case, this is carried out by choosing

the m × l matrix K so that the effect of the i th component of the reference

signal is strong on the i th component of the output y but weak on the j thcomponent of y when i ≠ j.

v Robust I-controller : The robust multivariable I-controller construction

follows Davison (1976) is a robust controller which eliminates the steady-

state error in the output.

v Robust PI-controller : The robust multivariable PI-controller can now be

constructed (if it exists) based on P-controller and I-controller in the case

that inverse of two important matrices CB and G(0) are exist.


Multivariable Tuning Regulators for Unknown Systems

(Penttinen and Koivo, 1980)

Design based on step response (Regular System)


Lecture 8

49


----------------------------------------------------------------------------------------Multivariable Tuning Regulators for Unknown Systems (Penttinen And Koivo, 1980)

)(

;;;;)()()()(

tCxy

mlRdRyRuRxtDdtButAxtx plmn

Consider the following system.

32211)()( BsCACABsCBsBAsICsG

Suppose:

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

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

* The order n is unknown. * The disturbance d(t) might not be measurable.

Design based on step response (Regular System)


Lecture 8

50

r(t)BKC) x(t)(A-BK(t)x

Cx(t)r(t)-BKBKAx(t))(r(t)-y(t)BKAx(t)Bu(t)Ax(t)(t)x

e(t)Ku(t)

pp

ppp

p

0if d(t)



K P

G(s)r(s) e(s) u(s)

P-controller Design

K I /S

Suppose : 0IK

-

y(s)


Lecture 8

51



K P

G(s)r(s) e(s) u(s)

P-controller Design

r(t)dτBKe e : x(t))if x( p

tC)τ-(A-BKC)t(A-BK pp

0

00

1

0

I]M[edτe Mt

t

Mτ

0

1

0

1 rBKC)(A-BKerBKC) (A-BKx(t) pp

C)t(A-BK

ppp

r(t)BKC) x(t)(A-BK(t)x pp

-

y(s)

0

0

0 r)BKde (e x(t) :r r(t) p

tC)τ-(A-BKC)t(A-BK pp


Lecture 8

52

trCBK : y(t)t

r]BKk!

tC)(A-BKC[trCBKy(t)

p

p

k

kk

pp

0

0

2

1

0

0

0

1

0

1 rBKC)(A-BKerBKC) (A-BKx(t) pp

C)t(A-BK

ppp

0

1

0

1 rBKC)(A-BKCerBKC) C(A-BKy(t) pp

C)t(A-BK

ppp

...An!

t...A

!

ttAIe n

nAt 2

2

2



P-controller Design


Lecture 8

53

trCBK : y(t) t p 00

12

1

00

0

0

00

)C(CBBCB;(CB)

k

k

k

(CB)K TTTT

m

p

C)]P(BKPC)PA-[(BKPAC)P (A-BKPC)(A-BK p

T

p

T

p

T

p

By following Kp , y(t) is diagonal.

For checking stability:



P-controller Design

r(t)BKC) x(t)(A-BK(t)x pp


Lecture 8

54

1

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

))=CBu( )+CBu()=CAx((y=> and d(t)=)=x(Suppose:

d(t)+CBu(t)+CD(t)=CAx(t)x(t)=Cy

0000000

… u uuy … y y

=CBuy

=CBuy

=CBuy

mm

mm

]CB[=] [ 2121

22

11

The following algorithm gives an experimental way for determining the matrix

CB for Unknown Systems :

)(

)()()()(

tCxy

tDdtButAxtx

Step 2. Repeat step 1 and let :

u1, u2 , … um are independent then:



P-controller Design

Step 1. Let . When the steady-state is achieved, choose a constant input

and apply it to the plant. Then ( this value can

be computed e.g. graphically.)

0u(t)01 uu(t) 110 =CBuy)(y


Lecture 8

55

;

00

0

0

00

2

1

m

p

k

k

k

(CB)K



P-controller Design

Remark. The existence of the P-controller follows if rank (CB)= k.

If the P-controller exists, the tuning parameters can be found

experimentally.


Lecture 8

56

PI control for MIMO systemsI-controller Design

The robust multivariable I-controller construction follows:

G(s)

K P

r(s) e(s) u(s)

K I /S-

Suppose : 0pK

d(s)

0

I )(K=u(t) dtte

Multivariable tuning regulators: The feedforward and robust control of a general

servomechanism problem (Davison ,1976).

BCA)G(

);(εG); K(Gε K; KK II

10

00

Let :


y(s)


Lecture 8

57

PI control for MIMO systemsI-controller Design

G(s)r(s) e(s) u(s)

K I /S-

d(s)

Davison shows that the controller where is the

scalar tuning parameter, is a robust controller which eliminates the steady-state

error in the output. He also shows that there exists so that the closed-

loop system is stable if satisfies .

][ 1BCAKKI

0*

*0

)(

)()()()(

tCxy

tDdtButAxtx

0

e(t)dtKu(t)=

0

e(t)dtz(t)=

d(t)D

r(t)z(t)

x(t)

C

εBKA

(t)z

(t)x

01

0

0

0limlim e(t)(t)z tt

y(s)



Lecture 8

58

steady state gain of system

1

2121 ]...][...[)0( mmssssss uuuyyyG

B;CA r

yG -ss 1

0

(0)



The next algorithm of Davison (1976) provides a way to experimentally

determine the parameter for Unknown Systems. Again assume the

disturbance d(t)=0.

)0(G

Step 2. Repeat step 1 and let :

u1, u2 , … um are independent then:

I-controller Design

Step 1. Let . When the steady-state is achieved, choose a constant

input and apply it to the open loop plant. Then

(Measure e.g. graphically.)

0u(t)

01 uu(t) 1ss1 (0)= uGy

ss1y

]...)[0(]...[

0)((0)

(0)

(0)

2121

22

11

mssmssss

mmssm

ss

ss

u uuG y yy

ut uu=Gy

u=Gy

u=Gy


Lecture 8

59

PI control for MIMO systemsPI-controller Design

G(s)r(s) e(s) u(s)

K I /S-

d(s)

;εGKI (0)


Remark. Davison's necessary and sufficient condition for the existence

of the robust I-controller which eliminates the steady-state error in the

output for all constant disturbances and constant reference signals is

rank (G(0))= k. The tuning parameter can be found experimentally.

y(s)


Lecture 8

60

PI control for MIMO systemsPI-controller Design

G(s)

K P

r(s) e(s) u(s)

K I /S-

d(s)

;εGKI (0)


;

k

k

k

(CB)K

m

p

00

0

0

00

2

1

y(s)


Lecture 8

61

v So we can determine the gain matrices and (if they exist) .

v The step-inputs needed can be the same for both algorithms.

v Tune the P-controller for the plant by adjusting the tuning parameters,

[starting with small positive values so that the closed-loop response for a

step-input reference signal is as good as possible.]

v Tune the PI-controller by adjusting the tuning parameter of the I-

controller [starting with small positive values so that the closed-loop (Pl-

controlled) response for a step-input reference signal has the maximum

speed of response.]

IKPK



PI-controller Design


Lecture 8

62

This proposed method is quite simple, no model is constructed, and one

only needs steady-state information and derivatives of the step-responses

which are easily evaluated graphically.

Example 8-5:

)(010

001)(

)(

1

5

1

1

1

3

)(

0

1

1

1

1

1

)(

300

020

001

)(

txty

tdtrtxtx

PI control for MIMO systemsDesign example


Lecture 8

63

11

11

10

01

11

111

CB

5.05.0

11

10

01

5.05.0

11)0(

1

G

Let u=[ 1 0]

1

1CB

5.0

1)0(G



Lecture 8

64

11

11

10

01

11

111

CB

5.05.0

11

10

01

5.05.0

11)0(

1

G

Let u=[ 1 0]

11

11CB

5.05.0

11)0(G

Let u=[ 0 1]



Lecture 8

65

m

p

k

k

k

(CB)K

00

0

0

00

2

1

B]CAε[)(εGKI

10

2

1

1

0

0

11

11

k

kK p

1

5.05.0

11

KK I

121 kk 5.0



Lecture 8

66

121 kk 5.0

5.1121 kk



Lecture 8

67


The second control design method is based on :

Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)This method can be applied on regular or irregular systems.

G(s)

K P

r(s) e(s) u(s)

K I /S-

d(s)

)(

)()()()(

tCxy

tDdtButAxtx

Consider the following system and assuming the same conditions exist :

R )0(KK I KK p

)()()( tKztKetu

0

e(t)dtz(t)=

By following Kp and KI

Design based on step response (Irregular System)


Lecture 8

Harris and Billings show that if

68

)()()( 1 sNsDsG

Let R )0(KK I KK p


----------------------------------------------------------------------------------------Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)

KsNsDsIsDs lc )()()1()()( 1

Remark. The existence of the PI-controller follows if rank (G(0))= k. If the

PI-controller exists, the tuning parameters can be found experimentally.

PI-controller Design

and for small value of the closed loop system will be stable.

the characteristic equation of the closed loop system is

1)( TT GGGK )0(;

000

0

00

00

2

1

i

m

)0()0()0( 1 NDGG

by following K


Lecture 8

69

33

33

5454

5454

11

11

35110

02

1

K,..

..αεK

Σ)(GGGK

,,ε.,αΣ

TT

Example 8-5:

)(010

001)(

)(

1

5

1

1

1

3

)(

0

1

1

1

1

1

)(

300

020

001

)(

txty

tdtrtxtx


----------------------------------------------------------------------------------------Self-tuning and adaptive control : theory and applications (Harris and Billings, 1981)

Design example


Lecture 8

70

Penttinen and Koivo’s Method

Harris and Billings’s Method



Lecture 8

71

Design based on step response(regular systems). rank full has CB

1

2

2

1

1 0

00

0

0

00

)(XXXX),(GεK,

k

k

k

(CB)K TT

m

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

mm k

k

k

)(GεK,

k

k

k

)(GεαK

00

0

0

00

0

00

0

0

00

02

1

2

2

1

1

Design based on step response.



Lecture 8

72

Design based on model.

Design based on the model (regular systems). rank full has CB

Exercise 8-1 : Design a PI controller for following system.

a) Derive the eigenvalues of the open loop system.

b) Derive the eigenvalues of the closed loop system.

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

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

xy

ux

0010

1101

0136.1

146.3136.1

0679.5

00

104.2343.1273.4048.0

893.5654.6273.4067.1

675.0029.45814.0

676.5715.62077.038.1

x



Lecture 8

73

Design based on model.

Design based on the model (irregular systems). rank fullnot has CB

Exercise 8-2: Design a PI controller for following system.

a) Derive the eigenvalues of the open loop system.

b) Derive the eigenvalues of the closed loop system.

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

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

xy

ux

011

001

284.742.11

143.0717.0

00

964.039.90

121.10

100

x



Lecture 8

74

Exercise 8-3: Design a PI controller for following system.

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

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

analysis the results.

xa

y

ux

11

111

10

11

00

300

016

113

x

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



Lecture 8

75

References

• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.

References

• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.

• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.

• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/

Web References

• http://www.um.ac.ir/~karimpor

• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق


Lecture 8

76

References

• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.

References

• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.

• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.

• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/

Web References

• http://www.um.ac.ir/~karimpor

• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق

Documents

Multivariable Control Systems - Personal Datakarimpor.profcms.um.ac.ir/imagesm/354/stories/mul_con/... · References are appeared in the last slide. ... Characteristic-locus method