Multivariable Control Systems - Ali Karimpour

Multivariable Control Multivariable Control SystemsSystems

Ali KarimpourAssociate Professor

Ferdowsi University of Mashhad

Lecture 6

References are appeared in the last slide.

Dr. Ali Karimpour Feb 2014

Lecture 6

2

Limitation on Performance in MIMO Systems

Topics to be covered include: Scaling and Performance Shaping Closed-loop Transfer Functions

Fundamental Limitation on Sensitivity

Limitations Imposed by Time Delays

Limitations Imposed by RHP Zeros

Limitations Imposed by Unstable (RHP) Poles

Fundamental Limitation on Performance (Frequency domain)

Fundamental Limitation on Performance (Time domain)


Lecture 6

3

Introduction

One degree-of-freedomconfiguration

nGKIGKdGGKIrGKIGKsy d111 )()()()(

nsTdGsSrsTsy d )()()()(

• Performance, good disturbance rejection LITS or or 0

• Performance, good command following LITS or or 0

• Mitigation of measurement noise on output 0or or 0 LIST


Lecture 6

4


Scaling and Performance Shaping Closed-loop Transfer Functions








Lecture 6

5

Scaling

ryedGuGy d ˆˆˆ;ˆˆˆˆˆ

rDreDeyDyuDudDd eeeud ˆ,ˆ,ˆ,ˆ,ˆ 11111

rDyDeDdDGuDGyD eeeddee ;

ddedue DGDGDGDG ˆ,ˆ 11 ryedGGuy d ;


Lecture 6

6

Scaling and Performance

For any reference r(t) between -R and R and any disturbance d(t)

between -1 and 1, keep the output y(t) within the range r(t)-1 to r(t)+1

(at least most of the time), using an input u(t) within the range -1 to 1.

For any disturbance |d(ω)| ≤ 1 and any reference |r(ω)|≤ R(ω),

the performance requirement is to keep at each frequency ω the

control error |e(ω)|≤1 using an input |u(ω)|≤1.


Lecture 6

7

Shaping Closed-loop Transfer Functions

Many design procedure act on the shaping of the open-loop transfer

function L.

An alternative design strategy is to directly shape the magnitudes of

closed-loop transfer functions, such as S(s) and T(s).

Such a design strategy can be formulated as an H∞ optimal control

problem, thus automating the actual controller design and leaving

the engineer with the task of selecting reasonable bounds “weights”

on the desired closed-loop transfer functions.


Lecture 6

8

The terms H∞ and H2

The H∞ norm of a stable transfer function matrix F(s) is simply defineas,

)(max)(

jFsF

We are simply talking about a design method which aims to press down the peak(s) of one or more selected transfer functions.

Now, the term H∞ which is purely mathematical, has now establisheditself in the control community.

In literature the symbol H∞ stands for the transfer function matrices with bounded H∞-norm which is the set of stable and proper transfer function matrices.


Lecture 6

9

The terms H∞ and H2

The H2 norm of a stable transfer function matrix F(s) is simply defineas,

djFjFtrsF H)()(

21)(

2

Similarly, the symbol H2 stands for the transfer function matrices with bounded H2-norm, which is the set of stable and strictly proper transfer function matrices.

Note that the H2 norm of a semi-proper transfer function is infinite, whereas its H∞ norm is finite.

Why?


Lecture 6

10

Weighted Sensitivity

As already discussed, the sensitivity function S is a very good indicator

of closed-loop performance (both for SISO and MIMO systems).

The main advantage of considering S is that because we ideally want

S to be small, it is sufficient to consider just its magnitude, ||S|| that is,

we need not worry about its phase.

Why S is a very good indicator of closed-loop performance in many

literatures?


Lecture 6

11


Typical specifications in terms of S include:

• Minimum bandwidth frequency ωB*.

• Maximum tracking error at selected frequencies.

• System type, or alternatively the maximum steady-state tracking error, A.

• Shape of S over selected frequency ranges.

• Maximum peak magnitude of S, ||S(jω)||∞≤MThe peak specification prevents amplification of noise at high frequencies, and also introduces a margin of robustness; typically we select M=2


Lecture 6

12


)(1swp

,)(

1)(jw

jSP

Mathematically, these specifications may be captured simply by an upper bound

1)()(,1)()(

jSjwjSjw PP

The subscript P stands for performance


Lecture 6

13

Weighted Selection

AsMssw

B

BP

/)(

plot of )(1jwP


Lecture 6

14

Weighted Selection

A weight which asks for a slope -2 for L at lower frequencies is

22/1

22/1/)(As

MsswB

BP

The insight gained from the previous section on loop-shaping design is very useful for selecting weights.

For example, for disturbance rejection

1)( jSGd

It then follows that a good initial choice for the performance weight is to let wP(s) look like |Gd(jω)| at frequencies where |Gd(jω)| >1


Lecture 6

15



Lecture 6

16

Stacked Requirements: Mixed Sensitivity

The specification ||wPS||∞<1 puts a lower bound on the bandwidth, but not an upper one, and nor does it allow us to specify the roll-off of L(s) above the bandwidth.

To do this one can make demands on another closed-loop transfer function

KSwTwSw

NjNN

u

T

P

,1)(max

For SISO systems, N is a vector and

222)( KSwTwSwN uTP


Lecture 6

17

Stacked Requirements: Mixed Sensitivity

The stacking procedure is selected for mathematical convenience as it does not allow us to exactly specify the bounds.

For example, TwKSwK TP )()( Let 21

We want to achieve

(I) 1 and 1 21

This is similar to, but not quite the same as the stacked requirement

(II)122

21

2

1

707.0 and 707.0 toLeads (II) If 2121

In general, with n stacked requirements the resulting error is at most n


Lecture 6

18

Solving H∞ Optimal Control Problem

After selecting the form of N and the weights, the H∞ optimal controller is obtained by solving the problem

)(min KN

K

Let denote the optimal H∞ norm. )(min0 KN

K

The practical implication is that, except for at most a factor the

transfer functions will be close to times the bounds selected by

the designer.

n

0

This gives the designer a mechanism for directly shaping the magnitudes of

)( and )( , )( KSTS


Lecture 6

19


Example 6-1110

100)(,)105.0(

1110

200)( 2

ssG

sssG d

The control objectives are:

1. Command tracking: The rise time (to reach 90% of the final value) should be less than 0.3 second and the overshoot should be less than 5%.

2. Disturbance rejection: The output in response to a unit step disturbance should remain within the range [-1,1] at all times, and it should return to 0as quickly as possible (|y(t)| should at least be less than 0.1 after 3 seconds).

3. Input constraints: u(t) should remain within the range [-1,1] at all timesto avoid input saturation (this is easily satisfied for most designs).


Lecture 6

20


Consider an H∞ mixed sensitivity S/KS design in which

KSwSw

Nu

P

It was stated earlier that appropriate scaling has been performed so thatthe inputs should be about 1 or less in magnitude, and we therefore

AsMssww

B

BPu

/)( and 1


Lecture 6

21


110100)(of diagram Bode theSee

s

sGd

-20

-10

0

10

20

30

40

Mag

nitu

de (d

B)

10-3

10-2

10-1

100

101

102

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

We need control till 10 rad/sec to reduce disturbance and a suitable rise time.

sec/10 let So radcB

Overshoot should be less than 5% so let MS<1.5


Lecture 6

22


410,10,5.1,/)(

AM

AsMssw B

B

BP

For this problem, we achieved an optimal H∞ norm of 1.37, so the weighted sensitivity requirements are not quite satisfied. Nevertheless,the design seems good with

rad/sec 22.5 and 2.71,04.8,0.1,30.1 cTS PMGMMM


Lecture 6

23


The tracking response is very good as shown by curve in Figure. However, we see that the disturbance response is very sluggish.


Lecture 6

24


If disturbance rejection is the main concern, then from our earlier discussion we need for a performance weight that specifies higher gains at low frequencies. We therefore try

622/1

22/1

10,10,5.1,/)(

AM

As

Mssw B

B

BP

For this problem, we achieved an optimal H∞ norm of 2.21, so the weighted sensitivity requirements are not quite satisfied. Nevertheless,the design seems good with

rad/sec 2.11 and 3.43,76.4,43.1,63.1 c PMGMMM TS


Lecture 6

25



Lecture 6

26










Lecture 6

27

Fundamental Limitation on Sensitivity(Frequency domain)

S plus T is the identity matrix

ITS

1)()(1)( STS

1)()(1)( TST


Lecture 6

28

Interpolation ConstraintsRHP-zero:

If G(s) has a RHP-zero at z with output direction yz then for internal stability of the feedback system the following interpolation constraints must apply:

In MIMO Case: Hz

Hz

Hz yzSyzTy )(;0)(

In SISO Case: 1)(;0)( zSzT

0)( zGy Hz

Proof:

0)( zLy Hz LST

0)( zTy Hz 0))(( zSIy H

z

S has no RHP-pole



Lecture 6

29

Interpolation ConstraintsRHP-pole:

If G(s) has a RHP pole at p with output direction yp then for internal stability the following interpolation constraints apply

In MIMO Case: ppp yypTypS )(;0)(

In SISO Case: 1)(;0)( pTpS

Proof:

0)(1 pypL SLT T has no RHP-pole S has a RHP-zero

1 TLS 0)()()( 1 pp ypSypLpT ppp yypSIypT )()(



Lecture 6

30

Sensitivity Integrals

pN

ii

ii pdjSdjS

100

)Re(.)(ln)(detln

pN

iipdjS

10

)Re(.)(ln

In SISO Case: If L(s) has two more poles than zeros (Bode integral)

In MIMO Case: (Generalize of SISO case)



Lecture 6

31

Fundamental Limitation: Bounds on Peaks

TMSM TS min,min min,min,

In the following, MS,min and MT,min denote the lowest achievable values

for ||S||∞ and ||T||∞ , respectively, using any stabilizing controller K.


Lecture 6

32


Theorem 6-1 Sensitivity and Complementary Sensitivity PeaksConsider a rational plant G(s) (with no time delay). Suppose G(s) has Nz

RHP-zeros with output zero direction vectors yz,i and Np RHP-poles with output pole direction vectors yp,i. Suppose all zi and pi are distinct.

Then we have the following tight lower bound on ||T||∞ and ||S||∞

2/12/12min,min, 1 pzpzTS QQQMM

ji

jpHiz

ijzpji

jpHip

ijpji

jzHiz

ijz pzyy

Qppyy

Qzzyy

Q

,,,,,, ,,


Lecture 6

33


Example 6-2

2)1)(2()3)(1()(

sssssG

Derive lower bounds on ||T||∞ and ||S||∞

2,3,1 121 pzz

11

,4/1,6/14/14/12/1

, pzpz QQQ

156786.129531.7

1 2min,min,

TS MM


Lecture 6

34


2/12/12min,min, 1 pzpzTS QQQMM

ji

jpHiz

ijzpji

jpHip

ijpji

jzHiz

ijz pzyy

Qppyy

Qzzyy

Q

,,,,,, ,,

One RHP-pole and one RHP-zero

22

22

min,min, cossinpz

pzMM TS

p

Hz yy1cos

Exercise6-1 : Proof the above equation.


Lecture 6

35


Example 6-3

3,2;

11.020

011.0

cossinsincos

310

01

)(

pz

ss

szs

s

pssG

U

)3)(11.0(20

0))(11.0()(,

1001

00

sss

psszs

sGandU

0)3)(11.0(

))(11.0(20

)(,0110

9090

sszs

psss

sGandU


Lecture 6

36


5.0,2,/,

BB

Pu MIs

MsWIW


Lecture 6

37


The corresponding responses to a step change in the reference r = [ 1 -1 ] , are shown

Solid line: y1

Dashed line: y2

1- For α = 0 there is one RHP-pole and zero so the responses for y1 is very poor.

2- For α = 90 the RHP-pole and zero do not interact but y2 has an undershoot since of …

3- For α = 0 and 30 the H∞ controller is unstable since of …


Lecture 6

38


ijji minmin

A lower bound on the time delay for output i is given by the smallestdelay in row i of G(s)

For MIMO systems we have the surprising result that an increased time delay may sometimes improve the achievable performance. As a simpleexample, consider the plant

1

11)( sesG


Lecture 6

39


The limitations of a RHP-zero located at z may also be derived fromthe bound (by maximum module theorem)

)())((.)(max)()( zwjSjwsSsw PPP

1)()(

sSswP1)( zwP


Lecture 6

40


Performance at Low Frequencies 1)()(

sSswP

1)( zwP

AsMssw

B

BP

/

)( 1/)(

AzMzzw

B

BP

Real zero: )(2

IzB

Imaginary zero

)11()1(M

zAB

)(87.0 IIzB 2

11M

zB

Exercise6-2 : Derive (I) and (II).


Lecture 6

41


Performance at High Frequencies 1)()(

sSswP

1)( zwP

11)( B

Pz

Mzw

Real zero: zB 2

BP

sM

sw

1)(

MzB /11

1

B

Ps

Msw

1)(


Lecture 6

42


Moving the Effect of a RHP-zero to a Specific Output

Example 6-4

22111

)1)(12.0(1)(

ssssG

which has a RHP-zero at s = z = 0.5

The output zero direction is

45.0

89.01

25

1zy

Interpolation constraint is

0)()(2;0)()(2 22122111 ztztztzt


Lecture 6

43


Moving the Effect of a RHP-zero to a Specific Output0)()(2;0)()(2 22122111 ztztztzt

zszs

zszs

sT0

0)(0

???01

)( 11

zszs

zsssT 41

45.0

89.0zy

???10

)(2

2

zs

szszs

sT

12


Lecture 6

44

Limitations Imposed by RHP Zero

Theorem 6-2 Assume that G(s) is square, functionally controllable and stable and has a single RHP-zero at s = z and no RHP-pole at s = z. Then if the k’th Element of the output zero direction is non-zero, i.e. yzk ≠ 0 it is possible toobtain “perfect” control on all outputs j ≠ k with the remaining output exhibiting no steady-state offset. Specifically, T can be chosen of the form

1...000...00........................

..............................0...000...100...000...01

)( 1121

zss

zss

zszs

zss

zss

zsssT nkk kjfor

yy

zk

zjj 2


Lecture 6

45


)())((.)(max)()( pwjTjwsTsw TTT

1)()(

sTswT 1)( pwT

TBTT M

ssw 1)(

Real RHP-pole

1

T

TBT M

Mp pBT 2

Imaginary RHP-pole pBT 15.1

pc 2


Lecture 6

46










Lecture 6

47

Fundamental Limitation on Sensitivity(Time domain)

Reference: “Interaction Bounds in Multivariable Control Systems” K H Johanson, Automatica, vol 38,pp 1045-1051, 2002

Consider the system:

)()()()()()()(

sYsRsCsUsUsGsY

Settling time is defined as:

ttrtyt kkmksi ,)()(:infmax

0,...,1

Rise time is defined as:

),0(,/ˆ)(:sup0

ttrtyt iri

Let a step response signal at i th input but other inputs are zero so

)(ˆ tur


Lecture 6

48


Settling time is defined as:

ttrtyt kkmksi ,)()(:infmax

0,...,1

ε is settling level.

0),()(sup0

trtyy iit

oi

Undershoot is defined as:

0),(sup0

tyy it

ui

The interaction from ri to output k is:

)(supˆ0

tyy kt

ki

Overshoot in output i is:


Lecture 6

49


0)( zTy Hz

Remember that if open loop transfer function G has a real RHP zero z > 0 with zero direction yz then we have:

Also we have

)(0 zTy Hz )()()( zYyzRzTy H

zHz

0

)( dttyey ztHz

Theorem6-3: Consider the stable closed loop system with zero initial conditions at t=0 and let for t>0. Assume that the open loop transfer function G has a real RHP zero z > 0 with zero direction yz and yz1 >0. then we have:

Trtr 0,...,0,ˆ)(

m

kzkzzt

m

kkzk

uz yry

eyyyy

s2

12

111 )ˆ(1

1ˆ1

undershoot

interaction settling time settling levelelements of zero direction


Lecture 6

50


Theorem6-3: Consider the stable closed loop system with zero initial conditions at t=0 and let for t>0. Assume that the open loop transfer function G has a real RHP zero z > 0 with zero direction yz and yz1 >0. then we have:

Trtr 0,...,0,ˆ)(

Remark1: For small settling level:

m

kzkzzt

m

kkzk

uz yry

eyyyy

s2

12

111 )ˆ(1

1ˆ1

1ˆˆ

1

1

2111

sztz

m

kkzk

uz e

yryyyy

• For small zero lower bound is large.

• There is compromise between undershoot and interactions.

Remark2: For SISO system there is always undershoot but in MIMO systems, however, we see that there is undershoot only if all but one element of zero direction are zero.


Lecture 6

51


Remember that if open loop transfer function G has a real RHP pole p > 0 with pole direction yp then we have:

Let

sr

srsr

sR

/ˆ...00

0.../ˆ00...0/ˆ

)(

0)( pypS

and

)()()()( sSRsYsRsE

so we have:

pypRpS )()(pypS )(0 p

ptp ydtteeypE .)()(

0


Lecture 6

52


Theorem6-4: Consider the stable closed loop system with zero initial conditions at t=0. Assume that the open loop transfer function G has a real RHP pole p > 0 with pole direction yp and yp1 >0. Consider m independent responses with for t>0. Then we have:

rtri ˆ)(

ppt

p ydtteeypE .)()(0

m

kkpk

ptp

rm

kkpk

op yyeyptryyyy r

211

1

2111 ˆ1

2ˆˆ 1

overshoot

interaction rise timeelements of pole direction

Remark1: If the pole direction is such that

mkyy pkp ,...,2for 1

Then a real RHP pole far from the origin must necessarily give large overshoot if the rise time is long.


Lecture 6

53


Example6-5: Experimental set-up for the quadratic-tank process.


Lecture 6

54


Example6-5(Continue): Experimental set-up for the quadratic-tank process.

pointset valve:1

Input#1Pump1

Input#2Pump2

Output#1 Output#2

pointset valve:2

Valve set points are used to make theprocess more or less difficult to control.

6.0,7.0 here zero, RHP no 2,1 If 2121

1

11

sss

ssssG

67.98124.3

)67.981)(90.381(71.1

)57.951)(05.321(04.2

57.95111.3

)(

045.0012.0 21 zz

34.0,43.0 here zero, RHP one 1,0 If 2121

sss

ssssG

55.111197.1

)55.1111)(3.561(11.3

)75.761)(30.521(33.3

75.76169.1

)(

051.0014.0 21 zz


Lecture 6

55


High undershoot for small interaction.

For a unit step in r1 we have:

For a settling time of ts1=100 we have:


Lecture 6

56



Lecture 6

57



Lecture 6

Exercises

58

6-3 Consider the following weight with f>1.

6-4 Consider the weight


Lecture 6

Exercises (Continue)

59

6-5 Consider the plant

6-6 Repeat 6-5 for following plant.


Lecture 6

60

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

• تحلیل و طراحی سیستم هاي چند متغیره، دکتر علی خاکی صدیق

Multivariable Control Systems - Ali Karimpour

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