multivariable control systems

Multivariable Control Multivariable Control SystemsSystems

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

2

Ali Karimpour Sep 2009

Chapter 3Chapter 3

Limitation on Performance in MIMO SystemsTopics to be covered include:

Scaling and Performance

Shaping Closed-loop Transfer Functions

Fundamental Limitation on Sensitivity

Functional Controllability

Fundamental Limitation: Bounds on Peaks

Limitations Imposed by Time Delays

Limitations Imposed by RHP Zeros

Limitations Imposed by Unstable (RHP) Poles

3


Chapter 3

Introduction

One degree-of-freedom configuration

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

4


Chapter 3

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.

5


Chapter 3

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.

6


Chapter 3

The terms H∞ and H2

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

as, )(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 established

itself 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.

7


Chapter 3

The terms H∞ and H2

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

as,

djFjFtrsF H)()(

2

1)(

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?

8


Chapter 3

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?

9


Chapter 3


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)||∞≤M

The peak specification prevents amplification of noise at high frequencies, and also introduces a margin of robustness; typically we select M=2

10


Chapter 3


)(

1

swp

,)(

1)(

jwjS

P

Mathematically, these specifications may be captured simply by an

upper bound

1)()(,1)()(

jSjwjSjw PP

The subscript P stands for performance

11


Chapter 3

Weighted Selection

As

Mssw

B

BP

/

)(

plot of )(

1

jwP

12


Chapter 3

Weighted Selection

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

22/1

22/1/)(

As

Mssw

B

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

13


Chapter 3


14


Chapter 3

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

KSw

Tw

Sw

NjNN

u

T

P

,1)(max

For SISO systems, N is a vector and

222)( KSwTwSwN uTP

15


Chapter 3

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)12

2

2

12

1

707.0 and 707.0 toLeads (II) If 2121

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

16


Chapter 3

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 )( , )( KSYS

17


Chapter 3


Example 3-1 110

100)(,

)105.0(

1

110

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 0

as 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 times

to avoid input saturation (this is easily satisfied for most designs).

18


Chapter 3


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

KSw

SwN

u

P

It was stated earlier that appropriate scaling has been performed so that

the inputs should be about 1 or less in magnitude, and we therefore

As

Mssww

B

BPu

/

)( and 1

19


Chapter 3


110

100)(of diagram Bode theSee

ssGd

-20

-10

0

10

20

30

40

Mag

nitu

de (

dB)

10-3

10-2

10-1

100

101

102

-90

-45

0

Pha

se (

deg)

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

20


Chapter 3


410,10,5.1,/

)(

AMAs

Mssw 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

21


Chapter 3


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

22


Chapter 3


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

23


Chapter 3


24


Chapter 3


S plus T is the identity matrix

ITS

1)()(1)( STS

1)()(1)( TST

25


Chapter 3


Interpolation Constraints RHP-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

26


Chapter 3


Interpolation Constraints RHP-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-pole

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

27


Chapter 3


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)

28


Chapter 3


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.

29


Chapter 3


Theorem 3-1 Sensitivity and Complementary Sensitivity Peaks Consider 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

jpH

iz

ijzpji

jpH

ip

ijpji

jzH

iz

ijz pz

yyQ

pp

yyQ

zz

yyQ

,,,,,, ,,

30


Chapter 3


Example 3-2

2)1)(2(

)3)(1()(

ss

sssG

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

2,3,1 121 pzz

1

1,4/1,

6/14/1

4/12/1, pzpz QQQ

156786.12

9531.71 2

min,min,

TS MM

31


Chapter 3


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

ji

jpH

iz

ijzpji

jpH

ip

ijpji

jzH

iz

ijz pz

yyQ

pp

yyQ

zz

yyQ

,,,,,, ,,

One RHP-pole and one RHP-zero

22

2

2min,min, cossin

pz

pzMM TS

p

Hz yy1cos

Exercise : Proof the above equation.

32


Chapter 3


Example 3-4

3,2;

11.0

20

011.0

cossin

sincos

3

10

01

)(

pz

s

ss

zs

s

pssG

U

)3)(11.0(

20

0))(11.0()(,

10

0100

ss

spss

zs

sGandU

0)3)(11.0(

))(11.0(

20

)(,01

109090

ss

zspss

s

sGandU

33


Chapter 3


5.0,2,/

,

BB

Pu MIs

MsWIW

34


Chapter 3


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 …

35


Chapter 3


Definition 3-1 Functional controllability.An m-input l-output system G(s) is functionally controllable if the

normal rank of G(s), denoted r, is equal to the number of outputs; that is,

if G(s) has full row rank. A plant is functionally uncontrollable if r < l.Remark 1: The minimal requirement for functional controllability is

that we have at least many inputs as outputs, i.e. m ≥ l

Remark 2: A plant is functionally uncontrollable if and only if

,0))(( jGl

Remark 3: For SISO plants just G(s)=0 is functionally uncontrollable.

Remark 4: A MIMO plant is functionally uncontrollable if the gain is identically zero in some output direction at all frequencies.

36


Chapter 3


BAsICsG 1)()( is functionally uncontrollable if An m-input l-output system

lBrank )(1- The system is input deficient or

2- The system is output deficient or lCrank )(

3- The system has fewer states than outputs lAsIrank )(

If the plant is not functionally controllable, i.e. lr then there are

l-r output directions, denoted y0 which cannot be affected.

0)()(0 jGjy H

From an SVD of G(jω) the uncontrollable output directions y0(jω) are

the last l-r columns of Y(jω).

37


Chapter 3


Example 3-5

2

4

2

21

2

1

1

)(

ss

sssG

This is easily seen since column 2 of G(s) is two times column 1.

The uncontrollable output directions at low and high frequencies are, respectively,

1

2

5

1)(,

1

1

2

1)0( 00 yy

38


Chapter 3

Limitations Imposed by Time Delays

ijj

i minmin

A lower bound on the time delay for output i is given by the smallest delay 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 simple

example, consider the plant

1

11)( se

sG

39


Chapter 3


The limitations of a RHP-zero located at z may also be derived from

the bound (by maximum module theorem)

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

1)()(

sSswP1)( zwP

40


Chapter 3


Performance at Low Frequencies 1)()(

sSswP

1)( zwP

As

Mssw

B

BP

/

)( 1/

)(

Az

Mzzw

B

BP

Real zero: 2

zB

Imaginary zero

)1

1()1(M

zAB

zB 87.02

11

MzB

41


Chapter 3


Performance at High Frequencies 1)()(

sSswP

1)( zwP

11

)( B

P

z

Mzw

Real zero: zB 2

B

P

s

Msw

1

)(

MzB /11

1

B

P

s

Msw

1

)(

42


Chapter 3


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

Example 3-6

221

11

)1)(12.0(

1)(

ssssG

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

The output zero direction is

45.0

89.0

1

2

5

1zy

Interpolation constraint is

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

43


Chapter 3


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

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

zs

zszs

zs

sT0

0)(0

10)(

01)(

2

211 zs

s

zs

zssT

zs

zs

zs

ssT

14 21

45.0

89.0zy

44


Chapter 3


Theorem 3-2 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 to obtain

“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...10

0...000...01

)( 1121

zs

s

zs

s

zs

zs

zs

s

zs

s

zs

ssT nkk kjfory

y

zk

zjj 2

45


Chapter 3


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

1)()(

sTswT1)( pwT

TBTT M

ssw

1)(

Real RHP-pole

1

T

TBT M

Mp pBT 2

Imaginary RHP-pole pBT 15.1

pc 2

multivariable control systems

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