jason gujasongu.org/4601/ch6.pdf2 vi.2. controllability a control system is said to be completely...

1

VI Pole Placement and Observer Design

1. Introduction

2. Controllability

3. Observability

4. Useful Transformation

5. Design Via Pole Placement

6. State Observers

7. Servo Systems

VI.1. Introduction Controllability and observability play an important role in the optimal control of multivariable systems. Pole placement will be used to design the controller and the observer.

2

VI.2. Controllability A control system is said to be completely state controllable if it is possible to transfer the system from any arbitrary initial state to any desired state in a finite time period. Consider the discrete time control system defined by kTHukTGxTkx 1 6.1

Where kTx =n-vector (state vector at kth sampling instant)

kTu (control signal at kth sampling instant) G = nn matrix H = 1n matrix T = Sampling Period We assume that kTu is constant for TktkT 1

Define controllability matrix HGGHH n 1 The condition for complete state controllability is that the nn matrix HGGHH n 1 be of rank n, or that

Rank nHGGHH n 1 It is possible to find n linearly independent scalar equations from which a sequence of unbounded control signals )1,2,1,0( nkkTu can be uniquely determined such

that any initial state 0x is transferred to the desired state in n sampling periods. Solutions of the system 6.1

0

2

1

0 1

u

Tnu

Tnu

HGGHHxGnTx nn

Alternative form of the condition for complete state controllability Consider the discrete time control system defined by kTHukTGxTkx 1 6.1

Where kTx =n-vector (state vector at kth sampling instant)

kTu =r-vector (control signal at kth sampling instant)

3

G = nn matrix H = rn matrix T = Sampling Period If the eigenvectors of G are distinct, then it is possible to find a transformation matrix P such that

n

GPP

0

0

2

1

1

Note: if the eigenvalues of G are distinct then the eigenvectors of G are distinct. The converse is not true. The condition for complete state controllability is that, if the eigenvectors of G are distinct, then the system is completely state controllable if and only if no row of

HP 1 has all zero elements. If the G matrix does not possess distinct eigenvectors, then diagnalization is impossible. We may transform G into a Jordan canonical form. Example for Jordan canonical form:

60

5

4

032

121

11

011

J

Assume we can find a transformation matrix S such that

JGSS 1 If we define a new state vector x̂ by kTxSkTx

, substitute into 6.1,

4

kTHuSkTxJ

kTHuSkTxGSSTkx1

111

6.2

The condition for the complete state controllability of the system of above equation is as follows: The system is completely state controllable if and only if 1) no two Jordan blocks in J of 6.2 are associated with the same eigenvalues, 2) The elements of any row of HS 1 that corresponds to the last row of each Jordan block are not all zero and 3) The elements of each row of HS 1 that correspond to distinct eigenvalues are not all zero. Example 6.1 Consider the system defined by

kukx

kx

dc

ba

kx

kx

1

1

1

1

2

1

2

1

kx

kxky

2

101

Determine the conditions on a,b,c and d for complete state controllability.

dcbadc

barankGHHrank

21

1

Complete Output Controllability kTHukTGxTkx 1 6.3

kTCxkTy 6.4 Where kTx =n-vector (state vector at kth sampling instant)

kTu (control signal at kth sampling instant)

kTy =m-vector (output vector at kth sampling instant) G = nn matrix H = 1n matrix C = nm matrix T = Sampling Period Define output controllability matrix HCGCGHHC n 1 The condition for complete output controllability is that the matrix

5

HCGCGHHC n 1 be of rank m, or that

Rank mHCGCGHHC n 1 The system defined by equation 6.3 and 6.4 is said to be completed output controllable, or simply output controllable, if it is possible to construct an unconstrained control signal kTu defined over a finite number of sampling periods nTkT 0 such that, starting

from any initial output 0y , the output kTy can be transferred to the desired point

fy in the output space in at most n sampling periods.

Thus

0

2

1

0 1

u

Tnu

Tnu

HCGCGHHCxCGnTCx nn

So:

0

2

1

0 1

u

Tnu

Tnu

HCGCGHHCxCGnTy nn

Next, consider the system defined by the equations: kTHukTGxTkx 1 6.5

kTDukTCxkTy 6.6 Where kTx =n-vector (state vector at kth sampling instant)


kTy =m-vector (output vector at kth sampling instant)

0

2

1

0 1

u

Tnu

Tnu

HGGHHxGnTx nn

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G = nn matrix H = rn matrix C = nm matrix D = rm matrix T = Sampling Period Define output controllability matrix HCGCGHHCD n 1 The condition for complete output controllability is that the matrix HCGCGHHCD n 1 be of rank m, or that

Rank mHCGCGHHCD n 1

0

2

1

0 1

u

Tnu

Tnu

HCGCGHHCxCGnTCx nn

0

21

0

0

2

1

0

1

1

u

TnuTnu

nTu

HCGCGHHCDxCG

nTDu

u

Tnu

nTTnu

HCGCGHHCxCG

nTDunTCxnTy

nn

nn

Controllability of a Linear Time-Invariant continuous time control system. Consider the system defined by

BuAxx

DuCxy x = state vector (n-vector) u = control vector (r-vector) y = output vector (m- vector)

7

A = nn matrix B = rn matrix C = nm matrix D = rm matrix Complete state controllability.

nBAABBrank n 1 Output controllability

mBCACABCBDrank n 1

8

VI.3. Observability

Consider the unforced discrete time control system defined by kTGxTkx 1 6.7


kTy =m-vector (output vector at kth sampling instant) G = nn matrix C = nm matrix T = Sampling Period The system is said to be completely observable if every initial state 0x can be

determined from the observation kTy over a finite number of sampling periods. Or

nCGCGCrank n ***** 1

Alternative form of the condition for complete observability kTGxTkx 1 6.9

kTCxkTy 6.10 Suppose the eigenvalues of G are distinct, and a transformation matrix P transforms G into a diagonal matrix, so that GPP 1 is a diagonal matrix. Define : kTxPkTx

Then 6.9 and 6.10 can be written as follows: kTxGPPTkx

11

0ˆ1 xGPPCPnTy

kTxCPkTCxkTyn

9

nn

n

n

nGPP

0

0

2

1

1

0

0

0

0

0

0

0 22

11

2

1

1

nnn

n

n

nn

n

n

n

x

x

x

CPxCPxGPPCPnTy

The system is completely observable if and only if none of the columns of the nm matrix CP consists of all zero elements. Note: if the ith column of CP consists of all zero elements, then the state variable 0ix

will not

appear in the output equation and therefore can not be determined from observation of kTy .

Thus x(0), which is related to 0x

by P, can not be determined. If the matrix G involves multiple engenvalues, then G may be transformed into Jordan canonical form:

JGSS 1 If we define a new state vector x̂ by kTxSkTx

, substitute into 6.7,

kTxJ

kTxGSSTkx

11

01 xGSSCSnTykTxCSkTyn

The condition for the complete observalibility of the system of above equation is as follows: The system is completely observable if and only if 1) no two Jordan blocks in J are associated with the same eigenvalues, 2) none of the columns of CS that corresponds to the first row of each Jordan block consists of all zero elements 3) The elements of each column of CS that correspond to distinct eigenvalues are not all zero.

Example 6.2 Consider the system defined by

kukx

kx

dc

ba

kx

kx

1

1

1

1

2

1

2

1

10

kx

kxky

2

101

Determine the conditions on a, b, c and d for complete observability.

020

1***

b

b

arankCGCrank

Condition for complete observability in the z plane: Note: a necessary and sufficient condition for complete observability is that no pole-zero cancellation occur in the pulse transfer function. If cancellation occurs, the canceled mode can not be observed in the output. Principle of Duality. Consider the system S1 defined by the equations

kTHukTGxTkx 1 6.11



kTy =m-vector (output vector at kth sampling instant) G = nn matrix H = rn matrix C = nm matrix T = Sampling Period

Consider S1 counterpart, S2 defined by the equations:

kTuCkTxGTkx

**1 6.13

kTxHkTy

* 6.14 Where kTx

=n-vector (state vector at kth sampling instant)

kTu

=m-vector (control signal at kth sampling instant)

kTy

=r-vector (output vector at kth sampling instant) G *= conjugate transpose of G

11

H *= conjugate transpose of H C * = conjugate transpose of C T = Sampling Period

Note: The analogy between controllability and observability is referred to as the principle of duality, due to kalman. The principle of duality states that the system S1 defined by equations 6.11-12 is completely state controllable (observable) if and only if system S2 defined by equation 6.13-14 is completely observable (state Controllable). For system S1: 1) A necessary and sufficient condition for complete state controllability is that

nHGGHHrank n 1 2) A necessary and sufficient condition for complete observability is that


For system S2: 1) A necessary and sufficient condition for complete state controllability is that


2) A necessary and sufficient condition for complete observability is that

nHGGHHrank n 1

Complete Observability Linear Time-Invariant Continuous-Time Control system. Consider the system defined by

Axx

Cxy x = state vector (n-vector) y = output vector (m- vector) A = nn matrix C = nm matrix Complete observability is that the rank of the . nmn matrix

12

nCACACrank n ***** 1

Effects of the Discretization of a continuous-time control system on controllability and observability.

Note: It can be shown that a system that is completely state controllable and completely observable in the absence of sampling remains completely state controllable and completely observable after introduction of sampling if and only if, for every eigenvalue of the characteristic equation for the continuous-time control system, the relationship

ji ReRe

Implies

T

nji

2Im

Where T is the sampling period and ,2,1 n It is noted that, unless the system contains complex poles, pole-zero cancellation will not occur in passing from the continuous-time to the discrete time case.

Example 6.3 Consider the continuous time control system defined by

ux

x

x

x

1

0

02

20

2

1

2

1

2

101x

xy

Determine the controllability and observability of the continuous time system and the corresponding discrete time control system.

lecontrollabrankABBrank

2

01

20

observablerankCACrank

2

20

01***

The eigenvalues of the state matrix are ji 22,

Discretizing the continuous time control system will be

13

,2cos2sin

2sin2cos

TT

TTeTG AT

T

T

TT

TTBAIeTH AT

2sin5.0

5.02cos5.0

1

0

02

20

4

1

12cos2sin

2sin12cos)( 1

ku

T

T

kx

kx

TT

TT

kx

kx

2sin5.0

5.02cos5.0

2cos2sin

2sin2cos

1

1

2

1

2

1

kTx

kTxkTy

2

101

T

nji

24Im

Check:

TT

TTTTrankGHHrank

2sin5.02sin5.0

2sin5.02cos5.02cos5.05.02cos5.0 22

Above matrix rank is 2 only if 2

02sin5.0n

TT

14

VI.4. Useful Transformation

Transforming state-space equations into canonical forms kHukGxkx 1 6.15

kDukCxky 6.16 We will transform the equations 6.15-6.16 into the following three canonical forms: 1) Controllable canonical form 2) observable canonical form, 3) diagonal or Jordan canonical form. A) controllable canonical form

MWT , HGGHHM n 1

0001

001

01

1

1

32

121

a

aa

aaa

Wnn

nn

The elements ia shown in matrix W are coefficients of the characteristic equation

011

1

nnnn azazazGzI

Now let us define kxTkx

Then eq. 15-16 become

kuHkxGkHuTkxGTTkx 111

kuDkxCkDukxCTky

Where, DDCTCHTHandGTTG ,,11 , or

15

ku

kx

kx

kx

kx

aaaakx

kx

kx

kx

n

n

nnnn

n

1

0

0

0

1000

0100

0010

1

1

1

1

1

2

1

121

1

2

1

kuD

kx

kx

kx

babbabbabky

n

nnnn

2

1

0110110

Where kb are those coefficients appearing in the numerator of the following pulse

transfer function.

nn

nnnn

nn

azazaz

bzbzbzbDHGZICDHGZIC

11

1

11

1011

B) Observable canonical form

1* WNQ , ***** 1CGCGCN n

1

2

11

100

010

001

000

a

a

a

a

GGQQ n

n

n

011

011

0

1

bab

bab

bab

HHQ nn

nn

And

1000 CCQ

By defining kxQkx


kuHkxGkx 1

kuDkxCky

16

Where, DDCQCHQHandGQQG ,,11 , or

ku

bab

bab

bab

bab

kx

kx

kx

kx

a

a

a

a

kx

kx

kx

kx

nn

nn

n

n

n

n

n

n

n

011

022

011

0

1

2

1

1

2

1

1

2

1

100

010

001

000

1

1

1

1

kuD

kx

kx

kx

ky

n

2

1

1000

C) Diagonal Jordan Canonical Form If the eigenvalues of the matrix G are distinct, the corresponding eigenvectors n 21 are distinct.

Define the transformation matrix nP 21

nP

P

P

GPP

00

00

000

00

2

1

1

By defining kxPkx


kuHkxGkx 1

kuDkxCky

Where, DDCPCHPHandGPPG ,,11 , or

17

ku

kx

kx

kx

kx

P

P

P

kx

kx

kx

kx

n

n

n

n

nn

n

1

2

1

1

2

1

2

1

1

2

1

000

0000

000

000

1

1

1

1

kuD

kx

kx

kx

ky

n

n

2

1

21

Where the i and i are constants such that i i will appear in the numerator of the term

ipz 1

when the pulse transfer function is expanded into partial fractions as follows:

Dpzpzpz

DHGZICDHGZICn

nn

2

22

1

1111

In many cases we choose 121 n .

Remarks: The sufficient and necessary conditions for the system to be completely state controllable is that 0i and sufficient and necessary conditions for the system to be

completely observable is 0i .

The case for multiple eigenvalues ip of matrix G then Jordan canonical form will be

formed. ( skipped) Invariance property of the rank condition for the controllability matrix and observability matrix. 1) For controllability matrix HGGHHM n 1

Let P be a transformation matrix and GGPP~1 , HHP

~1

Then:

111111

311131

21121

~

~

~

nn GGPGPPGPPPPGP

GGPGPPGPPPPGP

GGPGPPPPGP

18

M

HGHGH

HPPGPHGPPPHP

HGPGHPHP

HGGHHPMP

n

n

n

n

~

~~~~~ 1

111111

1111

111

Since matrix P is no singular, rank M= rank M~

Similarly, for the observability matrix

***** 1CGCGCN n

Let P be a transformation matrix and GGPP~1 , CCP

~

Then

N

CGCGC

CGPCGPCPNPn

n

~*

~*

~*

~*

~*

~*********

1

1

Since P is non singular, rank N= rank N~

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VI.5. Design Via Pole Placement

Remarks: if the system is completely state controllable, then poles of the closed-loop system may be placed at any desired locations by means of state feedback through an appropriate state feedback gain matrix. Necessary and sufficient conditions for arbitrary pole placement The state equation is kHukGxkx 1 6-17 Where kx state vector at kth sampling instant

ku control signal at kth sampling instant G nn matrix H 1n matrix

Figure 6.1

If the control signal is chosen as kKxku Where K is the feedback gain matrix. Then the system becomes a closed loop control system and its state equation becomes kxHKGkx 1 6-18

We will choose K such that the eigenvalues of HKG are desired closed-loop poles,

nuuu ,, 21

Remarks: We can approve that a necessary and sufficient condition for arbitrary pole placement is that the system be completely state controllable.

1) For necessary conditions, we can assume that 6-17 is not completely state controllable. Then the controllability matrix is not full rank, then we can find out that matrix K can not control all the eigenvalues.

20

2) The sufficient conditions. We will approve that if the system is completely state controllable, then we will find matrix K that make the eigenvalues of HKG as desired.

The desired eigenvalues of HKG are nuuu ,, 21 .

Noting the characteristic equation of the original system 6-17 is 01

11

nn

nn azazazGZI

Figure 6.2

We define a transformation matrix T as follows: MWT , where HGGHHM n 1 , which is of rank n,

And

0001

001

01

1

1

32

121

a

aa

aaa

Wnn

nn

nn aaa，a ,121 are from 011

1

nnnn azazazGZI

Then we will have

21

121

1

1000

0100

0010

aaaa

GGTT

nnn

And

1

0

0

0

1

HHT

Next we define 11

nnKTK

Then

121

11

0000

0000

0000

1

0

0

0

nnn

nnKH

The characteristic equation HKGZI

Becomes as follows:

0

1000

010

001

0000

0000

0000

1000

0100

0010

1000

0000

0010

0001

111

11

112211

121121

nnnnnn

nnnnnn

nnnnnn

azazaz

azaaa

z

z

aaaa

z

KHGZIHKGZI

The characteristic equation with the desired eigenvalues is given by

22

01

1121

nn

nnn zzzuzuzuz

111 a

222 a

nnn a

Hence, from the equation we have

1

1111

111

1

Taaa

T

TKK

nnnn

nn

Ackermann’s Formula By using the state feedback kKxku , we wish to place the closed loop poles at nuuu ,, 21 .

That is, we desire the characteristic equation to be 01

1121

nn

nnn zzzuzuzuzHKGZI

Let us define HKGG

Since Cayley-hamilton theorem states that G

satisfies its own characteristic equation, we have

011

1 IGGGG nn

nn

We now use this equation to derive ackerman’s formula. II

HKGG

GHKGHKGHKGHKGG

22

22323 GHKGGHKHKGGGHKGHKGHKGHKGHKGHKGG

121 nnnnn GHKGGHKHKGGHKGHKGHKGG

Multiply the above equations in order by 1,,, 11 nn and adding the results

Left side is G

=0. Right side can be written as

23

02

32

121

1

11221

K

GKGKaKa

GKGKaKa

HGGHHG

HKGGHKGHKaGHKaHKaG

nnn

nnn

n

nnnnn

Since the system is completely state controllable, the controllability matrix HGGHH n 1 is of rank n and its inverse exists. We can have

GHGGHH

K

GKGKaKa

GKGKaKa

nn

nn

nnn

112

32

121

Premultiplying both side of the equation by 100 We have

GHGGHHK n 11100 6-19

Above equation is the ackerman’s formula to determine the feedback K. Once the desired characteristic equation is selected, there are several different ways to determine the corresponding state feedback matrix K for a completely controllable system. Method 1 :

1

1111

111

1

Taaa

T

TKK

nnnn

nn

Where The characteristic equation of the original system is

01

11

nn

nn azazazGZI

The characteristic equation with the desired eigenvalues is given by 01

1121

nn

nnn zzzuzuzuz

24

MWT , where HGGHHM n 1 , which is of rank n, And

0001

001

01

1

1

32

121

a

aa

aaa

Wnn

nn

Method 2 : The desired state feedback gain matrix K can be given by Ackermann’s formula. We have

GHGGHHK n 11100

Method 3: If the desired eigenvalues nuuu ,, 21 are distinct, then the desired state feedback gain matrix K

can be given as follows:

121111 nK

Where n 21 satisfy the equation

,iii uHKG ni ,2,1 ， since they are the eigenvectors of matrix HKG

HIuG ii1 ， ni ,2,1

Special case: for deadbeat response, nuuu ,, 21 =0

K is simplified into

121001 nK

HG ii

， ni ,2,1

Method 4: if the order of the system is low, substitute K into the characteristic equation. HKGZI =0 and then matches the coefficients of powers in z of this characteristic equation

with equal powers in z of the desired characteristic equations.

Example 6.4 Consider the system defined by kHukGxkx 1

Where ,11

11

G ,2

0

H

Note that

25

2211

11 2

zzz

zGZI

Hence ,21 a 22 a Determine a suitable feedback gain matrix such that the system will have the closed loop pole at ,5.05.0 jz ,5.05.0 jz Method 1:

rankfullGHHM ,22

20

controllable

rankfulla

W ,01

12

01

11

22

02

01

12

22

20MWT

The desired characteristic equation for the desired system is 5.05.05.05.05.0 2 zzjzjzGZI

,11 5.02

5.025.05.05.0

05.015.1

22

02)2(125.0

1

11122

TaaK

Method 2: Referring to Ackermann’s formula

G

G

G

GGHH

GHGGHHK n

05.0

05.0

5.05.010

22

2010

10

100

1

1

11

5.01

15.05.0

11

11

11

115.0

2

2 IIGGG

5.025.05.01

15.005.005.0

GK

Method 3:

26

12111 K

HIuG ii1 ， 2,1i

jj

j

j

Ij

HIuG

1

2

5.01

1

2

0

5.05.01

15.05.0

2

05.05.0

11

11

1

1

111

jj

j

j

Ij

HIuG

1

2

5.01

1

2

0

5.05.01

15.05.0

2

05.05.0

11

11

1

1

122

jj

jjj

j

jj

j

j

jjj

5.01

2

5.01

15.01

2

5.01

1

2.3

1

5.01

1

5.01

15.01

2

5.01

21

121

5.025.0

5.01

2

5.01

15.01

2

5.01

1

2.3

111

11 121

jj

jjj

j

j

K

Method 4:

For lower order system, it will be simpler to substitute the K into the characteristic equation.

27

022222

2121

11

2

0

11

11

0

0

1222

21

21

kkzkz

kzk

z

kkz

zHKGZI

The desired characteristic equation for the desired system is 05.05.05.05.05.0 2 zzjzjzGZI

Thus: 122 2 k and 5.0222 12 kk

25.01 k , 5.02 k Deadbeat Response. Consider the system defined by kHukGxkx 1

With state feedback kKxku The state equation becomes kxHKGkx 1

Note: the solution of the last equation is given by

0xHKGkx k If the eigenvalues of HKG lie inside the unit circle, then the system is asymptotically stable. By choosing all eigenvalues of HKG to be zero, it is possible to get the deadbeat response, or nqqkforkx ,,0

Nilpotent matrix:

0000

1000

0100

0010

nnN , we have 0nN

28

Recall:

0

1000

010

001

0000

0000

0000

1000

0100

0010

1000

0000

0010

0001

111

11

112211

121121

nnnnnn

nnnnnn

nnnnnn

azazaz

azaaa

z

z

aaaa

z

KHGZIHKGZI

when nuuu ,, 21 =0 we can easily get

0000

1000

0100

0010

0000

0000

0000

1000

0100

0010

121121

nnnnnn aaaa

KHG

Which is a nilpotent matrix.

Thus we have 0n

KHG

In terms of original state , we have

00

0000

1

111

xTKHGT

xTKTHGTxTKTHGTxKHTTGTxHKGnxn

nnnn

29

Remarks: If the desired eigenvalues are all zeros then any initial state 0x can be brought to the origin in

at most n sampling periods and the response is deadbeat, provided the control signal ku is unbounded. In deadbeat response, the sampling period is the only design parameter. The designer must choose the sampling period carefully so that an extremely large control magnitude is not required in normal operation of the system. Trade off must be made between the magnitude of the control signal and the response speed.


Where ,11

11

G ,2

0

H

Note that

2211

11 2

zzz

zGZI

Hence ,21 a 22 a Determine a suitable feedback gain matrix such that the system will have the closed loop pole at 0z 0z , which is dead beat response.

22

02

01

12

22

20MWT

The desired characteristic equation for the desired system is

,01 02

105.05.0

05.022

22

02)2(020

1

11122

TaaK

kx

kx

kx

kx

kx

kx

kx

kxTT

kx

kxTT

kx

kx

2

1

2

1

2

1

2

11

2

11

2

1

00

10

22

02

20

00

5.05.0

05.0

22

02

11

11

5.05.0

05.0

102

0

11

11

1

1

For any initial state given by

b

a

x

x

0

0

2

1

30

000

10

1

1

2

1 b

b

a

x

x

and

0

0

000

10

2

2

2

1 b

x

x

Thus the state kX

for k=2,3,4… becomes zero and the response is indeed deadbeat. Control system with reference Input:

Figure 6.3

Consider the system in above figure. kHukGxkx 1

kCxky The control signal ku is given by

kKxkrKku 0

By eliminating ku from the state equation, we have

krHKkxHKGkKxkrKHkGxkx 001

The characteristic equation for the system is 0 HKGZI

If the system is completely state controllable, then the feedback gain matrix K can be determined to yield the desired closed-loop poles. Remarks: the state feedback can change the characteristic equation for the system, it also changes the steady state gain of the entire system. 0K can be adjusted such that the unit-step

response of the system at steady state is unity.

31

VI.6. State Observers

Note: 1) In practice, not all state variables are available for direct measurement. 2) In many practical cases, only a few state variables of a given system are measurable. 3) Hence, it is necessary to estimate the state variables that are not directly measurable.

Such estimation is commonly called observation.

A state observer, also called a state estimator, is a subsystem in the control system that performs an estimation of the state variables based on the measurements of the output and control variables. Full order state observation: estimate all n state variables regardless of whether some state variables are available for direct measurement. Minimum order state observation: observation of only the unmeasurable state variables. Reduced order state observation: observation of all unmeasurable state variables plus some of the measurable state variables.

Figure 6.4

Necessary and sufficient condition for state Observation The state equation is kHukGxkx 1 6-20

kCxky 6-21 Where

32

kx state vector at kth sampling instant

ku control vector at kth sampling instant

ky output vector (m-vector) G nn non singular matrix H rn matrix C nm matrix From 6-20, we have

kHuGkxGkx

kHuGkxkxG11

11

1

1

6-22

Shifting k by 1,

11

1111122

111111

kHuGkHuGkxG

kHuGkHuGkxGGkHuGkxGkx

Similarly 2112 1233 kHuGkHuGkHuGkxGkx

1111 11 nkHuGkHuGkHuGkxGnkx nnn

Substitute 6-22 into 6-21, we have kHuCGkxCGkCxky 11 1

1111 122 kHuCGkHuCGkxCGkCxky

21122 1233 kHuCGkHuCGkHuCGkxCGkCxky

11111 11 nkHuCGkHuCGkHuCGkxCGnkCxnky nnn

Combining them

1

10

00

1

1

1

11

1

12

1

2

1

nku

ku

ku

HCGHCGHCG

HCG

HCGHCG

HCG

kx

CG

CG

CG

nky

ky

ky

nnn

Or

33

1

10

00

1

11

11

1

12

1

2

1

nku

ku

ku

HCGHCGHCG

HCG

HCGHCG

HCG

nky

ky

ky

kx

CG

CG

CG

nnn

Note the right side of the equation is entirely known. Hence 1kx can be determined if and only if

nCG

CG

CG

2

1

is full rank. Or

C

CG

CGn

n

2

1

is full rank since G is not singular

Or ******* 12 CGCGCGC n is of full rank. If ky is scalar, and matrix C is a 1 by n matrix. Then 1kx can be obtained

1

10

00

1

11

11

1

12

11

2

11

2

1

nku

ku

ku

HCGHCGHCG

HCG

HCGHCG

HCG

CG

CG

CG

nky

ky

ky

CG

CG

CG

kx

nnnn

Remarks: 1) 1kx can be determined provided the system is completely observable.

2) in the presence of disturbance and measurement noise, 1kx can not be estimated accurately. 3) If C is not 1 by n matrix but is a m by n matrix, then the inverse of the matrix

nCG

CG

CG

2

1

is not defined. To cope with this, we will use dynamic model.

Consider the system is kHukGxkx 1 6-23

kCxky 6-24

We assume that the state kx is to be approximated by the state kx~ of the dynamic model kHukxGkx ~1~ 6-25

kxCky ~~ 6-26

34

Where matrices G, H, and C are the same as those original system. Let us assume that the dynamic model is subjected to the same control signal ku as the original system. If initial conditions are same for the original one as the dynamic one, then kx~ and kx

will be same. Otherwise, kx~ and kx will be different. If the matrix G is a stable one, however, kx~ will approach kx even for different initial conditions. If we denote the difference between kx~ and kx as kxkxke ~ Then subtract 6-25 by 6-20, we obtain kxkxGkxkx ~1~1 , or kGeke 1

ke will approach 0 if G stable. Remarks: although the state kx may not be measurable the output ky is measurable.

The dynamic model does not use the measured output ky . The dynamic model of equation 6-25 is modified into the following form: kxCkyKekHukxGkx ~~1~ , where matrix Ke serves as a weighting

matrix. Full-Order State Observer

Figure 6.5

Consider the state feedback control system above. The state equation is kHukGxkx 1 6-27

35

kCxky 6-28

kKxku Where kx state vector at kth sampling instant

ku control vector at kth sampling instant

ky output vector (m-vector) G nn non singular matrix H rn matrix C nm matrix K state feedback gain matrix We assume that the system is completely state controllable and completely observable, but kx is not available for direct measurement. Following figure shows a state observer incorporated into the system of previous figure.

Figure 6.6

kxKku ~

From above figure we have kxCkyKekHukxGkx ~~1~ 6-29

Which can be rewritten into kyKkHukxCKGkx ee ~1~ 6-30

36

Equation 6-30 is called a prediction observer since the estimate 1~ kx is one sampling

period ahead of the measurement ky . Error Dynamics of the full order state observer Notice that if kxkx ~

kHukxGkx ~1~ Which is identical to the state equation of the system. To obtain the observer error equation, let us subtract equation 6-30 from 6-27. kxkxCKGkxkx e

~1~1

Define kxkxke ~

keCKGke e1 6-31

From 6-31, we see that the dynamic behavior of the error signal is determined by the eigenvalues of CKG e . If CKG e is a stable matrix, the error vector will coverge

to zero for any initial error 0e . One way to obtain the fast response is to use deadbeat response. Remarks: 6-20, 6-21 are assumed to be completely observable, an arbitrary placement of the eigenvalues of CKG e is possible. Notice the eigenvalue of CKG e and

***eKCG are the same.

By use principle of duality, the condition for complete observability for the system defined by

kHukGxkx 1 6-32

kCxky 6-33 Is same as the complete state controllability condition for the system kuCkxGkx **1 , for this system by selecting a set of n desired eigenvalues

of KCG ** , the state feedback gain matrix K may be determined. The desired matrix

eK , such that the eigenvalues of CKG e are the same as those of KCG ** .

*KKe


kCxky

Where ,11

11

G ,

2

0

H ,20C

Design a full order state observer, assuming that the system configuration is identical to the in above figure. The desired eigenvalues of the observer matrix are

,5.05.0 jz ,5.05.0 jz and desired ch equation is

37

05.02 zz

Design of general prediction observers Consider the system defined by

kHukGxkx 1 6-34

kCxky 6-35 Where kx state vector

ku control vector

ky output vector G nn non singular matrix H rn matrix C n1 matrix The system is assumed to be completely state controllable and completely observable.

Thus the inverse of ******* 12 CGCGCGC n exists Assume the control law is kxKku ~ , where kx~ is the observed state and K is an

nr matrix. Assume the system configuration is in figure 6.6.

kKeCxkHukxKeCG

kxCkyKekHukxGkx

~

~~1~ 6-36

38

Define 1* WNQ , where ******* 12 CGCGCGCN n

0001

001

01

1

1

32

121

a

aa

aaa

Wnn

nn

Where 01

11

nn

nn azazazGZI are the ch equation of the original state

equation given by 6-34. Next define kQkx 6-37

kHuQkGQQk 111 6-38

kCQky 6-39

Where, DDCQCHQHandGQQG ,,11 , or

1

2

11

100

010

001

000

a

a

a

a

GQQ n

n

n

6-40

1000 CQ 6-41

Now define kQkx ~~

kKeCxkHukxKeCG

kxCkyKekHukxGkx

~

~~1~ will be changed into

kKeCQQkHuQkQKeCGQk 111 ~1

~ 6-42 Subtract 6-42 from 6-38, we have

kkKeCQQGQQkk ~1

~1 11 6-43

Define kkke ~

kQeKeCGQke 11 6-44 Remarks: Select the desired observer poles and then to determine matrix Ke . If we require ke to reach zero as soon as possible, then we require the error response to be

deadbeat, then all eigenvalues of KeCG must be zero.

39

Let us write

1

11

n

n

KeQ , then

1

2

1

1

11

000

000

000

000

1000

n

n

n

n

n

KeCQQ

And

11

22

11

1

111

100

010

001

000

1000

a

a

a

a

GQQQKeCGQ nn

nn

nn

n

n

The ch equation becomes

01 QKeCGQzI

0

100

010

01

00

11

22

11

az

a

az

az

nn

nn

nn

or

0111

111

nnnn

nn azazazQKeCGQzI 6-45

Suppose the desired characteristic equation for error dynamics is

011

1

nnnn zzz 6-46

Compare the coefficient of 6-45 and 6-46, we have

111 a

222 a

nnn a

40

Hence, from the equation we have

111 a

222 a

nnn a

Since

1

11

n

n

KeQ , we have

1

11*

1

1

n

n

n

n

WNQKe

Figure 6.7 Alternative representation of the observed-state feedback control system

For deadbeat response, the desired ch equation becomes 0nz

1

11*

1

1

a

a

a

WNQKe n

n

n

n

6-47

Ackermann`s Formula : procedure is same as for state feedback.

41

1

0

01

1

n

e

CG

CG

C

GK

Summary: The full order prediction observer is given by equation

kKeCxkHukxKeCG

kxCkyKekHukxGkx

~

~~1~

The observed state feedback is given by kxKku ~

If we have the feedback equation substituted into the observer equation, we obtain kyKkHukxHKCKGkx ee ~1~

Similar to the state feedback, four methods will be used to determine the observer feedback gain

eK .

Method 1 :

11

111

11

11 *

a

a

a

WN

a

a

a

QK nn

nn

nn

nn

e

Where 1* WNQ , ******* 12 CGCGCGCN n

0001

001

01

1

1

32

121

a

aa

aaa

Wnn

nn

i ’s are the coefficients of the desired characteristic equation

011

1

nnnn zzzGZI

The characteristic equation of the original system is 01

11

nn

nn azazazGZI

Note: if the system is already in an observable canonical form, then the matrix eK can be

determined easily, because matrix 1* WN becomes an identity matrix, thus IWN

1*

42

Method 2 : The desired observer feedback gain matrix eK can be given by Ackermann’s

formula. We have

1

0

01

1

n

e

CG

CG

C

GK

IGGGG nnnn

11

1

Method 3: If the desired eigenvalues nuuu ,, 21 of matrix CKG e are distinct, then the observer

feedback gain matrix eK may be given by the equation as follows:

1

1

11

2

1

n

eK

Where

n

2

1

satisfy the equation

1 IuGC ii ， ni ,2,1

Special case: for deadbeat response, nuuu ,, 21 =0

eK is simplified into

0

0

11

2

1

n

Ke

ii CG ， ni ,2,1

Method 4: If the order of the system is low, substitute eK into the characteristic equation.

CKGZI e =0 and then matches the coefficients of powers in z of this characteristic

equation with equal powers in z of the desired characteristic equations.

43


kCxky

Where ,11

11

G ,

2

0

H ,20C

Design a full order state observer, assuming that the system configuration is identical to the in above figure. The desired eigenvalues of the observer matrix are

,5.05.0 jz ,5.05.0 jz and desired ch equation is

05.02 zz 0222 zzGZI

Method 1: Method 2: Method 3:

Method 4:

Effects of the addition of the observer on a closed-loop system. Consider the completely state controllable and completely observable system defined by the equations

44

kHukGxkx 1

kCxky For the state feedback control based on the observed state kx~ , we have

kxKku ~ kxkxHKkxHKGkxHKkGxkx ~~1

Define kxkxke ~ kHKekxHKGkx 1

Also the observer was given by keCKGke e1

Combine this two:

ke

kx

CKG

HKHKG

ke

kx

e01

1

The characteristic equation for the system is

00

CKGZI

HKHKGZI

e

Pole placement design and the observer design are independent of each other. Remarks: The poles of the observer are usually chosen so that the observer response is much faster than the system response. A rule of thumb is to choose an observer response at least four to five times faster than the system response ( or deadbeat response). Current observer: In the prediction observer the observed state kx~ is obtained from measurements of the

output vector up to 1ky and of the control vector up to 1ku . A different

formulation of the state observer is to use ky for the estimation of kx~ . This can be

done in two steps. First step we determine 1kz , an approximation of 1kx based

on kx~ and ku . In the second step, we use 1ky to improve 1kx .

45

Minimum order observer:

State feedback control scheme where the feedback state consists of the measured portion of the state and the observed portion of the state obtained by use of the minimum order observer.

47

VI.7. Servo Systems

Note: It is generally required that the system has one or more integrators within the closed loop, to eliminate the steady state error to step inputs.

For the plant:

kHukGxkx 1 6-48

kCxky 6-49 Where kx Plant state vector

ku control vector

ky output vector G nn non singular matrix H mn matrix C nm matrix For the integrator :

kykrkvkv 1 6-50 Where kv actuating error vector

kr command input vector 6-50 can be rewritten as follows:

48

1

1

111

krkCHukvkCGx

kHukGxCkrkv

kykrkvkv

6-51

The control vector ku is given by

kvKkxKku 12 We then have

1

1

11

111

112122

12

12

12

krKkuCHKHKIkxCGKGKK

krkCHukvkCGxKkHukGxK

krkCHukvkCGxKkxK

kvKkxKku

m

6-52

Combine with kHukGxkx 1

We have

1

0

1

1

112122

krKku

kx

CHKHKICGKGKK

HG

ku

kx

m

6-53

Output equation can be written as

ku

kxCky 0

For the step input rkr

rKku

kx

CHKHKICGKGKK

HG

ku

kx

m 112122

0

1

1 6-54

rKu

x

CHKHKICGKGKK

HG

u

x

m 112122

0 6-55

Define the error vector xkxkxe , and ukukue

Subtracting the equation 6-55 from 6-54, we obtain

kw

Iku

kxHG

ku

kx

CHKHKICGKGKK

HG

ku

kx

me

e

e

e

me

e

0

00

1

1

12122 6-56

Where

ku

kxCHKHKICGKGKKkw

e

em 12122 6-57

Define

49

ku

kxk

e

e , n+m vector

00

HGG

, mnmn matrix

mIH

0, mmn matrix

mn

m

ICHCG

HIGKK

CHKHKICGKGKKK

012

12122

, mnm matrix

Then we have kwHkGk

1 6-58

And kKkw

Thus, 6-58 will be completely state controllable, K

can be designed, and 12 KK can be obtained using following equation:

mn

mn IK

CHCG

HIGKKI

CHCG

HIGKKK 00 1212

Example B-6-17 : Figure shows a servo system where the integral controller has a time delay of one sampling period. Determine the feedforward gain K1 and the feedback gain K2 such that the response to the unit step sequence input is deadbeat.

Solution:

jason gujasongu.org/4601/ch6.pdf2 vi.2. controllability a control system is said to be completely...

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