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MM13MM13 State-Space Method –– Feedback with EstimatorFeedback with Estimator

Reading Material:

DC: p.289-299, 302-310, 314-319

1. Observability2. Estimator Design3. Combined Control Law and Estimator

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PreconditionsThe considered system (A,B) is controllable; All the state variables are available for feedback.

MM12. Full State Feedback ControlMM12. Full State Feedback Control

X=AX+Bu C

u=-KX

0))(det(0)det(

)(..

=−−=−

⎪⎩

⎪⎨⎧

=−=⇒

⎪⎩

⎪⎨⎧

=+=

BKAsIAsICXY

XBKAXCXY

BUAXX

Open loop system Closed loop system

If some state variable can not be measured or the measuring is quite expensive... How about this situation?

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.

When All States are not available...When All States are not available...

X=AX+Bu C

u=-KX

If some state variable can not be measured or the measuring is quite expensive... How about this situation?

?state

Estimator

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How to get an estimator? How to get an estimator?

What kind of condition should be satisfied in order to get a correct estimator? If possible, what kind of structure should an estimator have? What’s the parameters inside the estimator should be adjust? And how? How to combine the estimator with the feedback controller? What’s the estimator design for discrete-time systems? How to get the reference input into the controlled loop?

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Observability:The considered system is called observable if and only if the observability matrix is full column rank:

1. Observability (1/2)1. Observability (1/2)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−1

:

nCA

CAC

OmatrixityobservabilM

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Observer canonical form: 1. Observability (2/2)1. Observability (2/2)

[ ]

nnnn

nnn

o

n

o

n

o

oo

oooo

asasassa

bsbsbsbwhere

Cb

bb

B

a

aa

A

XCYUBXAX

++++=

+++=

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−

=

⎪⎩

⎪⎨⎧

=+=

−−

−−

L

L

L

MOMM

ML

L

22

11

22

11

2

1

2

1

.

)(

,)(

001

,,

0001

001

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How to get an estimator? How to get an estimator?

What kind of condition should be satisfied in order to get a correct estimator?If possible, what kind of structure should an estimator have? What’s the parameters inside the estimator should be adjust? And how? How to combine the estimator with the feedback controller? What’s the estimator design for discrete-time systems? How to get the reference input into the controlled loop?

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ObjectiveEstimate the system state through output and input

Estimator Structure

2. Estimator Design (1/3)2. Estimator Design (1/3)

⎪⎩

⎪⎨⎧

=+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=−++=

CXYBUAXXand

l

lLwhereXCyLBuXAX

n

.1.

,)ˆ(ˆˆ M

(A,B)

(A,B)

C

L

C

u(t) Y(t)X(t)

+-

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Error Equation

2. Estimator Design (2/3)2. Estimator Design (2/3)

0))(det(,~)(~. =−−−= LCAsIXLCAX

)()())(det(

,

00)(100)(01)(

111

22

11

nnnn

nn

laslasLCAsI

la

lala

LCA

+++++=−−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+−+−

=−

− L

L

OMM

L

L

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Estimator’s Gain Design (Ackermann’s formula)Step 1: construct a polynominal of matrix variable

Step 2: calculate the observability matrix

Step 3: calculate the gain vector L= [ l1 l2 .... ln]T

2. Estimator Design (3/3)2. Estimator Design (3/3)

IAAAA nnnn

e ααασ ++++= −− L22

11)(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−1nCA

CAC

OM

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= −

1

0)( 1 MOAL eσ

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Control problemto select a row matrix K for satisfacotry placement of the poles of the system matrix A-BKControllability matrix TC=[B AB A2B .... An-1B ]Ackermann’s control formula

Estimation problemto select a column matrix L for satisfacotry placement of the poles of the system matrix A-LCObservability matrix OAckermann’s estimator formula

Duality of Estimation and Control (1/2)Duality of Estimation and Control (1/2)

)(]100[ 1 ATK cc σ−= L

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−1nCA

CAC

OM

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= −

1

0)( 1 MOAL eσ

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Duality:Control problem: A, B, C

Estimation prolem: AT, CT, BT

Duality of Estimation and Control (2/2)Duality of Estimation and Control (2/2)

L=acker(AT,CT,pe)’

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Principle rule: the estimator poles should be chosen to be faster than the control poles by a factor of 2 to 6.

A best estimator design is a balance between good transient response and low enough bandwidth

Pole selection methodsDominant second-order poles Prototype Design Symmetric Root Locus method

If sensor noise is a significant factor, the reduced-order estimator is less effective

Estimator Pole SelectionEstimator Pole Selection

See p.526-527...

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How to get an estimator? How to get an estimator?

What kind of condition should be satisfied in order to get a correct estimator? If possible, what kind of structure should an estimator have? What’s the parameters inside the estimator should be adjust? And how?How to combine the estimator with the feedback controller?

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3. Combined Control Law and Estimator3. Combined Control Law and Estimator

Closed loop system and poles

Plant Sensor Y=CX

ControllerK

Estimator

BuAXX +=.

)ˆ(

ˆˆ.

XCyL

BuXAX

−+

+=

0))(det())(det())(0

)(det(

~0~..

=+−+−=⎥⎦

⎤⎢⎣

⎡+−

+−

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

LCAsIBKAsILCAsI

BKBKAsI

XX

LCABKBKA

X

X Separation principleSeparation principle

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Dynamic Compensator Dynamic Compensator

State space description and transfer function

Plant Sensor Y=CX

ControllerK

Estimator

BuAXX +=.

)ˆ(

ˆˆ.

XCyL

BuXAX

−+

+=

Dynamic compensator

LLCBKAsIKsD

XKu

LyXLCBKAX

c1

.

))(()(

ˆ

ˆ)(ˆ

−−−−−=

⎪⎩

⎪⎨⎧

−=

+−−=

How about the relationship of the state space design with Frequency response and root locus design methods? How about the relationship of the state space design with Frequency response and root locus design methods?

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Example: Compensator DesignExample: Compensator Design

Consider a plant with transfer function G(s)=1/s2. Design a dynamic compensator such that the control poles at ωn=1rad/sec, ζ=0.707, and the estimator poles at ωn=5rad/sec, ζ=0.5. Step 1: get the state space description of the plant... Step 2: calculate the desired control poles and estimator poles..Step 3: calculate the state feedback gain K through the Ackermann’s control formula...Step 4: calculate the state feedback gain L through the Ackermann’s estimator formula...Step 5: Write out the dynamic compensator’s equation...

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% the original system can be found at FC p.530 example 7.19% system matricesA=[0 0; 1 0]; B=[1; 0]; C=[0 1]; D=0; sysP=ss(A,B,C,D); % desired control polesPc=[-sqrt(2)/2+j*sqrt(2)/2 -sqrt(2)/2-j*sqrt(2)/2]; %desired estimator poles wn=5; Pe=[-wn/2+j*sqrt(3)/2*wn -wn/2-j*sqrt(3)/2*wn]; K=acker(A,B,Pc); % calculate the control gainL=acker(A',C',Pe)'; % calculate the estimator gain% the compensator structure % dot(Xc)=(A-BK-LC)Xc+Ly = AcXc+Bcy

% u=-KXc = CcXcAc=A-B*K-L*C; Bc=L; Cc=K; Dc=0; sysD=ss(Ac,Bc,Cc,Dc); bode(sysD); grid; % the margins of the loop gain system

syscha=sysD*sysP; margin(sysP); hold; margin(syscha); hold; figure% Root locus analysis[Z,P,Kd]=zpkdata(sysD); sysDK=sysD/Kd; sysrl=sysP*sysDK; rlocus(sysrl); sgrid

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