observer based comp - Memorial University of Newfoundland · the eigenvalues ofA−LC to be about 10 times farther to the left in the complex plane than the (dominant) eigenvalues

1

ENGI 7825: Control Systems II

Instructor: Dr. Andrew Vardy

Adapted from the notes of Gabriel Oliver Codina

��

�� s u(t)! x(t)! y(t)!

+!

+!

+!+!

x(t)!x0!

��

��

Observer-Based Compensators

Observer-based Compensators Lets now investigate what happens if the estimated state, given by the full-order observer replaces the true state vector in the state feedback control law, provided (A,B) is controllable and (A,C) is observable: OBSERVER-BASED COMPENSATORS AND THE SEPARATION PROPERTY 325

+!

r(t) u(t)Plant x(t)

Observer x(t)

y(t)

K

^

x(t)^

FIGURE 8.6 High-level observer-based compensator block diagram.

+

+!

+!

+

+

Plant x(t)r(t)

x(t)

u(t) y(t)

K

L

A

B C^x(t) y(t)

x0^

FIGURE 8.7 Detailed observer-based compensator block diagram.

This interconnection of a state feedback control law and a state observerresults in a dynamic observer-based compensator given by the stateequation

˙x(t) = Ax(t) + Bu(t) + L[y(t) ! Cx(t)]

u(t) = !Kx(t) + r(t)

or˙x(t) = (A ! BK ! LC)x(t) + Ly(t) + Br(t)

u(t) = !Kx(t) + r(t) (8.4)

which is a particular type of dynamic output feedback compensator.A high-level observer-based compensator block diagram is shown inFigure 8.6; details are shown in Figure 8.7.

The feedback interconnection of the open-loop state equation (8.1) andthe observer-based compensator (8.4) yields the 2n-dimensional closed-loop state equation

!x(t)˙x(t)

"=

!A !BK

LC A ! BK ! LC

" !x(t)x(t)

"+

!BB

"r(t)

Note: These notes do not include the initial pre-multiplication of r(t) by G. This scaling matrix can easily be introduced if needed to adjust the steady-state value of y(t).

The input to the plant is not the same as for state feedback. It is now based on the state estimate vector, which evolves according to the observer: After substituting u(t) we see that the state equation for the observer is impacted by the inclusion of K:

(t)(t)t rKxu +−=)(

(t)(t)t rxKu +−= ˆ)(

( ) )()(ˆ)(ˆ tt(t)t BrLyxLCBKAx ++−−=

OBSERVER-BASED COMPENSATORS AND THE SEPARATION PROPERTY 325

+!

r(t) u(t)Plant x(t)

Observer x(t)

y(t)

K

^

x(t)^

FIGURE 8.6 High-level observer-based compensator block diagram.

+

+!

+!

+

+

Plant x(t)r(t)

x(t)

u(t) y(t)

K

L

A

B C^x(t) y(t)

x0^

FIGURE 8.7 Detailed observer-based compensator block diagram.

This interconnection of a state feedback control law and a state observerresults in a dynamic observer-based compensator given by the stateequation

˙x(t) = Ax(t) + Bu(t) + L[y(t) ! Cx(t)]

u(t) = !Kx(t) + r(t)

or˙x(t) = (A ! BK ! LC)x(t) + Ly(t) + Br(t)

u(t) = !Kx(t) + r(t) (8.4)

which is a particular type of dynamic output feedback compensator.A high-level observer-based compensator block diagram is shown inFigure 8.6; details are shown in Figure 8.7.

The feedback interconnection of the open-loop state equation (8.1) andthe observer-based compensator (8.4) yields the 2n-dimensional closed-loop state equation

!x(t)˙x(t)

"=

!A !BK

LC A ! BK ! LC

" !x(t)x(t)

"+

!BB

"r(t)

x(t) = (A −LC)x(t)+Bu(t)+Ly(t)

Combining the open-loop (OL) state equation and the observer-based compensator yields the 2n-dimensional closed-loop (CL) state equation: In state feedback design we could adjust K to place the eigenvalues of A – BK and for observer design we could adjust L to place the eigenvalues of A - LC. However, how do we place the eigenvalues for the following 2n dimensional system dynamics matrix?

[ ] ⎥⎦

⎤⎢⎣

⎡⋅=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−−

−=⎥

⎦

⎤⎢⎣

⎡

(t)(t)

(t)

(t)(t)(t)

(t)(t)

xx

0Cy

rBB

xx

LCBKALCBKA

xx

ˆ

ˆ

(t)(t)(t)(t)(t)

CxyBuAxx

=+= ( )

(t)(t)ttt(t)t

rxKuBrLyxLCBKAx

+−=++−−=

ˆ)()()(ˆ)(

⎥⎦

⎤⎢⎣

⎡−−

−LCBKALC

BKA

2

Using the state-vector we obtained an “A” matrix that brought no clear interpretation. So we transform to use the state vector

Using the coordinate transformation: the CL state equation becomes: The eigenvalues of a triangular matrix lie along the main diagonal. This extends to block triangular matrices such as the following:

Where ¾() is a function which extracts eigenvalues. Remembering that coordinate transformations do not affect the eigenvalues, we can conclude that: Which is known as the separation property of observer-based compensators. It means that we can design the state feedback (K) and observer components (L) separately.

⎥⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡(t)(t)

(t)(t)

(t)(t)

(t)(t)

xx

ΙΙ0I

xx

xx

ΙΙ0I

xx

~ˆˆ~

[ ] ⎥⎦

⎤⎢⎣

⎡⋅=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−

−−=⎥

⎦

⎤⎢⎣

⎡

(t)(t)

(t)

(t)(t)(t)

(t)(t)

xx

0Cy

r0B

xx

LCA0BKBKA

xx

~

~~

σ A −BK −BK

0 A −LC⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎞

⎠⎟=σ A −BK( )σ A −LC( )

( ) ( )LCABKALCBKALC

BKA−−=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−−

−σσσ

Example ►  Lets consider (again) the following system:

►  We already looked at designing an observer for this system and

obtained L = [5 2 -20]T

►  Now we just proceed to state feedback design---the separation principle tells us that we can design a K matrix independently of the observer (i.e. we proceed as if there was no observer). These are the desired eigenvalues and characteristic polynomial:

►  The system is CCF so we can easily obtain K:

306 OBSERVERS AND OBSERVER-BASED COMPENSATORS

along with the fact that

Q!1OCF =

!

""""""#

a1 a2 · · · an!1 1a2 a3 · · · 1 0...

... . .....

...

an!1 1 · · · 0 01 0 · · · 0 0

$

%%%%%%&= P !1

CCF

is symmetric, we obtain the Bass-Gura formula for the observer gainvector, that is,

L =

!

""""""#

!

""""""#

a1 a2 · · · an!1 1a2 a3 · · · 1 0...

... . .....

...

an!1 1 · · · 0 01 0 · · · 0 0

$

%%%%%%&Q(A,C)

$

%%%%%%&

!1 !

""""""#

!0 ! a0

!1 ! a1

!2 ! a2

...

!n!1 ! an!1

$

%%%%%%&

Example 8.1 We construct an asymptotic state observer for the three-dimensional state equation

!

"#x1(t)

x2(t)

x3(t)

$

%& =

!

#0 1 00 0 1

!4 !4 !1

$

&

!

"#x1(t)

x2(t)

x3(t)

$

%& +

!

"#001

$

%& u(t)

y(t) = [ 1 0 0 ]

!

"#x1(t)

x2(t)

x3(t)

$

%&

Since this state equation is in controller canonical form, we obtain theopen-loop characteristic polynomial by inspection, from which we deter-mine the polynomial coefficients and eigenvalues as follows:

a(s) = s3 + s2 + 4s + 4= (s + 1)(s ! j2)(s + j2)

"1,2,3 = !1, ±j2a2 = 1 a1 = 4 a0 = 4

We choose the following eigenvalues for the observer error dynamics:

{µ1, µ2, µ3} = {!1, !2, !3}


along with the fact that

Q!1OCF =

!

""""""#

a1 a2 · · · an!1 1a2 a3 · · · 1 0...

... . .....

...

an!1 1 · · · 0 01 0 · · · 0 0

$

%%%%%%&= P !1

CCF

is symmetric, we obtain the Bass-Gura formula for the observer gainvector, that is,

L =

!

""""""#

!

""""""#

a1 a2 · · · an!1 1a2 a3 · · · 1 0...

... . .....

...

an!1 1 · · · 0 01 0 · · · 0 0

$

%%%%%%&Q(A,C)

$

%%%%%%&

!1 !

""""""#

!0 ! a0

!1 ! a1

!2 ! a2

...

!n!1 ! an!1

$

%%%%%%&

Example 8.1 We construct an asymptotic state observer for the three-dimensional state equation

!

"#x1(t)

x2(t)

x3(t)

$

%& =

!

#0 1 00 0 1

!4 !4 !1

$

&

!

"#x1(t)

x2(t)

x3(t)

$

%& +

!

"#001

$

%& u(t)

y(t) = [ 1 0 0 ]

!

"#x1(t)

x2(t)

x3(t)

$

%&

Since this state equation is in controller canonical form, we obtain theopen-loop characteristic polynomial by inspection, from which we deter-mine the polynomial coefficients and eigenvalues as follows:

a(s) = s3 + s2 + 4s + 4= (s + 1)(s ! j2)(s + j2)

"1,2,3 = !1, ±j2a2 = 1 a1 = 4 a0 = 4

We choose the following eigenvalues for the observer error dynamics:

{µ1, µ2, µ3} = {!1, !2, !3}


Because of the block triangular structure of the transformed closed-loopsystem dynamics matrix, we see that the 2n closed-loop eigenvalues aregiven by

!

!"A ! BK BK

0 A ! LC

#$= ! (A ! BK ) " ! (A ! LC )

Because state coordinate transformations do not affect eigenvalues, weconclude that

!

!"A !BK

LC A ! BK ! LC

#$= ! (A ! BK ) " ! (A ! LC )

This indicates that the 2n closed-loop eigenvalues can be placed by sep-arately and independently locating eigenvalues of A ! BK and A ! LCvia choice of the gain matrices K and L, respectively. We refer to this asthe separation property of observer-based compensators.

We generally require the observer error to converge faster than thedesired closed-loop transient response dynamics. Therefore, we choosethe eigenvalues of A ! LC to be about 10 times farther to the left in thecomplex plane than the (dominant) eigenvalues of A ! BK .

Example 8.6 We return to the open-loop state equation in Example 8.1.The open-loop characteristic polynomial is a(s) = s3 + s3 + 4s + 4. Theopen-loop the eigenvalues "1,2,3 = !1, ±j2 indicate that this is amarginally stable state equation because of the nonrepeated zero-real-parteigenvalues. This state equation is not bounded-input, bounded outputstable since the bounded input u(t) = sin(2t) creates a resonance withthe imaginary eigenvalues and an unbounded zero-state response results.We therefore seek to design an observer-based compensator by combiningthe observer designed previously with a state feedback control law.

Given the following eigenvalues to be placed by state feedback, namely,

µ1,2,3 = ! 12 ± j

#3

2 , !1

the desired characteristic polynomial is #(s) = s3 + 2s2 + 2s + 1. Sincethe open-loop state equation is in controller canonical form we easilycompute state feedback gain vector to be

K = KCCF

= [( #0 ! a0) (#1 ! a1) (#2 ! a2) ]

= [ (1 ! 4) (2 ! 4) (2 ! 1) ]

= [ !3 !2 1 ]



!

!"A ! BK BK

0 A ! LC

#$= ! (A ! BK ) " ! (A ! LC )


!

!"A !BK

LC A ! BK ! LC

#$= ! (A ! BK ) " ! (A ! LC )





µ1,2,3 = ! 12 ± j

#3

2 , !1


K = KCCF

= [( #0 ! a0) (#1 ! a1) (#2 ! a2) ]

= [ (1 ! 4) (2 ! 4) (2 ! 1) ]

= [ !3 !2 1 ]



!

!"A ! BK BK

0 A ! LC

#$= ! (A ! BK ) " ! (A ! LC )


!

!"A !BK

LC A ! BK ! LC

#$= ! (A ! BK ) " ! (A ! LC )





µ1,2,3 = ! 12 ± j

#3

2 , !1


K = KCCF

= [( #0 ! a0) (#1 ! a1) (#2 ! a2) ]

= [ (1 ! 4) (2 ! 4) (2 ! 1) ]

= [ !3 !2 1 ]


On inspection of

A ! BK =! 0 1 0

0 0 1!1 !2 !2

"

we see from the companion form that the desired characteristic polynomialhas been achieved. The previously specified observer (from Example 8.1),together with the state feedback gain vector computed independently ear-lier yields the observer-based compensator (8.4) [with r(t) = 0], that is,

#

$%

˙x1(t)˙x2(t)˙x3(t)

&

'( =!!5 1 0

!2 0 119 !2 !2

"#

%x1(t)x2(t)x3(t)

&

( +! 5

2!20

"

y(t)

u(t) = [ 3 2 !1 ]

#

%x1(t)x2(t)x3(t)

&

(

yielding the six-dimensional homogeneous closed-loop state equation#

$$$$$$$%

x1(t)x2(t)x3(t)˙x1(t)˙x2(t)˙x3(t)

&

'''''''(

=

#

$$$$$%

0 1 0 0 0 00 0 1 0 0 0

!4 !4 !1 3 2 !15 0 0 !5 1 02 0 0 !2 0 1

!20 0 0 19 !2 !2

&

'''''(

#

$$$$$$%

x1(t)x2(t)x3(t)

x1(t)x2(t)x3(t)

&

''''''(

y(t) = [ 1 0 0 | 0 0 0 ]

#

$$$$$$%

x1(t)x2(t)x3(t)

x1(t)x2(t)x3(t)

&

''''''(

We investigate the closed-loop performance for the following initialconditions and zero reference input:

x0 =! 1

!11

"

x0 =

#

%000

&

( r(t) = 0 t " 0

These are the same initial conditions as in Example 8.1, for whichthe zero-input response of the closed-loop state equation is shown in


0 5 10 15!2

!1

0

1

Closed-LoopObserver

0 5 10 15!2

!1

0

1

time (sec)0 5 10 15

!2

!1

0

2

1

x 3 (

unit/

sec2 )

x 2 (

unit/

sec)

x 1 (

unit)

FIGURE 8.8 Closed-loop state response for Example 8.6.

Figure 8.8. Each state variable is plotted along with its estimate, and theresponses show that the estimates converge to the actual state variablesand that all six closed-loop state variables asymptotically tend to zero astime tends to infinity, as expected. !

Example 8.7 We return to the open-loop state equation!

"x1(t)x2(t)x3(t)

#

$ =% 0 1 0

0 0 1!18 !15 !2

&!

"x1(t)x2(t)x3(t)

#

$ +

!

"001

#

$u(t)

y(t) = [ 1 0 0 ]

!

"x1(t)x2(t)x3(t)

#

$

for which the state feedback gain vector

K = [ 35.26 24.55 14.00 ]

Observer-Based State Feedback with Integral Control

State feedback with integral control (a.k.a. robust steady-state tracking) provides the ability to eliminate steady-state error in systems lacking an inherent integrator. It also provides robustness against disturbances. This concept can be extended to system’s with inaccessible states, necessitating the inclusion of a state observer in the loop: STEADY-STATE TRACKING WITH OBSERVER-BASED COMPENSATORS 341

+!

kI A,B Plant

K

r(t) "(t)"(t) u(t)+!

x(t)C

y(t)

Observerx(t)^

.

FIGURE 8.11 Observer-based servomechanism block diagram.

instead. The Observer block represents

˙x(t) = (A ! LC)x(t) + Bu(t) + Ly(t)

which, as shown in the block diagram, has u(t) and y(t) as inputs.

Example 8.9 We construct an observer-based type I servomechanismfor the state equation of Example 8.2 by incorporating the observer de-signed in Example 8.2 into the state-feedback-based type 1 servomech-anism designed in Example 7.9. In addition to the assumptions veri-fied in Example 7.9, the state equation is observable as demonstratedin Example 8.2. We recall from Example 7.9 that the augmented statefeedback gain vector

[ K !kI ] = [ 3.6 !1.4 2.2| !16 ]

yields

!A ! BK BkI

!C 0

"=

#

$$%

0 1 0 00 0 1 0

!21.6 !13.6 !4.216!1 0 0 0

&

''(

having eigenvalues at !0.848 ± j2.53 and !1.25 ± j0.828. In addition,the observer gain vector

L =

#

%158

362443624

&

(

locates the eigenvalues of A ! LC at !13.3 ± j14.9, !133.3, which aremuch farther to the left in the complex plane. Everything is in place tospecify the four-dimensional observer-based compensator:

#

$$$%

! (t)˙x1(t)˙x2(t)˙x3(t)

&

'''(=

#

$$%

0 0 0 00 !158 1 00 !3624 0 116 !43645.6 !13.6 !4.2

&

''(

#

$$%

!(t)

x1(t)x2(t)x3(t)

&

''(

3

Some conditions have to be assumed about the open-loop state equation: 1.  System is SISO. 2.  (A, B) is controllable, and (A, C) is observable. 3.  It does not have poles at s=0. 4.  It does not have zeros at s=0.

Then, the observer-based compensator is of the form: Which can be rewritten as an (n+1)-dimensional state equation:

(t)k(t)t Iξ+−= xKu ˆ)(

)()( tytr(t) −=ξ

)()(ˆ)()(ˆ tt(t)t LyBuxLCAx ++−=

[ ] ⎥⎦

⎤⎢⎣

⎡⋅−=

)(ˆ)(

t(t)

kt I xKu

ξ

y(t)r(t)(t)t

k(t)t

I⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡L0xLCBKABx

11ˆ)(00

ˆ)( ξξ

Combine these with the open-loop equation to obtain the (2n+1)-dimensional closed-loop equation: Once again, a coordinate transformation can be performed to replace the observer state with the observer error . This leads to: The block triangular structure of this new system dynamics matrix indicates that the (2n+1) eigenvalues are:

a) the (n+1) eigenvalues of

which can be freely assigned by proper choice of the augmented state feedback gain vector K*=[K -kI] b) The n eigenvalues of (A-LC) which can be arbitrarily located by the observer gain vector L.

r(t)(t)t(t)

k

k

(t)t(t)

I

I

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

x

x

LCBKABLC0CBKBA

x

x1

ˆ)(0

ˆ)( ξξ

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅=(t)t(t)

tyx

x0C

ˆ)(0)( ξ

)(ˆ tx )(ˆ)()(~ txtxtx −=

r(t)(t)t(t)k

(t)t(t) I

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

x

x

LCA000CBKBBKA

x

x1

~)(0

~)( ξξ

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅=(t)t(t)

tyx

x0C

~)(0)( ξ

⎥⎦

⎤⎢⎣

⎡−−

0CBBKA Ik

y(t)r(t)(t)t

k(t)t

I⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡L0xLCBKABx

11ˆ)(00

ˆ)( ξξ

[ ] ⎥

⎦

⎤⎢⎣

⎡⋅−=

)(ˆ)(

t(t)

kt I xKu

ξ

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observer based comp - Memorial University of Newfoundland · the eigenvalues ofA−LC to be about 10 times farther to the left in the complex plane than the (dominant) eigenvalues