Combined State Feedback Controller and Observer

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Combined State Feedback Controller and Observer

• Combined State Feedback Controller and Observer
  Formulation
• Separation Principle
• Transfer Function Representation
• Combined State Feedback Controller and Reduced
  Order Observer
• Illustrative Examples

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Motivation

• In most control applications all state variables
  are not measurable
• A full or reduced order observer may be used to
  estimate needed states
• Separation principle allows independent
  controller and observer design

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Combined Controller-Observer Formulation
Plant
Controller
Observer
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State Feedback Observer Block Diagram
u
y
r
System
L
x
y
z-1
C
H
Observer
G
-K
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Closed-Loop System
Closed-Loop Control Subsystem
Closed-Loop Observer Subsystem
Overall Closed-Loop System
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Separation Principle
Eigenvalues of the closed loop systems are the
union of eigenvalues of

• Closed-loop poles, i.e., eigenvalues of G-HK and
• Observer poles GA-LC

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Separation Principle--Transfer Function
Closed-Loop System Laplace Transform
Closed-loop transfer function

• Closed-loop transfer function is the same as full
  state feedback
• Observer dynamics are canceled

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Laplace Domain Representation
Plant
Controller
Observer
Laplace Transform of Observer
Controller in z-domain
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Transfer Function Block Digaram Representation
Control Input
Solving for U gives
where
Y
R
G
F1
F2
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D.C. Motor Example
Consider a d.c. servomotor system given by the
transfer function y position output u input
voltage the motor. Motor parameters k/J1 and
b/J5.
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Feedback Control Design for D.C. Motor Example
Desired Closed-Loop System damping ratio ?0.8
and natural frequency ?n500 rad/sec (less than
3 maximum overshoot and settling time of 0.01
sec.)
System in Controllable Form
Control gain
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Servo Example Observer Design
Observer Dynamics (2 times faster than
controller) ?0.8 and ?n1000 rad/sec
Observer gain
Observer State Equations
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Equivalent Transfer Function of Servo Example
Feedback and Feedforward Block
where
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Closed-Loop Transfer Function
R
G
F1
F2
Pole-zero Cancellation
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Combined Controller Reduced Order Observer (CCRO)
Plant State Equation
Reduced Order Observer
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Transfer Function Representation of CCRO
Partition State Feedback Gain
Reduced Order Observer Transfer Function
Substitute Z in Control Law
where
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Transfer Function Block Digaram Representation of
CCRO
Y
R
G
F1
F2
F1 and F2 are of order n-q (lower than FOO)
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D.C. Motor Example
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Closed-Loop System of CCRO Example
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Matlab Solution
Simulation Example of Combined
Observer System G(s)b/(s2bs) Observable
form dx1/dtx2, dx2/dt-ax2bu System
Matrices b1 a5 A0 10 -a B0b C1
0 D0 plantss(A,B,C,D)
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MATLAB Example Control Design
Control Design desired closed-loop damping and
natural frequency zeta0.8 wn500 pd-zetawnsq
rt(zeta2-1)wn desired closed-loop
poles Ackermans's formula to find
gain Kacker(A,B,pdconj(pd))
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MATLAB Example FOO Design
Full-order observer r2 pdo-rzetawnrsqrt(z
eta2-1)wn desired observer poles L(acker(A',C
',pdoconj(pdo)))' AoA-LC ffss(Ao,B,K,0)
Part of feedforward block fbss(Ao,L,K,0)
Feedback Block g-(K/Ao)L
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MATLAB Example ROO Design
Reduced-order observer pr-rwn A11A(1,1)
A12A(1,2) B1B(1) A21A(2,1) A22A(2,2)
B2B(2) K1K(1) K2K(2) Lr(A22-pr)/A12 ArA22
-LrA12 ByA21-LrA11ArLr BuB2-LrB1 ffrss(
Ar,Bu,K2,0) fbrss(Ar,By,K2,K1K2Lr) gr-(K2/Ar
)ByK1K2Lr
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Simulink Block Diagram
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Simulation Parameters
Slow Observers r0.5, Fast Observers
r5 Mismatch b1.5 Random input and measurement
noise
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Slow Observer Plots
FOO,ROO
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Fast Observer Plots
FOO,ROO
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Comments on Simulation Results

• Slow ROO performs better than slow FOO
• High speed ROO and FOO are similar

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End of Lecture 18
Next Lecture Optimal Control