Easy design for observable form. 11. Observer Form. ⢠Observer gain vector in terms of coefficients vectors. Example: Observer Form. ⢠Desired poles -9 ± j 9.
State Observers • In practice, the state variables are not all available for measurement (impossible or expensive). • Need estimate of the state vector for state feedback estimate of the state vector
State Observers M, Sami Fadali Professor of Electrical Engineering University of Nevada
1
2
Estimation Error (Open-Loop)
State Observer/Estimator
estimation error
• Observer A dynamic system whose input includes the control and the output y and whose output is an estimate of the state vector • Open-loop Observer
• Error Dynamics: – Unbounded error for unstable state matrix A. – Does not correct for modeling imperfections.
Fails due to modeling errors and disturbances. Unstable for unstable plant. 3
4
Full-Order Observer
Block Diagram of Observer u(t)
• Estimates all the state variables.
, L = observer gain • Choose the matrix to assign eigenvalues. • For perfect estimation, the equation reduces to the open-loop observer.
x Ax Bu
y(t)
x(t) C
xˆ Aobs xˆ Bu Ly
xˆ ( t )
5
Theorem (Proved Earlier)
Procedure I
• The system is observable if and only if the dual system is controllable. • If is controllable then its eigenvalues can be arbitrarily assigned (stable) using state feedback.
same eigenvalues!
6
7
1. Check the observability (detectability) of the system. 2. Solve the pole assignment problem for the dual system to obtain (can use MATLAB commands acker or place). For a detectable system, the unobservable modes must be included in the set of eigenvalues to be assigned. 3. Transpose to obtain the observer gain matrix .
8
Example: Observer Design
Observable Form
Desired poles 9 j9 >> l=place(a',c',[-9+j*9,-9-j*9]) ' % Can use "acker" l= 9.0000 78.0000
• Observable forms are always observable. • Obtain observer form from controller form and interchanging by transposing and (similarly using phase variable form). • Easy design for observable form.
9
Observer Form
10
Example: Observer Form
• Desired poles 9 j 9 • Observer Design • Observer gain vector in terms of coefficients vectors. 11
12
Calculation of the Transformation Matrix (i) Using the controllability matrix with the dual
Observer: Bass-Gura Formula replaced by
• Controller gain formula (to be used for observer)
and design a controller CC
C C
C C
(later: equivalently use observability matrix of original pair)
• For observer design use the transformation
• Transpose the “controller” gain to get the observer gain 13
14
Transformation Matrix
Example
• A system can be transformed to observable form if and only if it is observable: