Digital Control Systems
STATE OBSERVERS
State Observers
State Observers
State Observers
State Observers
Observer design
(S1-Actual system)
(S2 - Dynamic Model)
State Observers
Observer design
:
State Observers
Observer design
S2
Correction term
Full Order State Observer
State Feedback Control System
Assume that the system is completely state controllable and completely observable, but x(k) is not available
for direct measurement.
Full Order State Observer
Observed State Feedback Control System
Called as prediction observer. The eigenvalues of G-KeC are observer poles
Full Order State Observer
Error Dynamics of the full order observer
That is, the response of the state observer system is identical to the response of the original system
Observer error equation
Full Order State Observer
Error Dynamics of the full order observer
The dynamic behaviour of the error signal is determined by the eigenvalues of G-KeC.
If matrix G-KeC is a stable matrix
• the error vector will converge to zero for any initial error e(0)
•
will converge to
regardless of the values of
• If the eigenvalues of G-KeC are located in such a way that the dynamic behaviour of the error vector is adequately fast,
then any error will tend to zero with adequate speed.
• One way to obtain fast response is deadbeat response which can be achieved if all eigenvalues of G-KeC are
chosen to be zero
Full Order State Observer
Example:
rank(
)=2
Full Order State Observer
Example:
Full Order State Observers
Design of full order state observer by using observable canonical form
The system is completely state controllable and completely observable
Control law to be used :
State observer dynamics:
Full Order State Observers
Design of full order state observer by using observable canonical form
State transformation to observable canonical form:
Full Order State Observers
Design of full order state observer by using observable canonical form
State transformation to observable canonical form:
Full Order State Observers
Design of full order state observer by using observable canonical form
State Observer Dynamics
Full Order State Observers
Design of full order state observer by using observable canonical form
(S1-Actual system)
(S2-Dynamic system)
Define
then state observer dynamics :
Full Order State Observers
Design of full order state observer by using observable canonical form
Desired characteristic equation for the error dynamics is
Full Order State Observers
Design of full order state observer by Ackermann’s formula
Assumption: System is completely observable and the output y(k) is scalar.
Full Order State Observers
Example:
rank(
)=2
The system is completely observable
Characteristic equation of the system:
Desired characteristic equation for the error dynamics
Full Order State Observers
Example:
Design of full order state observer by using observable canonical form
Full Order State Observers
Example:
Design of full order state observer by using Ackermann’s Formula
Full Order State Observers
Example:
Design of full order state observer by causal method
Desired characteristic equation
Full Order State Observers
Effects of addition of the observer on a closed loop system
Completely controllable and
completely observable system
Full Order State Observers
Effects of addition of the observer on a closed loop system
Minimum-Order Observer
Full order state observers are designed to reconstruct all the state variables. But some state variables may be accurately
Measured. Such accurately measurable state variables need not be estimated.
An observer that estimates fewer than n state variables, where n is the dimension of the state vector, is called
reduced order observer.
If the order of the reduced order observer is the minimum possible, the observer is called a minimum-order observer.
Note that if the measurement of output variables involves significant noises and is relatively inaccurate then the use of
full order observer may result in a better system performance
Minimum-Order Observer
Minimum-Order Observer
(
(
)
)
Minimum-Order Observer
The state and output equations for full order observer:
The state and output equations for minimum order observer:
known quantities
Minimum-Order Observer
List of necessary substitutions for writing the observer
equation for the minimum order state observer
Observer equation for for the full order observer
Observer equation for for the minimum order observer
Minimum-Order Observer
Minimum order observer equation:
Dynamics of minimum
order observer
Minimum-Order Observer
Observer error equation:
Minimum-Order Observer
Design of minimum order state observer
The error dynamics can be determined as desired by following the technique developed for the full order observer, that is:
Rank(
The characteristic equation for minimum order observer:
Ackermann’s formula:
)=n-m
Minimum-Order Observer
Summary:
Minimum order observer equations in terms of
Minimum order observer equations in terms of
Minimum-Order Observer
Summary:
Minimum order observer equations in terms of
Minimum order observer equations in terms of
Minimum-Order Observer
Effects of addition of the observer on a closed loop system
Completely state controllable
and completely observable
Minimum-Order Observer
Effects of addition of the observer on a closed loop system
Notice that:
Define
Minimum-Order Observer
Effects of addition of the observer on a closed loop system
State feedback &min.ord. observer equation:
Minimum order observer error equation:
Characteristic equation for the system:
Minimum-Order Observer
Example:
rank(
)=2
rank(
)=2
Minimum-Order Observer
Example:
Pole placement:
Minimum-Order Observer
Example:
Observer:
Minimum-Order Observer
Example:
Observer:
Minimum-Order Observer
Example:
Pulse transfer function of regulator
Pulse transfer function of original system
Minimum-Order Observer
Example:
Characteristic equation of observed state feedback system