Digital Control SystemsSTATE OBSERVERS
State Observers
State Observers
State Observers
State ObserversObserver design
(S1-Actual system)
(S2 - Dynamic Model)
State ObserversObserver design
:
State ObserversObserver 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 availablefor 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 ObserverExample:
rank( )=2
Full Order State ObserverExample:
Full Order State ObserversDesign 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 ObserversDesign of full order state observer by using observable canonical form
State transformation to observable canonical form:
Full Order State ObserversDesign of full order state observer by using observable canonical form
State transformation to observable canonical form:
Full Order State ObserversDesign of full order state observer by using observable canonical form
State Observer Dynamics
Full Order State ObserversDesign of full order state observer by using observable canonical form
(S1-Actual system)
(S2-Dynamic system)
Define then state observer dynamics :
Full Order State ObserversDesign of full order state observer by using observable canonical form
Desired characteristic equation for the error dynamics is
Full Order State ObserversDesign of full order state observer by Ackermann’s formula
Assumption: System is completely observable and the output y(k) is scalar.
Full Order State ObserversExample:
rank( )=2 The system is completely observable
Characteristic equation of the system:
Desired characteristic equation for the error dynamics
Full Order State ObserversExample:Design of full order state observer by using observable canonical form
Full Order State ObserversExample:Design of full order state observer by using Ackermann’s Formula
Full Order State ObserversExample:Design of full order state observer by causal method
Desired characteristic equation
Full Order State ObserversEffects of addition of the observer on a closed loop system
Completely controllable andcompletely observable system
Full Order State ObserversEffects 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 accuratelyMeasured. 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 offull 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 observerequation 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 ObserverDesign 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:
The characteristic equation for minimum order observer:
Ackermann’s formula:
Rank( )=n-m
Minimum-Order ObserverSummary:
Minimum order observer equations in terms of
Minimum order observer equations in terms of
Minimum-Order ObserverSummary:
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 ObserverExample:
Pole placement:
Minimum-Order ObserverExample:
Observer:
Minimum-Order ObserverExample:
Observer:
Minimum-Order ObserverExample:
Pulse transfer function of regulator
Pulse transfer function of original system
Minimum-Order ObserverExample:
Characteristic equation of observed state feedback system