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