Digital Control Systems STATE OBSERVERS

State Observers

Observer design (S 1 -Actual system) (S 2 - Dynamic Model)

State Observers Observer design :

State Observers Observer design S2S2 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-K e C 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-K e C. If matrix G-K e C 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-K e C 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-K e C 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 (S 1 -Actual system) (S 2 -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

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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: The characteristic equation for minimum order observer: Ackermann’s formula: Rank( )=n-m

Minimum-Order Observer Summary: Minimum order observer equations in terms of

Minimum-Order Observer Summary: 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

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