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Reduced Order Observer

Automatic Control & System Theory 1G. Palli (DEI)

AUTOMATIC CONTROL AND SYSTEM THEORYAUTOMATIC CONTROL AND SYSTEM THEORY

REDUCED ORDER OBSERVERREDUCED ORDER OBSERVER

Gianluca Palli

Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione (DEI)

Università di Bologna

Email: gianluca.palli@unibo.it

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Reduced Order Observer

G. Palli (DEI) Automatic Control & System Theory 2

Problem statement:

with q  outputs, full rank  C  matrix ( rank(C)=q  ) and ( A,C ) fullyobservable, provide an estimation of the system state by mean of adynamic system of order (n-q ).

Given an n -order continuous-time [discrete-time] linear system

Solution:

The output information about the q components of the state are directlyexploited and only the (n-q ) missing components are estimated.

By means of a state space transformation T=[T1 T2] where T1=C + (rightpseudoinverse of C ) and ima(T2)=ker(C) , an equivalent system ( A’,B’,C’,D’ )is obtained such that C’=[I q  0 (n-q)  ] .

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Reduced Order Observer

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Equivalent system

By means of the change of variable y 0 = y – D u = z 1 we obtain:

Defining as z the state of the equivalent system it follows:

where L is the (n-q) xq matrix of the reduced-order observer gains.

=0

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Reduced Order Observer

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Reduced-order observer designBy assuming w=z 2 + L y 0 we obtain:

This can be rewritten in more compact form as [dicrete-time case]:

where:

The n-q reduced-order observer eigenvalues can be arbitrarily assigned bymeans of a suitable choice of the matrix L  if the couple ( A’ 

22  , A’ 

12 ) is fully

observable, this condition is always verified if ( A,C ) is fully observable andC has rank q .

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Reduced Order Observer

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Reduced-order observer structure

The reduced-order observer isa (n-q )-order system that

estimates the components of the state that cannot bedirectly reconstructed fromthe output.In this way it is possible tofully exploit the system outputand to estimate only the

 “missing” information aboutthe state.

Separation property

The 2n-q eigenvalues of the system composed by the static state feedback K 

and by the reduced-order observer are the union (with repetition) of the n eigenvalues of  A + B K and of the n-q eigenvalues of  A’ 22 + L A’ 12 .

Integrator

State-SpaceInput Output

Stateestimation

Reduced Order ObserverReduced Order Observer

R d d O d Ob

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Reduced Order Observer

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Separation property for the reduced-orderobserver (continuous-time case)

The obtained dynamic system with state feedback can be then written as:

By means of the feedback and by assuming the error function :

By posing we obtain:

With the given assumption, the following properties hold:

Reduced Order ObserverReduced Order Observer