Descomposicion de kalman

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MECH550P: Foundations in Control Engineering
Lecture 14 Kalman decomposition

Dr. Ryozo Nagamune Department of Mechanical Engineering University of British Columbia

Spring 2008

MECH550P :Foundations in Control Engineering

1

Review & Today’s topic
So far, we have learned controllability and observability:
Condition Decomposition Duality

Today, we will study the combination ofdecompositions for controllability and observability, called Kalman decomposition.

Spring 2008

MECH550P : Foundations in Control Engineering

2

Rudolf Emil Kalman (1930-)American-Hungarian Mathematical system theorist During 1950s & 60s, he developed state-space control theory
Controllability, observability Linear quadratic regulator Kalman filter etc.

Most of the theory in thiscourse have been established by Kalman!
Spring 2008 MECH550P : Foundations in Control Engineering 3

Two decompositions
SS model

System Decomposition for controllability Controllable partUncontrollable part
Spring 2008

Decomposition for observability Observable part Unobservable part
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MECH550P : Foundations in Control Engineering

Kalman decomposition (idea)
SS modelSystem

Controllable & observable part Controllable & unobservable part Uncontrollable & observable part Uncontrollable & unobservable part
Spring 2008 MECH550P : Foundations in Control Engineering 5 Kalman decomposition (idea)
Conceptual figure (Not block diagram!) System

Spring 2008

MECH550P : Foundations in Control Engineering

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Kalman decomposition
Every SS model can betransformed by z=Tx for some appropriate T into a canonical form:

Spring 2008

MECH550P : Foundations in Control Engineering

7

Remarks
(Aco,Bco): controllable & (Aco,Cco): observable Transferfunction is determined by ONLY controllable & observable parts. (next slide)

If uncontrollable and/or unobservable parts are unstable, we need to change the structure (actuators/sensors) of the...
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