ss lec14 kalmandecomposition(1)

8/18/2019 SS Lec14 KalmanDecomposition(1)

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MECH468 : Modern Control Engineering

MECH522 : Foundations in Control Engineering

L14 : Kalman Decomposition

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

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Review & Today’s topic

• So far, we learned controllability and observability:

• Condition

• Decomposition

• Duality

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

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Rudolf Emil Kalman 1930-)

Hungarian-born American

Electrical engineer & mathematician

During 1950s & 60s, he developed

state-space control theory • Controllability, observability

• Linear quadratic regulator

• Kalman filter

Most of the theory in this coursehave been established by Kalman!

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Two decompositions

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SystemDecomposition for

controllability

Decomposition for

observability

Controllable part

Uncontrollable part

Observable part

Unobservable part


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Kalman decomposition idea)

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System

Controllable & observable part

Uncontrollable & observable part

Controllable & unobservable part

Uncontrollable & unobservable part


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Kalman decomposition

Conceptual figure Not block-diagram)

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System


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Kalman decomposition

• Every SS model can be transformed by z=Tx forsome appropriate T into a canonical form:

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Note the decomposition structure for controllability.


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Kalman decomposition 2

• If the second and the third state vectors areexchanged, one can get another form:

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Note the decomposition structure for observability.


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Remark

• It may happen that some states are missing.

• Ex. No controllable and unobservable part

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Remarks

• ( Aco ,Bco): controllable & ( Aco ,C co): observable

• Transfer function is determined by ONLYcontrollable & observable parts.

• If uncontrollable and/or unobservable parts areunstable, we need to change the structure

(actuators/sensors) of the system, because nooutput feedback control can stabilize the system.(next slide)

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Eigenvalues of A-matrix

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These have to be stable

for output feedback control.


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How to find ? review)

• For controllability, we used image space ofcontrollability matrix.

• For observability, we used kernel space ofobservability matrix.

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13/19

How to find for Kalman

decomposition?

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14/19

Very simple example: revisited

• Three cars with one input and two outputs

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Car1

Car2

Car3

x1: positionx2: velocity

x3: position

x4: velocity

x5: position

x6: velocity

Guess which state is

controllable and/or observable!


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SS model of three car example

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Velocity of

1st mass

Position of

3rd mass


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Derivation of T

• Controllability matrix

• Observability matrix

• (Inverse of T)=[e2,e1,{e5 ,e6},{e3 ,e4}](=T in this case)

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xample (cont’d)

• After transformation …

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Matlab command “minreal.m”

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Remark: Matlab will return matrices corresponding the

states in an order different from the order in Slide 7.


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Summary

• Kalman decomposition

• Combination of

• Decomposition for controllability

•

Decomposition for observability• How to find coordinate transformation matrix T

• Image space of controllability matrix

• Kernel space of observability matrix

•

Next, controllability and observability for discrete-time systems

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ss lec14 kalmandecomposition(1)

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