1. Controllability and Observability
Lecture 06 Analysis (II)
Controllability and Observability
6.1 Controllability and Observability
6.2 Kalman Canonical Decomposition
6.3 Pole-zero Cancellation in Transfer Function
6.4 Minimum Realization
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2. Modern Contral Systems
Motivation1
uncontrollable
controllable
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3. Modern Contral Systems
Controllability and Observability
Plant:
Definition of Controllability
A system is said to be (state) controllable at time , if there exists a finite such for any and any , there exist an input that will transfer the state to the state at time , otherwise the system is said to be uncontrollable at time
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Controllability Matrix
Example: An Uncontrollable System
※ State is uncontrollable.
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Proof of controllability matrix
Initial condition
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Motivation2
observable
unobservable
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Definition of Observability
A system is said to be (completely state) observable at time , if there exists a finite such that for any at time , the knowledge of the input and the output over the time interval suffices to determine the state , otherwise the system is said to be unobservable at
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Observability Matrix
Example: An Unobservable System
※ State is unobservable.
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Proof of observability matrix
Inputs & outputs
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Example
Plant:
Hence the system is both controllable and observable.
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Theorem I
Controllable canonical form
Controllable
Theorem II
Observable canonical form
Observable
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example
Controllable canonical form
Observable canonical form
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Linear system
1. Analysis
Theorem III
Jordan form
Jordan block
Least row has no zero row
First column has no zero column
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Example If
uncontrollable
unobservable
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controllable
observable
In the previous example
controllable
unobservable
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Example
L.I.
L.I.
L.D.
L.I.
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Kalman Canonical Decomposition
Diagonalization:
All the Eigenvalues of A are distinct, i.e.
There exists a coordinate transform (See Sec. 4.4)
such that
System in z-coordinate becomes
Homogeneous solution of the above state equation is
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How to construct coordinate transformation matrix for diagonalization
All the Eigenvalues of A are distinct, i.e.
The coordinate matrix for diagonalization
Consider diagonalized system
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Transfer function is
H(s) has pole-zero cancellation. ∑
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Kalman Canonical Decomposition
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Kalman Canonical Decomposition: State Space Equation
(5.X)
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Example
Plant:
is uncontrollable.
is unobservable.
The same reasoning may be applied to mode 1 and 2.
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Pole-zero Cancellation in Transfer Function
From Sec. 5.2, state equation
may be transformed to
Hence, the T.F. represents the controllable and observable
parts of the state variable equation.
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Example
Plant:
Transfer Function
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Example 5.6
Plant:
Transfer Function
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Minimum Realization
Realization:
Realize a transfer function via a state space equation.
Example
Realization of the T.F.
Method 1:
Method 2:
※There is infinity number of realizations for
a given T.F. .
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Minimum realization:
Realize a transfer function via a state space equation
with elimination of its uncontrollable and unobservable parts.
Example 5.8
Realization of the T.F.
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