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ECE Lecture 604 State Variable Analysis Doug Looze
Oct. 6, 2009 Linear Systems Douglas Looze 1
Announcements
PS3 available
Due Thursday
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Linear Systems Douglas Looze
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Last Time
Characterizations of
Controllability Observability
In terms of
Grammians Linear independence of vector time functions
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Conditions
Equivalent
(A(t), B(t)) controllable Independent rows of ( t , ) B ( ) Grammian invertible
n n matrix Full rank
W t0 , t f
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(
)
tf = ( t0 , t ) B ( t ) BT ( t ) T ( t0 , t ) dt t0
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Equivalent
(A(t), C(t)) observable Independent columns of C ( t ) ( t , t0 ) Grammian invertible
n n matrix Full rank
M t0 , t f
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(
)
tf T = ( t , t0 ) CT ( t ) C ( t ) ( t , t0 ) dt t0
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Exercise
Consider the linear, time-varying system
0 0 d x( t) = x( t) + b( t) u( t) dt 0 1
Specify whether the system is controllable on any interval for the following input matrices. If it is controllable on some interval, determine such an interval
1 a) b ( t ) = 1
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1 b) b ( t ) = t e
1 c) b ( t ) = -t e
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Linear Systems Douglas Looze
The transition matrix for this system is:
0 0 0 1 t 1 = e At = e
0 t 0 e
The controllability kernel matrix is:
0 1 1 1 ( t0 , t ) B ( t ) = = t0 - t 1 t0 - t 0 e e
Since the rows are linearly independent on any interval, the system is (C) on all intervals
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The transition matrix for this system is:
0 0 0 1 t 1 = e At = e
0 t 0 e
The controllability kernel matrix is:
0 1 1 1 ( t0 , t ) B ( t ) = = t0 - t t t0 0 e e e
Since the rows are not linearly independent on any interval, the system is not (C) on any interval
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The transition matrix for this system is:
0 0 0 1 t 1 = e At = e
0 t 0 e
The controllability kernel matrix is:
0 1 1 1 ( t0 , t ) B ( t ) = = t0 - t -t t0 - 2t 0 e e e
Since the rows are linearly independent on any interval, the system is (C) on all intervals
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Today
Characterization of linear independence for controllability and observability Exercises Reading
Rugh, Ch. 9 p. 144146, 149
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Alternative Tests
Controllability is a range space concept Observability is a null space concept Define
Then
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Question
Is there a test for linear independence of a finite number of vector time that functions is
Finite-dimensional Non-integral Independent of the transition matrix
Yes
If the vectors are sufficiently differentiable
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Controllability
Define
K0 ( t ) = B ( t ) d K j ( t ) = - A ( t ) K j -1 ( t ) + K j -1 ( t ) dt ( t , ) B ( ) j
Claim
( t , ) K j ( ) =
j
Proof (induction):
Check true for j = 0 ( t , ) K 0 ( ) = ( t , ) B ( )
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Assume for j Check true for j + 1
( t , ) K j ( )
j = [ ( t , ) B ( ) ] j
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Theorem: Suppose that, for some t, B(t) is q times differentiable and A(t) is (q1) times differentiable. Then the linear system is controllable over any interval containing t if
rank K 0 ( t ) L K q ( t ) = n Note: by the previous theorem K j ( t) =
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( t , ) B ( ) j t =
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j
Proof: Suppose the rank test holds but there is an interval [t0,tf] containing t for which the system is not controllable.
Then W(t0,tf) is not invertible
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T x a ( t0 , ) B ( ) = 0
=
T xb K 0 ( )
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Remarks
Test for linear independence of rows of ( t0 , t ) B ( t ) Same proof can be used for any matrix V(t) rank V ( t ) d dq V( t) L V( t) = n dt dt q
Can determine Kj(t) recursively Note strong conclusion: any interval containing t Sufficient only
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Recall
Test for independence of rows of a matrix V(t) d dq rank V ( t ) V( t) L V( t) = n q dt dt Can use transpose to present an equivalent test for independence of columns Recursion formula for derivatives
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Define
Observability
L j ( t) =
L0 ( t ) = C ( t )
j t
C ( t ) ( t , ) j
Theorem: Suppose that, for some t, C(t) is q times differentiable and A(t) is (q1) time differentiable. Then the linear system is observable over any interval containing t if
L0 ( t ) rank M = n Lq ( t )
Linear Systems Douglas Looze
d L j ( t ) = L j -1 ( t ) A ( t ) + L j -1 ( t ) dt
t =
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Remarks
Test for linear independence of columns of C ( t ) ( t , t0 ) Can determine Lj(t) recursively Note strong conclusion: any interval containing t Sufficient only
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Summary
Rank tests for (C) and (O) Exercises
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Next Time
Basis change Duality LTI systems Reading
Rugh, Ch. 9 p. 146149, 150152
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