Post on 28-Dec-2019
Lecture – 20
Controllability and Observability of Linear Time Invariant Systems
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
Evaluation of Matrix Exponential eAt
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
3
Method – 1: Power-series
This method is useful and accurate only if the series truncates naturally. Otherwise, series truncation introduces approximation error.
Direct computation of eAt as power series is computationally inefficient as well.
2 2 3 3
2! 3!At A t A te I At= + + + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Method – 2: Using Laplace Transform
This method results in closed form expressions for eAt, can be quite useful for small matrices.
Numerical algorithms exist to evaluate
. However, its inverse still need to be found.
Can be quite cumbersome for large matrices.
( ) 11Ate L sI A −− ⎡ ⎤= −⎣ ⎦
( ) 1sI A −−
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
5
Method – 3: Using Similarity Transform(Provided the matrix can be diagonalizable)
( )( )
1
2 2 3 3
1 1 21 1
2 2 3 31
1
2! 3!
2!
2! 3!
0 00 00 0 n
At
t
t
A t A te I At
PDP PDP tPP PDP t
D t D tP I Dt P
eP P
e
λ
λ
− −− −
−
−
= + + + +
= + + +
⎛ ⎞= + + + +⎜ ⎟
⎝ ⎠⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
1A PDP−=
SimilarityTransformation:
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
6
Method – 4: Sylvester’s FormulaCase – 1: Distinct Eigenvalues
eAt satisfies the following determinant equation:1
2
2 11 1 1
2 12 2 2
2 1
2 1
Ultimate aim
11
1 n
tn
tn
tnn n n
n At
ee
eI A A A e
λ
λ
λ
λ λ λλ λ λ
λ λ λ
−
−
−
−
= 0
( ) ( ) ( ) ( )2 10 1 2 1
At nne t I t A t A t Aα α α α −−= + + + +
i.e.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Method – 4: Sylvester’s FormulaCase – 1: Distinct Eigenvalues
( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1
2
0 1 1
2 10 1 1 2 1 1 1
2 10 1 2 2 2 1 2
2 10 1 2 1
The coefficients , , , can be determined from the following setof equations:
n
n
tnn
tnn
tnn n n n
t t t
t t t t et t t t e
t t t t e
λ
λ
λ
α α α
α α λ α λ α λα α λ α λ α λ
α α λ α λ α λ
−
−−
−−
−−
+ + + + =+ + + + =
+ + + + =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Method – 4: Sylvester’s FormulaCase – 2: Repeated Eigenvalues
eAt satisfies the following determinant equation:
( ) ( ) ( ) ( )2 10 1 2 1
At nne t I t A t A t Aα α α α −−= + + + +
i.e.
( )( )
( )
1
1
1
4
23
1 1
2 21 1 1
2 3 11 1 1 1
2 3 14 4 4 4
2 3 1
2 3 1
1 20 0 1 3
2 20 1 2 3 111
1 n
tn
tn
tn
tn
tnn n n n
n At
n n t e
n t eee
eI A A A A e
λ
λ
λ
λ
λ
λ λ
λ λ λλ λ λ λλ λ λ λ
λ λ λ λ
−
−
−
−
−
−
− −
−
= 0
1 1 1 4
3 times
Eigenvalues:, , , , , nλ λ λ λ λ
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
9
Method – 4: Sylvester’s FormulaCase – 2: Repeated Eigenvalues
( ) ( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1
1
1
0 1 1
23
2 3 1 1 1
2 21 2 1 3 1 1 1
2 10 1 1 2 1 1 1
2 10 1 4 2 4 1 4
The coefficients , , , can be determined from:
1 23
2 22 3 1
n
tnn
tnn
tnn
nn
t t t
n n tt t t e
t t t n t te
t t t t e
t t t t e
λ
λ
λ
λ
α α α
α α λ α λ
α α λ α λ α λ
α α λ α λ α λ
α α λ α λ α λ
−
−−
−−
−−
−−
− −+ + + =
+ + + + − =
+ + + + =
+ + + + =
( ) ( ) ( ) ( )
4
12 10 1 2 1
t
tnn n n nt t t t eλα α λ α λ α λ −
−+ + + + =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Method – 4: Sylvester’s FormulaExample
( ) ( )
1
2
1,2
12
2
2
22
2
0 1, 0, 2
0 2
To compute using Sylvester's formula, we have
1 1 0 11 1 2
Expanding the determinant2 2 0
11 11 2 22 0
At
t
t t
At At
At t
tAt t
t
A
e
ee e
I A e I A e
e A I Ae
ee A I Ae
e
λ
λ
λ
λλ −
−
−−
−
⎡ ⎤= = −⎢ ⎥−⎣ ⎦
= − =
− + + − =
⎡ ⎤−⎢ ⎥= + − =⎢ ⎥⎣ ⎦
0
Controllability of Linear Time Invariant Systems
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
12
Controllability• A system is said to be controllable at time t0 if
it is possible by means of an unconstrained
control vector to transfer the system from any
initial state to any other state in a finite
interval of time
• Controllability depends upon the system matrix
A and the control influence matrix B
0X
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Graphical Meaning
0X
fX
Must happen in finite time.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Condition for Controllability:(single input case)
System:
Solution:
Assuming
BuAXX +=
τττ dBueXetXt
tAAt )()0()(0
)(∫ −+=
1
1 1
1
( )
0
0
0 (0) ( )
(0) ( )
tAt A t
tA
e X e Bu d
X e Bu d
τ
τ
τ τ
τ τ
−
−
= +
=−
∫
∫
1( ) 0,X t =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Condition for Controllability:(single input case)
( )1
0( ) Sylvester's formula
nA k
kk
e Aτ α τ−
−
=
=∑
[ ]
1 11
00 01
0
10 1 1
(0) ( ) ( ) ( )t tn
A kk
k
nk
kk
Tnn
X e Bu d A B u d
A B
B AB A B
τ τ τ α τ τ τ
β
β β β
−−
=
−
=
−−
= − = −
= −
⎡ ⎤= − ⎣ ⎦
∑∫ ∫
∑1
0
( ) ( )t
k k u dβ α τ τ τ∫where
This system should have a non-trivial solution for [ ]0 1 1T
nβ β β −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Controllability1If the rank of is ,
then the system is controllable.
nBC B AB A B n−⎡ ⎤⎣ ⎦
Example:
uxx
xx
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
−=⎥
⎦
⎤⎢⎣
⎡12
2001
2
1
2
1
( )
2 1 0 2 2 21 0 2 1 1 2
2 The system is controllable.
B
B
C
rank C
⎡ − ⎤ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
= ∴
Result:
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
17
Output Controllability
1If the rank of is ,
then the system is output controllable.
nBC CB CAB CA B D p−⎡ ⎤⎣ ⎦
Result:
, ,n m p
X AX BUY CX DU
X U Y
= += +
∈ ∈ ∈R R R
Note: The presence of term in the output equation always helps to establish output controllability.
DU
Observability of Linear Time Invariant Systems
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
19
Observability• A system is said to be observable at time
t0 if, with the system in state X(t0) ,it is
possible to determine this state from the
observation of the output over a finite
interval of time
• Observability depends upon the system
matrix A and the output matrix C
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
20
Observability
( ) 1If the rank of is ,
then the system is observable.
nT T T T TBO C A C A C n
−⎡ ⎤⎢ ⎥⎣ ⎦
Example:
[ ]1 1 1
2 2 2
1 0 21 0
0 2 1x x x
u yx x x
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
( )
1 1 0 1 1 10 0 2 0 0 0
1 2 The system is NOT observable.
B
B
O
rank O
⎡ − ⎤ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
= ≠ ∴
Result:
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
21
Controllability and Observability in Transfer Function Domain
The system is both controllable and observable if there is no Pole-Zero cancellation.
Note: The cancelled pole-zero pair suppresses part of the information about the system
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Principle of DualitySystem S1:
System S2:
The principle of duality states that the system S1 is controllable if and only if system S2 is observable; and vice-versa!
Hence, the problem of observer design for a system is actually aproblem of control design for its dual system.
1
X AX BUY CX
= +=
2
T T
T
Z A Z C VY B Z= +
=
2 1
2 1
n
nB
T T T T T T TB
C B AB A B A B
O C A C A C A C−
−⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
2 1
2 1
nT T T T T T TB
nB
C C A C A C A C
O B AB A B A B
−
−
⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Stabilizability and Detectability
Stabilizable system: Uncontrollable system in which uncontrollable part is stable
Detectable system: Unobservable system in which the unobservable subsystem is stable
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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ExampleRef: B. Friedland, Control System Design, McGraw Hill, 1986
[ ]
1 1
2 2
3 3
4 4
System Dynamics
2 3 2 1 12 3 0 0 22 2 4 0 22 2 2 5 1
Output Equation
7 6 4 2
BXX A
C
x xx x
ux xx x
y X
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − − − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
=
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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( )( ) ( ) ( )( )( )
( )( )( )( ) ( )1
pole-zero cancellation
2 3 4 11 2 3 4 1
y s s s sC sI A B
u s s s s s s− + + +
= − = =+ + + + +
Transfer Function:
Implication: What appears to be a fourth-order system, isactually a first-order system! Hence, there is either loss of controllability or observability(or both).
Question: Is this system stabilizable?
ExampleRef: B. Friedland, Control System Design, McGraw Hill, 1986
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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( )( ) ( )1
1
Define . Then
Let 4 3 2 1 1 0 0 0 13 3 2 1 0 2 0 0 0
,2 2 2 1 0 0 3 0 11 1 1 1 0 0 0 4 0
X TX
X TX T AX Bu
X TAT X TB u
T TAT TB
−
−
=
= = +
= +
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⇒ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦
ExampleRef: B. Friedland, Control System Design, McGraw Hill, 1986
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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11
2 121 2
33
44
1
2
3
4
2,
34
Implications:
: Affected by the input; visible in the output: Unaffected by the input; visible in the output: Affecte:
x uxxx
y CX CT X x xx ux
xx
xxxx
−
− +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥= = = = +⎢ ⎥ ⎢ ⎥− +⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦
d by the input; Invisible in the outputUnaffected by the input; Invisible in the output
ExampleRef: B. Friedland, Control System Design, McGraw Hill, 1986
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Block Diagram:
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Where do uncontrollable or unobservable systems arise?
Redundant state variables
Physically uncontrollable system
Too much symmetry
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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References
K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999.
B. Friedland: Control System Design, McGraw Hill, 1986.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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