Controllability and Observability of Linear Time Invariant ... 20.pdfADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 4 Method – 2: Using Laplace Transform

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



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= + + + +


4

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 −−


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:


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.


7

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

−

−−

−−

−−

+ + + + =+ + + + =

+ + + + =


8

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


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




−

−−

−−

−−

−−

− −+ + + =

+ + + + − =

+ + + + =

+ + + + =

( ) ( ) ( ) ( )

4

12 10 1 2 1

t

tnn n n nt t t t eλα α λ α λ α λ −

−+ + + + =


10

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




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


13

Graphical Meaning

0X

fX

Must happen in finite time.


14

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 =


15

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 −


16

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:


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




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


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:


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


22

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

−

−

⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦


23

Stabilizability and Detectability

Stabilizable system: Uncontrollable system in which uncontrollable part is stable

Detectable system: Unobservable system in which the unobservable subsystem is stable


24

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

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − − − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

=


25

( )( ) ( ) ( )( )( )

( )( )( )( ) ( )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?



26

( )( ) ( )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

−

−

=

= = +

= +

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⇒ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦



27

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



28

Block Diagram:


29

Where do uncontrollable or unobservable systems arise?

Redundant state variables

Physically uncontrollable system

Too much symmetry


30

References

K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999.

B. Friedland: Control System Design, McGraw Hill, 1986.


31

Documents

Controllability and Observability of Linear Time Invariant ... 20.pdfADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 4 Method – 2: Using Laplace Transform