Class-Note-System and Control Theory 1.a (1)

8/10/2019 Class-Note-System and Control Theory 1.a (1)

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CONTENTS

Controllability

Range Space and Null Space

Reachability

Observability

Kalman Decomposition

Lyapunov Stability

Lyapunov Equation and solution, Sylvester Equation and its solution

Pole-PlacementI. Bass- Gura Method

II. Ackermans Formulae

Reduced Order Observer

LQR and LQG

LQG as a general H2 Problem


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CONTROLLABILITY :

Consider a system with n-dimensional equation

= + The system is controllable (or in short, pair(A, B) ) if for any initial state

state , there exits an input () such that (0) is transferred to () in fTHEOREM :

For the above system, the following statements are equivalent

1. The n-dimensional pair (A, B) is controllable.

2. The (n n) matrix()= = ()()

is non-singular for any > 0 .


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3. The n n p controllability matrixC = [ . . . ]

has rank n (full row rank).

4. The n ( n + p ) matrix | | has full row rank at every eigen valu5. If in addition, all eigenvalues of A have negative real parts, then the uniq+ =

is positive definite. The solution is called the controllability Gramian

and can be expressed as

=

Note:

Row rank means the left range space dimension.


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Proof :

34, Assumep is a eigenvector ofA corresponding to eigen value . Then = [0 ]

So for any

first part is always zero, but assume that

is also zero. Then

uncontrollable.

From (3) Rank of [ . . .] is n. That means there is no v such that ;[ . . . ] = [0 0 0]

but take v = p then [0 ] [0 ] [0 0 0].So system is un-controllable.

Assume system is uncontrollable. So there must exist a such that [ Next: Any matrix (A, B) can be decomposed into such that ; 0 B

0 Where controllable and , parts.

Now [ ] 0 0


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Take is a left eigen vector of and = 0 , [ ] =[0 0]

13, (t)= + Take Laplace on both sides

0 = + () = 0 + Take Laplace Inverse on both sides

= 0 + = 0 + [ + + /2!+]

= [ . . . ]()() () /2!.

.

.


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= [ . . . ]

...

C

The region in which () can be present is decided by C .. in the range spaceNOTE :

Range : By linear transformation the space we can reach is called rang

of a matrix.

Linear Vector Space : Assume, ( )If two operations (i) +

and (ii) , (scalar , then the set is called space. So set space.


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Example :

M =1 1 12 2 2

What is the Range ?

, with any x will give component = 0. So all the vector will be in plane is spanned by 12Null or kernel of a matrix : Set of independent vectors for which =00

kernel of a matrix.

y =

0

11,

1

01Similarly we have Left null space and left range space.Row space= Range of () =>

111

Left Null = Null of (

) => 2

1


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If the matrix is of here n=2, m=3NOTE :

1. Range and left-null space are orthogonal2. Analogously Row space and null space are orthogonal

3. Dimension of a space :

Number of independent vectors required to span a space is the dimensi

the vectors are called basis vectors.

3. Rank of a matrix : dimension of range space = dimension of row space.

4. In the example : Rank=1

5. Dimension of range+ Dimension of left-null space=n

6. Dimension of row space+ Dimension of null space=m


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Proof :

x t = [B AB AB . . . ]

...

d

Since (i.e., any where), the range space dimension should be n orhave rank n. End of proof.

3 2 Follow directly from above because

= =

()()= [ + + /2!+ ][ + + /2!+ ]dt

From the above, [ + + /2!+ ] has rank n.So there is no vector such that [ + + /2!+ ] 0So, = [ + + /2!+ ][ + + /2!+ ]for any

0


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35,=()()

=

is linear combination of

(

> 0). Since system is stable.

is finite.Consider+ =[+ ]

= []

= [

]

= 0-= So is a solution of+ += 0


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Uniqueness Say another solution exists + += 0 (1)+ += 0 (2)

Subtract equation (2) from (1);( ) + ()= 0

[( ) + ( )]= 0 ( )+ ( )= 0

[( )] = 0So [( )] constant

At t= 0, = constantt= , = 0 =

REACHABILITY


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REACHABILITY :

If the input of a system = + can take sates (solution) from zero to anyin finite time, then the system is called Reachable. The difference between ReachaControllability is explained next. Take the following example in discrete domain,

= 1 10 0

,

=10 = =1 10 0

So rank = 1 System is UncontrollableNow see even if the system is not controllable it can be taken to zero in finite time (S

= , () =00( + 1) =

A

() + ()00=1 10 0 + 10 u = ( + ), in one step we can reach 0 state.Note: Reachability is a typical property seen in discrete time system, for continuous Reachability and controllability are same

Class-Note-System and Control Theory 1.a (1)

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