Lumped Systems: Controllability and Observability Analysis, Lecture notes of Teaching method

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EE 510: Lumped Systems Theory

Fall 2016

Handout 4:

Controllability and Observability

Prof. Mohamed Zribi

Updated 2 October 2016

Outline of the Handout

1. Controllability

2. Obsevability

3. Duality

4. Properties of Controllability and Observability

5. Decomposition of Controllability and Observability Spaces

6. Minimal Realizations

7. Stabilizability and Detectability

8. Controllability and Observability for Discrete Time Systems

1. Controllability

 Controllability deals with whether or not the state of

a state-space equation can be controlled from the

input. Controllability is sometimes called reachability.

Controllability

Consider the state equation

where and

Definition: The above state equation or the pair (A,B) is

said to be controllable if initial state x 0 and any final

state , an input that transfers to in a finite time.

Otherwise (A,B) is NOT controllable.

x  = Ax + Bu n n A R

× ∈.

n p

B R

×

∀

x 1 (^) ∃ x 0 x 1

Controllability to the Origin and Reachability

Consider the following three controllability notions:

(1) Controllability: Transfer any state to any other state in

finite time

(2) Controllability to the origin: transfer any state to the

zero state in finite time

(3) Controllability from the origin (reachability): transfer

the zero state to any state in finite time

 In continuous time, the three definitions are equivalent.

Note that,

Let the control be in the form

Clearly, if is a

nonsingular matrix, then the control

will transfer system state from x (0) at t=0 to x (t1) at time t = t 1

1 1 (^1 ) ( ) 1 (0) 0 ( ).

t At A t x t e x e Bu d

τ τ τ

− = + ∫ ( 1 ) ( ) (^0)

T AT t u B e u

τ τ

−

⇓ 1 1 (^1 )^ (^1 ) 1 0 0

( ) (0) ( )

At t A t T A^ T t x t e x e BB e d u

τ τ τ

− − = + ∫

⇓

1 ( 1 ) ( 1 ) 1 0

( ) :

t (^) A t T AT t Wc t e BB e d

τ τ τ

− −

∫

( 1 ) (^11) ( ) ( )[ ( ) 1 1 (0)]

T A^ T t At u B e Wc t x t e x

τ τ

− − = −

Theorem: The following four statements are equivalent:

(1) (A,B) is controllable.

(2)

is nonsingular for any t > 0.

(3) The n by n*p controllability matrix

has rank n.

(4) rank for all

( ) ( ) 0 0

t (^) A T AT t A t T AT t

Wc t e BB e d e BB e d

τ τ τ τ τ τ

− −

( λ i (^) I − A B | ) = n (all eignvalues of A) λ i

or, if Bisn 1 det ( C ) det [B|AB A B] 0

n 1 × (^) m = ≠

− 

C [B|AB|A B| A B]

2 n 1 m

− = 

Theorem: Suppose that. Then (A,B) is

controllable iff the unique solution of

is positive definite. The unique solution

is called the controllability Gramian.

Note that is the set of all eignvalues of A.

λ ( A ) C

− ⊆

T T AWc + W Ac = − BB

0

A T A^ T Wc e BB e d

τ τ τ

∞

∫

λ ( A )

Method 1: Rank Test

Cm is called the controllability matrix with dimension n x pn.

 The state vector x(t) is controllable if and only if

Rank ( C m ) = n

 To find the rank, reduce the matrix Cm into reduced form and rank Cm is

equivalent to the number of non-zero rows.

 For single inputs systems, the rank test is equivalent to the determinant of

Cm being different than zero for controllable systems.

C [B^ AB A B A B]

2 n 1 m

− =     

Example 1: Rank test

Notethat(A,B) isin the CCF.

Therefore,thesystem is.

Notethatdet

rank rank

[B AB]

n 2 1

, B

A

x Ax Bu

Remark :

controllable

[ (C ) -1]

(C )

C

m

m

m

Example 3: Rank test

Consider the LTI system:

Controllability matrix:

u u

x x  = Ax + B

[ ]

Therefore,thesystemiscontrollab le.

Also, notethat det(C ) 2 0

rank(C ) 2 n 0 2

1 1 C

2

1

0

1

2 4

1 1

m

m m

= ≠

= =  

  

 − = =

 

  

−  = 

  

  

  



−

−

B AB

AB

Example 4: Rank test

Consider the system

C [B AB]

1

0 B 0 2

1 0 A

x Ax Bu

m =

 



 



 



 



−

−

 = +

C m

zero row so rank Cm = 1. rank so the system is not controllable

C (^) m ≠ n = 2

Consider the LTI system:

System is controllable

Example 6: Rank test

x x u Ax B u

 −^ −   

[ ] rank(C ) 2 n

C m  m = =

= B AB =

Remark: Controllable canonical form

y [ ... ] x

u

1

0

.

0

0

x

...

0 0 0 ... 1

.. .. .. ... ..

0 0 1 ... 0

0 1 0 ... 0

x

= × × × ×

































































× × × ×

 =

× × ×

× ×

×

C (^) m ⇒ C^ m ≠^0

A system whose state space representation is in the

CCF is controllable.