Lecture 10: Observability

Question 1

Observability Definition

Answer 1

1. Present state can be determined based on future outputs and inputs

Question 2

Unobservability Mathematical Definition

Answer 2

yzi = 0; C(t)Iota(t,t0)x(t0) = 0 for t >= t0

Question 3

Unobservable Subspace

Answer 3

Set of all unobservable states x at t0

Question 4

Completely Observable

Answer 4

Unobservable subspace has zero vector

Every vector can be estimated based on future outputs and inputs

Question 5

Constructability Definition

Answer 5

Present state can be determined based on past outputs and inputs

Question 6

Unconstructability Mathematical Definition

Answer 6

yzi = 0;C(t)Iota(t,t0)x(t0) t <= t1

Question 7

Unconstruable Subspace

Answer 7

Set of all unconstructable states at t1

Question 8

Completely Constructable

Answer 8

Unconstructable subspace has zero vector

Every vector can be constructed based on past outputs and inputs

Question 9

Observability and Constructability Relationship for LTI/V systems

Answer 9

1. If syste is observable is constructable

2. If system is constructable is observable if inv(Iota) exists

Question 10

Complete Observability for LTI Systems

Answer 10

1. Completely observable if Q has full rank
   Q = {C;CA;CA^2…CA^n-1}
2. PBH for every eigenvalue of lambda of A [A-lambda*I;C] has full rank
3. Grammian is non-singular for all values t>t0

Question 11

Duality Property for LTI systems (A,B,C,D)

Answer 11

Systems is reachable if and only if (transpose(A),transpose(C),transpose(B),transpose(D)) is observable

Question 12

Why is observability dual of reachability

Answer 12

Taking the transpose of dual system yields controllability matrix and has full rank, thus observability matrix has full rank

Question 13

Unobservable Subspace

Answer 13

Finding nullspace of observability matrix yields unobservable states

Question 14

A-Invariance of Unobservable Subspace

Answer 14

If x is an element of unobservable subspace then Ax is also an element of unobservable subspace

Question 15

Method for Finding Observability Canonical Form Transformation

Answer 15

U = (transpose(U1),transpose(U2))
inv(U) = (T1,T2)
U1 -> basis{span(Q)}
U2 -> n-p rows that make U2 non-singular

Question 16

Observability Canonical Form State Space

Answer 16

x' = [A'11 0; A'21 A'22]x + [B1';B2']u
y = [C'1 0]x + Du

Question 17

Properties of Observability Canonical Transformation

Answer 17

1. (A’11, C’1) is completely observable
2. A’11 and A’22 are invariant
3. A’11 -> observable eigenvalues
4. A’22 -> unobservable eigenvalues

Question 18

Observable for LTV Systems

Answer 18

1. If grammian is non-singular then LTV is observable

2. N-Test

Question 19

N-Test

Answer 19

N0 = C(t)
Nj = N(j-1) * A(t) + N’(j-1)(t), j = 1,…,n-1
System is observable if [N0;N1;…N(n-1)] has full rank

Question 20

Conditions for Observability of D.T.S

Answer 20

Observability Matrix has full rank

Question 21

Constructability Matrix for DTS

Answer 21

[C(k0);C(k0+1)Iota(k0+1,k0);…;C(k1-1)Iota(k1-1,k0)]