Lecture 9: Controllability

Question 1

Controllability

Answer 1

transfers state to origin

x(tf) = 0; x(tf) = -Iota(tf,t0)*x0

Question 2

Reachability

Answer 2

set of all states x1 reachable at t1

x0 = int(Iota(tf,Tau)B(Tau)u(Tau)dTau

Question 3

Relationship Between Reachability/Controllability

Answer 3

Completely reachable -> completely controllable

Conpletely controllable -> completely reachable if transitions matrix is non singular (CT systems always true)

Question 4

A-invariant

Answer 4

if x -> controllable subspace then Ax -> controllable subspace

Question 5

Reachability in Controllable Subspace

Answer 5

Any element of Controllable subspace can reach any other element within the controllable subspace

Question 6

Controllability Canonical Form

Answer 6

T = (T1,T2)
T1 = span of q-dimensional controllable subspace
T2 = form basis with T1

Question 7

Controllability Transformation Matrix applied to System

Answer 7

A' = [A'11 A'12; 0 A'22] = inv(T)*A*T
B' = [B'1; 0] = inv(T)*B

Question 8

Properties of Controllability Transformation Matrix

Answer 8

1. (A’11,B’1) is completely controllable
2. A’11 controllable eigenvalues
3. A’22 uncontrollable eigenvalues
4. x’ = inv(T) * x -> controllability canonical form

Question 9

Methods of determining Controllable for LTV

Answer 9

1. If grammian yields non-singular matrix

2. M-test must have rank = dim(A)

Question 10

Controllaibility for DT Linear systems

Answer 10

1. If Iota has full rank then reachability matrix must have full rank
2. If Iota has less than full rank then Rc rank must be < n

Question 11

DT Reachability Matrix

Answer 11

1. Rc = (B(k1 - 1), Iota(k1,k1-1)B(k1-2),…, Iota(k1, k0 + 1)B(k0))
2. Rc = (B, AB, A^2B, … , A^(n-1)*B) for time invariant

Question 12

M-Test

Answer 12

M0(t) = B(t)
Mj(t) = -A(T)*M(j-1)(t) + M'(j-1)(t), j = 1, ... , n-1