EE611

Solution to HW#22

1. (See Solution to Homework #23!!)

2a)

which is singular!! Therefore system in NOT controllable!

b)

system controllable W_c(t₀,t₁) is nonsingular rows of (t₀,t)B(t) are linearly independent. For our case, (t₀,t)B(t) = e^A(t0-t)B. But by Cayley-Hamilton,

If rows of (t₀,t)B are linearly independent, then (t₀,t)B = 0 iff = 0, where is nx1.

Since c_i's are arbitrary, (t₀,t)B = 0 AⁱB = 0, i=0,…,n-1 e^A(t0-t)B = 0

[B AB … A^n-1B] = 0 = 0.

Therefore, [B AB … A^n-1B] has full rank system controllable M and rank n

c)

d)

check:

f)

This is an open-loop control!! BAD BAD BAD!! Not robust to disturbances, noise, or mismatches in our model compared to the actual system dynamics!!