Homework 7 - Due: 03/07
Problem 1
Recall that for an LTI system,
- Prove that the matrix
defined below is nonsingular for some if and only if is.
- The result of (a) implies that the pair
is controllable if and only if the pair is controllable. Is this true for LTV systems? Prove or give a counter example.
Problem 2
For the scalar system
consider the problem of steering its state from
Since the system is controllable, we know that this transfer is possible for every value of
. Verify this by giving an explicit formula for a control that solves the problem.Is the control you obtained in part (a) unique? If yes, prove it; if not, find another control that achieves the transfer (in the same time
).Now suppose that the control values must satisfy the constraint
at all times. Is the above problem still solvable for every ? for at least some ? Prove or disprove.Answer the same questions as in part (c) but for the system
(again with ).
Problem 3
Let us revisit our consensus problem from the last homework. Assume that agent 1 is a the leader and knows a desired location
Problem 4
Consider the system
Problem 5
Do BMP Problem 5.5.6
Problem 6
Recall that the controllable subspace of an LTI system