Control Theory Homeworks

HW1, due on Sep 3,

1: Is (1 4 7)', (2 5 8)' and (3 6 9)' a basis in R^3?
2: Problem 3.5
3: Problem 3.6

HW2, due on Sep 10,

1: Problem 3.7
2: Problem 3.8 (parameters means free parameters.)

HW3, due on Sep 17,

1: Find the eigenvalues and eigenvectors of A_1 in Problem 3.13.
2: Calculate e^{At} for A_1 in Problem 3.13.
3: Find the Jordan form of A_1 in Problem 3.13.
4: Problem 3.19.
5: Problem 4.1
6: Calculate e^{At} for A in Problem 4.1

HW4, due on Sep 24,

1: Problem 4.2 (one method only). The step response is the response when initial condition=0 and the input is a step function.
2: Find the transfer function in Problem 4.4.
3: Find x(t) and y(t) for Problem 4.4 when initial condition=0 and u(t)=0. .
4: Problem 4.16
5: Problem 4.17
6: Problem 4.19

HW5, due on Oct 8,

1: Problem 6.1, controllability part only.
2: Problem 6.3
3: Problem 6.4, assuming B_1 is full row rank.
4: Problem 6.14, controllability part only.
5: Problem 6.15, controllability part only.
6: Problem 6.16, controllability part only.

HW6, due on Oct 29,

1:Problem 6.1, observability only.
2: Problem 6.14, obesrvability only.
3: Problem 6.15, observability only.
4: Problem 6.16, observability only.
5:Problem 6.1, find the controllable canonical form if it is controllable. Find the observable canonical form if it is obeservable.
6:Problem 6.2, find the controllable canonical form if it is controllable. Find the observable canonical form if it is obeservable. Change B=[0 1;1 0; 0 0] to B=[0;1;0].

HW7, due on Nov 5,

1: Let \dot{x}=[1 0;0 0] x. Find all the equilibrium points and discuss their stability i.s.l. and asymptotical stability.
2: Is the system in Problem 5.7 BIBO stable, stable i.s.l, asymptotically stab le? (you may set -2u=0 in the output equation.)
3: Is the system in Problem 5.10 stable i.s.l, asymptotically stable?
4: Is the system in Problem 5.11 stable i.s.l, asymptotically stable?

HW8, due on Nov. 12,

For this HW, you can use any method you prefer.

1: Problem 8.1.
2: Problem 8.4.
3: Problem 8.5.
4: Problem 8.6.
5: Problem 8.7.

HW9, due on Dec. 3 (this is the last HW),

1: Design a state observer for Problem 8.1 with eigenvalues {-2+2j, -2-2j}.
2: Design a state feedback using the observer for Problem 8.7 so that the eigenvalues of the feedback system are at -2, -1+j and -1-j, and the eigenvalues of the observer are at -1,-1 and -1.