Description
1: 15 points
Given
A =
1 1 0
0 0 1
0 0 1
.
(a) Find A10
.
(b) Find A103
.
(c) Find e
At
.
2: 20 points
Find a closed-form expression for Ak
for k ≥ 1 where
A =
0 0 0
9 23 30
−7 −18 −235
.
3: 20 points
For the following system, find an expression for y(n) if the input u(k) = 1 ∀k ≥ 0 starting at
x(0) =
0 0T
.
x(k + 1) =
0 1
−0.5 −1
x(k) +
1
1
u(k)
y(k) =
1 0
x(k)
4: 30 points
Let x1(t) be the water level in Tank 1 and x2(t) be the water level in Tank 2 . Let α be the rate
of outflow from Tank 1 and β be rate of outflow from Tank 2 . Let u be the supply of water to
the system. The system can be modelled into the following differential equations:
dx1
dt = −αx1 + u
dx2
dt = αx1 − βx2
Figure 1: Tank Problem
Given α = 0.1, β = 0.2, u = 1, x1(0) = 2, x2(0) = 1, find the water level in both tanks after 5s.
(a) Find y(5) for the CT system. Solve with the Cayley-Hamilton theorem. You may use a
calculator but do not directly use programming.
(b) Find the discretized state space representation using sample time T = 1s.
(c) Find y(5) of the discrete time system. Also plot signals y(t) for both CT and DT systems in
the same figure.
5: 20 points
A PID controller in its classical form is not realizable because it has more poles than zeros. To
implement the controller, we add a pole to the system at “high” frequency to get a bi-proper
system, i.e.
C(s) = Kds
2 + Kps + Ki
s (τs + 1)
with τ small.
(a) Find a state space realization for the modified (bi-proper) PID controller.
(b) Find and graph the solution for the controller response (starting at zero initial state) given an
input of u = 1 for Kd = 10, Kp = 100, Ki = 1, and τ = 0.001. The response should grow without
bound – why?
