Description
1. Show that the internal dynamics of the following system is unstable using the concept
of zero dynamics. The output y=x1. Suppose the controller u needs to drive y to yd.
ẋଵ
ẋଶ
൨ =
xଶ
ଷ + u
−u
൨
2. Design a controller for the following system to track a constant reference signal yd=1.
Assume that the model is accurate, and the states [x1, x2]
T
are measurable. Write the
full expression of the controller, as a function of the measured states [x1, x2]
T
. Please
simulate your control system in Matlab/Simulink and test the effectiveness of your
controller design. Plot time response y. Is y driven to yd in your simulation?
𝑥̇ଵ = sin(𝑥ଶ)
𝑥̇ଶ = 𝑥ଵ
ସ cos(𝑥ଶ) + 𝑢
𝑦 = 𝑥ଵ
3. (For EEC743/ESC794 students only) Find the internal dynamics and zero dynamics
of the following system where y is output and u is input. Is the zero dynamics stable?
𝑦̇ + 𝑧
ଷ𝑒
௬௭మ = 𝑢
𝑧̈− (𝑦̇ + 𝑦
ଷ)(𝑧̇
ସ + 1) + 𝑧
ହ + 𝑦𝑧 = 0
4. Globally stabilize the following nonlinear system, where u is the control input.
𝑦̇ + 𝑦
ଶ𝑒
௬
ర௭ = 𝑢
𝑧̈+ 𝑧̇
ଷ − 𝑧
+ 𝑦𝑧ଶ = 0
5. The following pendulum system has a control input u which is used to stabilize the
angular position (𝜃 ) of pendulum at 𝛿ଵ =
గ
ଶ
.
𝑥̇ଵ = 𝑥ଶ
𝑥̇ଶ = − ቀ
𝑔
𝑙
ቁ sin(𝑥ଵ + 𝛿) −
𝑘
𝑚
𝑥ଶ +
1
𝑚𝑙ଶ 𝑢
In the equations above, 𝑥ଵ = 𝜃 − 𝛿ଵ and 𝑥ଶ = 𝜃̇. The parameter values are m=0.1,
𝑙=1, and 𝑘=0.02. The controller is designed as 𝑢 = −𝑘𝑠𝑔𝑛(𝑎ଵ𝑥ଵ + 𝑥ଶ), where a1=1
and k=4. We suppose the output y=𝜃. Simulate the control system and show that the
output is driven to గ
ଶ
.
