Description
1. The system shown in the figure below represents a satellite control system. The block
diagram consists of a first-order system with a zero of –a, a sign function, and a
double integrator. Please note that the parameter “p” is equivalent to the Laplace
transform variable “s”. The parameter “a” is a positive real number. You may choose
an “a” at your convenience for a successful computer simulation. Draw the phase
portrait of the system using Matlab/Simulink, and determine the system’s stability
through the phase portrait.
2. (For EEC743 and ESC794 students only) The following nonlinear system has limit
cycles. Without solving the state equations explicitly, show that the number of limit
cycles is infinity. (Hint: Use polar coordinates)
𝑥ሶ ൌ𝑦𝑥ሺ𝑥ଶ 𝑦ଶ െ 1ሻ𝑠𝑖𝑛
1
𝑥ଶ 𝑦ଶ െ 1
𝑦ሶ ൌ െ𝑥 𝑦ሺ𝑥ଶ 𝑦ଶ െ 1ሻ𝑠𝑖𝑛
1
𝑥ଶ 𝑦ଶ െ 1
3. Using Bendixson’s theorem to show that the following system has no limit cycles.
𝑥ሶ
ଵ ൌ െ𝑥ଵ 𝑥ଶ
𝑥ሶ
ଵ ൌ 𝑔ሺ𝑥ଵሻ 𝑎𝑥ଶ, 𝑎്1
4. Using Poincare’s theorem to show that the following system has no limit cycles.
𝑥ሶ
ଵ ൌ1െ𝑥ଵ𝑥ଶ
ଶ
𝑥ሶ
ଶ ൌ 𝑥ଵ
5. Using Lyapunov’s direct method to prove the stability of the following system.
𝑥ሶ ൌ െ𝑦 െ 𝑥ଷ
𝑦ሶ ൌ𝑥െ𝑦ଷ
6. Consider the following autonomous system. Find the domain of attraction for the
stability of the system.
𝑥ሶ
ଵ ൌ ሺ𝑥ଵ െ 𝑥ଶሻሺ𝑥ଵ
ଶ 𝑥ଶ
ଶ െ 1ሻ
𝑥ሶ
ଶ ൌ ሺ𝑥ଵ 𝑥ଶሻሺ𝑥ଵ
ଶ 𝑥ଶ
ଶ െ 1ሻ
