Description
Problem 1 Consider one dimensional actuated rotational body:
ϕ¨ = u
Assume that we apply the switching and rate feedback as follows:
u = −sign(kϕ + ˙ϕ)
Draw the phase portrait of the system and determine convergence of the trajectories starting from
ϕ0 > 0, ϕ0 < 0 and ϕ0 = 0
Problem 2 Consider the linear two dimensional system in form:
x˙ = Ax, x ∈ R
2
Study how the eigenvalues of A affect the phase portraits. Plot the different types of phase portraits (node, saddle, focus, center) with asociated complex plane representation of eigenvalues.
Hint: Read the chapter 2.4 of Slotine’s textbook and repeat figure 2.9
Problem 3 Consider the following systems:
(a)
(
x˙ 1 = x2 − x1(x
2
1 + x
2
2 − 1)
x˙ 2 = −x1 − x2(x
2
1 + x
2
2 − 1)
(b)
(
x˙ 1 = x2 + x1(x
2
1 + x
2
2 − 1)
x˙ 2 = −x1 + x2(x
2
1 + x
2
2 − 1)
(c)
(
x˙ 1 = x2 − x1(x
2
1 + x
2
2 − 1)2
x˙ 2 = −x1 − x2(x
2
1 + x
2
2 − 1)2
Draw the associated phase portraits, and describe the behavior of system trajectories.
Hint: use the figure 2.10 from the textbook.
1
Problem 4 Consider damped nonlinear pendulum with described by:
x¨ = −x˙ − sin x
Find the equlibrium points and deduce their stability using Lyapunov linearization method.
Problem 5 Implement software routine that will implement the Lyapunov linearization method
including:
• Solving for equlibrium
• Symbolical linearization
• Checking for the stability of each equlibrium
• Drawing the phase portrait together with stable and unstable points.
Test developed routine on following system:
(
x˙ 1 = x1 − x
3
1 + 2x1x2
x˙ 2 = −x2 +
1
2
x1x2
2