Description
1. Exercise 1.2 (Khalil pg. 48). Consider a single-input-single-output system described by the nth-order differential
equation
y
(n) = g1
t, y, y, . . . , y ˙
(n−1), u
+ g2
t, y, y, . . . , y ˙
(n−2)
u˙
where g2 is a differentiable function of its arguments. With u as input and y as output, find a state-space model.
Hint: Take xn = y
(n−1) − g2
t, y, y, . . . , y ˙
(n−2)
u.
2. Exercise 1.7 (Khalil pg. 29). Figure 1 shows a feedback connection of a linear time-invariant system and a
nonlinear time-varying element. The variables r, u, and y are vectors of the same dimension, and ψ(t, y) is a
vector-valued function. With r as input and y as output, find a state-space model.
Linear System G(s) =
C(sI − A)
−1B
ψ(t, y)
r + u y
−
Figure 1: Exercise 1.7.
3. Exercise 1.11 (Khalil pg. 50). A phase-locked loop can be represented by the block diagram of Figure 2.
Let {A, B, C} be a minimal realization of the scalar, strictly proper transfer function G(s). Assume that all
eigenvalues of A have negative real parts, G(0) 6= 0, and θi = constant. Let z be the state of the realization
{A, B, C}.
(a) Show that closed-loop system can be represented by the state equations
z˙ = Az + B sin e, e˙ = −Cz
(b) Find all the equilibrium points of the system.
(c) Show that when G(s) = 1/(τs + 1), the closed-loop model coincides with the model of a pendulum
equation.
sin(·) G(s)
θi + e u y
R
θ0
−
Figure 2: Exercise 1.11.
