ECH267-Homework 2: Advanced Process Control Solved

Reading Assignment: Lecture notes; Khalil Chapter 4.1-4.7

Exercise 4.1 (Khalil pg. 181). Consider a second-order autonomous system. For each of the following types of equilibrium points, classify whether the equilibrium point is stable, unstable, or asymptotically stable:stable node
unstable node
stable focus
unstable focus
center
Justify your answer using phase portraits (sketches of phase portraits are acceptable).

Exercise 4.2 (Khalil pg. 181). Consider the scalar system x˙ = axp+g(x), where p is a positive integer and g(x) satisfies |g(x)| ≤ k |x|p+1 in some neighborhood of the origin x = 0. Show that the origin is asymptotically stable if p is odd and a < 0. Show that it is unstable if p is odd and a > 0 or p is even and a 6= 0.
Exercise 4.3 (Khalil pg. 181). For each of the following systems, use a quadratic Lyapunov function candidate to show that the origin is asymptotically stable. Then, investigate whether the origin is globally asymptotically stable.
Hint: For the g.a.s case, you may want to try non-quadratic Lyapunov functions, such as with a, b positive integers, and show that their derivatives are negative for all (x1,x2) 6= (0,0).

x˙1 = −x1 + x1x2
x˙2 = −x2

x˙1 = −x2 − x1(1 − x21 − x22)
x˙2 = x1 − x2(1 − x21 − x22)

x˙1 = x2(1 − x21)
x˙2 = −(x1 + x2)(1 − x21)

x˙1 = −x1 − x2
 

Exercise 3.4 (Khalil Second Edition pg. 155). Using, study stability of the origin of the system
x˙1 = x1(k2 − x21 − x22) + x2(x21 + x22 + k2) x˙2 = −x1(k2 + x21 + x22) + x2(k2 − x21 − x22)

when (a) k = 0 and (b) k 6= 0.

Exercise 3.8 (Khalil Second Edition pg. 155). Consider the system
x˙1 = x2 x˙2 = x1 − sat(2x1 + x2) (a) Show that the origin is asymptotically stable.

Show that all trajectories starting in the first quadrant to the right of the curve x1x2 = c (with sufficiently large c > 0) cannot reach the origin.
Hint: In part (b), consider V (x) = x1x2; calculate V˙ (x) and show that on the curve V (x) = c the derivative V˙ (x) > 0 when c is large enough.

Show that the origin is not globally asymptotically stable.
Exercise 4.13 (Khalil pg. 183). For each of the following systems, show that the origin is unstable:x˙1 = x31 + x21x2
x˙2 = −x2 + x22 + x1x2 − x31

x˙1 = −x31 + x2
x˙2 = x61 − x32

Hint: In part (2), show that  is a nonempty positively invariant set, and investigate the behavior of the trajectories inside Γ.

Exercise 3.22 (Khalil First Edition pg. 158). Consider the system
x˙1 = −x1 + x2 x˙2 = (x1 + x2)sinx1 − 3x2

Show that the origin is the unique equilibrium point.
Show, using linearization, that the origin is asymptotically stable. (c) Show that the origin is globally asymptotically stable.
Hint: Use a simple quadratic Lyapunov function and the inequality |sinx1| < 1.

Exercise 4.20 (Khalil pg. 185). Suppose the set M in LaSalle’s theorem consists of a finite number of isolated points. Show that limt→∞ x(t) exists and equals one of these points.
Exercise 4.26 (Khalil pg. 186). Let x˙ = f(x), where f : Rn → Rn. Consider the change of variables z = T(x), where T(0) = 0 and T : Rn → Rn is a diffeomorphism in the neighborhood of the origin; that is, the inverse map T−1(·) exists, and both T(·) and T−1(·) are continuously differentiable. The transformed system is
                                                                     z˙ = fˆ(z),      where

Show that x = 0 is an isolated equilibrium point of x˙ = f(x) if and only if z = 0 is an isolated equilibrium point of z˙ = fˆ(z).
Show that x = 0 is stable (asymptotically stable/unstable) if and only if z = 0 is stable (asymptotically stable/unstable).
Determine the three equilibrium points of the system
x˙1 = x2

.

Investigate the stability of the equilibrium point x = 0. Verify the conclusions about the phase portrait and region of attraction of the system. In particular, generate its phase portrait by simulations and plot the level sets of the Lyapunov function (if using Matlab, use the commands meshgrid and contour). Verify that the setis (approximately) the largest estimate of the region of attraction that can be generated from this Lyapunov function.

Exercise 3.15 (Khalil Second Edition pg. 157). Consider the system
x˙1 = x2

x˙2 = −x1 − x2sat(x22 − x23) x˙3 = x3sat(x22 − x23)

where sat(·) is the saturation function. Show that the origin is the unique equilibrium point, and use V (x) = xTx to show that it is globally asymptotically stable.

Exercise 3.16 (Khalil Second Edition pg. 157). The origin x = 0 is an equilibrium point of the system
x˙1 = −kh(x)x1 + x2 x˙2 = −h(x)x2 − x31

Let D = {x ∈ R2 | kxk2 < 1}. Using , investigate stability of the origin in each of the following cases.

k > 0, h(x) > 0, ∀x ∈ D.
k > 0, h(x) > 0, ∀x ∈ R2
k > 0, h(x) < 0, ∀x ∈ D (4) k > 0, h(x) = 0, ∀x ∈ D
k = 0, h(x) > 0, ∀x ∈ D
k = 0, h(x) > 0, ∀x ∈ R2
Exercise 3.19 (Khalil Second Edition pg. 158). Consider the system
x˙1 = x2 x˙2 = −asinx1 − kx1 − dx2 − cx3 x˙3 = −x3 + x2

where all coefficients are positive and k > a. Using

with some p > 0, show that the origin is globally asymptotically stable.

Exercise 4.18 (Khalil pg. 184). The mass-spring system of Exercise 1.11 is modeled by
My¨ = Mg − ky − c1y˙ − c2y˙ |y˙|

Show that the system has a globally asymptotically stable equilibrium point.

Exercise 4.23 (Khalil pg. 185). Consider the linear system, where (A,B) is controllable, P = PT > 0 satisfies the Riccati equation
PA + ATP + Q − PBR−1BTP = 0

R = RT > 0, and Q = QT ≥ 0. Using V (x) = xTPx as a Lyapunov function candidate, show that the origin is globally asymptotically stable when:

Q > 0.
Q = CTC and (A,C) is observable. Hint: Apply LaSalle’s theorem and recall that for an observable pair (A,C), the vector C exp(At)x ≡ 0 ∀t if and only if x = 0. 16. Exercise 4.28 (Khalil pg. 186). Consider the system
x˙1 = −x1

Show that x = 0 is the unique equilibrium point.
Show, by using linearization, that x = 0 is asymptotically stable.
Show that Γ = {x ∈ R2|x1x2 ≥ 2} is a positively invariant set. (d) Is x = 0 globally asymptotically stable?
Exercise 3.30 (Khalil Second Edition pg. 161). For each of the following systems, use linearization to show that the origin is asymptotically stable. Then, show that the origin is globally asymptotically stable.x˙1 = −x1 + x2
x˙2 = (x1 + x2)sinx1 − 3x2

x˙1 = −x31 + x2
x˙2 = −ax1 − bx2, a,b > 0

Exercise 4.36 (Khalil pg. 188). Is the origin of the scalar system x˙ = −x/(t + 1), t ≥ 0, uniformly asymptotically stable?
Exercise 4.37 (Khalil pg. 188). For each of the following linear systems, use a quadratic Lyapunov function to show that the origin is exponentially stable:
              (1)                                                                  

              (2)                                                                             

              (3)                                                                   

              (4)                                                                               

In all cases, α(t) is continuous and bounded for all t ≥ 0. 20. Exercise 4.41 (Khalil pg. 189). Consider the system

x˙1 = x2 x˙2 = 2x1x2 + 3t + 2 − 3x1 − 2(t + 1)x2

Verify that x1(t) = t, x2(t) = 1 is a solution.
Show that if x(0) is sufficiently close to , then x(t) approaches .
Exercise 4.44 (Khalil pg. 189). Consider the system
x˙1 = −x1 + x2 + (x21 + x22)sint x˙2 = −x1 − x2 + (x21 + x22)cost

Show that the origin is exponentially stable and estimate the region of attraction.

Exercise 4.48 (Khalil pg. 190). Consider two systems represented by x˙ = f(x) and x˙ = h(x)f(x) where f : Rn → Rn and h : Rn → R are continuously differentiable, f(0) = 0, and h(0) > 0. Show that the origin of the first system is exponentially stable if and only if the origin of the second system is exponentially stable.