[Solved] ECH267 Homework 2-Advanced Process Control

1. 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).

2. Exercise 4.2 (Khalil pg. 181). Consider the scalar system x = ax^p+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.
3. 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).

1. x1 = x1 + x1x2

x₂ = x₂

2. x1 = x2 x1(1 x21 x22)

x2 = x1 x2(1 x21 x22)

3. x1 = x2(1 x21)

x₂ = (x₁ + x₂)(1 x²₁)

4. x1 = x1 x2

4. Exercise 3.4 (Khalil Second Edition pg. 155). Using, study stability of the origin of the system

x1 = x1(k2 x21 x22) + x2(x21 + x22 + k2) x2 = x1(k2 + x21 + x22) + x2(k2 x21 x22)

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

5. Exercise 3.8 (Khalil Second Edition pg. 155). Consider the system

x₁ = x₂ x₂ = 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.

6. Exercise 4.13 (Khalil pg. 183). For each of the following systems, show that the origin is unstable:
   • x1 = x31 + x21x2

x2 = x2 + x22 + x1x2 x31

• x1 = x31 + x2

x2 = x61 x32

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

7. Exercise 3.22 (Khalil First Edition pg. 158). Consider the system

x₁ = x₁ + x₂ x₂ = (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.

8. Exercise 4.20 (Khalil pg. 185). Suppose the set M in LaSalles theorem consists of a finite number of isolated points. Show that lim_t x(t) exists and equals one of these points.
9. 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 T¹() exists, and both T() and T¹() are continuously differentiable. The transformed system is

z = f(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 = f(z).
• Show that x = 0 is stable (asymptotically stable/unstable) if and only if z = 0 is stable (asymptotically stable/unstable).

10. Determine the three equilibrium points of the system

x₁ = 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.

11. Exercise 3.15 (Khalil Second Edition pg. 157). Consider the system

x₁ = x₂

x₂ = x₁ x₂sat(x²₂ x²₃) x₃ = x₃sat(x²₂ x²₃)

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

12. Exercise 3.16 (Khalil Second Edition pg. 157). The origin x = 0 is an equilibrium point of the system

x₁ = kh(x)x₁ + x₂ x₂ = 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

13. Exercise 3.19 (Khalil Second Edition pg. 158). Consider the system

x₁ = x₂ x₂ = asinx₁ kx₁ dx₂ cx₃ x₃ = x₃ + x₂

where all coefficients are positive and k > a. Using

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

14. 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.

15. Exercise 4.23 (Khalil pg. 185). Consider the linear system, where (A,B) is controllable, P = P^T > 0 satisfies the Riccati equation

PA + A^TP + Q PBR¹B^TP = 0

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

• Q > 0.
• Q = C^TC and (A,C) is observable. Hint: Apply LaSalles 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

x₁ = x₁

• Show that x = 0 is the unique equilibrium point.
• Show, by using linearization, that x = 0 is asymptotically stable.
• Show that = {x R²|x₁x₂ 2} is a positively invariant set. (d) Is x = 0 globally asymptotically stable?

17. 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.
    • x1 = x1 + x2

x₂ = (x₁ + x₂)sinx₁ 3x₂

• x1 = x31 + x2

x₂ = ax₁ bx₂, a,b > 0

18. Exercise 4.36 (Khalil pg. 188). Is the origin of the scalar system x = x/(t + 1), t 0, uniformly asymptotically stable?
19. 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, (t) is continuous and bounded for all t 0. 20. Exercise 4.41 (Khalil pg. 189). Consider the system

x₁ = x₂ x₂ = 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 .

21. Exercise 4.44 (Khalil pg. 189). Consider the system

x₁ = x₁ + x₂ + (x²₁ + x²₂)sint x₂ = x₁ x₂ + (x²₁ + x²₂)cost

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

22. 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.