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Homework 3-2

The document outlines Homework 3 for ME5659 Fall 2024, consisting of three problems related to linear systems and stability analysis. Problem 1 involves characterizing stability through eigenvalues and Lyapunov analysis, along with MATLAB simulations. Problems 2 and 3 focus on local linearization of an inverted pendulum and conditions for asymptotic stability in linear systems, respectively.

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41 views3 pages

Homework 3-2

The document outlines Homework 3 for ME5659 Fall 2024, consisting of three problems related to linear systems and stability analysis. Problem 1 involves characterizing stability through eigenvalues and Lyapunov analysis, along with MATLAB simulations. Problems 2 and 3 focus on local linearization of an inverted pendulum and conditions for asymptotic stability in linear systems, respectively.

Uploaded by

lucky2.clashofclans
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Homework 3

ME5659 Fall 2024


Due: See Canvas, turn in on Gradescope

Problem 1 (9 points)
Consider the following linear systems ẋ = Ax, x(0) = x0 , where
   
0 1 −0.2
(i) A = , x0 = .
−14 −4 0.1
   
0 1 0.01
(ii) A = , x0 = .
−14 4 0.02
   
0 1 0
(iii) A = , x0 = .
−14 0 0

(a) 3 points. Characterize the stability of the equilibrium point from the eigenvalues of A.

(b) 3 points. Use Lyapunov stability analysis to determine whether the system equilibrium state xeq = 0
is asymptotically stable. Use Q = I in the Lyapunov equation. (Do all calculations by hand.)

(c) 3 points. Use MATLAB to plot the state trajectories x(t) vs. time t with the initial condition x0 .
Each plot has two trajectories x1 (t), x2 (t). Hand in your plots and your code.

1
Problem 2 (6 points)
Consider the inverted pendulum (Figure 1) which is characterized by

ml2 θ̈ = mglsinθ − bθ̇ + T


where T denotes a torque applied at the base and g is the gravitational acceleration. We assume that u = T
and y = θ are its input and output, respectively.

Figure 1: Simple pendulum

(a) (3 points) Perform local linearization of this system around the equilibrium point θ = π, derive the
linear state-space models and determine whether it is stable or not.

(b) (3 points) Perform local linearization of this system around the equilibrium point θ = 0, derive the
linear state-space models and determine whether it is stable or not.

2
Problem 3 (10 points)
Consider the following linear system ẋ = Ax, x(0) = x0 , where
 
0 1
A=
a b

(a) (4 points) Under what conditions on a, b is the system equilibrium, xeq = 0, asymptotically stable?
Use Lyapunov stability analysis.

(b) (3 points) Write the Lyapunov function V (x) and its time derivative V̇ (x). Use the P matrix
obtained in (a) to evaluate stability.

(c) (3 points) If b = 0, a < 0, show that the linear system is stable but not asymptotically stable.

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