Homework 1
ME5659 Fall 2024
Due: See Canvas, turn in on Gradescope
Problem 1 (6 points)
Describe the dynamical systems in state-space representations.
(a) 3 points. For the following system described by the given transfer function, derive valid state-space
realization (define the state variables and derive the state-space representation):
Y (s) 4s4 − s2 + 5s − 1
G(s) = = 4
U (s) 2s + 2s2 − 4s + 6
(b) 3 points. Given the following differential equations, derive valid state-space realization with u =
T T
u1 u2 and y = y1 y2 (define the state variables and derive the state-space representation):
ÿ1 (t) + 2ẏ1 (t) − 5(y2 (t) − y1 (t)) = u1 (t)
ÿ2 (t) + ẏ1 (t) − 4ẏ2 (t) − 3(y2 (t) − y1 (t)) = u2 (t)
1
Problem 2 (6 points)
Consider a pendulum as shown in Fig. 1. We assume that the mass m is concentrated at pendulum end,
with length as l. Gravity should be considered. The pendulum is driven by a torque input T at the base,
and the base rotation joint is subject to rotational damping b. The equation of motion of this pendulum is
ml2 θ̈ + bθ̇ + mgl sin θ = T,
where T is input and pendulum angle θ is output.
Figure 1: Simple pendulum
(a) 2 points. Define the 2 state variables of the system. The input is u = T . Put the equations of
motion in nonlinear state-space form, ẋ = f (x, u).
(b) 4 points. Consider the initial angle of pendulum is θ0 and initial torque T0 = 0 for a passive system.
A small toque input δT is at the base joint, leading to perturbation angle of δθ. Linearize the nonlinear
state-space model about θ0 and T0 to obtain the linear state-space models for θ0 = 0 and θ0 = π.
2
Problem 3 (6 points)
A single-wheel chair cart (unicycle) moving on the plane with linear velocity v and angular velocity ω can
be modeled by the nonlinear system
ṗx = vcosθ, ṗy = vsinθ, θ̇ = ω,
where (px , py ) denote the Cartesian coordinates of the wheel and θ its orientation. Regard this as a system
T
with input u = v ω
(a) (3 points) Construct a state-space model for this system with state
x1 (t) px cosθ + (py − 1)sinθ
x = x2 (t) = −px sinθ + (py − 1)cosθ
x3 (t) θ
T
and output y = x1 x2 .
(b) (3 points) Compute a linearization for this system around the equilibrium point xeq = 0, ueq = 0.
3
Problem 4 (7 points)
Consider the following model for a DC motor:
d2 θ dθ
J +b = Kt i (1)
dt2 dt
di dθ
L + Ri + Kb = Vs , (2)
dt dt
where J is the mass-moment of inertia of the load on the motor, which is damped by rotary damper with
linear damping constant b. The torque delivered by the motor is Kt i, where Kt is the motor torque constant
and i is the motor current. The motor has internal series resistance R and inductance L, and a motor speed
constant Kb . The voltage supplied to the motor is Vs . In all parts, consider the input u = Vs and the output
y = θ, motor shaft angle.
(a) (2 points) In a coupled system it may not be clear at first what the order of the system is. In this
problem, we have, effectively, a first-order system in i and a second-order system in θ, giving us three states.
Making the choice for states x1 = θ, x2 = θ̇, and x3 = i, calculate the A and B matrices for a state-space
representation.
(b) (2 point) To better illustrate that the system is third-order, find a single third-order differential
equation in terms of θ and its derivatives (the current will not appear in the equation). Laplace transforming
the ODEs for manipulation, or using a differential operator will make this easier.
(c) (2 points) State-space representations of dynamical systems are not unique. Making the choice for
states x1 = θ, x2 = θ̇, and x3 = θ̈, calculate the A and B matrices for a state-space representation.
(d) (1 point) While state-space representations of systems are not unique, they represent the same
systems if originating from the same set of differential equations. Assume that all constant parameters are
equal to 1, and use MATLAB to calculate the eigenvalues of both A-matrices (from parts (a) and (c) ), and
show that they are the same. List your MATLAB code, and the program/command outputs.
