University of Pennsylvania
ESE 500: Linear Systems Theory
Homework II
Due: on Wednesday 10/20 at 23:59 on Gradescope
INSTRUCTIONS
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e-
sponding to each problem. Failure to do so may result in your work not graded completely.
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E
ata (Last update 10/8)
1. Problem 8: The output should read y =
[
1 cos θ + (r2 − 1) sin θ
−r1 sin θ + (r2 − 1) cos θ
]
.
Problem 1 (Matrix differentiation - 13 points). Let A(t) be an n × n matrix, which depends
on t, i.e. its elements are functions of t:
[A(t)]ij = aij(t), for i, j = 1, . . . , n
The derivative Ȧ(t) of A(t) with respect to t, is also an n× n matrix and is defined by taking
element-wise derivatives:
[Ȧ(t)]ij = ȧij(t), for i, j = 1, . . . , n
a) (4 points) For n× n differentiable matrices A1(t), A2(t) prove:
d
dt
(
A1(t)A2(t)
)
= Ȧ1(t)A2(t) +A1(t)Ȧ2(t)
1
) (4 points) Using induction, prove that for n × n differentiable matrices A1(t), A2(t), ... ,
Ak(t), we have:
d
dt
(
A1(t)A2(t)...Ak(t)
)
= Ȧ1(t)A2(t)...Ak(t) +A1(t)Ȧ2(t)...Ak(t XXXXXXXXXXA1(t)A2(t)...Ȧk(t)
c) (5 points) The exponential of a n× n matrix A is defined as the following:
expA =
∞∑
i=0
Ai
i !
where A0 = I. Suppose A(t) and Ȧ(t) commute. Prove:
d
dt
expA(t) = Ȧ(t) expA(t)
Problem 2 (Functions of a matrix - 26 points). Let f, g be functions over matrices and A,B ∈
Rn×n. Suppose AB = BA.
a) (3 points) Prove f(A)g(B) = g(B)f(A).
) (3 points) Prove f(AT ) = f(A)T .
c) (3 points) Let A = QJQ−1 be any matrix decomposition. Prove f(A) = Qf(J)Q−1.
d) (4 points) eA+B = eAeB. Hint: if two functions satisfy the same differential equation, then the uniqueness
of solution of differential equations says they are equal.
e) (4 points) Prove det(eA) = etr(A). Note: you can use known facts about determinant and trace.
f) (4 points) Prove that BeAt = eAtB if and only if AB = BA.
g) (5 points) For the time invariant linear state equation ddtx(t) = Ax(t) show that given any
x0 there exists a constant α(x0) such that
det[x(t) Ax(t XXXXXXXXXXAn−1x(t)] = α(x0)e
tr(A)t
Problem 3 (State space representation - 8 points). Consider a system with input u(t) and
output y(t) which can be described using the following set of differential equations:
z̈1(t) = z1(t) + z2(t) + u̇(t)
ż2(t) = ż1(t) + z2(t) + u(t)
y(t) = ż1(t)
a) (4 points) Define the states of the system such that it can be represented as an 3-dimensional
LTI system, i.e., as the following:
ẋ(t) = Ax(t) +Bu(t)
y(t) = Cx(t) +Du(t)
where A,B,C,D are constant matrices.
2
) (4 points) Consider T defined below, as a new basis for the state space and let x̂(t) be the
epresentation of x(t) with respect to the basis T.
T =
10
1
,
01
1
,
11
1
Compute Â, B̂, Ĉ, D̂ in the new representation of the system with respect to T :
˙̂x(t) = Âx̂(t) + B̂u(t)
y(t) = Ĉx̂(t) + D̂u(t)
Problem 4 (10 points). Using the definition of transition matrix, prove that:
a) (2 points) φA(t2, t1)φA(t1, t0) = φA(t2, t0), for all t0, t1, t2.
) (1 point) φA(t, t) = I, for all t.
c) (2 points) φ−1A (t, τ) = φA(τ, t), for all t, τ .
d) (5 points)
d
dτ
φA(t, τ) = −φA(t, τ)A(τ)
Problem 5 (8 points). Find the solution x(t) of the following time variant system
ẋ(t) =
[
(t+ 1)2 t+ 1
t+ 1 t2 − 1
]
x(t), x(0) = x0
Problem 6 (Periodic System - 13 points). Consider the system
ẋ(t) = A(t)x(t)
where A(t) is a periodic matrix with period T . That means A(t+ T ) = A(t) for all t ∈ R.
a) (2 points) First, consider the state transition matrix Φ(t1, t0) for the system. Define the
matrix Ψ(t, 0) = Φ(t+ T, 0). Show the Ψ satisfies:
Ψ̇(t, 0) = A(t)Ψ(t, 0)
Ψ(0, 0) = Φ(T, 0)
) (2 points) Show that Φ(t+ T, 0) = Φ(t, 0)Φ(T, 0).
c) (3 points) Since we know that Φ(T, 0) is invertible, there exists some complex n× n matrix
R such that Φ(T, 0) = eTR. Define
P (t)−1 := Φ(t, 0)e−tR
Show that P (t)−1 is periodic with period T . This implies that P (t) is periodic with period
T . Also show that P (T ) = I.
d) (4 points) Show that
Φ(t, t0) = P (t)
−1e(t−t0)RP (t0)
Hint: Note that Φ(t, t0) = Φ(t, 0)Φ(0, t0)
e) (2 points) Express the system using the coordinate frame z(t) = P (t)x(t). What do you
notice about this new system?
3
Problem 7 (Discretization of continuous time LTI systems - 12 points). The dymanics of an
aircraft consist of a set of nonlinear coupled differential equations. Under certain assumptions,
though, these can be decoupled and reformed in a linear system. Aircraft pitch is governed
y the longitudinal dynamics. Let’s assume that the aircraft is in steady-cruise at constant
altitude and velocity (thus, the thrust, drag, weight and lift forces balance each other in the
x- and y-directions) and that a change in pitch angle will not change the speed of the aircraft
under any circumstance (unrealistic but simplifies the problem). Under these assumptions, the
longitudinal equations of motion for the aircraft are:
ȧ = µΩσ[−(CL + CD)a+
1
µ− CL
q − (CW sin γθ + CL]
q̇ =
µΩ
2iyy
[[CM − η(CL + CD)]a+ [CM + σCM (1− µCL)]q + (ηCW sin γ)δ]
θ̇ = Ωq
where a is the angle of attack, θ is the pitch angle and q the rate of pitch angle, δ the el-
evator deflection angle, µ = ρSc̄4m , ρ the density of air, S the platform’s area of the wing, c̄
the average chord length, m the mass of the aircraft, Ω = 2Uc̄ , U the equili
ium flight speed,
CT , CD, CL, CW , CM the coefficients of thrust, drag,lift, weight and pitch moment, γ the flight
path angle, σ = 11+µCL , iyy the normalized moment of inertia and η = µσCM .
Using the parameters from the data from one of Boeing’s commercial aircraft, the above
system becomes:
ȧ = −0.313a+ 56.7q + 0.232δ
q̇ = −0.0139a− 0.426q + 0.0203δ
θ̇ = 56.7q
In this problem we will transform this simple continuous time model in discrete time one
upon which one may design an autopilot that controls the pitch of an aircraft.
a) (2 points) Rewrite the model in state-space space form:
ẋ(t) = Ax(t) +Bδ(t), y(t) = Cx(t)
Use x(t) := [a, θ, q]T . The elevator deflection angle δ is considered the input of the system
and the pitch angle θ of the aircraft is the output.
) (5 points) Let the sampling period be Ts = 0.01s. Compute the discrete-time system. Then,
check your derivations using the c2d function of Matlab.
Note: You may use MATLAB for simple numerical calculations (e.g., matrix inversion,
matrix multiplication), but you should show all the steps in your derivations.
4
c) (3 points) Using δ(t) = 0.2, simulate the continuous time system using ode45 of Matlab fo
t ∈ [0, 10]. Plot the output y(t) with zero initial conditions, i.e. x(0) = [0, 0, 0]T .
d) (2 points) Simulate the discrete time system, for δ(k) = 0.2, k = 0, 1, . . . , N − 1, where
N = 10/Ts. In the same plot as before, plot the discretized y(k). Is the response of the
discrete-time system close to the one of the continuous time? Repeat the simulation fo
Ts ∈ {0.1, 0.5, 1, 2, 5, 10} and plot the responses in the previous plot (don’t forget to label
what’s what in the plot). For all of the simulations assume zero initial conditions, i.e.
x(0) = [0, 0, 0]T . Comment on the graph.
Problem 8 (Local linearization around a trajectory: unicycle - 10 points). A single-wheel cart
(unicycle) moving on the plane with linear velocity v and angular velocity ω can be modeled
y the following system of nonlinear of ODEs:
̇1 = v cos θ
̇2 = v sin θ
θ̇ = ω,
(1)
where (r1, r2) denotes the Cartesian coordinates of the wheel and θ its orientation. Regard this
as a system with input u =
[
v
ω
]
and output y =
[
1 cos θ + (r2 − 1) sin θ
−r1 sin θ + (r2 − 1) cos θ
]
.
a) (0.5 points) Prove that the map that takes input u(t) into y(t) is nonlinear.
) (2 points) Construct a state-space model for this system with state
x =
x1x2
x3
:=
r1 cos θ + (r2 − 1) sin θ−r1 sin θ + (r2 − 1) cos θ
θ
.
c) (0.5 points) Prove that xeq = 0, ueq = 0 is an equili
ium for the state-space system derived
in part 2. Is this the only equili
ium point?
d) (1.5 points) Compute a local linearization for the state-space system derived in part 2, around
the