24667-Homework 1 Solved

Exercise 1. Types of Systems (20 points)
A system has an input u(t) and an output y(t), which are related by the information provided below. Classify each system as linear or non-linear and time invariant or time-varying, and explain why.

y(t) = 0 for all t (4 points)
y(t) = u3(t) (4 points)
y(t) = u(3t) (4 points)
y(t) = e−tu(t − T) (4 points)
(4 points)
Exercise 2. State space representations (30 points)
A company deployed 3 teams of drones in a region. Each team consists of a pair of drones. One drone in the team carries a transmitter and the other one carries a receiver. Transmitter i transmits at power level pi (pi 0). The path gain from transmitter j to receiver i is Gij (Gij 0 for j 6= i, and Gii 0). The signal power at receiver i is given by si = Giipi. The noise plus interference power (caused by other transmitters j 6= i) at receiver i is given by

qi = σ2 + XGijpj,

j6=i

where σ2 0 is the self-noise power of the receivers.

Figure 1: The wireless network

The signal to interference plus noise ratio (SINR) at receiver i is defined as Si = si/qi. For signal reception to occur, the SINR must exceed some threshold value γ (i.e., Si ≥ γ). We assume p, q and S are discrete-time signals. For example, pi(k) represents the transmit power of transmitter i at time k (k = 0,1,2,...). We want to have a certain SINR, e.g.

Si(k) = si(k)/qi(k) = αγ,

where α 1 is an SINR safety margin. To achieve this goal, someone designed the following control rule pi(k + 1) = pi(k)(αγ/Si(k)).

Show that the power control update algorithm can be expressed as a linear dynamical system with constant input, e., in the form
p(k + 1) = Ap(k) + Bσ2,

where A ∈R3×3 and B ∈R3×1 are constant and p(k) = [p1,p2,p3]T. Describe A and b

explicitly in terms of σ,γ,α and the components of G. (10 points)

Use Python to simulate the power control algorithm. Use the problem data
Experiment with two different initial conditions: p1 = p2 = p3 = 0.1 and p1 = 0.1,p2 = 0.01,p3 = 0.02. Plot Si and p as a function of t, and compare it to the target value αγ. Repeat for γ = 5. Can the controller achieve the goal to make Si(t) → αγ? Plot

all the pi(k) as well. Submit your code to Gradescope. (20 points)

Exercise 3. Linearization (15 points)
Perform linearization on the given differential equation

y¨+ (1 + y)y˙ − 2y + 0.5y3 = 0

Exercise 4. Equilibrium (10 points)

The simplified dynamics of the vertical ascent of a Space X rocket can be modeled as

where D is the distance from earth to the surface of the rocket, m is the actual mass of the rocket, g is the gravity constant, and u is the thrust. During a short period of time, we can assume D, m, g, u are all constant. ln(∗) is the natural logarithmic,

Find the equilibrium states () of the above dynamic system. Perform linearization on the system.

Exercise 5. Linearization (25 points)
Model the earth and a satellite as particles. The normalized equations of motion, in an earth-fixed inertial frame, simplified to 2 dimensions (from Lagrange’s equations of motion, the Lagrangian , where r is the radius of the trajectory of the satellite, θ is the angle, k is the Newtonian constant:

with u1, u2 are control input, representing the radial and tangential forces due to the thrusters. The reference orbit with u1 = u2 = 0 is circular with r(t) ≡ p and θ(t) = ωt, where p is a representing the constant cruise radius, ω is the constant angular velocity of the satellite.

What’s the value of k expressed in terms of p and w, when the satellite is on the reference orbit? (10 points)
Obtain the linearized equation about this orbit. (Hint: we linearize on a trajectory, not a equilibrium point, so ˙x 6= 0) (15 points)