[SOLVED] 24667-Homework 1

Description

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.

1. y(t) = 0 for all t (4 points)
2. y(t) = u³(t) (4 points)
3. y(t) = u(3t) (4 points)
4. y(t) = e^−tu(t − T) (4 points)
5. (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 p_i (p_i > 0). The path gain from transmitter j to receiver i is G_ij (G_ij > 0 for j 6= i, and G_ii > 0). The signal power at receiver i is given by s_i = G_iip_i. 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 S_i = s_i/q_i. For signal reception to occur, the SINR must exceed some threshold value γ (i.e., S_i ≥ γ). We assume p, q and S are discrete-time signals. For example, p_i(k) represents the transmit power of transmitter i at time k (k = 0,1,2,…). We want to have a certain SINR, e.g.

S_i(k) = s_i(k)/q_i(k) = αγ,

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

1. 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 ∈R^3×3 and B ∈R^3×1 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)

2. 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 S_i 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 S_i(t) → αγ? Plot

all the p_i(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.

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