Exercise 1. Types of Systems

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

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 = 2 + XGijpj,

j6=i

where ² > 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 (i.e., S_i ). 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) = ,

where > 1 is an SINR safety margin. To achieve this goal, someone designed the following control rule p_i(k + 1) = p_i(k)(/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) + B²,

where A R³³ and B R³¹ are constant and p(k) = [p₁,p₂,p₃]^T. Describe A and b

explicitly in terms of ,, and the components of G.

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 . Repeat for = 5. Can the controller achieve the goal to make S_i(t) ? Plot

all the p_i(k) as well.

Exercise 3. Linearization

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

Model the earth and a satellite as particles. The normalized equations of motion, in an earth-fixed inertial frame, simplified to 2 dimensions (from Lagranges equations of motion, the Lagrangian , where r is the radius of the trajectory of the satellite, 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 (t) = t, where p is a representing the constant cruise radius, is the constant angular velocity of the satellite.

1. Whats 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