**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.

*y*(*t*) = 0 for all*t***(4 points)***y*(*t*) =*u*^{3}(*t*)**(4 points)***y*(*t*) =*u*(3*t*)**(4 points)***y*(*t*) =*e*^{−t}*u*(*t*−*T*)**(4 points)****(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*0). The path gain from transmitter

_{i }>*j*to receiver

*i*is

*G*(

_{ij }*G*> 0 for

_{ij }*j*6=

*i*, and

*G*0). The signal power at receiver

_{ii }>*i*is given by

*s*=

_{i }*G*. The noise plus interference power (caused by other transmitters

_{ii}p_{i}*j*6=

*i*) at receiver

*i*is given by

*q**i *= *σ*2 + X*G**ijp**j,*

*j*6=*i*

where *σ*^{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*. For signal reception to occur, the SINR must exceed some threshold value

_{i}/q_{i}*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*))

*.*

- 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*) + *Bσ*^{2}*,*

where *A *∈R^{3×3 }and *B *∈R^{3×1 }are constant and *p*(*k*) = [*p*_{1}*,p*_{2}*,p*_{3}]* ^{T}*. Describe

*A*and

*b*

explicitly in terms of *σ,γ,α *and the components of *G*.

- Use Python to simulate the power control algorithm. Use the problem data

Experiment with two different initial conditions: *p*_{1 }= *p*_{2 }= *p*_{3 }= 0*.*1 and *p*_{1 }= 0*.*1*,p*_{2 }= 0*.*01*,p*_{3 }= 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.

**Exercise 3. **Linearization

Perform linearization on the given differential equation

*y*¨+ (1 + *y*)*y*˙ − 2*y *+ 0*.*5*y*^{3 }= 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 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 *u*1, *u*2 are control input, representing the radial and tangential forces due to the thrusters. The reference orbit with *u*1 = *u*2 = 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**

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