Problem 1

We consider a simple quadrotor, such as that shown in Fig. 1. We are

Figure 1: quadrotor

interested in controlling the vertical motion of the machine. The altitude can be changed by acting on the angular speed of an electric motor that turns the propellers. In turn, the propellers produce an upwards vertical force by pumping air downwards. The quadrotor's altitude above the ground can be measured via a infrared laser probe (LIDAR) when the quadrotor is close to the ground (altitude less than 2 meters above the ground). When the quadrotor's altitude is higher, the only reliable sensor is a pressure altimeter. The LIDAR provides altitude information at a nominal rate of 50 updates per second (or one update every 20 msec). The pressure altimeter runs a lot slower: it can provide altitude information at nominally two updates per second. We consider two control algorithms: One for high altitudes, and one for low altitudes. To make matters more precise, the quadrotor is a dynamical system consisting of the following elements: First, there is the dynamics of the electric motor and its propellers. The force response f of the propeller to a desired force f_d satisfies the simplified differential equation

df/dt = 7.4f + 7.4f_d. (1)

1. Using the programming language of your choice, simulate the "response" of the propeller when f_d and f were originally at zero and f_d follows the "step" profile shown in Fig. 2. For this simulation, use

Figure 2: step

the approximation

df/dt ≈ (f(t + dt) - f (t))/ dt

with dt = 0.01 sec. Plot f (t) vs. time.

2. The rest of the vehicle is given by the differential equation

d/dt.z = v

d/dt.v = f,

where z is the altitude of the quadrotor, and v is its velocity.

A linear altitude controller for low altitude (altitude < 2m) was designed for the quadrotor as the differential equation:

dx_c/dt= -5x_c + 5(z - z_d)

f_d = 2.45x_c - 2.5(z - z_d),

where z_d is the desired altitude of the quadrotor. xc is the state of the controller, in other terms, the controller is a dynamical system. x_c has no particular significance a priori. Another linear altitude controller for high altitude (altitude > 2m) was designed for the quadrotor as the differential equation

dx_c/dt = -x_c + (z - z_d)

f_d = 99/800x_c - 0.125(z - z_d)

In this equation, x_c is, again, the state of the controller. Qualitatively discuss the properties of both controllers: For example, is one "faster" than the other?

3. A Matlab simulation is provided to you. This simulation contains the effects of both the "low altitude" controller and the "high altitude" controller. The main file is "exam.m" and there is a function named "propag.m". Run the program and explain what it does relative to the system description above.

4. Explain the meaning of the parameters "dt", "total time", "sample length", and "duration".

5. What "comfort zones" do you find for these parameters from the viewpoint of simulating the system? Are these "comfort zones" the same?

6. Based on the simulation given to you, write a simulation to get the response of the helicopter to a 2m step up input, knowing the helicopter is at 1 meter altitude to begin with. Be careful when writing your simulation as the helicopter goes through the 2m boundary and must therefore switch controller at that point.

Problem 2

Write a one-page essay on the following topic

In, the authors propose a survey of Security, Safety and Privacy issues for UAVs. They list a set of common threats to be addressed to allow for the safe operation of UAV in the civilian airspace. Yet, one may note there are several challenges when modelling and then analysing these systems. In [2], the authors report on dependency loops between analysis (see section 2). These means that one need to consider the system as a whole, and not as a separate set of concerns (e.g. safety, security, performance, etc).

Taking the topic of CPS at large, select for one class of systems, two concerns (and associated analysis). Illustrate how optimising the system for one concern affects the other. Impact may be positive or negative. You are encouraged to go beyond the provided references and propose your own. You may consider safety, security, performance, energy, accuracy in reaching a goal (e.g. a position, a speed, etc.), etc.