Control of Mobile Robots - - Week 1 Exercise

You got a score of 4.00 out of 5.00.

Question 1

One way of getting a general feeling for what a differential equation is up to is to look at the sign and magnitude of the derivative at different points for different values of x. Use this idea for the dynamics

x˙(t)=−x(t)3.

Which one of the plots below (where t is on the ``x-axis" and x(t) is on the ``y-axis") was generated by this system? Note that x(0)=10.

Your Answer
	Correct	1.00

Question ExplanationThe first thing to check is what the axes and the initial condition for each plot is; in our case, these are plots of x(t) that start at x0=10. Thus, the correct plot will (at least initially) have a negative rate of change (x˙(0)=−1000 to be exact). Working through using process of elimination we find the correct plot.

Question 2

One way of modeling epidemics is to describe how the fraction of infected individuals evolves over time. Let I be that fraction, with the model being

I˙=βI(1−I)−ρI.

Here, the constants β and ρ are the infection and recovery rates, respectively. What are the possible equilibrium points to this system (values for I when the fraction of infected individuals is not changing)?

Your Answer		Score	Explanation
When I=0 or I=(β−ρ)/β	Correct	1.00
Only when I=(β−ρ)/β
When I=0 or I=(1−β)/ρ
Only when I=(1−β)/ρ
Only when I=0
Total		1.00 / 1.00

Question ExplanationWe need to see what happens when I(t) does not change, i.e., when I˙=0. So, solve βI(1−I)−ρI=0. This is a 2nd-order polynomial equation, which means it has two solutions (although the two solutions may be the same).

Question 3

If someone gives you a possible solution to a differential equation, one of the checks needed to see if this is indeed a solution is by taking the required number of derivatives and seeing if the proposed solution does in fact satisfy the differential equation. Let

x¨(t)=−ω2x(t).

Which of the following options is not a possible solution to this equation?

Your Answer		Score	Explanation
x(t)=0
x(t)=cos(ωt)
x(t)=e−ωt	Correct	1.00
x(t)=sin(ωt)
x(t)=ωsin(ωt)−cos(ωt)
Total		1.00 / 1.00

Question ExplanationPlugging in each function and differentiating twice we get a second derivative that is not the same as the function in the question for one of the choices.

Question 4

We saw that the model of a cruise-controller could be given by

x˙=cmu−γx,

where u is the input, x is the speed of the car, and c,m,γ are constant parameters. If there was no wind resistance in the cruise-control model (γ=0), what would the steady-state values be for the velocity x when using a pure D-regulator, i.e., when u=ke˙=k(r˙−x˙)=−kx˙ (since r is constant)?

Your Answer		Score	Explanation
x(∞)=0
x(∞)=r
x(∞)>r
Impossible to say
x(∞) less than r	Inorrect	0.00
Total		0.00 / 1.00

Question ExplanationPlugging in our choice of controller u=−kx˙ and γ=0 as well we get: x˙=−kcmx˙, which is no longer a differential equation. Think in terms of discrete time: xk+1=xk+dt``x˙′′ but what is x˙ now? And how can we determine x now?

Question 5

Let a discrete-time system be given by

xk+1=max{0,5−xk}.

If this system starts at x0=10, what happens to the state of the system?

Your Answer		Score	Explanation
It keeps switching between 0 and -5
It keeps growing from 10 up to ∞
It jumps down to x1=0 and increases up by one until x5=5 and then jumps back to 0 again (and the process repeats)
It keeps switching between 0 and 5	Correct	1.00
It jumps down to x1=0 and remains at 0 for ever
Total		1.00 / 1.00

Question ExplanationStart by plugging in x0=10. Find x1. Keep plugging in, using this discrete update rule, until you see the pattern start to repeat. Be careful -- it's easy to make simple mistakes in your addition!