Rob’s Homework - or - a short story about Waldo

Fact: The following story is entirely fictitous. Any resemblance to real events or persons, past or present, is entirely coincidental.

There once was a young Cambridge graduate, called Waldo, who got a job in a big aerospace company that was making the next generation of supersonic airliners. He was attached to the control systems division and his first project was to design a control system for a component in the engine. Not having remembered much control from his E9 course he read the popular press and decided that neural fuzzy controllers would be the best possible. So he went away and built a mathematical model of his component, designed a neural fuzzy controller for it and took it to his boss.

“Hmmmmm,” said Alec his boss (a real engineer), “This all sounds very flash but I’d like to see it working before we put it on the plane.” So Waldo trundled off to the test bed and set it up. It worked well for a while but gradually seemed to go unstable until … “BANG!!”, there was a huge explosion. Waldo was distraught. What could have gone wrong? His mathematical model said the output should be the same as the input and these neural fuzzy controllers were the latest things. With his head drooped he went to see a wily old control engineer.

“Ah,” said Keith the wily old control engineer, “the problem’s probably your mathematical model. It’s not clear that your mathematical model will give the same output as your real system.” And he was right. When Waldo put inputs into his real system the output was two times the input, not one.

“So it was the model all along!” cried Waldo excitedly.

“Probably,” said Keith, “What you need is an uncertain model and a robust controller.” So Waldo went back to the text books and built a linear fractional transformation model, as shown below. But having learnt from the previous disaster he decided to validate the model before setting up on the test bed. Unfortunately none of the text books could tell him how to validate this type of model. Can you help?

[figure: v; w; y; u; P; Δ]

Figure 1: Waldo’s LFT model Waldo’s model was

P = [ \begin{matrix}P_{11} & P_{12} \\ P_{21} & P_{22} \end{matrix} ] = [ \begin{matrix}[1 0] & 2 \\ [1 2] & 1 \end{matrix} ]

with Δ ∈ {Δ ∈ IR^{2×1} : ∥Δ∥ < γ} and for his experiment u = 2, y = 4.

1. Given Waldo’s model (γ = 0.25) and his set of input/output data, could his model have produced this set of data? (Note that if you solve question 2 then this question is solved for free!)

2. What is the smallest γ for which Waldo’s model is consistent with the data?

3. Let w = P_{11}v + P_{12}u, where w ∈ IR^{n_{w}}, P_{11} ∈ IR^{n_{w}×n_{v}}, v ∈ IR^{n_{v}}, P_{12} ∈ IR^{n_{w}×n_{u}} and u ∈ IR^{n_{u}}. Show that if σ̄(P_{11}) < γ^{−1} then the set of v satisfying ∥v∥_{2}^{2} ≤ γ^{2}∥w∥_{2}^{2} is convex.

4. Now let v = [ \begin{matrix}v_{1} \\ v_{2} \end{matrix} ] and show that the set for v for which ∥v_{i}∥_{2}^{2} ≤ γ^{2}∥w_{i}∥_{2}^{2}, i = 1, 2 is

not necessarily convex.

5. For bonus point see if you can find another method of solving q2. (I know of three different methods of solution!)