We want to control our body weight (system output) by altering how much we eat (system input), based on measurements of our weight, so that it follows a particular trajectory.
We could simply try to calculate how much we should eat ahead of time based on this trajectory and our initial weight, and then follow this schedule without updating it based on any subsequent measurements ("open loop control"). However, this would require a very good mathematical model that can accurately predict future weight based on current weight and food intake. If we instead choose how much to eat based on continuous measurements of our weight (using feedback to achieve "closed loop control"), we can get away with using a much worse model; this is one of the key advantages of feedback.
The proposed control algorithm is simple: when eating a meal, eat a total weight of food and drink (F(t)) that is the weight you would like to be at the end of the meal (r(t+T)), minus your weight at the start of the meal (w(t)). This rule ignores metabolism, so you will fall below your target weight by the next meal, but you will eat accordingly more then.
Obviously, as the article notes, this requires that "the reference r(⋅) has to be chosen larger than w(⋅) at all times". You can see this in Figure 1: the measured mass starts below the reference, then increases up to the reference, then falls below it, then increases up to it at the end of each meal.
1) Having a BMI of 26.1 kg/m2, you don't lose 7.4 kg over one month just by eating less (unless you just stop eating), 2) I'm sure what the article explains goes a bit beyond "eat less"
Since I have the advantage of having read the article, I can say that your speculation is correct: the author describes a systematic way to control how much less to eat. But "eat less" is still the bottom line. He lost weight because he ate less over the month of the experiment that he had been eating previously. How else did he lose the weight? Note that he doesn't report increasing his caloric expenditure through exercise, for example.
I have occasionally remarked, you can't gain more weight than the weight of the food you eat, because conservation of mass. This is sometimes my response to people who don't eat a single donut on a special occasion. Naturally the pound or less of food you might eat in a day is not permanently conserved. I also think that a pound of food sounds like an enormous amount, but it really isn't. If you like hamburgers (and let's say you make your own so they are wholesome) you might cook a third of a pound of ground meat to make one.
> you can't gain more weight than the weight of the food you eat, because conservation of mass
Not, strictly speaking, true: not only must we consider mass-energy equivalence (the chemicals bonds in food vs metabolic byproducts could contribute unequally to mass), but more importantly, we inhale mass well as ingest. From the article, one mass-unit of inhaled oxygen produces less than one unit of exhaled carbon dioxide (although I suspect that more than enough water is lost though net exhalation alone to make up for it). So we have to be a little careful with this conservation of mass argument. It's still basically correct, though, and more-or-less blows the "weight gain is exactly proportional to excess calories" argument out of the water.
