Quiet down out there!
Wouldn't you like to fly?
Now, there's one place in the universe that's blissfully noise-free: the pristine world of mathematics. Add 2 and 3, and you always get 5. Exactly. For a given value of x, the function f(x) returns the same value. Every time. The same is true of applied mathematics. If I simulate, say, the motion of an airplane, I know it must obey Newton's laws of motion. And when I run the simulation, the airplane follows the same course. Exactly.
The real world is not so accommodating. A real airplane obeys the same laws of motion, but it's also subject to disturbances that aren't modeled in my simulation: wind gusts and updrafts; headwinds; changes in air temperature and density; pilot inputs. Even people moving around in the cabin.
Perhaps more importantly, every sensor measuring a flight-related parameter has errors of its own: electronic noise, scale factor errors, quantization errors, drift, temperature sensitivities, etc. All of these things introduce uncertainties into my perfect world of simulation.
But wait; there's more. Even if I could simulate all those effects, the simulated airplane is still not going to behave like the real thing, because there are system parameters whose values I don't know exactly. I may know the mass and inertia properties of the dry airplane, but not with an unknown number of passengers of unknown weights, or the mass and distribution of fuel in the tanks. I may have precise data sheets that give me the theoretical thrust of the engines, but not the actual thrust they're producing at any given time.
So there are three ways my simulation can never be perfect: I don't know the precise values of the system parameters, I don't have precise measurements of the system state, and my math model doesn't include all the possible effects. In the real world, our challenge is to make our best guess as to all these things, based on the noisy measurement data we have available.
Because the system is subject to unmodeled effects, it's not enough to just measure the state once or twice, and assume the system will obey its laws of motion from there on. I have to continue to take measurements, and constantly improve my best guess as to both the system state and the parameters that I thought I knew, but got wrong.
Parameter estimation is the determination of those system parameters. State estimation is the determination of the state of our system. A good data processing system can do both, and nothing does it better than the fabled Kalman filter.
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