Analytical tools are very important. It may be hard to check the correctness of a machine by trying it in several possible initial conditions with all possible inputs; sometimes it is more easier to create a mathematical model and then solve a theorem about the model.
For some systems, such as pure software calculations, or the addition circuitry in a computer, it is possible to find correctness or speed with just a model of the system in question. For other machines, such as fuel injectors, it is not possible to solve the correctness of the controller without also modelling the environment to which it is inter connected, and then check the behaviour of the coupled system of the environment and controller.
To provide some of these tradeoffs, we can do a very simple analysis of a robot moving toward a light. Imagine that we manage it so that the robot's speed at time t, V [t], is related to the difference between the real light level, X[t], and a given light level, Xdesired; that is, we may model our control method with the difference relation
Now, we require to model the world. For simplicity, we take the light level to the robot's position); in addition we consider the robot's position at time t is its position at time t - 1 plus its speed at time t - 1.
Now, for a required value of k, we can calculate how the system will act over time, by resulting the difference relation in X.
These same types of analyses may be given to robot control machine as well as to the temporal characteristics of voltages in a digital circuit, and even to problems as not related as the result of a monetary policy decision in economics. It is very important to check that treating the machine as moving in discrete time stamp is an approximation to underlying continuous dynamics; it may create models that are simpler to analyze, but it needs sophisticated understanding of sampling to check the effects of this approximation on the answer of the results.