โ Scribed by de Larminat, Philippe
No coin nor oath required. For personal study only.
Simulation and Implementation of Continuous Time Loops
This chapter deals with ordinary differential equations, as opposed to partial derivative equations. Among the various possible problems, we will consider exclusively the situations with given initial conditions. In practice, the other situations -fixed final and/or intermediary conditions -can always be solved by a sequence of problems with initial conditions that we try to, by optimization, determine so that the other conditions are satisfied. Similarly, we will limit ourselves to 1 st order systems (using only first order derivatives) as long as in practice we can always obtain such a system by increasing the number of equations.
We will study successively the linear and non-linear cases. Even though the linear case has by definition explicit solutions, the passage from formal expression to a virtual reality, with the objective of simulating, is not so trivial. On the other hand, in automatic control, Lyapunov or Sylvester's matrix equations, even if also linear, cannot be processed immediately, due to a prohibitive calculating time. For the non-linear case we will analyze the explicit approaches -which remain the most competitive for the systems whose dynamics remain of the same order of magnitude -and then we will finish by presenting a few explicit diagrams mainly addressing systems whose dynamics can significantly vary.
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