15. Dynamical systems#
We will now formally define dynamical system, i.e, what does it mean to be a dynamical system, what elements are in a dynamical system, and as well some axioms that a dynamical system has to satisfy.
Definition#
A dynamical system \(D(U, \Sigma, Y, s, r)\) with:
\(U\): input space. The input function \(u(\cdot): T \rightarrow U = \mathbb{R}^{n_i}\), with \(n_i\) the dimension of the input.
\(\Sigma\): state space. Function \(x(\cdot): T \rightarrow \Sigma: \mathbb{R}^n\).
\(Y\): output space. Output function \(y(\cdot): T \rightarrow Y = \mathbb{R}^{n_o}\) with \(n_o\) the dimension of the output.
\(s\): state transition function, defines as \(s(\cdot, \cdot, \cdot, \cdot): T \times T \times \Sigma \times U \rightarrow \Sigma\). An example is \(x(t) = s(t, t_0, x_0, u(\cdot))\Big|_{t_0}^t\)
\(r\): read-out map (output map), defines as \(r(t, x(t), u(t))\rightarrow y(t)\). With \(T\) be the set of time, it can either be \(\mathbb{R}_+\) or \(\{nT\}, n \in \mathbb{Z}\).
Axioms#
If a system satisfies the following axioms, then it is a dynamical system.
State transition axiom#
If the inputs are the same over a time interval, and we start at the same state, then the state transition function will give us the same result at the end of that time interval.
What is substantial about this is that we do not care what happen before and after the chosen time interval. We can think of it as at \(t_0\), whatever happened before has already been encapsulated into \(x_0\), regardless of the previous control sequences, and if \(u = \bar{u}\) over \([t_0, t_1]\), then the state transition function will go through the same sequence of states.
Semigroup axiom#
Given \(t^* \in [t_0, t_1]\)