This document gives a brief introduction to the mathematical modeling of hybrid systems and their solutions. A hybrid system \(\mathcal{H}\) with state \(x\in \mathbb{R}^n\) is modeled as follows:

\[\mathcal{H}: \left\{\begin{array}{rl} \dot x = f(x) & x \in C \\ x^+ = g(x) &x \in D \end{array} \right. \]

Throughout the HyEQ documentation, \(\mathbb{R}\) denotes the set of real numbers and \(\mathbb{N} := \{0, 1, \dots\}\) denotes the set of natural numbers. We define \(\mathbb{R}_{\geq 0} := [0, \infty).\)

The representation of \(\mathcal{H}\) , above, indicates that the state \(x\) can evolve or flow according to the differential equation \(\dot x = f(x)\) while \(x \in C,\) and can jump according to the difference equation \(x^+ = g(x)\) while \(x \in D.\) The function \(f : C \to \mathbb{R}^n\) is called the flow map , the set \(C \subset \mathbb{R}^n\) is called the flow set , the function \(g : D \to \mathbb{R}^n\) is called the jump map , and the set \(D \subset \mathbb{R}^n\) is called the jump set .

Roughly speaking, a function \(\phi : E \to \mathbb{R}^n\) is a solution to \(\mathcal{H}\) if the following conditions are satisfied:

  1. \(\phi(0, 0) \in \overline{C} \cup D.\)
  2. The domain \(E \subset \mathbb{R} \times \mathbb{N}\) is a hybrid time domain where for each \((t, j) \in E\) , the \(t\) component is the amount of ordinary time that has elapsed and the \(j\) component is the number \(j\) of discrete jumps that have occured.
  3. For all \(j\in \mathbb{N}\) such that \(I^j := \{t \mid (t, j) \in E\}\) has nonempty interior, we have that \(\phi(t, j) \in C\) for all \(t \in \mathrm{int}\: I^j\) and \(\frac{d\phi}{dt} = f(\phi(t, j))\) for almost all \(t \in I^j\) . We call \(I^j\) an interval of flow .
  4. At each \((t, j) \in E\) such that \((t, j+1) \in E\) , then \(\phi(t, j) \in D\) and \(\phi(t, j+1) = g(\phi(t, j))\) . We call such a \((t, j)\) a jump time .

For a rigorous definition of hybrid solutions, see Chapter 2 of Hybrid Dynamical Systems by Goebel, Sanfelice, and Teel [1]. See also Hybrid Feedback Control by Sanfelice [2].

References

[1] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid dynamical systems: modeling, stability, and robustness . Princeton University Press, 2012.

[2] R. G. Sanfelice, Hybrid Feedback Control . Princeton University Press, 2021.

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