Behavior Across Discontinuities

Before the derivation of the conservation equations, we first study the behavior of system variables across discontinuities. The set of system variables can be decomposed into those that appear differentiated, x, and those that only appear algebraically, y. When discontinuities occur, jumps in the state variables x cause impulses on the rate variables $\dot{x}$ due to the derivative nature, which may propagate along algebraic variables, y, and ultimately back to interact with other $\dot{x}$ variables. Impulses between rate variables have to satisfy the conservation constraint. For example, the two capacitor system can be described by Eq. (1) where $$x = \left[\begin{array} {c} v_1 \\ v_2\end{array}\right]; y = \left[\begin{array} {c} i\end{array}\right]$$
When the switch is closed and v₁ and v₂ have values inconsistent with v₁ = v₂, impulses occur based on the v₁ = v₂ and C₁ (v₁⁺ - v₁^-) = -C₂ (v₂⁺ - v₂^-) constraints. The latter can be derived from Eq. (1) by first eliminating y from all equations with x to yield $$\left[\begin{array} {cc} C_1 & C_2 \\ C_1 & C_2\end{array}\r... ...}\right] \left[\begin{array} {c} v_1 \\ v_2 \end{array}\right].$$
This is a system of dependent equations. Eliminating one and integrating the remaining equation results in $$\left[\begin{array} {cc} C_1 & C_2 \\ \end{array}\right] \in... ...[\begin{array} {c} \dot{v_1} \\ \dot{v_2}\end{array}\right] = 0$$
or,  $$\left[\begin{array} {cc} C_1 & C_2 \\ \end{array}\right] \le... ...array} {c} v_1^+ - v_1^- \\ v_2^+ - v_2^-\end{array}\right] = 0$$
The constraint that the capacitor voltages are equal when the switch is closed can be expressed as $$v_1^+ = v_2^+$$
and combined with Eq. (6) v₁⁺ and v₂⁺ can be derived.

In this example, impulses occur on all $\dot{x}$ and y variables. If this is not the case, $\dot{x}$ can be split into variables $\dot{\bar{x}}$ that cause impulses because of discontinuities in $\bar{x}$and variables $\dot{\hat{x}}$ that do not because of continuous behavior of $\hat{x}$. Note that the continuous character causes the latter to have the convenient property  $$\int_{t^-}^{t^+}\hat{x} dt = 0$$
as well as ($\hat{x}$ is continuous)  $$\int_{t^-}^{t^+} \dot{\hat{x}} dt = [\hat{x}]_{t^-}^{t^+} = 0$$
Also, $\bar{x}$ may have a step discontinuity at t but no impulse, and, therefore, the integral over an infinitesimal interval is as well,  $$\int_{t^-}^{t^+}\bar{x} dt = 0$$
These properties are used to derive a general methodology for finding the conservation constraints from a given matrix DAE.