eventTotalDerivative

Method of continuousTimeMdss.

Resolve the mode switch and consistent slopes of a hybrid implicit multilinear IVP at an event.

Source: src/simulator/@continuousTimeMdss/eventTotalDerivative.m

What is computed

At a single time $t_0$ , produce a consistent post-event point of a hybrid mdss model: the state derivative $\dot x_1$ , algebraic signals $y_1$ , binaries $z_1$ , algebraic slopes $\dot y_1$ , and the equality / inequality function values $F$ , $G$ .

Input value and slope

$u_0 = u(t_0),\qquad \dot u = \frac{u_0 - u(t_{\mathrm{near}})}{t_0 - t_{\mathrm{near}}}$

(finite difference against the nearest input sample $t_{\mathrm{near}}$ ).

Mode solve

The solving order is walked building the norm-1 monomial vector $m$ . Subproblems before the event are solved with the binaries fixed (solveSubProblemFixZ); explicit continuous variables use the affine closed form $v = -B/A$ (see solveStepBruteForce). When an event index ie is given, the binaries of the event’s subproblem are enumerated over all $2^{n_z}$ combinations, ordered by descending Hamming distance to the previous $z$ (largest mode change first), and each is propagated through the following subproblems.

Consistent slopes (implicit function theorem)

From the equality Jacobian $J_F$ , the state second derivative and algebraic slope follow from differentiating $F(\dot x,x,u,y)=0$ along the trajectory:

$\begin{bmatrix}\ddot x\\[2pt] \dot y\end{bmatrix} = -\,J_{F,[\dot x,\,y]}^{-1}\,J_{F,[x,\,u]} \begin{bmatrix}\dot x\\[2pt] \dot u\end{bmatrix}.$

Acceptance test

A candidate mode is accepted only if it is feasible and every active inequality is being crossed inward — its total time derivative is non-increasing:

$G \le \text{tol},\qquad \dot G_j = J_{G_j}\,[\,\ddot x;\ \dot x;\ \dot y;\ \dot u\,] \le 0 \quad\text{for active } j .$

A binary belonging to the enumerated set keeps its enumerated value rather than being re-derived from $\operatorname{sign}(F_0)$ — that re-derivation is degenerate at a coincident crossing (its switching surface sits at $G\approx 0$ ) and would otherwise pin the bit to its previous value. If no candidate passes both tests, the outputs are returned as NaN.

Input arguments

Argument	Description
`sys`	The `continuousTimeMdss` simulation object (carries the `mdss` model).
`ie`	Triggering inequality (event) index; empty for a pure consistent-slope solve.
`t0`	Event time.
`x0`	Current state values.
`y0`	Current algebraic-signal values.
`z0`	Current binary-signal values.
`u`	Input trajectory, one row per input-time sample, one column per input.
`ut`	Time samples of the input trajectory.
`solvingOrder`	The block solving order `[varIdx, eqIdx, type, subset, subproblem]`.
`opt`	Solver options.

Output arguments

Output	Description
`xp1`	Post-event state derivative $\dot x_1$ .
`y1`	Post-event algebraic signals.
`z1`	Post-event binary signals.
`dy1`	Post-event algebraic slopes $\dot y_1$ .
`F`	Equality function values.
`G`	Inequality function values.