solveSubProblemFixZ

Method of continuousTimeMdss.

Solve one coupled subproblem for its continuous unknowns with the binaries held fixed.

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

What is solved

Given a coupled subproblem (a block of the solving order) and the binaries fixed at $z$, solve for the block’s continuous unknowns $w=[\dot x_s;\, y_s]$ from its equality rows

$$F_s(w) = \Phi_{\mathrm{eq},s}\, m(w) = 0 ,$$

where $m$ is the norm-1 monomial vector with the fixed $z$ already folded in (substitution rule as in solveStepBruteForce).

Explicit (type = 1) rows never reach this function — they are solved directly by solveStepBruteForce while walking the solving order. This routine only handles coupled subproblems, which are either linear (type = -1) or nonlinear (type = 0), selected by the block’s type flag.

Linear (type = -1): $F_s$ is affine, $F_s = A\,w + b$, with

$$A_{:,j} = \Phi_{\mathrm{eq},s}\big[(t_j - f_j + c_j)\odot m\big], \qquad b = \Phi_{\mathrm{eq},s}\Big[m \odot \textstyle\prod_{j}(1-|c_j|)(1-t_j)\Big],$$

solved directly by $w = -A^{-1}b$ (backslash).

Nonlinear (type = 0): solved with lsqnonlin (Levenberg–Marquardt) on $F_s(w)=0$ with the analytic Jacobian from the product rule,

$$\frac{\partial F_s}{\partial w_o} = \Phi_{\mathrm{eq},s}\!\left[\frac{\partial m}{\partial w_o}\right], \qquad \frac{\partial m}{\partial w_o} = \Big(\textstyle\prod_{p\ne o}\varphi_p(w_p)\Big)\,\varphi_o'(w_o),$$

where $\varphi_o'$ is the derivative of the row-$o$ factor: $c_o$ from the continuous part plus $(t_o-f_o)$ from the true/false part.

Each solved $w_o$ is folded back into $m$, and the subproblem function values $F$ (equality) and $G$ (inequality) are returned.

Used by eventTotalDerivative for the subproblems that precede an event and for each fixed candidate of the enumerated event subproblem.

Input arguments

Output arguments

See also

continuousTimeMdss · solveSubProblemVariableZ · eventTotalDerivative · msim

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