solveSubProblemVariableZ

Method of discreteTimeMdss.

Solve one coupled subproblem for its continuous unknowns and its binary signals.

Source: src/simulator/@discreteTimeMdss/solveSubProblemVariableZ.m

What is solved

A coupled subproblem is a block of equations from the solving order that must be solved jointly. Its unknowns are some continuous signals $w=[\,x_{k+1,s};\, y_s]$ (successor states and algebraic variables) and some binary signals $z_s$. Let

$$n_z = \text{the number of binary signals in this subproblem.}$$

The block must satisfy its equality rows and stay inequality-feasible:

$$F_s = \Phi_{\mathrm{eq},s}\, m(w, z_s) = 0, \qquad G_s = \Phi_{\mathrm{ineq},s}\, m(w, z_s) \le 0,$$

where $\Phi=\Phi_c+\Phi_{+1}-\Phi_{-1}$ is the parameter-matrix split and $m$ is the norm-1 monomial vector — the running product obtained by folding each known signal value into the columns of the structure matrix (the update rule is described on solveStepBruteForce).

Algorithm

Each binary can only be $0$ or $1$, so there are $2^{n_z}$ possible binary vectors. The method enumerates all of them, ordered by ascending Hamming distance to the current $z$ (fewest switched bits first, since a mode usually changes as little as possible):

$$z^{(k)} \in \{0,1\}^{n_z},\qquad k = 1,\dots,2^{n_z}.$$

For each candidate $z^{(k)}$:

1. Fold $z^{(k)}$ into $m$ (the binaries are now known), leaving a continuous-only solve for $w$.

2. Solve the block’s equality rows $F_s(w)=\Phi_{\mathrm{eq},s}\,m(w)=0$ for $w$, according to the block’s type flag.

Linear (type = -1): $F_s$ is affine, $F_s = A\,w + b$, solved by $w=-A^{-1}b$, with

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

Nonlinear (type = 0): lsqnonlin (Levenberg–Marquardt) drives $F_s(w)\to 0$ using the analytic product-rule Jacobian.

3. Check inequality feasibility. The candidate is accepted as soon as

$$\max_i (G_s)_i < \sqrt{\varepsilon},$$

and its $x_{k+1,s}, y_s, z^{(k)}$ are written out.

($t_j, f_j, c_j$ are the true/false/continuous structure rows of unknown $j$.) If no binary vector yields a feasible solution, a No Solution found warning is raised.

This method is called by solveStepBruteForce whenever a coupled subproblem contains binary signals.

Inputs / outputs

[xp, y, z, G, F, EqXUZ] = solveSubProblemVariableZ(sys, solvingOrderSubProblem, EqXUZ, xpInit, yInit, y0, zInit) — returns the updated successor-state row xp ($x_{k+1}$), algebraic row y, binary row z, the subproblem inequality/equality function values G, F, and the running monomial vector EqXUZ ($m$).

See also

discreteTimeMdss · solveStepBruteForce · msim

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