jacobian

Method of mdss.

Evaluate the Jacobian of the model’s equations with respect to all signals.

Source: src/model/@mdss/jacobian.m

Syntax

jacobianMatrix = jacobian(sys, stateDeltaVector, stateVector, inputVector, algebraicVector, binaryVector)

Description

jacobian returns the matrix of partial derivatives of every equation with respect to every signal, evaluated at the supplied point. With each equation written as $f(\dot{x}, x, u, y, z)$ , the result is

$J = \frac{\partial f}{\partial [\dot{x};\, x;\, u;\, y;\, z]},$

with one row per equation and one column per signal (in structure-matrix row order).

The six signal vectors are assembled into the full signal vector via the model’s index vectors, and the evaluation is delegated to the underlying tensor. For a CPNTensor the unified jacobian kernel is called; for a TTTensor the tensor-train path computeJacobianMatrixMdss is used.

Input arguments

Argument	Description
`sys`	The `mdss` model.
`stateDeltaVector`	State-derivative values $\dot{x}$ (`nState` entries).
`stateVector`	State values $x$ (`nState` entries).
`inputVector`	Input values $u$ (`nInput` entries).
`algebraicVector`	Algebraic values $y$ (`nAlgebraic` entries).
`binaryVector`	Binary values $z$ (`nBinary` entries; `[]` if none).

Output arguments

Output	Description
`jacobianMatrix`	Jacobian matrix, rows = equations, columns = signals.

Example

sys = stringParser.symbolicToMdss(["0 = m*xp1 - y1 - u1"; ...
                                   "0 = y1 - k*x1"], 0, ...
                                  ["xp","x","u","y","z"], ["m","k"], [1200, 3]);

% Jacobian at xp1 = 0, x1 = 0, u1 = 1, y1 = 0 (no binaries)
J = jacobian(sys, 0, 0, 1, 0, []);

The equations here are $f_1 = 1200\,\dot{x}_1 - y_1 - u_1$ and $f_2 = y_1 - 3\,x_1$ . J has one row per equation and one column per signal. As with incidenceMatrix, slice it with the index vectors to get a labelled view — the two equalities against [xp1 x1 u1 y1]:

cols  = [sys.stateDeltaIndex; sys.stateIndex; sys.inputIndex; sys.algebraicIndex];
block = full(J(sys.equalityIndex, cols))

which yields

          xp1    x1    u1    y1
 eq1  ->  1200    0    -1    -1
 eq2  ->     0   -3     0     1