Jacobian — Control Systems/Symbolic

Jacobian

Control Systems / Symbolic

The Jacobian of a vector expression, evaluated at the input: J(i,k) = ∂f_i/∂x_k, an m×n matrix for m expression entries and n declared variables. This is MATLAB's jacobian(f, v); a scalar expression gives the row [1,n], as it does there.

It is the linearisation a hand-written plant model usually carries as hard-coded entries: write the nonlinear state derivatives as [f1; f2; ...] and the block derives the matrix, symbolically, once, when the configuration loads.

Ports

• x – the variables, a vector of n entries (a column [n,1] or a row [1,n]) where n is the number of names in Variables: entry k is the k-th name. A Mux in front builds it from scalar signals.
• y – the Jacobian, [m,n]: one row per expression entry and one column per declared variable.

Parameters

• Expression – the expression, in MATLAB syntax over the names in Variables: numbers, pi, + - * / ^, parentheses and the functions sin cos tan sec csc cot asin acos atan acot sinh cosh tanh asinh acosh atanh exp log log2 log10 sqrt. A matrix is written [f1; f2], with semicolons between rows and commas between the entries of a row. abs, sign and heaviside are refused here, unlike on Symbolic Expression: see Notes.
• Variables – the names of the input's entries, in order, separated by spaces or commas: x y z. Each is a MATLAB identifier, none may repeat, and none may be a function name or pi, e, i, j, Inf, NaN or eps. At most 12.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The calculus is done once, at configuration load; what each target carries is the RESULT, printed as one inline expression per entry with its constants folded to 17 significant digits, so a generated core differentiates nothing and has no parameter to tune. Where a target lacks a function it gets the identity – Java has no inverse hyperbolic functions and PLC Structured Text no hyperbolic functions at all, so those are written through log, exp and sqrt. The three HDL targets are simulation-only real arithmetic, quantized to Q16.16 only at the ports: a derivative of a transcendental expression is not a fixed-point datapath.

Simulink bridge

None (Support::None). jacobian is a Symbolic Math Toolbox function, and that toolbox ships no Simulink library at all, so there is no library path a diagram could name; the bridge reports this block rather than dropping it, and it therefore has no parity testbench. Code export verification covers it across all ten languages. No configuration crosses, including "Sampling Time (s)".

Notes

• The expression must be a VECTOR – [f1; f2; ...], a column or a row, or a scalar. A matrix expression has no Jacobian, and the block says so rather than flattening it.
• abs, sign and heaviside are refused in the expression. The symbolic engine answers diff(abs(x)) only under an assumption that x is real – console state a block may not depend on – and the derivative of sign or heaviside is an impulse no sample can carry. Symbolic Expression accepts all three, because it differentiates nothing.
• Algebraic, with no state: the output depends only on the current input.
• The calculus is the console's own symbolic engine's (the engine behind diff and jacobian on the command line), over exact rational arithmetic; constants are folded to doubles only after it.
• Size limits are on the generated code: 144 entries, 4000 operations per entry. An expression whose derivatives grow past that is refused with that reason rather than truncated.
• No state space: the map is nonlinear in general, so model reduction correctly declines the block.