Symbolic Matrix Inverse — Control Systems/Symbolic

Symbolic Matrix Inverse

Control Systems / Symbolic

Inverts a square matrix of expressions symbolically and evaluates the result on the input at every sample: Y = M(x)⁻¹. This is MATLAB's inv on a sym matrix.

The inverse is taken once, when the configuration loads, as adj(M) / det(M) over exact rational arithmetic. What runs per sample is n×n closed-form expressions of the input – no elimination, no pivot search, no branch. That fixed arithmetic is the reason to reach for this block instead of a numeric inverse: a numeric one chooses its pivots from the numbers, so its operation order changes with the signal, and three of the ten export targets cannot carry that.

The factory setting is [x+2, y; y, x+3], whose inverse is [x+3, −y; −y, x+2] / ((x+2)(x+3) − y²) – non-singular at the origin, so the block out of the library is a worked one.

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. ⚠ n is the number of VARIABLES, not the size of the matrix – the two are unrelated, and a 2×2 matrix over one variable takes a [1,1] input.
• y – the inverse, an [m,m] matrix, the shape of Expression.

Parameters

• Expression – a SQUARE matrix literal over the names in Variables, written with semicolons between rows and commas between the entries of a row: [x+2, y; y, x+3]. A non-square matrix is refused with its shape, and so is a scalar – for a scalar the inverse is 1/f, which Symbolic Expression already writes. Entries may use 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 abs sign heaviside.
• Variables – the names of the input's entries, in order, separated by spaces or commas: x y. 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 algebra 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 inverts 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: the entries share a determinant in the denominator, and a reciprocal across many orders of magnitude is not a fixed-point datapath.

Simulink bridge

None (Support::None). inv on a sym 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

• ⚠ A SINGULAR matrix is answered, not refused. Every entry of the adjugate is divided by a determinant that is exactly zero, so the output is infinities – which is the block saying the matrix has no inverse at that input. MATLAB warns and answers infinities in the entries whose adjugate is non-zero and 0 elsewhere; this block answers infinities everywhere, because the division happens entry by entry. Neither set of numbers means anything, and a matrix that is singular only at SOME inputs is the ordinary case: the determinant is an expression, and it has zeros.
• ⚠ abs, sign and heaviside ARE accepted here, unlike everywhere else in this family. Nothing is differentiated, so the engine's objection – that diff(abs(x)) needs an assumption the console would have to hold – does not arise, and an opaque function is carried the way a variable is.
• Algebraic, with no state: the output depends only on the current input.
• Size limits are on the generated code: 144 entries, 4000 operations per entry. The adjugate of an n×n matrix holds n² determinants of order n−1, so the expressions grow fast in n; a matrix whose inverse outgrows the limit is refused with that reason rather than truncated.
• No state space: the map is nonlinear in general, so model reduction correctly declines the block.