Polynomial Fit — Control Systems/Curve Fitting

Polynomial Fit

Control Systems / Curve Fitting

Fits a least-squares polynomial of degree N to the last W samples and emits its coefficients, in descending powers: ŷ(x) = c₀·x^N + c₁·x^N−1 + … + c_N. The newest sample sits at x = 0 and the sample i steps back at x = −i·h, so the curve is written in the window's own coordinates and the constant term is the fitted value at the present instant.

This is polyfit on a stream. It emits the curve rather than one point of it, which is what a slope, a curvature, an extrapolation or an area needs; the fitted VALUE alone is already Signal Smoothing / Savitzky-Golay Filter.

Ports

• u – the sampled signal being fitted. Scalar: one channel and its own window – see Notes.
• c – the coefficient column, [N+1, 1], in descending powers. Its height follows Polynomial Degree alone and does not depend on the input, which is scalar.

Parameters

• Window Length – W, how many samples the fit sees. A whole number from 2 to 101, and it must be greater than Polynomial Degree: a polynomial of degree N needs more than N points to be determined. Bounded above because the dot products are unrolled at export.
• Polynomial Degree – N, the degree fitted. A whole number from 0 to 6, and less than Window Length. Degree 0 makes the single coefficient the window's mean; degree 1 makes the pair a slope and an intercept.
• Abscissa Step – h, the spacing between consecutive samples on the fitted axis. A single positive number. Leave it at 1 and the curve is written per sample, so a derivative comes out per sample; set it to the block's sampling period and the curve is written in seconds, so a derivative comes out per second. It rescales the coefficients exactly – the coefficient of x^j carries h^−j – and changes no fitted value.
• 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 (N+1)×W coefficient bank is structural and is inlined into the arithmetic at export time rather than exposed as a tunable parameter: it follows from all three settings, and changing any of them changes how many multiplies and how many outputs the core contains, which no runtime parameter can do. Re-export after changing them.

The three HDL targets are genuine synthesizable Q16.16: a shift register and one fixed multiply-accumulate per coefficient, with the products accumulated at full width and shifted back once per row rather than per term. No solve reaches the hardware – it happened at export time. Note that a small Abscissa Step scales the higher coefficients by h^−j, so a step well below 1 at degree 5 or 6 can leave the top coefficient outside the Q16.16 range; leave the step at 1 for the hardware targets and rescale downstream if that matters.

Simulink bridge

None (Support::None). Neither the Simulink standard library nor the Curve Fitting Toolbox ships a block that fits a polynomial to a running window – polyfit and fit are MATLAB functions, and the toolbox has no Simulink library at all, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench; code export verification still covers it across all ten languages. No configuration of it crosses either, including "Sampling Time (s)", which has no counterpart to be written to.

Notes

• Stateful, and discrete by nature (setDiscreteOnlyBlock(true)): the window advances once per sample.
• One solve, then arithmetic. A least-squares coefficient vector is a linear function of the window, so the small normal-equation system is solved once when the configuration loads and every sample afterwards is (N+1) fixed dot products. Nothing inverts a matrix per sample, here or in any exported core.
• Verified against MATLAB. For the nine-sample window [1.2 −0.7 2.5 0.3 −1.9 0.8 1.1 −2.2 0.45] at degree 3 and step 0.4 this block's coefficients match polyfit to 4.4e−14 relative in R2026a; at degree 2 and step 1 over an eight-sample window, to 5.1e−15.
• The window is zero-prefilled, and the zeros count. The first W−1 coefficient vectors of a run are a startup transient in which the fit sees samples that were never measured. A batch polyfit has no such phase because it is handed the whole vector at once; a stream cannot. The convention matches Moving Median, Detrend and Savitzky-Golay Filter.
• Scalar only. One channel and its own history; wire one block per channel. A shared window would mix them.
• Fitting is not smoothing. A high degree over a short window follows the noise: at degree N over W samples the fit has W−N−1 degrees of freedom left to reject anything with, and at W = N+1 it interpolates the window exactly and rejects nothing.
• No state space. The block is linear in its input, but its output depends on W past samples through a fixed filter bank rather than through an A/B/C/D pair, so it carries none and model reduction correctly declines to merge it.