Surface Fit — Control Systems/Curve Fitting

Surface Fit

Control Systems / Curve Fitting

Fits a least-squares polynomial surface ẑ = f(x, y) to the last W samples of three signals and emits its coefficients and the fitted value at the newest point. The model is MATLAB's polyIJ: every term p_ab·x^a·y^b with a ≤ I, b ≤ J and a + b ≤ max(I, J), so poly23 is p00 + p10·x + p01·y + p20·x² + p11·x·y + p02·y² + p21·x²·y + p12·x·y² + p03·y³.

This is fit([x y], z, 'polyIJ') on a stream: at every sample, once the window is full, the output equals what that call returns for the last W samples.

Ports

• x – the first predictor, scalar [1,1].
• y – the second predictor, scalar [1,1].
• z – the response being fitted, scalar [1,1]. The three are sampled together: sample k is the point (x_k, y_k, z_k).
• c – the coefficient column, [n, 1], in MATLAB's order – by total degree, and within one total degree by descending power of x: p00, p10, p01, p20, p11, p02, p30, p21, p12, p03, … with the terms the model leaves out skipped. n follows X Degree and Y Degree alone (n = 9 for poly23, 7 for poly31, 21 for poly55), exactly numel(coeffnames(fittype('polyIJ'))).
• zf – the fitted surface evaluated at the newest (x, y), [1,1]: the value ẑ the fit predicts for the sample that just arrived, so z − zf is its residual.

Parameters

• X Degree – I, the highest power of x. A whole number from 0 to 5, MATLAB's own range (there is no poly60).
• Y Degree – J, the highest power of y. A whole number from 0 to 5.
• Window Length – W, how many samples the fit sees. A whole number from 2 to 64, and at least n, the number of coefficients: MATLAB refuses fewer points than coefficients ("Insufficient data") and so does this block. Leave it well above n – at W = n the surface interpolates the window exactly and rejects no noise at all.
• 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.

Unlike the other fits in this family, the least-squares problem is solved every sample in every generated core: both predictors arrive on ports, so the design matrix is data and nothing can be solved ahead of time. Each core keeps the three windows, rebuilds the W×n design matrix, factorizes it by modified Gram–Schmidt and back-substitutes – about W·n² multiply-adds and n square roots per sample. The degrees and the window are structural and are inlined at export; re-export after changing them.

The three HDL targets carry the solve in real arithmetic and are simulation-only, not synthesizable: a per-sample factorization with square roots and divisions of run-time quantities does not belong in a Q16.16 datapath. The inputs and outputs are quantized to Q16.16 at the ports, so their answers differ from the software targets' by what that quantization moves the fit; each coefficient must also stay inside the format's ±32768.

Simulink bridge

None (Support::None). The Curve Fitting Toolbox ships no Simulink library at all – fit and prepareSurfaceData are MATLAB functions – and no Simulink block fits a surface to a running window, 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. The output depends on the sample that just arrived, so a feedback loop through the block is an algebraic loop.
• Zero until the window is full. For the first W−1 samples of a run both outputs are 0: the block fits real samples only, and never the fictitious zeros a prefilled window would hold. From sample W on, the answer is the fit of exactly the last W samples.
• A window that cannot determine the model gives zeros. If the points do not pin the surface down – a predictor that does not move, points all on one line when the model needs more – both outputs are 0 until the window recovers. MATLAB answers such data with a warning and coefficients of order 10³¹; a stream cannot stop to warn, and zero is an answer a downstream block can test for. The test is on each pivot of the factorization against its own column's norm, at 10⁻¹⁰.
• No normalization. Like fit by default (Normalize off), the coefficients are in the units of x and y themselves. Data far from the origin – x around 1000, say – makes the powers of x nearly parallel and the fit ill-conditioned, exactly as it does in MATLAB; subtract an offset upstream first.
• Verified against MATLAB. Over 50 sliding 16-sample windows of a verification run, the coefficients this block emits agree with fit([x y], z, 'poly23') in R2026a to 2.5e−14 absolute (3.2e−15 of the largest coefficient) and the fitted value to 3.1e−15; at poly31, to 3.0e−14 and 1.2e−15. MATLAB solves by Householder QR and this block by modified Gram–Schmidt; both are orthogonal factorizations, so the error grows with the conditioning of the data rather than with its square.
• prepareSurfaceData has no counterpart. It reshapes gridded data into columns and drops non-finite points; samples here already arrive one point at a time, and a non-finite sample makes the outputs non-finite until it has left the window.
• Evaluate Fit cannot read c. That block evaluates a polynomial in ONE variable; this output is a surface's coefficients in MATLAB's two-variable order. Use zf for the fitted value, or evaluate the terms yourself.
• No state space. The output is a nonlinear function of the window (the predictors enter the design matrix), so the block carries none and model reduction correctly declines to merge it.