Weibull Fit — Control Systems/Curve Fitting

Weibull Fit

Control Systems / Curve Fitting

Fits the two-parameter Weibull curve to the last W samples by nonlinear least squares and emits its two parameters:

ŷ(x) = a·b·x^b−1·e^−a·xb, for x > 0

This is MATLAB's weibull library model, and the output carries a then b – coeffnames(fittype('weibull'))'s own order. b is the shape (b = 1 is a plain exponential decay, b > 1 a curve that rises from zero, peaks and falls) and a is the scale: the curve peaks at x = ((b−1)/(a·b))^1/b. Unlike every other fit in this family the model is not linear in its parameters, so the block runs a real iteration per sample – see Notes.

Ports

• u – the sampled signal being fitted; it is the ordinate y, one value per step. Scalar: one channel and its own window – see Notes.
• ab – the parameter column, [2, 1]: row 1 is a, row 2 is b. Its height is fixed and does not follow any setting.

Parameters

• Options – empty (default), or the path of a Fit Options block (Home/Fit Options), MATLAB’s fitoptions: for the run, its Start Point and Maximum Iterations replace this block’s Start Point and Iterations where it has them; an option it leaves at default changes nothing.
• Window Length – W, how many samples the fit sees. A whole number from 3 to 101. Two parameters need at least two points, and a third is the first that makes it a fit rather than an interpolation.
• Abscissa Start – x₀, where the oldest sample of the window sits on the fitted axis. A single strictly positive number: x^b−1 is undefined at zero and not a real number below it, which is why the Weibull model is only defined for a positive abscissa (fit refuses non-positive x outright).
• Abscissa Step – h, the spacing between consecutive samples on that axis, so the newest sample sits at x₀ + (W−1)·h. A single positive number. The axis runs FORWARDS, oldest sample first, as on Power Fit.
• Start Point – the pair [a₀ b₀] the iteration starts from, both strictly positive. It is fitoptions' StartPoint, and here it is required rather than optional: see Notes. A start far from the answer costs iterations, and on data with more than one local minimum it decides which one the block reports.
• Iterations – how many Levenberg-Marquardt sweeps each sample runs, a whole number from 1 to 400. There is no convergence test: every sample runs exactly this many sweeps, which is what keeps ten generated cores on the same path. Convergence here is linear rather than quadratic (the residual at the optimum is large), so the count buys accuracy at a constant rate – measured against a 400-sweep answer on the rig's own data: 20 sweeps 6.6e−4, 30 sweeps 4.7e−5, 40 sweeps 3.3e−6, 60 sweeps 1.3e−8 relative.
• 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.

What every target carries is the iteration itself, not a bank of coefficients: the window, the 2×2 normal system, the damping and the accept/reject test are all in the generated core, and the only things inlined from the configuration are the abscissa, the start point, the bounds and the sweep count. A core therefore costs about 2·W·Iterations exponentials per step; the Window Length and Iterations settings are what a user trades against that.

The three HDL targets are simulation-only real arithmetic, quantizing only at the port boundary: an exponential, a division and a 2×2 solve have no Q16.16 form. They additionally clamp both parameters to the ±32767 that format can carry, which the seven software targets and this block's own simulation do not.

Simulink bridge

None (Support::None). Fitting a Weibull curve to a running window is a Curve Fitting Toolbox function (fit with the weibull model), and that toolbox ships 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.

Notes

• Stateful, and discrete by nature (setDiscreteOnlyBlock(true)): the window advances once per sample.
• ⚠ MATLAB's own start point for this model is RANDOM, and this block's is not. weibull is one of the two library models with no start-point heuristic, so fit(x, y, 'weibull') draws rand(2,1) and warns "Start point not provided, choosing random start point" – two calls on the same data need not agree. A generated core has no such draw and a stream cannot reproduce one, so Start Point is configuration here. Set it deliberately; it is the one parameter of this block that changes which answer you get rather than how fast you get it.
• ⚠ The fit is a fixed number of sweeps, not a converged solve. There is no tolerance test, because a tolerance makes the number of sweeps depend on arithmetic that differs by one unit in the last place between backends, and the ten cores would then take different paths. Raise Iterations rather than expecting a convergence flag; the block reports no residual.
• ⚠ Both parameters are bounded below by zero, as in MATLAB, and the bound is approached by halving. A step that would carry a or b to or past zero is replaced by a step to halfway between the current value and zero. Clamping exactly onto the bound would be worse than it sounds: at a = 0 the model's sensitivity to b is identically zero, so the 2×2 system would be singular from that iteration on. The parameters are also bounded above, at a ≤ 10⁶ and b·|ln x| ≤ 50, so that no intermediate can overflow – a VHDL real that overflows aborts the simulation rather than returning an infinity.
• ⚠ The abscissa runs forwards here, as on Power Fit, and backwards on the rest of this family. Polynomial Fit, Fourier Fit, Gaussian Fit, Sum of Sines Fit and Exponential Fit put the newest sample at x = 0. A Weibull has no value at zero and none below it, so this block's window is oldest at x₀ and increases towards the newest sample – an elapsed-time axis rather than a look-back one.
• ⚠ Its output is NOT a polynomial, and Evaluate Fit cannot read it. Evaluate Fit, Fit Derivative and Fit Integral all speak coefficients in descending powers; these are the two parameters of a Weibull curve. Wiring this block into any of the three produces a number rather than an error, so the mistake is silent.
• The window is zero-prefilled, so the first W−1 outputs of a run are a startup transient – the convention Polynomial Fit, Fourier Fit, Gaussian Fit, Exponential Fit and Power Fit all follow. On an all-zero window the least-squares answer drives a towards its lower bound and leaves b wherever the start point left it: a curve of zero height says nothing about its shape.
• Scalar only. One channel and its own history; wire one block per channel.
• No state space. The model is nonlinear in both parameters, so there is no A/B/C/D pair to seed and model reduction correctly declines to merge it.