Rational Fit — Control Systems/Curve Fitting

Rational Fit

Control Systems / Curve Fitting

Fits a ratio of two polynomials to a frame of ordinates by nonlinear least squares and emits its coefficients:

ŷ(x) = (p₁xⁿ + p₂xⁿ⁻¹ + … + p_n+1) / (x^m + q₁x^m−1 + … + q_m)

These are MATLAB's ratnm library models, and the output carries p₁…p_n+1 then q₁…q_m – coeffnames(fittype('rat21'))'s own order. The denominator is monic: its leading coefficient is fixed at 1, as in MATLAB, because scaling numerator and denominator together leaves the curve unchanged and the fit would otherwise be undetermined. So a fit of degrees (n, m) has n + m + 1 coefficients, not n + m + 2.

Ports

• y – the frame of ordinates to fit, a column of [W, 1]. W is read off the wire, not configured: whatever feeds this port sets how many points the fit sees, and it must be at least n + m + 2 – one more point than there are coefficients – and at most 64. Entry k is the ordinate at x = Abscissa Start + k·Abscissa Step, counting from zero.
• pq – the coefficient column, [n + m + 1, 1]: the n + 1 numerator coefficients in descending powers, then the m denominator ones, also in descending powers, with the leading 1 left implicit. Its height follows Numerator Degree and Denominator Degree.

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.
• Numerator Degree – n, a whole number from 0 to 5, MATLAB's own range. At n = 0 the numerator is the single constant p₁.
• Denominator Degree – m, a whole number from 1 to 5. It starts at one because a rational model with a constant denominator is a polynomial, and ratn0 does not exist in MATLAB either – use Polynomial Fit for that.
• Abscissa Start – x₀, where the first entry of the frame sits on the fitted axis. Any single number; the window must stay inside ±1000, which is what keeps the arithmetic of the iteration inside the range a real can carry on the hardware targets.
• Abscissa Step – h, the spacing between consecutive entries, so entry k sits at x₀ + k·h. A single positive number.
• Iterations – how many Levenberg-Marquardt sweeps the fit runs, a whole number from 1 to 400. There is no convergence test: every frame runs exactly this many sweeps, which is what keeps ten generated cores on the same path. On the verified configurations 30 sweeps and 400 agree to 2.4e−16.
• 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 linearized start, the normal equations, the elimination, the damping and the accept/reject test are all in the generated core, and the only things inlined from the configuration are the two degrees, the abscissa, the bounds and the sweep count. A core costs about (n+m+1)²·W·Iterations multiply-accumulates per step; Iterations and the frame length are what a user trades against that.

The three HDL targets are simulation-only real arithmetic, quantizing only at the port boundary: a division per point per sweep and a dense elimination have no Q16.16 form. They additionally clamp every coefficient 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 rational model is a Curve Fitting Toolbox function (fit with a ratnm 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

• Algebraic and stateless: the answer is a function of the frame on the port and nothing else. Unlike the rest of this family it keeps no window of its own, so it has no startup transient.
• ⚠ It takes a FRAME where the rest of this family takes a stream, and the model is why. A rational curve is identified by its curvature: as the pole runs off to infinity the model degenerates into a polynomial of degree n−m with the coefficients growing without bound, so data with no curvature has no bounded best fit. A sliding window over a noisy signal meets that corner constantly – measured on this block's own export rig, a streamed window drove the coefficients onto the internal ±10⁶ bound and left the hardware targets 1.4 % and 142 % away from the reference – while a frame is what a rational fit is for: a curve sampled on a fixed abscissa, arriving whole. Feed it from a Tapped Delay if a running window really is what you want, and expect the degeneracy.
• ⚠ MATLAB's own start point for this model is RANDOM, and this block's is not. ratnm is one of the two library models with no start-point heuristic, so fit(x, y, 'rat21') draws rand(4,1) and warns "Start point not provided, choosing random start point"; measured, two such calls on one dataset answered [−254.4 2230.7 4908.0] and [0.30 −0.31 −0.92]. This block starts from the linearized fit instead – the least-squares solution of P(x) − y·(Q(x) − x^m) = y·x^m, which is linear in every coefficient – so the start is a function of the data and reproduces on any machine. It is a different estimator from the one the iteration then improves (it weights each point by its own denominator), which is exactly why it is used as a start and not as the answer.
• ⚠ The fit is a fixed number of sweeps, not a converged solve, and the block reports no residual. 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.
• ⚠ A pole inside the frame is refused, not fitted. A trial whose denominator comes within 10⁻⁹ of zero at any sample (in the abscissa's own scale) is rejected and the iteration damps instead, and if the linearized start itself has a pole in the window the fit restarts from a denominator whose roots all sit two window-widths to the left. Every coefficient is also held inside ±10⁶, a step that would leave that box stopping halfway to it. Both bounds exist so that no intermediate can overflow: a VHDL real that overflows aborts the simulation rather than returning an infinity.
• ⚠ Its output is NOT a single polynomial, and Evaluate Fit reads only half of it. Evaluate Fit, Fit Derivative and Fit Integral speak one coefficient vector in descending powers; this is two of them, and the denominator's leading 1 is implicit. Split the column with a Demux and evaluate the halves separately if you need the curve – remembering to prepend the 1.
• Several channels need several blocks. One frame, one fit.
• No state space. The model is nonlinear in the denominator coefficients, so there is no A/B/C/D pair to seed and model reduction correctly declines to merge it.