Nonlinear Curve Fit — Control Systems/Optimization

Nonlinear Curve Fit

Control Systems / Optimization

Fits a model you type to a window of measurements that arrive on a port, and answers the fitted parameters – MATLAB's lsqcurvefit, solved again on every sample:

[x, resnorm, exitflag] = lsqcurvefit(F, x₀, xdata, ydata),   minimizing  Σ_i ( F(x, xdata_i) − ydata_i )² over the parameters x.

The abscissa is configuration and the ordinates are the signal: the model is sampled at fixed sites and the measurements at those sites arrive on the wire, which is the shape a streaming fit has – a spectrum at fixed wavelengths, a sensor bank at fixed positions, a decay sampled at fixed delays.

Ports

• ydata – the measured ordinates, one per abscissa site, oldest site first. A COLUMN [W,1], where W is the number of entries in Abscissa. A size that disagrees stops the run and says both numbers.
• x – the fitted parameters, a column [n,1] in the order they are named in Parameters. n and its size follow from the configuration alone, never from the input.
• resnorm – the sum of squared residuals at the answer, [1,1]; MATLAB's second output.
• exitflag – MATLAB's third output, [1,1]. See Notes for what each value means.

Parameters

• Options – empty (default), or the path of a Solver Options block (Home/Solver Options), MATLAB’s options argument: for the run, every option it sets replaces this block’s parameter of the same name; one it leaves at default, or one this block does not have, changes nothing.
• Model – the expression fitted, over the parameter names and the independent variable: a*exp(b*t), a/(1 + exp(-b*(t - c))), a*sin(b*t + c). It is read by the console's own symbolic engine and accepts that engine's function set – sin cos tan sec csc cot asin acos atan acot sinh cosh tanh asinh acosh atanh exp log log2 log10 sqrt, the operators + − × / ^, and pi. abs and sign are refused by name, because the engine will not differentiate them without an assumption no block may depend on, and this block needs the derivative.
• Parameters – the names fitted, separated by spaces or commas: a b. One to 4 of them, and the output column carries them in this order.
• Independent Variable – the name the abscissa takes in the model, t by default. It must not be one of the parameter names.
• Abscissa – the xdata, a row or a column of 2 to 40 numbers. Its length is W, the height the ydata port must have, and it need not be evenly spaced – that is the whole difference from the presets in Curve Fitting. It must have at least as many entries as there are parameters, because lsqcurvefit refuses a problem with fewer measurements than unknowns.
• Start Point – x₀, one entry per parameter. The search is LOCAL, so this chooses which minimum is found.
• Optimality Tolerance – MATLAB's OptimalityTolerance, default 1e-6. The search stops with flag 1 when the largest gradient entry falls below it.
• Function Tolerance – MATLAB's FunctionTolerance, default 1e-6; flag 3.
• Step Tolerance – MATLAB's StepTolerance, default 1e-6; flag 2.
• Maximum Iterations – default 400, MATLAB's. The loop is emitted with a FIXED trip count, so a large value here costs every target the same run time whether the search needs it or not.
• Maximum Function Evaluations – default 400. Reaching either limit is flag 0.
• 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. The model and its derivatives are printed into each one in that target's own syntax, with the abscissa folded in as literals, so a generated core carries no expression reader and no table of sites. The three HDL targets – VHDL, Verilog and SystemVerilog – run the search in real arithmetic and are simulation-only, quantized at the port boundary: a trust region with a secular equation and a Brent root is not a Q16.16 datapath. Every setting above is baked in at export time; none of them is tunable on a generated core.

Simulink bridge

None. The Optimization Toolbox ships no Simulink library at all, so there is no block to map onto and no library path a diagram could name; lsqcurvefit is a MATLAB function. The block is reported rather than dropped when a model crosses. Sampling Time (s) → SampleTime, as on every block, does not apply here for the same reason.

Notes

• Algebraic. No state: the whole fit is re-run from the start point on every sample, so the answer depends on the window and never on what came before it.
• The Jacobian is exact, not a finite difference. The symbolic engine differentiates the model with respect to each parameter when the configuration loads. MATLAB's own default is the opposite – Jacobian is 'off', so lsqcurvefit estimates it by forward differences unless told otherwise – and the two take different steps on the same problem. Compare against Jacobian = 'on'.
• Exit flags, MATLAB's: 1 the gradient is small enough, 2 the step is small enough, 3 the change in the sum of squares is small enough, 0 an iteration or evaluation limit was reached.
• ⚠ LOCAL, and a small gradient is not a certificate. A different Start Point can reach a different fit, and a model that cannot describe the data walks the parameters outwards until a limit stops it – exit flag 0 with a large x is that case, not a defect.
• ⚠ A model that can overflow is a software-only export. An exp can overflow at a parameter the search passes through, and lsqcurvefit handles that: the trial point is rejected and the trust region is cut by twenty instead of by four. The seven software targets do the same. The three HDL targets do not get that far – a real that overflows stops the simulation rather than carrying an infinity – so keep the abscissa and the model inside the range a real can hold if the generated core is going to hardware.
• ⚠ The whole fit runs once per sample, at a fixed trip count. This is a real solver on the wire, not a recursion – budget for it.
• The nine preset fits in Control Systems / Curve Fitting are cheaper where they apply: they fit a FIXED model on an EVENLY SPACED abscissa, and most of them collapse into a bank of constants at configuration load. Reach for this block when the model is not one of theirs, or the sites are not evenly spaced.