Recursive Output Error — System Identification/Recursive

Recursive Output Error

System Identification / Recursive

Streaming identification of an output-error model – the parallel-model recursion, in which the regressor carries the model's own past outputs rather than the measured ones:

φ[k] = [ u[k−d] … u[k−d−m], −ŷ[k−1] … −ŷ[k−n] ]
ŷ[k] = φ[k]′θ[k−1],   e[k] = y[k] − ŷ[k]

Why it exists beside the ARX block. With measured y in the regressor, output noise is correlated with the regressor and biases θ no matter how long the run is. With ŷ, it is not, and the estimate stays unbiased. That single substitution is the whole difference, and it is also the source of this block's one difficulty – see Predictor Bound.

Stateful and discrete by nature: it declares itself discrete-only and always steps at its own rate.

Ports

• u – the plant input, a scalar.
• y – the measured plant output, a scalar.
• B – the numerator estimate, [m+1, 1], descending [b₀ … b_m].
• F – the denominator estimate, [n+1, 1], descending and monic: the leading 1 is a constant, and the estimated coefficients follow it.
• e – the a-priori prediction error y − ŷ, the residual a Residual Whiteness Monitor downstream is meant to consume.
• stable – 1 on a normal sample, 0 on any sample where the predictor clamp engaged and the update was frozen. Per sample, not latched; put a Debounced Decision after it if a sticky verdict is wanted.

Parameters

• Numerator Order (m) and Denominator Order (n) – m must be ≤ n, or the model is noncausal and the block refuses to run. The convention matches the rest of the library: an intrinsic input delay of d = n − m samples.
• Forgetting Factor (λ) – in (0, 1]. Not a free tuning knob: the recursion divides the covariance by λ every step, so an unexcited direction grows as λ^−k and the per-step Lyapunov exponent is ln(1/λ). At 0.97 that is e³⁰ over a 1000-sample run; at 0.997 it is e³. Prefer 0.995–0.999.
• Initial Covariance – the diagonal P is seeded with. Large means "trust the data, not the seed" and adapts fast.
• Predictor Bound – the magnitude ŷ is clamped to. This is the stability guard, and what it promises is narrow: it keeps the RECURSION bounded, NOT the estimated denominator stable. Because ŷ feeds back into its own regressor, a momentarily unstable F̂ would otherwise make ŷ diverge, corrupt the regressor and diverge further. When the clamp engages the parameter update is frozen for that sample – adapting on saturated data is worse than not adapting – and stable drops to 0 so the intervention is visible rather than silent. Set it a few times larger than the plant output's working range.
• Sampling Time (s) – zero or less inherits the solver's rate.

Code export

All ten targets. The orders, and therefore every array bound, are fixed at export time, so no backend allocates or sizes dynamically. Each emits the same six phases as the live solver: shift the u history, build φ from the u and ŷ histories, predict and clamp, form e, run the gated update, then shift the ŷ history and write the ports.

The warm-up gate is real and is not inherited. The update is withheld until both histories are full; without it the first n samples would adapt against zero padding. Recursive Least Squares needs no such gate because its regressor arrives on a port – this block builds its own, exactly as Recursive IIR Identification does.

The three HDL targets are SIMULATION-ONLY, by this family's standing rule: the recursion divides once per sample, and the covariance spans roughly twelve decades between its seed and its settled value. Neither is carryable in the Q16.16 signal format. They run the recursion in real arithmetic and quantize only on the way out.

Simulink bridge

None. The System Identification Toolbox's recursive estimators are MATLAB objects driven from a script, not a Simulink block with this port shape, and every block in this family registers Support::None for the same reason. It therefore has no parity testbench, which is the documented consequence; code export verification still covers it across all ten languages.

Notes

• No state space: the recursion is nonlinear in its own state, so no A/B/C/D describes it and model reduction correctly refuses the block.
• The seed is read from the config in every backend, never from the running estimate – export happens after the model build has already solved a step.
• F is reported monic, with the constant 1 leading, so the port width is a function of the configured order alone and never of the values.