Recursive ARMAX — System Identification/Recursive

Recursive ARMAX

System Identification / Recursive

Identifies an ARMAX model online by extended least squares:

A(q)y[k] = B(q)u[k] + C(q)e[k]

Unlike an ARX estimator, this one also models the colour of the noise through C(q). It does so by putting its own past prediction errors into the regressor:

φ[k] = [ −y[k−1] … −y[k−na], u[k] … u[k−nb], e[k−1] … e[k−nc] ]

which is why it is called pseudo-linear regression: the last segment is not measured data but a quantity the estimator itself produces. The parameter vector is then moved by an ordinary forgetting-factor recursive least-squares step.

Reach for it when the disturbance on your plant is not white – when an ARX fit leaves structure in its residuals. Where ARX would bias the A and B estimates to explain that structure, ARMAX gives the noise its own polynomial to live in.

Ports

• u – the input, a [1,1] scalar per step.
• y – the measured output, a [1,1] scalar per step.
• A – the estimated [na,1] denominator coefficients a₁…a_na. The leading 1 of A(q) is not estimated and is not on the port.
• B – the estimated [nb+1,1] numerator coefficients b₀…b_nb. nb+1, not nb: b₀ is the direct feedthrough term.
• C – the estimated [nc,1] noise coefficients c₁…c_nc. As with A, the leading 1 is not estimated.
• e – the current a priori prediction error, a [1,1] scalar. It is the residual the regressor will carry forward, so watching it is how you see the estimator converge.

Parameters

• A Order – na ≥ 1, the number of estimated denominator coefficients.
• B Order – nb ≥ 0. The B port is one longer than this.
• C Order – nc ≥ 1, the length of the noise polynomial. This is the parameter that distinguishes the block from an ARX estimator; at nc = 0 it would be one.
• Forgetting Factor – λ in (0, 1]. Below 1 the estimator discounts old data and can track a changing plant; at exactly 1 it weights the whole history equally and settles. ⚠ A small λ on quiet data makes the covariance grow without bound – the classic estimator wind-up.
• Initial Covariance – P₀, the diagonal the covariance starts from. Large means "distrust the initial guess", so the first samples move θ a long way; small means the opposite.
• Initial Parameters – the seed for θ. A single value broadcasts to every entry; a vector of the full length na+nb+1+nc is taken in order.
• 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. Each generator re-implements the recursion step for step, with the three orders – and therefore every array bound – fixed at export time from the configuration. θ is seeded from the config in every backend, never from the running estimate, which by export time has already moved.

The three HDL targets are simulation-only, this family's standing rule: the update divides once per sample and the covariance spans the decades between its seed and its settled value, and neither belongs in a Q16.16 datapath. They carry the recursion in real arithmetic and quantize only at the port boundary.

Simulink bridge

None. Simulink's recursive estimators belong to the System Identification Toolbox, which is not installed here, so a mapping could not be parity-verified even if it were written. The bridge reports the block rather than dropping it silently, and it has no parity testbench, which is the documented consequence of Support::None. Code export verification still covers it across all ten languages.

Notes

• Stateful and inherently discrete: θ, the covariance and three history buffers advance once per sample, so the block declares setDiscreteOnlyBlock(true).
• ⚠ The update ORDER is part of the definition. u[k] enters the regressor at once (b₀ is a feedthrough term) while y and e enter one step later, so the histories are pushed at two different points in the step. A backend that shifted one of them early or late still produces a well-formed recursion – it just estimates a different system.
• Convergence is not guaranteed, and that is a property of extended least squares rather than of this implementation: the regressor contains a quantity the estimator generates, so the recursion is only pseudo-linear. It behaves when the input is persistently exciting and C(q) is stably invertible.