NCbT Overview

Non-iterative Correlation-based Tuning (NCbT) is a direct data-driven control method that tunes fixed‑structure controllers from a single batch of input–output data. It requires no explicit plant model and no iterative optimization.

Problem setting

In model reference control, the goal is to make the closed‑loop system behave like a desired reference model \( M(q^{-1}) \). The ideal cost is

\[ J_{\mathrm{MR}}(\rho) = \left\| M - \frac{K(\rho)G}{1+K(\rho)G} \right\|_2^2, \]

where \( G \) is the unknown plant and \( K(\rho) \) is a fixed‑structure controller linearly parameterized by \( \rho \). Direct minimization of \( J_{\mathrm{MR}} \) is non‑convex and requires a model of \( G \). NCbT avoids these difficulties through a correlation‑based approximation.

Correlation approximation

NCbT exploits the approximation

\[ \frac{1}{1+K(\rho)G} \approx 1-M, \]

which holds near the optimal solution. This leads to the approximate error:

\[ \varepsilon(t,\rho) = M u(t) - (1-M) K(\rho) y(t). \]

The key idea is to choose \( \rho \) so that \( \varepsilon(t,\rho) \) becomes uncorrelated with external instrumental signals, ensuring asymptotic noise rejection. The resulting estimator is solved via a single least‑squares problem.

Weighting filter \( W \)

To improve conditioning and efficiency, NCbT introduces a filter \( W \) that “whitens” the input spectrum. In the frequency domain:

\[ W(e^{-j\omega}) = \frac{1 - M(e^{-j\omega})}{\Phi_u(\omega)}, \]

where \( \Phi_u(\omega) \) is the power spectral density of \( u(t) \). In discrete time, \( W \) is implemented using spectral factorization (e.g., an AR model of \( u \)).

Instrumental variables and least‑squares solution

Let \( u_W = W * u \) be the filtered input. Instruments are built by stacking a symmetric window of \( 2\ell+1 \) lags:

\[ \zeta_w(t) = \big[u_W(t+\ell),\, u_W(t+\ell-1),\, \dots,\, u_W(t-\ell)\big]^T. \]

Regressors are defined as

\[ \phi(t) = \beta(q^{-1})\big[(1-M) y(t)\big], \]

where \( \beta(q^{-1}) \) is a vector of stable basis functions that define the controller structure (e.g., integrator, delays, FIR). The independent term is \( u_M = M u \).

After truncating the signals to avoid boundary effects, the sample matrices are

\[ Q = \frac{1}{N}\sum_{t} \zeta_w(t) \phi^T(t), \qquad Z = \frac{1}{N}\sum_{t} \zeta_w(t) u_M(t). \]

The controller parameters are obtained by solving the least‑squares problem:

\[ \hat{\rho} = (Q^T Q)^{-1} Q^T Z. \]

Hyperparameter choices

• Window length \( l \) – controls the bias‑variance trade‑off. Typical values: 10–50. Larger \( l \) reduces variance but may increase bias and degrade conditioning.
• Basis \( \beta \) – must be stable and capture the expected controller dynamics.
• Reference model \( M \) – stable, strictly proper.