Noise-Aware PID Tuning

• Noise Aware PID Tuning is a methodology that combines Jacobi-based differentiation with PID control to mitigate noise amplification and delay.
• It leverages extended parameter domains, including negative values, to balance estimation bias and noise variance effectively.
• The approach enables robust, real-time control in applications like robotics and industrial automation by ensuring precise derivative estimation.

Noise Aware Proportional Integral Derivative Tuning is an advanced set of methodologies and principles designed to improve the performance of PID controllers in the presence of measurement and process noise, particularly focusing on the crucial derivative action. The objective is to minimize noise amplification, achieve robust control, and tune the controller systematically by leveraging both analytic error models and parameter selection techniques that explicitly account for stochastic noise characteristics. This area builds on recent developments in algebraic parametric estimation and orthogonal polynomial expansions—most notably the Jacobi-based estimators—and extends classical integer parameterizations to include negative real-valued parameters, offering enhanced flexibility for noise attenuation and real-time control precision.

1. Algebraic Jacobi-Based Derivative Estimators

Central to noise aware PID tuning is the use of algebraic Jacobi polynomial-based differentiation, where the $n$-th derivative of a measured signal %%%%1%%%% is estimated by the projection onto a truncated Jacobi orthogonal series. Estimators depend on three main parameters: shape parameters $\kappa$ and $\mu$ (traditionally limited to nonnegative integers, but extended to $\kappa,\mu \in (-1,+\infty)$ in the cited work), and the integration window $T$.

The “minimal” Jacobi estimator for the $n$-th derivative is: $$D^{(n)}_{(\mu,\kappa)}(t_0) = \frac{\gamma^{(\mu,\kappa)}_n}{(\beta T)^n} \int_{0}^1 w^{(\mu,\kappa)}(\tau) P_n^{(\mu,\kappa)}(\tau) y(t_0 + \beta T \tau) d\tau,$$ where $w^{(\mu,\kappa)}(\tau) = \tau^\kappa (1-\tau)^\mu$, $P_n^{(\mu,\kappa)}(\tau)$ is the Jacobi polynomial, and $\gamma^{(\mu,\kappa)}_n$ is a normalization constant. This approach generalizes beyond integer values, enabling new regimes for tuning the estimator’s response to noise.

Additionally, “affine” estimators are constructed as convex combinations of several minimal estimators using weighting functions and an additional parameter $\xi$ to further reduce time delays.

2. Statistical Error Analysis and Noise Propagation

The estimator error comprises:

• A bias term from truncating the orthogonal series
• A noise term, reflecting the impact of additive stochastic noise $\omega(t)$ in the measured signal

The noise error’s mean and variance are rigorously characterized: $$E[e_\omega^{(\beta T)}(t_0)] = \int_0^1 p^{(\mu,\kappa)}_{(\beta T)}(\tau)\,E[\omega(t_0 + \beta T\tau)]\,d\tau,$$ with the variance, for minimal first-order estimators ($n=1$), given by: $$\operatorname{Var}[e_\omega^{(\beta T)}(t_0)] = \frac{2\eta}{T} \cdot \frac{\mu+1}{2\mu+2\kappa+5} \cdot \frac{B(2\mu+2,2\kappa+3)}{B^2(\kappa+2,\mu+2)},$$ where $\eta$ is the noise intensity (as variance for Wiener noise), and $B(\cdot,\cdot)$ is the Beta function.

Chebyshev-type inequalities, e.g.,

$$\Pr\left(\left|e_\omega^{(\beta T)}(t_0) - E[e_\omega^{(\beta T)}(t_0)]\right| < \gamma\sqrt{\operatorname{Var}[e_\omega^{(\beta T)}(t_0)]}\right) > 1 - \frac{1}{\gamma^2},$$

provide explicit probabilistic error bounds.

Critically, the orthogonality properties of the Jacobi polynomials ensure that for a wide noise class—those with mean and covariance as polynomials of degree less than $n$—the bias can be completely eliminated, and the noise error mean vanishes.

3. Parameter Tuning Strategies for Noise Minimization

Allowing $\kappa$ and $\mu$ to take negative real values within $(-1,0]$ is shown to dramatically reduce the variance of the noise error compared to the original integer-only setting, as evidenced by simulation results and variance formulas in the paper. This is achieved without significantly sacrificing dynamic performance or increasing time delay—indeed, certain negative settings enable nearly delay-free derivative estimates.

Tuning is thus a multiobjective process: decrease the bias term (by minimizing $$C_T^{(\mu,\kappa)} = \beta T\cdot\frac{\kappa+n+1}{\mu+\kappa+2n+2}$$), minimize the variance (via careful selection of $\kappa$, $\mu$, and $T$), and, for affine estimators, optimally select the weighting parameter $\xi$.

Table: Parameter Effects in Jacobi-Based Differentiators

Parameter	Primary Effect	Tuning Range
$\kappa$, $\mu$	Noise/bias trade-off	$(-1,+\infty)$
$T$	Noise/sensitivity delay	$>0$
$\xi$ (affine)	Delay compensation	Jacobi root

Simulations confirm that with $\kappa,\mu\in(-1,0]$, total error variance is greatly reduced and time delay is similarly minimized.

4. Application to Noise Aware PID Controllers

The results apply directly to PID controller design, specifically the derivative action which is highly susceptible to high-frequency sensor noise. Classical approaches use finite-difference or filtered schemes, but the Jacobi-based algebraic estimator enables a direct, tunable mapping between estimator parameters and noise performance. By substituting the standard derivative approximation with a minimally delayed, noise-attenuating Jacobi differentiator (with $\kappa, \mu$ negative), the resulting PID controller becomes significantly less sensitive to noise.

The procedures for implementation are systematic:

• Use the statistical error formulas to specify the desired variance/bias trade-off
• Select $\kappa, \mu, T$ (and, if needed, $\xi$) to meet both performance and robustness requirements
• Validate delay characteristics through simulation or frequency response analysis

This approach generalizes for control of robot joints, electro-mechanical processes, and feedback systems in which real-time, robust noise tolerance is required.

5. Theoretical and Practical Significance

The extension from integer to real, especially negative, parameter domains fundamentally enhances the noise-aware tuning landscape for derivative estimation in PID design. Theoretical advances include:

• Closed-form expressions for estimator bias and variance as continuous functions of $(\kappa,\mu)$
• Explicit Chebyshev-type probabilistic error bounds
• Characterization of estimator delay as a function of parameter selection

In practical terms, this methodology:

• Leads to delay-free or minimally delayed, low-noise derivative estimates when negative real parameters are chosen appropriately
• Provides explicit error quantification, enabling rigorous controller tuning in stochastic environments
• Ensures adaptability of PID controllers for industrial scenarios with significant, structured noise

These contributions offer a systematic, analytic alternative to heuristic and empirical noise-reduction techniques, ensuring that noise-aware PID tuning is compatible with modern robustness and safety requirements.

6. Real-World Implementation Considerations

For effective deployment:

• The integration window $T$ should be selected based on the bandwidth of the process and the noise spectrum; too small a window increases bias, too large raises noise sensitivity
• Integral windup protection may be required for the PID integral term if large, persistent disturbances or noise spikes are present
• Parameter adaptation can be automated via real-time monitoring of estimator error statistics, further enhancing robustness

In summary, leveraging Jacobi-based algebraic differentiators with noise-aware parameter tuning provides a rigorous, tunable framework for improving PID controllers in noisy environments. By modulating shape and scaling parameters beyond the confines of traditional integer domains, practitioners can realize controllers with lower estimation variance, reduced time delay, and enhanced overall robustness—features essential for advanced automation and feedback systems operating under realistic, noisy conditions (Liu et al., 2011).