EC-PIDUNN Architecture

• EC-PIDUNN is a hybrid control architecture that combines an untrained MLP with a modified PID controller to generate control signals based solely on error-centric inputs.
• The design discards the need for plant modeling or supervised training by using fixed random weights and an adaptive gain update mechanism stabilized by a factor τ.
• Empirical validations on nonlinear robotics benchmarks show nearly critically damped responses, zero overshoot, and improved stabilization over classical PID controllers.

The Error-Centric PID Untrained Neural-Net (EC-PIDUNN) is a hybrid control architecture designed to address the limitations of classical Proportional-Integral-Derivative (PID) controllers and conventional PIDNN (PID Neural Network) schemes in nonlinear and interconnected dynamic systems. EC-PIDUNN integrates a randomly-initialized, untrained multilayer perceptron (MLP) with a modified PID controller stabilized by an explicit factor, τ, to generate control signals based solely on error-centric input without requiring any predefined plant model or network training. Empirical results on nonlinear robotics scenarios demonstrate EC-PIDUNN’s capacity to achieve nearly critically damped responses and robust convergence that outperform both classical PID and traditional PIDNN approaches (Razzaq, 6 Dec 2025).

1. Architectural Principles and Distinctiveness

EC-PIDUNN structurally diverges from classical PID and PIDNN frameworks by leveraging only the steady-state error and recent control history as inputs. The architecture comprises two main functional blocks: an untrained feed-forward MLP that transforms a low-dimensional, error-centric vector into a parameter vector, ρₜ; and a dynamic postprocessing layer that computes and applies adaptive PID gains. Unlike PIDNNs, which require careful, large-scale supervised training over plant and feedback variables, EC-PIDUNN operates with a fixed, random set of network weights and dispenses with any form of plant modeling or online learning. The synergy of this architecture derives from the internal feedback of both its control outputs and gain-perturbation vector, enabling online adaptive gain shaping while constraining instabilities through the stabilizing factor τ.

2. Error-Centric Input Processing and Parameterization

At each time step, EC-PIDUNN constructs a compact input state for its untrained network:

• One-step error: $eₜ = yₜ^{des} - yₜ^{act}$
• Previous control signal: $u_{t-1}$
• Error-difference: $Δεₜ = eₜ - u_{t-1}$
• Prior gain-perturbation vector: $ρₜ = \left[ρ_p, ρ_i, ρ_d\right]$

These six quantities are concatenated as $$I_{ρ} = [eₜ, u_{t-1}, Δεₜ, ρ_p, ρ_i, ρ_d] \in \mathbb{R}^6$$ and mapped through a single hidden-layer MLP (typically with 10–50 ReLU or tanh units), yielding a new gain adjuster, $ρ_{t+1} = f(I_{ρ}; w^0) \in \mathbb{R}^3$, where $w^0$ are the fixed, randomly initialized network weights. This high-dimensional, nonlinear mapping is key to enriching the representation of the recent error trajectory beyond scalar or tuple-based error encoding. A plausible implication is that the random features produced by the untrained network provide a diverse basis for gain shaping.

3. Gain Adaptation, Stabilizing Factor, and Improved PID Law

To mitigate sensitivity from the random feature generator and ensure robust output, EC-PIDUNN introduces a stabilizing hyperparameter τ directly into the PID formulation. The improved control law is:

$$uₜ = K_p eₜ + \frac{K_i}{τ} \int_0^T e(τ)\, dτ + τ K_d \dot{e}_t$$

where τ ($1 \leq τ \leq 5$ in practice) damps the integral term and amplifies or attenuates the derivative action accordingly. The update rules for the time-varying PID coefficients are:

$$\begin{align*} K_p &= K_{p0} + ρ_p \cdot \frac{Δεₜ}{Δt} \ K_i &= K_{i0} + ρ_i \cdot \frac{Δεₜ}{Δt} \ K_d &= K_{d0} + ρ_d \cdot \frac{Δεₜ}{Δt} \end{align*}$$

where $K_{p0}, K_{i0}, K_{d0}$ are coarse baseline gains from an initial PID tuning. In the $\Delta\epsilon_t\to0$ regime, the control law seamlessly reduces to a baseline PID. This enhanced control update ensures boundedness even in the presence of irregular ρₜ outputs.

4. Feedback Loop, Hyperparameters, and Algorithmic Flow

The architecture is internally closed by feeding back both $uₜ$ and $ρₜ$ into subsequent inputs, anchoring the dynamic behavior of the controller. The standard control iteration consists of:

1. Measurement of $eₜ$ and calculation of $Δεₜ$
2. Formation of the 6-dimensional input $I_{ρ}$
3. Propagation through fixed-weight MLP: $ρ_{t+1} = f(I_{ρ}; w^0)$
4. Gain adaptation using the dynamic compute formulae
5. Computation of $uₜ$ using the improved PID law (with τ)
6. Looping forward with updated $u_{t-1}$ and $ρₜ$

Network and tuning hyperparameters include a single hidden-layer MLP (10–50 units), ReLU/tanh activations, Gaussian or Xavier initialization, and τ selected via offline experimentation in the $[1, 5]$ range. Sensitivity to individual hyperparameters is demonstrably low, as the stabilizing factor τ smooths over potential irregularities in random gain updates.

5. Empirical Validation on Nonlinear Robotics Systems

EC-PIDUNN was evaluated across two nonlinear robotic control benchmarks:

• Ackermann-Steered Unmanned Ground Vehicle (UGV):
  • Steering ($u_\varphi$): Rise time $9.65$ s (EC-PIDUNN) vs. $9.8$ s (PID); settling time $1.9$ s vs. $14.5$ s; overshoot $0\%$ vs. $66\%$.
  • Speed ($u_v$): $0\%$ overshoot vs. $19.4\%$ for classical PID.
• Pan-Tilt Camera Tracking:
  • Objective: Maintain a moving target centered via pan $(θ)$ and tilt $(φ)$ control.
  • EC-PIDUNN achieved near-critical damping and zero overshoot, while classical PID yielded oscillatory or slow convergence.

The results reflect the architecture’s primary advantage: robust stability and rapid convergence without model-based tuning or training dataset generation. The error-centric, feedback-driven design produces closed-loop behaviors indistinguishable from near-optimal damped systems.

Benchmark	Rise Time (s)	Settling Time (s)	Overshoot (%)
UGV Steering (EC-PIDUNN)	9.65	1.9	0
UGV Steering (Classical PID)	9.8	14.5	66

6. Stability Considerations and Theoretical Insights

A formal Lyapunov proof is absent; however, two engineered stabilizing mechanisms are employed: (i) τ explicitly bounds the integral and derivative action irrespective of gain perturbation trajectory, and (ii) the mutual reinforcement of $uₜ$ and $ρₜ$ within the network’s input curtails unbounded gain drift, ensuring asymptotic reversion to baseline PID for vanishing error-difference. Empirical evidence from all tested nonlinear scenarios confirms zero overshoot and nearly critically damped dynamics, suggesting an inherent stabilizing bias in the error-centric feedback loop when augmented by τ.

7. Comparative Analysis and Significance

The principal distinction between EC-PIDUNN and prevailing PIDNNs is the abandonment of network training and the exclusive reliance on error-centric, model-free adaptation. By confining updates to a dynamic neighborhood around coarse PID baselines and employing random nonlinear gain shaping, EC-PIDUNN sidesteps the extensive computational and data burdens associated with supervised neural control. This approach achieves comparable or superior performance in nonlinear, high-variance environments, notably in scenarios where model uncertainties or prohibitive training data requirements limit the deployment of conventional adaptive or neural-enhanced controllers. A plausible implication is that random-feature-driven gain adaptation, stabilized by error history and explicit damping/amplification, suffices for many practical nonlinear control contexts (Razzaq, 6 Dec 2025).