⚡ electrical engineering

Optimal input signal design for closed loop errors-in-variables system

This paper proposes nonparametric and harmonic design approaches to optimize input signals for closed-loop errors-in-variables systems, addressing the limitations of traditional output-error-only assumptions and validating the method's effectiveness through simulations.

Original authors: Wen Xiangyun, Wang Xiangjun, Guo Xiaoyong

Published 2026-09-23

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CC BY 4.0

Original authors: Wen Xiangyun, Wang Xiangjun, Guo Xiaoyong

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of engineering, systems are rarely perfect. Whether it is a factory robot, a power grid, or a medical device, the signals used to control them and the data they send back are almost always tainted by noise. Sensors drift, actuators wobble, and the environment interferes. For decades, many theoretical models simplified this reality by assuming that only the output data was dirty, while the input signal remained perfectly clean. This assumption made the math easier, but it often failed to reflect the messy truth of the physical world. When both the command sent to a machine and the response it gives back are corrupted by errors, the system is known as an errors-in-variables system. Designing the right input signal for such a system is a delicate balancing act. Engineers must choose a signal strong enough to reveal the machine's hidden characteristics, yet careful enough not to amplify the noise or damage the equipment. If the signal is too weak, the machine's true nature remains hidden; if it is too aggressive, the noise drowns out the useful information.

Researchers Wen Xiangyun, Wang Xiangjun, and Guo Xiaoyong from institutions in Zhengzhou, China, tackled this challenge by focusing on closed-loop systems where errors exist in both input and output. Their work, published in a technical paper, explores how to construct the ideal input signal to make these noisy systems easier to understand and control. Instead of relying on the old assumption that inputs are clean, they developed two distinct strategies to find the best possible signal. The first approach, called nonparametric design, involves a deep mathematical analysis of how the noise in the input and output interact. By treating the prediction errors from both sides as a single, combined problem, the team derived a specific formula that describes the optimal shape of the input signal. This method does not assume a fixed structure for the signal but rather calculates the ideal power distribution across different frequencies to minimize the impact of the noise.

The second strategy, known as harmonic design, takes a more constructive approach. Instead of calculating a broad formula, the researchers built a specific type of signal made up of multiple pure tones, or harmonics, layered together. They used a technique called the Schroeder scan, which carefully arranges the timing and phase of these tones. This arrangement ensures that the signal has a high average power—enough to excite the system effectively—without creating dangerous spikes in volume that could exceed safety limits. By tuning four key elements: the frequency, the number of tones, the loudness of each tone, and their timing, the team created a waveform that acts like a precise probe, revealing the system's behavior without being overwhelmed by interference.

To test whether these methods actually worked, the team ran a computer simulation. They created a virtual model of a closed-loop system and introduced random noise to both the input and output, mimicking real-world conditions. They then applied their newly designed input signals to this noisy environment. The results were clear: the parameters of the system, which represent its internal mechanics, gradually settled into their correct values as the simulation progressed. The output generated by the identified model matched the actual measured output with high similarity. This confirmed that the designed signals successfully filtered out the confusion caused by the noise, allowing the system to be identified accurately even when both the command and the response were imperfect.

The study concludes that these two design methods are effective tools for handling the complexity of real-world systems where errors are unavoidable. While the current work proves successful in simulations for linear systems, the authors acknowledge that the real world is often more complicated, with systems that change over time or behave in non-linear ways. They note that their current findings are specific to the conditions they tested and that future research will need to address these more complex, shifting behaviors. For now, however, the work provides a solid, verified path for engineers to design better input signals, ensuring that even in a noisy world, machines can be understood and controlled with greater precision.

Technical Summary: Optimal Input Signal Design for Closed Loop Errors-in-Variables System

Problem Statement
In many theoretical studies of system identification, it is common to assume that measurement errors exist only in the output. However, in engineering practice, both input and output signals are frequently contaminated by noise (e.g., sensor noise, actuator drift, environmental interference). Systems where both inputs and outputs contain errors are known as Errors-in-Variables (EIV) systems. The presence of input errors complicates the design of optimal input signals, as simply increasing input amplitude may not improve accuracy and can even exacerbate noise interference. Furthermore, in closed-loop EIV systems, limited data and input errors can render certain parameters unidentifiable. The challenge lies in designing input signals that maximize information acquisition and parameter estimation accuracy while adhering to physical constraints (amplitude, power, time) and mitigating the impact of noise in both channels.

Methodology
The paper proposes two distinct approaches to design optimal input signals for closed-loop EIV systems: Nonparametric Design and Harmonic Design.

Nonparametric Design:
- The authors derive the prediction error for the system considering noise in both the input ( $r(t)$ ) and output ( $y(t)$ ).
- Two prediction errors are formulated: one ( $\epsilon_1$ ) considering only output noise, and a second ( $\epsilon_2$ ) incorporating both input and output noise.
- A composite cost function is constructed by combining these two prediction errors.
- By applying power spectrum correlation theory and minimizing this cost function, the authors derive a nonparametric expression for the optimal input signal power spectrum. This derivation accounts for the variances of the input and output noise ( $\sigma_r^2$ and $\sigma_y^2$ ) and the system transfer function $G(q, \theta)$ .
Harmonic Design:
- This approach focuses on constructing time-domain waveforms by manipulating four key parameters: frequency, harmonic order, amplitude, and phase.
- The Schroeder scan method is employed to generate the optimal input signal. This method utilizes a deterministic, periodic multi-harmonic signal where the initial phase of each harmonic component is optimized.
- The phase recurrence relation is designed to shift the peaks of individual harmonics in the time domain, resulting in a composite signal with a low peak-to-average power ratio (PAPR). This allows for the injection of maximum average power within device safety limits, thereby enhancing the signal-to-noise ratio.
- The amplitude of each harmonic is set to ensure uniform energy distribution across the frequency domain.
- Parameter identification is performed using the conjugate gradient method to minimize a cost function $V(\theta)$ derived from the prediction error power spectrum.

Key Contributions

Dual-Approach Framework: The paper presents a comprehensive framework utilizing both nonparametric spectral analysis and parametric harmonic waveform construction to address the specific challenges of closed-loop EIV systems.
Composite Cost Function Derivation: A novel derivation of a composite cost function that jointly considers prediction errors from both input and output noise channels, leading to a specific nonparametric expression for the optimal input signal.
Application of Schroeder Scan: The adaptation of the Schroeder scan method for EIV closed-loop systems to generate input signals that maximize system excitation while minimizing peak power constraints and disturbance effects.
Simulation Validation: The study validates the proposed methods through simulation, demonstrating that the designed input signals allow system parameters to converge to target values despite the presence of white noise in both input and output channels.

Results
The effectiveness of the proposed input signal designs was verified through a simulation experiment using a specific linear time-invariant transmission model.

Parameter Convergence: Simulation results (Figures 2 and 3) showed that the estimated parameters for polynomials $A(q)$ and $B(q)$ gradually converged to their target values as simulation time increased, remaining stable thereafter.
Output Matching: Figure 4 demonstrated a high degree of similarity between the measured output and the identified output, indicating that the designed input signal effectively captured the system's characteristics.
Noise Robustness: The results confirmed that the optimal input signals enabled effective and accurate parameter identification even in the presence of external noise interference (mean 0, variance 0.5 for both input and output).

Significance and Claims
The paper claims that the proposed design methods successfully address the limitations of traditional approaches that ignore input errors. By employing nonparametric and harmonic designs, the study achieves optimal performance in parameter estimation for closed-loop EIV systems. The authors assert that their approach maximizes information acquisition and reduces experimental costs by avoiding blind experimentation and under-excitation.

The paper maintains a modest scope, noting that the current research is based on the assumption of linear time-invariant (LTI) parameters. The authors acknowledge that nonlinear and time-varying characteristics in actual engineering systems may degrade the performance of these methods. Consequently, the paper identifies the investigation of nonlinear systems as the primary direction for future work, rather than claiming immediate applicability to all complex, nonlinear industrial scenarios.

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