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Stability and Controllability of Revenue Systems via the Bode Approach

Yichuan Niu, Jianhui Chen

TL;DR

This work formulates budget pacing in online advertising as a linear time-invariant control problem and applies Bode-based stability analysis to design robust compensators. By modeling bidding, auction dynamics, and pacing as an integrated plant with ZOH discretization, the authors derive PI and zero-pole compensators to achieve positive gain and phase margins under varying auction intensity. Simulations and real-world tests show improved pacing accuracy and resilience to traffic fluctuations compared with legacy pacing. The approach offers a principled, scalable framework that bridges control theory and modern advertising platforms for stable, precise budget allocation.

Abstract

In online revenue systems, e.g. an advertising system, budget pacing plays a critical role in ensuring that the spend aligns with desired financial objectives. Pacing systems dynamically control the velocity of spending to balance auction intensity, traffic fluctuations, and other stochastic variables. Current industry practices rely heavily on trial-and-error approaches, often leading to inefficiencies and instability. This paper introduces a principled methodology rooted in Classical Control Theory to address these challenges. By modeling the pacing system as a linear time-invariant (LTI) proxy and leveraging compensator design techniques using Bode methodology, we derive a robust controller to minimize pacing errors and enhance stability. The proposed methodology is validated through simulation and tested by our in-house auction system, demonstrating superior performance in achieving precise budget allocation while maintaining resilience to traffic and auction dynamics. Our findings bridge the gap between traditional control theory and modern advertising systems in modeling, simulation, and validation, offering a scalable and systematic approach to budget pacing optimization.

Stability and Controllability of Revenue Systems via the Bode Approach

TL;DR

This work formulates budget pacing in online advertising as a linear time-invariant control problem and applies Bode-based stability analysis to design robust compensators. By modeling bidding, auction dynamics, and pacing as an integrated plant with ZOH discretization, the authors derive PI and zero-pole compensators to achieve positive gain and phase margins under varying auction intensity. Simulations and real-world tests show improved pacing accuracy and resilience to traffic fluctuations compared with legacy pacing. The approach offers a principled, scalable framework that bridges control theory and modern advertising platforms for stable, precise budget allocation.

Abstract

In online revenue systems, e.g. an advertising system, budget pacing plays a critical role in ensuring that the spend aligns with desired financial objectives. Pacing systems dynamically control the velocity of spending to balance auction intensity, traffic fluctuations, and other stochastic variables. Current industry practices rely heavily on trial-and-error approaches, often leading to inefficiencies and instability. This paper introduces a principled methodology rooted in Classical Control Theory to address these challenges. By modeling the pacing system as a linear time-invariant (LTI) proxy and leveraging compensator design techniques using Bode methodology, we derive a robust controller to minimize pacing errors and enhance stability. The proposed methodology is validated through simulation and tested by our in-house auction system, demonstrating superior performance in achieving precise budget allocation while maintaining resilience to traffic and auction dynamics. Our findings bridge the gap between traditional control theory and modern advertising systems in modeling, simulation, and validation, offering a scalable and systematic approach to budget pacing optimization.

Paper Structure

This paper contains 22 sections, 23 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Real-time Revenue System
  4. Bidding and Auction
  5. Pacing
  6. Spending Signal Sensing
  7. The Plant
  8. Modeling Methodology
  9. Laplace Transformation
  10. Bode Method
  11. Compensator Design
  12. Practitioner Methodology: PID Compensator
  13. Theorist Methodology: Zero-Pole Compensator
  14. Experiments
  15. Evaluation Metrics
  16. ...and 7 more sections

Figures (8)

  • Figure 1: Feedback system diagram. $G_c(s)$ is compensator to be designed in later section, $G(s)$ is the modeled bidding and auction with $W_n$ gain, $H(s)$ is the LPF feedback path.
  • Figure 2: Bode plot of the open-loop transfer function. $W_n = 13.52$, $T_{ps} = 10$, $T_f = 2 / 2\pi$, aliasing effects for $f > 5 \times 10^{- 2}$ Hz.
  • Figure 3: PI compensator Bode plot. $K_p = 5 \times 10^{-5}$, $K_i = 1 \times 10^{-5}$.
  • Figure 4: Upper: Zero with $K_c = 20, z_1 = 10 \times 2\pi$. Lower: Pole with $K_c = 20, p_1 = 10 \times 2\pi$
  • Figure 5: $K_p$ and $K_i$ vs GM and PM regarding maximum $W_n$ and minimum $W_n$
  • ...and 3 more figures