Stability and Controllability of Revenue Systems via the Bode Approach

In an advertising system, the probability of conversion is often used to represent the likelihood of a desired event. In this work, we use the probability of an ad click as the conversion event. This probability ${p(click)}$ is provided in real-time by ad-ranking machine learning models based on the ad and the impression opportunity. Assuming ${p(click)}$ is perfectly calibrated, an ad will eventually receive a click when the cumulative summation of ${p(click)}$ across multiple impressions reaches $1.0$.

To participate in an auction, advertisers must place a bid, ${max\_bid}$, representing the amount they are willing to pay for a click. For a single impression opportunity, each ad submits its final bid as follows [20, 21]:

The ad with the highest bid is shown to the user, and the advertiser is charged either the final bid in a first-price auction or the runner-up’s bid with second-price [22]. At this stage, the revenue system is not “paced” because total ad spend is entirely dependent on auction traffic. In practice, however, advertisers operate within daily or lifetime budget constraints, ensuring that cumulative spending on all winning bids does not exceed these limits.

To enforce such constraints, a pacing control variable is introduced into the final bid calculation:

where $\lambda\in(0,~{}1]$ acts as a bid modifier. This type of pacing is also known as discount pacing. The final auction and pricing process are then governed by the $\lambda$ stochastically.

From bidding and auction to actual budget spending, the pacing control variable $\lambda$ influences the budget spent by rescaling bids, thereby increasing or decreasing the win rate in auctions. In most cases, $\lambda$ updates are scheduled at fixed intervals using a cron job.

Under ideal conditions $\lambda$ remains constant as long as auction intensity and the total number of impression opportunities remain unchanged. However, in practice, $\lambda$ must adapt to environmental changes to ensure that the remaining budget is spent appropriately throughout the day or campaign lifetime, following the prescribed budget spending plan. The ultimate objective of this work is to design a suitable $\lambda$ controller, based on principles from Classical Control Theory in Section 5, to achieve precise and robust pacing.

Spending signal sensing involves detecting the actual spending velocity for a specific pacing cohort while simultaneously performing denoising. Although it may appear unrelated to system performance, the sensing and smoothing algorithm plays a critical role in ensuring system responsiveness and stability because it interacts with the desired spending velocity in the feedback loop. The commonly used methods for signal sensing include (details in Appendix):

• Exponential Smoother: Exponentially smooth the signal by applying a weighted sum bewteen past values and current values.

• Low-Pass Filter (LPF): Canonical first-order or second-order filtering to reject high frequency noises.

Occasionally, moving average smoothing may be employed yet less favorable, since the moving window functions like a high-order integrator which often introduces significant lag into the control loop. In practice, we prefer a first-order low-pass filter, as it effectively removes most noise while maintaining sufficient responsiveness without a resonance frequency. Increasing the time constant $T_{f}$ provides stronger and smoother filtering yet introduces additional lag into the feedback loop.

where $s$ is the Laplace complex frequency variable which will be introduced in later sections. $f_{c}$ is the filter cutoff frequency which may be the inverse of minimum spending noise period in seconds.

More advanced yet uncommonly used filtering methods, such as the Kalman filter or G-H filter, require additional complexity of modeling the physical system and may also experience risks of diverging from the ground truth in some cases.

To model the plant under control, i.e. the ad spending process, it is necessary to transform the discrete, non-linear, and time-varying processes of bidding, auction, and pricing into continuous linear time-invariant (LTI) components. This transformation can be done with following assumptions:

1. Auction occurs evenly and continuously producing a stream of ad spend.

2. The system operates in real-time from change of $\lambda$ to change of spend with negligible lag

3. At any given time mark, the relationship between $\lambda$ and budget spending velocity is monotonic and free of hysteresis.

With these assumptions, the plant can be represented as a black box that provides a pointwise mapping from $\lambda$ to ad spending velocity. Assuming positive correlation between spending velocity and $\lambda$

satisfy $sv\propto\lambda$, where $\Theta$ denotes other unknown factors relate to $sv$. At $\lambda=\lambda_{n}$, we have

This simplified modeling is only for controllability perspective, and our focus lies on the maximum $W_{n}$ and minimum $W_{n}$. If the controller can regulate the plant robustly within these bounds, the entire system will remain stable across its operational range. By analyzing historical auction data, the $W_{n}$ values corresponding to each $\lambda$-level can be determined. Typically, larger pacing cohorts (e.g., campaign level or ad account level) result in a smoother $\lambda$-to-spending velocity curve.

While the discreteness of pricing and auctions can be ignored due to their high frequency, the discreteness introduced by pacing intervals cannot be neglected. There are two choices to model this interval: work out the whole design in sampling Z-domain or employ a signal holding mechanism. In this work, we opt for the later option by placing a Zero-Order Hold (ZOH) discretization at the input of the plant. The lag introduced by the ZOH mimics the pacing interval, adding a slight “inertia” to an otherwise non-inertial plant. The system diagram is illustrated in Figure 1.

The Laplace transform is one of the most commonly used tools for analyzing and designing control systems. In the absence of a compensator, the open-loop transfer function of the ad ranking and auction system can be represented as the product of all components within the feedback loop.

$T_{ps}$ represents the sampling rate of ZOH, which is defined by the pacing cycle. $T_{f}$ denotes the tunable decay parameter of the first-order LPF used to smooth the observed spending velocity. If the spend is measured at the impression level, $T_{f}$ may be smaller than $T_{ps}$.

To convert the system into a standard form for easier analysis, it is necessary to resolve the term $e^{-sT_{ps}}$. The Taylor expansion proves useful for this approximation, and we consider terms up to the 10th order.

Note that choice of the approximation order affects the number of zeros in the system but does not influence the poles. The open-loop transfer function of the system is finally expressed as follows:

The plant has a single pole arising from the observed spending low-pass filter, which introduces a lag. Due to the high value of $W_{n}$, typically resulting from the intensity of impression opportunities, the uncompensated plant becomes naturally unstable in response to the high-frequency (short-term) range due to the pole, meaning the ad spend will be diverging from the desired budget plan. Consequently, a compensator with at least one integrator is required to increase the phase margin and attenuate the gain contributed by $W_{n}$.

In classical control theory, several methods are used to evaluate system stability and dynamic performance, such as Nyquist stability criterion, root locus analysis, and Bode plot [6, 7, 8, 9]. In this work, we rely on the Bode stability criterion, specifically gain margin and phase margin, to quantitatively measure system stability. Additionally, the Bode method aids in compensator design from desired performance perspective, constructing compensators, and analyzing open-loop and closed-loop stability and dynamics, offering more intuitive visualization compared to other methods.

• Gain margin (GM): The gain margin measures how much the system gain can be increased before it becomes unstable. A larger gain margin indicates a more robust system capable of tolerating variations in gain. In our context, “robust” refers to the system’s resilience to fluctuations in request traffic and auction intensity.

• Phase margin (PM): The phase margin represents the additional input-to-output lag that can be introduced into the ad system before instability occurs. A larger phase margin indicates a more stable system with reduced oscillatory behavior. Understanding phase margin is critical for assessing how spending noise affects stability.

By analyzing the Bode plot of the open-loop plant in Figure 2, we observe that both the phase margin and gain margin are negative, indicating instability in the absence of a compensator. The high gain further reflects an extremely sensitive $\lambda$-to-spending response, emphasizing the need for stabilization.

The system gain dropped drastically above $5\times 10^{-2}$ Hz. This behavior is consistent with the Nyquist–Shannon sampling theorem, which states that for a sampling period of $T_{ps}=10$ seconds, the maximum frequency for lossless information representation must be below $5\times 10^{-2}$ Hz. Any gain increase observed beyond $1\times 10^{-1}$ Hz is unreliable due to aliasing effects, which shall be ignored.

With the practitioner approach, a compensator is preselected, and its parameters are adjusted to align with the plant’s stability and dynamic performance requirements. Among compensators, the Proportional–Integral–Derivative (PID) controller is arguably the most widely used due to its simplicity and effectiveness [9].

The plant exhibits low inertia and high gain, making the differential term $K_{d}s$ potentially hazardous, as it would further amplify the gain at high frequencies. Staying with P, and I terms, the best $K_{p}$ and $K_{i}$ can be determined through a grid search, balancing GM and PM considerations. Figure 3 shows the PI compensator in the Bode plot. The characteristics of lowing gain and increasing phase angle at high frequency is desirable for our plant.

A more mathematical approach to design a compensator is to define a generic compensator with zeros and poles. The formulation of the generic compensator is as follows [9]:

where $K_{c}$ is the compensator global gain, bringing $+20log_{10}K_{c}$ dB lift with no change in phase angle in the Bode plot. $z_{1}$, $z_{2}$, $\cdots$ are the locations of zeros and $p_{1}$, $p_{2}$, $\cdots$ are the locations of poles. Each zero brings a $+20$ dB/decade slope starting at $f=z_{i}/2\pi$ Hz with a $90^{\circ}$ leading phase at $[z_{i}-decade,z_{i}+decade]$, whereas each pole decays gain by a $-20$ dB/decade slope starting at $f=p_{i}/2\pi$ Hz with a $90^{\circ}$ lagging phase at $[p_{i}-decade,p_{i}+decade]$ on the Bode plot.

Therefore, the compensator can be crafted using the building components $K_{c}$, $z_{i}$, and $p_{i}$ to shape the final system transfer function and achieve the desired Bode curve.

Note that once the generic compensator has been parameterized, there is still no straightforward method for implementing it in computer programming. To overcome this, the compensator transfer function must be converted from the continuous Laplace domain to the discrete Z-domain using the Tustin (Bilinear) Approximation. This involves substituting the frequency variable s with:

where $T_{z}$ is the sampling period that could be the same as the plant $T_{ps}$. By replacing $G_{c}(s)$ with the discrete output-to-input ratio at step $k$, $y[k]/u[k]$, we may further convert Z-domain representation into a recurrence relation of the following form:

Programming this recurrence relation, with memorization of previous inputs and outputs, can ultimately implement the desired compensator in a digital system.

To evaluate the performance of the proposed compensator, we use the stability and dynamic criteria outlined in Section 4.2. Additionally, we adopted the pacing error metrics from [23] measuring the normalized system pacing error through simulation. The pacing error is defined as:

where $N$ is the number of pacing cohorts, $M$ is the number of measurements, $dS_{ij}$ is the desired spending velocity for cohort $i$ at pacing period $j$ and $aS_{ij}$ is observed actual spending velocity. $PE$ represents the total relative execution error from the desired spending velocity.

From a business perspective, we can further weight $PE$ by the cohort spend as follows [23]:

where $W_{i}$ represents the cohort’s daily spend weight.

The compensator and ad system, initially represented in the Laplace domain, are converted and simulated in the time domain (top graph of Figure 7). Key simulation parameters (Ad Set 1 in Appendix) are derived from our in-house data:

• $max(W_{n})=13.52$ $/($\lambda\cdot$ minute)

• $min(W_{n})=1.707$ $/($\lambda\cdot$ minute)

• $T_{ps}=10$ second

• $T_{as}=0.87$ second

• $T_{f}=10/2\pi$ second

• Daily budget: $ $387.5$

• Traffic curve pattern: Normalized ads request curve per ad set

• Desired spending velocity: $\frac{remaining\_budget\times current\_traffic}{remaining\_traffic}$

To be more realistic, random noise is drawn from a Gaussian distribution with a mean equal to $G(s)$ output and a variance equal to $5$% of the output value, which is then added to the actual spending velocity. The maximum $W_{n}$ represents the lower bound for stability, while the largest PM at the minimum $W_{n}$ provides the best dynamics. An ideal compensator should exhibit a positive GM and PM for maximum $W_{n}$, as well as the largest PM for the minimum $W_{n}$.

PI compensator: In the compensated pacing system, the open-loop transfer function is given by $G_{c}(s)\cdot ZOH\cdot G(s)\cdot H(s)$. It is easy to prove that P control alone is insufficient to stabilize the system (proof in the Appendix), and D control is undesirable as it amplifies feedback noises and thereby reducing the signal-to-noise (S/N) ratio. In the Practitioner method, we select a PI compensator and perform a grid search for parameters that satisfy the Bode stability criteria. To prevent integrator windup, the integrator is limited to the range $[0,0.5]$ and is only activated when the lumped output is within the range $(0,1)$. Nine combinations of $K_{p}$ and $K_{i}$ with their stability performance can be found in Table 1. We can see that a stronger integrator (larger $K_{i}$) must match with a higher gain (larger $K_{p}$) to maintain stability. However, $K_{p}$ will find its limitation to stabilize the system once $K_{i}$ goes beyond a certain point. Their corresponding plots are shown in Figure 5.

Zero-pole compensator: As a demonstration, three zero-pole compensators are designed following the methodologies outlined in Section 5.2 with $K_{c}=1.0$ In these compensators, zeros capture the input error history, while poles capture the output ($\lambda$) history. Therefore, adding more zeros improves the dynamics with respect to the spending error, whereas poles help maintain steady pacing and slow down the response. We can artificially create zeros to cancel the poles from the ad system modeling and improve dynamics. Also found in Table 1, there are three examples of zero-pole compensators. Poles trade PM for GM while zeros behaves vice versa.

Additionally, we measure the traffic curve spectrum using the Fast Fourier Transform (FFT) in Figure 6. Since pacing functions as a tracking system, we expect the closed-loop system to have a negative gain for frequencies above the highest traffic spectrum at $9.3\times 10^{-5}$ Hz, indicating resilience to noise. For the compensators tested Table 1, the lowest closed-loop bandwidth is $9.68\times 10^{-5}$ Hz, which is sufficient to adapt the traffic fluctuation pattern.

To evaluate the validity of the designed compensator, we tested seven top-spending ad sets (listed in the Appendix) from our Click ad product. These ad sets were selected from the 1/20 budget split group and tested for one day on January 12, 2025 (PT). Due to the short testing period, we preloaded the integrator by an initial $\lambda$ from the last value prior to the test to ensure the system reached equilibrium faster, mimicking continuous operation. Each ad set was paced according to its targeted daily traffic curve by its own independent $\lambda$ compensator. The pricing employed a second-price auction model following the legacy business logic.

In Figure 7, we compare time-domain simulated spending curve and real-world auction spending curve from our methodologies against the legacy pacing system, which employs a step-based control mechanism. The simulated pacing closely aligns with real-world auction outcomes, supporting the effectiveness of the control system modeling. Our method demonstrates a more stable spending velocity, with reduced overshoot and undershoot compared to the legacy system. PE and SWPE comparisons are both improved dramatically in Table 2. Figure 8 shows the control variables and signals of Ad Set 1 during the 1-day auction.