Sensor accuracy is pivotal for reliable environmental assessment, yet maintaining data fidelity is particularly challenging for cost-effective miniature sensors deployed on mobile platforms. Zou et al. (2021) showed that particulate-matter sensors suffer from pronounced temperature- and humidity-induced bias, underscoring the need for climate-adaptive calibration. Liang (2021) compared statistical and machine-learning calibration frameworks and found that multivariate models incorporating relative humidity, temperature, and pressure markedly outperformed simple regressions. These data-driven approaches, however, depend on extensive labeled datasets and frequent retraining, which limits their utility for sensors that traverse steep vertical gradients aboard UAVs.

Allka et al. (2023) enhanced the accuracy of internet of things (IoT) temperature nodes through temporal-pattern denoising, but their validation was restricted to quasi-stationary conditions and did not address the rapid ambient changes typical of UAV missions. Complementary physics-informed techniques have also been pursued: Liu et al. (2023) applied feed-forward thermal compensation to micro-accelerometers, stabilizing bias across wide temperature excursions, and Lundstrom and Mattsson (2020) demonstrated that enclosure radiation and airflow can distort ambient readings, indicating that disturbance terms should be estimated rather than merely filtered. Gu et al. (2020) extended these ideas to aerial IoT networks, using neural networks to correct multi-sensor drift during maneuvering flight.

The accuracy of sensor data becomes particularly critical when sensors are deployed on moving platforms, where rapid variations in temperature, pressure, and airflow can degrade measurement fidelity compared with stationary observatories. This distinction is also essential in the context of the present work, which specifically targets the development of ultra-miniaturized, lightweight, and low-cost instrumentation for airborne vehicles. In such scenarios, accuracy can be quantitatively expressed by the coefficient of determination (R²) relative to reference sensors, as reported in (Ali et al., 2021; Zafra-Perez et al., 2023; Yeom, 2021; Saha et al., 2021; Pohorsky et al., 2024; Nadzir et al., 2025). The consolidated results are summarized in Table 1, providing a basis for comparing the estimated performance levels of the proposed system.

DOBs offer a model-based route to real-time error mitigation when the system dynamics are reasonably known (Iwata et al., 2020; Santina et al., 2020; Daoud et al., 2020). They have been applied to battery state estimation (Messier et al., 2020) and industrial temperature control (Tang and Xu, 2023), yet their use with multi-sensor payloads undergoing rapid thermal excursions remains limited. To address this gap, the present study embeds a DOB-based error-mitigation scheme in a compact sensor suite and evaluates its efficacy during a rocket-assisted UAV mission in which the payload encounters abrupt tropospheric and lower-stratospheric temperature transients (Figure 1).

Embedding a disturbance observer in the sensor MCU allows the payload to infer the thermal and electrical states of both peer sensors and the sensor itself without additional reference hardware. This enables anomaly-aware fusion by withholding inconsistent sensors and supports on-board bias correction of calibration-relevant states, while keeping the system minimal.

The main contributions of this study are as follows.

The remainder of this paper is organized as follows. Section 2 presents the mathematical models, control architecture, implementation of the DOB, and the sensor-compensation strategy. Section 3 describes the system-identification procedure and parameter estimation, validating the theoretical model with the proposed multi-sensor design. Section 4 outlines the simulation and chamber-test setup, including scenarios and protocols used to evaluate the system under controlled conditions. Section 5 reports the experimental observations, with a linear-regression analysis comparing the estimated temperature against reference measurements. Finally, Section 6 summarizes the conclusions and outlines future research directions.

A multi-sensor setup was designed to maintain the operational temperature range of the sensors by adjusting the heater power. It operates as part of a feedback loop, where the controller modifies its output based on the sensor data and system requirements. The following equations describe the general state-space representation of the plant:

where A is the state matrix, B is the input matrix, Bd is the disturbance input matrix, C is the output matrix, d(t) is the disturbance, u(t) is the plant input, x(t) is the state vector of the system, and y(t) is the output.

The DOB detects and compensates for disturbances that are not directly observed by sensors, such as temperature variations and voltage fluctuations caused by the operation of processors and communication devices. By providing real-time disturbance compensation, the DOB ensures that the sensor data remain accurate despite external environmental variations. The disturbance d^(t) obtained from the DOB was used to adjust the sensor data. This yields the estimated output y^(t) of the system after disturbance compensation. The plant model with the estimated disturbance is represented as:

Assuming that x(t)=x^(t), the disturbance d^(t) is calculated directly from Equations 1 and 2, as shown in Equation 3:

However, this approach assumes that the model effectively matches a real system. Model errors can generate inaccuracies in observed disturbances. The differential equation of the disturbance relative to the estimated disturbance d^(t) where defined in the Equation 18 in (Chen et al., 2016) as:

where Gq(s) is the Q-filter that substitutes for the products of Bd and C accounting for model uncertainty. Thus, if estimation fast enough x˜≈0, by estimating the state disturbance using Equations 1, 2 and 4, we obtain as shown in Equation 5:

where y^(t) is the observed output from (2) and y(t) is the system output measured directly by the sensors, and G(s) is the high-pass filter as a matrix-valued transfer function designed to ensure that d˜(t) converges to zero.

To accurately predict the sensor output ys(t), it is important to measure the state of the sensor. Therefore, estimating the state xs(t) of the sensor and calculating an accurate measurement can be expressed as:

where As represents the state-transition matrix of the sensors, Bs is the input matrix of the sensors affected by the sensor input w(t), Cs is the output matrix of the sensors, and Ds is the feed-through of the sensors. The estimated sensor input w^(t) is defined as shown in Equation 7:

where F3 and F4 are the matrices designed for the sensor estimation from and d^(t), y(t) respectively. The sensors measured y(t) with a measurement noise n(t).

The system controller plays a crucial role in maintaining the operational stability of the sensors by controlling their temperature. The heating elements in the system were dynamically adjusted to maintain the desired temperature range. This is essential for accurate data collection under dynamic environmental conditions. Proactive temperature management helps mitigate the effects of rapid temperature variations owing to external factors such as UAV operations.

To optimize the control and minimize the energy consumption, the controller uses the reference input r(t)∈[T0(t),…,Ti(t)]. This is the set of target temperatures. Here, each Ti(t) corresponds to a specific ith component within the system. The general proportional-integral-derivative controller can be expressed as shown in Equation 8:

where c(t)∈[Q˙0(t),…,Q˙i(t)], Q˙i(t), represents the designed controllable heat generation in each ith element, e(t) represents the control error, and controllers gain matrix Kp, Ki, and Kd are designed to e(∞)=0. In Equation 9, e(t) and v(t) are defined as:

where F₁ is the matrix designed for controller from w(t).

Finally, the controller output u(t) is defined in Equation 10:

where F2 is a matrix designed to compensate for d^(t) in u(t). The integration of this control mechanism with sensors ensured that each component operated within its threshold. This enhances the overall robustness and reliability of the system. The following sections discuss the advanced functionalities and integration of the DOB and sensors within this refined control framework.

A mathematical model was required to apply the DOB, based on which we developed a mobile multi sensor system and temperature model. Figure 3 illustrates the multi-sensor system and its components. This figure shows the temperature states of the ith component, denoted as Ti(t). For the identification, we define the output matrix C, state x(t), and sensor output ys(t) as follows:

where i∈[0,1,2,3,4] represents the heater, microcontroller unit with the mainboard, enclosed air within the multi-sensor, surface temperature outside the multi-layered insulation (MLI), and atmosphere, respectively. Although the atmosphere is not a component of a multi-sensor system, it is expressed as the fourth component to derive the equations. Each of T0(t)−T3(t) were measured using dedicated temperature sensors. However, no sensor was installed to measure the atmospheric temperature T4 in the multi-sensor system. In this system, u(t) and d(t) are defined as the heat transfer between each component 208 and the atmosphere. These are expressed as follows:

where Q˙c(t) represents the electric heater power.

The total heat transfer rate.

Q˙i(t)  for each component can be expressed in terms of the heat capacity and 211 heat transfer between i and j as follows:

where ci is the heat capacity of the ith component. Q˙ij(t) is the heat-transfer rate between components i and j, as follows:

where gij [W/K] represents the thermal conductance.

From Equations 13 and 14, the differential equation of the plant model is expressed as.

The relationships between Qij(t) and Qji(t),  gij(t) and gji(t), and the self-heat transfer terms Q˙ii and giiare expressed as follows:

From Equations 11, 12, 15 and 16, we obtain the following temperature ODEs:

These equations can be compactly expressed in state-space form as.

where matrices A, B, and B_d are defined in Equation 18 as the compact state-space representation of Equation 17.

As the gain G of the DOB increased, the rate of decrease in the disturbance error decreased. However, if G is excessively high, the sensor noise may be amplified and the quality of the estimation results may be degraded. We selected a G value that was ten times higher than the sensor data collection frequency (0.25Hz) to balance error reduction and noise amplification in this system.

In addition, the optimal G can be obtained by defining the estimation error at the most rapidly changing point of the linearized model as the cost function and solving it through simulation and experiments using a sequential quadratic programming (SQP) algorithm. In this study, we employed the optimization results derived from both the external temperature simulation and the experimental data described in Section 4. The optimized G for this model is constant, consistent with the state estimation framework, and all experiments were conducted under identical conditions.

The DOB-based sensor estimator predicts the states of sensors that cannot be measured directly. In this study, a virtual atmospheric temperature sensor was designed to verify the performance of the sensor estimator. Assuming a multi-sensor without an external conductive medium, all thermal disturbances originate externally. From Equations 13, 14 and 16, we obtain as shown in Equation 19:

Thus, the estimated sensor output y^s(t) from Equation 6 where designed:

where the sensor model input vector w(t) in Equation 20 is defined as:

Therefore, from Equation 21 and the block diagram in Figure 2, the designed matrices F₃ and F₄ are defined in Equation 22.

Parameter estimation was performed to determine the thermal conductance and heat capacity of the system. Although some thermal conductance g_ij and heat capacity c_i values can vary with atmospheric pressure, in this study, we assumed a constant linear system, considering that the DOB was designed to operate even with model uncertainties. The trust-region-reflective least-squares algorithm is commonly used for nonlinear optimization. This was employed for the estimation. The parameters obtained through the optimization process are listed in Table 3.

The simulation process was based on a point-mass model capable of simulating nonlinear variations in the drag and mass. Using the mission settings specified in Table 4, we simulated the UAV trajectories as shown in Figure 4.

To evaluate the proposed system, we established a series of real mission scenarios for the multi-sensor system, simulating temperature and pressure conditions. The UAV-attached sensor rapidly increases to its maximum altitude via rocket propulsion. After attaining the maximum altitude, the UAV deploys its wings and transitions to gliding after it attains the winged flight altitude. This continues until the ballistic flight time ends and then decreases during the total flight time. Each scenario varied in terms of the maximum altitude attained and descent speed. The results are listed in Table 4.

Using the mission settings specified in Table 4, we simulated the UAV trajectories as shown in Figure 4. The trajectories were generated with a point-mass profile generator capable of simulating nonlinear variations in drag and mass, implemented within a MATLAB/Simulink based simulation model reused from our prior works. The dynamic model comprises a 6-DOF ECEF rigid-body with gravity, ground, mass, wind/atmosphere (ISA), and NED model based controller blocks; propulsion and airframe submodels for rocket booster ascent and turbojet/UAV flight with aerodynamics; and actuator subsystems for servo and gimbal. We did not modify the governing equations for this study and only adjusted the scenario parameters to match Table 4. Full details of the simulation and settings are provided in Kim et al (2024, 2025).

Using the international standard atmosphere model, we converted the altitude variations over time into corresponding temperature and pressure variations, as illustrated in Figure 5. The temperature and pressure data for each scenario derived from these simulations served as the basis for setting the conditions for the subsequent experiments, enabling rigorous and realistic performance testing of the sensor system.

We conducted experiments according to these steps to accurately implement the temperature and pressure variations set by the simulation and compared them with those of the reference sensor. We used the given (a) scenario I and (b) scenario II to set up the experimental environment by utilizing the high-altitude test chamber configured in Figure 6. Temperature and pressure were controlled in a vacuum chamber.

However, owing to the limitations of the experimental apparatus, the accurate replication of the experimental environment is challenging. Therefore, the scenarios were restructured as follows: Test I, Test II, and Test III to adjust the time required to attain the target temperature. Each test followed the step-by-step target temperature and pressure of the underlying scenario, although marginally different times were required to attain the targets.

The experiments were performed three times to detect the disturbances caused by internal heat generation and compare the performance of the sensor estimator. Specifically, in Test III, arbitrary disturbances were introduced to include temperature variations from the internal heater, external temperature variations, and pressure variations derived from a modification of Test I. The resulting variations in temperature, pressure, and heater operation Q˙o(t) in Test III are detailed in the Results and Evaluation section.

The experimental setup included a shelf for mounting the multi-sensor and a reference sensor that measured atmospheric temperature. According to the simulation, the temperature and pressure varied from over 20°C to −40°C and from 100kPa to 20kPa within 1min. The chamber was cooled to the target temperature and experiments were conducted after installing the multi-sensor. This is illustrated in Figure 7. These adjustments caused differences in the target temperature, pressure, and timing compared with the original scenario. However, these variations can be detected by a DOB, which is designed to detect abrupt variations in the environment. This highlights the accuracy of sensor estimation and demonstrates the robustness of the system.