Ensemble-Based PID Gain Learning Framework for Adaptive Temperature Control in Micro-Thermoelectric Systems

A.RESEARCH STATUS OF ADAPTIVE TEMPERATURE CONTROL IN MICRO-TEC SYSTEMS

Micro-TECs are solid-state Peltier devices that provide compact, vibration-free, and bidirectional thermal control. They are used in polymerase chain reaction (PCR) thermal cyclers, biomedical analyzers, laser diode and photonic stabilization, and the localized cooling of microelectronics. The thermal response of a Micro-TEC is nonlinear. The Peltier heat, the Joule self-heating, and the conductive exchange with the hot side together link the drive current to the temperature. This coupling changes with the operating point, with the heat-sink and ambient conditions, and with the thermal load. For this reason, fixed or heuristically tuned PID gains that work well at one setpoint often perform poorly at another.

Reported methods for Micro-TEC temperature control include fixed-gain PID with anti-windup, gain scheduling, fuzzy and adaptive PID, and model predictive control (MPC). More recent work applies machine learning to PID design. Representative examples are the learning-based controller reported for the Micro-TEC of a PCR device [6] and the machine learning performance assessment of PID loops [7]. The present work differs in that it does not retune the controller online. The improved GA and PSO are run once, offline, over a fixed set of operating scenarios. The gains they produce are stored and used to train a single ensemble model. This model then returns suitable gains for new operating conditions without any further search.

B.MOTIVATION AND CONTRIBUTION OF RESEARCH WORK

Despite their long-standing adoption, PID controllers remain widely used in modern control systems because of their simplicity, interpretability, and compatibility with resource-constrained platforms. As control applications increasingly involve nonlinear dynamics and variable operating conditions, the demand for PID tuning strategies that go beyond fixed heuristics has become more pronounced. In practice, maintaining consistent control performance across different regimes often requires repeated manual retuning or reliance on computationally intensive optimization procedures, both of which limit scalability and practical deployment. Recent advances in machine learning have created new opportunities for enhancing classical control frameworks without abandoning their structural advantages. In particular, learning-based approaches enable the extraction of patterns from historical or synthesized data, allowing controller parameters to be selected in a systematic and automated manner. When applied to PID tuning, such approaches offer the potential to reduce dependence on trial-and-error methods while improving consistency in performance metrics such as overshoot, response time, and stability. Among data-driven techniques, ensemble learning has demonstrated strong generalization capability in regression problems involving nonlinear relationships and uncertainty. Its ability to aggregate multiple learners makes it well suited for predicting continuous-valued controller gains under varying conditions while remaining computationally efficient once trained. This characteristic is especially attractive for PID tuning scenarios, where rapid gain selection is required but online optimization is impractical. The motivation for this work therefore arises from the need to bridge classical PID control and modern data-driven methods. Ensemble learning is used to automate PID parameter selection. This keeps the simplicity, transparency, and reliability that underpin the widespread use of PID controllers in practical control systems.

A.PID CONTROL LAW AND PROBLEM FORMULATION

Traditionally, PID controllers are widely employed in Micro-TEC systems because of their simplicity and ease of implementation. The PID control law in eq. (2) and eq. (3) are given as:

The PID control law regulates the current supplied to the TEC module, considering the temperature error between the desired setpoint and the output. The classical tuning model mainly struggles with issues like overshoot, thermal lag, parameter drift, and nonlinear heat transfer. Recent research has focused on optimization through PSO- and GA-based optimization to minimize steady-state error and thermal settling and state error rate.

Figure 1 shows the resulting closed-loop control architecture, in which the ensemble-predicted gains (Kp, Ki, Kd) drive the PID controller regulating the Micro-TEC plant of Eq. (1) against thermal-load and ambient-temperature disturbances.

Fig. 1. Control system.

B.OPTIMIZATION-BASED PID TUNING

The research proposes ensemble learning inside the workflow that tunes PID controllers. A GA and a particle swarm routine still run, but the added software learns from every completed tuning job and reuses that knowledge. As a result, the next tuning converges faster, keeps the loop stable under wider plant conditions, and needs fewer objective function calls. A plain GA evolves strings of three numbers (the PID gains) by breeding, mutating, and simulating each child. One simulation per child is slow. The study trains a network on all earlier simulations. The network then acts as a surrogate. It reads a set of gains and immediately outputs an estimated closed-loop score. Only the most promising children go to the full simulator. The surrogate therefore trims computation time. When the setpoint temperature or the thermal load shifts, a classical GA would discard the old population as well as restart. The machine learning layer avoids this restart. It retains the mapping between past operating points and the gain sets that worked. The search begins inside a smaller, better-informed region. The layer also watches population diversity or short-term history and then raises or lowers mutation probability and crossover probability on the fly. The code keeps the genetic module and adds the surrogate next to shorten each new tuning campaign without sacrificing robustness. The machine learning (ML) model h(R) is trained to get the approximate true fitness function, denoted as h(R), where R(k)=(Mr(k),Mk(k),Mf(k)). The optimization is shown in eq. (4):

The GA replaces the usual full simulation step with a value called h that a machine learning model provides, whereas this value further guides the search. Machine learning also improves PSO. A surrogate model estimates how well each set of PID gains will perform. The swarm does not need a complete simulation for every candidate. Because the model gives fast feedback, the swarm keeps moving and does not settle too early, and the optimizer shortens the overall run time.

Standard PSO revises every particle velocity with two constant factors: one for the particle’s own memory and the other for the group influence. Different parts of the search space, however, demand different amounts of exploration. A small neural network reads the present state of the particle and of the whole swarm and then outputs new values for those two factors on every step. The velocity update rule becomes eq. (5). The eq. (6) is further modified as:

In eq. (6), e1(v) and e2(v) denote the acceleration coefficients that a neural network predicts at each iteration v from the current state of the particle and the swarm, replacing the fixed constants used in standard PSO. A neural network learns the patterns and adapts to the swarm’s behavior. The machine learning model that guides each particle takes the particle’s present position as its starting input Zk(v)=(Mr,Mk,Mf) to accommodate the velocity xk(v+1) and improve the fitness function’s negative eq. (7):

The ML system also learns how to pick gain values for a schedule. The neural network is trained so that it links each set of operating conditions to a set of gains. To build the training data, they run GA or PSO many times, while they vary the plant conditions and store the gains that each run returns. Once training ends, the ML model receives new conditions and outputs the gains that suit them as shown in eq. (8):

The ML model is a neural network, a decision tree, or another regression method that links the current state of the plant to the best PID gains. When ML joins GA besides PSO, the system adapts faster and needs fewer calculations while it tracks the nonlinear behavior. A neural ML routine searches for the PID set (Mr,Mk,Mf) that still works best when driven to its limits. Those limits appear when the plant parameters, the ambient temperature, or the thermal load reach values that push the Micro-TEC outside the range where normal controllers keep it steady. It includes the reaction rate constant m0, activation energy Gc, heat transfer coefficient W, and inflow concentration EC0; further, the optimization is structured as eq. (9):

Here, L(Mr,Mk,Mf,∅︀c) defines the cost function based on the error metric. The main aim is to find the PID parameters that minimize the error while considering the parameters ∅︀c, accommodating m0, GC, W, which could make the system difficult to control.

The study uses neural networks to improve two tasks: setting PID gains and finding the parameter values that cause the largest control error. Two separate neural models carry out the work. The first model predicts the parameters that create the worst-case behavior. It learns to output the parameter set that produces the greatest tracking error for a fixed set of PID gains. Training this model supplies a fast way to estimate which settings yield the hardest plant to regulate. In the GA optimization, every individual is a single vector that holds both the PID gains in eq. (10):

The study uses neural networks to improve two tasks: setting PID gains and the parameter values that cause the largest control error. Two separate neural models carry out the work. The first model predicts the parameters that create the worst-case behavior. It learns to output the parameter set that produces the greatest tracking error for a fixed set of PID gains. Training this model supplies a fast way to estimate which settings yield the hardest plant to regulate. In the GA optimization, every individual is a single vector that holds both the PID gains in eq. (11):

This cuts the number of full simulations required for every candidate. The GA updates the PID gains. Mutation and selection drive down the error that the neural network forecasts for the most extreme values. In the particle swarm, every particle holds the PID gains together in eq. (12):

The velocity and position of each particle change according to the predictions of the neural network outputs. The swarm reaches good PID settings. Neural networks learn from records of many earlier runs. In each run, either a GA or particle swarm algorithm tunes the PID settings at different operating points, (V,EC0,m0,Gc), with the corresponding optimal gains Mr,Mk,Mf. Every record lists an input side and an output side. The input side holds the operating conditions. The neural network mapping is shown in eq. (13):

The trained model takes the parameters and returns the best PID gains, or it takes the gains and returns the parameters. After training, it captures how the plant conditions map to good gains, and the optimizer covers the search space quickly. It also exposes situations that drive the PID to the edge of stability. Adversarial machine learning is used to create those extreme cases on demand. A neural network is taught to pick the set of plant numbers that gives the poorest closed-loop response for a fixed PID. Those hostile cases are injected into the tuning run, forcing the algorithm to pick gains that still work when the plant is at its nastiest. Because the predictor is neural, it remembers earlier runs and needs fewer plant simulations next time.

This cuts the CPU budget, and in the adversarial loop, the task is to choose Mr,Mk,Mf so that the controller survives the worst plantGc that the inner loop can invent. The inner loop chooses reaction rate constant m0, activation energy Gc, heat transfer coefficient W, and related terms to maximize control error.

The outer loop adjusts the same three PID numbers to drive that error back down. Equation (2) states the problem: maximize error inside and minimize it outside. Two optimizers run side by side. One GA or Particle Swarm search hunts for the plant the other searches for the PID.

In the GA version, each chromosome is one vector that holds both the PID triplet and the plant parameters, exactly as written in eq. (14). The outer GA minimizes error, while the inner GA maximizes it. The single fitness score tells how well the controller tracks and how the plant acts.

The algorithm treats the PID gains and the parameters as two separate groups of genes. It uses crossover, mutation, and selection on both groups in the same cycle. The PID gains change to lower the error signal. These parameters are changed to push the plant dynamics toward conditions that are hard for any controller to handle, as shown in eq. (15):

The evolutionary process adjusts both PID gains besides parameters through crossover, mutation, and selection steps that move toward an optimal outcome. PID gains change, so the error drops. The parameters change, so the system becomes harder to control. Each particle in the PSO swarm holds a single vector that contains the three PID gains and every uncertain parameter, as listed in eq. (15).

At every iteration, the algorithm changes the velocity and then the position of each particle. One part of the search looks for the NCS (Nonlinear Control Systems) parameter set that gives the largest tracking error. The other part looks for the trio (Mr,Mk,Mf) that gives the smallest error.

Each particle moves toward its own best previous combination of gains and parameters and also toward the swarm’s best combination. The parameter set produces the worst error and the gain set that produces the least error. Equation (15) provides the fitness value for any candidate vector.

The task is a min–max problem. The outer loop minimizes the cost with respect to Mr,Mk,Mf, where the inner loop maximizes the same cost. The compact statement of this min–max outcome appears in eq. (16). Running the PID controller for every proposed parameter vector demands a large amount of CPU time. A surrogate model trained offline with neural networks replaces most of those runs.

Once trained, the surrogate gives an approximate value of (Mr,Mk,Mf,∅︀c) almost instantly – GA or PSO can test far more candidates while searching the combined space of gains. The surrogate function Ĵ therefore stands in for the exact cost during the optimization:

Each adversarial training run serves as a separate experiment. The agent receives a single numerical reward after each worst-case test, and this reward equals the measured performance of the PID loop during that test. The sequence of rewards drives the agent’s learning in eq. (17). Adversarial machine learning methods create extreme situations on purpose. Each situation is built to stress a PID controller as much as possible. A model trained with those methods learns which settings give the poorest controller response. After training, the model offers the hardest cases while the controller is tuned, and the final PID settings withstand tougher conditions:

C.ENSEMBLE MODEL

Figure 2 illustrates the complete offline-to-online pipeline, from scenario generation and IGA/IPSO optimization through repository construction and stacking-ensemble training, to online gain prediction at deployment.

Fig. 2. Proposed ensemble.

The proposed ensemble learning module replaces repeated evolutionary optimization by learning the mapping between operating scenarios and near-optimal PID gains. First, improvised IGA and IPSO are executed offline across multiple scenarios to construct a PID tuning knowledge repository. In case of each scenario Tj=[g0,g1,g2,g3], the optimal performance gain Cj*=[Mr,Mk,Mf] are obtained through control objective K; this gives te supervised pairs (Tj,Cj*).

To approximate G(.), ensemble learning techniques (Random Forest, Extra Trees, and Gradient Boosting) are trained, and their predictions are combined through stacking:

Thus, the ensemble distills optimization knowledge generated by IGA and IPSO into a fast predictive model, enabling real-time, scenario-aware PID tuning without online evolutionary search.

A.EXPERIMENTAL SETUP

All controllers are tested on the Micro-TEC plant of Section III. Temperature is given in degrees Celsius and time in seconds, so the results describe Micro-TEC thermal behavior rather than a generic step response. Five controllers are compared: GA-tuned PID, PSO-tuned PID, ML-assisted GA-PID, ML-assisted PSO-PID, and the proposed ensemble-scheduled PID. The reported metrics are averaged over 375 test scenarios that were held out from training and validation; the repository holds 2,500 scenarios, split into 1,750 for training, 375 for validation, and 375 for testing. Setpoint tracking is tested at 25, 40, 60, and 95 C. Disturbance rejection uses a thermal-load step of +1 W at 100 s and −1 W at 200 s, and robustness is checked for an ambient change from 25 C to 35 C.

Table I summarizes the resulting control-performance metrics, and Table II reports the ensemble’s gain-prediction accuracy and offline efficiency.

Table I. Micro-TEC control performance on held-out test scenarios (mean ± std)

Table II. Ensemble prediction accuracy and repository/efficiency metrics

B.RESULTS

Figure 3 illustrates the tracking performance of the optimized gain-scheduled PID controller under varying reference conditions. The target output represents the desired setpoint trajectory, while the system response shows the actual plant output obtained using the tuned PID gains. As observed, the controller is able to closely follow setpoint changes with minimal overshoot and smooth transient behavior. The response demonstrates fast convergence to the desired value after each setpoint transition, indicating effective tuning of the PID parameters. The absence of sustained oscillations and the reduced settling time highlight the stability and robustness achieved through the learning-assisted PID tuning approach. The results confirm that the optimized gain-scheduled PID controller maintains accurate tracking performance across multiple operating regions.

Fig. 3. Target and system response.

Fig. 4 illustrates the system response obtained using a GA-optimized PID controller in the presence of external disturbances. Despite the injected disturbances, the controller maintains stable tracking of the reference signal with limited deviation from the desired output. Transient fluctuations are observed during disturbance intervals; however, the response quickly recovers and converges to the target value. This behavior demonstrates the robustness of the GA-tuned PID controller and its ability to mitigate disturbance effects while preserving overall stability.

Fig. 4. GA-optimized PID controller in the presence of external disturbances.

Fig. 5 presents the response of the PSO-optimized PID controller subjected to external disturbances. Compared to the reference trajectory, the plant output exhibits larger oscillations and increased sensitivity to disturbances, indicating reduced disturbance rejection capability. Although the controller responds rapidly to setpoint changes, sustained oscillatory behavior suggests that the PSO-tuned gains are less effective in maintaining stability under adverse conditions. This result shows the importance of careful tuning strategies when applying optimization-based PID controllers in disturbance-prone environments.

Fig. 5. PSO-optimized PID controller subjected to external disturbances.

Fig. 6 presents a comparative evaluation of PID tuning strategies under identical reference conditions. The GA- and PSO-optimized PID controllers exhibit noticeable oscillations and sensitivity to disturbances, particularly during setpoint transitions. In contrast, the ensemble learning-based PID controller demonstrates significantly improved tracking accuracy, reduced oscillatory behavior, and faster convergence to the desired setpoint. The smoother response and minimal steady-state deviation indicate superior robustness and generalization capability of the ensemble approach, confirming its effectiveness for data-driven PID tuning across varying operating regions.

Fig. 6. PID tuning strategies under identical reference conditions.

Fig. 7 compares the tracking performance of ML-assisted GA-optimized PID, ML-assisted PSO-optimized PID, and the proposed ensemble learning-based PID controller under identical reference conditions. While both ML-GA and ML-PSO controllers improve tracking accuracy compared to classical tuning, noticeable oscillations and transient fluctuations remain during setpoint transitions. In contrast, the ensemble learning-based PID controller exhibits smoother response characteristics, minimal overshoot, and faster settling across all operating regions. The close alignment between the ensemble response and the target trajectory highlights its superior generalization capability and robustness, validating the effectiveness of the ensemble approach for data-driven PID tuning.

Fig. 7. GA-optimized PID, ML-assisted PSO-optimized PID, and the proposed ensemble learning-based PID controller under identical reference conditions.