MODULE 01 / PROBLEM FORMULATION
The Multi-Pollutant Indoor Blind-Spot Phenomenon
Conventional National Air Quality Indices (CPCB India, US-EPA, EU CAQI) are engineered strictly for outdoor ambient macro-environments. They evaluate air quality via an upper supremum (maximum operator) over outdoor criteria pollutants:
$$\text{AQI}_{\text{standard}} = \max\left(I_{\text{PM2.5}},\, I_{\text{PM10}},\, I_{\text{NO2}},\, I_{\text{SO2}},\, I_{\text{CO}},\, I_{\text{O3}},\, I_{\text{NH3}},\, I_{\text{Pb}}\right)$$
The Mechanical Filtration Paradox:
In modern air-conditioned classrooms, lecture halls, and corporate offices, mechanical HVAC systems and HEPA filters rapidly eliminate outdoor coarse and respirable dust (PM2.5 < 15 µg/m³, PM10 < 25 µg/m³). Consequently, standard equations calculate an AQI of 22–35 ("Good").
However, when 50 occupants congregate in a closed room, respiratory metabolic exhalation produces carbon dioxide at approximately 0.005 L/s per person. Within 40 minutes, indoor CO₂ surges past 2,500 ppm and bio-effluent Total VOCs exceed 2.0 ppm. Occupants experience acute cognitive impairment, lethargy, decreased cerebral oxygenation, and increased pathogen transmission risk. Standard AQI is completely blind to CO₂ and VOCs.
MODULE 02 / STANDARD EQUATION
CPCB Table S3 Piecewise Breakpoint Model
The Central Pollution Control Board (CPCB) calculates sub-indices for individual pollutants using linear piecewise interpolation across empirical concentration breakpoints:
$$I_p = \left[ \frac{I_{\text{HI}} - I_{\text{LO}}}{BP_{\text{HI}} - BP_{\text{LO}}} \right] \cdot \left(C_p - BP_{\text{LO}}\right) + I_{\text{LO}}$$
Where \(C_p\) is the measured pollutant concentration, \([BP_{\text{LO}},\, BP_{\text{HI}}]\) is the enclosing concentration breakpoint bracket, and \([I_{\text{LO}},\, I_{\text{HI}}]\) is the corresponding AQI category interval.
| AQI Category |
Index Range |
PM 2.5 (µg/m³) |
PM 10 (µg/m³) |
NO₂ (µg/m³) |
CO (mg/m³) |
| Good |
0 – 50 |
0 – 30 |
0 – 50 |
0 – 40 |
0.0 – 1.0 |
| Satisfactory |
51 – 100 |
31 – 60 |
51 – 100 |
41 – 80 |
1.1 – 2.0 |
| Moderate |
101 – 200 |
61 – 90 |
101 – 250 |
81 – 180 |
2.1 – 10.0 |
| Poor |
201 – 300 |
91 – 120 |
251 – 350 |
181 – 280 |
10.1 – 17.0 |
| Very Poor |
301 – 400 |
121 – 250 |
351 – 430 |
281 – 400 |
17.1 – 34.0 |
| Severe |
401 – 500 |
251+ |
431+ |
401+ |
34.1+ |
Mathematical Characteristic: Discontinuous first derivatives at boundary thresholds (\(dI/dC\) steps abruptly). Totally non-responsive to sub-threshold synergistic pollutant combinations.
MODULE 03 / NON-LINEAR INFERENCE
Mamdani Fuzzy Inference Engine (Continuous Centroid)
The Mamdani model maps input parameters into fuzzy linguistic partitions through continuous trapezoidal membership functions:
$$\mu_A(x; a, b, c, d) = \max\left( 0,\, \min\left( \frac{x - a}{b - a},\, 1,\, \frac{d - x}{d - c} \right) \right)$$
Four atmospheric inputs are partitioned into linguistic sets {Low, Moderate, High}:
- PM2.5: Low[0,0,30,60], Mod[30,60,90,120], High[90,120,400,400] µg/m³
- PM10 : Low[0,0,50,100], Mod[50,100,150,250], High[150,250,600,600] µg/m³
- CO₂ : Low[0,0,500,800], Mod[700,1000,1400,1800], High[1500,2000,5000,5000] ppm
- VOC : Low[0,0,0.3,0.6], Mod[0.4,0.8,1.5,2.2], High[1.5,2.2,10,10] ppm
Output categories are integrated across continuous domain \(u \in [0, 500]\) using Center of Gravity (CoG) defuzzification:
$$\text{AQI}_{\text{Mamdani}} = \frac{\int_{0}^{500} u \cdot \mu_C(u) \, du}{\int_{0}^{500} \mu_C(u) \, du} \approx \frac{\sum_{i=1}^{N} u_i \cdot \mu_C(u_i)}{\sum_{i=1}^{N} \mu_C(u_i)}$$
Mathematical Characteristic: Infinite differentiability across transitions; captures non-linear physiological interactions where mild particulate load combines with elevated carbon dioxide to accelerate pollutant uptake.
MODULE 04 / EDGE COMPUTING EFFICIENCY
Takagi-Sugeno Zero-Order Inference Engine
While Mamdani defuzzification requires computational integration over hundreds of spatial points, the Takagi-Sugeno ("Takaki") model defines consequents as deterministic crisp mathematical constants (singletons):
$$R_k: \text{IF } x_1 \text{ is } A_1^k \land x_2 \text{ is } A_2^k \dots \text{ THEN } y_k = c_k, \quad c_k \in \{25, 75, 150, 250, 350, 450\}$$
The overall index is defuzzified using the normalized firing-strength weighted average:
$$\text{AQI}_{\text{Sugeno}} = \frac{\sum_{k=1}^M w_k \cdot y_k}{\sum_{k=1}^M w_k}, \quad w_k = \min_{i=1}^n\left(\mu_{A_i^k}(x_i)\right)$$
Microcontroller Execution Advantage:
On the ESP32 dual-core Xtensa 32-bit LX6 processor (240 MHz), the Takagi-Sugeno engine completes execution in < 15 µs with deterministic O(M) instruction cycles and zero numerical Riemann sum loops.
MODULE 05 / MASTER OPERATIONAL INDEX
Dynamic Multi-Regime Effective AQI (AQI_eff)
To achieve holistic environmental protection without compromising outdoor compliance standards, the station computes an adaptive multi-regime fusion index:
$$\text{AQI}_{\text{eff}} = \begin{cases}
\text{AQI}_{\text{cpcb}} & \text{CO}_2 < 1000\text{ ppm} \land \text{VOC} < 0.6\text{ ppm} \quad [\text{Nominal Regime}] \\[6pt]
(1 - w)\cdot\text{AQI}_{\text{cpcb}} + w\cdot\text{AQI}_{\text{mamdani}} & 1000 \le \text{CO}_2 < 2000\text{ ppm} \quad [\text{Crowd IAQ Coupling}] \\[6pt]
(1 - k)\cdot\text{AQI}_{\text{mamdani}} + k\cdot\text{AQI}_{\text{sugeno}} & 2000 \le \text{CO}_2 < 3500\text{ ppm} \quad [\text{Severe Stagnation}] \\[6pt]
\text{AQI}_{\text{sugeno}} & \text{CO}_2 \ge 3500\text{ ppm} \lor \text{VOC} \ge 3.0\text{ ppm} \quad [\text{Hypoxia Overload}]
\end{cases}$$
$$\text{where } w = \text{clamp}\left(\max\left(\frac{\text{CO}_2 - 1000}{1000}, \frac{\text{VOC} - 0.6}{1.2}\right), 0, 1\right), \quad k = \text{clamp}\left(\max\left(\frac{\text{CO}_2 - 2000}{1500}, \frac{\text{VOC} - 1.8}{1.2}\right), 0, 1\right)$$
Outdoor Smog Invariant Override:
$$\text{AQI}_{\text{eff}} = \max\left(\text{AQI}_{\text{eff}},\, \text{AQI}_{\text{cpcb}}\right)$$
This mathematical invariant guarantees that an acute outdoor smog event (e.g., brick-kiln emissions, wildfire, agricultural stubble burning) can never be masked or suppressed by indoor ventilation weighting.
MODULE 06 / COMPARATIVE DOMINANCE
Pollutant Dominance & Environmental Regime Matrix
The table below summarizes which physical factor dominates under specific operational scenarios, which algorithmic engine governs the decision boundary, and the corresponding physiological mechanism:
| Environmental Scenario |
Dominant Pollutant Factor |
Governing Engine |
Regime Classification |
Physiological / Atmospheric Mechanism |
Outdoor Urban Highway
Peak vehicular transit |
NO₂ & CO
NO₂ > 180 µg/m³ |
CPCB Standard |
Outdoor Dominant |
Photochemical oxidation of fuel combustion byproducts; causes acute bronchial irritation. |
Brick-Kiln / Winter Smog
Thermal inversion layer |
PM 2.5 & PM 10
PM2.5 > 120 µg/m³ |
CPCB Standard |
Smog Override |
Fine respirable carbonaceous soot penetrates deep alveolar capillaries; cardiovascular stress. |
Clean Rural Ambient
Open agricultural fields |
Clean Background
CO₂ ~ 410 ppm |
All Concordant |
Nominal Baseline |
Optimal atmospheric equilibrium; all 4 algorithmic engines report AQI < 35 ("Good"). |
15-Person Office Room
Standard corporate space |
CO₂ & Organic Vapors
CO₂ ~ 1,400 ppm |
Mamdani CoG |
Transitional IAQ Coupling |
Exhaled bio-effluents begin accumulating; onset of mild drowsiness and reduced alertness. |
50-Person Closed Classroom
Air conditioner running |
CO₂ & High VOCs
CO₂ > 2,600 ppm |
Mamdani-Sugeno Fusion |
Critical Stagnation (Blind Spot) |
AC traps dust (CPCB says "Good"), but hypercapnic air causes acute cognitive drop, headaches, and viral transmission risk. |
Extreme Chemical Spill
Poor industrial ventilation |
High VOC / Gas
VOC > 4.0 ppm |
Takagi-Sugeno |
Hazardous Overload |
Volatile chemical vapors trigger neurotoxicity thresholds; demands immediate forced ventilation. |