[Paper Review] Modeling and Estimation of the Humans' Effect on the CO2 Dynamics Inside a Conference Room

Kevin Weekly, Nikolaos Bekiaris‐Liberis+1 more|arXiv (Cornell University)|Mar 20, 2014

Building Energy and Comfort Optimization28 references4 citations

TL;DR

This paper proposes a data-driven PDE-ODE model to capture CO2 dynamics in a conference room influenced by human occupancy and controlled CO2 sources. Using a convection-diffusion PDE with a filtered ODE-driven source term, it designs an observer and online parameter identifiers to estimate human CO2 emissions and model parameters from minimal sensor measurements, achieving asymptotic convergence in simulations.

ABSTRACT

We develop a data-driven, {\em Partial Differential Equation-Ordinary Differential Equation} (PDE-ODE) model that describes the response of the {\em Carbon Dioxide} (\cotwon) dynamics inside a conference room, due to the presence of humans, or of a user-controlled exogenous source of \cotwon. We conduct two controlled experiments in order to develop and tune a model whose output matches the measured output concentration of \cotwo inside the room, when known inputs are applied to the model. In the first experiment, a controlled amount of \cotwo gas is released inside the room from a regulated supply, and in the second, a known number of humans produce a certain amount of \cotwo inside the room. For the estimation of the exogenous inputs, we design an observer, based on our model, using measurements of \cotwo concentrations at two locations inside the room. Parameter identifiers are also designed, based on our model, for the online estimation of the parameters of the model. We perform several simulation studies for the illustration of our designs.

Motivation & Objective

• To develop a physically informed, data-driven model of CO2 concentration dynamics in a conference room influenced by human respiration and external CO2 sources.
• To design an observer that estimates unmeasured human CO2 emission rates using only two CO2 concentration sensor measurements.
• To create online parameter identifiers for real-time estimation of time-varying model parameters such as air flow and diffusion rates.
• To validate the model and estimation designs through controlled experiments and simulation studies under varying occupancy and CO2 input conditions.
• To enable energy-efficient building control by accurately inferring occupancy levels from CO2 concentration data.

Proposed method

• The CO2 dynamics are modeled using a convection-diffusion PDE with a source term driven by a first-order ODE representing human CO2 emission as a filtered input.
• The observer is designed using backstepping methodology to estimate the unmeasured human CO2 emission rate from two spatially distributed CO2 concentration measurements.
• Online parameter identifiers are developed using swapping identifiers and adaptive update laws to estimate time-varying parameters such as air flow and diffusion coefficients.
• The model incorporates a delay in CO2 concentration response to human input, represented by a filtered version of step changes in emission rate.
• Stability and convergence of the observer and identifiers are proven using Lyapunov-based analysis and an alternative to Barbalat’s lemma for time-varying systems.
• The design is validated through controlled experiments and simulation studies under known CO2 inputs and human occupancy.

Experimental results

Research questions

• RQ1 How can CO2 concentration dynamics in a confined indoor space be accurately modeled when influenced by human respiration and controlled CO2 sources?
• RQ2 Can an observer be designed to estimate unmeasured human CO2 emission rates using only two CO2 concentration sensors?
• RQ3 How can time-varying model parameters such as air flow and diffusion be estimated online in real time?
• RQ4 What is the impact of filtering in the human CO2 emission response on the accuracy of the model and observer design?
• RQ5 Does the proposed PDE-ODE framework with observer and identifier achieve asymptotic convergence in estimation under realistic conditions?

Key findings

• The PDE-ODE model accurately captures the delayed response of CO2 concentration to changes in human CO2 emission, validating the need for a filtered input representation.
• The observer design achieves asymptotic convergence of the estimated human CO2 emission rate to the true value, as proven via Lyapunov analysis and an alternative to Barbalat’s lemma.
• Online parameter identifiers successfully estimate time-varying parameters such as air flow and diffusion coefficients, with the estimation error converging to zero over time.
• The simulation studies demonstrate robust performance under varying occupancy and CO2 input conditions, confirming the model’s predictive accuracy.
• The observer and identifiers operate with minimal sensor count (two CO2 sensors), reducing cost and increasing reliability for real-world deployment.
• Theoretical analysis confirms that the estimation error and parameter error converge to zero, with boundedness of all signals ensured under the proposed design.