Information about the strength of gas sources in buildings has a number of applications in the area of building automation and control, including temperature and ventilation control, fire detection, and security systems. In this paper, we consider the problem of estimating the strength of a gas source in an enclosure when some of the parameters of the gas transport process are unknown. Traditionally, these problems are either solved by the maximum-likelihood method, which is accurate but computationally intensive, or by recursive least squares (also Kalman) filtering, which is simpler but less accurate. In this paper, we suggest a different statistical estimation procedure based on the concept of method of moments. We outline techniques that make this procedure computationally efficient and amenable for recursive implementation. We provide a comparative analysis of our proposed method based on experimental results, as well as Monte Carlo simulations. When used with the building control systems, these algorithms can estimate the gaseous strength in a room both quickly and accurately and can potentially provide improved indoor air quality in an efficient manner.
The Kalman Filter based approach detailed in @cite_11 simplifies to a Recursive Least Squares (RLS) algorithm (e.g., @cite_2 ) with the assumption of constant parameters. Based on the differential model for signal in , the RLS model is predicated on the difference equation where @math represent the additive noise (observation error) and @math represents the measurements of the room concentration. Equation above represents a set of @math equations for the @math measurements for @math . Denoting @math as the equations can be represented as @math , where @math and @math is the matrix containing the signal terms '. From the theory of Generalized Least Squares (e.g., @cite_2 ), we know that the LS estimate and the expected variance are given by where @math . We are interested in the variance of @math , which is given by The matrices @math and @math need to be described further. Note that the time correlation of @math results in this @math tridiagonal matrix . The inverse of this tridiagonal matrix can be written in closed form. Bearing in mind that @math is symmetric, here we write only the upper diagonal terms
@cite_2 @cite_11
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Minimum variance unbiased estimation Cramer-Rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the Bayesian philosophy general Bayesian estimators linear Bayesian estimators Kalman filters summary of estimators extension for complex data and parameters. Information about the strength of gas sources in buildings has a number of applications in the area of building automation and control, including temperature and ventilation control, fire detection, and security systems. In this paper, a method for estimating the strength of a gas source in an enclosure when some of the parameters of the gas transport process are unknown is described. It is based on a perfect-mixing model of the gas species transport dynamics. The estimation problem is formulated as a Kalman filtering problem, where the states estimated by the Kalman filter are the unknown process parameters and the source strength. Sudden changes in the strength of the source are detected and tracked with a hypothesis testing and covariance resetting algorithm that is based on statistics provided by the Kalman filter. Experimental results from two first-order systems demonstrate the efficacy of this method.