# The Online Identification of the Behaviour of Pollutants inside the Tunnel Tube
## 2 Parametric Identification of the Behaviour of Pollutants in Individual Parts of the Tunnel Tube  
Methods for parameter estimation allow determining the parameters of a model for a chosen structure from a sequence of measured input and output values. Identification methods assume the discrete measurement of signals, resulting in corresponding numerical sequences of input \(u(k)\) and output \(y(k)\) values. This article focuses on the analysis and identification of the movement of harmful substances in the eastern tunnel section of the Mrazovka road tunnel located in Prague. Data analysis was carried out using modelling in the Matlab software environment (version 2022b, Mathworks, Natick, MA, USA). Offline stochastic models were created, and simulations and predictions of harmful substance concentrations in the tunnel were performed to select an appropriate parametric model structure. Data sources include personal and freight vehicles' traffic intensity, speeds, concentrations of CO and NOx, opacity (OP), airflow, and air speed. These are variables measured directly inside the tunnel via sensors. The system is described through relationships between input and output processes. The chosen procedure for estimating the dynamic system model involves several steps: determining the model's structure, parameter estimation, and model verification. The program utilizes a database from measured traffic intensity, vehicle speeds, tunnel air speeds, harmful substance concentrations, and opacity inside the tunnel. The processed data were recorded over several days in the Mrazovka tunnel, with measurements taken twice a minute. The two-day database contains 5760 samples for each variable, respectively. Sensing principles are discussed in [12; 13; 14; 15; 16; 17]. Various methods can be used to compute model parameters. Iterative and recursive least squares algorithms form the basis of parameter estimation [18]. Linear regression models describe the dynamical systems. There are many derived identification algorithms for pseudo-linear regression models, such as the Recursive Extended Least Squares algorithm (RELS) for models with Controlled Autoregressive Moving Average (ARMA) [19; 20], the Least Squares Filtering Algorithm for Controlled ARMA (CARARMA) systems [21], Recursive Generalized Least Squares Parameter Estimation [22], Adaptive Forgetting Factor Recursive Least Square Algorithm [23], and others. The term parametric identification refers to creating a model describing the dynamic behaviour of pollutants in a tunnel. The term tunnel section means a tunnel section with defined input and output variables. The system identification procedure is depicted in Figure 1.  
The selection of an appropriate model structure is the most crucial part. This selection must be based on understanding the identification method and the information about the identified system. Once the model structure has been determined, the chosen identification method will provide us with the precise form of this structure and the exact procedure for determining its parameters. A fundamental question is whether it is good enough for the intended use. Testing the suitability of the model is referred to as model validation. System identification is an integral part of every control design and deals with the problem of creating reliable mathematical models of dynamic processes based on observed input-output data (Figure 2). The input variable is denoted as \(u(t)\), the output variable as \(y(t)\), the output predictions as \(\hat{y}(t)\), and the estimated parameters as \(\hat{\theta}(t)\). The results of the identification determine the achieved quality of control. Figure 1: System identification procedure. General aspects of model selection include the following:
* Flexibility: Using a model structure that provides good properties for describing the systems. It can be achieved using more parameters or their \"strategic\" placement.