Calibration of the Parameters of a Model of an Engineering System Using the Global Optimization Method

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1. Introduction

Many engineering system models rely on several unknown parameters. Their values need to be identified by direct or indirect measurements. However, in many cases, this is not possible (Wang, 1997). In the case of hydro-electromechanical systems treated here, some of the unknown parameters vary with time and the system consumption (efficiency, size of the cylinder, etc…). Other parameters, given in the technical documentation of a component, represent the values ​​when the component works separately. Therefore, they do not reflect the actual operating conditions of the system, the interactions between the components and their influence during operation. For these reasons, the identification of the system’s real values becomes a necessity, hence the importance of calibrating the studied system. The calibration is defined as the identification of the model's unknown parameters and/or structure based on measurements, and prior knowledge (Dai & al., 2009). This technique allows us to estimate the model parameters that cannot be measured. This is achieved by comparing the model results with measurements and adjusting the parameters of the model to obtain a match between the model's predictions and the experimental results (Dai & al., 2009). The purpose of the calibration is ultimately to find the values of the unknown parameters, which minimize the deviation between the simulated and experimental values.

These unknown parameters are calibrated by solving a constrained optimization problem for which an OBJ is to be minimized or maximized subject to the constraints. In our case, the OBJ is defined as the sum of the squared differences between the simulated and experimental outputs. This function should be minimized so that the simulation outputs converge to the experimental ones. The OBJ can be expressed as follows (1):OBJ = * ) (1) where s is the total number of evaluated simulation outputs, n is the total count of simulation steps, is the simulated value of the k^th measured simulation physical output at the i^th simulation step and is the experimental value of the k^th measured physical output at the i^th simulation step.

This OBJ is subject to two constraints:

• 1.

  Inequality constraint: the calibrated parameters are bounded as follows (2):

  
  
  O_{i min} ≤ O_i ≤ O_{i max}(2)

where O_i is the i^th calibrated parameter, O_{i min} and O_{i max} are the lower and upper bounds respectively of O_i.

• 2.

  Equality constraint: the variation of outputs depends on the values of the calibrated parameters and the time such as (3):

  
  (3)

where X_k is the k^th physical output, O_i is the i^th calibrated parameter and t is the time.