# 9

# Process Identification Methods for Continuous-Time Differential Equation Models

Two groups of process identification methods to obtain the process model in the form of continuous-time differential equation are introduced in this chapter. The first group estimates the model parameters using the least-squares method after it converts the continuous-time differential equation to the algebraic equation using the time-weighted integral transform. The second group estimates the model parameters by solving the nonlinear multivariable optimization problem of which the objective function is the norm of the modeling error. Numerical examples and MATLAB codes for each process identification method are provided.

## 9.1 Identification Methods Using Integral Transforms

The identification method using the integral transform provides the following continuous-time time-invariant linear model:

where the transfer function of *G*(*s*) is strictly proper; that is, *m* < *n*. *u*(*s*) and *y*(*s*) denote the Laplace transforms of the process input (i.e. controller output) and the process output respectively. Two identification methods are introduced in this section. The first method can be applied to the case that the process is initially in a steady state. The second method can incorporate the case of the initially unsteady state.

### 9.1.1 Identification Method for the Case of an Initially Steady State

The identification ...