## 3 Parameter Identification of the System
### Identification of Delay Time  
Parameter identification and control of industrial processes with time delay are always the main problems in the control field. In recent years, some scholars have studied the parameter identification of time delay processes extensively and deeply. For identification, the estimation of delay time is very important. Whether the delay time can be accurately estimated is related to the identification of other parameters of the model. Correlation analysis is a widely used identification method, which has the advantages of strong anti-interference ability and high identification accuracy. In the actual production process, there are almost no exceptions to the interference of random noise. Therefore, the time delay of the system was estimated according to the correlation of two signals.  
The discrete time cross-correlation function of two signals \(x(t)\) and \(y(t)\) can be expressed as  
\[R_{xy}(\tau)=\frac{1}{N}\sum\limits_{k=1}^{N}x(k)y(k+\tau) \tag{21}\]  
where \(k=1,2,\cdots N\), \(\tau=1,2,\cdots n<<N\).  
According to the definition of cross-correlation function, the time delay of two signals can be measured by detecting the peak value of the correlation function, and the maximum value of \(R_{xy}\) can be obtained. Then, \(\tau\) corresponds to the maximum value of \(R_{xy}\) is the estimated value of the time delay of the identified object. It can be expressed as  
\[R_{xy}(\overset{\wedge}{d})=Max\big{\{}R_{xy}(\tau)\big{\}} \tag{22}\]  
For many practical industrial objects, the input and output signals are not only simple pure lag relations, but they are also generally accompanied by inertial links. If the time delay of the object is obtained directly by the cross-correlation function, the error will be large, so the input and output signals should be processed to increase the correlation and improve the identification accuracy.  
In general, the relationship between input signal and output signal of the subject can be expressed as  
\[y(k)=\frac{B(z^{-1})}{A(z^{-1})}u(k)z^{-d} \tag{23}\]  
To increase the correlation between input and output signals, the following processing needs to be performed.  
\[u_{f}(k)=\frac{\overset{\wedge}{B}(z^{-1})}{\overset{\wedge}{A}(z^{-1})}u(k) \tag{24}\]However, \(y(k)\) satisfies the following formula.  
\[y(k)=y_{f}(k)=\frac{B(z^{-1})}{A(z^{-1})}u(k)z^{-d} \tag{25}\]  
So, it is easy to draw a conclusion that, if \(\frac{\hat{\frac{\hat{B}}{B}(z^{-1})}}{\hat{A}(z^{-1})}=\frac{B(z^{-1})}{A(z^{-1 })}\), then \(y_{f}(k)=u_{f}(k-d)+n(k)\), where \(n(k)\) is the system noise. Obviously, the relationship between the signal \(u_{f}(k)\) and signal \(y_{f}(k)\) after proper filtering is pure delay and its correlation will be greatly increased. Therefore, the actual delay time of the identified object can be determined according to the cross-correlation function of \(u_{f}(k)\) and \(y_{f}(k)\).  
If the delay time is directly estimated by using the cross-correlation function, the amount of the calculation will be large and time-consuming, due to the excessive times of additions and multiplications. Therefore, in this paper, a fast and efficient discrete Fourier transform and its inverse transform are used to calculate the cross-correlation function by references [16] to improve the computational efficiency of the algorithm. The calculation steps are as follows.  
1. Calculate the length \(n\) of output sequence \(y(k)\);
2. Calculate the discrete Fourier transform of \(u(k)\) and \(y(k)\) by using fast Fourier transform (FFT);
3. Calculate \(R_{xy}(k)=u(k)\times y(k)\);
3. Calculate \(R_{xy}(k)\) by using inverse fast Fourier transform (IFFT), finally \(R_{xy}(n)\) can be obtained.