Research on Multivariable System Identification and PID Decoupling Control

With the development of modern industry, more and more industrial systems, social and economic systems are no longer limited to single-variable systems, but multivariable systems with complex structures and uncertain models. Although traditional control methods can meet the control requirements of industrial systems to a large extent, they cannot achieve satisfactory control effects on some industrial control systems with characteristics such as strong coupling, uncertainty, nonlinearity, incomplete information and large lag. Therefore, the research on multivariable system control has received more and more attention. To control a multivariable system, especially to implement some advanced control algorithms such as predictive control and internal model control, they are all based on models, so the system model is a prerequisite for implementing multivariable control.

1 Model Identification Method

Figure 1 is a typical two-variable control system block diagram. As can be seen from Figure 1, the identification of the model is to identify the four transfer functions G11(s), G21(s), G12(s), and G22(s). Here, the frequency domain-based step response method is used for identification. For the PID control system, the transfer function G(s) between the controller output u and the process output y is expressed as:

After discretization, it is replaced by jω instead of s to become

For the process frequency response, taking the range of ωi as [-π, 0] can fully reflect the frequency characteristics of the system. In order to obtain more accurate results, π is divided into M intervals. Calculate the ωi value

Its corresponding phase angle

Since most systems in the control process can be replaced by a second-order plus lag model, the model transfer function is set to Thus
, the transfer function model parameters are obtained.

For the typical system in Figure 1, when the system is stable, disconnect u1 and u2, add a step signal to u2, record the values ​​of y1 and y2, and then substitute them into equations (2) to (12) to identify G11(s) and G21(s). Similarly, add a step signal to u2, set u1=0, identify G12(s) and G22(s), and then the transfer function matrix of the system is obtained.


2 Lag link approximation
Since the obtained model contains a lag link, and the lag link cannot be directly decoupled, various approximation methods are compared. The usual approximation methods are: first-order Pade approximation, second-order symmetric Pade approximation, and second-order asymmetric Pade approximation. The literature has conducted experiments on it many times and found that the first-order Pade approximation has fluctuations at the initial moment, but the approximation effect is better when the lag is large. This is because the Pade approximation introduces zero points. The second-order symmetric Pade approximation has the worst effect, and the second-order symmetric Pade approximation not only has fluctuations at the initial moment but also produces overshoot. The second-order asymmetric Pade approximation has a short adjustment time and no obvious overshoot, but the fluctuation is large. Therefore, the shift processing and second-order Taylor series expansion, namely the full-pole approximation method, are used.

Through simulation verification, it is found that the full-pole approximation method has the smallest error because it avoids introducing zero points. It has a shorter adjustment time than the Pade approximation and has no overshoot, that is, it can better obtain the step response characteristics.

3 Decoupling control
The internal structure of the multi-input and multi-output system is complex, and there is a certain degree of coupling. For such a coupled object, industrial process control requires the system to be able to operate safely and stably, and have good adjustment performance, and can track the changes of the set value with a small error, and make the steady-state error zero. In order to achieve high-quality control performance, decoupling design must be carried out. How to remove the coupling between them and turn them into an independent single-variable system for control is an important method to solve multivariable control. The process of removing coupling is decoupling. Among them, the commonly used decoupling methods are diagonal matrix method, inverse Nyquist curve method and characteristic curve method. Among them, the diagonal matrix method plays a great role in the field of process control.

Formula (15) is a multivariable system transfer function matrix. Diagonal matrix decoupling is to transform the transfer function matrix of the coupled object into a diagonal matrix, as shown in Formula (16). Except for the elements on the main diagonal, all other elements are zero. In this way, the input U(s) and the output Y(s) become a one-to-one correspondence, so as to achieve the purpose of easy control.

Assume that in order to transform the transfer function matrix into a diagonal matrix, an n×n matrix D(s) is added to the output end of U(s).

Since the probability that the model identified by the method mentioned above is a singular matrix is ​​very small, taking a two-input and two-output system as an example, assuming that G(s) is a non-singular square matrix, there is an inverse matrix. The block diagram of the decoupling control system for the PID controller is shown in Figure 2.

After adding the decoupling controller, the system is transformed into

For equation (20), it can be controlled by a single variable control method.

4 Decoupling system simulation
The decoupling control is simulated and verified using MATLAB software. Assume that the transfer function matrix is

First, the full-pole approximation is used to transform it into a linear system. After the full-pole approximation is used, equation (21) becomes

Taking G11 and G22 in equation (21) as examples, the full-pole approximation is performed, and the simulation results are shown in Figure 3.

From the above simulation results, it can be seen that the output curves before and after approximation are basically the same, indicating that the full-pole approximation method can well reflect the performance of the original system.
In order to obtain the decoupling matrix, the inverse of equation (22) is taken and multiplied with to obtain D(s)

to obtain the decoupling matrix. The coupling degree between the systems after decoupling is analyzed by simulation. After adding step signals from the input ends u1 and u2, the system output curves before and after decoupling are shown in Figure 4. It can be seen from the figure that the coupling degree between the two loops is greatly reduced after decoupling, which effectively reduces the interference between the loops and greatly improves the performance of the control system.

5 Conclusions
Theoretical and simulation experiments have proved that the control method using frequency domain identification and diagonal matrix decoupling has achieved good results, providing a guarantee for the long-term stable operation of the system.