introduction
PID control is currently the most widely used control technology. It is a simple control algorithm with a long history of application and familiarity in industry. Since Hagglund proposed the idea of a predictive PI controller in 1992 (Hagglund, 1992), the predictive PID algorithm has been gradually developed and improved, and has been successfully applied to the control of some complex objects. Control theory has attracted increasing attention due to its enormous economic benefits, and the application of numerous advanced control algorithms in complex industrial processes has narrowed the gap between theory and practice.
This controller combines predictive algorithms and PID control. PID controllers are independent of process lag time, while predictive control relies primarily on process lag time, using previous control actions to determine the current control action. This particular PID control algorithm organically combines the simplicity, practicality, and robustness of PID with the predictive capabilities of model predictive control algorithms.
This paper uses the Toeplitz equation to solve the Toeplitz equation, which reduces the computational burden of predictive control, shortens the online optimization time of the predictive controller, solves the control problem caused by system time delay, tunes the PID control parameters, and achieves the expected results.
Problem Statement
In recent decades, control theory has attracted increasing attention due to its significant economic benefits. Numerous advanced control algorithms have been applied to complex industrial processes, bridging the gap between theory and practice. On the other hand, traditional PID controllers, due to their simplicity, stability, and ease of operation, still hold a considerable share of the control market. Therefore, in today's increasingly competitive global market, improving traditional controllers and optimizing traditional control methods through advanced control technologies to achieve economic benefits and enhance corporate competitiveness has become a trend.
However, complex industrial processes are characterized by difficulties in modeling, complex correlations, time-varying object structures and parameters, uncertainties in disturbances and environments, and diverse requirements and constraints. Traditional optimal control, based on precise mathematical models of the object, is not applicable in industrial environments, as has been proven in industrial practice. Optimization-based control is clearly superior to simple regulation. This raises the question: how to appropriately integrate optimization into dynamic control to form an optimal control mode adapted to complex industrial processes? Predictive control fulfills this requirement.
This research combines generalized predictive control (GPC) and classical PID control methods to solve the control challenges of systems with large time delays using predictive optimization principles. By rapidly solving the Diophantine equation, the cumbersome recursive solution process of the Diophantine equation in traditional GPC algorithms is avoided.
GPC based on the Toeplitz method
2.1 Basic Expression of GPC
First, the performance metric J function is expressed as follows:
(1)
Where e(i) is the error between the object output and the reference smooth curve, i.e., N is the prediction time domain, M is the control time domain, and is the control weighting constant.
The above equations can be written in vector form:
(2)
Where is the predicted output error vector, Y is the future output vector, and is the future control gain vector.
2.2 Introduction to the Toeplitz Equation
Given a single-input, single-output controlled object transfer function model:
(3)
Where, and are the polynomials of the difference shift operator:
(4)(5)
Introducing a gain model:
(6)
in,
Introducing convolution matrices and Hankel matrices,
in,
Therefore, according to the definition of sum, the expression can be rewritten as:
(7)
Similarly, the right side of the equation can also be transformed to obtain:
PID parameter design
3.1 Description of the Generalized Prediction Model
Generalized predictive control is described by the following discrete difference equation, also known as the CARIMA model:
(12)
Use the following Diophantine equation
3.2 Combination of PID and GPC
The specific algorithm of PID control is as follows: It constructs the control deviation based on the given value r(t) and the actual output value y(t), and then uses the proportional (P), integral (I), and derivative (D) of the deviation to form the control quantity through a linear combination, thereby controlling the controlled object, as shown in the following formula:
(twenty three)
Experimental simulation and result analysis
Select a simulation model, as follows:
Using the same parameters, the simulation results of the traditional PID algorithm and the improved GPC-PID algorithm are shown in the figure below. The red curve represents the traditional PID algorithm, and the blue curve represents the improved GPC-PID algorithm.
Figure 1 - Control Output
As can be seen from Figure 1, the GPC-PID prediction algorithm used in this paper is smoother than the traditional PID controller. The new prediction algorithm takes longer to reach stability than the traditional algorithm. The Toeplitz matrix effectively demonstrates this characteristic, saving online calculation time, while the traditional algorithm does not have this advantage.
method | Online time calculation |
Traditional PID algorithm | 0.11068s |
This article's method | 0.05749s |
Table 1 - Comparison of Calculation Time
This table shows that the improved GPC-PID algorithm takes less time and exhibits significantly reduced output fluctuations. The improved algorithm's shorter online computation time effectively reduces the complexity of solving the Dioppanto equations online, thus alleviating the system's burden. The final curve is also smoother, achieving the expected results.
Conclusion
PID control is currently the most widely used control technology. This project, while ensuring the performance of classic PID control and leveraging its simplicity and practicality, tunes the PID control parameters based on the rolling optimization principle. The proposed method avoids the weakness of existing predictive PID control methods that require recursive solving of the Diophantine equation, thus improving the running speed of the predictive PID algorithm and broadening its engineering application scope.
Author's personal information:
Ren Junru: School of Information Science and Engineering, Wuhan University of Science and Technology, specializing in control theory and control engineering, with a focus on predictive control.
P.O. Box 161, School of Information Science and Technology, Wuhan University of Science and Technology, No. 947 Heping Avenue, Qingshan District, Wuhan, 430081, China. Tel: 13207148738, Email: [email protected]
