﻿ 基于理想Bode传递函数的分数阶PID频域设计方法及其应用
 控制与决策  2019, Vol. 34 Issue (10): 2198-2202 0

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NIE Zhuo-yun, ZHU Hai-yan, LIU Jian-cong, LIU Rui-juan, ZHENG Yi-min. Fractional order PID controller design in frequency domain based on ideal Bode transfer function and its application[J]. Control and Decision, 2019, 34(10): 2198-2202. DOI: 10.13195/j.kzyjc.2018.0186.
[复制英文]

### 文章历史

1. 华侨大学 信息科学与工程学院，福建 厦门 361021;
2. 厦门理工学院 应用数学学院，福建 厦门 361024

Fractional order PID controller design in frequency domain based on ideal Bode transfer function and its application
NIE Zhuo-yun 1, ZHU Hai-yan 1, LIU Jian-cong 1, LIU Rui-juan 2, ZHENG Yi-min 1
1. School of Information Science and Engineering, National Huaqiao University, Xiamen 361021, China;
2. School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
Abstract: A simple fractional-order PID controller design method based on the ideal Bode transfer function is proposed in this paper. Model matching and identification are introduced, such that the design problem of the fractional-order PID controller, with five variables involved, is transformed into a simple one dimensional search problem. A short-memory method is used for the implementation of the fractional-order PID controller. The proposed design method is applied to the speed tuning of a DC motor with good performance and robustness. Experimental results are provided to illustrate the effectiveness and availability of the proposed method.
Keywords: fractional-order PID    ideal Bode transfer function    model matching and identification    DC motor
0 引言

1 分数阶微积分与分数阶PID控制

 (1)

 (2)

 (3)

 图 1 分数阶PID单位反馈控制系统
2 控制器设计 2.1 理想Bode传递函数

 (4)

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 (6)
 (7)

2.2 模型匹配与辨识

 (8)

1) Gp(s)稳定, 且存在非零稳态值Gp(j0);

2) Gp(s)无时滞, 使得Gp(s)Gc(s)与式(4)无时滞保持一致.

Step 1: 零频ω=0模型匹配.当且仅当λα时, 存在非零稳态.故选取λ=α, 式(8)表示为

 (9)

 (10)

Step 2: ω=ωx模型匹配.考虑理想对象ω=ωx的频率响应, 有

 (11)

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 (13)

 (14)

Step 3: 在(0, ωx)范围内的模型优化匹配.设定频率步长Δω, 在(0, ωx)范围内取Gp(jω)的频率响应数据; 针对每个迭代的μ值, 取式(9)中(jω)的频率响应数据, 其中α=λ, ki满足式(10), kd(μ)和kp(μ)满足式(14)计算误差平方和, 建立频域响应误差指标

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3 仿真实例

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 图 2 阶跃响应及误差曲线

 图 3 Bode图
4 运动控制平台 4.1 实验平台

 图 4 电机调速控制实验平台

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4.2 控制器实现

G-L定义的分数阶微积分, 其本质在于对历史输入信号进行加权求和, 可用于FOPID控制器的数字实现.控制器的时域输出表达式为

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4.3 实验结果

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 图 5 调速控制效果

5 结论

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