基于多主导极点法的时滞积分系统PID控制器设计

doi:10.16180/j.cnki.issn1007-7820.2023.08.009

摘要/Abstract

摘要：

伺服与扰动抑制是时滞积分系统最基本的控制问题,对其进行控制难度较大。文中提出一种基于直接综合法和多主导极点配置法的微分先行PID(Proportional -Integral-Derivative)整定方法,这种方法通过比较串联滤波器与时滞积分被控对象组成的特征方程与实际期望的特征方程的系数,将三阶主导极点置于-1/λ处,并将二阶非主导极点置于-5/λ处(λ为调整参数),从而获得期望的特征方程。以实现期望的鲁棒性方式获得设计的控制器参数,通过选择不同的调优参数获取相应的Ms(Maximum sensitivity)值,在参数具有标称性的限定条件下拟合出关于Ms和调优参数的关系曲线,给出整定规则的解析形式。PIPTD(Pure Integral Plus Time Delay system)、DIPTD(Double Integral Plus Time Delay system)和FOPTDI(First-Order Plus Time Delay Integral System)系统的仿真结果表明,IAE(Integral Absolute Error)指标平均可降低35.79%,TV(Total Variation)指标平均可降低18.97%。

关键词: 直接综合法, 微分先行PID, Ms, 阶跃响应, 方波响应, 积分系统, 时滞, 多主导极点配置法

Abstract:

As the most basic control problem of the time delay integral system, it is difficult to control servo and disturbance rejection. In this study, a differential forward PID tuning method based on direct synthesis method and multiple dominant pole-placement method is proposed. This method mainly compares the coefficients of the characteristic equation composed of series filter and the time-delay integral controlled process with the actual desired characteristic equation. The third-order dominant pole is placed at -1/λ, and the second-order non-dominant pole is placed at -5/λ (λ is the adjustment parameter), so as to obtain the desired characteristic equation. Parameters of the controller are tuned to achieve the desired robustness. The corresponding Ms values can be obtained by selecting different tuning parameters, and the relationship curves between Ms and the tuning parameters can be fitted under the limited condition that the parameters are nominal, and the analytical form of the tuning rules can be given. The simulations of PIPTD, DIPTD and FOPTDI systems show that the IAE index can be reduced by average 35.79% and the TV index can be reduced by average 18.97%.

Key words: direct synthesis method, differential forward PID, Ms, step response, square wave response, integral system, time delay, multiple dominant pole-placement method

中图分类号:

TP273

丁祥鑫,张伟,王亚刚. 基于多主导极点法的时滞积分系统PID控制器设计[J]. 电子科技, 2023, 36(8): 56-64.

DING Xiangxin,ZHANG Wei,WANG Yagang. PID Controller Tuning of Time Delay Integral System Based on Multi Dominant Pole Method[J]. Electronic Science and Technology, 2023, 36(8): 56-64.

图/表 24

图1

图2

图3

表1

PIPTD系统下的PID整定规则"

被控对象	PID控制器	PID参数及滤波器参数
$k p e - L s s$	G_c=k_c(1+1/τ_is) G_d=1+τ_ds 1.5≤Ms≤3.0	$λ L$ =9.398Ms^-3.677+0.973 9
		$τ i L$ =2.136 $(λ L) 1.072$ +0.548 8
		$τ d L$ =-0.001 739 $(λ L) 4.012$ +0.158
		k_pk_cL²=-0.495 5+1.497 $(λ L) - 0.467 4$

表1

表2

DIPTD系统下的PID整定规则"

被控对象	PID控制器	PID参数及滤波器参数
$k p e - L s s$	$G c = k c (1 + 1 τ i s) (α s + 1 β s + 1) G d = 1 + τ d s$ 2.0≤Ms≤4.0	$λ L$ =10.86Ms^-2.359+0.988 9
		$τ i L$ =2.179 $(λ L) 0.963 1$ -0.107 3
		$τ d L = 0.55 (λ L) - 0.082 22 - 0.000 556 2 (λ L > 2.5) τ d L = 1.237 e - 6 (λ L) 8.458 + 0.505 5 (λ L < 2.5)$
		k_pk_cL²=0.662 3 $(λ L) - 1.566$ -0.022 94
		$α L$ =2.169 $(λ L) 0.965 7$ -0.096 38
		$β L$ =0.029 44 $(λ L) 2.124$ -0.003 975

表2

表3

FOPTDI系统下的PID整定规则"

被控对象	PID控制器	PID参数及滤波器参数
$k p e - L s s (τ s + 1)$	G_c=k_c(1+1/τ_is) G_d=1+τ_ds F(s)= $(α s + 1) (β s + 1)$	$λ τ$ =-0.000 820 8 $(L τ) - 5.154$ +1.29
		k_pk_cτ=1.007 $(λ τ) - 1.686$ +0.151 8
		$τ i τ$ =1.891 $(λ τ) 0.878 6$ -0.118 9
		$τ d τ$ =2.156 $(λ τ) 0.760 2$ -0.382 1 $L τ$ <0.3
		$τ d τ$ =0.026 57 $(λ τ) - 4.672$ 0.30≤ $L τ$ ≤0.305
		$τ d τ$ =0.118 4 $(λ τ) 10.47$ +1.626 $L τ$ >0.305
		$α τ$ =0.273 7-4.058 $(L τ) 2.392 L τ$ <0.3
		$α τ$ =1.759-2.854×10¹⁵ $(L τ) - 26.22$ 0.30≤ $L τ$ ≤0.305
		$α τ$ =4.968×10^-9 $(L τ) - 13.47$ $L τ$ >0.305
		$β τ$ =0.249 1 $(λ τ) 1.19$ -0.087 1

表3

表4

PID参数和滤波器参数(PIPTD)"

被控对象	方法	τ_i	τ_d	k_c	Ms
G_p= $0.05 e - 5 s s$	本文	16.876 5	2.124 4	3.296 0	2.0
	CH	18.940 0	1.890 0	3.727 0	2.0
	AA	-	2.378 0	2.993 3	2.0
	AA	20.575 0	1.770 0	3.320 9	2.0
	VV	18.660 0	1.890 0	3.727 0	1.9
	ROP	20.200 0	1.800 0	3.408 0	2.0

表4

图4

表5

阶跃响应的IAE和TV性能指标的比较(PIPTD)"

被控对象	方法	追踪			扰动抑制
被控对象	方法	IAE	TV	OS	IAE	TV	OS
G_P= $0.05 e - 5 s s$	本文	10.13	5.007	0.000	5.68	2.25	0.296
	CH	11.40	3.950	0.001	5.08	2.18	0.304
	AA	7.154	39.18	0.075	6.19	2.77	0.323
	VV	11.54	3.864	0.028	5.94	1.95	0.343
	ROP	12.15	9.106	0.001	1.23	1.96	0.316

表5

图5

图6

表6

方波响应下根据IAE指标和TV指标对比图(PIPTD)"

被控对象	方法	IAE	TV
G_p= $0.05 e - 5 s s$	本文	20.265 0	9.912 200
	CH	22.788 5	3.098×10¹³
	AA	7.154 40	136.205 4
	VV	11.545 9	1.555×10¹³
	ROP	24.290 2	2.697 9×10¹³

表6

表7

PID参数和滤波器参数(DIPTD)"

被控对象	方法	τ_i	τ_d	k_c	α	β	Ms
G_p= $e - s s 2$	本文	6.180	0.52	0.09	6.18	0.30	2.0
	CH	9.726	3.82	0.14	1.08	1.04	2.0
	JL	21.280	7.25	0.05	-	-	2.0
	JL	-	10.70	0.03	-	-	2.0
	AA	22.380	7.84	0.04	-	-	2.0

表7

图7

表8

阶跃响应的IAE和TV性能指标的比较(DIPTD)"

被控对象	方法	追踪			扰动抑制
被控对象	方法	IAE	TV	OS	IAE	TV	OS
G_p= $e - s s 2$	本文	3.708	1.111	0.00	63.33	2.2	5.259
	CH	5.989	0.149	0.01	70.61	2.3	6.127
	AA	5.466	3.284	0.22	541.10	4.0	20.660
	JL	7.768	0.212	0.00	461.90	2.1	18.650

表8

图8

图9

表9

方波响应下根据IAE指标和TV指标对比图(DIPTD)"

被控对象	方法	IAE	TV
G_p= $e - s s 2$	本文	7.388 10	3.776 6
	CH	11.901 0	2.398 0×10¹³
	AA	10.924 5	12.166 5
	JL	15.440 4	0.727 1

表9

图10

表10

PID参数和滤波器参数(FOPTDI)"

被控对象	方法	τ_i	τ_d	k_c	α	β	Ms
G_p= $0.2 e - s s (4 s + 1)$	本文	3.879	0.50	3.41	3.87	0.18	2.0
	CH	5.904	1.95	5.74	0.63	0.49	2.0
	AA	10.420	2.47	3.67	-	-	2.0
	JL	10.390	2.47	3.68	-	-	1.9
	ROP	8.263	2.16	5.15	-	-	2.0

表10

表11

阶跃响应的IAE和TV性能指标的比较(FOPTDI)"

被控对象	方法	追踪			扰动抑制
被控对象	方法	IAE	TV	OS	IAE	TV	OS
G_p= $0.2 e - s s (4 s + 1)$	本文	2.080	53.200	0.017	1.13	2.99	0.137
	CH	4.506	9.350	0.089	1.09	2.31	0.174
	AA	3.130	42.130	0.256	2.84	1.83	0.259
	JL	4.133	19.420	0.000	2.82	1.84	0.259
	ROP	2.745	5.175	0.002	1.48	2.10	0.196

表11

图11

表12

方波响应下根据IAE指标和TV指标对比"

被控对象	方法	IAE	TV
G_p= $0.2 e - s s (4 s + 1)$	本文	135.403 8	4.715 7
	CH	1.331 5×10³	8.993 7
	AA	144.413 3	6.260 4
	JL	66.002 5	8.266 6
	ROP	1.026 8×10³	9.982 5

表12

图12

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