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

Abstract

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

CLC Number:

TP273

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.

Figures/Tables 24

Figure 1.

Figure 2.

Figure 3.

Table 1.

PID tuning rules for PIPTD"

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 $(λ L) - 0.467 4$

Table 1.

Table 2.

PID tuning rules for DIPTD"

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

Table 2.

Table 3.

PID tuning rules for FOPTDI"

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

Table 3.

Table 4.

PID settings and filter parameters (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

Table 4.

Figure 4.

Table 5.

Performance comparison in terms of IAE and TV for step response (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

Table 5.

Figure 5.

Figure 6.

Table 6.

Performance comparison in terms of IAE and TV for square wave response (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

Table 6.

Table 7.

PID settings and filter parameters (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

Table 7.

Figure 7.

Table 8.

Performance comparison in terms of IAE and TV for step response (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

Table 8.

Figure 8.

Figure 9.

Table 9.

Performance comparison in terms of IAE and TV for square wave response (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

Table 9.

Figure 10.

Table 10.

PID settings and filter parameters (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

Table 10.

Table 11.

Performance comparison in terms of IAE and TV for step response (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

Table 11.

Figure 11.

Table 12.

Performance comparison in terms of IAE and TV for square wave responses"

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

Table 12.

Figure 12.

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PID Controller Tuning of Time Delay Integral System Based on Multi Dominant Pole Method

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