7 9 1 9 : 1 815(1)9 14 5 Control Theory & Applications Vol. 7 No. 9 Sep. 1 PI,, (, 5164) : PI,,.,, (SOPDT).,, PI ;,, : ; PI ; ; : TP73 : A A new practical method for closed-loop identification with PI control ZHU Xue-feng, XIAO Shu-jun, WANG Xiu (College of Automation Science and Technology, South China University of Technology, Guangzhou Guangdong 5164, China) Abstract: For a closed-loop system with a PI controller, we propose a method to identify the transfer function of the controlled object. From the under-damped-response of the closed-loop system, we approximate the closed-loop system by a second-order-plus-dead-time(sopdt) transfer function. The transfer function of the PI controller is then removed from the approximate closed-loop transfer function. From the remaining transfer function, the open-loop transfer function of the controlled object is determined through a coefficient-comparison process. Both the simulations and the control experiments on a pilot-equipment show advantages of the proposed method, such as the high identification-accuracy, the low computational complexity and the easy online implementation, thus demonstrating its application potential. Key words: closed-loop identification; proportional-integral control; second-order-plus-dead-time; coefficientcomparison 1 (Introduction),..,, [1].,., R. Manat P.J. Fleming [], (FOPDT) FOPDT. [3,4] R. Manat P.J. Fleming PI PID (SOPDT). Wonhui Cho [5] PI, FOPDT. [6] N (Laguerre) [7,8],,., : 1,. 1 [ 4, 9],. (Identification principle) : Tp s + ζ p T p s + 1. (1) : 9 7 15, : 9 1. : 8 (8B134).
9 : PI 141 PI G c (s) = K c(t i s + 1), () T i s M(s) = G p(s)g c (s) 1 + G p (s)g c (s). (3) PI, 1. K cl e d cls M(s) = Tcl s + ζ cl T cl s + 1. (4) Fig. 1 1 PI Typical closed-loop step response under PI control [1],. [1,4,7] : K cl = 1, ζ cl = ρ/(1 + ρ), T cl = t p1 1 ζ cl, π S c d cl = ζ cl T cl. C ss C ρ = ( 1 π ln C p1 C ss ), C ss C S c = (C ss C(t))dt. (5) (6) [9] PI (4) [6], M(s) C(s) = (1 M(s))G c (s). (7) (3) (7) C(s) = G p (s). (8) Fig. Equivalent block diagram of the process Pade [1] : e θs = 1 θs/ 1 + θs/, ()(4)(8) (7), : s(s /d cl ) E s 4 + Ds 3 + Cs + Bs + A, (9) : A = (1 K cl) T i Tcl d =, cl B = 1 ( + 4ζ cl ), T i Tcl T cl d cl C = + 4ζ cl + ( ζ cl + 1 ), Tcl T cl d cl T i T cl d cl D = 1 + ζ cl +, T i T cl d cl E = 1. K c Tcl (1) (1) (9), (1) : (Tp s + ζ p T p s + 1) (T p s + ζ p T p s + 1) s (s + ξ). (11) (11) ξ, [13] (11), ξ,. Pade (11) (9), K p, T p, ζ p, d p : d p = d cl, T p = d p B, K p = T p E, ζ p = 1 T p(d ). d p (1), (1) ζ p, λ(λ 1.1 1. ), λ = 1.13. : ζ p = 1 T p(d d p )λ.
14 7 3 (Simulation experiments),, PI PI,. 1. 3 4. Table 1 1 Simulation examples and the results of identification G p(s) PI K c T i K p T p ζ p d p IAE Rel IAE.8 1.9 [3] 11.7 1.9 6.4 53.48.496 1.798 1.95 1.37 6.43 8.36.63 1.8e 1s 5s + 1.8 13 6.1.6 13 7.7 [3] 4.88 1.15 1.18 18.1.1188 1.8 4.68 1.6 1.63 4.15.391 [3] 9.5 1.17 4.67 17.55.997 1.8 9.5 1.3 5.5 5.53.18 3.6 15 1. [3] 1.8 38.15 1.6 7.4.54 5.5 3 3.7 [3].796 1..4.18.4.5.76.91.4.1.8.5e.5s (s + 1)(3s + 1) 5.5 4 5.6 3.8 3 7.6 [3].6459 1.7.49.3.76.5.6999.93.5.11.7 [3].67 1..38.11.7.5.6931.99.38.1.4 4 4 3.5 9. [3].663 1.3.43.17.41.5.6556.99.43.1.3 3 4 Fig. 3 Unit step response of FOPDT process and the result of Fig. 4 Unit step response of SOPDT process and the result of the identification the identification : 1), PI,,. ) : d cl,, d cl (1)(13) K p, T p, ζ p, d p d p (
9 : PI 143 4 PI, d cl 4.5484). 3) 1 IAE : IAE = ŷ(t) y(t) dt, (13) ŷ, y. IAE, IAE, IAE,,, IAE ;,,, IAE. IAE. IAE(Relative IAE, Rel IAE), IAE Rel IAE = y ss y(t) dt, (14) y ss Rel IAE,. Rel IAE IAE, 4) 1, 4 PI, [3],, 5) [3],,. [3] ), 4 PI. 6), PI, 5%,,3, ). 4 (Identification of the experimental hyperthermia instrument),, 5,., T 3,, T 3. SSR, T 3. Fig. 5 5 Schematic diagram of the experimental hyperthermia instrument,, T 3 PI : K p =, T i = 3, PI 6. Fig. 6 6 Closed-loop response of heat treatment instrument : K p =.199, T p = 87.5, ζ p = 1.875, d p = 1.6,
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