Application of H Compensator to a First-order lag plus deadtime System with Changes in Characteristics õ Yuji YAMAKAWA*, Yutaka IDE**, Yoshiyuki NODA*
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1 Application of H Compensator to a First-order lag plus deadtime System with Changes in Characteristics õ Yuji YAMAKAWA*, Yutaka IDE**, Yoshiyuki NODA***, Takanori YAMAZAKI****, Kazuyuki KAMIMURA***** and Shigeru KUROSU****** This paper concerns a new design method of robust controllers which achieve robust stability as well as adequate disturbance rejection and tracking performance. In a previous paper, the robust PID controller has been designed for a plant which was commonly approximated by a first-order lag plus deadtime system. The PID parameters obtained by solving a two-disk type mixed sensitivity problem have been modified by contrast to those tuned by the traditional Ziegler-Nichols rule. The previous results showed that the proposed PID controller was effective enough for practical applications. In this paper we redesign a robust controller for the same plant with changes in characteristics of the plant through H control theory. A practical and systematic design method for the H compensator is proposed. This paper demonstrates a practical applicability of the H compensator to a typical plant in process industries comparing with the control performance obtained by the robust PID controller. Key Words: process control, PID control, H control *The University of Tokyo, Bunkyo-ku **Tokyo A. & T. University, Koganei, Tokyo ***Toyohashi University of Technology, Toyohashi ****Oyama National College of Technology, Oyama *****Yamatake Co., Ltd., Ota-ku ******Crotech, Shimodate (Received April 1, 2003) (Revised January 29, 2004)
2 W(s)-g(s+h)=Km(Tm's+1) (Tm'> Tm)7)s+fT ms+1 1-LPs+LP2s21-LPs elps~ 2 12 ~ 2 (8) LPs+LP2s21+LPs p1=6/lp+1/tp(=2/lp+1/tp), p2=12/l2p+6/(lptp)(=2/(lptp)), P(s)_KPe-LpS1 p3=12/(l2ptp)(=0), q1=kp/tp(=-kp/tp), KPH=Kp(1+8K) TP'=TP(1+OT) (2) Lp'=LP(1+OL) ƒâk=-ƒât=-ƒâl=ƒâ (3) P(s)tigist+q2s+q3tgis+g2(9) s3+pis2+p2s+p3 q2=-6kp/(lptp)(=2kp/(lptp)), q3=12kp/(l2ptp)(=0) s2+pis+p2 P Œ(s)=(1+ƒ (s))p(s) (4) Frequency [rad/min] ƒ (jƒö): modeling error ƒ P1 (jƒö), ƒ P2(jƒÖ): Pade approximated modeling error W (jƒö): weighting function <1(5) Fig. 1 Frequency responses Table 1 Km, Tm, and Tm Œ ƒ (jƒö) < W(jƒÖ) (6) s(s)_fix+1)(tps+1)e-lps`s-1
3 K(s)=K(s)(11) zy -b1cyp(ew)(12) wl1+p(jw)k(jw) zy/w1 0 zy/w1 = P(jƒÖ) 0 maxbicyp~w)<y(13)1+p(j w)k(jw)
4 (a) Effects of cy on frequency responses of closed loop (zy/w1) ATX+XA+X(y-2B1B1T -B2B2T )X + C1TC1=0 AY+ YAT + Y(y-2C1TC1-C2TC2) Y+B1B1T -0 (14) (b) Effects of cy on step responses (zy) to disturbance input Fig. 3 Effects of cy u,-cojy (15) A=A+ y-2b1b1tx +B2F+ZLC2,B=-ZL C=F=-B2TX, L=-YC2T,Z=(I-y-2YX)-1 (16) K Œ(s)=C(sI-A)B (17) (a) Effects of c4 on frequency responses of closed loop (zy/w1) (b) Effects of c4 on step responses (zy) to disturbance input Fig. 4 Effects of c4
5 K(S)=121S4+922S3+ 923S f5~~~~S(d 1S5+d2S4+d3S3+d4S2+d5S+d6) (b) Step responses to disturbance input (1) Plant with Pade approximation (2) Plant with pure deadtime Fig. 5 Effects of variable ć (in case of a second-order Pade approximation)
6 (a) Location of poles and zeros ( : zeros ~: poles) (b) Step responses to disturbance input (1) Plant with Pade approximation (2) Plant with pure deadtime Fig. 6 Effects of variable ć (in case of a first-order Pade approximation)
7
8 (b) Step responses to reference input Fig. 9 Comparison of characteristics between H compensator and PID controller (b) PID controller Fig. 11 Observation noise (in case of that ĉ=80%)
9 2) G.D. Shilling: Process Dynamics and Control, Holt, Rinehart and Winston, Inc., 196/201 (1963) 6) M. Kasahara et al.: Design and Tuning of Robust PID Controller for HVAC systems, ASHRAE Transactions, 105-2, 154/166 (1999) 12) K. Glover and J.C. Doyle: State-space Formulae for All Stabilizing controllers that satisfy an H -norm Bound and Relations to Risk Sensitivity, System & Control Letters, 11, 167/172 (1988) 13) J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis: State-Space Solutions to Standard H2 and H Control Problems, IEEE Transactions on Automatic Control, 34-8, 831/847 (1989)
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