64 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 Fig. 1. Conﬁguration of the cascade control system.ment requires prior information of the process. the phase lag of the closed inner loop will be muchFurthermore, the ultimate frequency used for outer less than that of the outer loop. This feature leadsloop design is based on initial ultimate frequency to the rationale behind the use of cascade control.without considering changes in inner loop control The crossover frequency for the inner loop isparameters. higher than that for the outer loop, which allows This paper presents a novel auto-tuning method higher gains in the inner loop controller in order tofor the cascade control system. By utilizing the regulate more effectively the effect of a distur-fundamental characteristic of cascade control sys- bance occurring in the inner loop without endan-tems, a simple relay feedback test is applied to the gering the stability of the system.outer loop to identify simultaneously both innerand outer loop process model parameters. A modelmatching the PID controller tuning method based ´on Pade coefﬁcients and the Markov parameter is 3. Relay feedback test for cascade controlproposed to control the overall system perfor- systemsmance. Two examples are given to illustrate theeffectiveness of the proposed method. The Astrom-Hagglund relay feedback test is based on the observation: when the process output2. Fundamentals of cascade control systems lags behind the input by Ϫ radians, the closed- loop system may generate sustained oscillation The conﬁguration of the cascade control scheme around the ultimate frequency ͑the frequencyis shown in Fig. 1, where an inner loop is embed- where the phase lag is Ϫ͒. The proposed relayded within an outer loop and the outer loop output feedback test for the auto-tuning cascade controlvariable is to be controlled. The control system system is shown as in Fig. 2. When the relay feed-consists of two processes and two controllers with back test begins, switch A points to position 2,outer loop transfer function G p1 , inner loop trans- switch B points to position 4, and switch C pointsfer function G p2 , outer loop controller G c1 , and to position 5. After the test, switch A points toinner loop controller G c2 , respectively. position 1, switch B points to position 3, and The two controllers of cascade control systems switch C points to position 6. As the inner loopare standard feedback controllers ͑i.e., P, PI, or process acts much faster than the outer loop pro-PID͒. Usually, a proportional controller is used for cess, output u of the inner loop process in Fig. 2the inner loop, integral action is needed when the under the relay feedback test acts as a step re-inner loop process contains essential time delays, sponse in half of the period of the stationary os-and the outer process is such that the loop gain in cillation, as shown in Fig. 3. Therefore a singlethe inner loop must be limited ͓1͔. relay feedback test can be used to obtain simulta- To serve the purpose of reducing or eliminating neously both the inner loop and outer loop processthe inner loop disturbance d 2 before its effect can models parameters.spill over to the outer loop, it is essential that the In practice, the real process model is usuallyinner loop exhibit a faster dynamic response than represented by low order plus dead-time model.that of the outer loop ͑as industry rule of thumb, it Here, the transfer function with the following formshould be at least ﬁve times ͓1͔͒. Consequently, ͑ﬁrst-order plus dead time͒ is adopted:
Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 65 Fig. 2. Conﬁguration of the proposed identiﬁcation model. ki tϭ0 in the process input; the process input and G pi ͑ s ͒ ϭ e ϪL i s , ͑1͒ output are collected until process enters a new T i sϩ1 steady state again. The process response after deadwhere iϭ1 stands for the outer process model and time tϭL 2 is described byiϭ2 stands for the inner process model, respec-tively. This model is characterized by three param- u ͑ t ͒ ϭk 2 ͑ 1Ϫe ͑ tϪL 2 ͒ /T 2 ͒ ϩw ͑ t ͒ , tуL 2 , ͑2͒eters: the static gain k, the time constant T, and the where w ( t ) is the white noise in measurement ofdead time L. It describes a linear monotonic pro- u ( t ) . It follows from the above relation thatcess quite well in many industrial applications andis often sufﬁcient for PID controller tuning. ͫ ͬT2 ͓ u ͑ t ͒ k 2 ͔ L ϭk 2 tϪA ͑ t ͒ ϩ ␦ ͑ t ͒ , tуL 2 , 23.1. Inner loop process model identiﬁcation ͑3͒ As the inner loop output u can be considered as where A ( t ) is the area under the process responsea step response in half period of the relay feedback and ␦͑t͒ is the integration of measurement noise;test, some well-developed step testing methods they are given as following, respectively:͓2,6,7͔ can be readily applied to identify param-eters of the inner loop. In this paper, the methodproposed by Ref. ͓2͔ is adopted due to its robust- A͑ t ͒ϭ ͵ u͑ t ͒dt, 0 tness; it is brieﬂy described as follows: Suppose that the inner process model is repre-sented by Eq. ͑1͒, and a unit step change occurs at ␦͑ t ͒ϭ ͵ w͑ t ͒dt. t 0 ͑4͒ Fig. 3. Inner loop and outer loop response under the proposed relay feedback test.
66 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72The inner process model’s static gain k 2 is com- where T u is the period of the stationary oscillation,puted from the process steady states of input and andoutput, 2 uϭ . ͑10͒ ⌬u Tu k 2ϭ , ͑5͒ ⌬u d As the overall open loop transfer model function iswhere ⌬u denotes the change of process output G p ͑ s ͒ ϭG p1 ͑ s ͒ G p2 ͑ s ͒and ⌬u d stands for the deviation in the manipu-lated input. Eq. ͑2͒ falls into a system of linear k 1k 2 ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s ,equations, ͑ T 1 sϩ1 ͒͑ T 2 sϩ1 ͒ ⌿Xϭ⌫ϩ⌬ for tуL 2 , ͑6͒ the outer loop process model transfer functionwhere k1 G p1 ͑ s ͒ ϭ e ϪL 1 s ͫ ͬ T 1 sϩ1 T2 Xϭ L , can be obtained by the following steps. ͫ ͬ 2 ͑1͒ Read off the overall system time delay L u ͓ mT s ͔ k2 ϭL 1 ϩL 2 of G p in the transfer function from the u ͓͑ mϩ1 ͒ T s ͔ k2 initial part of the relay feedback test, since the ⌿ϭ , inner loop transfer function delay L 2 is already ] ] available, the time delay L 1 can be computed as u ͓͑ nϩ1 ͒ T s ͔ k2 ͫ ͬ L 1 ϭLϪL 2 . ͑11͒ k 2 t ͓ mT s ͔ ϪA ͓ mT s ͔ ͑2͒ Obtain the frequency response of G p1 ( j ) k 2 t ͓͑ mϩ1 ͒ T s ͔ ϪA ͓͑ mϩ1 ͒ T s ͔ at ϭ u from ⌫ϭ ] , G p͑ j u ͒ k 2 t ͓͑ nϩ1 ͒ T s ͔ ϪA ͓͑ nϩ1 ͒ T s ͔ G p1 ͑ j u ͒ ϭ , ͑12͒ ͫ ͬ G p2 ͑ j u ͒ ␦ ͓ mT s ͔ and calculate ␦ ͓͑ mϩ1 ͒ T s ͔ G p1 ͑ j u ͒ ⌬ϭ ] . ͑7͒ k1 GЈ ͑ j u͒ϭ ϭ Ϫ jL 1 u ϭ ␣ ϩ j ␤ , p1 jT 1 u ϩ1 e ␦ ͓͑ nϩ1 ͒ T s ͔ ͑13͒T s is the sampling interval, and mT s уL 2 . The which is the frequency response for G p1 ( j )best estimation X * of X can be obtained using the without delay, where ␣ has positive sign and ␤ hasstandard least-square method as negative sign. X * ϭ ͑ ⌿ T ⌿ ͒ Ϫ1 ⌿ T ⌫. ͑8͒ ͑3͒ Calculate T 1 and k 1 , respectively, by ␤The best estimation of T 2 and L 2 can then be ob- T 1 ϭϪ , ͑14͒tained from X * . ␣ϫu ␣ 2ϩ ␤ 23.2. Outer loop process model identiﬁcation k 1ϭ . ͑15͒ ␣ By relay feedback test, the frequency response 4. Controller designof overall process model G p ( s ) at the ultimate fre-quency u is estimated as As the main purpose of inner loop control is to ͵ y ͑ t ͒e0 Tu Ϫ j ut dt eliminate the input disturbance, a P or PI control- ler using widely accepted model based tuning G ͑ j ͒ϭ ͑9͒ rules such as Ziegler-Nichols ͓8͔, Chien-Hrones- ͵ u ͑ t ͒e p u Tu , Ϫ j ut dt Reswick ͑CHR͒ ͓9͔, or Cohen-Coon ͓10͔ tuning d 0 rules will sufﬁce. This feature makes it very easy
Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 67to integrate the tuning method into the existing where stands for the desired damping ratio, usu-auto-tuning systems. Without loss of generality, ally selected as 0.707, the natural frequency nthe PI control structure of the form can be chosen between 0.5 and 1.0 times the ulti- mate frequency u from the relay feedback test K i2 ͓13͔. An alternative of desired closed-loop transfer G c2 ͑ s ͒ ϭK p2 ϩ s function for large dead-time systems can be ex- pressed as ͓14͔and the Chien-Hrones-Reswick ͑CHR͒ ͓9͔ tuningrule ͑20% overshoot͒ will be used for comparison 2 nstudy. The controller parameters are given, respec- H͑ s ͒ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s . s 2 ϩ2 n sϩ 2 ntively, by 0.7T 2 If the control speciﬁcations are not available, de- K p2 ϭ , ͑16͒ fault settings for the parameter ϭ0.707 and k 2L 2 u ( L 1 ϩL 2 ) ϭ2 can be used, which implies that the overshoot of the objective step response is 0.304T 2 K i2 ϭ . ͑17͒ about 5%, the phase margin is 60°, and the gain k 2L 2 2 margin is 2.2. For simplicity, Eq. ͑20͒ can be re- written as the parametric form:With the PI controller, the closed-loop transferfunction G 2 ( s ) of inner loop and the open loop d0transfer function G 1 ( s ) are then obtained as H͑ s ͒ϭ , ͑21͒ e 0 ϩe 1 sϩe 2 s 2 G p2 ͑ s ͒ G c2 ͑ s ͒ where d 0 ϭ 2 , e 0 ϭ 2 , e 1 ϭ2 n , and e 2 ϭ1. G 2͑ s ͒ ϭ n n 1ϩG p2 ͑ s ͒ G c2 ͑ s ͒ ͑2͒ Approximating the time delays in G 1 ( s ) of Eq. ͑19͒, since the dead time L 2 of the inner loop k 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s ϭ , process model is very small, it is always approxi- s ͑ 1ϩT 2 s ͒ ϩk 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s mated as 1 or ͑18͒ e ϪL 2 s Ϸ1ϩ ͑ ϪL 2 s ͒ , ͑22͒ G 1 ͑ s ͒ ϭG 2 ͑ s ͒ G p1 ͑ s ͒ e Ϫ ( L 1 ϩL 2 ) s is approximated as k 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s ͑ L 1 ϩL 2 ͒ 2 s 2 ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s Ϸ1Ϫ ͑ L 1 ϩL 2 ͒ sϩ s ͑ 1ϩT 2 s ͒ ϩk 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s 2 ͑23͒ k1 • e ϪL 1 s . ͑19͒ in order to gain a more accurate approximation. T 1 sϩ1 Substitute Eqs. ͑22͒ and ͑23͒ to Eq. ͑19͒, and re-As G 1 ( s ) is not a standard transfer function, it is write G 1 ( s ) asdifﬁcult to directly apply existing tuning rules.Therefore a model-matching algorithm ͓11,12͔ is g 0 ϩg 1 sϩg 2 s 2 ϩg 3 s 3 G 1͑ s ͒ ϭ , ͑24͒proposed to obtain the PID control parameters for h 0 ϩh 1 sϩh 2 s 2 ϩh 3 s 3overall system performance. The brief theoreticalbackground of a controller design based on model wherematching is attached in the Appendix; for more g 0 ϭk 1 k 2 K i2 ,details please refer to Refs. ͓11,12͔. Its applicationto this particular problem is given as follows. g 1 ϭk 1 k 2 ͓ K p2 ϪK i2 ͑ L 1 ϩL 2 ͔͒ , ͑1͒ Assuming that the process dead time issmall, the desired reference model H ( s ) is chosenas g 2 ϭk 1 k 2 ͫ K i2 ͑ L 1 ϩL 2 ͒ 2 2 ϪK p2 ͑ L 1 ϩL 2 ͒ , ͬ 2 n k 1 k 2 K p2 ͑ L 1 ϩL 2 ͒ 2 H͑ s ͒ϭ , ͑20͒ g 3ϭ , s 2 ϩ2 n sϩ 2 n 2
68 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72and ͑4͒ The PID parameters can be computed as h 0 ϭk 2 K i2 , a 1 b 1 Ϫa 0 b 2 K p1 ϭ , ͑27͒ b2 h 1 ϭk 2 T 1 K i2 ϩ1ϩk 2 K p2 Ϫk 2 K i2 L 2 , 1 h 2 ϭT 1 ͑ 1ϩk 2 K p2 Ϫk 2 K i2 L 2 ͒ ϩT 2 Ϫk 2 K p2 L 2 , a0 K i1 ϭ , ͑28͒ b1 h 3 ϭT 1 ͑ T 2 Ϫk 2 K p2 L 2 ͒ . a 2 b 2 Ϫa 1 b 1 b 2 ϩa 0 b 2 1 2As indicated in Refs. ͓11,12͔, the Pade coefﬁcients ´ K d1 ϭ , ͑29͒and the Markov parameters characterize, respec- b3 1tively, the low- and high-frequency responses of a b2system, or the responses of the steady state and T n1 ϭ . ͑30͒transition state. Pϭ5 and M ϭ0 are selected in b1order to get a good system response approxima-tion in a steady state ͓11͔. Pade coefﬁcients of ´ 5. Comparison studyH ( s ) are estimated as Two examples are presented here to illustrate c 0 ϭ1, the effectiveness of the proposed tuning method for cascade control systems. In order to show the c 1 ϭ ͑ d 1 Ϫe 1 c 0 ͒ /e 0 , accuracy of the proposed identiﬁcation method in c 2 ϭ ͑ d 2 Ϫe 1 c 1 Ϫe 2 c 0 ͒ /e 0 , a noisy environment, the noise-to-signal ratio ͑NSR͒, deﬁned as ͓15͔ NSRϭmean͓abs͑noise͔͒/ c 3 ϭ ͑ d 3 Ϫe 1 c 2 Ϫe 2 c 1 Ϫe 3 c 0 ͒ /e 0 , mean͓abs͑signal͔͒ is introduced. In this paper, the proposed identiﬁcation method is applied to both c 4 ϭ ͑ d 4 Ϫe 1 c 3 Ϫe 2 c 2 Ϫe 3 c 1 Ϫe 4 c 0 ͒ /e 0 . processes with noise level 10% NSR. Example 1. Consider a cascade control system ͑3͒ The PID controller with pϭmϭ2 ͓11͔ is discussed by Hang ͓4͔ and Tan ͓5͔ with plant mod-given as els of K i1 K d1 s e Ϫs e Ϫ␣s G c1 ͑ s ͒ ϭK p1 ϩ ϩ G p1 ͑ s ͒ ϭ , G p2 ͑ s ͒ ϭ , s 1ϩT n1 s ͑ 1ϩs ͒ 2 1ϩ ␣ s a 0 ϩa 1 sϩa 2 s 2 where ␣ϭ0.1. After the relay feedback test, the ϭ , ͑25͒ b 0 ϩb 1 sϩb 2 s 2 parameters for the inner loop process model is ob- tained aswhere b 0 ϭ0 and a 0 , a 1 , a 2 , b 1 , b 2 can be com-puted by solving the following linear matrix equa- e Ϫ0.115sͫtions: G p2 ͑ s ͒ ϭ0.9563 1ϩ0.0947s ΅ g 0c 1 0 0 h0 0 and the outer loop process model is estimated as g 0 c 2 ϩg 2 c 0 g 0c 1 0 h 0 c 1 ϩh 1 h 0c 0 g 0 c 3 ϩg 1 c 2 g 0c 2 g 0c 1 h 0 c 2 ϩh 1 c 1 h 0 c 1 e Ϫ1.63s ϩg 2 c 1 ϩg 1 c 1 ϩh 2 ϩh 1 G p1 ͑ s ͒ ϭ0.965 . 1ϩ1.908s g 0 c 4 ϩg 1 c 3 g 0 c 3 ϩg 1 c 2 g 0 c 2 h 0 c 3 ϩh 1 c 2 h 0 c 2 ϩg 2 c 2 ϩg 3 c 1 ϩg 2 c 1 ϩg 1 c 1 ϩh 2 c 1 ϩh 3 ϩh 1 c 1 ϩh 2 The parameters calculated for the inner loop PI ͫ ͬͫͬ 0 0 g3 0 h3 controller G c2 using the Chien-Hrones-Reswick ͑CHR͒ tuning rule ͑20% overshoot͒ are obtained a0 0 as K p2 ϭ0.603, K i2 ϭ2.277. The overall system a1 0 reference model of the cascade control system is ϫ a2 ϭ 0 . ͑26͒ chosen to be in the form of Eq. ͑20͒ with ϭ0.8 b1 0 and u ϭ0.6. The outer loop controller’s param- b2 1 eters are obtained as K p1 ϭ0.6592, K i1 ϭ0.3536,
Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 69 Fig. 4. Tuning procedure and step response.K d1 ϭ0.2886, and T n1 ϭ1.4392. The proposed the inner loop are obtained as K p2 ϭ2.895, K i2auto-tuning procedure and step response in the ϭ0.147. The overall system reference model ofnoisy environment are shown in Fig. 4. The result the cascade control system is chosen to be in theis also compared to the methods proposed by form of Eq. ͑20͒ with ϭ0.707 and u ϭ0.03, theHang and Tan. Fig. 5 shows the closed-loop per- outer loop controller’s parameters are obtained asformance from tϭ0 s to tϭ80 s; a large step load K p1 ϭϪ4.9225, K i1 ϭϪ0.1105, K d1 ϭϪ51.1205,disturbance seeps into the process for all cases at and T n1 ϭ6.4596. The control performance com-tϭ50 s. parison study is carried out from tϭ0 s to Example 2. Consider a process model of the tϭ2000 s, a large step load disturbance is addedpacked-bed reactor provided by Ref. ͓16͔. The into the process at time tϭ1000 s, as shown ingoal is to tightly control the exit concentration Fig. 6.temperature, and the most signiﬁcant disturbance From the examples, the improved performanceis the heating medium temperature. The inner and of the proposed tuning method is clearly evident.outer loop process models are given by e Ϫ20s 6. Conclusion G p1 ͑ s ͒ ϭϪ0.19 , 1ϩ50s This paper developed a novel auto-tuning e Ϫ8s method for the cascade control system. As the in- G p2 ͑ s ͒ ϭ0.57 , 1ϩ20s ner loop process acts much faster than the outer loop in the cascade control system, both inner looprespectively. To reduce the disturbance, a cascade and outer loop process model parameters can becontrol strategy is adopted. Using the relay feed- identiﬁed using one relay feedback test by utiliz-back test, the parameters for the inner and outer ing this physical property under the proposedprocess models from the experiment are estimated structure. Consequently, well-established modelas based PI tuning rules can be applied to tune the e Ϫ22.6s inner loop, and a model matching the PID control- G p1 ͑ s ͒ ϭϪ0.192 , ler design method was proposed to tune the outer 1ϩ47.5s loop. Finally, two examples were given to show e Ϫ8.5s the effectiveness of the proposed method. The G p2 ͑ s ͒ ϭ0.558 . method is very straightforward and has been inte- 1ϩ19.7s grated into an existing auto-tuning system. It isThe PI controller using the Chien-Hrones- now being tested in a centralized HVAC systemReswick ͑CHR͒ tuning rule ͑20% overshoot͒ for and the ﬁeld results will be reported soon.
70 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 Fig. 5. Performance comparison.Appendix and a controller’s transfer function model Suppose we have a process model a 0 ϩa 1 sϩ¯ϩa p s p C͑ s ͒ϭ . ͑A2͒ b 0 ϩb 1 sϩ¯ϩb m s m g 0 ϩg 1 sϩ¯ϩg q s q G͑ s ͒ϭ , ͑A1͒ h 0 ϩh 1 sϩ¯ϩh n s n It is desired that C ( s ) be obtained in such a way Fig. 6. Performances of the proposed method ͑solid line͒ and of Ref. ͓16͔ ͑dashed line͒.
Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 71 ´that the overall system matches a set of Pade co- iefﬁcients and Markov parameters of a given a ref- x i ϭ ͚ a j g iϪ j , iϭ0,1, . . . ,qϩp, ͑A9͒ jϭ0erence model i d 0 ϩd 1 sϩ¯ϩd r s r H͑ s ͒ϭ , ͑A3͒ y i ϭ ͚ b j h iϪ j , iϭ0,1, . . . ,nϩm. ͑A10͒ e 0 ϩe 1 sϩ¯ϩe r s r jϭ0and a simple controller C ( s ) can be found such Using the constraint PϩM ϭpϩmϩ1, where P Pthat ´ is the number of Pade coefﬁcients, M is the num- ber of Markov parameters, p is the numerator’s C͑ j ͒G͑ j ͒ order of C ( s ) , and m is the denominator’s order of ХH ͑ j ͒ . ͑A4͒ 1ϩC ͑ j ͒ G ͑ j ͒ C ( s ) . The parameters a, b i of C ( s ) can be ob- tained uniquely by solving the set of linear Eqs.The model matching of time moments and Mar- ͑A5͒–͑A10͒.kov parameters is very effective in model reduc-tion for obtaining approximate models, because ´the Pade coefﬁcients and the Markov parameters Referencescharacterize, respectively, the low- and high- ͓1͔ Astrom, K. J. and Hagglund, T., PID Controller: ´frequency responses of a system. The Pade coefﬁ- Theory, Design and Tuning. Instrument Society ofcients of H ( s ) are deﬁned as America, Research Triangle Park, NC, 1995. ͓2͔ Bi, Q., Cai, Wenjian, Lee, E. L., Wang, Qingguo, c 0 ϭd 0 /e 0 , Hang, C. C., and Zhang, Y., Robust identiﬁcation of ﬁrst order plus dead-time model from step response. ͫ ͬͲ k Control Eng. Pract. 1, 71–77 ͑1999͒. ͓3͔ Li, M. X., Bruijn, P. M., and Verbruggen, H. B., Tun- c k ϭ d k Ϫ ͚ e j c kϪ j e0 , ͑A5͒ ing of cascade PID controller with fuzzy inference. jϭ1 Asia-Pac. Eng. Part A, Electr. Eng. 2, 65–70 ͑1994͒. ´ ͓4͔ Hang, C. C., Loh, A. P., and Vasnani, V. U., Relaywhere c k is the kth Pade coefﬁcient. feedback auto-tuning of cascade controllers. IEEE The Markov parameters of H ( s ) are deﬁned as Trans. Control Syst. Technol. 2, 42– 45 ͑1994͒. ͓5͔ Tan, K. K., Lee, T. H., and Ferdous, R., Simultaneous m 0 ϭd r /e r , online automatic tuning of cascade control for open ͫ ͬͲ loop stable process. ISA Trans. 39, 233–243 ͑2000͒. k ͓6͔ Shinskey, F. G., Process Control System: Application, m k ϭ d rϪk Ϫ ͚ e rϪ j m kϪ j er , ͑A6͒ Design, and Tuning. 3rd ed., McGraw-Hill, New York, jϭ1 1998. ͓7͔ Marlin, T. E., Process Control: Designing Process andwhere m k is the kth Markov parameter. Control System for Dynamic Performance. McGraw- A set of liner equations as shown in the follow- Hill, New York, 1995. ´ing can be gained in order to match the Pade co- ͓8͔ Ziegler, J. G. and Nichols, N. B., Optimum settings for automatic controllers. Trans. ASME 64, 759–768efﬁcients and Markov parameters: ͑1942͒. ͓9͔ Chien, K. L., Hrones, J. A., and Reswick, J. B., On the ␣ 0 c 0 ϭx 0 , automatic control of generalized passive systems. k Trans. ASME 74, 175–185 ͑1952͒. ͓10͔ Cohen, G. H. and Coon, G. A., Theoretical consider- ␣ 0 c k ϭx k Ϫ ͚ ␣ j c kϪ j , kϭ1,2, . . . , PϪ1, ation of retarded control. Trans. ASME 75, 827– 834 jϭ1 ͑1953͒. ͓11͔ Aguirre, L. A., New algorithm for closed-loop model 1ϭ ␣ nϩm , ͑A7͒ matching. IEEE Electron Device Lett. 27, 2260–2262 ͑1991͒. k ͓12͔ Aguirre, L. A., PID tuning on model matching. IEEE m k ϭx nϩmϪk Ϫ ͚ ␣ nϩmϪ j m kϪ j , Electron Device Lett. 28, 2269–2271 ͑1992͒. jϭ1 ͓13͔ Wang, Q. G., Hang, C. C., and Biao, Zou, A frequency response approach to autotuning of multivariable con- kϭ1,2, . . . ,M Ϫ1, trollers. Trans. Inst. Chem. Eng., Part A 75, 64 –72 ͑1997͒.where ͓14͔ Kiong Tan, K. K., Wang, Qing-Guo, and Hang, Chang Chien, Advances in Industrial Control. Springer- ␣ i ϭx i ϩy i , iϭ0,1, . . . ,nϩm, ͑A8͒ Verlag, London, 1999.
72 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72͓15͔ Haykin, S., An Introduction to Analog & Digital Com- pany, U. S. A.; Research Scientist at National University of Singapore; munication. Wiley, New York, 1989. Principal Engineer and R&D Manager at Supersymmetry Services Pte͓16͔ Marlin, Thomas E., Process Control: Designing Pro- Ltd, Singapore; Senior Research Fellow at Environmental Technology Institute, Singapore, respectively. He is currently serving as Associate cesses and Control Systems for Dynamic Perfor- Professor at Nayang Technological University, Singapore. Dr. Cai’s mance. 2nd ed., McGraw-Hill, New York, 2000. current research interests include multivariable control and HVAC sys- tem optimization. Sihai Song was born in 1974, Zhejiang, P. R. China. He Ya-Gang Wang was born in graduated from Zhejiang Uni- 1967, Shanxi, P. R. China. He versity with a bachelor degree received his B.Eng., M.Eng., in electrical engineering in and Ph.D. from Department of 1997, and a second bachelor Automation, China University degree of international com- of Mining and Technology, modities inspection in 1998. In Taiyuan University of Technol- 2001, he started his postgradu- ogy and Shanghai Jiao Tong ation studies at Nanyang Tech- University, P. R. China, in nological University, Sin- 1988, 1991, and 2000, respec- gapore. He is interested in PID tively. After graduation, he auto-tuning and computer- worked as a lecturer in Taiyuan aided control system design. University of Technology, P. R. China, and a Postdoctoral Re- search Fellow in Nanyang Technological University, Singapore. He is interested in process control and instrumentation, PID auto-tuning, and Wenjian Cai received his multivariable control. B.Eng., M.Eng., and Ph.D. from Department of Precision Instrumentation Engineering, Department of Control Engi- neering, Harbin Institute of Technology, P. R. China, and Department of Electrical Engi- neering, Oakland University, U. S. A., in 1980, 1983, and 1992, respectively. After graduation, he worked as a Postdoctoral Research Fellow at the Center for Advanced Ro-botics, Oakland University, U. S. A.; Engineer at CEC Controls Com-