Advaned PID Controller Synthesis using Multisale Control Sheme Bejay Ugon a,*, Jobrun Nandong a, and Zhuquan Zang b a Department of Chemial Engineering, Curtin University, 989 Miri, Sarawak, Malaysia b Department of Eletrial and Computer Engineering, Curtin University, 989 Miri, Sarawak, Malaysia {bejay@urtin.edu.my, jobrun.n@urtin.edu.my, zqzang@urtin.edu.my} Abstrat The PID ontroller has been widely applied in industries sine many deades ago despite the advanement in many advaned ontrol tehniques. Proess models suh as the First-Order plus Deadtime (FOPDT) has often been used to design or tune the PID ontroller. A large number of PID ontroller tuning formulas have been established sine the wellknown Ziegler-Nihols formula introdued in the 94s. In this paper, we present a new approah based on the Multi-sale Control sheme to onstruting a PID ontroller tuning formula whih is appliable to the FOPDT model. The effetiveness of the proposed PID tuning formula is ompared with some of the best PID tuning formulas reported in the literature. eywords PID Tuning; -Delay; Multisale Control I. INTRODUCTION In proess industry, the presene of time delay or deadtime often auses poor ontrol performane or even losed-loop instability. In fat, the presene of deadtime has been reognized as one of the limiting fators for the losed-loop performane []. Interestingly, the model whih inorporates deadtime, i.e., the First-Order plus Deadtime (FOPDT) model has been used rather extensively in the PID ontroller design, i.e., PID ontroller tuning formulas or rules. Sine the wellknown Ziegler-Nihols PID tuning formula introdued in the 94s, a large number of PID tuning formulas based on the FOPDT model have been developed; see the summary report in [2]. In this brief paper, we shall demonstrate a new approah to devising an advaned PID ontroller augmented with a filter; this PID formula is derived based on the priniple of the Multisale Control (MSC) sheme reently introdued by Nandong and Zang [3]. The rest of this paper is organized as follows. Setion II provides a brief overview of the MSC sheme and a PID tuning formula derivation based on the FOPDT model. Setion III demonstrates the effetiveness of the proposed PID tuning formula using two ase studies: Steam Superheater and Cement Cooler Grate systems. Finally, Setion IV highlights some onlusions and future works. II. MULTI-SCALE CONTROL SCHEME A. Multi-sale Plant Deomposition The details about the Multi-sale Control (MSC) sheme an be found in [3]. Here, we only provide a brief overview of the MSC sheme. The priniple of the MSC sheme is first to deompose a given plant into a sum of basi fators or modes with distint speed responses. For a general ase, onsider a rational transfer funtion P whih an be deomposed into a sum of n + basi fators or modes as follows: N ( P ( + P ( + P2 ( +... + Pn ( () D( Here, P i, i {,,2,..., n} is the plant fator, whih is either first or seond order system with real oeffiients. The dynami of P i is slower than that of P i+ for i,,2,..., n ; P is alled the outermost fator while P i, i {,2,3..., n} the innerlayer fator. Notie that this general ase results in a n + - layer MSC sheme. In real appliation, we might just need a 2- or 3-layer MSC sheme. In this present paper, we demonstrate the appliation of the 2-layer MSC sheme to onstruting a PID tuning formula based on the First-Order plus Deadtime (FOPDT) model. B. Realization of the 2-Layer Multi-sale Control Sheme The blok diagram of a 2-layer multi-sale ontrol (MSC) sheme is shown in Figure ; i.e., a given plant P an be deomposed into 2 modes ( P P + P ). Here, Wi is alled the multi-sale preditors; i the multi-sale sub-ontrollers; P the augmented overall plant transfer funtion. The outermost sub-ontroller ( ) orresponds to the fator with the slowest dynami ( P ) and orresponds to the fastest dynami ( P 2 ). In Figure, the losed-loop transfer funtion for the inner layer is given by: s G s ( ) ( ) (2) + ( W (
The augmented overall plant transfer funtion is expressed as P ( G ( (2) The overall MSC ontroller is then given by ms ( G ( ( (3) τ + α A p (7) τ α p α B 2 (8) α τ The multi-sale preditor is hosen as the inner mode, i.e. B W ( α s + (9) Assuming P-only ontroller with gain is used in the inner-loop, the following losed-loop setpoint transfer funtion is obtained H r ( G ( ) s () + W ( The equation () an be simplified to Figure. Blok diagram of 2-layer multi-sale ontrol sheme: (a) full 2- loop, (b) redued single-loop [3] C. Derivation of PID Tuning Formula Let us onsider a proess whih an represented by the First-Order Plus Deadtime (FOPDT) model as follows θs pe τs + where p,τ, θ denote the proess gain, time onstant and deadtime respetively. By using / Padé formula to approximate the delay omponent in (4) gives (4) p ( α (5) ( τs + )( + α where α. 5θ. Assuming that τ > α, and after applying partial fration expansion to (5), we obtain 2 modes as follows A B + τs + αs + (6) G ( o ( αs + ) τ s + () where the overall gain and losed-loop time onstant are given respetively by o (2) + B α τ (3) + B Now, onsider the ratio of open-loop to losed-loop time onstant as given by α λ (4) τ From (3) and (4), an be expressed as follows λ (5) B Note that, the greater the value of λ, the faster is the losed-loop response, i.e., more aggressive ontroller ation.
Next, let us onsider that a PI ontroller is hosen to ontrol the outermost mode, i.e. ( + τ I s (6) where and τ I indiate the sub-ontroller gain and reset time respetively. To determine assume first that the outermost is ontrolled using a P-only ontroller. Then it an be determined in the same way as the P ontroller for the inner mode, i.e. λ (7) A where λ is the ratio of the open-loop to losed-loop time onstant, i.e., defined in the same way as (4). It follows that, the reset time for the PI ontroller is set based on a fration of the open-loop time onstant for the outermost mode as follows τ I γτ,.3 γ.2 (8) The overall MSC ontroller (3) an now be arranged in the form of ms ( ) s τ I o S( τ + + ) I s αs s τ s + (9) Here, S ( ) denotes the sign of ontroller gain, whih needs to be inluded in (9) to get the orret sign for the overall ontroller gain. It an be easily shown that the overall MSC ontroller (8) an be expressed as a pratial PID ontroller augmented with a filter given by τ I α τ D (23) τ + α I The parameters above an be further expressed in terms of model and our speifiation parameters θ θ θ ( λ )( λ ) γτ + τ τ 2 2 2 S( ) θ γ θτ p τ + λ 2 Remarks: τ I γτ +. 5θ (25) γθτ τ D 2 γτ + θ (26) θ τ f (27) 2λ (24) The ontroller speifiations are λ,λ and γ in order to obtain the MSC-PID parameters via (24) - (27). The λ and λ are adjusted first while keeping γ until gain margin (GM) approximately 7 db is reahed. Then, γ is slightly adjusted to get a desired final response. III. ILLUSTRATIVE EXAMPLES Example - Steam Superheater A linearized model for the Steam Superheater main temperature (i.e., from Loop-pro Control Station, In.) is obtained as follows: ms ( + + τ Ds τ I s τ f s + The equivalent MSC-PID tuning parameters τ D are expressed as o S( )( τ I + α) (2) τ I τ I τ I + α (22) (2), τ I and 27.7s 7.6e 27.7s + (28) Additionally, a perturbed model for the Steam Superheater temperature is given by P 35s 6.9e ( ( 7.5s + ) 2 (29) Upon approximating the time-delay omponent in (28) using the / Padé formula, the nominal model (28) an be written as follows:
7.6( 3.85 P ( (3) ( 27.7s + )(3.85 s + ) Next, we an deompose (3) into two fators using partial fration expansion:.2 3 P ( 27.7s+ P 4.32 ( 3.85s+ (3) where the multi-sale preditor is hosen as the inner mode 4.32 W ( (32) 3.85s + For performane omparison, two other PID ontroller base on FOPDT model are designed with minimum IAE formula of Rovira et al. [4] and Suyama [5] for ideal PID ontroller given by: G ( + Td (33) T s + i Note that, the Rovira et al. [4] PID tuning formula is based on servo ontrol while that of Suyama [5] is based on diret synthesis ontrol. The tuning parameter for the MSC-PID ontroller is based on the settings: λ 3. 9, λ 2 and γ. 69, whih gives GM 9.6bB, PM 6 and DM 43.9 seonds. Tuning parameter for Rovira et al.[3] give GM 6.42, PM 7.3 and DM 46.3 seonds and tuning parameter for Suyama[5] give GM 7.9dB, PM 6, DM 39.9 seonds. The performanes of the 3 different PID ontrollers are evaluated against unit step hange in the setpoint,. unit in input disturbane and followed by unit in output disturbane. Figures 2 and 3 show the disturbane rejetion performanes at the nominal ondition (28) for the 3 different PID ontrollers. For the input type disturbane, it is obvious (Figure 2) that the MSC-PID ontroller shows superior performane over the PID ontrollers tuned using Rovira et al. [4] and Suyama [5]. Figures 4 and 5 show the losed-loop responses to input and output disturbanes at the perturbed ondition given in (29). In term of the IAE value, the MSC-PID provides better performane than the Rovira et al. and Suyama PID ontrollers. TABLE I. PID TUNING PARAMETERS FOR STEAM SUPERHEATER. PID Formula τ I τ D f Rovira et al.[4] -.522 45.498 9.695 - Suyama [5] -.397 27.7 3.85 - Proposed MSC -.55 32.963 8.37.54 τ.2 Suyama: IAE 44.9 Rovira: IAE 55.6 MSC: IAE 26.5 5 5 2 25 3 35 Figure 2. Input disturbane rejetion response at nominal ondition for the Steam Superheater 2.8.6.4.2 Suyama: IAE 44.9 Rovira: IAE 55.6 MSC: IAE 26.5 9 95 2 25 2 25 22 225 23 235 Figure 3. Output disturbane rejetion response at nominal ondition for the Steam Superheater.2.2 Suyama: IAE 275.7 Rovira: IAE 265.5 MSC: IAE 223 2 3 4 5 6 7 8 Figure 4. Input disturbane rejetion performane at perturbed ondition for the Steam Superheater
2.8.6 For performane omparison, 3 PID ontrollers based on the FOPDT model (34) are designed using formulas: (a) Witt & Waggoner [6], (b) aya & Sheib [7], and () proposed MSC- PID. The PID tuning formulas of Witt & Waggoner [6] and aya & Sheib [7] are developed for the lassial PID ontroller of the form.4.2 Suyama: IAE 275.7 Rovira: IAE 265.5 MSC: IAE 223 9 2 2 22 23 24 25 26 27 Figure 5. Output disturbane rejetion response at perturbed ondition for the Steam Superheater. Overall, in term of the total Integral Absolute Error (IAE) value, the proposed MSC-PID ontroller gives the best performane both at the nominal and perturbed onditions; the Rovira et al. PID ontroller gives better performane than the Suyama PID ontroller at the perturbed ondition, and vie versa at the nominal ondition. Example 2 Cement Cooler Grate A linearized model for the Cement Cooler Grate pressure (i.e., from Loop-pro Control Station, In.) is given by: G + T s + d (39) Ti s T s + d N Note that, the Witt and Waggoner formula [6] is developed based on the proess reation ontrol, while the aya and Sheib formula [7] is based on servo ontrol. In both the Witt & Waggoner and aya & Sheib formulas, N is used. For the proposed MSC-PID sheme, we use these settings: λ 3.8, λ and γ. 8, whih lead to GM 6.55 db and PM 56.2 o, DM 5.65 minutes. TABLE II. PID CONTROLLER PARAMETERS FOR THE CEMENT COLLER GRATE. PID Formula τ I τ D f Witt & Waggoner [6] -.78 4.78 4.78 - aya & Sheib [7] -.95 4.2954 4.49 - Proposed MSC -.36 7.334.6.99 τ P ( 4.78s 43.7e 6.8s + A perturbed linearized model is given by (34).4.2 P 42.2e 7. 2 ( 5.s + s (35) The appliation of / Padé formula to (34) yields an approximated model as follows: 43.7( 2.39 (36) ( 6.8s + )( + 2.39.2 Witt & Waggoner: IAE 3.77 aya & Sheib: IAE 3.2 MSC: IAE 2.4 5 5 2 The appliation of partial fration expansion to (36) leads to 2.26 55.5 P ( + (37) 6.8s + 2.29s + The multi-sale preditor is hosen as the inner mode 55.5 W ( (38) 2.29s + Figure 6. Closed-loop response at norminal ondition for the Cement Coller Grate pressure The omparative performanes resulting from the 3 different PID ontroller tunings at nominal ondition for a onseutive unit step hange and.2 units hange in input disturbane are shown in Figure 6. It is learly shown that the setpoint traking performane for the MSC-PID ontroller is better than the other two ontrollers tuned using Witt & Waggoner and aya & Sheib formulas. The disturbane rejetion performanes for the 3 PID ontrollers are quite the
same. Overall, the MSC-PID gives the best performane in term of the IAE value, i.e., the smallest IAE value. Figure 7 demonstrate the losed-loop responses for the 3 different PID ontrollers under a perturbed ondition (35). Again, the MSC- PID ontroller provides the best performane. Conferene on Industrial Eletronis, Control, Instrumentation and Automation, pp. 7-22, 992. [6] S.D. Witt and R.C. Waggoner "Tuning parameters for non-pid three mode ontrollers," Hydroarbon Proessing, pp. 74-78, 99. [7] A. aya and T.J. Sheib, "Tuning PID ontrols of different strutures," Control Engineering, pp. 62-65, 988. 2.8.6 Witt & Waggoner: IAE 33.4 aya & Sheib: IAE 45.44 MSC: IAE 26.5.4.2.2 5 5 2 Figure 7. Closed-loop responses at perturbed ondition for the Cement Cooler Grate pressure IV. CONCLUSIONS Over the last few deades, a large number of PID tuning formulas based on the First-Order plus Deadtime (FOPDT) model have been developed. In this paper, we have presented a new approah based on the Multi-sale Control (MSC) sheme to onstruting a PID tuning formula for a proess represented by the FOPDT model. Based on the two industrial proesses (Steam Superheater and Cement Cooler Grate), we have shown the superiority of our new PID tuning formula based on the MSC sheme (MSC-PID) over some of the best PID tuning formulas established over the last several deades, e.g., Rovira et al. [4], Suyama [5], Witt & Waggoner [6] and aya &Sheib [7]. In future works, we will further extend the appliation of the MSC sheme to onstruting a few other PID tuning formulas based on the Seond-Order plus Deadtime (SOPDT) and Seond-Order Integrating plus Deadtime (SOIPDT) models. REFERENCES []. J. Astrom, "Limitation on ontrol system performane," Eur. J. Control, vol. 6, no., pp. 2-2, 2. [2] O. Aidan, A summary of PI and PID ontroller tuning rules for proesses with time delay. Part 2: PID ontroller tuning rules, Proeedings of PID: IFAC Workshop on Digital Control, pp. 242-247, Terrasa, Spain April 4-7, 2. [3] J. Nandong and Z. Zang, "High-performane multi-sale ontrol sheme for stable, integrating and unstable time-delay proesses," vol. 23, pp. 333-343. [4] A.A. Rovira, P.W. Murril and C.L. Smith, "Tuning ontroller for set point hanges," Intruments and Control Systems, pp.67-69, Deember 969. [5]. Suyama, "A simple design method for sampled-data PID ontrol systems with adequate step responses," Proedings of the International