DESIGN OF PI CONTROLLERS FOR MIMO SYSTEM WITH DECOUPLERS

Transcription

1 Int. J. Chem. Sci.: 4(3), 6, ISSN X DESIGN OF PI CONTROLLERS FOR MIMO SYSTEM WITH DECOUPLERS A. ILAKKIYA, S. DIVYA and M. MANNIMOZHI * School of Electrical Enineerin, VIT Universit, VELLORE 634 (T.N.) INDIA ABSTRACT In this paper, Proportional Interal (PI) Controllers are desined for unstable Multi Input Multi Output sstem with dela based on the equivalent transfer function (ETF) model. Decouplin method is proposed for multivariable processes. The simplified decoupler is used to eliminate the process interactions between the controlled and manipulated variables. PI controllers are desined for the diaonal elements of ETFs b snthesis method meant for unstable first order plus time dela (FOPTD) sstems to ive satisfactor performances. Since the unstable sstem has lare overshoot. To reduce the overshoot, a double loop PI controller is desined. Ke words: Two input two output control, Unstable sstem, PI controllers, Simplified decoupler, Robustness, Double loop control. INTRODUCTION Multi-loop proportional interal (PI) controllers, have been widel used for processes with small interactions because of their man practical advantaes such as their simple control structure and fewer tunin parameters. Basicall desin of Multi input multi output (MIMO) controllers is more complex than sinle input sinle output (SISO) controllers because of the loop interactions. The MIMO process can be controlled b decoupled controllers. Unstable SISO sstems are more difficult to control than that of the stable sstems. Control of unstable sstems is iven b Padmasree and Chidambaram. There are several methods to desin the unstable SISO Proportional Interal controllers. Luben et al. have iven the desin of decentralized PI controllers for a stable MIMO sstem. A comparative stud of some multivariable PI controller tunin methods for stable sstems iven b Tanttu et al. 3 It requires the transfer function matrix. Onl a few methods are * Author for correspondence; mmanimozhi@vit.ac.in

2 Int. J. Chem. Sci.: 4(3), available for unstable multivariable processes. Govindhaannan and Chidambaram 4 have iven the method of desinin multivariable PI controllers for unstable sstem. Decouplin is used to reduce the control loop interactions. Ideal, simplified and inverted are the three tpes of decuplin techniques. Ideal decouplin has the complicated decoupler elements. It is rarel used in practice, reatl facilitates the tunin of the controller transfer matrix. Simplified decouplin is b far the most popular method. Its main advantae is simplicit of its elements. Inverted decouplin, which is also rarel implemented. Ideal and inverted decouplin is sensitive to the modellin error. The decentralized controller wors well when the interaction amon the loop is not lare. If there is more interaction it won t ive acceptable response. The decentralized PI controllers do not stabilize the sstem if the unstable pole is present in each of the transfer function. Centralized controller reduces the interaction better than the decentralized controller. Kumar et al. 5 have iven a snthesis method to centralized PI controller for interactin multivariable processes. Desin of double loop PI controllers for the unstable sstems based on the Tanttu and Lieslehto method was iven b Govindhaannan and Chidambaram 6. The concept of equivalent transfer function/effective open-loop transfer functions (ETFs/EOTF to desin the multi-loop control sstem 7,8. Recentl Rajapandian and Chidambaram 9 have iven the simple decoupled equivalent transfer function method to Desinin controller for MIMO processes. This method is extended to unstable multivariable sstem b Hazaria and Chidambaram. However, assumption involved in this method is that decoupler should be stable. Simplified decoupler method ives the less interactions and better performances when compared to the ideal and inverted decouplin methods. Methodolo Desin based on ETF model Consider the control problem shown in Fi.. Because to completel characterize the process dnamics, there are two controlled variables and two manipulated variables, four process transfer functions are necessar. ( ( = Gp,( = G p,( u ( u ( ( ( = G p,( = G ( () p, u ( u (

3 6 A. Ilaia et al.: Desin of Pi Controllers for. Fi. : Bloc diaram for TITO sstem The transfer function in eq () can be used to determine the effect of a chane in either u or u on and. Simultaneous chane in u and u have an additive effect on each controlled variable b the principle of superposition. = Gp ( u( + Gp ( u( () ( p + p s = G u G u ( ) (3) These input-output relations can also be expressed in vector-matrix notation as = G u( (4) Where ( and u( are vectors with two elements G p ( is the process transfer function matrix, p ( u( = ; u =...(5) u G p( G p G = (6) p G p( G p The process transfer function models are expressed as FOPDT models G = p e θs ( τ s + ) (7) i =, ; j =,

4 Int. J. Chem. Sci.: 4(3), 6 6 If the second feedbac controller is the automatic mode, with, then the closed-loop transfer function between and u. r = u = (8) p p c p + c p This can be written as u p p( c p ) = p (9) ( + ) p c p Similarl for the second loop, u p p( c p) = p () ( + ) p c p The complicated relation of eqs (9) and () can be simplified b assumin two assumptions: First, the perfect controller approximation for the other loop was used to simplif the eqs (9), () that is i =, ( ci + ci ) = () Second, ETFs have the same structure of the correspondin open-loop model. B usin the perfect controller approximation, eqs (9) and () can be approximated as follows: eff p p p = = p () u p eff p p p = = p (3) u p Here, p eff and p eff are the effective open-loop transfer functions (EOTF). The formulation of EOTFs for the hiher dimension sstems is difficult. On the other, the derivation of ETFs is eas for hiher dimension sstems.

5 6 A. Ilaia et al.: Desin of Pi Controllers for. The basic steps required to obtain ETF for a TITO sstem are as follows:- To describe the dnamic properties of a transfer function, the normalized ain ( N ) for a particular transfer function, ( is defined as N = = (4) σ τ + θ Normalized ain matrix is expressed as N N τ + θ τ + θ N = KΘTar = = (5) N N τ + + θ τ θ Where Θ denotes Hadamard division, is the stead state ain, and T ar = τ + θ is defined as the averae residence time which sinifies the response speed of the controlled variable i to manipulated variable u j. Hence, the RNGA (denoted as φ) can be obtained as T φ = N N (6) Where φ φ φ = (7) φ φ Where denotes Hadamard multiplication. Relative averae residence time arra (RARTA), which is defined as the ratio of loop i u j averae residence times, when other loops are closed and when other loops are open, is iven b γ γ Γ = φ Θ Λ = (8) γ γ Hence for an unstable sstem, the ETF can be expressed as ˆ p = ˆ p ˆ θ s e ˆ τ s (9)

6 Int. J. Chem. Sci.: 4(3), 6 63 p ˆ p =, Λ τˆ = γτ, and θˆ = γθ It should be noted that the method is applicable when EOTF = ETF. Pairin criteria For multivariable controllers, parin is required. The problem of loop interaction can be minimized b a proper choice of input output pairins. If there is n input and n output in the control sstem, then n! was of pairin the controlled and manipulated variables will be there. Some of these control confiurations would be immediatel rejected as bein impractical and unworable. Bristol s relative ain arra method. It is a sstematic approach. Relative Gain Arra is a measure of process interactions. Where is the stead state ain. Desin of controller λ λ T RGA ( Λ) = = () λ λ Fi. constitutin of input variable u, output variable, process ain p, decoupler element d and controller ain c can be modelled as follows: The relationship between the input and process output is iven b = ( d( u( () Fi. : TITO process with simplified decoupler (d = ; d = )

7 64 A. Ilaia et al.: Desin of Pi Controllers for. The output of the TITO sstem with decoupler is ( = d d u( u * d( = * () Desin of simplified decoupler, d = ; d = d = p p ; d = p p Desin of double loop control structure to reduce the overshoot G p, process ain; G d, decoupler element; G c,i inner-loop diaonal proportional controllers; G c,o outer-loop diaonal proportional interal controllers;, output variable; r set point values; v, load variable. Fi. 3: Bloc diaram for the double loop control scheme Desin of inner loop diaonal P controller p c = (3) ξ.5 Desin of outer loop diaonal PI controller G p = p e θ s τ s + (4)

8 Int. J. Chem. Sci.: 4(3), 6 65 Here p is the stead state value reached, θ is the initial value noted, and τ = t θ, where t is the time taen to (.6 p ) from fiure of proportional controller. Desin of outer loop diaonal PI controller c τ = + θ p τ c.5 p I, τ τ = (5) Based on these values diaonal PI controller matrix for the outer loop can be obtained. Snthesis method In direct snthesis (DS) method, the controller desin is based on a process model. This method is used achieve the desired closed loop transfer function for MIMO processes with multiple time delas. c, = c, + τ I, (6) c, p, =.8668 ξ.888 for. ξ.7 τ τ I, =.53 e ξ for ξ.7 Where ξ = θ τ Simulation example Let us consider the transfer function matrix and it is expressed as s.3s.e.3e (7s + ) (7s + ) G = (7) p.8 s. 35s.8e 4.3e (9.5s + ) (9.s + )

9 66 A. Ilaia et al.: Desin of Pi Controllers for. From the iven transfer function matrix, stead state ain,. = Relative ain arra,.654 RGA( Λ) = Averae residence time, T ar 8 = Normalized ain, N.75 = Relative normalized ain arra,.5537 φ = Relative averae residence time arra,.9559 Γ = The ETF model matrix is iven b e.786e ˆ ( ) = 6.69s 6.97s p s (8) e.6455e 8.43s 8.794s

10 Int. J. Chem. Sci.: 4(3), 6 67 The simplified decoupler matrix is iven b d = (5.9s ) e (9.5s + ).45s.599e.7s (9) b The diaonal elements of PI controller is obtained b snthesis method is iven G c s = (3) s Effective open-loop transfer function (EOTF) diaonal elements is obtained as eff = u s.e = (7s + ) (7.7878s ) e s + 6.5s +.75s.35s.s eff 4.3e.65e = = (3) u (9.s + ) (9.5s + ) eff, eff are the diaonal elements of EOTF. The closed loop response of ETF is same as EOTF.. Y, Y Response, Interaction, Y, Y.5.5 Response, Interaction, Time( Time( Fi. 4: Response of the PI controllers for controlled variables with interaction separatel (, )

11 68 A. Ilaia et al.: Desin of Pi Controllers for U,U Response,u Interacion,u U,U - Response,u Interaction,u Time( Time( Fi. 5: Response of the PI controllers for manipulated variables with interaction separatel (u, u ) To reduce the overshoot dual loop control scheme is used. Here, the inner loop proportional controller tunin parameters are c, =.8 and c, = The fitted parameter are p, =.695, τ =.5, p, =.9999, τ =.5 and θ = θ =. Which is calculated from the response of P controller with decoupler. For the stabilized FOPTD sstem, diaonal PI controllers b the snthesis method and placed in the outer loop. Y Time( Y Time( Fi. 6: Response of proportional controllers with decoupler Hence the outer loop diaonal PI controller matrix is iven b G c s = (3) s

12 Int. J. Chem. Sci.: 4(3), 6 69 Fi. 7 shows that the response of the double-loop controller is much superior to that of a sinle loop PI controller. Fi. 8 shows the responses of sinle and double-loop controllers when the load variables v and v enter the sstem alon with the inputs u and u..5.5 Y Y.5 double loop, sinle loop,.5 double loop, sinle loop, Time( Time( Fi. 7: Response of sinle-loop and double loop two input two output PI controllers with decoupler for servo problems.. Y sinle loop, double loop, Y double loop, sinle loop, Time( Time( Fi. 8: Response of sinle-loop and double loop two input two output PI controllers with the decoupler for reulator problems; (left) v =., v = ; (riht) v =, v =. The robustness of the double-loop sstem is evaluated b disturbin each process ain, time dela (θ) and time constant (τ), ± % of the normal value in process. The robustness performance is shown in Fi. 9. Table shows the IAE/ISE values for the above robustness studies for the servo and reulator problems.

13 6 A. Ilaia et al.: Desin of Pi Controllers for Y Y.4. p p*. p*.9.4. p p*. p* Time( Time( Y Y p p*. p*.9.4. p p*. p* Time( Time( Fi. 9: Robustness of double-loop controllers for perturbation in each p (%) in the process: (top panel servo problem; (bottom panel reulator problem Table : Comparison of IAE and ISE values for disturbance in the model parameters for the double-loop control sstem Perturbation Servo () Servo () Reulator () Reulator () IAE ISE IAE ISE IAE ISE IAE ISE p p p θ Cont

14 Int. J. Chem. Sci.: 4(3), 6 6 Perturbation Servo () Servo () Reulator () Reulator () IAE ISE IAE ISE IAE ISE IAE ISE. θ θ τ τ τ CONCLUSION Based on the equivalent transfer function (ETF) model, multivariable PI controllers are desined for unstable multivariable sstems with time dela. The simplified decoupler reduces the loop interaction. The advantae of double-loop method is used to reduce the overshoots and eliminate the interactions. REFERENCES. R. Padmasree and M. Chidambaram, Control of Unstable Sstems, st Ed., Narosa Publishin House: New Delhi, India (6).. Luben, Michael L., and William L. Luben, Essentials of Process Control, McGraw- Hill Int., Sinapore (997). 3. T. Tanttu, Juha and Jua Lieslehto, A Comparative Stud of Some Multivariable PI Controller Tunin Methods, Intellient Tunin and Adaptive Control, (99). 4. J. Govindhaannan and M. Chidambaram, Multivariable PI Control of Unstable Sstems, Process Control and Qualit,, (997). 5. V. V. Kumar, V. S. R. Rao and M. Chidambaram, Centralized PI Controllers for Interactin Multivariable Processes b Snthesis Method, ISA Transactions, 5(3), 4-49 (). 6. J. Govindaannan and M. Chidambaram, Two-Stae Multivariable PID Controllers for Unstable Plus Time Dela Sstems, Indian Chem. En., 4(), (). 7. E. Ganon, A. Pomerleau and A. Desbiens, Simplified, Ideal or Inverted Decouplin? ISA Transactions, 37(4), (998).

15 6 A. Ilaia et al.: Desin of Pi Controllers for. 8. H. P. Huan, J. C. Jen, C. H. Chian and W. A. Pan, Direct Method for Multi Loop P/PID Controller Desin, J. Process Control, 3, (3). 9. C. Rajapandian and M. Chidambaram, Controller Desin for MIMO Processes Based on Simple Decoupled Equivalent Transfer Functions and Simplified Decoupler. Ind. En. Chem. Res., 5, ().. S. Hazaria and M. Chidambaram, Desin of Proportional Interal Controllers with Decouplers for Unstable Two Input Two Output Sstems, Industrial En. Chem. Res., 53(5), (4). Revised : Accepted :.6.6