International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 64...
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Design of multiloop controller for multivariable system using coefficient 2

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Design of multiloop controller for multivariable system using coefficient 2

  1. 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 253 DESIGN OF MULTILOOP CONTROLLER FOR MULTIVARIABLE SYSTEM USING COEFFICIENT DIAGRAM METHOD M. Senthilkumar a and S.Abraham Lincon b Department of Electronics and Instrumentation Engineering, Annamalai University, Annamalai Nagar, Tamilnadu, India ABSTRACT In this study the controller for coupled tank multivariable system is designed using coefficient Diagram method. Coefficient Diagram Method is one of the polynomial methods in control design. The controller design by CDM method is based on the choice of coefficients of the characteristics polynomial of the closed loop system according to the convenient performance such as equivalent time constant, stability indices and stability limit. Controller is designed for the coupled tank system by using CDM method; it is shown that CDM design is fairly stable and robust whilst giving the desired time domain system performance. Keywords: coefficient diagram method, coupled tank, multiloop, CDM-PI. I. INTRODUCTION Many processes in power plants, refinery process, aircrafts and chemical industries are multivariable or multi-input multi-output (MIMO). The control of MIMO processes are more complicated than SISO processes. The methodology used to design a controller for the SISO process cannot be applied for MIMO process because of the interaction exhibit between the loops. Many methods have been presented in the literature for control of MIMO process. Proportional-integral-derivative (PID) or Proportional-Integral (PI) based controllers are used very commonly to control TITO systems. Generalized Ziegler-Nichols method [2], feedforward method [3], decentralized relay autotuner method [4 , ISTE optimization method [5]] are among them. Usually two types of control schemes are available to control MIMO processes. The first is decentralized control scheme or multiloop control scheme, where single loop controllers are used (the controller matrix is a diagonal one). The second scheme is a full multivariable controller (known as the centralized controller), where the controller INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 4, May – June 2013, pp. 253-261 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 254 matrix is not a diagonal one. Multiloop controllers do not explicitly consider the decoupling of the inter-loop interactions unlike full multivariable controllers. In this work, the Coefficient Diagram Method (CDM) is used to design a controller for MIMO process. CDM is now a well established approach to design controllers that provide outstanding time domain characteristics in closed-loop (Manabe, 1994; Manabe, 1998S; Budiyono and Sutarto, 2004; Cahyadi, et al, 2004).Basically CDM is based on pole assignment, where the locations of the closed-loop system are obtained using predetermined templates. Although it has been demonstrated that the designs based on CDM has some robustness, it is possible to show that some of the nice characteristics of the design can be lost if large perturbations in the model of the system exist. II. COEFFICIENT DIAGRAM METHOD The design of controller is not a difficult except the robustness issue, if the denominator and numerator of the transfer function of the system are determined independently according to stability and response requirements. But this is also addressed by coefficient diagram method [7].CDM is a polynomial algebraic method which uses characteristics polynomial for controller design which also gives sufficient information with respect to stability, response and robustness in a single diagram. When the plant dynamics and the performance specifications are given, one can find the controller under some practical limitations together with the closed-loop transfer function satisfactorily. As a first step, the CDM approach specifies partially the closed-loop transfer function and the controller, simultaneously; then decides on the rest of parameters by design. The parameters are stability index ߛ௜ , equivalent time constant τ , and stability limit ߛ௜ ‫כ‬ , which represent the desired performance. The choice of the stability indices affect the stability and unsteady-state behavior of the system, and can also be used for the robustness investigation. As for the equivalent time constant, which specifies the response speed, hence the settling time [7]. The basic block diagram of CDM control system is shown in Fig 1. Where y is the output, r is the reference input, u is the controller signal and d is the external disturbance signal. N(s) and D(s) are the numerator and Denominator of the plant transfer function. A(s) is the denominator polynomial of the controller transfer function while F(s) and B(s) are called the reference numerator and the feedback numerator polynomials of the controller transfer function.CDM controller structure resembles to a Two Degree of Freedom (2DOF) control structure because it has two numerators in controller transfer function. The system output is given by yሺ‫ݏ‬ሻ ൌ ேሺ௦ሻிሺ௦ሻ ௉ሺ௦ሻ ‫ݎ‬ ൅ ஺ሺ௦ሻேሺ௦ሻ ௉ሺ௦ሻ ݀ (1) where P(s) is the characteristics polynomial of the closed loop system. ܲሺ‫ݏ‬ሻ ൌ ‫ܣ‬ሺ‫ݏ‬ሻ‫ܦ‬ሺ‫ݏ‬ሻ ൅ ‫ܤ‬ሺ‫ݏ‬ሻܰሺ‫ݏ‬ሻ ൌ ∑ ܽ௜‫ݏ‬௜௡ ௜ୀ଴ (2) CDM needs some design parameters with respect to the characteristic polynomial coefficient which are τ equivalent time constant, ߛ௜ stability index, and ߛ௜ ‫כ‬ stability limit. The relations between these parameters and the coefficients of the characteristic polynomial ( ai ) are shown in (3).
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 255 ߬ ൌ ௔భ ௔బ (3a) ߛ௜ ൌ ௔೔ మ ௔೔శభ௔೔షభ , ݅ ൌ 1~ሺ݊ െ 1ሻ, ߛ଴ ൌ ߛ௡ ൌ ∞ (3b) ߛ௜ ‫כ‬ ൌ ଵ ఊ೔షభ ൅ ଵ ఊ೔శభ (3c) From eq. 3a-c, the coefficients ai can be written as ܽ௜ ൌ ఛ೔ ∏ ఊ೔షೕ ೕ೔షభ ೕసభ ܽ଴ୀܼ௜ܽ଴ (4) The design parameters are substituted in eq (2) and the target characteristic polynomial is obtained as ܲ௧௔௥௚௘௧ሺ‫ݏ‬ሻ ൌ ܽ଴ ቈቊ∑ ቆ∏ ଵ ఊ೔షೕ ೕ ௜ିଵ ௝ୀଵ ቇ ሺ߬‫ݏ‬ሻ௜௡ ௜ୀଶ ቋ ൅ ߬‫ݏ‬ ൅ 1቉ (5) The equivalent time constant specifies the time response speed. The stability indices and the stability limit indices affect the stability and the time response. The variation of the stability indices due to plant parameter variation specifies the robustness. Fig 1. standard block diagram of CDM III. CONTROLLER DESIGN USING CDM Most of the processes encountered in industry are described as FOPTD ‫ܩ‬௣ሺ‫ݏ‬ሻ ൌ ௞೛ ఛ೛௦ାଵ ݁ିఏ௦ (6) Where kp is process gain τ is time constant and θ is time delay. Since the transfer function of the process is of two polynomials, one is numerator polynomial N(s) of degree m and other is the denominator polynomial D(s) of degree n where m n, the CDM controller polynomial A(s) and B(s) of structure shown in Fig 1are represented by ‫ܣ‬ሺ‫ݏ‬ሻ ൌ ∑ ݈௜‫ݏ‬௜ and ‫ܤ‬ሺ‫ݏ‬ሻ ൌ ∑ ݇௜‫ݏ‬௜௤ ௜ୀ଴ ௣ ௜ୀ଴ (7) + - y(s) F(s) + )s(D )s(N )s(A 1 B(s) r(s) CDM controller d(s)
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 256 For the controller to be realized the condition p q must be satisfied. For a good performance the degree of controller polynomial chosen is important. in this paper the controller polynomial for FOPTD process with numerator Taylor’s approximation is chosen as A(s)=s (8a) B(s) =k1s+k0 (8b) For computation of the coefficient of the controller polynomial in CDM pole placement method is used. A feedback controller is chosen by pole-placement technique and then, a feedforward controller is determined so as to match the steady-state gain of closed loop system. According to this, the controller polynomials which are determined by Eq. 8a and 8b are replaced in Eq. 2. Hence, a polynomial depending on the parameters ki and li is obtained. Then, a target characteristic polynomial Ptarget(s) is determined by placing the design parameters into Eq. 5. Equating these two polynomials. A(s) D(s) + B(s) N(s) =Ptarget(s) (9) Is obtained, which is known to be Diophantine equation. Solving these equations the controller coefficient for polynomial A(s) and B(s) is found .The numerator polynomial F(s) which is defined as pre-filter is chosen to be ‫ܨ‬ሺ‫ݏ‬ሻ ൌ ܲሺ‫ݏ‬ሻ ܰሺ‫ݏ‬ሻ⁄ |௦ୀ଴ ൌ ܲሺ0ሻ ܰሺ0ሻ⁄ (10) This way, the value of the error that may occur in the steady-state response of the closed loop system is reduced to zero. Thus, F(s) is computed by ‫ܨ‬ሺ‫ݏ‬ሻ ൌ ܲሺ‫ݏ‬ሻ ܰሺ‫ݏ‬ሻ⁄ |௦ୀ଴ ൌ 1 ݇௣⁄ (11) IV. CDM-PI The transfer function of conventional PI controller is ‫ܩ‬௖ሺ‫ݏ‬ሻ ൌ ݇௖ ቀ1 ൅ ଵ ்೔௦ ቁ (12) The controller gain kc and integral time Ti are related with polynomial coefficient as k1=kc and k0=kc/Ti . By using the CDM, the values of k1, and K0 can be designed as follows: 1) Find the equivalent time constant τ 2) The stability index γ2 = 2, γ1 = 2.5 [7] are used. 3) From eq. (2), derive the characteristic polynomial with the PI controller stated in eq (12) and equates to the characteristic polynomial obtained from eq (14). Then the parameters k1 and K0 of the PI controller are obtained. 4) Set the pre-filter B,(s) = K0. And Gf(s) is feed forward controller ‫ܩ‬௙ሺ‫ݏ‬ሻ ൌ ிሺ௦ሻ ஻ሺ௦ሻ ൌ ௞బ ௞భୱା௞బ ൌ ଵ ்೔௦ାଵ (13)
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 257 Fig 2. equivalent block diagram of CDM V. COUPLED TANK PROCESS Coupled tank is used in Petro-chemical industries, paper making industries and water treatment industries for processing chemicals or for mixing treatment .the control of level of fluid in tanks are a challenging problem due to interactions between tanks and also serves as a MIMO process. The schematic diagram of coupled tank process is shown in Fig 3. The controlled variables are levels of tank1 (h1) and( h2). The levels of the tank are maintained by manipulating the inflow to the tanks. βx is the valve ratio of the pipe between tank1 and tank2, β1 and β2 are the valve at the outlet of tank1 and tank2 respectively. The mass balance equation of the coupled tank process is ‫ܣ‬ଵ ௗ௛భ ௗ௧ ൌ ݇ଵ‫ݑ‬ଵ െ ߚଵܽଵඥ2݃‫ܪ‬ଵ െ ߚ௫ܽଵଶඥ2݃ሺ‫ܪ‬ଵ െ ‫ܪ‬ଶሻ (14) ‫ܣ‬ଶ ௗ௛మ ௗ௧ ൌ ݇ଶ‫ݑ‬ଶ ൅ ߚ௫ܽଵଶඥ2݃ሺ‫ܪ‬ଵ െ ‫ܪ‬ଶሻ െ ߚଶܽଶඥ2݃‫ܪ‬ଶ (15) Where A1 and A2 are the cross sectional area of the tanks, a1 and a2 is the cross sectional area of outlet pipe in tank1 and tank2 respectively, K1 and K2 are the gains of pump1 and pump2 and g is the specific gravity. The parameters of the process and its operating points are listed in Table I and Table II Fig 3. schematic diagram of coupled tank process GC(s) Gp(s) + - r(s) y(s) Gf(s) d(s) + h1 Fin1 Fin2 h2 Fout1 Fout2 ββββx ββββ1 ββββ2 A1 A2
  6. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 258 Table .I Parameters of Coupled Tank Process Table .II Operating Conditions of Coupled Tank Process VI. SIMULATION AND EXPERIMENTAL RESULTS OF COUPLED TANK PROCESS The transfer function model for the coupled tank process is identified using reaction curve method. ‫ܩ‬௣ሺ‫ݏ‬ሻ ൌ ቎ ଵ଺.ଽଽ ௘షభమ.ఴవೞ ሺଶଵସ.଴ଷ௦ ା ଵሻ ଺.଺ଽ ௘షళమ.ఱళೞ ሺଶ଴ସ.ଽଷ௦ ା ଵሻ ଽ.ଶଷ ௘షయఱ.బభೞ ሺଶହ଺.ସସ௦ ା ଵሻ ଵଵ.ଷ଼ ௘షమఱ.బరೞ ሺଵ଺ଽ.ଵହ௦ାଵሻ ቏ (16) The controller parameter are obtained using the above identified model.The multiloop PI controller for the coupled tank process is obtained using BLT method as for loop1 Kc1=0.2080 ,Ki1=0.5826 and for loop2 Kc2=0.1309 , Ki2=0.6926.The multloop CDM- PI controller is obtained with τ=25 and found to be Kc1=0.5300 ,Ki1=0.0590 and Kc2=0.7900 , Ki2=0.0880 for loop 1 and loop2 respectively The closed loop response of the coupled tank process for a setpoint change in tank 1 from its operating value of 18.32 to 25 is shown in Fig. 4and its coressponding interaction effect in tank2 is shown in Fig 5. Fig 6 shows the closed loop reponse of tank2 for a setpoint change of 17 cm from its operating value and its corresponding effect of interaction is shown in Fig 7.simulink oriented VDPID control system is used for the real time control of coupled tank process.VDPID is high speed two input two output interface card. A1, A2 a1,a2, a12 β1 β2 βx 154 0.5 0.7498 0.8040 0.2245 u1 u2 h1 h2 k1 k2 2.5 2.0 18.32 12.23 33.336 25.002
  7. 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 259 Fig.4 closed loop response of coupled tank for Fig.7 closed loop interaction response of setpoint change in tank1 coupled tank for setpoint change in tank2 Fig.5 closed loop interaction response of coupled Fig.8 closed loop response of coupled tank tank for setpoint change in tank1 for setpoint change in tank1 (real time) Fig.6 closed loop response of coupled tank for Fig.9 closed loop interaction response of setpoint change in tank2 coupled tank for setpoint change in tank1(real time) 0 100 200 300 400 500 600 11 12 13 14 15 16 17 18 Timein seconds Levelincm 0 50 100 150 200 250 300 350 400 450 500 12 13 14 15 16 17 18 19 20 21 Time in seconds Levelincm BLT CDM-PI 0 50 100 150 200 250 300 350 400 450 500 18 20 22 24 26 28 30 32 Time in seconds Levelincm BLT CDM-PI 0 50 100 150 200 250 300 350 400 450 500 18.1 18.15 18.2 18.25 18.3 18.35 18.4 18.45 18.5 18.55 Time in seconds Levelincm BLT CDM-PI 0 50 100 150 200 250 300 350 400 450 500 12.15 12.2 12.25 12.3 12.35 12.4 12.45 Time in seconds Levelincm BLT CDM-PI 0 100 200 300 400 500 600 18 20 22 24 26 28 30 Timein seconds Levelincm
  8. 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 260 Fig.10 closed loop response of coupled tank for Fig.11 closed loop interaction response of setpoint change in tank2 (real time) coupled tank for setpoint chang in tank2(real time) Fig.12 Block diagram of multiloop CDM-PI control system Table III Performance measure of controllers controller Setpoint change in tank1 Setpoint change in tank2 LOOP1 LOOP2 LOOP1 LOOP2 ISE IAE ISE IAE ISE IAE ISE IAE BLT 803.97 277.94 O.485 9.44 1.69 15.31 286.44 15.383 CDM-PI 257.97 89.33 0.37 4.123 0.793 5.95 93.1 45.63 0 100 200 300 400 500 600 10 12 14 16 18 20 Timein seconds Levelincm 0 100 200 300 400 500 600 15 20 25 Time inseconds Levelincm Gc1 Gc2 + + - - + + + + r1 y1 r2 y2 g11 g12 g21 g22 Gf1 Gf2
  9. 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME 261 VII. CONCLUSIONS In this paper Multiloop CDM-PI controller is designed for coupled tank process and compared with the controller designed by Biggest Log Modulus method through simulation and experimentation. The ISE and IAE are taken as performance indices. Results shows supremacy of the multi loop CDM-PI controller and the ease in design of Multiloop controller REFERENCES [1] M. Zhaung and D.P. Atherton (1994), “PID controller design for TITO system,” IEEProc- control Theory Appl., Vol. 141 no.2pp 111-120. [2] A. Niderlinski (1971), “A heuristic approach to the design of linear multivariable integrating control system,” Automatica, vol.7, 691-701. [3] P.J. Cawthrop (1987), “Continuous-Time self-Tuning Control”, Vol.1-Design, Research Studies Press, Letchworth, UK. [4] Q.G. Wang, B. Zou, T.H. Lee andQ. Bi (1996),”Autotuning of multivariable PID controllers from decentralized relay feedback,”Automatica, vol.33, no.3, pp49-55. [5] S. Majhi (2001), "SISO controller for TITO systems", presented at the Int. Canf on Energy. Automation and information Tech. [6] S. Skogestad ,I. Postlehwaite (1996 ), “Multivariable Feedback Control Analysis and Design ,” John Wiley & Sons Chichester. [7] S. Manbe(1998), “Coefficient Diagram Method,” Proceedings of the 14th IFAC Symposium on Automatic Control in Aerospace, Seoul. [8] K Astrom and T. Hagglund (1995) , “PID Controllers : Theory Design and Tuning ,”Instrument Society of America,2nd Edition. [9] I. Kaya (1995), “Autotuning of a new PI-PD Smith predictor based on time domain specifications,” ISA Transactions, vol.42 no.4,pp.559-575. [10] S.Manabe (1994), “A low cost inverted pendulum system for control system education,” Proc. Of the 3rd IFAC Symp. On advances in control Education, Tokyo. [11] A.Desbiens,A. Pomerlau and D. Hodouin (1996), “Frequency based tuning of SISO controller for two –by-two processes,” IEE Proc. Control Theory Appl., Vol .143 ,pp 49-55. [12] M.Senthilkumar and S. Abraham Lincon (2012), “Design of stabilizing PI controller for coupled tank MIMO process,” International Journal of Engineering Research and Development , Vol.3 no.10:pp.47–55. [13] S.E Hamamci,I. Kaya and M.Koksal (2001), “Improving performance for a class of processes using coefficient diagram method ,” presented at the 9th Mediterran conference on control , Dubrovnik . [14] S.E. Hamamci (2004) , “Simple polynomial controller design by the coefficient diagram method ,” WSEAS Trans. on Circuits and Systems, v.3, no.4, pp.951-956. [15] S.E. Hamamci and M. Koksal (2003) , “Robust controller design for TITO systems with Coefficient Diagram Method s,” CCA 2003 IEEE Conference on Control Applications . [16] W.L Luyben (1986),”simple methods for tuning SISO controllers in multivariable systems,”Ind.Eng.Chem. Process Des. Dev.25 654-660. [17] VenkataRamesh.Edara, B.Amarendra Reddy, Srikanth Monangi and M.Vimala, “Analytical Structures for Fuzzy PID Controllers and Applications”, International Journal of Electrical Engineering & Technology (IJEET), Volume 1, Issue 1, 2010, pp. 1 - 17, ISSN Print : 0976-6545, ISSN Online: 0976-6553.

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