Relay Auto Tuning Of Parallel Cascade Controller

Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA Relay Auto Tuning Of Parallel Cascade Controller Sathe Vivek and M. Chidambaram Abstract The present work is concerned with relay auto tuning of parallel cascade controllers. The method proposed by Srinivasan and Chidambaram [10] to analyze the conventional on-off relay oscillations for a single loop feedback controller is extended to the relay tuning of parallel cascade controllers. Using the ultimate gain and ultimate cross over frequency of the two loops, the inner loop (PI) and outer loop (PID) controllers are designed by Ziegler-Nichols tuning method. The performances of the controllers are compared with the results based on conventional relay analysis. The improved method of oscillation is that all higher order harmonics (of the relay output) are filtered by the system. Li et al. [7] have pointed out that error of –18% to +27% is obtained in the calculation of ku by this method. An excellent review on the relay feedback method is given by Yu [13]. Srinivasan and Chidambaram [10] have improved the conventional relay auto tune method by proposing a method to calculate the value of ku by using appropriate value of number of harmonics coming out of the system output. This method gives an accurate value of ku. Shen et al.[9] have used a biased relay for getting the model parameters of a FOPTD model. In the method the process gain analyzing biased auto tune method proposed for single is calculated from kp = ∫e(t) d(ωt) / ∫y(t) d(ωt), the limits of feedback controller by Srinivasan and Chidambaram [11] is integration are from 0 to 2π. Since the value of ku is calculated also applied to relay auto tune of parallel cascade controllers. form ku = 4h/(πa0), the method also does not give good results. The proposed methods give an improved performance over that Recently Srinivasan and Chidambaram [11] have proposed an of the conventional on-off relay tune method. improved analysis of the biased relay auto tune method. Key words: Parallel cascade, relay, PI controllers, asymmetric Parallel cascade control scheme utilizes two control loops: the secondary (or ‘slave’) or inner loop receives its set point from relay Introduction the primary (or ‘master’) or outer loop. In parallel cascade control manipulated variable affects both variables directly [6]. Åström and Hägglund [1] have suggested the use of an ideal Due to two control loops present, there are two controllers to be (on-off) relay to generate sustained closed loop oscillations. The tuned. Hang et al.[3] have proposed a relay auto tuning of series ultimate gain can be found using ku=4h/(πa0) [where h is the cascade control loops. They have used conventional on-off relay relay height and ‘a0’ is the amplitude of the closed loop testing and using the value of k from 4h/(πa ) and using u oscillation]. PID controllers can then be designed by using Ziegler-Nichols method. Luyben (1987) has employed the relay 0 Ziegler-Nichols tuning formulae, the controllers are tuned. With the inner loop under PI control action, the relay test is repeated feedback method to identify a first order plus time delay transfer for the outer loop. Vivek and Chidambaram [8] extended function (FOPTD) model. Once, ku and ωu are known, then the Srinivasan and Chidambaram [10] method to tune series cascade amplitude criterion and phase angle condition can be written controllers to get an improved performance. In the present work, down. To get the three parameters of FOPTD model, knowledge the methods of Srinivasan and Chidambaram [10],[11] are of the process gain or delay should be known. Luyben noted the applied to tune parallel cascade controllers. The improved delay form the initial portion of the relay oscillation. In deriving performance of the proposed parallel controller is compared the relation ku , an assumption made in the conventional relay with that of the conventional relay analysis. ISBN:978-988-98671-6-4 WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA 1. Propose method-1 ωu=0.67. Once PI controller is designed based on these values, The parallel cascade control scheme is considered here. First the conventional on-off relay is considered. The relay is used in the inner loop and the outer loop is kept under manual mode. The the inner loop is kept under PI and then the relay is kept in the outer loop. From the relay oscillation, the value of ku=1.13 is obtained by conventional method. Srinivasan and Chidambaram relay oscillations are noted. For simulation study, the process [10] method gives ku as 1.09 and ωu=0.47. A PID controller is mode is assumed are (k G ) =k exp(-4.0)/(s+1) and designed based on this value. The details of the results are given p p 2 p2 (kpGp)1=kp1exp(-6.0)/(s+1). Where kp1=1.0 and kp2=1.0. Using a in Table I. The servo response is evaluated for a step change in symmetric relay height of 1, the oscillation in the inner loop the set point. The response by the conventional analysis using output variable y2 is noted. The amplitude and frequency of single harmonics gives oscillatory response. Whereas, using the oscillations are noted as 0.98 and 0.67 respectively. Using the method of Srinivasan and Chidambaram [10] gives an excellent relation ku =4h/(πa0), the value of ku is obtained as 1.29. Based response. The IAE values for servo and regulatory performance are given in Table II. on the transfer function model, the exact value of ku is calculated as 1.18. Thus significant error is obtained in ku by the 2. Proposed method –2 conventional relay analysis. Using the results of relay testing, the PI settings are calculated by using Ziegler and Nichols continuous cycling tuning method as kc = 0.57 and τI = 7.8. Using these PI settings in the inner loop (instead of the relay) and introducing a relay in the outer loop, the oscillation in the outer loop is noted with amplitude of 1.12 and frequency of 0.48. The value of ku =1.13 is obtained from 4h/(πa0). Based on the relay test results, the outer loop PID controller is designed by using Ziegler-Nichols method as kc=0.68, τI =6.49 and τD =1.63. The closed loop servo response is evaluated for a unit step change in the set point and the response is shown in Fig. 1. An oscillatory response is obtained. Similar response is obtained for a regulatory problem also [with (kLGL)2 = 1] as shown in Fig. 2. In this section, the biased relay auto tune is applied with a relay height of +2 and –1. In literature for single loop asymmetric relay tuning method, the value of γ = 2 is used. Therefore, the relay height of +2 and –1 is used in the present study also. However, simulations studies are carried out with different relay heights of γ=2, 2.5, 3 and 4. The results are summarized in Table 3. It is observed that the resulted PI controller settings are not changed significantly. The method proposed by Srinivasan and Chidambaram [11] is extended here to parallel cascade systems. The value of ku for the inner loop based on the identified FOPTD model (kP=0.99,τ=0.97, τD =4.01) obtained by the relay method is 1.20 and frequency of oscillation ω=0.66. PI controller is designed The method of Srinivasan and Chidambaram [10] is applied by Ziegler-Nichols method. The inner loop is kept under PI and now. Srinivasan and Chidambaram [10] have given a method to the asymmetric relay is then used in the outer loop. Based on the find out the value of ku by considering the higher order oscillation obtained in y1 and hence based on the identified harmonics. The initial portion of the relay output gives an FOPTD model (kP =1.0, τ=0.99, τD =6.0), the value of ku is indication of how many higher order harmonics present in the obtained for the outer loop as 1.1 with ω=0.47. PID controller is relay output. A value of 5 higher order harmonics (N=5) is designed for the outer loop based on the values of ku and ωu. recommended. Let us use their method for analyzing the parallel The results of the relay test are given in Table I. The servo cascade auto tuning. response in y1 for a unit step change in the set point is shown in For the system under study, the value of ku in the inner loop is Fig. 4. The performance is as good as the proposed method-1. In obtained as 1.17 for the inner loop and frequency of oscillations the proposed method-1, the value of order (N) of higher order ISBN:978-988-98671-6-4 WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA harmonics is to be selected. Whereas in the asymmetric method obtained as 1.17 and 0.67. Using Tyreus-Luyben settings, the PI of Srinivasan and Chidambaram [11], there is no such value of settings are calculated as kc = 0.36 and τI = 20.59. Using this PI N required. In the asymmetric method, the model is to be setting in the secondary loop and a relay in the outer loop, an identified and then the controller settings are calculated. In the oscillations in the outer loop are noted with amplitude of 0.51 symmetrical method, the controller settings are calculated based and ωu = 0.48 respectively. By using the proposed method-1 on the ultimate values obtained from the relay test. Fig. 2 shows with N=9, the value of ku is calculated as 2.0. By using Tyreusthe regulatory response for step a change in the inner loop Luyben tuning formulas, the PID settings are calculated as kc = disturbance. 0.91, τI = 28.67 and τD =2.068. Table V shows the details of For a single loop system, it is known that Ziegler-Nichols tuning controller settings for conventional and proposed method-1. The formulae give an oscillatory response. An attempt is made here servo response in y1 for a unit step change in the set point is to use the tuning formulas other than Ziegler-Nichols tuning shown in Fig. 6. Fig. 7 shows the regulatory response for step a formulae. For a single loop control system, Tyreus and Luyben change in the inner loop disturbance. The performances are [12] and Luyben [6] have suggested improved tuning formulas. found to be sluggish. Basically Tyreus-Luyben method detunes the proportional gain and increases integral time. The performance of the Tyreus- The sluggish response observed because of detuning the Luyben tuning formula for the cascade control system has not proportional gain and increasing integral time for secondary been reported so far. An attempt is made here to compare the loop. In cascade control system the response of the secondary performance of the Tyreus-Luyben tuning formulas for a loop should be faster so as to take quick action on disturbance parallel cascade control system. In the present work, Tyreus- entering in the secondary loop before its effect is felt by main Luyben tuning formulae are applied to design the controller control variable. If Fig. 4 and Fig. 6 are compared then, the settings for the inner loop and outer loop. Since the relay test for response (rise time and settling time) using Ziegler-Nichols the outer loop depends on the settings for the inner loop, the tuning formulae, is faster than the response using Tyreusrelay tuning for the outer loop has to be repeated. As stated Luyben tuning formulae. In this view, the performance analyzed earlier, using a symmetric relay height of ±1, the oscillation in using Ziegler-Nichols tuning formulae is preferred. the inner loop output variable y2 is recorded. The amplitude and frequency of oscillations are noted as 0.98 and 0.67 Conclusions respectively. Based on the principle harmonics, the value of ku The modified analysis of relay auto tuning proposed for single is obtained as 1.29. The PI settings are calculated using Tyreusfeedback system by Srinivasan and Chidambaram [10] and Luyben tuning formula as kc = 0.4 and τI = 20.59. Using the PI modified analysis of asymmetric auto tuning by Srinivasan and settings in the inner loop and introducing a relay in the outer Chidambaram [11] are extended to tune parallel cascade loop, the oscillation in the outer loop is noted with a amplitude controllers. Both the methods effectively take care of higher of 0.55 and frequency of 0.49. The value of ku =2.3 is obtained order harmonics. The performances of the PI-PID controllers from 4h/(πa0). Based on the relay test results, the outer loop PID are compared with the conventional relay analysis (principle controller settings are designed as kc=1.04, τI =27.9 and τD =2.0 harmonic analysis). The present methods give a better Srinivasan and Chidambaram method (proposed method-1) for performance than that of the conventional analysis. The the improved auto tuning is applied now. For the inner loop, controller settings using the Tyreus-Luyben tuning formulae considering higher order harmonics (N=5) ku and ωu are ISBN:978-988-98671-6-4 WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA proposed for a single loop system gives a sluggish response References when applied to the parallel cascade control systems. Table I Controller setting comparisons using Ziegler-Nichols settings Loop Inner Outer Controller settings kc τI kc τI τD Asymmetric Relay 0.54 7.9 Symmetric relay N=1 N=5 0.5837 0.5294 7.8000 7.8000 N=1 N=9 0.6818 0.6568 6.4915 6.5660 1.6229 1.6415 0.664 6.6097 1.6524 [1] Åström, K.J. and T. Hägglund ; “Automatic tuning of simple regulators with specification on phase and amplitude margin,” Automatica, 20, 645 (1984) [2] Chidambaram, M., Applied Process Control, Allied Pub, New Delhi, pp 123-125 (1998) [3] Hang, C.C., A.P.Loh and V.U. Vasnani; “Relay feedback auto tuning of cascade controllers,” IEEE Control sys. Tech, CST-2,42 [4] Kreysizig, E. Advanced Engineering Mathematics, John Wiley, New York, 5th ed. 235, (1996) Table II. Performance Comparison of proposed methods and conventional method for (τd/τ)inner-loop= 4.0, (τd/τ)outer-loop= 6.0 Comparison parameters Overshoot Settling time Symetric Relay Con. PM-1 0.214 0.1196 98 68 Servo Regu. Servo Regu. 16.98 15.18 15.0 11.6 IAE Asy. Relay PM-2 0.1314 74 Servo Regu. 15.07 11.89 Con.: Conventional PM-1: Proposed method-1 PM-2: Proposed method-2 Table III. Effect of change in relay height (asymmetric relay testing) on PI settings Asymmetric relay PI controller [5] Luyben, W.L.; “Derivation of transfer function model for highly nonlinear distillation column,” Ind. Eng. Chem. Res. 26, 2490, (1987) [6] Luyben, W.L. and M. L. Luyben; Essentials of process control, McGraw-Hill international edition 1997, 301-308., (1997) [7] Li, W., E. Eskinat. and W.L. Luyben; “An improved autotune identification method,” Ind. Eng. Res. Des. 30, 1530, (1991) [8] Sathe Vivek and M. Chidambaram 2003; “Cascade controller tuning auto tune method,” Proceedings of 3rd International conference on Chemical and Bioprocess H γ ku kc τI 0.5 4.0 1.1859 0.5336 0.2050 1.0 1.0 1.0 2.0 2.5 3.0 1.1972 1.1902 1.1837 0.5387 0.5356 0.5327 0.1975 0.2000 0.2008 Engineering, (ICCBPE),, 27 - 29th August, 2003, Malaysia, pp 851-857 [9] Shen, S.H., J-S. Wu. and Yu, C-C.; “Use of biased relay feedback for system identification,” AIChE J., 42, 1174, (1996) Table IV. Controller setting comparisons using Tyreus-Luyben settings Loop Inner Outer Controller settings kc τI kc τI τD ISBN:978-988-98671-6-4 Conventional method N=1 0.4054 20.592 N=1 1.0548 27.984 2.019 Proposed method-1 N=5 0.3676 20.592 N=9 0.91 28.67 2.068 [10] Srinivasan, K. and M. Chidambaram; “An improved auto tune identification method,” Proc. Int. Conference on digital modeling & simulation (DAMS-2003), Jan 6-8, Coimbatore, India., (2003a) [11] Srinivasan, K. and M. Chidambaram; “Modified relay feedback method for improved system identification,” Comp. & Chem. Engg, 27, 727-732, (2003b) WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA [12] Tyreus, B.D. and W.L.Luyben, “Tuning PI controllers for [13] Yu, C-C.; Auto tuning of PID controllers: Relay feedback integrator-dead time ptocesess’, Ind. Eng. Chem. Res. 31, approach, Springer- Verlag, Berlin, (1999) 2625-2628 (1992) Fig. 1 Servo response in y1 using Ziegler-Nichols settings Outer oscillatory response – Conventional analysis Inner solid – Proposed method -1 Inner dash – Proposed method -2 Fig. 3 Servo response in y1 using Tyreus-Luyben settings Legends: Solid – Conventional analysis Inner dash – Proposed method -1 Fig. 2 Regulatory response in y1 for a disturbance in inner loop using Ziegler- Fig. 4 Regulatory response in y1 for a Nichols settings PID controller for outer disturbance in inner loop using Tyreus- loop and PI for inner loop (Legends: as in Luyben settings Fig. 1.) PID controller for outer loop and PI for inner loop (Legends: as in Fig. 3) ISBN:978-988-98671-6-4 WCECS 2007