CHAPTER 3 TUNING METHODS OF CONTROLLER

Transcription

1 57 CHAPTER 3 TUNING METHODS OF CONTROLLER 3.1 INTRODUCTION This chapter deals with a simple method of designing PI and PID controllers for first order plus time delay with integrator systems (FOPTDI). Controllers have been designed using various tuning methods like equating coefficient method (EQ), direct synthesis method (DS), model reference control (MRC) method and dual loop control (DLC) method. In order to demonstrate the effectiveness of the developed methods to tune PI and PID controller for setpoint tracking and disturbance rejection, the IMC and ZN tuning methodologies are chosen for comparison. The performance of the controller under uncertainty in model parameters has been analyzed and its robustness is verified. 3.2 EQUATING COEFFICIENT METHOD System whose dynamics are slow with large time constant can be approximated as integrating systems. Integrating processes are frequently encountered in the process industries. The process would be non-minimumphase if it contains time delays and/or right-half-plane zeros. The design of controllers for such integrating processes is challenging and an interesting problem. Conventional PI and PID methods with unity feedback control structure have been developed by several authors (Lee et al 2003) have

2 58 proposed unity feedback control structure for the control of integrating processes with time delay. In this equating coefficient method, the closed loop transfer function of the given process along with the given controller is being found for the servo problem and the coefficients of the corresponding s, s 2, s 3 in the numerator and that in the denominator are equated to find the tuning parameters. Tuning parameters can be of single or more than one. In this work, tuning of both PI and PID controllers has been done. The equations for the controller settings are simple. More the tuning parameter, the complexity of the tuning is increased. Closed loop identification method is preferred over that of the open loop method since the former is insensitive to disturbances. Transfer function models are used for designing PI/ PID controllers. The methods for designing PID controllers for unstable FOPTD systems have been developed by DePaor and O Malley (1989), Ho and Xu (1998), optimization method by Cheng and Hwang (1998), Manoj and Chidambaram (2001) and synthesis method by and Jung et al (1999). An excellent review of the work reported on the design of PID controllers is given by Astrom and Hagglund (1995). Transfer function models are used for designing PI/PID controllers. In many of these methods, one or two adjustable parameters are used to calculate the PID settings but the design procedure is complicated. In this work, a simple method proposed by Sakthe Vivek and Chidambaram (2005) to design PI and PID controllers for both stable and unstable First Order Plus Time Delay(FOPTD) system have been developed for FOPTDI systems.

3 EQ- PI Controller The Equating Coefficient (EC) method developed to design PI controllers for FOPTD system is extended for FOPTDI systems. The equating coefficient method gives simple equations for the controller settings. This method is based on matching the coefficient of corresponding powers of s in the numerator and that in the denominator of the closed loop transfer function, since the objective of the controller is to make y/y r =1. Controllers are designed using single tuning parameter, and to improve the performance of the controller under uncertainty in model parameters, concept of two tuning parameters is used. The performance of the closed loop system is evaluated for both the original and the approximated model. The controllers are also tuned using IMC and ZN, and its performance has been compared by simulation. The following sets of linear algebraic equations are obtained for PI controller for FOPTDI system, (1-a)k1+ 0.5(1 +a )k2 = 0 (3.1) 0.5(1 +a )k 1+ (1 -a= )k2 a (3.2) derived: By solving the equations (3.1) and (3.2) the following equations are k 1 =k c k p t d (3.3) k 2 = æ k1 t 1 ö ç ètd ø (3.4) q =st d (3.5)

4 60 The value of a is greater than one and this parameter is considered as tuning parameter. It has been found by simulation that a = 1.01 gives best result for Rmodel1, Rmodel2 and Rmodel3. The value of a less than 1.01 doesn t yield better Integral Square Error (ISE) and Integral Absolute Error (IAE) values. Solving the equations (3.3) to (3.5), the values of k 1 and k 2 are obtained. Using the definitions of k 1 and k 2, the PI controller settings are obtained as, kckpt = (3.6) d t t I d = (3.7) From the equations (3.6) and (3.7), the controller settings has been obtained by substituting the k p and t d of the given model. For k p =1.667, t d = and t = 0.558, the value of k c = 1.7 and t i = are obtained. By using these controller parameters, the transfer function model is simulated for both servo and regulatory responses and its performance has been compared with IMC and ZN methods EQ - PID Controller A stable FOPTD system with an Integrator is represented by -td ke s p s t s+ 1, where k is the process gain, t p d is the time delay and t is the time ( ) constant of the process. For the purpose of designing controllers, the dynamics of many processes can be described adequately by a FOPTD model. The method gives simple set of equations for the controller settings. The performance of the control system is compared with that of the IMC method

5 61 and ZN methods. The closed loop transfer function relating the output(y) to the set point (y r ) is given by, y(q) (k q + k + k q )e y (q) [q [( )q 1] (k q k k q )e ] 2 -q = 2 2 -q r t t d (3.8) where q = t s, and s - is the laplace operator. Using pade s approximation for q e - as [(1-0.5q)/ (1+0.5q)] in the denominator of the equation (3.8), the numerator and the denominator terms are expanded using the taylor series for 0.5q e and 0.5q e -. The coefficient of q in the numerator is equated to a 1 times that of the denominator of the closed loop transfer function. The coefficients of q 2 and q 3 of the numerator are equated to a 1 times that of the denominator. The following sets of linear algebraic equations are obtained for PID controller. Since the objective of the control system is to make y to follow y r, the corresponding coefficients of q, q 2 and q 3 of the numerator with that of the denominator are equated. Since the presence of integral model makes the offset zero, the constant term in the numerator and that in the denominator is the same. (1 -a )k + 0.5(1 +a )k = 0 (3.9) (1 +a )k + (1 -a )k + (1 -a=)k a (3.10) (1 -a )k (1 )k [0.5 ( )] a= t t d a 2 (3.11) By solving these linear algebraic equations 3.9, 3.10 and 3.11, PID controller settings are obtained. It has been found by simulation that a 1=1.1, a 2 =0.8 a 1 gives best result for Rmodel1, Rmodel2 and Rmodel3. The ISE

6 62 and IAE values are also obtained to evaluate the performance criteria. The performance of the EQ-PID controller is compared with the controller designed by the EQ-PI method. The Figure 3.1 and Figure 3.2 show the servo and regulatory response of EQ-PI controller respectively and it has been compared with the IMC and ZN methods of controller. The simulation shows that the IMC and ZN method produces a response having large undershoots on the other hand the EQ-PI method shows marginally quicker settling time than IMC and ZN method. EQ-PI IMC ZN Figure 3.1 Servo response of Rmodel 3

7 63 EQ-PI IMC ZN Figure 3.2 Regulatory response of Rmodel 3 The PI controller settings obtained by using the equations 3.6 and 3.7 for the FOPTDI system with k p = ,t =1.0659, t d =0.673 are k c =0.296, t i = By using these controller settings, the process has been simulated and it shows a response having large overshoot. With the same controller settings, the model3 has been simulated and response is shown in Figure 3.3 where IMC and ZN controller gives sluggish response with offset.

8 64 EQ-PID EQ-PI Figure 3.3 Regulatory response of model 3 The tuning parameters obtained using EQ-PID method are k c = , t I = and t d =0.4 for the average FOPTDI transfer function (Rmodel3) model. The same controller settings give better response for the Rmodel1 and Rmodel2 also.

9 65 EQ-PID EQ-PI Figure 3.4 Servo response under parameter uncertainty +12% in process gain k p The robustness of the EQ method is studied by using +12% perturbation in process gain k p from the nominal value is shown in Figure 3.4, whereas the controller settings are those calculated for the process with nominal parameters. The servo response of the system under +12% uncertainties in time constant t and +16% uncertainties in time delay t d are shown in Figure 3.5 and Figure 3.6 respectively. From these results, it is observed that the EQ-PI method produce an oscillatory response with overshoot and longer settling time following a setpoint.

10 66 EQ-PI EQ-PID Figure 3.5 Servo response under parameter uncertainty of +12% in time constant t EQ-PID EQ-PI Figure 3.6 Servo response under parameter uncertainty of +16% in time delay t d

11 67 Analyses of the results indicate that the tuning method IMC referring to the response curves of Figures 3.4 to 3.6 under uncertainty in process gain, time constant and time delay respectively, the EQ-PID method settles with less oscillation and offset. The ISE and IAE value comparison for EQ-PI and EQ-PID tuning method are shown in Table 3.1. Table 3.1 ISE, IAE value comparison for EQ-PI and EQ-PID controller (k c = , t I =1.0659,t d = 0.673); PI(k c =0.296, t i =0.002) Models for simulation RModel 1 RModel 2 RModel 3 Method Servo Response Regulatory Response ISE IAE ISE IAE PID PI PID PI PID PI The controller parameters obtained are k c = ,t i =0.0454, t d = 0.4 for the FOPTDI system with k p =3.1818,t =1.0659,t d = The performance evaluation criteria ISE and IAE values for EQ-PID method gives less value compare to EQ-PI controller. The performance under model parameter uncertainty is better for EQ-PID method compared to EQ-PI method. Also the Table 3.2 shows the ISE and IAE values comparison for the transfer function of the system under uncertainty in process gain k p, time constant t and time delay t d.

12 68 Table 3.2 ISE, IAE value comparison for system under parameter uncertainty (Parameters for controller design K p = , t =1.0659, t d = 0.673) S.No Method ISE values for uncertainty IAE values for uncertainty +12% in k p +12% in t +16% in t d +12% in k p +12% in t +16% in t d 1. EQ-PI EQ-PID For uncertainty in process parameters, the performance evaluation criteria such as ISE and IAE are comparatively less for EQ-PID method. 3.3 DIRECT SYNTHESIS METHOD Time delay is common in all process industries due to transportation delay, recycle loops, composition analysis loops and the like. These types of processes can be approximated as integrating processes with time delay for the purpose of designing controllers instead of controlling in the original form. Using direct synthesis method, a PID controller in series with a lead/lag compensator is designed for control of closed loop FOPTD processes with time delay. Guidelines are provided for selection of the desired closed loop tuning parameter. The method gives significant load disturbance rejection performance. With the reference to the literature studied, several PI and PID controllers design methods have been proposed by Bhattacharya et al (1993 and 1995) for control of integrating processes with time delay. However, when there is large time delay, control of integrating processes is

13 69 difficult because of the limitations imposed by the time delay on system performance and stability. The controller for the processes such as Integrating Plus Unstable First-Order Plus Time Delay (IUFOPTD), Double Integrating Plus Time Delay(DIPTD) and Double-Integrating Plus First-Order Plus Time Delay(DIFOPTD) have been addressed by Skogested (2003) and Hang et al (2003). But most of the method uses modified form of Smith Predictor whose structure is complicated. The number of tuning parameters required is also more. Practically a simple control structure with a simple controller is desirable as it is very easy for the operator to tune. In general G p is the process transfer function and G c is the controller transfer function. For designing the controller G c, the method developed by Seshagiri Rao et al (2008) has been considered. In this DS method for set point tracking, a simple controller design method with only one controller in a single feedback loop for all classes of integrating processes has been considered. The desired output behavior of the closed loop can be specified as a trajectory model based on the process to design the required form of the controller. With the conventional controllers, there may be problems like overshoot and settling time. In this work, based on the nature of the integrating process, the desired closed loop transfer function is chosen and correspondingly the controller structure is derived. In this case PID controller in series with lead / lag compensator is obtained. The controllers are also tuned using ZN method (Ziegler 1942) and Rivera et al (1986), and the performance has been compared by simulation. The direct synthesis method gives simple equations for the controller settings. The performance of the closed loop system has

14 70 been evaluated for both original and approximated model. The transfer function model is of the type, G p -qs ke = (3.12) s( t s + 1) The desired closed loop transfer function is considered as, y ( h s +h s + 1)e y ( l s + 1) r 2 -qs 2 1 (3.13) This design method is chosen here because the desired output behavior of the closed loop can be specified as a trajectory model based on the process to design the required form of the controller. The closed loop relation for setpoint changes is given by, y GG y 1 GG c p = (3.14) + r c p from equation (3.14) the controller is given by, 1 (y/y) r Gc = G p 1 - (y/y) r (3.15) According to the direct synthesis method, the closed loop trajectory model should be specified for designing the controller. The controller can be written as G c 1 (y/y) r d = G 1 -(y/y) p r d (3.16)

15 71 where (y/y r ) d is the desired closed loop trajectory for set-point changes. The PID controller is designed in series with the lead/ lag compensator. If the process is G p -qs ke =, the desired closed loop transfer function is s( t s + 1) 2 -qs æ y ( h 2s +h 1s + 1)e ö considered as, ç = 3. Using first-order pade èy r ( l s + 1) ø approximation for the time delay, after simplification the controller is obtained as, c c d i ( a s+ 1) 1 G = k (1 + +t s) ts ( b s + 1) (3.17) where k h1 = k(3l + 1.5ql qh - h ) c (3.18) t= h 1 1 h 2 t d = h ql a= 0.5, qb= t l + ql+ qh -h 2 ( ) (3.19) in which h 1 =3l+ q and h = 2 ( 0.5 q-tl ) 3 + ( 3t 2-1.5qt) l 2 + 3qtl qt 2 2 ) t(0.5 q+t) (3.20) The tuning parameter l should be selected in such a way that the resulting controller gains should be positive for positive values of k. Hence to get positive values of controller gain (k c ), the constraint to be followed is,

16 72 h < l + ql - h + qh (3.21) In addition, lshould be selected in such a way that the resulting controller gives good robust control performance. The initial value of the tuning parameter can be taken as equal to half of the time delay of the process to get good control performance. If not, then, the tuning parameter can be increased from this value till good nominal and robust control performance are achieved. For suitable value of l and b, the controller designed on DS method gives good control performances. However for high value of b, the phase lag imposed by the term (bs+1) in the controller is more, thus the designed controller with this value of b is not able to give robust control performances which results in low gain and phase margins of the open loop system than the required values (gain margin should be >1.7 and phase margin should be >35 0 for robust control of a process). Based on many simulation studies, it is observed that taking 0.1 b instead of b gives good compromise between nominal performance and robust control performance. Thus, in the present work, the value of b obtained is modified as 0.1b for simulation studies. The desired closed loop transfer function is chosen based on the nature of the integrating process and correspondingly the controller structure is derived in this method.

17 73 Rmodel1 Rmodel2 Rmodel3. Figure 3.7 Closed loop performance comparison of Rmodel 1, 2 and 3. Controller designed on Rmodel 2 In Figure 3.7, performance of the closed loop system are evaluated by giving a unit step input in the set point and a negative step input of 0.1 in the load at t= 25s. In this work, the controller is designed using DS method for Rmodel2 and simulated for other models such as Rmodel1 and Rmodel3. From the response curves, it is clearly observed that the controller gives better performance for the other models such as Rmodel1 and Rmodel2 too. Referring to Figure 3.8, the IMC method settles at a faster rate than that of DS method for model3 and the ZN gives a very sluggish response confirming that it may not be suited for process of this nature. Whereas the same IMC controller does not give good performance for model1 and model 2.

18 74 DS IMC ZN Figure 3.8 Closed loop performance comparison of model 3 Table 3.3 gives the ISE and IAE values for the closed loop response of the system. It is clearly seen that the DS method gives less ISE and IAE values comparatively Table 3.3 DS, IMC and NZ comparison of ISE and IAE values for servo problem Process Controller ISE IAE RModel 1 DS IMC ZN RModel 2 RModel 2 DS IMC ZN DS IMC ZN

19 75 For any closed loop system it is necessary to analyze the stability and robustness for uncertainties in the process. The controller parameters obtained for the FOPTDI system of Rmodel3 using the equations ( ) are K c =1.25; t i =0.125; t D = and b = with the tuning parametera = and q =0.67. Using these controller settings the closed loop performance of the system is evaluated. Figure 3.9 shows the response of the system for +34% perturbations in time delay and the DS method gives the best performance for upto ± 28 % uncertainty in K p, ± 26% uncertainty in t and ± 34% uncertainty in time delay t d whereas other two methods IMC and ZN gives unstable performance for such uncertainty values. DS IMC ZN Figure 3.9 Responses for a perturbation of +34 % in process time delay t for the Rmodel 2 Table 3.4 gives the corresponding ISE and IAE values for the closed loop response of the system under +34% uncertainty in time delay. IMC, ZN gives more higher error value.

20 76 Table 3.4 DS method comparison of ISE and IAE values for uncertainty in time delay Process Controller ISE IAE DS RModel 1 IMC Z-N DS RModel 2 IMC Z-N DS RModel 2 IMC Z-N By giving positive and negative step input at certain point of time, among the comparison of DS and EQ-PID, DS method gives faster settling time for both servo and regulatory response. Figure 3.10 and Figure 3.11 shows the closed loop response comparison for the methods DS and EQ-PID under uncertainty of +23% in process gain k p and +24% of time constantt. Whereas the controller has been designed for the nominal values. With such percentage of uncertainty the IMC method takes more settling time and gives undershoot also.

21 77 DS EQ-PID Figure 3.10 Closed loop response of RModel2 for uncertainty of 23% of process gain k p DS EQ-PID Figure 3.11 Closed loop response of RModel 2 for uncertainty of +24% times constant

22 78 Referring to the Table 3.5, the performance evaluation criteria ISE and IAE are low for DS method of controller when compared to the other method. The EQ-PID method produce large undershoot for the negative input and takes more time to settle than DS method. The response has been obtained after applying negative step input, The DS controller settles faster comparatively. PID controller tuning parameters have been obtained using different methods like EQ-PI, EQ-PID, optimization method, and direct synthesis method. In the equating coefficient method single and two tuning parameters have been used to design the controller. Table 3.5 Performance comparison of DS and EQ-PID method Models for Simulation Model 1 Method Evaluation Criteria ISE IAE DS EQ-PID DS Model 2 EQ-PID DS Model 3 EQ-PID This method gives simple equations and the performance of the system is evaluated by simulation comparing with IMC and ZN method. Among EQ-PI and EQ-PID, EQ-PID performance is better in terms of evaluation criteria like ISE and IAE. The controllers have also been tuned using and direct synthesis method. Comparing the DS with EQ-PID, DS gives less error value and overshoot.

23 MODEL REFERENCE CONTROL For controller design purposes, the dynamics of the processes are described by first order time delay model. In this model reference control Jacob and Chidambaram (1996) have designed PI controller to force the error to follow the given error dynamics for unstable first order plus time delay system. In this work, for stable first order plus time delay system with integrator, the model reference control method has been extended to design PID controller. By specifying the settling time, t s, the value of the controller gain is obtained. Thus the controller is robust. For nonlinear systems, the values of time constant and gain will be varying. Hence, the controller settings based on fixed values of time constant and gain have to be detuned. However, the controller is shown to be robust for perturbation in time constant and gain. Normally due to certain non linearity of the systems, many real systems exhibit multiple steady states. Sometimes for safety reasons, it may also be necessary to operate the system at unstable steady state. In this work PID controller for stable first order plus time delay systems has been designed. The performance of the closed loop system is evaluated for both the actual and the approximated model. The controllers are also tuned using Internal Model Control (IMC) and the performance is compared by simulation. The general transfer function of the process to be controlled is given by, k p - Ls e s( t s + 1) (3.21) where k p is the process gain,t is the time constant, and L is the process lag. The corresponding time domain description is given by,

24 80 && y = ((- y& + k u(t -L)) t (3.22) p with t=0, y& =0 Let the error can be defined as, e= yr - y (3.23) where given by, yr is the setpoint value. Hence, the double derivative of the error is && e= && y -&& y r Figure 3.12 Block diagram representation of the process with the control law equation && e = && y -[- y& + k u(t -L)] t (3.24) r p equated to To get the expression for u(t), let the right side of equation 3.24 be c I t 0 D. - k [e + (1 t ) ò edt +t e] (3.25)

25 81 Then, the control law equation u (t) is given by, t t u(t) = {y && r+ (y& t ) + k [e + (1 t c I) ò 0edt +tdedt]} t+ L (3.26) k p u(t) t = pt+ L, where t L kp p + = p + Lp& by solving the equation 3.26, the tuning parameters for PID controller are obtained as, k 25 = (3.27) z t c 2 s 1 t= 0.4t z (1 + +t s) 2 I s D tis (3.28) t D = 0.4ts (3.29) Thus specifying the settling time and damping coefficient ( z ) for the error system, controller gain and integral time can be calculated. To use the control law equation (3.26), it required the prediction of y at (t+ t d ). A simple prediction formula for y(t+ t d ) can be obtained from the truncated Taylor s series expansion of y(t+ t d ) as y(t+ t d )=y(t)+ t. ' d y(t). The block diagram representation of the feedback system of the control law is shown in Figure The closed loop system is thus stable and for a step input in y r, we get y equals y r as tà. For nonlinear systems, the value of time constant and gain will be varying. However, the present controller is shown to be robust for perturbations in the values of time constant and gain. Hence, it is expected that the present linear controller will give good performance on the nonlinear system also.

26 82 The performance of the control system is evaluated for step change in y r at the input and the load and the simulated results are shown in Figures 3.13 and 3.14 respectively. Referring to the servo response of Rmodel2 in Figure 3.13, MRC gives less overshoot and faster settling time. In the regulatory response of the same model, MRC gives less undershoot comparatively. MRC IMC Figure 3.13 Servo response of Rmodel 2 MRC IMC Figure 3.14 Regulatory response of Rmodel2

27 83 With the same controller settings, the other first order plus time delay with integrator models like Rmodel1 and Rmodel2 and the original models Model1, Model2 and Model3 are simulated for both servo and regulatory responses. The values of the error in both cases give very less value. The ISE and the IAE performance indices have been calculated and is shown in Table 3.6. The parameters of the open loop system are k p =2.1818, t = ; t d = The parameters obtained for PID controller are k c = 0.452; t I = 0.96 and t D =0.366 for MRC method and kc= 0.12;t I = and t D = for IMC method. The controller settings are obtained for the average model using MRC and IMC method. Table 3.6 Performance comparison of MRC with IMC(K c =0.452, t i 0.96 t d =0.36); IMC (K c =0.12, t i =0.012; t d =0.131) Models for simulation RModel 1 RModel 2 RModel 3 Method Servo Response Regulatory Response ISE IAE ISE IAE MRC IMC MRC IMC MRC IMC Figure 3.15 shows the step response of average original model. The IMC controller shows offset whereas the model reference controller gives faster settling time. The regulatory response of the same model is shown in Figure 3.16 demonstrate the improvement obtained with the MRC controller. The response of the MRC controller was comparatively better in terms of reducing the overshoot and settling time.

28 84 MRC IMC Figure 3.15 Servo response of model 3 MRC IMC Figure 3.16 Regulator response of model 3

29 85 The behavior of manipulated variable versus time behavior is shown in Figure The performance in terms of ISE and IAE is listed in Table 3.7. The performance of the model reference control is significantly better than that of the IMC controller. MRC IMC Figure 3.17 Manipulated variables versus time behavior Table 3.7 Performance comparison for actual model 2 Method Servo Response ISE IAE MRC IMC Regulatory Response ISE IAE MRC IMC

30 86 The simulation results demonstrate the capability of the MRC controller in accommodating uncertainty in the process model. Uncertainty of about -19% in time constantt, -29% in process lag L and -21% in process gain k p are separately introduced and the response is shown in the Figures 3.18 to 3.20 respectively. When the IMC controller almost fails to cope with the model-process mismatch, the MRC controller provides acceptable results. It can also be noticed that the response of the MRC controller settles at the desired setpoint with minimum oscillations compared to IMC controller where oscillations persist for quite long time. MRC IMC Figure 3.18 Comparison of MRC and IMC for uncertainty of -19% in time constant t for Rmodel 3

31 87 MRC IMC Figure 3.19 Comparison of MRC and IMC for uncertainty of -29% in process lag L for Rmodel 3 MRC IMC Figure 3.20 Comparison of MRC and IMC for uncertainty of -21% in process gain k p for Rmodel 3

32 88 The results of the above simulation examples indicate, in general, that the MRC controller reduced the sensitivity to modeling errors comparatively. In addition, both the overshoot and settling time have been considerably reduced. 3.5 DUAL LOOP PID CONTROL In the Dual loop control, the process to be controlled is controlled using dual loops. The general transfer function of the process to be controlled is given by equation (3.21), where k p is the process gain,t is the time constant and L is the process lag. The general block diagram of the dual loop PID is shown in the Figure Figure 3.21 Block diagram of the dual loop control PID The inner loop consists of PI controller and the outer loop consists of PID controller designed using direct synthesis method developed by Sheshagiri Roa et al (2009) for unstable system. In this work, it has been extended for stable system. The closed loop transfer function of the inner loop is given by y kke c p = y s + kke in c p -Ls -Ls The delay is approximated as resultant equation is given by, e Ls 1 0.5Ls = Ls - - and the

33 89 k ck p ( 1-0.5Ls) ( + ) + ( - ) y = y s 1 0.5Ls k k 1 0.5Ls in c p (3.30) c p( + ) ( ) -Ls y k k 1 0.5Ls e = y s 1 0.5L s k k 1 0.5L s ( + ) + ( -( ) ) in c p ( + ) -Ls y kckp 1 0.5Ls e = y 0.5Ls k k L s + k k ( ) 2 in c p c p The closed loop transfer function of the inner loop is given by, ( + ) -Ls y k ck p 1 0.5Ls e = y 0.5Ls k k L s + k k ( ) 2 in c p c p (3.31) Figure 3.22 Block diagram of the simplified dual PID Figure 3.22 shows the simple block diagram of the simplified dual PID controller. After simplification, the closed loop transfer function of the controller is obtained as, 1 ( a s + 1) Gc = k c(1 + +tds) tis ( b s + 1) (3.32)

34 90 where 3 0.5ql b= t l + ql+ h -h 2 ( ) t= i2 h1 t d2 h = h 2 1 a 0.5 =q h 1 =3l+q and (0.5 q-tl ) + (3t -1.5 qtl ) + 3qtl+ 0.5qt h 2 = t(0.5 q+t) The performance of the closed loop control system is evaluated for a step input y r. The response of the dual loop controller is compared with model reference controller. Simulation of the plant models have been conducted to study the performance of the dual loop controller comparing with MRC type controller. Referring to the figures 3.23 and 3.24, it is observed clearly that the dual loop PID controller provides improved performance and settles at a faster rate both in servo and regulatory responses whereas the MRC controller shows sluggish response.

35 91 Dual MRC Figure 3.23 Servo response of Rmodel2 Dual MRC Figure 3.24 Regulatory response of Rmodel 2

36 92 The sensitivity of the controller to model inaccuracies is also considered and is shown in the Figures 3.25 to The robustness of the controller is evaluated using ± 23% perturbations in the process gain and the controller used was the one designed for the actual value of the process parameters. Similarly robust response for perturbation in time delay and time constant is obtained. Dual IMRC Figure 3.25 The regulatory performance of controller under -23% uncertainty in process gain for Rmodel2 It is clear from the figures 3.26 and 3.27 that the MRC controller has reduced the effects of mismatch between the process and the model on the system performance and it is less sensitive to model errors. It provides stable response up to ± 20% variation in time delay and ± 26% variation in time constant.

37 93 Dual MRC Figure 3.26 Comparison of MRC and dual loop for uncertainty of -20% in time delay for Rmodel2 Dual MRC Figure 3.27 The servo performance of controller under -26% uncertainty in time constant for Rmodel 2

38 94 The simulation results demonstrate the capability of the dual loop controller in accommodating uncertainty in the process model. When the MRC controller almost fails to cope with model-process mismatch, the dual controller provides acceptable results. The response of the dual loop controller settles at the desired setpoint with minimum oscillations compared to the MRC where oscillations persist for quite long time. The ISE and the IAE performance indices have been calculated, and presented in Table 3.8. The content of the table shows the improvement in performance in terms of the ISE and IAE. Table 3.8 ISE, IAE value comparison of MRC and dual loop PID controller S.No Models Servo Response Regulator Response ISE IAE ISE IAE Model 1 MRC Dual Loop Model 2 MRC Dual Loop Model 3 MRC Dual Loop

39 95 Dual MRC Figure 3.28 The regulatory performance of controller under -34% uncertainty in time delay in the Rmodel2 Dual MRC Figure 3.29 Servo response of dual loop and MRC for RModel 2

40 96 The comparison of dual loop and MRC controller for both servo and regulatory response are shown in figures 3.28 and 3.29 respectively. Dual loop PID shows good response compared to MRC controller. Table 3.8 gives the calculated performance evaluation criteria. For all types of FOPTDI systems and the actual models like model1, model2 and model3 responses, the Dual loop PID method gives comparatively less error value. Dual MRC Figure 3.30 Servo Response of MRC, DS and Dual loop Figure 3.30 shows the overall comparison of servo response of MRC, DS and dual loop controller. The DS and Dual loop controller settles at the faster rate without any overshoot whereas the MRC controller settles with bit oscillation. Perturbation in process gain, time delay and time constant has also been simulated separately with the same controller parameters. Kharitonov s theorem has been used to determine the range at which the controller attains instability.

41 SUMMARY The transfer function model of the web guide used to control the position of the web in cold rolling mill is controlled using different tuning methods and its simulation results have been compared. The performance indices such as ISE and IAE values have been obtained and compared. The process parameters are perturbed for certain percentage of process gain, time constant and time delays separately and simulated with nominal controller parameters in order to check the range at which the controller attains instability.