A Novel PID Controller Cascaded With Higher Order Filter for FOPDT With Real Time Implementation

This work describes the design of Proportional-Integral-Derivative(PID) controller for stable and unstable First Order Processes with Dead Time(FOPDT). PID controller is augmented with a higher order filter. The PID and the filter parameters are analytically derived using polynomial method. Higher order time delay approximation has been considered for improved accuracy. Maximum sensitivity(MS) based analytical tuning procedure is presented and concrete tuning guidelines are derived. The suggested technique is validated against delay dominant and nonlinear processes. In terms of the various performance indices, the suggested technique is compared to existing methodologies. The proposed method is tested on a real time process to verify the practical application.

, [7], [8], 23 [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], 24 [20], [21]. When a process associated with dead time (time The associate editor coordinating the review of this manuscript and approving it for publication was Zhong Wu . responses independently. This feature enables for indepen-30 dent tuning of servo and regulatory responses. But, it is 31 difficult to analyse multi-loop control structures and tuning 32 also becomes cumbersome process. 33 The goal of this study is to provide a simple and effective 34 control method for unstable and stable FOPDT. A few of 35 the remarkable control strategies that are aimed to control 36 the Unstable First order Process with Dead time (UFOPDT) 37 are shortly covered here. For unstable processes, the Smith 38 predictor in its pristine form is not felicitous. Paor [2] has 39 developed a new variant of smith predictor based control 40 strategy for unstable processes. This modified smith predictor 41 has been amended by several of researchers since then [2], 42 [3], [4], [7], [8], [9], [10], [11], [13], [15], and [16]. In addi-43 tion to the modified Smith predictor control strategies, several 44 authors [5], [6], and [21] have developed internal model 45 control (IMC) based control techniques. The two degrees of 46 freedom (2 DOF) control structure that has been developed 47 by Ajmeri and Ali [18] can control the set point tracking 48 and disturbance rejection responses independently. However, 49 this approach is acceptable for the diminutive time delay to 50 time constant ratios (θ/τ ) only. Furthermore, it utilizes three 51 controllers that leads to intricacy in achieving tuning rules.

97
CONTROLLER 98 The method of deriving PID parameters is not unique. Sev-99 eral researchers have employed model based techniques in 100 deriving PID parameters ([5], [6], [14], [19], [20], [21], [22]). 101 Optimization based derivation of PID parameters is also pop-102 ular among the researchers( [23], [24], [25], [26]).The present 103 method has employed a model based technique i.e. poly-104 nomial technique to derive the unknown controller param-105 eters. Figure 1 shows the proposed control structure which 106 employs a conventional feedback structure with a set point 107 filter. In Figure 1, r indicates set point, F is set point filter, 108 Gc represents PID controller cascaded with fourth order filter, 109 d is process input disturbance, Gp indicates process and y 110 is process output. To reduce servo response overshoot and 111 alter servo response speed, the set point filter is utilised. 112 The generalised transfer function of UFOPDT is shown in 113 Equation 1.
where k, θ,τ represents gain of the process, dead time, pro-116 cess time constant respectively. Equations 2 and Equation 3 117 represent the servo and regulatory responses respectively.
The current study considers a PID controller cascaded with 122 fourth order filter as shown in Equation 4.
As illustrated in Equation 5, the process without delay can be 128 expressed as a ratio of two polynomials. where, VOLUME 10, 2022 in Equation 8 are key to describe the closed loop stability 140 dynamics of the proposed control scheme.
To obtain the unknown controller settings, it is customary .
Additional simplification leads to the CE as illustrated in 164 Equation 12, as shown at the bottom of the page. The PID 165 values are obtained by solving Equation 12 against a target 166 CE. The authors initial assumption is to have the target CE 167 as illustrated in Equation 13. This choice sets the location 168 of the closed loop poles at s = − 1 λ . However, the authors 169 have identified that this choice does not provide a effective 170 c 6 s 6 + c 5 s 5 + c 4 s 4 + c 3 s 3 + c 2 s 2 + c 1 s + 1 = 0 (12a) The chosen target CE can nullify the effect of few of the zeros  Table 1 shows the lowest possible minimum MS 213 that can be achieved w.r.t θ/τ . To alter the speed of response, 214 a specific range of tuning parameters is always required.
215 Table 2 shows different tuning ranges based on the θ/τ value.

216
In the case of a θ/τ value larger than 1, it is recommended to 217 keep to λ, which corresponds to the lowest MS.

218
If the values of k and τ in Equation 5 are negative, the 219 process is FOPDT. Figure 3 shows the effect of the θ/τ ratio 220   on minimum MS values for stable FOPDT. Table 3 shows 221 relation between λ and minimum possible MS value w.r.t θ/τ . 222 Table 4 shows the range of MS and λ values that are possible 223 for the given θ/τ .

231
The suggested filter suppresses the effect of zeros intro-232 duced by the controller, and the servo response speed can 233 be adjusted using the newly introduced tuning parameter γ . 234 However, because it is outside the closed loop, this tuning 235 parameter has no effect on the stability of the closed loop 236 system. As a result, when it comes to the closed loop stability, 237 γ selection is a trivial process.

239
In this section, various subsisting control strategies are 240 considered and compared to the suggested solution. Equa-241 tions 18-21 provide a mathematical representation of a variety 242 of widely used assessment parameters.        as shown in Figure 4 and the assessment reported in Table 7.

274
In the case of regulatory response, the suggested method 275 produces superior performance in IAE and settling time.  settings are listed in Table 6. For analysing the no model

319
The above process is previously studied by Muro-Cuellar 320 et al. [16] and Rao et al. [13]. Del Muro-Cuellar et.al [16] 321 method proved to be superior than other method. Table 5 322 shows the controller and its parameters for this method. 323 The controller parameters are calculated using the proposed 324 approach at λ = 2.75 and γ = 4. Table 6 shows the resulting 325 controller settings. Figure 8 depicts the nominal response for 326 a unit step as a set point at t = 0 sec and a perturbance of 327 0.003 units at t = 80 sec. In the case of servo response, it can 328 be concluded from the response and analysis shown in Table 7 329 that the other technique performs superior than the proposed. 330 However, in the instance of regulatory action, the suggested 331 method is capable of producing similar performance.

332
To examine the robust performance, + 2.5% variation in θ, 333 −2.5% variation in τ and +1.67% variation in k are used. 334 By observing the response shown in Figure 9 and perfor-335 mance comparison presented in Table 8, it is clear that the 336 proposed method performs better than the other technique. 337 The opposite control method is delivering relatively high 338 oscillations, settling times and large overshoot.

339
Example 4: 340 Control strategies developed for linear processes should 341 be applied to a higher order or nonlinear processes in 342 practicality. Nonlinear processes are linearized at an opera-343 tional point in order to obtain a linear model. The present 344 study considers a previously studied isothermal reactor 345 [11], [18], [19]. Cholette's model describing an isothermal 346 chemical reactor with non-ideal mixing [28] is shown in 347 Equation 25.
where, C A0 ,C A represents input, output concentrations 350 respectively. V is reactor volume and F is input flow rate. 351 The different operating parameters considered previously by 352 several authors [11], [20], and [18] are k 1 = 10 l/sec, k 2 = 353 10 l/mol, V = 1 l. This model is investigated at an operating 354 point F = 0.0333 l/sec, C A0 = 3.288 mol/l, C A = 355 1.316 mol/l. The measurement lag caused by the concentra-356 tion transducer is assumed to be the source of a dead period 357 of 20 seconds. The linearized model considered by [11], [20], 358 and [18] is shown in Equation 26 .
According to the data in Table 1, the value of λ is adjusted 361 to 21 (θ/τ is nearer to 0.2), and the value of γ is tuned 362 to 15. Tables 5 and 6 show the controller parameter settings 363 for the methods of Ajmeri and Ali [18] and the proposed 364 method, respectively. At t = 0 sec, the set point is varied 365 from 1.316 mol/l to 5 mol/l. Disturbance is assumed to be a 366 change in F, which is changed from 0.0333 l/sec to 0.4 l/sec at 367 t = 2000 sec. Figure 10 depicts the simulation results related 368 to nominal conditions.

369
Robust performance is examined by taking into account 370 a +30% variation in measurement lag, a −25% variation 371 in k 1 and a +25% variation in k 2 . Figure 11 depicts the 372 perturbed response. The suggested method has an overall 373 VOLUME 10, 2022 superior performance, especially in case of disturbance rejec-374 tion which is evident from the performance indices shown in 375   Tables 7 and 8.
the above process and the method of Vijayan et al. [17] 382 has shown superior performance than other method. 383 Vijayan et al. [17] method's controller parameter settings are 384 shown in Table 5. The proposed method's controller param-385 eters are obtained at λ is equal to 1.5 and γ is assumed 386 to be 0.5. Table 6 shows the controller parameters of the 387 proposed method. For the simulation study, a unit step input 388 is applied at t = 0 sec to analyse servo response and a 389 negative disturbance of magnitude 0.5 is applied at t = 50 sec 390 to analyse the regulatory response. Figure 12 illustrates the 391 nominal response. Table 7 shows a performance comparison 392 between two controllers. According to the responses, the pro- in k and θ, and a −10% variation in τ . Figure 13 depicts 398 the perturbed condition's response and Table 8 depicts the 399 performance analysis of two control strategies. Table 8 shows In a real-time scenario, the proposed method is compared 421 with Zhao et al. [22] method.    Figure 16 and performance matrix is represented in Table 10. 429 From Table 10, it is clear that the proposed method performs 430 better in terms of all performance indices both in servo 431 and regulatory conditions. In servo response, the proposed 432 method performs marginally better in terms of rise time but 433 significantly better in terms of all other performance indices. 434 In regulatory response, the proposed method performs 435 marginally better in terms of rise time and overshoot, but 436 significantly better in terms of all other performance indices. 437 In order to analyse the robustness of the proposed con-438 troller, a +20% perturbation is considered in θ. The perfor-439 mance curves of both methods are presented in Figure 17. 440 The comparative analysis of both methods are presented in 441  Table 10. The proposed method outperforms the other method 442 in perturbed conditions also which is evident from Table 10. 443 In servo response, the proposed method performs marginally 444 better in terms of rise time but significantly better in terms 445 of all other performance indices. In regulatory response, the 446 proposed method performs marginally better in terms of rise 447 time and overshoot, but significantly better in terms of all 448 other performance indices. The response of time varying set point profile is shown 455 in Figure 18. From the performance curves in Figure 18, 456 it is observed that the suggested control technique not only 457 tracking setpoint changes but also provided a better response 458 compared to the method of Zhao et al. [22]. 459 In order to assess the disturbance rejection performance, 460 multiple input disturbances are considered at different 461         The disturbance rejection case response of both methods is 465 depicted in Figure 19. From Figure 19, it is observed that the 466 proposed method gave a superior response.