A simple nonlinear PD controller for integrating processes

A simple nonlinear PD controller for integrating processes

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  Research Article A simplenonlinear PD controller for integrating processes Chanchal Dey a , Rajani K. Mudi b,n , Dharmana Simhachalam b a Department of of Applied Physics, University of Calcutta, 92, A.P.C. Road, Calcutta 700009, West Bengal, India b Department of Instrumentation & Electronics Engineering, Jadavpur University, Sector III, Block LB/8, Salt-lake, Calcutta 700098, West Bengal, India a r t i c l e i n f o Article history: Received 2 September 2012 Received in revised form 3 September 2013 Accepted 4 September 2013 Available online 2 October 2013 This paper was recommended for publication by Prof. A.B. Rad Keywords: PID control Nonlinear PD control Integrating process a b s t r a c t Many industrial processes are found to be integrating in nature, for which widely used Ziegler–Nichols tuned PID controllers usually fail to provide satisfactory performance due to excessive overshoot with large settling time. Although, IMC (Internal Model Control) based PID controllers are capable to reduce the overshoot, but little improvement is found in the load disturbance response. Here, we propose an auto-tuning proportional-derivative controller (APD) where a nonlinear gain updating factor α con- tinuously adjusts the proportional and derivative gains to achieve an overall improved performance during set point change as well as load disturbance. The value of α is obtained by a simple relation based on the instantaneous values of normalized error (eN) and change of error (ΔeN) of the controlled variable. Performance of the proposed nonlinear PD controller (APD) is tested and compared with other PD and PID tuning rules for pure integrating plus delay (IPD) and first-order integrating plus delay (FOIPD) processes. Effectiveness of the proposed scheme is verified on a laboratory scale servo position control system. & 2013 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Proportional-integral-derivative (PID) controllers are widely used in various industrial process control applications due to their simplicity and effectiveness [1–6]. But, performances of model free PID controllers are not usually satisfactory due to their oscillatory responses and large settling time for integrating processes with time delay [7–9]. For example, Fig. 1 shows such poor perfor- mances of the well known Ziegler–Nichols tuned PID (ZNPID) [10] controllers for IPD and FOIPD processes. In Fig. 1, initially a step set point change is applied and when the process reaches its steady state, an impulse load disturbance is introduced at the process input. Here, overshoots are found to be more than 60%, which is not acceptable in most of the applications [6]. On the other than hand, model based PID control techniques, like IMC can provide lower overshoot with faster settling on proper selection of tuning parameters (close-loop time constant) [11–15]. Fig. 2 shows the responses of model based IMC-PID controllers [11] for IPD and FOIPD processes. It is found that lower overshoot with faster settling is achieved in the set point response for IMC-PID com- pared to ZNPID but no significant improvement is observed during load rejection. Due to the presence of integral action, PID controllers are likely to produce oscillations for integrating plus dead time processes [1]. In general, proportional-derivative (PD) controllers, if properly designed [6], are capable of providing reasonable performances compared to PID controllers for integrating or zero-load processes with delay [16]. Robots and manipulators are extensively used in automation based manufacturing processes where any type of overshoot and/or undershoot is highly undesirable [17] in position- ing their arms. In spite of noise sensitivity, PD controllers help to reduce the overshoot by introducing higher damping [18]. So, there is a scope for designing improved PD controllers to achieve desired performance for integrating processes with dead time. But, till today, unlike PID controllers, probably there are less running schemes for PD controllers [19]. For IPD processes, Chidambaram and Padma Sree [16] used equating coefficient method to find the parameters of a PD controller, which will be denoted here as CPPD. Kristiansson and Lennartson [20] reported that the derivative action can signifi- cantly improve the control performance compared to PI control with equal stability margin for most of the plants including those with noticeably large time delay. Authors in [21] proposed a Lyapunov based approach to obtain PD parameters. Xu et al. [22] designed a nonlinear PD controller with increased damping corresponding to its linear counterpart. Its proportional and derivative gains are modulated nonlinearly based on the instanta- neous value of error (e) and the sign of change of error (Δe), and its various tunable parameters are chosen heuristically maintaining the stability of the system. Visioli [23] used genetic algorithm Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.09.011 n Corresponding author. Tel.: þ91 33 23352587; fax: þ91 33 23357254. E-mail addresses: chanchaldey@yahoo.co.in, cdaphy@caluniv.ac.in (C. Dey), rkmudi@yahoo.com, rkmudi@iee.jusl.ac.in (R.K. Mudi), chalamju10@gmail.com (D. Simhachalam). ISA Transactions 53 (2014) 162–172
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  based optimization schemeto minimize various integral errors like ISE and ITSE and finally got PD settings (here termed as VPD) for set point tracking. For FOIPD processes, Vítečková et al. [24] and O'Dwyer [25] suggested PD controllers towards achieving improved responses. There is a possibility to enhance the perfor- mance of a PD controller by extending its integer order of the derivative element to fractional order [26]. Fuzzy controllers have been successfully designed with improved performances compared to their conventional counter- parts [27–33]. In [28], the nonlinear gain modification scheme following an operator's strategy continuously adjusts the output scaling factor (considered to be the close-loop gain) of a fuzzy PD controller with the help of 49 fuzzy If-Then rules, defined on the current process states, i.e., e and Δe. Su et al. [29] developed a hybrid fuzzy PD controller by combining two nonlinear tracking differentiators to a conventional fuzzy PD controller. In spite of a number of merits, there are many limitations while designing a fuzzy controller, since, till now there is no standard methodology for its various design steps. Moreover, no clear guidelines are available for selecting appropriate values of its large number of design parameters. The above discussion and our literature survey reveal that compared to the well established PI and PID control techniques, less importance/attention has been given for the development of conventional PD control. Hence, there is a good prospect for further development of PD controllers with enhanced perfor- mance. With this perspective in mind, and encouraging results of [22,28], in this study, we are motivated to introduce a real time nonlinear gain modification scheme for a PD controller. Due to lack of a suitable auto-tuning scheme, here, we consider the most widely accepted Ziegler–Nichols ultimate cycle based PID (i.e., ZNPID) tuning rules [10] by ignoring its integral part for the initial setting of the proposed PD controller (APD). Note that, Ziegler– Nichols rules were originally developed for the tuning of P, PI, and PID controllers, but not for a PD controller. In the proposed APD, the proportional and derivative gains are continuously adjusted depending on the instantaneous process trend by introducing a nonlinear gain updating parameter α. The basic idea behind this real time gain adjustment mechanism is that when the process is moving towards the set point, control action will be conservative to avoid possible large overshoots and/or undershoots in the subsequent operating phases, and when the process is moving away from the set point, control action will be aggressive to bring it back quickly to its desired value. It is to be mentioned that, our proposed scheme is different from others [22,28] as far as its design simplicity and practical implementation are concerned. Nonlinear gain variation in [22] involves a number of heuristically chosen tunable parameters along with an exponential function in its gain update rules, whereas, in [28] the output scaling factor is adjusted through a number of fuzzy conditional rules derived from experts’ knowledge. The performance of the proposed PD controller is tested and compared with a large number of PID and PD tuning rules reported over the last decade for IPD and FOIPD processes. In addition, real time experimentation is also performed on a laboratory scale DC servo position control system. Performance analysis with respect to a number of performance indices reveals that APD is capable of providing an overall improved performance in comparison with PID settings given by AHPID [2], DMPID [3], ZNPID [10], SLPPID [11], CPPID [16], VPID [23], PCPID [34], ACPID [35], RRCPID [36], AMPID [37], HXCPID [38], RPID [50], and PD settings by CPPD [16], VPD [23], and LLPPD [39] for the IPD process under both set point change and load disturbance. Similarly, enhanced performance is also observed for the FOIPD process in comparison with PID tuning rules of DMPID [3], ZNPID [10], SPID [12], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42], SLPID [44], KLPID [47], RPID [50] and PD settings of VVSPD [24], DPD [25], EOMPD [45], and SJLPD [46]. Performance robustness of the proposed APD is studied with þ25% perturbations in process dead time and also in presence of measurement noise. Its stability robustness is established from the gain margin (GM) and phase margin (PM) values as well as through Kharitonov polynomials [48]. The rest of the paper is divided into three sections. In Section 2, we describe the various steps of the controller design, its nonlinear gain variation mechanism for different operat- ing points during transient phase, and its stability and robustness issues. Section 3 presents simulation study with detailed perfor- mance analysis as well as real time experimentation on a DC servo position control system modeled as a FOIPD process. There is a conclusion in Section 4. 2. The proposed controller A simplified block diagram of the proposed APD is shown in Fig. 3. In order to achieving a faster convergence of the system with smaller overshoot and undershoot, both the proportional and derivative gains are modified at each sampling instant based on the instantaneous values of normalized error eN and change of 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time(sec) Response IPD FOIPD Fig. 1. Responses for IPD ½GpðsÞ ¼ 0:0506eÀ 6s =sŠ and FOIPD ½GpðsÞ ¼ eÀ 4s =sð4sþ1ÞŠ processes under ZNPID [10]. 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response IPD FOIPD Fig. 2. Responses for IPD ½GpðsÞ ¼ 0:0506eÀ6s =sŠ and FOIPD ½GpðsÞ ¼ eÀ 4s =sð4sþ1ÞŠ processes under IMC-PID [11]. - r Process + + Normalization Ne - y Z −1 + e e (1 + )αpK + +NeΔ - × Load disturbance + + + + Noise Kd(1+γ )α Z −1 Fig. 3. Block diagram of the proposed auto-tuning PD controller (APD). C. Dey et al. / ISA Transactions 53 (2014) 162–172 163
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  error ΔeN ofthe controlled variable. In this study, load disturbance is applied at the process input and white noise is introduced at the process output as shown in Fig. 3. 2.1. Design of the proposed controller APD Discrete form of a PD controller at kth sampling instant can be described as: uðkÞ ¼ Kp eðkÞþ Td Δt ΔeðkÞ ð1Þ or uðkÞ ¼ KpeðkÞþKdΔeðkÞ; ð2Þ when Kd ¼ Kp Td Δt Here, Kp is the proportional gain, Td is the derivative time, Kd is the derivative gain, and Δt is the sampling interval. Error e(k) and change of error Δe(k) at kth sampling instant are defined by eðkÞ ¼ rÀyðkÞ; and ΔeðkÞ ¼ eðkÞÀeðkÀ1Þ: ð3Þ In Eq. (3), r is the set point and y(k) is the process output at kth instant. It has already been mentioned that, due to absence of any simple auto-tuning scheme for the initial setting of the propor- tional and derivative gains of a PD controller, we have chosen the widely practiced Ziegler–Nichols PID tuning rules [10] by dropping its integral term. Therefore, the values of Kp and Td of the initial PD controller are obtained by the following relations: Kp ¼ 0:6Ku; ð4Þ and Td ¼ 0:125Tu: ð5Þ where Ku is the ultimate gain and Tu is the ultimate period obtained by relay-feedback test [51]. These initial values of Kp and Kd (Eq. (2)) get modified continuously by the nonlinear gain updating factor α, which is defined by αðkÞ ¼ eNðkÞ Â ΔeNðkÞ; ð6Þ where eNðkÞ ¼ eðkÞ=jrj and ΔeNðkÞ ¼ eNðkÞÀeNðkÀ1Þ. In Eq. (6), eN(k) and ΔeN(k) are considered as the normalized values of e(k) and Δe(k), respectively. Eq. (6) indicates that the instantaneous value of α(k) may vary between À1 and þ1, since the usual range of eN(k) or ΔeN(k) is [À1, 1] for an acceptable close- loop performance where the peak overshoot remains within 100%. Observe that the gain updating parameter α changes with the instantaneous values of eN and ΔeN, hence, it contains the information regarding the current position and direction of move- ment of the process in the response trajectory. We utilize this intelligence property of α to realize the desired gain variation strategy mentioned earlier, i.e., control action will be made conservative or aggressive depending on the movement of the process towards or away from its set point, respectively. Keeping in mind such a real time gain adjustment in APD, we propose the following update rules: Ka pðkÞ ¼ Kpð1þαðkÞÞ; ð7Þ Ka dðkÞ ¼ Kdð1þγjαðkÞjÞ: ð8Þ Here, Ka p and Ka d are the time varying nonlinear proportional and derivative gains. γ is a positive constant for providing an appropriate variation of damping to get the desired close-loop response. In steady state condition αðkÞ ¼ 0, hence Ka pðkÞ ¼ Kp and Ka dðkÞ ¼ Kd. During transient phases, Ka p may be higher or lower than Kp depending on the sign of α but Ka d will always be higher than Kd. Such nonlinear gain variations are expected to provide necessary damping required for achieving an enhanced control performance. Detailed discussion about this online gain modifica- tion mechanism at different operating points and its influence on the close-loop performance is provided in the following Section 2.2. Thus, the proposed APD can be expressed as: ua ðkÞ ¼ Ka pðkÞ eðkÞþKa dðkÞ ΔeðkÞ: ð9Þ For proper tuning of the controller, suitable value of γ is to be selected, which may be done either using operator's knowledge or by trial depending on the process dynamics. In addition, for specific performance based applications where some constraints are imposed on the system behavior, the value of γ may be obtained through some optimization technique, so that the result- ing system response meets those specified performance indices. Through an extensive simulation study on a large number of integrating (IPD and FOIPD) processes, we suggest the following simple empirical relation for γ, which gives an overall satisfactory performance for all such cases. γ ¼ 2  Ku  Tu  K: ð10Þ In Eq. (10), K is the open-loop gain of the related process. Observe that γ depends on the dynamics of the process under control, which may be characterized by its critical point (Ku, Tu). Moreover, γ does not increase the design complexity as all the parameters, i.e., Ku, Tu, and K are obtained from the relay-feedback test [51], which is also used for setting the initial parameters of the proposed APD. We use the same relation of γ, i.e., Eq. (10), for both the IPD and FOIPD processes in our simulation as well as in experimental study on a DC servo system. In (Section 3) to come, we will observe that considerable deviations of γ from its respec- tive nominal values bring only a little change in the close-loop performance of IPD as well as FOIPD processes. 2.2. Online gain modification From the dynamic proportional and derivative gain expres- sions, as given by Eqs. (7) and (8), it is evident that α makes real time variations in Ka p and Ka d. The gain update rules (Eqs. (7) and (8)) are chosen in such a way that the proposed APD will be able to provide an improved performance under both set point change and load disturbance. Note that, unlike the linear control surface of PD controller with static gains, control surface of APD as shown in Fig. 4 is highly nonlinear in nature. Now, we explain how the proportional and derivative gains of APD are modified by the proposed scheme for providing appropriate control action under different operating conditions for achieving the desired perfor- mance. For a better understanding, we refer Fig. 5, which represents -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 u a ΔeN eN Fig. 4. Control surface of APD. C. Dey et al. / ISA Transactions 53 (2014) 162–172164
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  a typical close-loopresponse of an under-damped second-order process due to set point change and load disturbance. (i) During the operating stage, when the process is moving fast towards the set point (e.g., point like A or C in Fig. 5), there is a possibility of larger overshoot or undershoot in the subsequent operating phase. To avoid such situations, a considerable amount of damping should be present in the control action. This may be possible either by increasing the derivative action or by decreas- ing the proportional action separately, or by making such changes simultaneously. In this case, since e and Δe are of opposite sign, α will be negative, as a result, Ka p oKp (Eq. (7)). At the same time Eq. (8) indicates that Ka d 4Kd. A combined effect of the increased damping and reduced proportional gain will make the speed of response slow, which is expected to reduce the overshoot or undershoot. Thus, the proposed gain variation mechanism fulfills the requirement of appropriate control action for achieving an improved transient response. (ii) Opposite to the previous operating stage, when the process is far from the set point and moving away very fast from it (e.g., point B in Fig. 5), both the proportional and derivative gains should be large enough to restore the controlled variable quickly to its desired value. Under such situations, both e and Δe have large values with the same sign, thereby making α large and positive according to Eq. (6). Such a large positive value of α makes Ka p 4Kp and Ka d 4Kd respectively according to Eqs. (7) and (8). Therefore, ua 4u (i.e., control action becomes more aggressive) according to Eqs. (2) and (9). Similar situa- tion is also observed during load disturbance. Immediately after a load disturbance, e may be small, but Δe will be sufficiently large (e.g., point D in Fig. 5) and they are of the same sign, as a result α becomes positive. Therefore, according to Eqs. (7) and (8) both the proportional and derivative gains of APD will be higher than those of conventional PD controller. These higher gains will make APD to generate the required strong control action for restoring the process quickly to its desired value. Thus, APD is capable of providing the required variation in control action to improve the process recovery. From the above discussion, it appears that our simple gain modification scheme always attempts to generate an appropriate control action towards achieving an enhanced performance both under set point change and load disturbance. Next, we mention how the proposed scheme is tested for good stability margins as well as robustness against parametric variations. 2.3. Stability and robustness The proposed APD reduces to a simple nonlinear controller due to nonlinear gain variation through α. We know that the stability analysis for nonlinear control systems is not straight forward. Here, we study the relative stability for APD by calculating the stability margins along with their corner frequencies for two boundary values of α, i.e., at its maximum (αmax) and minimum (αmin) values. Under close-loop operation of the process, gain margin (GM) and phase margin (PM) values are calculated at αmax and αmin. For both the IPD and FOIPD processes, APD will be found to provide good stability margins in terms of GM and PM, which will be justified in the result section. In addition to the relative stability margins, stability robustness of the close-loop system for a given process is tested by Kharitonov's interval polynomials [48] at the boundary values of α (αmax and αmin) along with 725% simultaneous perturbations in process parameters. Stability robustness is ensured as all the roots of Kharitonov's polynomials will be found to be negative. 3. Results In this section, we present the detailed performance analysis of our proposed APD through simulation as well as real time implementation. For simulation study, we consider two well known integrating process models—pure integrating process with delay (IPD) [12,23,49] and first-order integrating process with delay (FOIPD) [43,44]. For the IPD process, performance of the proposed APD is compared with the reported PID tuning relations of AHPID [2], DMPID [3], ZNPID [10], SLPPID [11], CPPID [16], VPID [23], PCPID [34], ACPID [35], RRCPID [36], AMPID [37], HXCPID [38], and RPID [50], and PD settings of CPPD [16], VPD [23], and LLPPD [39]. Similarly, for the FOIPD process, performance of APD is compared with PID tuning rules reported in DMPID [3], ZNPID [10], SPID [12], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42], SLPID [44], KLPID [47], and RPID [50], and PD settings in VVSPD [24], DPD [25], EOMPD [45], and SJLPD [46]. For an in- depth comparison, in addition to the response characteristics, several performance indices, such as percentage overshoot (%OS), rise time (tr), settling time (ts), integral absolute error (IAE) and integral time absolute error (ITAE) are calculated for each setting. Here, ts is calculated following 2% criterion. To verify the robust- ness of the proposed APD, performance indices are also evaluated at the nominal as well as 25% increased value of process dead time. Experimental verification of the proposed APD along with all the above reported PID and PD settings is made on a laboratory scale DC servo position control system [52] identified to be a FOIPD model under both set point change and load disturbance. To verify the noise sensitivity of the proposed controller, white noise (with mean¼0 and variance¼0.1) is added to the measured controlled variable during simulation as well as experimental verification. For all the simulation and experimental studies, the only additional tuning parameter γ is estimated using Eq. (10). During perfor- mance study, first we apply a step set point change followed by an impulse load disturbance as shown in Fig. 3. 3.1. Integrating process with delay (IPD) Integrating processes are frequently encountered in process industries. Chien and Freuhauf [49] suggested that a number of chemical processes can be approximated by a pure integral plus time delay (IPD) model, defined by Gp ¼ KeÀ θs s : ð11Þ Here, we consider the well known IPD process model with open loop gain K¼0.0506 and dead time θ¼6 s [12,23,49]. Performance of APD for this IPD process is investigated along with other model based PID tuning rules proposed in AHPID [2], SLPPID [11], 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time(sec) Response A C D e Δe A B C D B Fig. 5. Typical close-loop response of an under-damped second-order process. C. Dey et al. / ISA Transactions 53 (2014) 162–172 165
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  CPPID [16], VPID[23], PCPID [34], ACPID [35], RRCPID [36], AMPID [37], HXCPID [38], and RPID [50], and PD settings in CPPD [16], VPD [23], and LLPPD [39]. In addition, model free PID tuning rules of DMPID [3] and ZNPID [10] are also considered for performance comparison. Close-loop responses with the nominal value of dead time (i.e., θ¼6 s) are shown in Fig. 6 for different controllers. The related performance indices are given in Table 1a. It is found that most of the model based and model free PID controllers fail to provide satisfactory performance under set point change and load variation simultaneously. In case of VPID [23], the process shows highly oscillatory response and for AHPID [2] it completely diverges. In comparison with VPD [23] and CPPD [16], APD offers lower overshoot with faster settling during set point change and at the same time process recovery is found to be improved under load variation as shown in Fig. 7(a). In case of LLPPD [39] due to over damped response no overshoot is found but load rejection is quite poor compared to APD. Performance robustness of the proposed APD is tested with 25% increased process dead time, i.e., θ¼7.5 s, and the respective performance indices are listed in Table 1b. From Table 1b it is found that under VPID [23], CPPID [16], and SLPPID [11] the process fails to settle within the simulation period and AMPID [37] shows a large overshoot in the set point response as well as poor load regulation. On the other hand, our proposed APD shows consistently improved performance compared to VPD [23] and CPPD [16]. Thus, from the process responses as shown in Figs. 6 and 7 and performance indices of Tables 1a and 1b it is found that the proposed APD shows an overall improved perfor- mance compared to others tuning relations. Moreover, response characteristics of APD shown in Fig. 7(c) reveal that the proposed scheme can overcome the effects of measurement noise. Fig. 7(b) shows the variation of proportional and derivative gains of the proposed APD with the gain updating factor α for θ¼6 s, which shows a considerable change in the derivative gain Ka d during transient phases of the close-loop operation. To check the sensitivity of the tuning parameter γ, performance indices are evaluated with 725% perturbations in its initial value (γ¼12.49) for both the nominal (i.e., θ¼6 s) and increased (i.e., θ¼7.5 s) values of dead time (Tables 1a and 1b). No considerable deviation is observed in the performance of APD due to such variations of γ, which justifies the robustness of our proposed scheme. Stability analysis of APD is provided in Table 1c in terms of GM and PM values with their corner frequencies. Stability robustness is observed (Table 1c) from the Kharitonov polynomials for two 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 2 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 -100 -50 0 50 100 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response CPPID [16] PCPID [34] SLPPID [11] RRCPID [36] HXCPID [38] VPD [23] RPID [50] ZNPID [10] VPID [23] AHPID [2] ACPID [35] DMPID [3] AMPID [37] CPPD [16] LLPPD [39] Fig. 6. Responses for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. C. Dey et al. / ISA Transactions 53 (2014) 162–172166
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  extreme values ofα (αmax and αmin) along with 725% simulta- neous perturbations in process parameters (K and θ). 3.2. First-order integrating process with delay (FOIPD) First-order integrating process with delay can be described by Gp ¼ KeÀ θs sðτ sþ1Þ : ð12Þ Here, we consider K¼1, θ¼4 s, and τ¼4 s [43,44]. Performance of APD is compared with a number of model free and model based PID tuning relations given by DMPID [3], ZNPID [10], SPID [12], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42], SLPID [44], KLPID [47], and RPID [50], and model based PD settings of VVSPD [24], DPD [25], EOMPD [45], and SJLPD [46]. Responses of (12) with θ¼4 s are shown in Fig. 8 and the performance indices for θ¼4 s and 5 s (with þ25% perturbation) are given in Tables 2a and 2b respectively. Fig. 8 shows that the process either goes to the verge of instability for RRCPID [42] or completely diverges under HXCPID [38], SLPID [44], and RPID [50]. In comparison with VVSPD [24] and DPD [25], our proposed APD exhibits lower overshoot with faster recovery as depicted in Fig. 9(a). Fig. 9 (c) justifies the robustness of APD against measurement noise. Fig. 9(b) shows the variations of proportional and derivative gains of APD with gain updating factor α. Under both set point change Table 1a Performance indices for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. Controller %OS tr ts IAE ITAE Model based PID tuning methods VPID [23] (2001) 118.20 9.81 143.75 37.36 2899.0 CPPID [16] (2003) 61.26 10.54 88.75 24.14 1545.0 AHPID [2] (2004) Diverges completely PCPID [34] (2005) 34.47 12.27 83.56 25.6 1597.0 ACPID [35] (2007) 27.51 21.99 144.21 39.45 3020.0 SLPPID [11] (2008) 2.85 18.29 59.03 14.67 1508.0 RPID [50] (2008) 40.46 19.56 133.27 39.73 3053.0 RRCPID [36] (2009) 0.00 64.12 64.12 29.33 1963.0 AMPID [37] (2010) 56.96 11.34 62.70 24.92 1478.0 HXCPID [38] (2011) 7.05 63.08 148.34 41.40 2749.0 Model free PID tuning methods ZNPID [10] (1942) 67.63 11.23 90.15 30.8 1919.0 DMPID [3] (2009) 20.96 11.80 107.64 24.71 1642.0 Model based PD tuning methods VPD [23] (2001) 16.67 11.64 61.48 16.41 988.4 CPPD [16] (2003) 13.77 12.05 58.62 15.99 946.4 LLPPD [39] (2006) 0.00 53.72 53.72 23.80 1440.0 Proposed auto-tuning PD method APD γ ¼ 12:49 4.01 12.86 41.87 15.06 859.6 γþ25% ¼ 15:61 3.28 13.27 41.87 15.09 863.1 γÀ25% ¼ 9:37 4.73 12.86 41.87 15.04 856.5 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 -2 0 2 4 Time(sec) Controlaction 0 50 100 150 200 250 -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 3 3.05 3.1 0 50 100 150 200 250 9 10 11 12 13 Time(sec) 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 -15 -10 -5 0 5 10 15 Time(sec) Controlaction APD APD α Kd a Kp a APD APD Fig. 7. (a) Response and the corresponding control action for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. (b) Variation of α, Ka p, and Ka d under APD for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. (c) Response and the corresponding control action with measurement noise for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. Table 1b Performance indices for the IPD process with 25% increased dead time GpðsÞ ¼ 0:0506eÀ 7:5s =s. Controller %OS tr ts IAE ITAE Model based PID tuning methods VPID [23] (2001) Unstable CPPID [16] (2003) 103.80 11.83 Not settled 52.37 4445.0 AHPID [2] (2004) Completely diverges PCPID [34] (2005) 67.14 13.55 80.25 30.11 1889.0 ACPID [35] (2007) 35.08 21.73 145.65 43.23 3296.0 SLPPID [11] (2008) 36.12 17.43 Not settled 52.51 6895.0 RPID [50] (2008) 50.22 23.19 121.98 44.94 3489 RRCPID [36] (2009) 0.00 74.66 74.66 30.81 2226.0 AMPID [37] (2010) 97.96 12.69 107.36 36.05 2560.0 HXCPID [38] (2011) 7.46 60.77 147.75 42.04 2853.0 Model free PID tuning methods ZNPID [10] (1942) 115.28 12.69 98.32 43.51 2943.0 DMPID [3] (2009) 49.15 13.54 110.80 29.58 2001.0 Model based PD tuning methods VPD [23] (2001) 42.35 13.12 109.08 28.07 2056.0 CPPD [16] (2003) 38.94 13.55 107.36 26.36 1877.0 LLPPD [39] (2006) 0.00 37.65 37.65 23.93 1432.0 Proposed auto-tuning PD method APD γ ¼ 12:49 27.14 13.55 66.48 22.68 1534.0 γþ25% ¼ 15:61 26.19 13.98 66.48 22.62 1537.0 γÀ25% ¼ 9:37 28.10 13.98 66.48 22.74 1531.0 Table 1c Stability and robustness analysis for the IPD process GpðsÞ ¼ 0:0506eÀ 6s =s. APD (αmax) APD (αmin) Kharitonov's Polynomials with αmax and αmin and 725% variation in K and θ GM (dB) 4.68 (Inf rad/s) 3.83 (Inf rad/s) s2 þ0.51sþ0.15¼0 s2 þ1.49sþ0.14¼0 PM (deg) 64.9 (0.19 rad/s) 68.7 (0.20 rad/s) s2 þ1.28sþ0.14¼0 s2 þ0.59sþ0.15¼0 C. Dey et al. / ISA Transactions 53 (2014) 162–172 167
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  and load disturbance,APD shows an overall improved perfor- mance than other PID and PD settings as revealed by Tables 2a and 2b. Its robustness is observed with þ25% perturbation in dead time as well as 725% variation in γ from its nominal value 16.51 (Tables 2a and 2b). GM and PM along with their corner frequencies are listed in Table 2c, which represents good stability margins. Kharitonov's polynomials of Table 2c also confirm its stability robustness under maximum and minimum values of α as well as 725% simultaneous variations in K, τ, and θ. 3.3. Real time implementation Servo position control system is a typical example of integrat- ing process. Here, performance of the proposed APD is verified on a DC servo position control system. The schematic block diagram of the position control system is shown in Fig. 10(a) and its experimental setup is shown in Fig. 10(b). The hardware setup is a Quanser make DC Motor Control Trainer (DCMCT) [52] and it has been identified as a FOIPD model. A small delay of 0.01 s is introduced by the Simulink delay block in the forward path of the process loop. This DC servo motor is a high quality 18-Watt motor of Maxon brand. This is a graphite brush DC motor with low inertia rotor. It has zero cogging and very low unloaded running friction. The transfer function of this servo motor (as provided by 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 -1000 -500 0 500 1000 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 1.5 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 -1000 -500 0 500 1000 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 0 0.5 1 Time(sec) Response 0 50 100 150 200 250 -50 0 50 Time(sec) Response WCPID [41] KLPID [47] RRCPID [42] AMPID [37] DPD [25] RPID [50] ZNPID [10] YPID [40] SPID [12] SLPID [44] DMPID [3] HXCPID [38]VVSPD [24] EOMPD [45] SJLPD [46] 1.5 Fig. 8. Responses for the FOIPD process GpðsÞ ¼ eÀ 4s =sð4sþ1Þ. Table 2a Performance indices for the FOIPD process GpðsÞ ¼ eÀ 4s =sð4sþ1Þ. Controller %OS tr ts IAE ITAE Model based PID tuning methods WCPID [41] (2002) 22.19 17.56 57.89 29.70 2561.0 SPID [12] (2003) 34.47 15.27 101.09 35.32 3017.0 KLPID [47] (2006) 34.88 17.57 87.27 39.64 3390.0 RRCPID [42] (2007) 30.26 12.96 Not settled 56.03 6836.0 SLPID [44] (2008) Completely diverges RPID [50] (2008) Completely diverges AMPID [37] (2010) 58.83 11.81 69.41 28.51 2122.0 HXCPID [38] (2011) Completely diverges Model free tuning PID tuning methods ZNPID [10] (1942) 69.74 12.96 87.27 38.0 3083.0 YPID [40] (1999) 67.63 17.57 Not settled 62.78 6304.0 DMPID [3] (2009) 25.06 14.11 127.59 31.30 2654.0 Model based PD tuning methods VVSPD [24] (2000) 16.46 12.96 49.83 18.01 1263.0 DPD [25] (2001) 26.29 11.81 67.68 20.52 1564.0 SJLPD [46] (2006) 0.00 36.0 36.0 14.94 1164.0 EOMPD [45] (2009) 1.73 28.51 41.76 24.74 1882.0 Proposed auto-tuning PD method APD γ ¼ 16:51 6.64 15.84 37.73 17.91 1237.0 γ þ25% ¼ 20:64 5.82 15.84 38.31 17.80 1226.0 γÀ25% ¼ 12:39 7.05 15.84 38.31 18.03 1248.0 C. Dey et al. / ISA Transactions 53 (2014) 162–172168
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  Quanser [52]) is GpðsÞ¼ 19:9eÀ0:01s sð0:09sþ1Þ : ð13Þ Quanser-Q8 DAQ board interfaces the DCMCT with the PC through USB port. With the help of QuaRC soft-ware based on Matlab–Simulink we implement the proposed auto-tuning PD con- troller. Similarly, other reported PID and PD controllers are also implemented for their performance evaluation on DCMCT. Real Time Workshop (RTW) and Real Time Windows Target (RTWT) generate C code using Microsoft Cþ þ Professional from the QuaRC block diagram, and the Quanser-Q8 board acts as the intermediary for two way data flow from the physical servo system to and from the QuaRC model. A high resolution encoder is used for position sensing of the DC motor. Performances of the proposed APD and other PID/ PD tuning rules (DMPID [3], ZNPID [10], SPID [12], VVSPD [24], DPD [25], AMPID [37], HXCPID [38], YPID [40], WCPID [41], RRCPID [42], SLPID [44], EOMPD [45], SJLPD [46], KLPID [47], and RPID [50]) are tested on the DCMCT. Responses of the DC servo motor for different controllers other than APD during set-point tracking and load variation for the nominal value of dead time (θ¼0.01 s) are shown in Fig. 11. We observe that even with this nominal value of dead time VVSPD [24], AMPID [37], HXCPID [38], RRCPID [42], SLPID [44], SJLPD [46], and RPID [50] provide unstable performances. The close-loop response and its corresponding control action for the proposed APD is shown in Fig. 12(a). Performance of APD has also been tested with measurement noise as shown in Fig. 12(b). Fig. 13 shows only stable performances of different controllers with 25% higher value of dead time, i.e., θ¼0.0125 s. Fig. 14(a) and (b), respectively, shows the responses of APD without and with measurement noise. Thus, results obtained from both nominal and increased values of dead time (Figs. 11–14) reveal that the proposed APD exhibits an overall improved performance compared to other PID and PD tuning rules as well as robustness against measurement noise. To summarize, from the simulation as well as real time experi- mentation, it is observed that for both IPD and FOIPD processes, the proposed APD shows consistently improved overall performance under set point change and load disturbance. APD is also found to provide good stability margins and performance robustness. Stability robustness of the proposed scheme is also verified at the boundary values of the gain modifying parameter α (i.e., αmax and αmin) along with 725% simultaneous perturbations in process parameters. 4. Conclusion We proposed a real time gain modification scheme through nonlinear parameterization of a PD auto-tuner. The proportional and derivative gains of the proposed auto-tuning PD controller (APD) have been modified in each instant by a nonlinear gain updating factor α defined on the instantaneous process states. Fig. 9. (a) Response and the corresponding control action for the FOIPD process GpðsÞ ¼ eÀ4s =sð4sþ1Þ. (b) Variation of α, Ka p, and Ka d under APD for the FOIPD process GpðsÞ ¼ eÀ 4s =sð4sþ1Þ. (c) Response and the corresponding control action with measurement noise for the FOIPD process GpðsÞ ¼ eÀ 4s =sð4sþ1Þ. Table 2c Stability and robustness analysis for the FOIPD process GpðsÞ ¼ eÀ 4s =sð4sþ1Þ. APD (αmax) APD (αmin) Kharitonov's polynomials with αmax and αmin and 725% variation in K, τ, and θ GM (dB) 9.07 (0.51 rad/s) 9.17 (0.52 rad/s) s3 þ0.61s2 þ0.57sþ0.04¼0 s3 þ0.52s2 þ0.17sþ0.01¼0 PM (deg) 53.1 (0.18 rad/s) 54.2 (0.17 rad/s) s3 þ0.52s2 þ0.57sþ0.01¼0 s3 þ0.61s2 þ0.17sþ0.04¼0 Table 2b Performance indices for the FOIPD process with 25% increased dead time GpðsÞ ¼ eÀ 5s =sð4sþ1Þ. Controller %OS tr ts IAE ITAE Model based PID tuning methods WCPID [41] (2002) 33.11 17.57 48.68 32.61 2834.0 SPID [12] (2003) 46.27 15.84 106.28 38.56 3307.0 KLPID [47] (2006) 42.32 17.57 83.81 42.78 3674.0 RRCPID [42] (2007) Completely diverges SLPID [44] (2008) Completely diverges RPID [50] (2008) Completely diverges AMPID [37] (2010) 87.43 12.38 125.29 43.10 3908.0 HXCPID [38] (2011) Completely diverges Model free tuning PID tuning methods ZNPID [10] (1942) 91.52 13.54 97.06 47.14 4025.0 YPID [40] (1999) 78.68 18.15 Not settled 73.18 7531.0 DMPID [3] (2009) 39.39 14.69 127.02 34.65 2957.0 Model based PD tuning methods VVSPD [24] (2000) 34.47 13.54 100.52 27.88 2341.0 DPD [25] (2001) 49.47 12.38 Not settled 43.83 4378.0 SJLPD [46] (2006) 10.32 11.81 42.34 15.74 1170.0 EOMPD [45] (2009) 4.59 24.48 45.22 26.41 2049.0 Proposed auto-tuning PD method APD γ ¼ 16:51 18.92 15.26 52.71 23.14 1782.0 γ þ25% ¼ 20:64 18.51 15.26 52.13 22.98 1769.0 γÀ25% ¼ 12:39γ 19.74 15.26 52.71 23.31 1796.0 C. Dey et al. / ISA Transactions 53 (2014) 162–172 169
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  DCMCT Hardware Quanser DAQ Board (Q8) PCwith Maltab and RTW and RTWT PC Inertial Load Encoder DAQ D/A Converter Amplifier DC Motor PCI Link Motor Load Encoder Fig. 10. (a) Schematic diagram of DCMCT. (b) Experimental setup of DC servo rig (Quanser DCMCT). 0 1 2 3 4 5 0 5 10 Time(sec) Response 0 1 2 3 4 5 0 5 10 Time(sec) Response 0 1 2 3 4 5 0 2 4 6 8 Time(sec) Response 0 1 2 3 4 5 0 2 4 6 8 Time(sec) Response 0 1 2 3 4 5 0 2 4 6 8 Time(sec) Response 0 1 2 3 4 5 0 5 10 Time(sec) Response 0 1 2 3 4 5 -10 -5 0 5 10 Time(sec) Response 0 1 2 3 4 5 0 5 10 Time(sec) Response ZNPID [10] YPID [40] WCPID [41] SPID [12] KLPID [47] DMPID [3] DPD [25] EOMPD [45] Fig. 11. Responses with nominal value of dead time for DCMCT. Fig. 12. (a) Response and the corresponding control action for the proposed APD with nominal value of dead time for DCMCT. (b) Response and the corresponding control action for the proposed APD under measurement noise with nominal value of dead time for DCMCT. C. Dey et al. / ISA Transactions 53 (2014) 162–172170
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  An empirical relationhas also been suggested for the selection of the single additional tuning parameter γ. The proposed gain modification scheme can be easily incorporated in conventional control loops. Performance analysis for integrating processes with varying dead time under both set point change and load distur- bance has revealed that the proposed APD is capable of providing an improved overall performance compared to other tuning rules reported in the literature. Performance robustness of the close- loop system under APD has been observed with considerable variations in process dead time as well as with measurement noise. Stability robustness has also been established by applying 725% simultaneous perturbations in process parameters. Further works may be done for setting more appropriate parameters of the initial PD controller, and finding more suitable value of γ for a given process. Similar gain adjustment schemes may be tried for other real world problems, which exhibit inverse response. All these possibilities are under investigation. References [1] Åström KJ, Hägglund T. The future of PID control. Control Engineering Practice 2001;9(11):1163–75. [2] Åström KJ, Hägglund T. Revisiting the Ziegler–Nichols step response method for PID control. Journal of Process Control 2004;14(6):635–50. [3] Dey C, Mudi RK. An improved auto-tuning scheme for PID controllers. ISA Transactions 2009;48(4):396–409. [4] Tang KM, Lo WL, Rad AB. Adaptive delay compensated PID controller by phase margin design. ISA Transactions 1998;37(3):177–87. [5] Lo HL, Rad AB, Chan CC, Wong YK. Comparative studies of three adaptive controllers. ISA Transactions 1999;38(1):43–53. [6] Ang KH, Chong GCY, Li Y. PID control system analysis, design, and technology. IEEE Transactions on Control Systems Technology 2005;13(4):559–76. [7] Lo WL, Rad AB, Li CK. Self-tuning control of systems with unknown time delay via extended polynomial identification. ISA Transactions 2003;42(2):259–72. [8] Vijayan V, Panda RC. Design of simple set point filter for minimizing overshoot for low order processes. ISA Transactions 2012;51(2):271–6. [9] Vijayan V, Panda RC. Design of PID controllers in double feedback loops for SISO systems with set-point filters. ISA Transactions 2012;51(4):514–21. [10] Ziegler JG, Nichols NB. Optimum setting for automatic controllers. ASME Transactions 1942;64(11):759–68. [11] Shamsuzzoha Md, Lee M. PID controller design for integrating processes with time delay. Korean Journal of Chemical Engineering 2008;25(4):637–45. [12] Skogestad S. Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control 2003;13(4):291–309. [13] Ali A, Majhi S. PI/PID controller design based on IMC and percentage overshoot specification to controller setpoint change. ISA Transactions 2009;48(1):10–5. [14] Selvi JAV, Radhakrishnan TK, Sundaram S. Performance assessment of PID and IMC tuning methods for a mixing process with time delay. ISA Transactions 2007;46(3):391–7. [15] Chia TL, Lefkowitz I. Internal model-based control for integrating processes. ISA Transactions 2010;49(4):519–27. [16] Chidambaram M, Sree R Padma. A simple method of tuning PID controllers for integrator/dead time processes. Computers Chemical Engineering 2003;27(2): 211–5. [17] Atia KR, Cartmell MP. A new methodology for designing PD controllers. Robotica 2001;19(3):267–73. [18] Ogata K. Modern control engineering. New Jersey: Prentice-Hall; 2002. 0 1 2 3 4 5 0 5 10 Time(sec) Response 0 1 2 3 4 5 -5 0 5 10 15 Time(sec) Response 0 1 2 3 4 5 0 2 4 6 8 Time(sec) Response 0 1 2 3 4 5 0 2 4 6 8 Time(sec) Response 0 1 2 3 4 5 -10 -5 0 5 10 Time(sec) Response 0 1 2 3 4 5 0 5 10 Time(sec) Response ZNPID [10] YPID [40] KLPID [47] DMPID [3] DPD [25] EOMPD [45] Fig. 13. Responses with 25% increased value of dead time for DCMCT. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 Time(sec) Response 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -20 0 20 Controlaction 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 Time(sec) Response 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -20 0 20 Time(sec) Controlaction APD APDAPD APD Time(sec) Fig. 14. (a) Response and the corresponding control action for the proposed APD with 25% increased value of dead time for DCMCT. (b) Response and the corresponding control action for the proposed APD under measurement noise with 25% increased value of dead time for DCMCT. C. Dey et al. / ISA Transactions 53 (2014) 162–172 171
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