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ISA Transactions 65 (2016) 394–406 Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Modiﬁed parallel cascade control strategy for stable, unstable and integrating processes G. Lloyds Raja n, Ahmad Ali Department of Electrical Engineering, Indian Institute of Technology Patna, Amhara, Bihta 801103, Bihar, India art ic l e i nf o a b s t r a c t Article history: Received 21 November 2015 Received in revised form 4 June 2016 Accepted 21 July 2016 Available online 9 August 2016 This manuscript presents a modiﬁed parallel cascade control structure (PCCS) for a class of stable, unstable and integrating process models with time delay. The proposed PCCS consists of three controllers. Internal Model Control (IMC) approach is used to design the disturbance rejection controller in the secondary loop. Parameters of the proportional-integral (PI) controller which is used for setpoint tracking is obtained by equating the ﬁrst and second derivatives of desired and actual closed loop transfer functions at the origin of s-plane. Routh Hurwitz stability criterion is used to design the proportional-derivative (PD) controller which stabilizes the unstable/integrating primary process model. An analytical expression is proposed for computing the desired closed loop time constant of the primary loop in terms of plant model parameters so as to achieve an user-deﬁned maximum sensitivity. Based on extensive simulation studies, a suitable value for the secondary closed loop time constant is also recommended. This is an advantage of the present work over the reported parallel cascade control schemes where authors provide a suitable range of values for the closed loop time constants. The proposed tuning strategy requires tuning of four/six controller parameters for stable/unstable and integrating process models which is less compared to the reported strategies. Simulation results illustrate that the proposed method yields signiﬁcant improvement in closed loop performance compared to some of the recently reported tuning strategies for both nominal and perturbed process models. & 2016 ISA. Published by Elsevier Ltd. All rights reserved. Keywords: Parallel cascade control Disturbance rejection Stabilizing controller Robustness 1. Introduction In cascade control structure (CCS), an intermediate sensor and controller are used to reject the disturbances before the controlled variable deviates from the setpoint which results in improved closed loop performance compared to the unity feedback scheme [1]. Luyben [2] was the ﬁrst to use parallel cascade control structure (PCCS). Overhead composition control of distillation column and temperature control of subcooled reﬂux are some of the practical scenarios where parallel cascade control is used. Fig. 1 shows the block diagram of PCCS in which the manipulated variable (u2) and disturbance (d) simultaneously affect primary and secondary outputs (y1 and y2). In Fig. 1, GP1 and GP2 denotes the transfer functions of primary and secondary process models whereas GC1 and GC2 denotes the primary and secondary controllers, respectively. The setpoint of the primary loop is represented by r 1 whereas r 2 denotes the setpoint of the secondary loop. Gpd1 and Gpd2 represent the transfer functions of disturbances entering the primary and secondary process outputs. n Corresponding author. E-mail addresses: lloyd.raja@gmail.com (G.L. Raja), ali@iitp.ac.in (A. Ali). http://dx.doi.org/10.1016/j.isatra.2016.07.008 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved. The performance of PCCS when the actual load disturbance is different from the nominal one was studied by Shen and Yu [3]. A PCCS that decouples the actions of primary and secondary loops has been reported in [4]. A PCCS for regulating arterial blood pressure of a biological system was proposed by Pottmann et al. [5] using H2 optimal control theory. The conventional PCCS was modiﬁed by Lee et al. [6] by including setpoint ﬁlters in primary and secondary loops. In the above work, an IMC based analytical approach was used to obtain the primary and secondary controller settings. Nandong and Zang [7] have reported that their multiscale control strategy yields signiﬁcant improvement in closed loop performance and robustness compared to the strategy reported in [6]. It is to be noted that none of the above mentioned parallel cascade schemes have considered unstable and integrating process models. Rao et al. [8] have reported that satisfactory closed loop performance is achieved for processes with large time delay by including a Smith predictor in the primary loop. Recently, a number of Smith predictor based parallel cascade control structures have been reported in literature for unstable and integrating process models [9–12]. A modiﬁed PCCS with two controllers and a setpoint ﬁlter was reported in [9] for stable and integrating process models. The same authors have reported another modiﬁed PCCS in [10] for stable

G.L. Raja, A. Ali / ISA Transactions 65 (2016) 394–406 395 d d Gpd2 Gpd1 Gpd2 Gp y1 y1 Gp1 r1 r2 Gc2 Gc1 - - u2 Gp1 n Gp2 Secondary loop y2 Gpd1 r'2 r1 u2 r2 - - Primary loop y2 Gp2 Gc1 - Gp2m - Gcd1 Fig. 1. General block diagram of parallel cascade control structure. Gcd2 and unstable process models. In the above cited work, the secondary controller was designed using IMC approach whereas a PID in series with lag-lead ﬁlter was used as a primary controller. Vanavil et al. [11] have modiﬁed the PCCS reported in [8] to control an unstable bioreactor. A modiﬁed PCCS for a class of stable, unstable and integrating process models was proposed in [12]. The primary and secondary controllers of the above mentioned work were designed using loop shaping technique whereas the primary setpoint ﬁlter was designed by minimizing integral squared error (ISE) performance criterion. From the above literature survey, it is observed that the recently reported works pertaining to PCCS [9–12] require tuning of a large number of controller/ﬁlter parameters. Hence in this work, a parallel cascade control strategy with less number of controller/ﬁlter parameters is proposed for a class of stable, unstable and integrating process models. The proposed PCCS consists of PI–PD control structure in the primary loop and an IMC based disturbance rejection controller in the secondary loop. Routh–Hurwitz stability criterion is used to tune the PD controller whereas the settings of the PI controller are obtained by equating the ﬁrst and second derivatives of desired and actual closed loop transfer functions at the origin of s-plane. Since the process models that are used to obtain the controller parameters are approximations of the actual dynamics, it is necessary that the controller settings must be robust. Maximum sensitivity is a measure of system robustness and is deﬁned as the inverse of the shortest distance from the Nyquist curve of the loop transfer function to the critical point ‘ 1’. For stable systems, it is desirable to have maximum sensitivity between 1.2 and 2 [13]. Analytical expression is proposed for the closed loop time constant of the primary loop so as to achieve an user-deﬁned maximum sensitivity for the primary loop. Suitable value of secondary closed loop time constant is also recommended after studying its effect on system performance and robustness for a wide range of process models. Simulation studies show that the proposed method yields improved and robust closed loop performance compared to the recently reported tuning strategies. The advantages of the proposed parallel cascade control strategy are as follows: 1. It requires tuning of less number of controller/ﬁlter parameters 2. It does not require a hit and trial approach for selecting the primary and secondary closed loop time constants (λ1 and λ2 ). This paper is organized as follows: the proposed PCCS and process models that are considered in the present work are discussed in Section 2. Controller settings of the proposed PCCS are derived in Section 3. Section 4 discusses the conditions for closed loop robust stability whereas guidelines for selecting the closed loop time constants are given in Section 5. Section 6 presents the results of simulation studies. Concluding remarks are given in Section-7. Fig. 2. Proposed parallel cascade control structure. 2. Theoretical developments The proposed parallel cascade control structure is shown in Fig. 2. Gc1 and Gcd1 are setpoint tracking and stabilizing controllers in the primary loop. The secondary disturbance rejection controller is denoted by Gcd2 . Gc1 and Gcd1 are assumed as PI and PD controllers with the following transfer functions: 1 ð1Þ Gc1 ðsÞ ¼ K c1 1 þ T i1 s Gcd1 ðsÞ ¼ K p ð1 þ T d sÞ ð2Þ From Fig. 2, the closed-loop transfer function for servo response of the primary loop (GA1 ) is given by GA1 ðsÞ ¼ Gc1 ðsÞGp1 ðsÞ y1 ðsÞ ¼ r 1 ðsÞ 1 þ Gp1 ðsÞðGc1 ðsÞ þ Gcd1 ðsÞÞ þ Gcd2 ðsÞðGp2 ðsÞ Gp2m ðsÞÞ ð3Þ where Gp2m denotes the transfer function of the secondary process model. Similarly, the closed-loop transfer function for regulatory response of the primary loop is obtained as follows: Gpd1 ðsÞ 1 þ Gp2 ðsÞGcd2 ðsÞ Gpd2 ðsÞGcd2 ðsÞGp1 ðsÞ y1 ðsÞ ¼ ð4Þ dðsÞ 1 þ Gp1 ðsÞðGc1 ðsÞ þ Gcd1 ðsÞÞ þ Gcd2 ðsÞðGp2 ðsÞ Gp2m ðsÞÞ If Gp2 ¼ Gp2m (under perfect model conditions), (3) and (4) reduces to (5) and (7) which are given below: GA1 ðsÞ ¼ Gc1 ðsÞGp1 ðsÞ Gc1 ðsÞGp ðsÞ ¼ 1 þ Gp1 ðsÞðGc1 ðsÞ þ Gcd1 ðsÞÞ 1 þ Gc1 ðsÞGp ðsÞ ð5Þ where Gp ðsÞ ¼ Gp1 ðsÞ y1 ðsÞ ¼ r ;2 ðsÞ 1 þ Gp1 ðsÞGcd1 ðsÞ y1 ðsÞ Gpd1 ðsÞ 1 þ Gp2 ðsÞGcd2 ðsÞ Gpd2 ðsÞGcd2 ðsÞGp1 ðsÞ ¼ 1 þ Gp1 ðsÞðGc1 ðsÞ þ Gcd1 ðsÞÞ dðsÞ ð6Þ ð7Þ From (5), it is observed that the desired servo response can be achieved by tuning Gc1 and Gcd1 . Once Gc1 and Gcd1 are tuned, satisfactory regulatory performance can be achieved by tuning Gcd2 . The dynamics of secondary process model is usually stable whereas that of primary process model may be stable, unstable or integrating in nature [5–10,12,14]. Hence, in the present work, the secondary process model is assumed as stable ﬁrst order plus time delay with positive zero (FOPTDPZ) or ﬁrst order plus time delay (FOPTD) transfer function as given below: K 2 1 p2 s θ2 s e ð8aÞ Gp2 ðsÞ ¼ τ2 s þ 1 Gp2 ðsÞ ¼ K2 τ2 s þ1 e θ2 s ð8bÞ

396 G.L. Raja, A. Ali / ISA Transactions 65 (2016) 394–406 Furthermore, the dynamics of the primary process model is represented by any one of the following transfer functions: Gp1 ðsÞ ¼ Gp1 ðsÞ ¼ Gp1 ðsÞ ¼ Gp1 ðsÞ ¼ K 1 ð1 p1 sÞ θ1 s e τ1 s þ 1 K1 e θ1 s τ1 s þ 1 K 1 ð1 p1 sÞ θ1 s e τ1 s 1 K1 e θ1 s τ1 s 1 ð9aÞ ð9bÞ ð10aÞ ð10bÞ ð11aÞ Gp1 ðsÞ ¼ K 1 θ1 s e s ð11bÞ Gp1 ðsÞ ¼ K 1 ð1 p1 sÞ θ1 s e sðτ1 s þ 1Þ ð12aÞ Gp1 ðsÞ ¼ K1 e θ1 s sðτ1 s þ 1Þ ð12bÞ ð13aÞ K1 e θ1 s c2 s2 þ c1 s þ 1 ð13bÞ Eqs. (10a)-(13a) represent unstable ﬁrst order plus time delay with positive zero (UFOPTDPZ), integral plus time delay with positive zero (IPTDPZ), second order integral plus time delay with positive zero (SOIPTDPZ) and stable second order plus time delay with positive zero (SOPTDPZ) process models, respectively. Moreover, (10b)–(13b) represent unstable ﬁrst order plus time delay (UFOPTD), integral plus time delay (IPTD), second order integral plus time delay (SOIPTD) and stable second order plus time delay (SOPTD) process models, respectively. 3. Controller design The proposed PCCS consists of three controllers (Gc1 , Gcd1 and Gcd2 ). Controller parameters are derived in terms of known plant model parameters in this section. 3.1. Secondary disturbance rejection controller Gcd2 IMC approach is used to design the controller Gcd2 . The process model Gp2 can be factorized into inverting and non-inverting parts as given below: ð14Þ þ GP2 where consists of time delays and right half-plane zeros (if any) whereas GP2 denotes the delay free part of the secondary process model. The IMC controller is given by ð15Þ In the above equation, M is a low-pass IMC ﬁlter whose transfer function is 1= λ2 s þ1 . Using (8), (14) and (15), Gcd2 is obtained as Gcd2 ðsÞ ¼ ðτ 2 s þ 1Þ K 2 λ2 s þ 1 where λ2 represents the adjustable tuning parameter. ð17Þ Gp ðsÞ ¼ ð16Þ K 1 ð1 p1 sÞe θ1 s ðτ1 þ 0:5K p K 1 p1 θ1 Þs2 þ ð1 0:5K p K 1 θ1 K p K 1 p1 Þs þ K p K 1 ð18Þ Assuming K p ¼ β =ðK 1 ð0:5θ1 þ p1 ÞÞ, (18) satisﬁes Routh–Hurwitz stability criterion if 0 o β o1. Based on extensive simulation studies given in Appendix A, the recommended value of β is 0.01. Since IPTDPZ process model is a limiting case of SOIPTDPZ process model (with τ1 ¼ 0), the above controller settings are also valid for IPTDPZ process models. Substituting K p ¼ β =ðK 1 ð0:5θ1 þ p1 ÞÞ in (18), we get the following SOPTDPZ transfer function: Gp ðsÞ ¼ K 1 ð1 p1 sÞ θ1 s e Gp1 ðsÞ ¼ c2 s2 þ c1 s þ 1 1 Gcd2 ðsÞ ¼ MðsÞ Gp2 ðsÞ K 1 ð1 p1 sÞe θ1 s ðsð1 þ τ1 sÞÞ þ K p K 1 ð1 p1 sÞð1 þT d sÞ 1 0:5θ1 s = 1 þ 0:5θ1 s Assuming T d ¼ 0:5θ1 , the above equation reduces to K 1 ð1 p1 sÞ θ1 s e s þ Gp2 ðsÞ ¼ Gp2 ðsÞGp2 ðsÞ 3.2.1. For SOIPTDPZ/IPTDPZ primary process model Substituting Gcd1 ðsÞ ¼ K p ð1 þ T d sÞ and Gp1 ðsÞ ¼ ðK 1 ð1 p1 sÞe θ1 s Þ =ðsðτ1 s þ 1ÞÞ in (6) and approximating the time delay term in the denominator by a ﬁrst order Padé approximation R1;1 ðsÞ [15], we get Gp ðsÞ ¼ Gp1 ðsÞ ¼ Gp1 ðsÞ ¼ 3.2. Stabilizing controller Gcd1 K 1 ð1 p1 sÞe θ1 s a2 s2 þ a1 s þ a0 ð19Þ where, a2 ¼ ðτ1 þ 0:5K p K 1 p1 θ1 Þ, a1 ¼ ð1 0:5K p K 1 θ1 K p K 1 p1 Þ and a0 ¼ K p K 1 . If Gp1 follows IPTD dynamics, then (19) reduces to a FOPTD transfer function given by ðK 1 e θ1 s Þ=ða1 s þ a0 Þ. 3.2.2. For UFOPTDPZ primary process model Substituting Gcd1 ðsÞ ¼ K p ð1 þ T d sÞ and (10a) in ð6Þ and approximating the time delay term in the denominator by 1 0:5θ1 s = 1 þ0:5θ1 s , we get Gp ðsÞ ¼ K 1 ð1 p1 sÞe θ1 s τ1 s 1 þ K p K 1 ð1 þ T d sÞð1 p1 sÞ 1 0:5θ1 s = 1 þ 0:5θ1 s ð20Þ θ1 s =(ð0:5θ1 p1 K p K 1 Þs2 Assuming T d ¼ 0:5θ1 , we get Gp ðsÞ ¼K 1 e þ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þs þ K p K 1 1). Using Routh–Hurwitz stability criteria, we get the minimum and maximum values of K p as K pmin ¼1=K 1 and K pmax ¼ τ1 =ðK 1 ðp1 þ0:5θ1 ÞÞ, respectively. In the present work, K p is assumed as rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m τ1 ð21Þ K p ¼ m K pmin K pmax ¼ K 1 ðp1 þ 0:5θ1 Þ Assuming m¼ 1 results in large overshoot and settling time for processes with θ1 =τ1 o 1. Therefore, the following strategy is used to obtain a suitable value for m: starting from m ¼ 1, m is reduced in steps of 0.01. For each value of m, b1 ¼ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þ and b0 ¼ ðK 1 K p 1Þ are computed until either b1 or b0 becomes negative. The least value of m for which b1 and b0 remains positive is used to obtain K p using (21). Substituting (21) and T d ¼ 0:5θ1 in (20) results in the following SOPTDPZ transfer function: Gp ðsÞ ¼ K 1 ð1 p1 sÞe θ1 s ðb2 s2 þ b1 s þ b0 Þ ð22Þ where b2 ¼ 0:5θ1 p1 K p K 1 , b1 ¼ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þ and b0 ¼ ðK 1 K p 1Þ. Tuning rules that are derived in this subsection for Gcd1 are summarized in Table 1. Remark-1. Since the design of Gcd1 involves ﬁrst order Padé approximation R1;1 ðsÞ [15], the controller settings obtained in Section 3.2.2 is recommended only for UFOPTDPZ process models with 0 o ðp1 þ θ1 Þ=τ1 r 1 and UFOPTD process models with 0 o θ1 =τ1 r 1. If the proposed method needs to be applied for UFOPTD process models with θ1 =τ1 4 1, it is recommended to use

G.L. Raja, A. Ali / ISA Transactions 65 (2016) 394–406 Table 1 Summary of controller settings for Gcd1 . Table 2 Summary of controller settings for Gc1 . Gp1 Gcd1 Gp1 Gc1 UFOPTD qﬃﬃﬃﬃﬃﬃﬃﬃﬃ τ1 K p ¼ Km1 0:5θ ; T d ¼ 0:5θ1 ; 1 UFOPTD b1 1 K c1 ¼ K 1 ðbθ01Tþi1 λ1 Þ; T i1 ¼ 2ðθ1 þ λ1 Þ þ b 0 IPTD 0:01 ; T d ¼ 0:5θ1 K p ¼ 0:5K 1 θ1 IPTD a1 1 K c1 ¼ K 1 ðaθ01Tþi1 λ1 Þ; T i1 ¼ 2ðθ1 þ λ1 Þ þ a 0 SOIPTD K c1 ¼ K 1 ðθa10 Tþi12λ1 Þ; T i1 ¼ 2ðθ11 þ 2λ11 Þ þ aa10 ¼ 0:5θ1 SOPTD K c1 ¼ K 1 ðθ1T i1þ 2λ1 Þ; T i1 ¼ 2ðθ11 þ 2λ11 Þ þ c1 ¼ 0:5θ1 FOPTD 1 K c1 ¼ K 1 ðθT1i1þ λ1 Þ; T i1 ¼ 2ðθ1 þ λ1 Þ þ τ 1 0:01 0:5K 1 θ1 ; T d SOIPTD Kp ¼ SOPTD FOPTD UFOPTDPZ K p ¼ 0; T d ¼ 0 K p ¼ 0; T d ¼ 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ; T d ¼ 0:5θ1 K p ¼ Km1 ðp þτ0:5θ 1Þ θ2 where b1 ¼ ðτ1 0:5K p K 1 θ1 Þ and b0 ¼ ðK 1 K p 1Þ. ¼ 0:5θ1 θ2 where a1 ¼ ð1 0:5K p K 1 θ1 Þ and a0 ¼ K p K 1 . θ2 2λ2 where a1 ¼ ð1 0:5K p K 1 θ1 Þ and a0 ¼ K p K 1 . 1 IPTDPZ 0:01 ðK 1 ð0:5θ1 þ p1 ÞÞ; T d 0:01 ðK 1 ð0:5θ1 þ p1 ÞÞ; T d Kp ¼ SOIPTDPZ Kp ¼ SOPTDPZ FOPTDPZ K p ¼ 0; T d ¼ 0 K p ¼ 0; T d ¼ 0 θ2 2λ2 θ2 2 θ 2λ2 UFOPTDPZ K c1 ¼ K 1 ðbθ0þT i12λ1 Þ; T i1 ¼ 2ðθ þ 2λ11 Þ þ bb10 where θ ¼ θ1 þ p1 , b1 ¼ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þ and b0 ¼ ðK 1 K p 1Þ. the Padé approximation R0;1 ðsÞ (e θ1 s ¼ 1=ð1 þ θ1 s)) [15] and assume K p ¼2=K 1 and T d ¼ θ1 . Accordingly, we get b1 ¼ τ1 and b0 ¼ 1. Remark- 2. If the primary process model is stable, Gcd1 is not required which results in Gp ¼ Gp1 . θ2 2λ2 K c1 ¼ K 1 ðaθ0þT i12λ1 Þ; T i1 ¼ 2ðθ þ 2λ11 Þ þ aa10 IPTDPZ where θ ¼ θ1 þ p1 , a1 ¼ ð1 0:5K p K 1 θ1 K p K 1 p1 Þ and a0 ¼ K p K 1 . θ2 2λ2 K c1 ¼ K 1 ðaθ0þT i12λ1 Þ; T i1 ¼ 2ðθ þ 2λ11 Þ þ aa10 SOIPTDPZ where θ ¼ θ1 þ p1 , a1 ¼ ð1 0:5K p K 1 θ1 K p K 1 p1 Þ and a0 ¼ K p K 1 . θ2 2λ2 SOPTDPZ K c1 ¼ K 1 ðθTþi1 2λ1 Þ; T i1 ¼ 2ðθ þ 2λ11 Þ þ c1 FOPTDPZ K c1 ¼ K 1 ðθT i1þ λ1 Þ; T i1 ¼ 2ðθθþ λ1 Þ þ τ1 where θ ¼ θ1 þ p1 2 3.3. Design of setpoint tracking controller Gc1 where θ ¼ θ1 þ p1 3.3.1. Stable FOPTDPZ/FOPTD/IPTD/UFOPTD primary process model The desired closed loop transfer function of the primary loop is assumed as Gd1 ðsÞ ¼ e θs λ1 s þ 1 ð23Þ where λ1 is the desired closed loop time constant of the primary loop and θ ¼ θ1 þ p1 . The actual closed loop transfer function of the primary loop is given by (5). The transfer function of Gc1 which is given by (1) is rearranged as Gc1 ðsÞ ¼ K c1 ~ G c1 ðsÞ s G0 p ð 0 Þ θ T i1 ¼ θ K p1 2 θ þ λ1 2 3.3.2. IPTDPZ/ UFOPTDPZ/SOIPTDPZ/SOIPTD/SOPTD primary process model 2 Assuming Gd1 ðsÞ ¼ e θs = λ1 s þ 1 and working on similar lines as in the previous sub-section, the controller settings for Gc1 are obtained as follows: K c1 ¼ 1 G~ c1 ðsÞ ¼ s þ T i1 ð25Þ Substituting (24) in (5), we get K c1 Gp ðsÞ̃G c1 ðsÞ s þ K c1 Gp ðsÞ̃G c1 ðsÞ ð26Þ ð29bÞ where G0 p ð0Þ is the ﬁrst derivative of Gp at s¼ 0. ð24Þ where GA1 ðsÞ ¼ 397 T i1 K p1 θ þ 2λ1 G0 p ð0Þ θ 2λ1 θ T i1 ¼ K p1 2 θ þ 2λ1 2 ð30aÞ 2 ð30bÞ Controller settings of Gc1 corresponding to various process models are summarized in Table 2. Computing the ﬁrst and second derivatives of (26) at s ¼ 0, we obtain the following: G0A1 ð0Þ ¼ 4. Conditions for closed loop robust stability T i1 K c1 K p1 G00 A1 ð0Þ ¼ 2 T i1 K c1 K p1 ð27aÞ 2 K c1 G p ð0Þ 1 þ K c1 K p1 þ T i1 0 ð27bÞ where K p1 ¼ Gp ð0Þ. Similarly, the ﬁrst and second derivatives of (23) at s ¼ 0 are obtained as follows: ð28aÞ G0d1 ð0Þ ¼ θ þ λ1 G00 d1 ð0Þ ¼ θ þ 2λ1 θ þ 2λ1 2 2 ð28bÞ By equating the corresponding derivatives of actual and desired closed loop transfer functions ((27) and (28)), the parameters of Gc1 are obtained as follows: K c1 ¼ T i1 K p1 θ þ λ1 ð29aÞ The plant model Gpm that is used for obtaining the controller settings is only an approximation of the actual plant dynamics Gp . Hence, it is necessary to assume the time constants (λ1 and λ2 ) such that the closed loop system is robust to uncertainties in estimated process dynamics. The condition for closed loop robust stability is given in [16] as follows: lm ðsÞT d ðsÞ o 1 8 ω A ð 1; 1Þ ð31Þ where T d is the closed loop complementary sensitivity function and lm is the process multiplicative uncertainty which is given by Gp ðsÞ Gpm ðsÞ lm ðsÞ ¼ ð32Þ Gpm ðsÞ If uncertainties exist in process gain, time delay and time constant of primary process model, λ1 should be selected such that

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