ISA Transactions 65 (2016) 394–406
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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:
[email protected] (G.L. Raja),
[email protected] (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
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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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the following condition is satisfied:
½ ð τ Δ τ Þs þ 1
1 1 ‖T d ‖1 o
; 8 ω 40
ðτ s þ 1Þ 1 þ ΔK 1 e Δθ1 s ððτ Δτ Þs þ 1Þ
1 1
1
K1
ð33Þ
where ΔK 1 , Δθ1 and Δτ1 represents uncertainties in gain, time delay and time constant of primary process model.
5. Guidelines for selecting closed loop time constants The primary and secondary closed loop time constants (λ1 and λ2 ) play an important role in setpoint tracking and load disturbance rejection performance. Smaller values of λ1 and λ2 yields better servo and regulatory performance but degraded robust stability. On the other hand, increasing λ1 and λ2 improves system robustness at the cost of closed loop performance. Hence, the choice of time constants is a tradeoff between performance and robustness.
Fig. 4. Plot of maximum sensitivity (M s1 ) versus normalized primary closed loop time constant λ1 =θt .
5.1. Choice of λ1 As discussed in Section 3, Gp is obtained as a FOPTD/SOPTD/ FOPTDPZ/SOPTDPZ transfer function depending on the dynamics of Gp1 . Using the model reduction techniques given in Appendix B, the SOPTD/FOPTDPZ/SOPTDPZ transfer function is approximated as Gp ðsÞ ¼ ðK t e θt s Þ=ð1 þ τt sÞ. Eventually, the primary loop of the proposed PCCS shown in Fig. 2 reduces to a unity feedback structure with a PI controller in series with a FOPTD transfer function as shown in Fig. 3. The primary loop transfer function is given by L1 ðsÞ ¼ Gc1 ðsÞGp ðsÞ
ð34Þ
Substituting Gc1 and Gp in the above equation gives 1 Kt e θt s L1 ðsÞ ¼ K c1 1 þ T i1 s ð1 þ τ t sÞ
Fig. 5. (a) Closed loop responses for different values of λ2 corresponding to a secondary process model with θ2 =τ2 ¼ 1 (b) control efforts for different values of λ2 corresponding to a secondary process model with θ2 =τ2 ¼1.
ð35Þ
Maximum sensitivity of primary loop of the proposed PCCS is given by 1 ð36Þ M S1 ¼ 1 þG ðsÞG ðsÞ p c1 1 In order to study the variation in maximum sensitivity (M S1 ) with respect to process dynamics and λ1 , a FOPTD process model with transfer function Gp ðsÞ ¼ ðK t e θt s Þ=ð1 þ τt sÞ is considered. Assuming K t ¼ τt ¼1, the time delay 0 θt 0 is varied and the corresponding M S1 values are plotted versus the normalized closed loop time constant ðλ1 =θt Þ as shown in Fig. 4. An analytical expression relating λ1 and M S1 is obtained using curve fitting toolbox of MATLAB as given below: M s1 C 1=B θt r8 ð37Þ λ1 ¼ θt 0:01 r A τt
d Gpd1
r1
y1 Gc1
Gp
Primary loop Fig. 3. Approximation of the primary loop of proposed PCCS shown in Fig. 2 to obtain λ1 .
where A ¼ 1:578e θ
e
0:04442 τ t t
θ
1:468 τ t t
þ 0:4108e
þ 0:9923e θ
1:255 τ t t
and
θ
0:01277 τ t t
,
B ¼ 0:5532
C ¼ 0:6355e
θ
0:006015 τ t t
θ
1:692 τ t
t . In the present work, it is recommended to 1:612e assume M S1 as 1.6.
5.2. Choice of λ2 The following primary and secondary process models are considered to illustrate the effect of λ2 =θ2 on the regulatory performance of the proposed PCCS: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
2e 2s 4e θ2 s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ s sþ1
ð38Þ
Proposed method gives the following controller settings for the above primary and secondary process models: K p ¼0.005, T d ¼ 1, K c1 ¼ 0.1241 and T i1 ¼99.499. A negative step load disturbance (d) of magnitude 0.1 at t ¼20 s is considered. To illustrate system robustness, a perturbation of þ20% is considered in gain and time delay of primary and secondary process models and disturbance transfer functions. Moreover, the time constants of the secondary process model and disturbance transfer functions are perturbed by 20%. For θ2 ¼1, the closed loop responses and control efforts corresponding to four different values of λ2 =θ2 are shown in Fig. 5 (a) and (b), respectively. It is evident that the choice of λ2 affects only the regulatory performance. Regulatory responses for θ2 ¼0.1, 0.5, 1, 1.5 and 2 are shown in Fig. 6(a)–(e) by considering a unit step load disturbance at t¼ 0. From Fig. 6(a)–(e), it can be observed
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Fig. 6. (a) Closed loop regulatory responses corresponding to a secondary process model with θ2 =τ2 ¼0.1 (b) closed loop regulatory responses corresponding to a secondary process model with θ2 =τ2 ¼ 0.5 (c) closed loop regulatory responses corresponding to a secondary process model with θ2 =τ2 ¼ 1.0 (d) closed loop regulatory responses corresponding to a secondary process model with θ2 =τ2 ¼ 1.5 (e) closed loop regulatory responses corresponding to a secondary process model with θ2 =τ2 ¼2.
that λ2 =θ2 ¼ 0:25 yields satisfactory regulatory performance and robustness. Increasing λ2 =θ2 beyond 0:25 degrades the regulatory performance whereas decreasing λ2 =θ2 below 0:25 results in oscillatory system output. Therefore, λ2 is assumed as 0.25 θ2 in the present work. Remark-3. For process models with θ2 ¼ 0, it is recommended to assume λ2 ¼ 0.01.
6. Simulation study In this section, seven typical examples are considered to illustrate the effectiveness of the proposed tuning strategy. The proposed tuning strategy is compared with that of Pottmann et al. [5], Lee et al. [6] and absolute error
Padhan and Majhi [12]. Integral R1 R1
2 error (ISE ¼ eðtÞ dt), Integral IAE ¼ 0 eðtÞ dt , integral square 0
R1
eðtÞ dt , peak value of the system time absolute error ITAE ¼ t 0
output PV ¼ max y1 and settling time (t s ) are the performance measures that have been considered for comparing the proposed method with some of the recently reported tuning strategies. Instantaneous error eðtÞ is the difference between setpoint (r 1 ) and the controlled variable (y1 ) at time ‘t’. A small value of the above mentioned performance measures indicate better servo and regulatory performance. Smoothness of the control signal u2 ðtÞ is evaluated by computing ! total variation of the manipulated variable 1
P
ui þ 1 ui whereas u2max ¼ max {ju2 ðt Þj} gives the TV ¼ i¼1
maximum magnitude of the control signal. It is desirable to have TV and u2max values as small as possible. In order to compare servo performance, a unit step change in setpoint is considered at t ¼ 0 for Examples 1–6. In the simulation studies, PD controller is
Fig. 7. Nominal closed loop responses for Example 1: a – Proposed, b – Padhan and Majhi [12].
implemented as follows: T ds Gcd1 ðsÞ ¼ K p 1 þ f T ds þ 1
ð39Þ
where f is the derivative filter parameter which is set as 0.1 in this work. Example-1. Padhan and Majhi [12] have studied the following primary and secondary process models: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
e 3s 2e 2s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ 10s 1 sþ1
ð40Þ
The proposed controller settings are Gcd2 ðsÞ ¼0:5ðs þ1Þ= ð0:5s þ1Þ, K p ¼ 1.007 (for μ ¼ 0:39Þ, T d ¼ 1:5, K c1 ¼ 1.4258 and T i1 ¼ 1217.8 whereas the tuning strategy reported by Padhan and Majhi [12] yields the following controller settings: Gcs ðsÞ¼ 10s=ð4s þ 1Þ, K c2 ¼ 0.1282, T i2 ¼0.9614, T d2 0.3717, K c1 ¼ 2.6536, T i1 ¼700.1482 and T d1 ¼1.4766. Regulatory performance of the closed loop system is illustrated by introducing a unit negative step load disturbance at t¼ 30 s. In order to illustrate that the proposed method is robust to uncertainties in estimated process dynamics, the gain
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and delay of Gp1 ; Gp2 , Gpd1 and Gpd2 are perturbed by þ20% whereas the corresponding time constants are perturbed by 20%. The nominal and perturbed responses and the corresponding control efforts are shown in Figs. 7–10. From Tables 3, 4
and Figs. 7–10, it is observed that the proposed tuning strategy is robust and shows improved closed loop performance. Moreover, the proposed tuning strategy requires tuning of less number of controller parameters as compared to the tuning strategy reported in [12]. Also Padhan and Majhi’s method fails to yield robust closed loop performance for the assumed perturbation in plant parameters. Example-2. The IPTD primary process model and stable FOPTD secondary process model studied by Padhan and Majhi [12] are as follows: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
Fig. 8. Nominal control efforts for Example 1: a – Proposed, b – Padhan and Majhi [12].
Fig. 9. Perturbed closed loop responses for Example 1: a – Proposed, b – Padhan and Majhi [12].
Fig. 10. Perturbed control efforts for Example 1: a – Proposed, b – Padhan and Majhi [12].
2e 2s 4e s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ s sþ1
ð41Þ
Controller settings obtained using the proposed method are Gcd2 ðsÞ¼ 0:25ðs þ 1Þ=ð0:25s þ 1Þ, K p ¼ 0.005, T d ¼ 1, K c1 ¼ 0.1241, T i1 ¼ 99.499. The tuning strategy reported in [12] yields the following controller settings: Gcs ðsÞ ¼ ðs þ 2Þ=ð4s þ 2Þ, K c2 ¼ 0.2677, T i2 ¼3.1796, T d2 ¼0.1317, K c1 ¼ 0.195, T i1 ¼ 9.9246 and T d1 ¼ 0.6486. In order to evaluate the closed loop regulatory performance, a negative step load disturbance (d) of magnitude 0.1 is introduced at t¼20 s. A perturbation of þ20% is considered in gain and time delay of primary and secondary process models and disturbance transfer functions whereas the time constants of Gp2 and Gpd2 are perturbed by 20% to illustrate the robustness of the proposed method. The closed loop responses and control efforts corresponding to the nominal and perturbed plants are shown in Figs. 11–14. From Figs. 11–14 and the performance measures given in Tables 3 and 4, it is observed that the proposed method yields improved regulatory performance compared to the tuning strategy reported in [12] for both nominal and perturbed cases. Since regulatory performance is more important than servo performance in process industries [17], the proposed method is more suitable for industrial applications. In practical scenarios, noise arises from measuring devices, control valves or the process itself. Therefore, in this simulation study an additive Gaussian noise (n) of zero mean and variance of 1 10 4 is considered at the input of primary and secondary processes as shown in Fig. 2. The primary loop output and corresponding control efforts in the presence of Gaussian noise are shown in Figs. 15 and 16, respectively. Thus it is illustrated that the
Table 3 Performance measures of nominal system. Example
Tuning methods
IAE
ISE
ITAE
TV
ts
PV
u2max
Servo response (y1/r1) 1 Proposed Padhan and Majhi [12] 2 Proposed Padhan and Majhi [12] 3 Proposed 4 Proposed Padhan and Majhi [12] 5 Proposed Lee et al. [6] Pottmann et al. [5] 6 Proposed
6.44 9.65 4.325 4 23.65 12.96 13.98 8.76 14.65 9 0.947
5.09 6.33 3.36 3 17.78 10.35 9.99 6.78 10.47 6.5 0.931
24.83 68.59 11.68 10 343.7 102.1 129.6 50.59 169.7 53 0.441
2.67 2.5 0.138 0.25 0.045 1.86 2.25 2.58 7.98 3 3.16
9.6 23.1 6.7 8.07 52.22 27 30 13.9 46 18.94 1.56
1.04 1 1.04 1 1.01 1.06 1 1.03 1.14 1 1
1.429 1.504 0.126 0.25 0.044 0.59 1.25 2.95 2.81 4 1.71
Regulatory response (y1/d) 1 Proposed Padhan and Majhi [12] 2 Proposed Padhan and Majhi [12] 3 Proposed 4 Proposed Padhan and Majhi [12] 5 Proposed Lee et al. [6] Pottmann et al. [5] 6 Proposed
1.91 2.22 1.1 1.978 11.99 5.24 14.34 0.0014 0.2351 0.9765 0.073
0.407 0.596 0.193 0.254 3.89 1.38 3.09 8.3x10 8 0.002 0.0238 0.009
73.1 84.93 37.2 63.03 2129 452 1907 0.0287 5.177 22.65 0.2562
2.64 3.44 0.166 0.199 0.241 2.89 7.33 12.07 1.46 1 3.3
42.43 42.42 28.1 40 199 83.6 111.8 4 4 4 3.72
0.25 0.295 0.249 0.295 0.46 0.41 0.6 1.25x10 4 0.019 0.0427 0.418
1.78 2.18 0.132 0.147 0.220 1.76 2.32 1.005 1.109 1 1.098
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proposed PCCS is able to track the setpoint and reject the load disturbances even in the presence of noise. Example-3. In this example, we consider the following SOIPTD primary and stable FOPTD secondary process models: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
e 6:5672s 2e 2s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ sð3:4945s þ 1Þ sþ1
ð42Þ
Proposed method yields the following controller settings: Gcd2 ðsÞ ¼ 0:5ðs þ 1Þ=ð0:5s þ 1Þ, K p ¼ 0.003, T d ¼ 3.2836, K c1 ¼ 0.042 and T i1 ¼ 322.964. Regulatory performance of the closed loop system is evaluated by introducing a negative step disturbance of magnitude 0.2 at t ¼150 s. To illustrate the robustness of the proposed method, perturbations of þ 40% and 40% are considered in the time delays of Gp1 and Gpd1 . The closed loop responses and control efforts for nominal and perturbed plant models are shown in Figs. 17 and 18. Performance measures corresponding to nominal plant model and þ40% perturbation in time delay are shown in Tables 3 and 4, respectively. Fig. 19 shows the magnitudes of complementary sensitivity function and the bound corresponding to þ 40% per
uncertainty turbation in θ1 1= e 2:6269s 1 . It is observed that the proposed tuning strategy satisfies the robust stability condition given by (31). Example-4. The following primary and secondary process models are considered: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
e 6s 2e 2s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ 10s 1 sþ1
401
t¼60 s. The closed loop responses and control efforts are shown in Figs. 20 and 21, respectively. In order to show that the proposed method is robust to uncertainties in process dynamics, the time delay of the primary, secondary process models and the corresponding disturbance transfer functions are perturbed by þ30%. Plots of the controlled and manipulated variables corresponding to the perturbed plant are shown in Figs. 22 and 23, respectively. From Tables 3, 4 and Figs. 20–23, it is observed that the proposed tuning strategy yields better closed loop performance and robustness compared to the tuning strategy reported in [12]. It is to be noted that Padhan and Majhi [12] have used a Smith predictor based cascade control structure with eight controller and filter parameters whereas the proposed method requires tuning of just six controller parameters. Moreover, it is to be noted that the tuning strategy reported in [12] is not robust. Example-5. Consider the following primary and secondary process models studied by Lee et al. [6]: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
e 4s 1 ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ 10s þ 1 20s þ 1
ð44Þ
By assuming λ2 ¼1 and λ1 ¼ 4, the tuning strategy reported by Lee et al. [6] yields K cs ¼19, τIs ¼ 1.9, τDs ¼0, qf 2 ðsÞ ¼1=ð1:9s þ 1Þ, K cp ¼ 4.41, τIp ¼ 10.9, τDp ¼1.24 and qf 1 ðsÞ ¼1=ð9:52s þ1Þ whereas the tuning strategy reported by Pottmann et al. [5] yields q1 ðsÞ¼ ð20s þ 1Þ=ð2s þ 1Þ, q2 ðsÞ ¼ ð10s þ1Þ=ðs þ 1Þ; q3 ðsÞ ¼ ð20s þ 1Þ= ð5s þ 1Þ. The proposed method gives the following controller settings:
ð43Þ
Tuning rules derived in Section 3 and Section 5 yields the following controller settings: Gcd2 ðsÞ ¼0:5ðs þ 1Þ=ð0:5s þ 1Þ, K p ¼ 1.0042, T d ¼ 3, K c1 ¼ 0.5866 and T i1 ¼ 1682. Assuming λcs ¼ 7:2, the tuning strategy reported by Padhan and Majhi [12] gives the following controller settings:Gcs ðsÞ ¼ 10s=ð7:2s þ 1Þ, K c2 ¼ 0.128, T i2 ¼0.9614, T d2 ¼0.3716, K c1 ¼ 1.3296, T i1 ¼ 779.77 and T d1 ¼5.11. With these controller settings, performance of closed loop system is compared by introducing negative unit step disturbance at
Fig. 11. Nominal closed loop responses for Example 2: a – Proposed, b – Padhan and Majhi [12].
Table 4 Performance measures of perturbed system. Example
Tuning methods
IAE
ISE
ITAE
TV
ts
PV
u2max
Servo response (y1/r1) 1 Proposed Padhan and Majhi [12] 2 Proposed Padhan and Majhi [12] 3 Proposed 4 Proposed Padhan and Majhi [12] 5 Proposed Lee et al. [6] Pottmann et al. [5] 6 Proposed
7.97 – 4.57 4.03 26.94 23.69 45 10.8 13.07 8.8 0.941
5.43 – 3.56 3.15 20.21 13.87 13.93 7.61 26.43 6.78 0.976
53.68 – 13.47 11 479.4 1076 7657 95.4 505.5 56.69 0.43
24.5 – 0.827 0.809 0.051 9.68 30.22 66.56 12.81 7.266 5.02
23.32 – 10.9 6.8 66.74 68.19 353 25 132 13.64 1.35
1.36 – 1.06 1 1.08 1.43 1.24 1.14 1.31 0.998 1.01
2.21 – 0.193 0.25 0.044 0.996 1.2531 2.48 2.29 4 1.67
Regulatory response (y1/d) 1 Proposed Padhan and Majhi [12] 2 Proposed Padhan and Majhi [12] 3 Proposed 4 Proposed Padhan and Majhi [12] 5 Proposed Lee et al. [6] Pottmann et al. [5] 6 Proposed
3.798 – 1.33 2.06 13.46 11.66 – 0.0017 0.55 0.974 0.085
1.109 – 0.217 0.317 4.482 3.077 – 1.03x10 7 0.0036 0.0264 0.013
158.4 – 38.67 65.24 2452 1150 – 0.032 14.01 22.24 0.2979
12.41 – 0.599 0.274 0.245 87.85 – 67.73 1.55 2.01 7.01
51.6 – 28.03 39.13 201 134 – 4.8 4.8 10.79 3.74
0.52 – 0.314 0.356 0.48 0.51 – 1.25x10 4 0.0204 0.054 0.484
2.51 – 0.167 0.168 0.221 1.99 – 1.007 1.078 1.06 1.216
(–) Unstable response.
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Fig. 12. Nominal control efforts for Example 2: a – Proposed, b – Padhan and Majhi [12].
Fig. 13. Perturbed closed loop responses for Example 2: a – Proposed, b – Padhan and Majhi [12].
Fig. 17. Nominal and perturbed closed loop responses for Example 3: a – Proposed, b – Proposed with þ 40% perturbation in θ1 , c – Proposed with 40% perturbation in θ1 .
Fig. 18. Nominal and perturbed control efforts for Example 3: a – Proposed, b – Proposed with þ 40% perturbation in θ1 , c – Proposed with 40% perturbation in θ1 .
Fig. 14. Perturbed control efforts for Example 2: a – Proposed, b – Padhan and Majhi [12]. Fig. 19. Magnitude plots for complementary sensitivity function and uncertainty norm bound for Example 3: a – uncertainty norm bound for þ 40% perturbation in θ1 2:6269s 1= e 1 , b – complementary sensitivity function for λ1 ¼ 8.2015 (M s1 ¼ 1:6Þ.
Fig. 15. Closed loop responses in the presence of Gaussian noise for Example 2: a – Proposed, b – Padhan and Majhi [12].
Fig. 20. Nominal closed loop responses for Example 4: a – Proposed, b – Padhan and Majhi [12].
Fig. 16. Control efforts in the presence of Gaussian noise for Example 2: a – Proposed, b – Padhan and Majhi [12].
Gcd2 ðsÞ ¼ ð10s þ 1Þ=ð0:01s þ 1Þ, K c1 ¼ 2.4799 and T i1 ¼20.9471. An unit step load disturbance (d) is introduced to evaluate the regulatory performance of the closed loop system. To illustrate the robustness of the proposed method, a perturbation of þ 20% is assumed in the gain of Gp1 , Gp2 , Gpd1 and Gpd2 whereas a perturbation of 20% is assumed in the time constant of Gp1 , Gp2 , Gpd1 and Gpd2 . Moreover, the time delays of Gp1 and Gpd1 are perturbed by þ20%. The closed loop responses and corresponding control efforts of nominal and perturbed plant models are shown in Figs. 24–27. It is observed that the proposed method
Fig. 21. Nominal control efforts for Example 4: a – Proposed, b – Padhan and Majhi [12].
yields improved and robust closed loop performance compared to the tuning strategy reported in [6] for both nominal and perturbed process models. One point worth noticing is that the deviation of y1 from the setpoint due to load disturbances is almost negligible in the proposed method. Moreover, it is to be noted that the proposed tuning strategy
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requires tuning of only four controller parameters whereas the tuning strategy reported in [6] requires tuning of eight controller and filter parameters. From the performance measures given in Tables 3–4, it is observed that the regulatory performance of the proposed method is superior to that of [5,6] .
Fig. 22. Perturbed closed loop responses for Example 4: a – Proposed, b – Padhan and Majhi [12].
Fig. 23. Perturbed control efforts for Example 4: a – Proposed, b – Padhan and Majhi [12].
Example-6. In order to illustrate that the proposed method can also be used for inverse response processes, the following process models are considered: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
ð 0:418s þ 1Þe 0:07s ; s
Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼
0:547e 0:03s sþ1
ð45Þ
Tuning rules derived in Sections 3 and 5 yields the following controller settings: Gcd2 ðsÞ ¼ ðs þ 1Þ=ð0:547ð0:0075s þ 1ÞÞ, K p ¼ 0.0221, T d ¼0.035, K c1 ¼ 1.0483 and T i1 ¼46.2233. Regulatory performance of the closed loop system is illustrated by introducing a negative unit step disturbance at t¼ 3 s. In order to show that the proposed method is robust to uncertainties in process dynamics, the gain and time delay of Gp1 , Gpd1 , Gp2 , Gpd2 and zero of Gp1 , Gpd1 are perturbed by þ 5% whereas the time constant of Gp2 and Gpd2 are
Fig. 26. Perturbed closed loop responses for Example 5 (a) servo response (b) regulatory response: x – Proposed, y – Pottmann et al. [5], z – Lee et al. [6]. Fig. 24. Nominal closed loop responses for Example 5 (a) servo response (b) regulatory response: x – Proposed, y – Pottmann et al. [5], z – Lee et al. [6].
Fig. 25. Nominal control efforts for Example 5 (a) during servo response (b) during regulatory response: x – Proposed, y – Pottmann et al. [5], z – Lee et al. [6].
Fig. 27. Perturbed control efforts for Example 5 (a) during servo response (b) during regulatory response: x – Proposed, y – Pottmann et al. [5], z – Lee et al. [6].
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Fig. 28. Closed loop responses of the proposed method for Example 6: a – nominal, b – perturbed.
Fig. 31. Real time system response for step input: a – desired ball position, b – magnetic levitation ball position.
Fig. 29. Control efforts of the proposed method for Example 6: a – nominal, b – perturbed.
Fig. 32. Control effort for step input.
Fig. 33. Real time system response for sinusoidal input: a – desired ball position, b – magnetic levitation ball position.
Fig. 30. Real time experimental platform.
perturbed by 5%. The closed loop responses and control efforts corresponding to the nominal and perturbed plants are shown in Figs. 28 and 29. From Tables 3, 4 and Figs. 28 and 29, it is observed that the proposed tuning strategy yields robust closed loop performance even for inverse response systems. In [12], examples with positive zeros are not considered. Therefore, no comparison is shown for this example. Example-7. In this example, we have performed a hardware-inthe-loop real-time simulation to illustrate that the proposed method yields satisfactory closed loop performance in practical scenarios. The magnetic levitation experimental setup of Feedback Instruments (Model No. 33-210) is shown in Fig. 30. Gc1 , Gcd1 , Gcd2 and Gp2 are implemented in Simulink whereas the above mentioned magnetic levitation setup is used instead of Gp1 . Our aim is to tune the controllers of the proposed PCCS so that the metal ball of the magnetic levitation setup tracks the reference signal within an operating region. Using the model identification procedure given in [18] and system identification toolbox of Matlab, the transfer function of the magnetic levitation system is estimated as Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼ 0:32662e0:023364s =ð 0:01056s þ 1Þ. Moreover, the transfer functions of Gp2 ðsÞ and Gpd2 ðsÞ are assumed as (2e 0:25s Þ=(s þ 1).
Fig. 34. Control effort for sinusoidal input.
Fig. 35. Real time system response for square input: a – desired ball position, b – magnetic levitation ball position.
The proposed controller settings are Gcd2 ðsÞ¼ 0:5ðs þ1Þ= ð0:0625s þ 1Þ,K p ¼6.1233, T d ¼ 0:0234, K c1 ¼ 1.0523 and T i1 ¼ 0.0163. A negative step change of magnitude ‘0.5’ is introduced at t ¼20 s to illustrate the servo response of the closed loop system. Regulatory performance is illustrated by introducing a negative step load disturbance (d) of magnitude '1.5' at t=40 s. Moreover, sinusoidal and square signals of
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Fig. A.1. Closed loop response of the proposed method for (A:1) if θ1 ¼ 4 (large time delay).
Fig. 36. Control effort for square input.
magnitude 0.5 and frequency 0.5 Hz are also considered as reference signals. Real-time system output and control efforts corresponding to the proposed method for step, sine and square reference inputs are shown in Figs. 31–36. It is observed that the metal ball satisfactorily tracks the user-defined reference signals even in the presence of disturbances. Since the tuning strategy reported by Padhan and Majhi [12] fails to levitate the steel ball, no comparison is shown for this example.
Fig. A.2. Control efforts of the proposed method for (A:1) if θ1 ¼ 4 (large time delay).
7. Conclusions In this manuscript, a modified parallel cascade control structure (PCCS) is proposed for stable, unstable and integrating systems. The proposed PCCS consists of a primary setpoint tracking controller, stabilizing controller and secondary disturbance rejection controller. Setpoint tracking controller is tuned by equating the first and second derivatives of desired and actual closed loop transfer functions whereas Routh Hurwitz stability criterion is used to design the stabilizing controller. Internal Model Control approach is used for obtaining the secondary disturbance rejection controller. Analytical expressions and guidelines are provided to help the user in obtaining the closed loop time constants. This makes the proposed method more advantageous than the reported tuning strategies in which the authors recommend a range of values for the closed loop time constants. The proposed tuning strategy is simple and requires tuning of just four or six controller parameters. Simulation studies show that the proposed tuning strategy yields improved closed loop performance and robustness.
Fig. A.3. Closed loop response of the proposed method for (A:1) if θ1 ¼ 1.
Fig. A.4. Control efforts of the proposed method for (A:1) if θ1 ¼ 1.
Appendix A. Choice of β In order to recommend a suitable value for β, the following primary and secondary process models are considered: Gp1 ðsÞ ¼ Gpd1 ðsÞ ¼
2e θ1 s 4e s ; Gp2 ðsÞ ¼ Gpd2 ðsÞ ¼ s sþ1
ðA:1Þ
For θ1 ¼ 4, 1 and 0.5 the closed loop responses and control efforts corresponding to the proposed method are plotted for three different values of β (0.001, 0.01, and 0.1) in Figs. A.1–A.6. From Figs. A.1 to A.6, it is noted that as ‘β’ approaches ‘0’, the closed loop performance is improved in the following ways: (1) less overshoot and settling time during regulatory response and (2) smoother control signals with lower peak value. It is recommended to assume β as 0.01 because it is evident from Figs. A.1 to A.6 that there is no siginificant change in closed loop response while decreasing β below 0.01.
Fig. A.5. Closed loop response of the proposed method for (A:1) if θ1 ¼ 0:5.
Appendix B. Model reduction Half-rule [19] is used in the present work for approximating higher order transfer function by a FOPTD transfer function. This approximation is essential to obtain the time delay to time
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Fig. A.6. Control efforts of the proposed method for (A:1) if θ1 ¼ 0:5.
constant ratio ðθt =τt ) required to compute the primary closed loop time constant (λ1 ) using (37). The SOPTDPZ transfer function to be approximated is first expressed in the following form: Gp ðsÞ ¼
K 1 ð1 p1 sÞ e θ1 s ðτa s þ1Þðτb s þ 1Þ
ðB:1Þ
In the above equation, τa Z τ b . Using half rule, the FOPTD transfer function is obtained as follows: Gp ðsÞ ¼
Kt e θt s ðτt s þ 1Þ
ðB:2Þ
where K t ¼ K 1 , τt ¼ τa þ τ2b and θt ¼ θ1 þ τ2b þp1 . Similarly, Gp ðsÞ ¼ (K 1 ð1 p1 sÞe θ1 s )/ðτ1 s þ 1Þ can be approximated by ðK t e θt s Þ= ðτ1 s þ 1Þ, where θt ¼ θ1 þ p1 .
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