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A novel dead time compensator for stable processes with long dead times
Journal of Process Control 22 (2012) 612–625
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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
A novel dead time compensator for stable processes with long dead times Kawnish Kirtania, M.A.A. Shoukat Choudhury ∗ Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
a r t i c l e
i n f o
Article history: Received 20 December 2010 Received in revised form 19 October 2011 Accepted 9 January 2012 Available online 2 February 2012 Keywords: Dead time compensator Stable processes Smith predictor Two-degrees-of-freedom controller Set-point tracking Regulatory control
a b s t r a c t This paper presents a new and simplified approach for the design of dead time compensators for processes with long dead times. The approach is based on a modified structure of the Smith predictor that allows the user to isolate the disturbance and set-point responses and thereby, provide a two-degrees-of-freedom control scheme. The proposed structure is easy to analyze and tune. Using an estimation of the dead time and process model of the plant, the proposed compensator is left with two tuning parameters that determine the closed-loop performance and robustness. The performance of the proposed compensator is compared with the most recent dead time compensator appeared in the literature. The method is evaluated on two simulated processes and a computer-interfaced pilot-scale two tank heating system to demonstrate the practicality and utility of the proposed scheme. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Dead times are present in systems where any kind of mass, energy or momentum is transferred. These terms are represented as exponential terms in the transfer function. The presence of delays typically results in: • Oscillatory behavior of the process output or instability. • Poor tracking of set points and poor regulatory functions. The effect of dead time can be compensated by removing the exponential term from the characteristic equation of the process. This was first done by Smith [1] and the dead time compensator (DTC) is named as the Smith predictor (SP). The shortcomings of the Smith predictor can be noted as – (a) sensitivity of the loop for error in dead time and process model estimation, (b) not applicable for integrating processes. To overcome these difficulties, several research studies have been reported to improve the performance of the Smith predictor. Wantanabe and Ito [2] suggested a modification in the process model to overcome the difficulties of using the Smith predictor for integrating plants. To make the Smith predictor more stable and faster H¨agglund [3] proposed use of a PPI (Predictive Proportional Integral) controller. This controller has all the features of Smith predictor with the advantage of tuning only three parameters as in a PID controller. To improve the response of the closed loop,
∗ Corresponding author. Tel.: +88 02 9665650/7960; fax: +88 02 9665609. E-mail address: [email protected] (M.A.A.S. Choudhury). 0959-1524/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2012.01.003
the structure proposed by Wantanabe and Ito [2] was modified by Normey-Rico and Camacho [4] by introducing the concept of two-degrees-of-freedom (2DOF) structure. They proposed a prefilter after the set point while the disturbance response remains unchanged. In 2002, Normey-Rico and Camacho [5] came up with modification of the pre-filter. A higher order pre-filter was proposed instead of a first order pre-filter. Torrico and Chang [6] studied the discrete implementation of the 2 DOF dead time compensator. Zhang et al. [7] determined different parameters of the design by Normey-Rico and Camacho [5] through optimization of some differential equations. Another 2 DOF dead time compensator was proposed for unstable systems by Liu et al. [11]. Various approaches were taken to improve the response of the processes with long dead time. Ingimunderson and H¨agglund [8] described the robust tuning procedures for the dead time compensators by frequency domain analysis. Huzmezan et al. [9] designed an adaptive control scheme to compensate the dead time for the batch processes. Tan et al. [10] applied the IMC design method to design a dead time compensator. A Unified approach was proposed by Normey-Rico and Camacho [12] where one filter was used for first order plus dead time (FOPDT) stable processes and two filters were used for FOPDT integrating or unstable processes to decouple the set-point tracking and disturbance rejection properties. This compensator provides also two-degrees-of-freedom control strategy to tune the set point and disturbance rejection response separately. A robust control design was suggested by Albertos and Garcia [13] which is as good as the unified approach by Normey-Rico and Camacho [12]. A new dead time compensator was proposed in [15] where two filters were used for the FOPDT stable processes which was found as good in simulation as in real time application.
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2.1. Set-point tracking response When the actual process P(s) is same as the process model Pn (s), the closed loop transfer function of the process for set-point tracking Y (s) F1 (s)C(s)Pn (s) = = R(s) Ysp (s) 1 + C(s)Gn (s) Fig. 1. Block diagram of proposed modified Smith predictor (MSP).
It is impossible to eliminate the effect of dead time completely. But the effect is minimized by the compensators. All compensators need some prior information about the process or system. The difference lies in the performance as well as in the robustness of the compensator. Each work cited above is somehow better than the previous one. Chronologically, the modified Smith predictor became versatile and applicable for different kind of processes. Eventually schemes with improved stability and robustness for cases with estimation error in process model and dead time were suggested. A simple and elegant dead time compensator is designed in this paper for the processes that can be approximated by first order plus dead time (FOPDT) model. This paper is organized as follows. Section 2 presents the process models and the 2 DOF DTC structure. In Section 3, the proposed tuning rules of the PI controller and the filter parameters are presented. Section 4 shows uncertainty and robustness analysis. A comparative simulation study is given in Section 5. Section 6 describes some experimental results followed by concluding remarks.
The proposed modified Smith predictor control structure is shown in Fig. 1. As can be seen, the structure is the same as the Smith Predictor but with two additional filters. F1 (s) is a traditional filter which improves the set point response and F2 (s) is a predictor filter that improve the disturbance rejection response. The proposed modified Smith predictor (MSP) can be used to take into account the robustness of the loop, improving disturbance rejection properties and decoupling the set-point and disturbance rejection responses. Therefore, the main drawback of the SP is considered in this design. Normey-Rico and Camacho [12] proposed a similar structure for dead time compensator. But the difference between the proposed one and the dead time compensator of Normey-Rico and Camacho [12] relies in the number of filters. Normey-Rico and Camacho proposed single filter for stable processes and two filters for integrating or unstable processes with dead time. The process type considered here is stable and two filters have been suggested. The next difference is in the filter structure. The structure of the filter in dead time compensator of Normey-Rico and Camacho [12] was of second order whereas the proposed one consists of two first order filters. In the proposed structure, Pn (s) = Gn (s)e−Ln s is the process model, Gn (s) is the delay free part of the model, Ln is the estimated time delay and C(s) is the controller. Here, the assumed process model is a FOPDT model. The FOPDT model can be written as
Pn (s) = Gn (s)e−Ln s =
Kp e−Ln s Ts + 1
The proposed model is based on the concept that the process can be made faster and robust by setting a pseudo set-point which will vary with time and remains stable when the process reaches the desired output. The set point passes through the filter F1 (s) and varies with time so that the process responses very fast to reach the desired value. To attain this type of response, the proposed structure of the filter F1 (s) is F1 (s) =
(1)
Zs + 1 T1 s + 1
(3)
The parameters Z and T1 are to be tuned. A simple tuning rule is described in Section 3 for this filter. 2.2. Disturbance rejection response The closed loop transfer function for disturbance rejection is expressed as
Y (s) C(s)Pn (s)F2 (s) = Pn (s) 1 − D(s) 1 + C(s)Gn (s)
= Pn (s)(s)
(4)
where (s) = [1 − (C(s)Pn (s)F2 (s))/(1 + C(s)Gn (s))]. From the above equation, it can be seen that if the term (s) is close to zero, the response will be better and improved. To attain this, the second filter F2 (s) can be designed as (more discussion in Section 3.3): F2 (s) =
2. Process model and dead time compensator (DTC) structure
(2)
Zs + 1 T2 s + 1
(5)
Z has the same parameter value as in the previous filter. But the parameter T2 is to be tuned to have a robust closed loop response as well as the best disturbance rejection response. This filter refines the signal in both feedback loops. It only affects the disturbance rejection response and robustness of the loop but not the set-point response. The parameter T2 is responsible for the robustness of the loop. Its tuning and properties are discussed in more detail in Section 3. 3. Tuning rules The proposed dead time compensator has three parameters to tune namely Z, T1 and T2 . The controller for the proposed structure is a PI controller. So the controller has two more parameters to be tuned. The tuning rules for the compensator filter parameters and the controller parameters are described in this section. 3.1. PI controller parameters The PI controller needs to be tuned to have a stable closed loop response. The controller can be tuned to have the best response by using the predictive structure. The rule suggested by [3] is used to calculate the controller parameters. If the controller gain cancels out the process gain and the time constant of the controller is equal to the process time constant, the controller and the process will converge to a first order transfer function of gain 1 and time constant T (if there is no dead time present). Therefore, controller parameters are tuned using IMC rules.
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3.2. Set-point tracking filter (F1 (s)) parameters
rejection becomes perfect. Therefore, the filter structure for F2 can be chosen as F2 (s) = (Zs + 1)/(T2 s + 1). Now
The filter for the set point tracking is tuned by setting the parameters Z and T1 . To analyze this, the transfer function between the process response and set-point change must be taken care of. The transfer function is R(s) =
Y (s) F1 (s)C(s)Pn (s) F1 (s)C(s)Gn (s)e−Ln s = = Ysp (s) 1 + C(s)Gn (s) 1 + C(s)Gn (s)
(6)
It is clear from the transfer function that filter F1 does not affect the characteristic equation. So it does not affect the stability of the process. Let set-point response without the filter be denoted as G1 (s). By applying IMC tuning rules G1 (s) =
C(s)Gn (s)e−Ls 1 + C(s)Gn (s)
(7)
e−Ls Ts + 1
F2 (s) =
e−Ls Zs + 1 e−Ls × = Ts + 1 T2 s + 1 T2 s + 1
(14)
Asymptotically, Eq. (14) is equal to 1. As the time propagates, the disturbance will be nullified. This is interesting to note that with the change of T2 , only the disturbance response will be affected. The parameter T2 can be chosen on the basis of how fast the disturbance rejection is required. From authors’ experience, a value of T/14 is a good choice to start with. Then it can be tuned as required by robustness and stability. 4. Robustness and stability analysis 4.1. Robustness analysis
((Ke−Ls )/(Ts + 1))Kc (1 + (1/(TI s))) 1 + (K/(Ts + 1))Kc (1 + (1/(TI s))) ((TI s + 1)/(Ts + 1))e−Ls (1/(TI s)) = 1 + ((TI s+ 1)/(Ts + 1))(1/(TI s)) e−Ls = Ts + 1
=
(8)
From Eq. (8), it is found that with the proposed controller, the process has the same time constant as for the open loop case but the gain changes to 1 so that the set point can be tracked properly. By combining Eqs. (8) and (6), the set-point response can be found as
R(s) = F1 (s)
e−Ls Ts + 1
(9)
Now, the filter is to be designed in a way so that it cancels the time constant of the process and converts it to a faster process. Therefore, the filter which will serve the purpose can be designed as F1 (s) =
Zs + 1 T1 s + 1
(10)
The parameter Z is chosen to be equal to the time constant T. So the transfer function becomes R(s) =
e−Ls T1 s + 1
(11)
It will essentially cancel out the long time constant from the system and respond quickly. Now the set point tracking monotonically depends on the value of T1 . The parameter T1 can be chosen on the basis of desired set point response. From the experience of the authors, the value of T1 can be initially chosen as T/7, i.e., 7 times faster than the original process time constant.
The filter used for disturbance rejection has two parameters Z and T2 . Z is chosen as the same as that in the set point tracking filter. Therefore, the only parameter to be tuned is T2 . The disturbance rejection response of the system can be characterized from the term (s) of Eq. (2). Ideally, (s) should be derived equal to zero. Therefore (12)
If C(s) is tuned as proposed, the resultant value of (s) becomes
(s) = 1 −
e−Ls 1 + Ts
F2 (s)
P(s) = Pn (s)[1 + ıP(s)]
(15)
As shown in [14], the multiplicative norm-bound uncertainty, ıP(ω), is |ıP(ω)| ≤ ıP(ω) ∀ω > 0
(16)
From the above consideration the characteristic equation for the system presented in Fig. 1 becomes 1 + C(s)Gn (s) + C(s)Gn (s)Pn (s)F2 (s)ıP(s) = 0
(17)
As the nominal system is stable, robust stability condition can be found for the proposed compensator as:
1 + C(jω)Gn (jω) , ∀ω > 0 C(jω)G (jω)F (jω)
ıP(ω) < dP(ω) =
n
(18)
2
where, (P(s) − Pn (s)) = Pn (s)ıP(s) = dP(s). Note that the filter only affects the disturbance rejection response and the robustness of the loop. Substituting the transfer function elements into Eq. (18) ıP(ω) < dP(ω)
1 + Kc (1 + (1/(TI jω)))(K/(Tjω + 1)) = Kc (1 + (1/(TI jω)))(K/(Tjω + 1))((Zjω + 1)/(T2 jω + 1))
(19)
Based on tuning rules suggested in Section 3, Kc = 1/K, TI = T. Thus Eq. (19) simplifies to
3.3. Disturbance rejection filter (F2 (s)) parameters
C(s)Pn (s)F2 (s) (s) = 1 − 1 + C(s)Gn (s)
The effect of the changes in filter parameters on the robustness and the stability of the process needs to be investigated to choose proper values of these parameters. Stability of the FOPDT process is not affected but the robustness is greatly influenced with the change of the filter parameters. The family of plants P(s) having the following relationship are under consideration
(13)
The second term is to be made close to 1 so that the response to disturbance change becomes zero and essentially the disturbance
ıP(ω) < dP(ω) =
1 + (1/(TI jω)) (1/(TI jω))((Zjω + 1)/(T2 jω + 1))
= |T2 jω + 1|
(20) (21)
Therefore, the robustness of the loop can be increased by increasing the value of T2 . However, by increasing T2 , the disturbance rejection response will become slower. This is the classical trade off between robustness and performance. For plants that can be approximated by a stable FOPDT model, the value of T2 can be increased gradually until a satisfactory regulatory and robustness trade-off is achieved. The robustness ‘margin’ desired would depend on the level of uncertainty in the plant. The greater the uncertainty, larger the value of T2 and therefore slower the regulatory performance.
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4.2. Uncertainty analysis By introducing some error in the model estimation, this uncertainty analysis is carried out. The analysis covers the uncertainty in dead time, gain and time constant estimation. As the dead time compensator depends on the model of the process, the uncertainties will affect the performance. But as long as the system is stable, no major problem occurs. By measuring the gain margin, the stability limit of a process can be understood. At the value of gain margin 1 the process becomes marginally stable. On further decreasing the gain margin value, the system becomes unstable. The analysis takes into account both set-point change and disturbance rejection responses. 4.2.1. Set-point responses • Uncertainty in dead time estimation: At first, the process was checked with some error in dead time estimation for the set-point change. For this, a new equation was derived with an estimation error of L for the delay L. Thus the transfer function between the set-point and output becomes Zs + 1 Y (s) e−Ls = × T1 s + 1 Ysp (s) (Ts + 1) + e−Ls (1 − e−Ls )
(22)
To analyze it further, the delay term is approximated by 1/1 Pade approximation. After some simplifications, Eq. (22) becomes Zs + 1 Y (s) = T1 s + 1 Ysp (s)
×
1 − (L/2)s ((TLLs3 )/4) + ((T (L + L)s2 )/2) + s(T + (L/2) + L) + 1
(23) With a 20% error in estimating dead time was considered for a process and then the gain margin was investigated. For this case the following model used by [12] was considered: 0.12 −3s P(s) = e 6s + 1
(24)
Now with an error of 20%, the estimated delay becomes 3.6 s. As the filter parameters depend on the process time constant, the filter parameters are not affected. The value of Z was equal to the time constant and the other parameter T1 was taken as 6/8. So according to Eq. (23), the transfer function for set-point change becomes Y (s) 6s + 1 × = 0.8s + 1 Ysp (s)
1 − 1.5s 2.7s3 + 10.8s2 + 8.1s + 1
Y (s) K(1 − (L/2)s) = D(s) (Ts + 1)(1 + (L/2)s)
(25)
2.7s3
For this case, the crossover frequency from Bode plot was 0.9862 and gain margin was 1.74. So the value of the gain margin is reached to practical limit by increasing the parameter T1 . The uncertainty in estimating the dead time is also compensated by a higher value of T1 . The Bode plots for the two cases are shown in Fig. 2. • Uncertainty in process gain estimation: for set point change, the transfer function with an error in gain estimation becomes Y (s) (1 − Ls/2) = × Ysp (s) (1 + Ls/2)((T/(KK c ))s + 1)
Zs + 1
(27)
T1 s + 1
As there is an error in estimation of K, the value of Kc will be different from K. So the value of Kc would be equal to K + K where K is the error in estimation Y (s) (1 − Ls/2) = × Ysp (s) (1 + Ls/2)(Ts/(K(K + K)) + 1)
Zs + 1
T1 s + 1
(28)
A 20% error was considered in process gain estimation. To understand the effect quantitatively, the transfer function for the set-point change for the process under consideration found to be Y (s) (1 − 1.5s) × = Ysp (s) (1 + 1.5s)(6s/(0.12(0.12 + 0.024)) + 1)
6s + 1
0.8s + 1 (29)
The Bode plot for the above transfer function is shown in Fig. 3. The crossover frequency was found to be 1.32 and the corresponding gain margin was 1/0.01198 = 83.47 which is quite high and away from instability. So further change in parameter was not needed. The general idea is that the error in the gain estimation does not lead to instability of the process. • Uncertainty in estimation of the process time constant: for this error, the parameter Z in the filter and TI in the controller would not be perfect. With a deviation of T from the time constant T is considered. So the resulting transfer function for set-point change Y (s) (Ts + 1)(1 − Ls/2) × = Ysp (s) ((T 2 + TT )s2 + 2Ts + 1)(1 + L/2)
Zs + 1
T1 s + 1
(30)
For a 20% estimation error, the process model under consideration provides the following transfer function Y (s) (6s + 1)(1 − 1.5s) × = Ysp (s) (43.2s2 + 12s + 1)(1 + 1.5s)
6s + 1
(31)
0.8s + 1
Fig. 4 shows the Bode plot of the transfer function which clearly shows that for the crossover frequency 1.408, the gain margin is 1.8. So the system is stable. 4.2.2. Disturbance rejection response • Uncertainty in dead time estimation: the transfer function between the output and disturbance is derived as
((T2 LL)/4)s3 + (((T2 (L + L) + L)s2 )/2) + (T2 + L + (L/2))s ((T2 LL)/4)s3 + (T2 (L + L) − ((LL)/2))(s2 /2) + (T2 + L + ((L + L)/2))s + 1
From the Bode plot, gain margin of the transfer function was determined to see whether it crosses the stability limit. The crossover frequency for this process was found to be 1.158 rad/s and the gain margin was 1.4 which is away from instability. However, the standard value of gain margin is in between 1.7 and 4.0. To achieve the limit, the parameter T1 is increased to 1.4 while other parameters remaining the same. In this case the transfer function for set-point change is 6s + 1 Y (s) = × 1.4s + 1 Ysp (s)
615
1 − 1.5s + 10.8s2 + 8.1s + 1
(26)
(32)
A 1/1 Pade approximation was considered for the Bode plot analysis in this case and L is the dead time estimation error. The parameter of disturbance rejection filter would be 6 and 0.2 for Z and T2 respectively for the process under analysis. For a 20% error in estimation, the resulting transfer function for disturbance rejection would be Y (s) 0.12(1 − 1.5s) = D(s) (6s + 1)(1 + 1.5s)
0.09s3 + 1.86s2 + 2.3s 0.09s3 − 0.09s2 + 2.6s + 1
(33)
Here from the Bode plot, no crossover frequency was found as the curve never reaches −180◦ of phase angle. Fig. 5 shows that
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Fig. 2. Bode plots for error in dead time estimation.
the values of amplitude ratio always remains below 1. So the gain margin never reaches the value below 1. • Uncertainty in estimating the process gain: the equation derived for the disturbance response is given below:
rejection for the process becomes Y (s) 0.12(1 − 1.5s) = D(s) (6s + 1)(1 + 1.5s)
× 1− K(1 − (L/2)s) Y (s) = D(s) (Ts + 1)(1 + (L/2)s)
(1 − (L/2)s)(Zs + 1) × 1− ((T/(KK c ))s + 1)(T2 s + 1)(1 + (L/2)s)
(34)
Here, K and Kc are not equal and thereby no cancelation occurs. Considering an error of 20%, the transfer function for disturbance
(1 − 1.5s)(6s + 1) ((6/(0.01728))s + 1)(0.2s + 1)(1 + 1.5s)
(35)
The Bode plot for the gain estimation error consideration is shown in Fig. 6. The stability of the process is not affected because the amplitude ratio never comes close to 1. • Uncertainty in estimating the time constant of the process: the equation derived for the uncertainty in estimating the time constant of the process and the corresponding equation for a 20% error for the process under consideration are as follows
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Fig. 3. Bode plot for the estimation error in process gain.
Fig. 4. Bode plot for the estimation error in process time constant.
Y (s) K(1 − (L/2)s) = D(s) ((T + T )s + 1)(1 + (L/2)s)
× 1−
=
(Ts + 1)(1 − (L/2)s)(Zs + 1) (Ts[(T + T )s + 1] + Ts + 1)(T2 s + 1)(1 + (L/2)s)
0.12(1 − 1.5s) (7.2s + 1)(1 + 1.5s)
−54s3 + 18s2 + 10.5s + 1 × 1− 12.96s4 + 73.04s3 + 63.9s2 + 13.7s + 1
(36)
There is no crossover frequency for the response as the phase angle never reaches the value of −180◦ in Bode plot. Moreover, the value of amplitude ratio always remains below 1. So the system is perfectly stable with the error. The plot is shown in Fig. 7. From this analysis, it is evident that some estimation errors in model estimation do not make the process unstable. Also with increasing the values of T1 and T2 , the robustness of the process may be increased as desired.
5. Simulation study (37)
As has been mentioned, several papers have been appeared in the literature with modifications to the original Smith predictor,
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Fig. 5. Bode plot for the estimation error in dead time.
Fig. 6. Bode plot for the estimation error in process gain.
some concerning the tuning of parameters of the original Smith predictor and others proposing the modification of the control structure. In general, the later modifications have demonstrated better performance than the previous works. For the case of stable plants, the dead time compensator (DTC) proposed by [12] can be considered as the best solution among the studies published so far. Normey-Rico and Camacho [12] have chosen to define their scheme as the ‘filtered Smith predictor’ (FSP). The proposed MSP results of this paper have been compared to the FSP results.
5.1. Example 1 A heat exchanger shown in Fig. 8 is considered by [12] where the output temperature T of the cold water is controlled using valve V
that manipulates the input flow of the hot water. The temperature of the hot water is controlled by an independent controller. As described by [12], the process can be represented by the following stable first order model when it is close to the nominal operating point P(s) =
0.12 −3s e 6s + 1
(38)
A dead time estimation error of 10% is considered here. The same model is used here for comparison by exciting it with a unit step change at t = 1 and a disturbance change of 0.1 at t = 70. Only one filter was suggested for stable processes by [12] and their proposed filter was used in place of F2 . The filter they proposed for the process was ((1 + 6s)(1 + 4.38s))/((1 + 2s)2 ). The proposed controller gain and the integral constant are 8.33 and 6 s respectively. For the
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Fig. 7. Bode plot for the estimation error in process time constant.
Fig. 8. Water heat exchanger used by [12].
proposed modified Smith predictor (MSP) in this paper, the controller gain and integral time constant is found to be the same as [12] and the filters designed as per the rules described in Sections 3.2 and 3.3 are F1 (s) =
6s + 1 , 0.8s + 1
F2 (s) =
6s + 1 0.2s + 1
(39)
Here, the value of T1 and T2 are chosen to be 0.8 and 0.2 respectively. The set-point response is shown in Fig. 9 along with the controller output. It can be observed from Fig. 9 that the set point response of the proposed MSP is better than the FSP proposed by [12]. The disturbance rejection response is shown separately in Fig. 10(a). The disturbance response has also been improved slightly. The controller output in Fig. 10(b) shows the increased performance for the disturbance rejection. This increase in performance can be seen in Fig. 11 by the comparison of dP(ω) values for the proposed MSP and FSP by [12]. Though the values of dP(ω) became similar with the increase in frequency, the robustness is compromised for the proposed MSP due to the faster disturbance rejection. As the robustness only depends on the value of T2 for the proposed MSP, it is easy to increase the robustness by increasing its value. This is shown in Fig. 12. 5.2. Example 2 Another process with a long dead time relative to the process time constant can be considered for evaluating the proposed method. The process has the following transfer function P(s) =
1.02 −8.2s e 1.7s + 1
(40)
Fig. 9. Comparative set point response and controller output of proposed MSP (solid line) and FSP (dashed line) by [12].
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Fig. 12. Comparative values of dp(ω) for T2 = 0.8 (solid line) and T2 = 2 (dashed line) in the proposed MSP.
Fig. 10. Comparative disturbance rejection response and controller output of proposed MSP (solid line) and FSP (dashed line) by [12].
Fig. 11. Comparative values of dp(ω) for proposed MSP (solid line) and FSP (dashed line) by [12].
The dead time for the system is almost five times the time constant of the process. A PI controller is not capable of controlling this particular process. From the proposed method, the controller is tuned to have the proportional gain of 0.9804 and the integral time
Fig. 13. Comparative response and controller output of proposed MSP (solid line) with FSP (dashed line) for long dead time system.
constant equal to the process time constant. The filters obtained for this system are F1 (s) =
1.7s + 1 , 0.6s + 1
F2 (s) =
1.7s + 1 s+1
(41)
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Fig. 15. Photograph of the set-up.
With the same controller tuning, the filter F2 can be constructed using the FSP criterion of [12] as: F2 (s) =
(1.7s + 1)(1.65s + 1) (5s + 1)2
(42)
It is to be noted that for stable processes, [12] has used only one filter, i.e. filter F2 (s). The output responses of the proposed modified Smith predictor and filtered Smith predictor are shown in Fig. 13(a). It shows that the proposed method has performed significantly better than FSP for systems with long dead time. Fig. 13(b) also shows how the controller output varies to track the set-point and compensate the disturbance. 5.3. Example 3 Fig. 14. Process response and controller output of proposed MSP (solid line) with FSP (dashed line) for second order plus dead time system.
The next example is chosen to be a second order plus dead time (SOPDT) model. The considered process is P(s) =
1 e−25s (50s + 1)(5s + 1)
Fig. 16. Schematic diagram of the pilot plant.
(43)
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Fig. 17. Comparative response of calculated model (dotted line) and actual process (solid line).
So one of the time constant of the process is 10 times of the other one. Also the dead is quite long which is half of the larger time constant. To apply the proposed method on the SOPDT model, the process has to be approximated to a first order plus dead time model. A popular and well known strategy, Skogestad’s half rule [16], is applied on the second order model to approximate it to a first order plus dead time process. By applying the half rule, the process becomes P(s) =
1 e−27.5s 52.5s + 1
(44)
As the approximation is complete, this model can be used to find out the filter parameters to design the filters. The tuned filters used for this process are found to be 52.5s + 1 F1 (s) = , 17.5s + 1
52.5s + 1 F2 (s) = 4s + 1
(45) 6. Experimental evaluation
The thumb rule is used for the disturbance rejection filter only and a conservative approach is taken in case of the filter F1 as it is an approximated process. The filter for FSP proposed by Normey-Rico and Camacho [12] is obtained based on the approximated FOPDT process which is expressed as F2 (s) =
(52.5s + 1)(30.49s + 1) (13.75s + 1)2
Fig. 18. Oscillatory set-point response of PI controller for transmitter TT-05.
(46)
For both the dead time compensators, the controller gain and integral time constant are found to be 1 and 52.5 respectively. A unit step change is made at t = 20 s and then a disturbance of size −0.2 is introduced into the system at t = 350 s for the evaluation. The comparative response of the proposed modified Smith predictor and filtered Smith predictor by Normey-Rico and Camacho [12] with respect to set-point tracking and disturbance rejection is shown in Fig. 14 along with the controller output. The set-point response is much slower in case of FSP than the proposed dead time compensator. The disturbance rejection response shows that the FSP of Normey-Rico and Camacho [12] is slightly better which may be a consequence of using a second order filter. Still the proposed method is generating almost same response in case of disturbance rejection whereas it is simpler than FSP. The controller output in Fig. 14(b) shows the aggressive behavior of the proposed modified Smith predictor in case of set-point tracking. But the value is reduced to the desired value by the time the output response crosses the dead time. So the set-point is tracked and disturbance is rejected by the proposed method satisfactorily for a second order plus dead time process.
6.1. Description of the experimental set-up The proposed method was evaluated on a two tank heating system pilot plant located in the Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. A photograph of the set-up is shown in Fig. 15 and the schematic diagram of the pilot plant is shown in Fig. 16. These tanks have large time constants because they have a diameter of 24 inches. Both of them have a steam-coil for heating the water. Tank-1 has the facility to introduce dead time to the temperature control loop. So the experiments were performed on the temperature control loop of Tank-1. The temperature was controlled by the steam control valve TCV-01. The level of the tank was kept constant at 60% during all experiments. For proper uniform heating of the tank water by steam, an air bubbling system was used as a stirring mechanism. 6.2. Process schematic From Fig. 16, it can be seen that the temperature transmitters, TT-02, TT-03, TT-04, TT-05, of Tank-1 can be used to introduce variable time delay. The transmitter TT-05 would yield the longest dead time. The distance of the transmitters from the water outlet of the tank results in transportation delays and the delay becomes larger when the temperature sensor selected is farthest from the tank. The transmitter from the tank outlet that results in the largest delay is clearly TT-05. If the proposed method works for this transmitter,
K. Kirtania, M.A.A.S. Choudhury / Journal of Process Control 22 (2012) 612–625
Fig. 19. Comparative set-point response of proposed MSP with T1 = 100, T2 = 50 (dashed line) and T1 = 150, T2 = 75 (solid line).
then it should work for other selections of the temperature sensors as well. The experiments were designed to check the method on TT-05. 6.3. Model identification To apply the method, a process model is needed. The process model would be same for all the transmitters except the delay which will vary. Open loop step tests were performed on Tank-1 temperature control loop and the model was identified from the step response data. The identified model is Gn (s)e−Ln s =
0.2 e−110s 1020s + 1
(47)
The model response was compared with the experimental response and shown in Fig. 17. As evident from the time constant of the model, the system is very slow. The delay, Ln , was found to be 110 s for TT-05. As per the rules suggested in Section 3, the controller gain and integral time constant are obtained as 5 and 1020 s, respectively. 6.4. Experimental evaluation Performance of a traditional PI controller, performance of the newly proposed modified Smith predictor (MSP) and the filtered Smith predictor (FSP) of [12] were compared. The PI controller was tuned as per the well known IMC (Internal Model Control) method by assuming the value of c = dead time. So it should provide a fast and strong control action on the process. The value of the proportional controller gain and the integral time constant for the process
623
Fig. 20. Comparative disturbance rejection response of proposed MSP with T1 = 100, T2 = 50 (dashed line) and T1 = 150, T2 = 75 (solid line).
was found to be 23.2 and 1020 s, respectively. The PI controller was implemented in the system and the response as shown in Fig. 18 was found to be oscillatory. The operating range of the control valve was 4–20 mA. The controller output in Fig. 18 shows that the valve was not saturated but unstable in the operating region. The proposed modified Smith predictor (MSP) would have the following filters as per the design rules described in Sections 3.2 and 3.3 F1 (s) =
1020s + 1 , 150s + 1
F2 (s) =
1020s + 1 75s + 1
(48)
The proposed modified Smith predictor was tested on the pilot plant using different filter parameters. To achieve a faster response, the values of T1 and T2 were decreased to 100 and 50 respectively. The comparative results of the proposed method for different filters are shown in Figs. 19 and 20. Both figures show that the control valve did not reach the saturation point during set-point tracking and disturbance rejection. For conservative control, T1 and T2 should be larger and for better performance, they should be smaller. For the same controller tuning, the filter F2 (s) according to [12] was found to be: ((1020s + 1)(208.84s + 1))/((60s + 1)2 ). The respective response is shown in Fig. 21(a). The response of the process was static initially and then it began to rise monotonically crossing the set-point and did not come back to the set-point. The filter parameter was changed to make the compensator less sensitive in which case the filter becomes F2 (s) = ((1020s + 1)(217.27s + 1))/((65s + 1)2 ). With the implementation of the changed filter, the actuator is stuck at the same point and the response did not reach the set-point as shown in Fig. 21(b).
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the value of IAE of the response obtained for the proposed modified Smith predictor for the parameter values of T1 = 150 and T2 = 75. The value of IAE for FSP was not calculated because the response was not satisfactory and IAE calculation in this case would not make sense. A step type disturbance was introduced in the system at 8890 s. The IAE value for disturbance rejection is calculated for the range of 8500–12,000 s. In case of disturbance rejection, IAE is not calculated for PI controller because the PI controller could not control the process and it became unstable. The proposed compensator shows the best performance in terms of IAE value. The proposed MSP was evaluated for various values of tuning parameter. To achieve a faster response, both the filter parameters are decreased by 50% that is, T1 = 100, T2 = 50. Two responses for the two sets of parameters such as T1 = 100, T2 = 50 and T1 = 150, T2 = 75 are shown in Figs. 19 and 20. Both responses are satisfactory. Therefore, the proposed method is robust and can handle dead time in real processes well. 7. Conclusion
Fig. 21. Experimental results of filtered Smith predictor proposed by [12]. Table 1 Table for the IAE for TT05 for set-point change. Time span (s) 4500–8000 PI controller FSP Proposed MSP (T1 = 150 and T2 = 75) Proposed MSP (T1 = 100 and T2 = 50)
221.39 – 107.98 99.47
Table 2 Table for the IAE for TT-05 for disturbance rejection. Time span (s) 8500–12,000 PI controller FSP Proposed MSP (T1 = 150 and T2 = 75) Proposed MSP (T1 = 100 and T2 = 50)
– – 107.41 94.47
6.5. Quantitative comparisons In order to quantify the response of the controllers and the DTCs, the Integral of Absolute Error (IAE) was calculated for each case. The IAE calculated for the controller and compensators at various conditions for TT05 are listed in Tables 1 and 2. The set point was changed at 5000 s for all cases. In Table 1, the value of IAE for 4500–8000 s reflects the slow response of PI controller and it is almost double
A simple and intuitive approach to design dead time compensators for stable processes with long dead time has been developed in this study. The proposed method is easy to understand and the tuning rules are straightforward. Two simple first order filters are used in the control loop to compensate the dead time and make the process response faster. The controller decouples the disturbance and set point response properties via the two-degrees of freedom control scheme. Two parameters namely the process gain and process time constant along with dead time are required to be estimated for this method. The comparative simulation study showed that the performance of the proposed dead time compensator is better than that of the previous dead time compensators appeared in the literature. Most of the DTCs reported in the literature were evaluated by simulation only. The proposed DTC has been successfully evaluated and compared with the best DTC reported in the literature [12] both in a simulation scenario and via an experimental study. The practical implementation on a pilot scale process with relatively large time constant and realistic operating conditions, demonstrates the practicality and utility of the proposed method. References [1] O.J.M. Smith, Closed control of loops with dead time, Chemical Engineering Progress 53 (1957) 217–219. [2] K. Wantanabe, M. Ito, A process-model control for linear systems with delay, IEEE Transactions on Automatic Control 6 (6) (1981) 1261–1268. [3] T. Hägglund, An industrial dead-time compensating PI controller, Control Engineering Practice 4 (6) (1996) 749–756. [4] J.E. Normey-Rico, E.F. Camacho, Robust tuning of dead time compensators for processes with an integrator and long time delay, IEEE Transactions on Automatic Control 44 (8) (1999) 1597–1603. [5] J.E. Normey-Rico, E.F. Camacho, A unified approach to design dead-time compensators for stable and integrative processes with dead-time, IEEE Transactions on Automatic Control 47 (2) (2002) 299–305. [6] B.C. Torrico, J.E. Normey-Rico, 2DOF discrete dead-time compensators for stable and integrative processes with dead time, Journal of Process Control 15 (2005) 341–352. [7] W. Zhang, J.M. Rieber, D. Gu, Optimal dead-time compensator design for stable and integrating processes with time delay, Journal of Process Control 18 (2008) 449–457. [8] A. Ingimundarson, T. Hägglund, Robust tuning procedures of dead time compensating controllers, Control Engineering Practice 9 (2001) 1195–1208. [9] M. Huzmezan, W.A. Gough, G.A. Dumont, S. Kovac, Time delay integrating systems: a challenge for process control industries. A practical solution, Control Engineering Practice 10 (10) (2002) 1153–1161. [10] W. Tan, H.J. Marquez, T. Chen, IMC design for unstable processes with time delays, Journal of Process Control 13 (2003) 203–213. [11] T. Liu, W. Zhang, D. Gu, Analytical design of two-degree-of-freedom control scheme for open-loop unstable processes with time delay, Journal of Process Control 15 (2005) 559–572. [12] J.E. Normey-Rico, E.F. Camacho, Unified approach for robust dead time compensator design, Journal of Process Control 19 (2009) 38–47.
K. Kirtania, M.A.A.S. Choudhury / Journal of Process Control 22 (2012) 612–625 [13] P. Albertos, P. Garcia, Robust control design for long time-delay systems, Journal of Process Control 19 (2009) 1640–1648. [14] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs, NJ, 1989. [15] K. Kirtania, M.A.A. Shoukat Choudhury, A two-degree-of-freedom dead time compensator for stable processes with long dead time, in: Proceedings of the
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Contents lists available at SciVerse ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
A novel dead time compensator for stable processes with long dead times Kawnish Kirtania, M.A.A. Shoukat Choudhury ∗ Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
a r t i c l e
i n f o
Article history: Received 20 December 2010 Received in revised form 19 October 2011 Accepted 9 January 2012 Available online 2 February 2012 Keywords: Dead time compensator Stable processes Smith predictor Two-degrees-of-freedom controller Set-point tracking Regulatory control
a b s t r a c t This paper presents a new and simplified approach for the design of dead time compensators for processes with long dead times. The approach is based on a modified structure of the Smith predictor that allows the user to isolate the disturbance and set-point responses and thereby, provide a two-degrees-of-freedom control scheme. The proposed structure is easy to analyze and tune. Using an estimation of the dead time and process model of the plant, the proposed compensator is left with two tuning parameters that determine the closed-loop performance and robustness. The performance of the proposed compensator is compared with the most recent dead time compensator appeared in the literature. The method is evaluated on two simulated processes and a computer-interfaced pilot-scale two tank heating system to demonstrate the practicality and utility of the proposed scheme. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Dead times are present in systems where any kind of mass, energy or momentum is transferred. These terms are represented as exponential terms in the transfer function. The presence of delays typically results in: • Oscillatory behavior of the process output or instability. • Poor tracking of set points and poor regulatory functions. The effect of dead time can be compensated by removing the exponential term from the characteristic equation of the process. This was first done by Smith [1] and the dead time compensator (DTC) is named as the Smith predictor (SP). The shortcomings of the Smith predictor can be noted as – (a) sensitivity of the loop for error in dead time and process model estimation, (b) not applicable for integrating processes. To overcome these difficulties, several research studies have been reported to improve the performance of the Smith predictor. Wantanabe and Ito [2] suggested a modification in the process model to overcome the difficulties of using the Smith predictor for integrating plants. To make the Smith predictor more stable and faster H¨agglund [3] proposed use of a PPI (Predictive Proportional Integral) controller. This controller has all the features of Smith predictor with the advantage of tuning only three parameters as in a PID controller. To improve the response of the closed loop,
∗ Corresponding author. Tel.: +88 02 9665650/7960; fax: +88 02 9665609. E-mail address: [email protected] (M.A.A.S. Choudhury). 0959-1524/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2012.01.003
the structure proposed by Wantanabe and Ito [2] was modified by Normey-Rico and Camacho [4] by introducing the concept of two-degrees-of-freedom (2DOF) structure. They proposed a prefilter after the set point while the disturbance response remains unchanged. In 2002, Normey-Rico and Camacho [5] came up with modification of the pre-filter. A higher order pre-filter was proposed instead of a first order pre-filter. Torrico and Chang [6] studied the discrete implementation of the 2 DOF dead time compensator. Zhang et al. [7] determined different parameters of the design by Normey-Rico and Camacho [5] through optimization of some differential equations. Another 2 DOF dead time compensator was proposed for unstable systems by Liu et al. [11]. Various approaches were taken to improve the response of the processes with long dead time. Ingimunderson and H¨agglund [8] described the robust tuning procedures for the dead time compensators by frequency domain analysis. Huzmezan et al. [9] designed an adaptive control scheme to compensate the dead time for the batch processes. Tan et al. [10] applied the IMC design method to design a dead time compensator. A Unified approach was proposed by Normey-Rico and Camacho [12] where one filter was used for first order plus dead time (FOPDT) stable processes and two filters were used for FOPDT integrating or unstable processes to decouple the set-point tracking and disturbance rejection properties. This compensator provides also two-degrees-of-freedom control strategy to tune the set point and disturbance rejection response separately. A robust control design was suggested by Albertos and Garcia [13] which is as good as the unified approach by Normey-Rico and Camacho [12]. A new dead time compensator was proposed in [15] where two filters were used for the FOPDT stable processes which was found as good in simulation as in real time application.
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2.1. Set-point tracking response When the actual process P(s) is same as the process model Pn (s), the closed loop transfer function of the process for set-point tracking Y (s) F1 (s)C(s)Pn (s) = = R(s) Ysp (s) 1 + C(s)Gn (s) Fig. 1. Block diagram of proposed modified Smith predictor (MSP).
It is impossible to eliminate the effect of dead time completely. But the effect is minimized by the compensators. All compensators need some prior information about the process or system. The difference lies in the performance as well as in the robustness of the compensator. Each work cited above is somehow better than the previous one. Chronologically, the modified Smith predictor became versatile and applicable for different kind of processes. Eventually schemes with improved stability and robustness for cases with estimation error in process model and dead time were suggested. A simple and elegant dead time compensator is designed in this paper for the processes that can be approximated by first order plus dead time (FOPDT) model. This paper is organized as follows. Section 2 presents the process models and the 2 DOF DTC structure. In Section 3, the proposed tuning rules of the PI controller and the filter parameters are presented. Section 4 shows uncertainty and robustness analysis. A comparative simulation study is given in Section 5. Section 6 describes some experimental results followed by concluding remarks.
The proposed modified Smith predictor control structure is shown in Fig. 1. As can be seen, the structure is the same as the Smith Predictor but with two additional filters. F1 (s) is a traditional filter which improves the set point response and F2 (s) is a predictor filter that improve the disturbance rejection response. The proposed modified Smith predictor (MSP) can be used to take into account the robustness of the loop, improving disturbance rejection properties and decoupling the set-point and disturbance rejection responses. Therefore, the main drawback of the SP is considered in this design. Normey-Rico and Camacho [12] proposed a similar structure for dead time compensator. But the difference between the proposed one and the dead time compensator of Normey-Rico and Camacho [12] relies in the number of filters. Normey-Rico and Camacho proposed single filter for stable processes and two filters for integrating or unstable processes with dead time. The process type considered here is stable and two filters have been suggested. The next difference is in the filter structure. The structure of the filter in dead time compensator of Normey-Rico and Camacho [12] was of second order whereas the proposed one consists of two first order filters. In the proposed structure, Pn (s) = Gn (s)e−Ln s is the process model, Gn (s) is the delay free part of the model, Ln is the estimated time delay and C(s) is the controller. Here, the assumed process model is a FOPDT model. The FOPDT model can be written as
Pn (s) = Gn (s)e−Ln s =
Kp e−Ln s Ts + 1
The proposed model is based on the concept that the process can be made faster and robust by setting a pseudo set-point which will vary with time and remains stable when the process reaches the desired output. The set point passes through the filter F1 (s) and varies with time so that the process responses very fast to reach the desired value. To attain this type of response, the proposed structure of the filter F1 (s) is F1 (s) =
(1)
Zs + 1 T1 s + 1
(3)
The parameters Z and T1 are to be tuned. A simple tuning rule is described in Section 3 for this filter. 2.2. Disturbance rejection response The closed loop transfer function for disturbance rejection is expressed as
Y (s) C(s)Pn (s)F2 (s) = Pn (s) 1 − D(s) 1 + C(s)Gn (s)
= Pn (s)(s)
(4)
where (s) = [1 − (C(s)Pn (s)F2 (s))/(1 + C(s)Gn (s))]. From the above equation, it can be seen that if the term (s) is close to zero, the response will be better and improved. To attain this, the second filter F2 (s) can be designed as (more discussion in Section 3.3): F2 (s) =
2. Process model and dead time compensator (DTC) structure
(2)
Zs + 1 T2 s + 1
(5)
Z has the same parameter value as in the previous filter. But the parameter T2 is to be tuned to have a robust closed loop response as well as the best disturbance rejection response. This filter refines the signal in both feedback loops. It only affects the disturbance rejection response and robustness of the loop but not the set-point response. The parameter T2 is responsible for the robustness of the loop. Its tuning and properties are discussed in more detail in Section 3. 3. Tuning rules The proposed dead time compensator has three parameters to tune namely Z, T1 and T2 . The controller for the proposed structure is a PI controller. So the controller has two more parameters to be tuned. The tuning rules for the compensator filter parameters and the controller parameters are described in this section. 3.1. PI controller parameters The PI controller needs to be tuned to have a stable closed loop response. The controller can be tuned to have the best response by using the predictive structure. The rule suggested by [3] is used to calculate the controller parameters. If the controller gain cancels out the process gain and the time constant of the controller is equal to the process time constant, the controller and the process will converge to a first order transfer function of gain 1 and time constant T (if there is no dead time present). Therefore, controller parameters are tuned using IMC rules.
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3.2. Set-point tracking filter (F1 (s)) parameters
rejection becomes perfect. Therefore, the filter structure for F2 can be chosen as F2 (s) = (Zs + 1)/(T2 s + 1). Now
The filter for the set point tracking is tuned by setting the parameters Z and T1 . To analyze this, the transfer function between the process response and set-point change must be taken care of. The transfer function is R(s) =
Y (s) F1 (s)C(s)Pn (s) F1 (s)C(s)Gn (s)e−Ln s = = Ysp (s) 1 + C(s)Gn (s) 1 + C(s)Gn (s)
(6)
It is clear from the transfer function that filter F1 does not affect the characteristic equation. So it does not affect the stability of the process. Let set-point response without the filter be denoted as G1 (s). By applying IMC tuning rules G1 (s) =
C(s)Gn (s)e−Ls 1 + C(s)Gn (s)
(7)
e−Ls Ts + 1
F2 (s) =
e−Ls Zs + 1 e−Ls × = Ts + 1 T2 s + 1 T2 s + 1
(14)
Asymptotically, Eq. (14) is equal to 1. As the time propagates, the disturbance will be nullified. This is interesting to note that with the change of T2 , only the disturbance response will be affected. The parameter T2 can be chosen on the basis of how fast the disturbance rejection is required. From authors’ experience, a value of T/14 is a good choice to start with. Then it can be tuned as required by robustness and stability. 4. Robustness and stability analysis 4.1. Robustness analysis
((Ke−Ls )/(Ts + 1))Kc (1 + (1/(TI s))) 1 + (K/(Ts + 1))Kc (1 + (1/(TI s))) ((TI s + 1)/(Ts + 1))e−Ls (1/(TI s)) = 1 + ((TI s+ 1)/(Ts + 1))(1/(TI s)) e−Ls = Ts + 1
=
(8)
From Eq. (8), it is found that with the proposed controller, the process has the same time constant as for the open loop case but the gain changes to 1 so that the set point can be tracked properly. By combining Eqs. (8) and (6), the set-point response can be found as
R(s) = F1 (s)
e−Ls Ts + 1
(9)
Now, the filter is to be designed in a way so that it cancels the time constant of the process and converts it to a faster process. Therefore, the filter which will serve the purpose can be designed as F1 (s) =
Zs + 1 T1 s + 1
(10)
The parameter Z is chosen to be equal to the time constant T. So the transfer function becomes R(s) =
e−Ls T1 s + 1
(11)
It will essentially cancel out the long time constant from the system and respond quickly. Now the set point tracking monotonically depends on the value of T1 . The parameter T1 can be chosen on the basis of desired set point response. From the experience of the authors, the value of T1 can be initially chosen as T/7, i.e., 7 times faster than the original process time constant.
The filter used for disturbance rejection has two parameters Z and T2 . Z is chosen as the same as that in the set point tracking filter. Therefore, the only parameter to be tuned is T2 . The disturbance rejection response of the system can be characterized from the term (s) of Eq. (2). Ideally, (s) should be derived equal to zero. Therefore (12)
If C(s) is tuned as proposed, the resultant value of (s) becomes
(s) = 1 −
e−Ls 1 + Ts
F2 (s)
P(s) = Pn (s)[1 + ıP(s)]
(15)
As shown in [14], the multiplicative norm-bound uncertainty, ıP(ω), is |ıP(ω)| ≤ ıP(ω) ∀ω > 0
(16)
From the above consideration the characteristic equation for the system presented in Fig. 1 becomes 1 + C(s)Gn (s) + C(s)Gn (s)Pn (s)F2 (s)ıP(s) = 0
(17)
As the nominal system is stable, robust stability condition can be found for the proposed compensator as:
1 + C(jω)Gn (jω) , ∀ω > 0 C(jω)G (jω)F (jω)
ıP(ω) < dP(ω) =
n
(18)
2
where, (P(s) − Pn (s)) = Pn (s)ıP(s) = dP(s). Note that the filter only affects the disturbance rejection response and the robustness of the loop. Substituting the transfer function elements into Eq. (18) ıP(ω) < dP(ω)
1 + Kc (1 + (1/(TI jω)))(K/(Tjω + 1)) = Kc (1 + (1/(TI jω)))(K/(Tjω + 1))((Zjω + 1)/(T2 jω + 1))
(19)
Based on tuning rules suggested in Section 3, Kc = 1/K, TI = T. Thus Eq. (19) simplifies to
3.3. Disturbance rejection filter (F2 (s)) parameters
C(s)Pn (s)F2 (s) (s) = 1 − 1 + C(s)Gn (s)
The effect of the changes in filter parameters on the robustness and the stability of the process needs to be investigated to choose proper values of these parameters. Stability of the FOPDT process is not affected but the robustness is greatly influenced with the change of the filter parameters. The family of plants P(s) having the following relationship are under consideration
(13)
The second term is to be made close to 1 so that the response to disturbance change becomes zero and essentially the disturbance
ıP(ω) < dP(ω) =
1 + (1/(TI jω)) (1/(TI jω))((Zjω + 1)/(T2 jω + 1))
= |T2 jω + 1|
(20) (21)
Therefore, the robustness of the loop can be increased by increasing the value of T2 . However, by increasing T2 , the disturbance rejection response will become slower. This is the classical trade off between robustness and performance. For plants that can be approximated by a stable FOPDT model, the value of T2 can be increased gradually until a satisfactory regulatory and robustness trade-off is achieved. The robustness ‘margin’ desired would depend on the level of uncertainty in the plant. The greater the uncertainty, larger the value of T2 and therefore slower the regulatory performance.
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4.2. Uncertainty analysis By introducing some error in the model estimation, this uncertainty analysis is carried out. The analysis covers the uncertainty in dead time, gain and time constant estimation. As the dead time compensator depends on the model of the process, the uncertainties will affect the performance. But as long as the system is stable, no major problem occurs. By measuring the gain margin, the stability limit of a process can be understood. At the value of gain margin 1 the process becomes marginally stable. On further decreasing the gain margin value, the system becomes unstable. The analysis takes into account both set-point change and disturbance rejection responses. 4.2.1. Set-point responses • Uncertainty in dead time estimation: At first, the process was checked with some error in dead time estimation for the set-point change. For this, a new equation was derived with an estimation error of L for the delay L. Thus the transfer function between the set-point and output becomes Zs + 1 Y (s) e−Ls = × T1 s + 1 Ysp (s) (Ts + 1) + e−Ls (1 − e−Ls )
(22)
To analyze it further, the delay term is approximated by 1/1 Pade approximation. After some simplifications, Eq. (22) becomes Zs + 1 Y (s) = T1 s + 1 Ysp (s)
×
1 − (L/2)s ((TLLs3 )/4) + ((T (L + L)s2 )/2) + s(T + (L/2) + L) + 1
(23) With a 20% error in estimating dead time was considered for a process and then the gain margin was investigated. For this case the following model used by [12] was considered: 0.12 −3s P(s) = e 6s + 1
(24)
Now with an error of 20%, the estimated delay becomes 3.6 s. As the filter parameters depend on the process time constant, the filter parameters are not affected. The value of Z was equal to the time constant and the other parameter T1 was taken as 6/8. So according to Eq. (23), the transfer function for set-point change becomes Y (s) 6s + 1 × = 0.8s + 1 Ysp (s)
1 − 1.5s 2.7s3 + 10.8s2 + 8.1s + 1
Y (s) K(1 − (L/2)s) = D(s) (Ts + 1)(1 + (L/2)s)
(25)
2.7s3
For this case, the crossover frequency from Bode plot was 0.9862 and gain margin was 1.74. So the value of the gain margin is reached to practical limit by increasing the parameter T1 . The uncertainty in estimating the dead time is also compensated by a higher value of T1 . The Bode plots for the two cases are shown in Fig. 2. • Uncertainty in process gain estimation: for set point change, the transfer function with an error in gain estimation becomes Y (s) (1 − Ls/2) = × Ysp (s) (1 + Ls/2)((T/(KK c ))s + 1)
Zs + 1
(27)
T1 s + 1
As there is an error in estimation of K, the value of Kc will be different from K. So the value of Kc would be equal to K + K where K is the error in estimation Y (s) (1 − Ls/2) = × Ysp (s) (1 + Ls/2)(Ts/(K(K + K)) + 1)
Zs + 1
T1 s + 1
(28)
A 20% error was considered in process gain estimation. To understand the effect quantitatively, the transfer function for the set-point change for the process under consideration found to be Y (s) (1 − 1.5s) × = Ysp (s) (1 + 1.5s)(6s/(0.12(0.12 + 0.024)) + 1)
6s + 1
0.8s + 1 (29)
The Bode plot for the above transfer function is shown in Fig. 3. The crossover frequency was found to be 1.32 and the corresponding gain margin was 1/0.01198 = 83.47 which is quite high and away from instability. So further change in parameter was not needed. The general idea is that the error in the gain estimation does not lead to instability of the process. • Uncertainty in estimation of the process time constant: for this error, the parameter Z in the filter and TI in the controller would not be perfect. With a deviation of T from the time constant T is considered. So the resulting transfer function for set-point change Y (s) (Ts + 1)(1 − Ls/2) × = Ysp (s) ((T 2 + TT )s2 + 2Ts + 1)(1 + L/2)
Zs + 1
T1 s + 1
(30)
For a 20% estimation error, the process model under consideration provides the following transfer function Y (s) (6s + 1)(1 − 1.5s) × = Ysp (s) (43.2s2 + 12s + 1)(1 + 1.5s)
6s + 1
(31)
0.8s + 1
Fig. 4 shows the Bode plot of the transfer function which clearly shows that for the crossover frequency 1.408, the gain margin is 1.8. So the system is stable. 4.2.2. Disturbance rejection response • Uncertainty in dead time estimation: the transfer function between the output and disturbance is derived as
((T2 LL)/4)s3 + (((T2 (L + L) + L)s2 )/2) + (T2 + L + (L/2))s ((T2 LL)/4)s3 + (T2 (L + L) − ((LL)/2))(s2 /2) + (T2 + L + ((L + L)/2))s + 1
From the Bode plot, gain margin of the transfer function was determined to see whether it crosses the stability limit. The crossover frequency for this process was found to be 1.158 rad/s and the gain margin was 1.4 which is away from instability. However, the standard value of gain margin is in between 1.7 and 4.0. To achieve the limit, the parameter T1 is increased to 1.4 while other parameters remaining the same. In this case the transfer function for set-point change is 6s + 1 Y (s) = × 1.4s + 1 Ysp (s)
615
1 − 1.5s + 10.8s2 + 8.1s + 1
(26)
(32)
A 1/1 Pade approximation was considered for the Bode plot analysis in this case and L is the dead time estimation error. The parameter of disturbance rejection filter would be 6 and 0.2 for Z and T2 respectively for the process under analysis. For a 20% error in estimation, the resulting transfer function for disturbance rejection would be Y (s) 0.12(1 − 1.5s) = D(s) (6s + 1)(1 + 1.5s)
0.09s3 + 1.86s2 + 2.3s 0.09s3 − 0.09s2 + 2.6s + 1
(33)
Here from the Bode plot, no crossover frequency was found as the curve never reaches −180◦ of phase angle. Fig. 5 shows that
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Fig. 2. Bode plots for error in dead time estimation.
the values of amplitude ratio always remains below 1. So the gain margin never reaches the value below 1. • Uncertainty in estimating the process gain: the equation derived for the disturbance response is given below:
rejection for the process becomes Y (s) 0.12(1 − 1.5s) = D(s) (6s + 1)(1 + 1.5s)
× 1− K(1 − (L/2)s) Y (s) = D(s) (Ts + 1)(1 + (L/2)s)
(1 − (L/2)s)(Zs + 1) × 1− ((T/(KK c ))s + 1)(T2 s + 1)(1 + (L/2)s)
(34)
Here, K and Kc are not equal and thereby no cancelation occurs. Considering an error of 20%, the transfer function for disturbance
(1 − 1.5s)(6s + 1) ((6/(0.01728))s + 1)(0.2s + 1)(1 + 1.5s)
(35)
The Bode plot for the gain estimation error consideration is shown in Fig. 6. The stability of the process is not affected because the amplitude ratio never comes close to 1. • Uncertainty in estimating the time constant of the process: the equation derived for the uncertainty in estimating the time constant of the process and the corresponding equation for a 20% error for the process under consideration are as follows
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617
Fig. 3. Bode plot for the estimation error in process gain.
Fig. 4. Bode plot for the estimation error in process time constant.
Y (s) K(1 − (L/2)s) = D(s) ((T + T )s + 1)(1 + (L/2)s)
× 1−
=
(Ts + 1)(1 − (L/2)s)(Zs + 1) (Ts[(T + T )s + 1] + Ts + 1)(T2 s + 1)(1 + (L/2)s)
0.12(1 − 1.5s) (7.2s + 1)(1 + 1.5s)
−54s3 + 18s2 + 10.5s + 1 × 1− 12.96s4 + 73.04s3 + 63.9s2 + 13.7s + 1
(36)
There is no crossover frequency for the response as the phase angle never reaches the value of −180◦ in Bode plot. Moreover, the value of amplitude ratio always remains below 1. So the system is perfectly stable with the error. The plot is shown in Fig. 7. From this analysis, it is evident that some estimation errors in model estimation do not make the process unstable. Also with increasing the values of T1 and T2 , the robustness of the process may be increased as desired.
5. Simulation study (37)
As has been mentioned, several papers have been appeared in the literature with modifications to the original Smith predictor,
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Fig. 5. Bode plot for the estimation error in dead time.
Fig. 6. Bode plot for the estimation error in process gain.
some concerning the tuning of parameters of the original Smith predictor and others proposing the modification of the control structure. In general, the later modifications have demonstrated better performance than the previous works. For the case of stable plants, the dead time compensator (DTC) proposed by [12] can be considered as the best solution among the studies published so far. Normey-Rico and Camacho [12] have chosen to define their scheme as the ‘filtered Smith predictor’ (FSP). The proposed MSP results of this paper have been compared to the FSP results.
5.1. Example 1 A heat exchanger shown in Fig. 8 is considered by [12] where the output temperature T of the cold water is controlled using valve V
that manipulates the input flow of the hot water. The temperature of the hot water is controlled by an independent controller. As described by [12], the process can be represented by the following stable first order model when it is close to the nominal operating point P(s) =
0.12 −3s e 6s + 1
(38)
A dead time estimation error of 10% is considered here. The same model is used here for comparison by exciting it with a unit step change at t = 1 and a disturbance change of 0.1 at t = 70. Only one filter was suggested for stable processes by [12] and their proposed filter was used in place of F2 . The filter they proposed for the process was ((1 + 6s)(1 + 4.38s))/((1 + 2s)2 ). The proposed controller gain and the integral constant are 8.33 and 6 s respectively. For the
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Fig. 7. Bode plot for the estimation error in process time constant.
Fig. 8. Water heat exchanger used by [12].
proposed modified Smith predictor (MSP) in this paper, the controller gain and integral time constant is found to be the same as [12] and the filters designed as per the rules described in Sections 3.2 and 3.3 are F1 (s) =
6s + 1 , 0.8s + 1
F2 (s) =
6s + 1 0.2s + 1
(39)
Here, the value of T1 and T2 are chosen to be 0.8 and 0.2 respectively. The set-point response is shown in Fig. 9 along with the controller output. It can be observed from Fig. 9 that the set point response of the proposed MSP is better than the FSP proposed by [12]. The disturbance rejection response is shown separately in Fig. 10(a). The disturbance response has also been improved slightly. The controller output in Fig. 10(b) shows the increased performance for the disturbance rejection. This increase in performance can be seen in Fig. 11 by the comparison of dP(ω) values for the proposed MSP and FSP by [12]. Though the values of dP(ω) became similar with the increase in frequency, the robustness is compromised for the proposed MSP due to the faster disturbance rejection. As the robustness only depends on the value of T2 for the proposed MSP, it is easy to increase the robustness by increasing its value. This is shown in Fig. 12. 5.2. Example 2 Another process with a long dead time relative to the process time constant can be considered for evaluating the proposed method. The process has the following transfer function P(s) =
1.02 −8.2s e 1.7s + 1
(40)
Fig. 9. Comparative set point response and controller output of proposed MSP (solid line) and FSP (dashed line) by [12].
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Fig. 12. Comparative values of dp(ω) for T2 = 0.8 (solid line) and T2 = 2 (dashed line) in the proposed MSP.
Fig. 10. Comparative disturbance rejection response and controller output of proposed MSP (solid line) and FSP (dashed line) by [12].
Fig. 11. Comparative values of dp(ω) for proposed MSP (solid line) and FSP (dashed line) by [12].
The dead time for the system is almost five times the time constant of the process. A PI controller is not capable of controlling this particular process. From the proposed method, the controller is tuned to have the proportional gain of 0.9804 and the integral time
Fig. 13. Comparative response and controller output of proposed MSP (solid line) with FSP (dashed line) for long dead time system.
constant equal to the process time constant. The filters obtained for this system are F1 (s) =
1.7s + 1 , 0.6s + 1
F2 (s) =
1.7s + 1 s+1
(41)
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Fig. 15. Photograph of the set-up.
With the same controller tuning, the filter F2 can be constructed using the FSP criterion of [12] as: F2 (s) =
(1.7s + 1)(1.65s + 1) (5s + 1)2
(42)
It is to be noted that for stable processes, [12] has used only one filter, i.e. filter F2 (s). The output responses of the proposed modified Smith predictor and filtered Smith predictor are shown in Fig. 13(a). It shows that the proposed method has performed significantly better than FSP for systems with long dead time. Fig. 13(b) also shows how the controller output varies to track the set-point and compensate the disturbance. 5.3. Example 3 Fig. 14. Process response and controller output of proposed MSP (solid line) with FSP (dashed line) for second order plus dead time system.
The next example is chosen to be a second order plus dead time (SOPDT) model. The considered process is P(s) =
1 e−25s (50s + 1)(5s + 1)
Fig. 16. Schematic diagram of the pilot plant.
(43)
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Fig. 17. Comparative response of calculated model (dotted line) and actual process (solid line).
So one of the time constant of the process is 10 times of the other one. Also the dead is quite long which is half of the larger time constant. To apply the proposed method on the SOPDT model, the process has to be approximated to a first order plus dead time model. A popular and well known strategy, Skogestad’s half rule [16], is applied on the second order model to approximate it to a first order plus dead time process. By applying the half rule, the process becomes P(s) =
1 e−27.5s 52.5s + 1
(44)
As the approximation is complete, this model can be used to find out the filter parameters to design the filters. The tuned filters used for this process are found to be 52.5s + 1 F1 (s) = , 17.5s + 1
52.5s + 1 F2 (s) = 4s + 1
(45) 6. Experimental evaluation
The thumb rule is used for the disturbance rejection filter only and a conservative approach is taken in case of the filter F1 as it is an approximated process. The filter for FSP proposed by Normey-Rico and Camacho [12] is obtained based on the approximated FOPDT process which is expressed as F2 (s) =
(52.5s + 1)(30.49s + 1) (13.75s + 1)2
Fig. 18. Oscillatory set-point response of PI controller for transmitter TT-05.
(46)
For both the dead time compensators, the controller gain and integral time constant are found to be 1 and 52.5 respectively. A unit step change is made at t = 20 s and then a disturbance of size −0.2 is introduced into the system at t = 350 s for the evaluation. The comparative response of the proposed modified Smith predictor and filtered Smith predictor by Normey-Rico and Camacho [12] with respect to set-point tracking and disturbance rejection is shown in Fig. 14 along with the controller output. The set-point response is much slower in case of FSP than the proposed dead time compensator. The disturbance rejection response shows that the FSP of Normey-Rico and Camacho [12] is slightly better which may be a consequence of using a second order filter. Still the proposed method is generating almost same response in case of disturbance rejection whereas it is simpler than FSP. The controller output in Fig. 14(b) shows the aggressive behavior of the proposed modified Smith predictor in case of set-point tracking. But the value is reduced to the desired value by the time the output response crosses the dead time. So the set-point is tracked and disturbance is rejected by the proposed method satisfactorily for a second order plus dead time process.
6.1. Description of the experimental set-up The proposed method was evaluated on a two tank heating system pilot plant located in the Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. A photograph of the set-up is shown in Fig. 15 and the schematic diagram of the pilot plant is shown in Fig. 16. These tanks have large time constants because they have a diameter of 24 inches. Both of them have a steam-coil for heating the water. Tank-1 has the facility to introduce dead time to the temperature control loop. So the experiments were performed on the temperature control loop of Tank-1. The temperature was controlled by the steam control valve TCV-01. The level of the tank was kept constant at 60% during all experiments. For proper uniform heating of the tank water by steam, an air bubbling system was used as a stirring mechanism. 6.2. Process schematic From Fig. 16, it can be seen that the temperature transmitters, TT-02, TT-03, TT-04, TT-05, of Tank-1 can be used to introduce variable time delay. The transmitter TT-05 would yield the longest dead time. The distance of the transmitters from the water outlet of the tank results in transportation delays and the delay becomes larger when the temperature sensor selected is farthest from the tank. The transmitter from the tank outlet that results in the largest delay is clearly TT-05. If the proposed method works for this transmitter,
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Fig. 19. Comparative set-point response of proposed MSP with T1 = 100, T2 = 50 (dashed line) and T1 = 150, T2 = 75 (solid line).
then it should work for other selections of the temperature sensors as well. The experiments were designed to check the method on TT-05. 6.3. Model identification To apply the method, a process model is needed. The process model would be same for all the transmitters except the delay which will vary. Open loop step tests were performed on Tank-1 temperature control loop and the model was identified from the step response data. The identified model is Gn (s)e−Ln s =
0.2 e−110s 1020s + 1
(47)
The model response was compared with the experimental response and shown in Fig. 17. As evident from the time constant of the model, the system is very slow. The delay, Ln , was found to be 110 s for TT-05. As per the rules suggested in Section 3, the controller gain and integral time constant are obtained as 5 and 1020 s, respectively. 6.4. Experimental evaluation Performance of a traditional PI controller, performance of the newly proposed modified Smith predictor (MSP) and the filtered Smith predictor (FSP) of [12] were compared. The PI controller was tuned as per the well known IMC (Internal Model Control) method by assuming the value of c = dead time. So it should provide a fast and strong control action on the process. The value of the proportional controller gain and the integral time constant for the process
623
Fig. 20. Comparative disturbance rejection response of proposed MSP with T1 = 100, T2 = 50 (dashed line) and T1 = 150, T2 = 75 (solid line).
was found to be 23.2 and 1020 s, respectively. The PI controller was implemented in the system and the response as shown in Fig. 18 was found to be oscillatory. The operating range of the control valve was 4–20 mA. The controller output in Fig. 18 shows that the valve was not saturated but unstable in the operating region. The proposed modified Smith predictor (MSP) would have the following filters as per the design rules described in Sections 3.2 and 3.3 F1 (s) =
1020s + 1 , 150s + 1
F2 (s) =
1020s + 1 75s + 1
(48)
The proposed modified Smith predictor was tested on the pilot plant using different filter parameters. To achieve a faster response, the values of T1 and T2 were decreased to 100 and 50 respectively. The comparative results of the proposed method for different filters are shown in Figs. 19 and 20. Both figures show that the control valve did not reach the saturation point during set-point tracking and disturbance rejection. For conservative control, T1 and T2 should be larger and for better performance, they should be smaller. For the same controller tuning, the filter F2 (s) according to [12] was found to be: ((1020s + 1)(208.84s + 1))/((60s + 1)2 ). The respective response is shown in Fig. 21(a). The response of the process was static initially and then it began to rise monotonically crossing the set-point and did not come back to the set-point. The filter parameter was changed to make the compensator less sensitive in which case the filter becomes F2 (s) = ((1020s + 1)(217.27s + 1))/((65s + 1)2 ). With the implementation of the changed filter, the actuator is stuck at the same point and the response did not reach the set-point as shown in Fig. 21(b).
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the value of IAE of the response obtained for the proposed modified Smith predictor for the parameter values of T1 = 150 and T2 = 75. The value of IAE for FSP was not calculated because the response was not satisfactory and IAE calculation in this case would not make sense. A step type disturbance was introduced in the system at 8890 s. The IAE value for disturbance rejection is calculated for the range of 8500–12,000 s. In case of disturbance rejection, IAE is not calculated for PI controller because the PI controller could not control the process and it became unstable. The proposed compensator shows the best performance in terms of IAE value. The proposed MSP was evaluated for various values of tuning parameter. To achieve a faster response, both the filter parameters are decreased by 50% that is, T1 = 100, T2 = 50. Two responses for the two sets of parameters such as T1 = 100, T2 = 50 and T1 = 150, T2 = 75 are shown in Figs. 19 and 20. Both responses are satisfactory. Therefore, the proposed method is robust and can handle dead time in real processes well. 7. Conclusion
Fig. 21. Experimental results of filtered Smith predictor proposed by [12]. Table 1 Table for the IAE for TT05 for set-point change. Time span (s) 4500–8000 PI controller FSP Proposed MSP (T1 = 150 and T2 = 75) Proposed MSP (T1 = 100 and T2 = 50)
221.39 – 107.98 99.47
Table 2 Table for the IAE for TT-05 for disturbance rejection. Time span (s) 8500–12,000 PI controller FSP Proposed MSP (T1 = 150 and T2 = 75) Proposed MSP (T1 = 100 and T2 = 50)
– – 107.41 94.47
6.5. Quantitative comparisons In order to quantify the response of the controllers and the DTCs, the Integral of Absolute Error (IAE) was calculated for each case. The IAE calculated for the controller and compensators at various conditions for TT05 are listed in Tables 1 and 2. The set point was changed at 5000 s for all cases. In Table 1, the value of IAE for 4500–8000 s reflects the slow response of PI controller and it is almost double
A simple and intuitive approach to design dead time compensators for stable processes with long dead time has been developed in this study. The proposed method is easy to understand and the tuning rules are straightforward. Two simple first order filters are used in the control loop to compensate the dead time and make the process response faster. The controller decouples the disturbance and set point response properties via the two-degrees of freedom control scheme. Two parameters namely the process gain and process time constant along with dead time are required to be estimated for this method. The comparative simulation study showed that the performance of the proposed dead time compensator is better than that of the previous dead time compensators appeared in the literature. Most of the DTCs reported in the literature were evaluated by simulation only. The proposed DTC has been successfully evaluated and compared with the best DTC reported in the literature [12] both in a simulation scenario and via an experimental study. The practical implementation on a pilot scale process with relatively large time constant and realistic operating conditions, demonstrates the practicality and utility of the proposed method. References [1] O.J.M. Smith, Closed control of loops with dead time, Chemical Engineering Progress 53 (1957) 217–219. [2] K. Wantanabe, M. Ito, A process-model control for linear systems with delay, IEEE Transactions on Automatic Control 6 (6) (1981) 1261–1268. [3] T. Hägglund, An industrial dead-time compensating PI controller, Control Engineering Practice 4 (6) (1996) 749–756. [4] J.E. Normey-Rico, E.F. Camacho, Robust tuning of dead time compensators for processes with an integrator and long time delay, IEEE Transactions on Automatic Control 44 (8) (1999) 1597–1603. [5] J.E. Normey-Rico, E.F. Camacho, A unified approach to design dead-time compensators for stable and integrative processes with dead-time, IEEE Transactions on Automatic Control 47 (2) (2002) 299–305. [6] B.C. Torrico, J.E. Normey-Rico, 2DOF discrete dead-time compensators for stable and integrative processes with dead time, Journal of Process Control 15 (2005) 341–352. [7] W. Zhang, J.M. Rieber, D. Gu, Optimal dead-time compensator design for stable and integrating processes with time delay, Journal of Process Control 18 (2008) 449–457. [8] A. Ingimundarson, T. Hägglund, Robust tuning procedures of dead time compensating controllers, Control Engineering Practice 9 (2001) 1195–1208. [9] M. Huzmezan, W.A. Gough, G.A. Dumont, S. Kovac, Time delay integrating systems: a challenge for process control industries. A practical solution, Control Engineering Practice 10 (10) (2002) 1153–1161. [10] W. Tan, H.J. Marquez, T. Chen, IMC design for unstable processes with time delays, Journal of Process Control 13 (2003) 203–213. [11] T. Liu, W. Zhang, D. Gu, Analytical design of two-degree-of-freedom control scheme for open-loop unstable processes with time delay, Journal of Process Control 15 (2005) 559–572. [12] J.E. Normey-Rico, E.F. Camacho, Unified approach for robust dead time compensator design, Journal of Process Control 19 (2009) 38–47.
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