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Ind. Eng. Chem. Res. 2008, 47, 4784–4790
Simple Robust Dead-Time Compensator for First-Order Plus Dead-Time Unstable Processes Julio E. Normey-Rico*,† and Eduardo. F. Camacho‡ Departamento de Automação e Sistemas, UniVersidade Federal de Santa Catarina, Floriano´polis, Santa Catarina 88040-900, Brazil, and Departamento de Ingeniería de Sistemas y Automa´tica, UniVersidad de SeVilla, SeVilla 41092, Spain
This paper presents a simple and efficient solution to control first-order unstable processes with dead time. The controller is based on a simple modified structure of the Smith predictor that allows to cope with unstable processes and to tune the controller for a robust behavior. The proposed structure is simple to analyze and tune and gives totally decoupled disturbance rejection and set-point responses. Comparative results with some controllers recently presented in literature show the advantages of the proposed strategy. 1. Introduction Many processes in industry, as well as in other areas, exhibit dead times in their dynamic behavior. Dead times are mainly caused by mass, energy, or information transportation phenomena, but they can also be caused by processing time or by the accumulation of time lags in a number of simple dynamic systems connected in series. Processes with significant dead times are difficult to control using standard feedback controllers. Dead-time compensators (DTC) include a model of the process in the structure of the controller in order to cope with the dead time. The first structure of predictive control, the Smith predictor (SP), presented at the end of the 1950s,1 was used to improve the performance of classical controllers (PI or PID controllers) for plants with dead time. However, for open loop unstable dead-time processes, the original SP is unstable.2,3 Over the past 25 years, numerous extensions and modifications of the SP have been proposed in order to allow its use with unstable plants. See for instance the works presented in the 1980s by De Paor and coauthors4,5 and the newer results obtained in refs 6–11 for the general case and in refs 12 and 13 for integrative plants. In particular, in ref 10 a 2DOF structure is proposed based on four controllers and in refs 8 and 9 the structure has three controllers. For the particular case of first order plus dead time (FOPDT) unstable processes Rao and Chidambaram14 have proposed a simpler controller based on the modified SP ideas presented by Astro¨m et al.15 and Normey-Rico et al.16 The structure proposed by Rao and Chidambaram has three controllers; the first one is tuned for the set-point tracking, the second is tuned for the disturbance rejection and the third is tuned to increase robustness. Comparative results presented in ref 14 show that the proposed controller performs better than the controllers in refs 8 and 9. This paper presents a simpler solution than the one in ref 14. The proposed dead-time compensation structure uses only two controllers to control the unstable FOPDT process. The analysis, design, and tuning of the controller are simple and allow to tune separately the set point and disturbance rejection response. The robustness is also considered in the design. Furthermore, the discrete implementation of the controller is analyzed and comparative results are given. The next section presents the controller structure and tuning while sections 3 and 4 are dedicated to analyzing the robustness * To whom correspondence should be addressed. E-mail: julio@ das.ufsc.br. † Universidade Federal de Santa Catarina. ‡ Universidad de Sevilla.
Figure 1. FSP: structure for analysis.
and section 5 describes the discrete implementation. Comparative results are presented in section 6. The paper ends with some conclusions. 2. Filtered Smith Predictor The proposed controller is shown in Figure 1. As can be seen, the structure corresponds to the basic SP structure with an additional filter (Fr(s)). The nominal model of the process is Pn(s) ) Gn(s)e-Lns. For the FOPDT unstable model Gn(s) ) K/(Ts - 1) is the dead-time-free model. The primary controller C(s) is a traditional PI controller with set-point weighting to improve the set-point response and Fr(s) is a predictor filter used to improve the predictor properties. The idea of using a predictor filter was introduced in ref 16 to improve the robustness of the SP for FOPDT stable models and also used in other controllers with the same objective.11,14,17,18 First consider the PI controller C(s) ) {Kc(1 + sTi)}/{Tis} without the set-point weighting. Thus, the nominal closed-loop transfer function is -Lns KcK(1 + sTi) Y(s) C(s)Gn(s)e ) ) e-Lns R(s) 1 + C(s)Gn(s) (Ts - 1)Tis + KcK(1 + Tis)
Defining T1 as the desired closed-loop time constant the closed-loop system has two real poles at s ) -1/T1, giving:
(
Ti ) T1 2 +
T1 T
)
Kc )
T1 + 2T KT1
The controller zero introduces an overshoot in the set-point response that can be eliminated using the set-point weighting b in the PI control law:19 u(t) ) Kc[br(t) - yp(t)] +
10.1021/ie0713487 CCC: $40.75 2008 American Chemical Society Published on Web 06/19/2008
Kc Ti
∫ (r(τ) - y (τ)) dτ t
0
p
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4785
Figure 2. FSP: structure for implementation.
In this case the nominal closed-loop system has the following transfer function: 1 + bTis -Lns Y(s) ) e Hr(s) ) R(s) (1 + T s)2 1
(1)
Note that this result can also be obtained with the use of a reference filter F(s) ) {bTis + 1}/{Tis + 1}. The zero of the transfer function Hr(s) can be tuned to speed up the response maintaining the overshoot less than 0.5% for every value of T1.20 This gives b ) 1.3(T1/Ti). Note that the weighting factor is not a new tuning parameter contrary to the tuning procedure in ref 14 where it is necessary to chose b ∈ [0.4, 0.6]. For the nominal disturbance rejection response Hq(s) )
1 + sTi -Lns Y(s) ) Pn(s)[1 e Fr(s)] Q(s) (1 + sT )2 1
Thus, Fr(s) is computed so as to (i) have a zero at s ) 0 in Hq(s) to reject steps; (ii) to cancel the dynamics defined by T1 and Ti to decouple the disturbance and set-point responses; and (iii) to obtain an internal stable system. The simplest Fr(s) is given by (1 + sT1)2 1 + sa Fr(s) ) 1 + sTi (1 + sT )2 0
thus: Hq(s) )
1 + sa -Lns Ke-Lns [1 e ] Ts - 1 (1 + sT )2
(2)
0
that is, Hq(s) has a zero at s ) 0 and cancels the dynamics defined by T1 and Ti, and T0 can be tuned to define the settling time of the disturbance rejection response. To analyze condition (iii), an equivalent block diagram is considered as in Figure 2. This scheme also shows the filter block F(s) associated to the set-point weighting. The internal stability condition is obtained if Ceq(s) does not cancel the unstable pole of the model at s ) 1/T. As Ceq(s) is Ceq(s) )
C(s) , 1 + C(s)M(s)
M(s) ) Gn(s)[1 - e-LnsFr(s)]
Figure 3. Rao and Chidambaram’s controller.
parameter T0 defines de disturbance rejection response and T1 the set-point response. In practice, the structure in Figure 2 is used for implementation. This implementation is done computing an stable M(s) in order to obtain the desired Ceq(s) for internal stability. This is obtained eliminating the common root at s ) 1/T from the denominator and numerator of M(s). Since digital computers are used for real-time implementation, discrete domain is used to describe the controller. This point is analyzed with details in section 5. As has been pointed out by Morari and Zafiriou21 the Smith predictor (SP) can be considered as a particular case of the IMC. Also, all the modified versions of the SP presented in the past years, such as the FSP presented in this paper, can also be represented as a 2 degrees of freedom IMC.3 Normally, in the IMC approach the tuning of the controllers is done using optimal procedures considering a defined type of input signals. However, we have preferred to use the proposed structure because it is similar to the traditional SP that is the most used dead-time compensation structure used in industry. Optimal PID controllers have also been proposed to control unstable dead-time processes.22 Because of the dead-time compensation, the responses obtained with the proposed approach are better than the ones obtained with the PID controller, mainly when the dead time can be estimated with small error.3 As will be shown in the following examples, the 2DOF tuning capability is an advantage of the proposed controller that allows to tune the desired responses for every FOPDT unstable process. In the controller presented in ref 14, (a) the parameter used to tune the set-point response also affects the disturbance rejection response, and (b) for long dead times the disturbance rejection response is sluggish and oscillatory for all the suggested values of the tuning parameters. Example 1. Consider the same process and model presented in ref 14: P(s) ) (4e-4s)/(4s - 1), Pn(s) ) (4e-4s)/(4s - 1). The proposed controller is compared to Rao and Chidambaram’s controller shown in Figure 3 where Cr(s) is a PI controller Cr(s) ) kc(1 + (1/tis)), Cd(s) is a PD controller Cd(s) ) kd + 0.7Ts/ K, and Ff(s) is a low-pass filter Ff(s) ) 1/(1.4Lns + 1). The tuning is given by
and M(s) ) K
kc )
-Lns
(1 + sTi)(1 + T0s) - (1 + sT1) (1 + as)e 2
2
(Ts - 1)(1 + sTi)(1 + T0s)2
in order to obtain an internally stable system M(s) cannot have a pole at s ) 1/T. Thus, a has to verify a ) T[(1 + T0/T)2eLn/T - 1] As can be seen from eqs 1 and 2 the controller obtains totally decoupled set-point and disturbance responses. The tuning
kd ) kd )
λ + 2T Kλ
ti ) λ(2 + λ/T )
[
]
1 0.533T + 0.746 K Ln
[
]
1 0.49T + 0.694 K Ln
if
Ln e 0.7 T
if 0.7
0 that is, δP(ω) is the multiplicative norm bound uncertainty. Thus, the robust stability condition for the FSP is δP(ω) < dP(ω) ) Figure 4. Plant output and control action for example 1. Proposed controller for T1 ) 4, T0 ) 5.2 (solid) and Rao and Chidambaram’s controller for λ ) Ln ) 4 and ) 0.4 (dashed). Case a, upper panels; case b, lower panels.
|1 + C( jω)Gn( jω)| |C( jω)Gn( jω)Fr( jω)|
∀ω>0
(3)
and with the proposed tuning dP(ω) )
|(1 + jωT0)2| |1 + jωa|
∀ω>0
(4)
Note that parameter T0 can be used to improve the robustness (increasing T0) or the disturbance rejection capabilities of the system (decreasing T0) without affecting the nominal set-point response (defined only by T1). However, because of the unstable characteristics of the process, robustness cannot be increased arbitrarily. This is an expected result because certain feedback action is needed to maintain stability and thus, the detuning of the controller (with high values of T0) has a limit. Note that a also increases with T0 and may reach high values, principally when Ln/T . 1; thus, dP(ω) cannot be increased arbitrarily. This limit can be computed for the case where only dead-time errors are considered: δP(jω) ) 1 - ejω∆L. The analysis of the maximum admissible modeling errors can be made as follows. Using the normalized frequency ωn ) ωLn, it follows that δP( jωn) ) 1 - e jωnδL Figure 5. Plant output and control action for example 1. Proposed controller for T1 ) 4, T0 ) 2.6 (solid) and Rao and Chidambaram’s controller for λ ) Ln ) 4, ) 0.4 (dashed).
The proposed controller is tuned using T1 ) Ln ) 4 and T0 ) 1.3Ln ) 5.2 to obtain the same set-point response as Rao and Chidambaram’s controller, which is tuned using λ ) Ln ) 4 and ) 0.4 as proposed in ref 14. In this case (case (a) in Figure 4), the set point response is the same with both controllers but the disturbance rejection is better in the FSP. Note that there is no oscillation in the transients of the FSP’s response. If a different tuning is used in the two controllers to speed up the set-point response (T1 ) λ ) Ln/5) the disturbance rejection of Rao and Chidambaram’s controller also changes (the new response is slower than in case (a)) while in the FSP the same disturbance rejection response as in case (a) is obtained (see case (b) in Figure 4). In the simulations a unitary step is applied at t ) 1 and a -0.05 step disturbance is introduced at t ) 20. To speed up the disturbance rejection response in the proposed controller a new T0 is used (T0 ) 2.6) in the simulations shown in Figure 5 maintaining T1 ) 4 and the same input signals. With the tuning proposed in Rao and Chidambaram’s controller it is not possible to achieve this type of response. Note that the set-point response is unchanged. Finally a bigger dead time is used in the process and model L ) Ln ) 6 and the tuning is T1 ) λ ) Ln ) 6 and T0 ) 1.3Ln ) 6.9. In this case the proposed controller has much more better disturbance rejection performance (see Figure 6).
and dP(ωn) )
|(1 + jωnT′ 0)2| |1 + jωna′|
where T′0 ) T0/Ln, a′ ) a/Ln, and a′ ) T′[(1 + T′ 0/T ′ )2e1/T′ - 1] The shapes of |δP| and dP for different values of T′0 are shown in Figure 7a. The robust stability condition can be obtained tuning T′0 until the two curves are tangent. Note that the intersection point is in the region where the two curves are almost linear and that, for these frequencies, the effect of T′0 on robustness is saturated. At these frequencies we can approximate |δP( jωn)| = | jωnδL| ) ωnδL As ωnT′0 . 1 and ωna′ . 1 it follows that dP(ωn) =
|( jωnT′0)2| | jωna′|
Noting that for the previous situations T′0/T′ + 1 = T′0/T′ a ′ e1/T′
(T′0)2 T′
⇒ dP(ωn) = ωn
T′ ′
e1/T
) dPmax(ωn)
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4787
Figure 6. Plant output and control action for example 1 for Ln ) 6. Proposed controller for T1 ) 6, T0 ) 6.9 (solid) and Rao and Chidambaram’s controller for λ ) Ln ) 6, ) 0.4 (dashed).
This gives an approximate upper bound for the maximum of the dead-time estimation error δLmax =
T′ ′
e1/T which shows the limitation imposed by the dead time and the unstable time constant of the system. Figure 7a shows that when high values of T′0 are used, the linear part of dP is equal to the bound obtained. This bound is independent of T0. Figure 7b shows this bound for different values of the relative unstable time constant. Note that for T/Ln ) T′ ) 0.2 (an unstable deadtime-dominant system), the closed-loop system can be unstabilized with an infinitesimal value of δL. On the other hand, if T/Ln ) T′ . 1, the stability can be achieved for high values of the dead-time uncertainties. In industrial processes we always expect to have dead-time estimation errors which implies that we always must try to increase the relationship T/Ln in the process design. This also implies that for FOPDT unstable systems we may consider that the process is dead-time dominant for smaller values of Ln/T than in the stable case. This is used in the next section to derive practical tuning rules. 4. Robust Tuning in Practice In the previous section the achievable robustness was analyzed. This analysis is useful when we have an estimation of the dead-time uncertainties and we are only analyzing stability. Thus, in this case, T0 can be tuned using δP(jω). On the other hand, because of the 2DOF structure T1 can be chosen to arbitrarily speed up the set-point response, as this parameter does not affect the robust stability condition. However, in practice, the objective is to achieve a certain robust performance and T0 and T1 must be tuned looking for this objective. In this case, a greater value of T0 must be used in order to obtain a lower oscillatory response when the model differs from the real process. To define the robust performance for the disturbance response, a simple rule can be used. For example, a factor of 0.5 (this value has been chosen empirically by simulation) can be applied to the robust stability condition for the case of dead-time estimation error dP(ω) > 2|1 - e-∆Ljω| f
|1 - e-∆Ljω| < 0.5 dP(ω)
∀ω
(5)
Figure 7. Robust stability analysis for unstable plants (δP(jωn) shown by thick line).
This can be solved numerically for each case to find T0. A similar procedure can be used to compute T1 using the real setpoint transfer function. In practice, it is often difficult to compute δP(jω) and a simple tuning procedure should be available. As pointed out in ref 14, in the literature of unstable FOPDT systems the closed-loop tuning parameters are related to the nominal dead-time Ln (see the suggested values of λ and Tf in section 2). In this paper, we use a similar idea but we also relate the tuning to the information given by the relationship Ln/T. A rule of thumb for tuning C(s) is to choose T1 ) Ln which implies that we can speed up the set-point response when the dead time is nondominant and we use a more conservative design when the dead-time is dominant. Note that this is the same conclusion obtained in ref 14. For T0 we have to consider the value of the dead time and the relationship Ln/T. The results in Figure 7b show that for T/Ln < 1 δLmax grows very slowly with T/Ln. However, for T/Ln g 1 the relationship between δLmax and T/Ln is approximately linear. Thus, we propose two tuning situations: (i) if the dead time is nondominant Ln e T, then T0 ) RLn; and (ii) is if the dead time is dominant Ln > T, then T0 ) RLn(Ln/T). Using simulations and robustness analysis we found R ) 1.3 for an initial tuning. This tuning must always take into account that smaller modeling errors will be accepted as Ln/T increases. For manual tuning the recommendation is to start with the recommended values and increase T0 and T1 to obtain better robustness and slower responses. Table 1 shows the suggested tuning rule and the following example illustrates the effect of T0 and T1 in the performance and robustness. Example 2. Consider the same model used in Example 1 and the process P(s) ) (3.5e-4.4s)/(3.8s - 1), that is, errors in the estimation of the gain, the dead time and the pole are considered. The proposed controller is tuned first using the values suggested in Table 1: T0 ) 5.2, T1 ) 4. Then, T0 is increased and decreased to analyze the effect on robustness. Figure 8 shows the modeling error and dP for three cases: T0 ) 5.2 (solid), T0 ) 7.8 (dashed), and T0 ) 4.16 (dashed-dotted). As can be seen, as T0 increases the distance between dP and the modeling error increases, giving better robustness. Note that the closed-loop system is stable in all cases.
4788 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 Table 1. Simple Initial Robust Tuning of the Controller model -Lns
Ke
/(sT - 1)
controller Kc(1+Tis)/Tis
filter 2
2
[(1+T1s) (1+as)]/[(1+T0s) (1+Tis)]
Kc
Ti
a
(T1 + 2T)/(KT1)
T1(2 + T1/T)
T[(1 + T0/T) e /T - 1]
The effect on the closed-loop responses is shown in Figure 9 for different values of T0 and T1. Case (a) show the responses for T0 ) 5.2 and case (b) for T0 ) 4.16. In the two cases three different values of T1 are used: T1 ) 4 (solid), T1 ) 8 (dashed) and T1 ) 2 (dashed-dotted). Note that the closed-loop responses deteriorates when T1 and T0 decrease and that T1 only affects the set-point response. In the simulations, a unitary step is applied at t ) 5 and a -0.05 step disturbance is introduced at t ) 100. 5. Discrete Implementation Because of implementation problems, only the discrete versions of the dead-time compensators are used in practice. In spite of the fact that all these algorithms are implemented on digital platforms, most works analyze only the continuous case. Some of the particular properties of the digital version of the SP and modifications are discussed in refs 3 and 23–25.
T1
T0
Ln
1.3(Ln/T)max(Ln,T) b
2 Ln
1.3T1/Ti
In the discrete domain, the FSP is implemented with a discrete 2DOF control structure equivalent to the one shown in the block diagram of Figure 2 with a sampling time Ts. The equivalent controller Ceq(z-1) is computed using, as in the continuous case, M(z-1) ) Gn(z-1)[1 - z-dFr(z-1)]. Note that the numerator Nm(z-1) and denominator Dm(z-1) of M(z-1) are polynomials and the elimination of the common roots is solved by directly dividing Nm(z-1) and Dm(z-1). Also note that for the imposed conditions M(1) ) 0, thus the equivalent controller has, as in the continuous case, integral action to reject the step disturbances. The discrete implementation of the control law allows for an exact computation of the dead-time compensation to cancel the unstable roots of the numerator and denominator of the controller. This procedure is simple and avoids polynomial approximations. Note that in the continuous domain the implementation of the controller for unstable plants can be done using: (a) the polynomial approximation of the dead time or (b) using the non dynamical form presented in refs 2 and 26. The discrete solution gives a better and simple solution. 6. Comparative Examples
Figure 8. Modeling error (bold) and dP for T0 ) 5.2 (solid), T0 ) 7.8 (dashed), and T0 ) 4.16 (dashed-dotted) for example 2.
Some examples taken from ref 14 are used here to show the good properties of the proposed controller and to compare the results with the ones obtained in Rao and Chidambaram’s paper. It must be noted that for the simulations a discretization of Rao and Chidambaram’s controller is used with the same sampling time as in the proposed controller. Example 3. Consider the same process and model used in Example 1. The proposed controller is tuned using Table 1 while Rao and Chidambaram’s controller is tuned using λ ) 4 and ) 0.4 as proposed in ref 14. Figure 10 (case (a)) shows the nominal responses and Figure 10 (case (b)) shows the case where the real dead time is L ) 4.4 and the real T ) 3.7. Note that both controllers have similar nominal set-point responses but the nominal disturbance rejection is better in the FSP, as it has no oscillations. Also the FSP maintains a good response when parameter uncertainties are considered while Rao and
Figure 9. Plant output and control action for example 2: (a, upper panels) T0 ) 5.2 (b, lower panels) T0 ) 4.16. T1 ) 4 (solid), T1 ) 8 (dashed) and T1 ) 2 (dashed-dotted).
Figure 10. Plant output and control action for Example 3 with Ln ) 4. Proposed controller (solid) and Rao and Chidambaram’s controller (dashed). Case a, upper panels; case b, lower panels.
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4789
t ) 1 and a -1 step disturbance is introduced at t ) 10. The sampling time is Ts ) 0.02. As can be seen in all the examples, the proposed controller allows to achieve a better compromise between performance and robustness. 7. Conclusions
Figure 11. Plant output and control action for example 3 with Ln ) 6. Proposed controller (solid) and Rao and Chidambaram’s controller (dashed). Case a, upper panels; case b, lower panels.
A simple robust dead-time compensator structure has been proposed to control FOPDT unstable processes. The proposed controller only introduces a prediction filter in the original structure of the Smith predictor. The correct tuning of this filter allows to control FOPDT unstable processes, obtaining a totally decoupled set-point and disturbance rejection responses. The tuning is simple; the controller has only two tuning parameters: the time constant of the closed-loop set point T1 and of the disturbance rejection response T0. T0 also defines the robustness of the close loop. Simulation results show the good qualities of the controller and also demonstrate that it performs better than other controllers recently presented in the literature. Acknowledgment Partially supported by CAPES-BRASIL Contract BEXO 0828/05-0 and CICYT Contract DPI 2005-4568. Literature Cited
Figure 12. Plant output and control action for Example 4. Proposed controller (solid) and Rao and Chidambaram’s controller (dashed). Case a, upper panels; case b, lower panels.
Chidambaram’s controller cannot give a stable closed-loop system. In the simulations, a unitary step is applied at t ) 1 and a -0.05 step disturbance is introduced at t ) 20. The sampling time is Ts ) 0.1. If a different dead time is used in the same process L ) 6 and the two controllers are tuning accordingly, the FSP shows better responses and better robustness, as can be seen in figure 11. In the simulations a unitary step is applied at t ) 1 and a -0.01 step disturbance is introduced at t ) 30. Figure 11 (case (a)) shows the nominal behavior; note that the FSP gives a better disturbance rejection response. Figure 11 (case (b)) shows the responses for a 5% error in the dead time (L ) 6.3 and Ln ) 6) where only the FSP maintains the stability. Example 4. Consider the same process and model presented in ref 14: P(s) ) e-0.4s/s - 1, Pn(s) ) e-0.4s/(s - 1), that is, a process with a small dead time. The proposed controller is tuned using T1 ) T0 ) Ln ) 0.4 while Rao and Chidambaram’s controller is tuned using λ ) 0.7Ln ) 0.28 and ) 0.5 to achieve the same set point response. Note that both controllers have similar disturbance rejection response for the nominal case although in the proposed controller the response has no oscillations (see Figure 12, case (a)). In the second simulation (see Figure 12, case (b)) the process is simulated with K ) 1.05, T ) 0.9, and L ) 0.46. Note that the proposed controller performs better. In the simulations a unitary step is applied at
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ReceiVed for reView October 6, 2007 ReVised manuscript receiVed February 11, 2008 Accepted March 28, 2008 IE0713487
