Simple Tuning Rules for Dead-Time Compensation of Stable

Article pubs.acs.org/IECR

Simple Tuning Rules for Dead-Time Compensation of Stable, Integrative, and Unstable First-Order Dead-Time Processes Bismark C. Torrico,*,† Marcos U. Cavalcante,† Arthur P. S. Braga,† Julio E. Normey-Rico,‡ and Alberto A. M. Albuquerque† †

Departamento de Engenharia Elétrica, Universidade Federal do Ceará, 60455-760 Fortaleza, CE, Brazil Departamento de Automaçaõ e Sistemas, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil

‡

ABSTRACT: This work proposes a new and simple design for the filtered Smith predictor (FSP), which belongs to a class of dead-time compensators (DTCs) and allows the handling of stable, unstable, and integrative processes. For this purpose, first, to use lower-order controller and filters, it is shown that it is not necessary to use the integral action in the primary controller, which is used to tune the set-point response; then, the FSP filters are designed to obtain the desired disturbance rejection, robustness, and noise attenuation. Using this procedure, it is possible to obtain a better compromise between performance and complexity than other solutions in the literature. Two simulation case studies are used to compare the obtained solution with some recently published results. A practical experiment involving a neonatal intensive care unit is also presented to illustrate the usefulness of the proposed DTC.

1. INTRODUCTION Dead-time processes are found in many industrial applications. Dead times are mainly caused by the time required to transport mass, energy, or information, 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.1−4 Conventional controllers such as proportional−integral− derivative (PID) controllers can be used when the dead time is small, but they exhibit poor performance in the case of large dead times.5−7 The difficulties of controlling this type of process can be explained in the frequency domain: The dead time introduces a further decrease in the phase of the system that makes the process more difficult to control.8 To solve these problems, dead-time compensators (DTCs) can be used.2 The Smith predictor (SP),9 proposed in 1957, was the first DTC strategy formulated to improve the performance of classical PI or PID controllers for processes with dead time. However, this strategy cannot be used in processes that have an unstable or integrative model, and its disturbance rejection response cannot be faster than the open-loop one.10 Several research studies involving attempts to overcome these difficulties have been reported in the past 25 years.2 Two main DTC groups have been received special attention from the academic community: DTC for integrative processes5,11−15 and DTC for unstable processes.16−19 Wide reviews of these modifications are presented in refs 2, 8, and 10. Unified solutions for dead-time processes, including robustness and disturbance rejection specifications that can handle stable or unstable processes, were proposed in refs 20 and 21. However, these strategies have limitations in the case of unstable deadtime process with measurement noise. Despite the importance of noise, it was not a common practice to analyze noise effects in previous DTC works. In dead-time process control, it is a common practice to use low-pass filters as a unique tuning option to detune an initial controller setup to achieve loop requirements (robustness, sensitivity to noise, and so on). For © 2013 American Chemical Society

example, in refs 2 and 14, disturbance-observer dead-time compensators (DODTCs) and filtered Smith predictor (FSP) robustness filters are presented; in ref 22, an internal model control (IMC) filter was used, and in refs 20, 23, and 24, prediction error filters were used. Nevertheless, in these works, noise filtering was not studied. In a recent study, 25 it was shown that previously cited works did not properly filter the noise. To overcome this problem, the authors used an extra parameter in the FSP filter, which increased the tuning complexity. In ref 26 was presented an applicable solution of the problems in refs 12 and 13 [known as the modified Smith predictor (MSP)] to control stable, integrative, and unstable dead-time processes, and it was shown that the MSP is a PID controller in series with a second-order filter defined by the dead time and an adjustable parameter. Nevertheless, an optimization tool is needed to set constraints on the robustness and sensitivity to measurement noise. In this article, a simplified and new tuning procedure of the FSP for first-order stable, integrative, and unstable dead-time processes is proposed. It shown that, for these simple cases, it is not necessary to increase the order and complexity of the FSP filter to deal with the noise if the primary controller, used to tune the set-point response, is properly chosen. To illustrate this effect, the proposed controller is tested in simulations and then for the temperature control of a neonatal incubator of the Electrical Engineering Department at the Federal University of Ceará, Fortaleza, Brazil. The next section reviews the unified DTC approach, whereas section 3 describes the new simplified FSP tuning. Section 4 presents two simulation case studies. An experimental case Received: Revised: Accepted: Published: 11646

May 1, 2013 July 15, 2013 July 25, 2013 July 25, 2013 dx.doi.org/10.1021/ie401395x | Ind. Eng. Chem. Res. 2013, 52, 11646−11654

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study is presented in section 5, and the work ends with some conclusions (section 6).

2. UNIFIED DTC APPROACH: A REVIEW In this section, we summarize the filtered Smith predictor (FSP), which is one of the most popular DTC structures2,21,25 and can be used to control stable, unstable, and integrative dead-time processes. The FSP control structure is shown in Figure 1. As can be seen, the structure is the same as that of the

H yr(z) =

F(z) C(z) Pn(z) Y (z ) = R (z ) 1 + C(z) Gn(z)

(3)

H yq(z) =

⎡ C(z) Pn(z) Fr(z) ⎤ Y (z ) = Pn(z)⎢1 − ⎥ 1 + C(z) Gn(z) ⎦ Q (z ) ⎣

(4)

Hun(z) =

C(z) U (z ) = −Fr(z) 1 + C(z) Gn(z) N (z )

(5)

where R(z), Q(z), N(z), U(z), and Y(z) represent the z transforms of the set point r(t), input disturbance q(t), measurement noise n(t), control signal u(t), and measurement output y(t), respectively. The implementation structure for unstable and integrative process, also called the unified dead-time compensator, is

Figure 1. FSP conceptual structure.

Smith predictor (SP) with two additional filters. F(z) is a reference filter to improve the set-point response, and Fr(z) is a predictor filter that improves the predictor properties, avoiding the appearance of slow or unstable poles of the plant in the disturbance rejection response and including extra parameters to improve robustness. In addition, Pn(z) = Gn(z)z−dn is the nominal process discretized with a zero-order hold, where Gn(z) is the dead-time-free model and dn is the nominal discrete dead time. The equivalent control two-degree-of-

Figure 3. FSP implementation structure.

presented in Figure 3, where S(z) is a stable transfer function computed with the equation21 S(z) = Gn(z)[1 − z −dnFr(z)]

Obtaining a stable function S(z) is equivalent to avoiding unstable pole−zero cancellations between Ceq(z) and Pn(z) or, equivalently, having a stable function Hyq(z). The filter F(z) and the primary controller C(z) are used to obtain the desired set-point response, and Fr(z), which does not modify the nominal set-point tracking, is used to the change disturbance rejection response and filter the noise. Note that C(z) can increase the tuning difficulty of Fr(z) because it affects eqs 4 and 5. 2.1. Tuning of the FSP. This subsection presents the main ideas and a brief analysis of the tuning procedure for the FSP presented in refs 2 and 25. The tuning of C(z) and F(z) is presented first, followed by the tuning of Fr(z). 2.1.1. Tuning of C(z) and F(z). Although C(z) and F(z) are used to define the set-point response, it is important to notice that C(z) also affects the disturbance rejection (eq 4) and the noise filtering (eq 5). Therefore, special attention must be paid to its tuning, mainly in the case of unstable processes. In refs 2, 21, and 25, C(z) and F(z) were designed using a traditional 2DOF approach. For example, in the case of a first-order plus dead-time (FOPDT) model, C(z) is a PI controller, and F(z) is a first-order filter.2 In general, F(z) is used to avoid the overshoot caused by the zeros introduced by C(z). 2.1.2. Tuning of Fr(z). Initially, the design of Fr(z) follows two objectives: to decouple the disturbance rejection from the set-point response and to avoid the appearance of slow or unstable plant poles in the disturbance rejection response [giving an internally stable system when P(z) is unstable].

Figure 2. Two-degree-of-freedom (2DOF) structure.

freedom (2DOF) structure for the FSP is shown in Figure 2, where Ceq(z) =

Feq(z) =

Fr(z) C(z) 1 + C(z) Gn(z)[1 − Fr(z)z −dn]

(1)

F (z ) Fr(z)

(2)

(6)

Note that Ceq(z) must have at least one pole at z = 1 to reject steplike disturbances and Feq(1) = 1 to guarantee set-point tracking. In all previous works on FSP, this was achieved by using a pole at z = 1 in C(z), and the filters were tuned to guarantee F(1) = 1 and Fr(1) = 1. In this article, we use a simpler C(z) without a pole at z = 1 and lower-order filters with F(1) = Fr(1) = kr, where kr is a constant. Moreover, to obtain an internally stable system, Ceq(z) cannot have zeros at the slow or unstable poles of the plant. Using the FSP structure, the nominal closed-loop transfer functions [when Pn(z) = P(z)] are 11647

dx.doi.org/10.1021/ie401395x | Ind. Eng. Chem. Res. 2013, 52, 11646−11654


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To achieve this goal, consider the nominal model written explicitly in terms of numerators and denominators as in the equation N (z ) N (z ) Pn(z) = n z −dn = − n + z −dn Dn(z) Dn (z) Dn (z)

Pn(z) = Gn(z)z −dn =

=

(7)

Dr (z) − z −dnNr(z) Dr (z) (z − z 0)(z − z1)···(z − zn)p(z) Dr (z)z dn

(8)

H̅ yr(z) =

where z0 = 1 and p(z) is an unknown polynomial. Note that the term (z − z0) appears only in the case that C(z) has a pole at z = 1. Thus, it will not appear in the simplified FSP (SFSP) proposed in this work. From eqs 8 and 6, we obtain S(z ) =

Nn(z) p(z)(z − z 0) Dn−(z) Dr (z)

|1 + C(e jω) Gn(e jω)| |C(e jω) Gn(e jω) Fr(e jω)|

(1 − zc) −dn z z − zc

(12)

To achieve this objective, the primary controller and the reference filter are proposed as C(z) = kc and F(z) = kr. Thus, using kc, kr, and eq 11 in eq 3, we obtain

(9)

H yr(z) =

Now, S(z) is stable and does not have any of the undesired poles of Pn(z). As the roots of Dr(z) are poles of S(z) and Hyq(z), they define the disturbance rejection response and also have connection with robustness. If desired, it is possible to obtain an ideal decoupling between the disturbance rejection and the step response. For this condition, Fr(z) should have fast poles, and Dr(z) and p(z) should be computed to eliminate the poles of Hyr(z) from Hyq(z).21 2.2. Robustness and Stability Analysis. The robustness analysis was performed considering that the process modeling errors can be represented as unstructured uncertainties, that is, P(z) = Pn(z) + ΔP(z) = Pn(z)[1 + δP(z)], where P(z) represents the real process. Also, let us assume that an upper bound for the norm of δP(z), z = ejω′Ts for 0 < ω′ < π/Ts, is given by δP (ejω′Ts). By definition, δP (z) is the norm-bound multiplicative uncertainty term, and Ts is the sampling period. In this case, considering ω = ω′Ts, the robust stability condition is22 δP(e jω) ≤ Ir(e jω) =

(11)

To perform control tuning, it is assumed that the following specifications are desired: (i) set-point following of steps with a defined settling time, (ii) steady-state rejection of step disturbances with the same time constant as the set-point response, and (iii) noise filtering and robust stability. Note that, in these specifications, the same closed-loop time constant is defined for all responses, which is a good solution in practical applications where robustness is an important issue.2 The tuning of the SFSP is performed in two steps: First, C(z) and F(z) are tuned for a desired step response, and then the filter Fr(z) is tuned considering both steplike disturbance rejection at steady state and the tradeoff between robustness and disturbance rejection. 3.1. Tuning of C(z) and F(z). Consider the desired closedloop transfer function

where the roots of D+n(z) are the undesired poles of the plant, represented as D+n(z) = (z − z1)···(z − zn). The same is done with the predictor filter, which is written as Fr(z) = [Nr(z)]/ [Dr(z)]. In ref 21, Fr(1) = 1 was considered. Thus, the first phase of the predictor filter design problem can be rewritten to find Nr(z) in such a way that 1 − z −dnFr(z) =

b0 −dn z z − a1

k rkcb0 z −d n z − a1 + kcb0

(13)

Then, to make H̅ yr(z) = Hyr(z), the controller parameters must be kc = (a1 − zc)/b0 and kr = (1 − zc)/(a1 − zc). 3.2. Tuning of the Filter Fr(z). For the proposed SFSP, the filter Fr(z) has three objectives: (i) to guarantee step disturbance rejection at steady state; (ii) to eliminate the open-loop pole from Hyq(z), which implies internal stability for unstable plants; and (iii) to reach a compromise between robustness and disturbance rejection speed. To achieve these goals, first, the noise attenuation (eq 5) is rewritten using C(z) and F(z) from subsection 3.1 as Hun(z) = −Fr(z)

kc(z − a1) z − zc

(14)

As can be observed, a first-order low-pass filter can be used to attenuate the noise at high frequencies. Nevertheless, to guarantee step input disturbance rejection and set-point tracking, a second-order filter is used14 Fr(z) =

(10)

b1z 2 + b2z (z − α)2

(15)

where 0 < ω < π and Ir(e ) is defined as the robustness index of the controller. Note that Dr(z) affects the numerator of Ir(ejω), thus slow poles of Fr(z) give better robustness. However, in the case of unstable processes, the 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 has a limit.21 Therefore, in practice, Fr(z) tuning is done to solve the tradeoff between robustness and disturbance rejection.

where b1 and b2 are used for the first two objectives and α for the third. To achieve objective i, Ceq(z) must have a pole at z = 1, and to achieve objective ii, the plant pole should not appear in the disturbance response, or equivalently, this pole should not be a zero of Ceq(z). Using eq 1, Ceq(z) can be written as

3. NEW SIMPLIFIED FSP (SFSP) TUNING This section presents a new simple method of tuning the FSP for stable, integrative, and unstable first-order plus dead-time (FOPDT) models. Consider the following FOPDT model

This implies that the term in the denominator of the second fraction on the right-hand side of eq 16, [1 + C(z) Gn(z)]/ [C(z) Gn(z)] − Fr(z)z−d, must be computed to cancel the plant pole (z = a1). The condition for the steady-state step

jω

Ceq(z) =

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Fr(z) 1 1 + C ( z ) G ( z ) ⎡ −d ⎤ Gn(z) n ⎢⎣ C(z) Gn(z) − Fr(z)z ⎥⎦

(16)



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disturbance rejection of the system is that the same factor has a zero at z = 1. For stable and unstable processes (a1 ≠ 1), this implies ⎧⎡ ⎤ ⎪ ⎢ 1 + C(z) Gn(z) − Fr(z)z −d ⎥ =0 ⎪ ⎣ C(z) Gn(z) ⎦ ⎪ z=1 ⎨ ⎤ ⎪ ⎡ 1 + C(z) Gn(z) − Fr(z)z −d ⎥ =0 ⎪⎢ ⎦ ⎪ ⎣ C(z) Gn(z) z = a1 ⎩

(17)

and for integrative process (a1 = 1), it implies ⎧ ⎡ ⎪ ⎢1 ⎪ ⎪ ⎣ ⎨ ⎪ d ⎡1 ⎪ ⎢ ⎪ dz ⎣ ⎩

⎤ + C(z) Gn(z) − Fr(z)z −d ⎥ C(z) Gn(z) ⎦ ⎤ + C(z) Gn(z) − Fr(z)z −d ⎥ C(z) Gn(z) ⎦

=0

Figure 4. Bound of noise attenuation |H̅ un|, three-dimensional plot.

=0

response and a short dead time contribute to better noise attenuation. Figure 5 shows numerical values of H̅ un for

z=1

z=1

(18)

The solution for eq 17 (a1 ≠ 1) is b1 =

1 [(1 − α)2 k r − a1d − 1(a1 − α)2 ] 1 − a1

b2 =

1 [a1(1 − α)2 k r − a1d − 1(a1 − α)2 ] a1 − 1

and for eq 18 (a1 = 1), the solution is ⎛ 1 ⎞ b1 = (1 − α)2 ⎜d − 1 + ⎟ + 2(1 − α) kcb ⎠ ⎝ ⎛ 1 ⎞ b2 = (1 − α)2 ⎜2 − d − ⎟ − 2(1 − α) kcb ⎠ ⎝

Figure 5. Bound of noise attenuation |H̅ un|, two-dimensional plot.

different values of zc and d. As can be observed, for a given value of H̅ un, there is a limit on the achievable closed-loop response (defined by zc) imposed by the dead time d: If the dead time is large, then zc must be chosen high to keep the desired H̅ un value. As the filter cannot be tuned arbitrarily for a desired H̅ un value, then special attention must be paid to the desired closed-loop response by choosing an appropriate value of zc.

where α is the robustness tuning parameter. The SFSP can be implemented similarly to the FSP explained in the previous section. 3.3. Noise Attenuation Analysis. The worst case for the analysis of noise attenuation is the unstable case, where harder constraints are imposed on Fr(z). Therefore, in this subsection the noise attenuation is analyzed for this case. However, conceptually, the analysis is valid for the other cases as well. For simplicity, the following two assumptions are considered: (i) Hun(ejω) is analyzed at ω = π, because Fr(ejω) is a low-pass filter and the noise can be interpreted as a high-frequency disturbance, and (ii) α tends toward 1 (α → 1) to obtain the maximum attenuation bound of Fr(ejω) at ω = π. Thus, using eqs 14 and 15 and the plant model, the bound of noise attenuation can be written as H̅ un =

|a1d − 1(a12 − 1)(a1 − zc)| |2b0(1 + zc)|

4. SIMULATION CASE STUDIES In this section, two case studies are presented, one for an unstable process and the other for an integrative process. The results obtained with the proposed SFSP are compared with those obtained with the FSP proposed in ref 25. 4.1. Unstable Process. The chemical reactor concentration control problem that was also used in refs 20, 21, and 25 is analyzed in this case study. This problem is used to show that correct tuning of the SFSP robustness filter can effectively reduce the effect of noise, maintaining a good tradeoff between robustness and performance. Moreover, a comparative analysis between the SFSP and FSP is presented. The linear concentration control process model is given by 3.433 P(s) = e−20s (20) 101.1s − 1 The same sampling period as proposed in ref 20 was used here, Ts = 0.5 s. Thus, the discrete model parameters are a1 = 1.00486, b0 = 0.016689, and d = 40 (see eq 11).

(19)

As can be observed, H̅ un depends on the desired closed-loop pole zc and dead time d. To analyze the effects of zc and d on H̅ un, H̅ un was computed for several values of zc and d in the ranges of [0.4−0.99] and [5−190], respectively, considering a plant with b0 = 0.01 and a1 = 1.01, as illustrated in Figure 4. The following remarks are based on Figure 4: (i) Low values of H̅ un occur in the region with high values of zc and low values of d and (ii) high values of H̅ un occur in the region with low values of zc and high values of d. In other words, a slow closed-loop 11649



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The FSP control parameters C(z), F(z), and Fr(z) were defined in ref 25 considering a closed-loop time constant of τ = 20 s and 30% dead-time estimation error as follows C(z) = 3.2501

z − 0.98876 z−1

(21)

F(z) = 0.4552

z − 0.9753 z − 0.98876

(22)

2

Fr(z) = 0.03535

z (z − 0.9968) (z − 0.995)(z − 0.85)2

(23)

On the other hand, the proposed SFSP was tuned following the sequence of section 3. First, considering the same closed-loop time constant as used for the FSP, the primary control is a − zc C(z) = 1 = 1.7678 b0 (24) F (z ) =

1 − zc = 0.83522 a1 − zc

b1z 2 + b2z (z − α)2

= 0.10204

attenuate the noise at high frequencies. Note that the FSP attenuates the noise more than the SFSP at high frequencies, which is an expected result because the order of the FSP filter is higher. Note also that the cutoff frequency of the SFSP is at lower frequencies than that of the FSP. To compare the performances of the FSP and SFSP, five scenarios of dead-time uncertainties were simulated: (i) 0% (nominal case), (ii) +30%, (iii) −30%, (iv) +60%, and (v) −60%. In all five cases, a unity input step disturbance was added. Figure 8 shows the closed-loop responses for the nominal case. In the same simulation, a measurement noise was added at

(25)

where zc = e−Ts/τ = 0.9753. Note that, if point response is required, then the decreased or increased, respectively. Second, the robustness filter Fr(z) is dead-time estimation error by using α stability condition (see eq 10) Fr(z) =

Figure 7. Output−input noise gain.

a faster or slower setconstant τ must be tuned based on 30% to satisfy the robust

z(z − 0.99567) (z − 0.977)2

(26)

Note that b1 and b2 depend on α and the process model parameters (see subsection 3.2). Observe that α is the unique free tuning parameter of Fr(z) that can be tuned (i) with lower values to obtain faster disturbance rejection and lower robustness or (ii) with higher values to obtain higher robustness but lower disturbance rejection. Figure 6 shows the values of the robustness index Ir(ejω) for both the FSP and the SFSP. A norm-bound multiplicative

Figure 8. Nominal system response with disturbance and noise.

t = 600 s. The noise was generated by means of a Simulink band-limited white noise with Ts = 0.5, noise power = 0.05, and seed = 0. As can be seen, the SFSP rejected the disturbance faster than the FSP, and the two controllers attenuated the noise in similar ways. Figure 9 shows the closed-loop response for the case with the dead-time estimation error of +30%. Note that the SFSP takes slightly longer to stabilize but its control signal is less oscillatory. Figure 10 shows the closed-loop response for the case with the dead-time estimation error of −30%. Note that both the output and the control of the FSP are more oscillatory than those of the SFSP; in addition, the SFSP rejects the disturbance faster. Figure 11 shows the closed-loop response for the case with the dead-time estimation error of +60%. Note that the FSP becomes unstable whereas the SFSP, despite a very oscillatory response, is stable.

Figure 6. Robustness index: Unstable plant.

uncertainty term, δP (ejω), defined in subsection 2.2, is also depicted as a loop specification considering ±30% dead-time estimation error. As can be observed, the SFSP has greater robustness than the FSP at midfrequencies, precisely where it is more important because the robustness stability condition (see eq 10) is closer to being violated. In Figure 7, the noise attenuation |Hun(ejw)| is compared for the FSP and SFSP. It can be observed that both controllers 11650



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Figure 9. System response with +30% dead-time error.

Figure 12. System response with −60% dead-time error.

Table 1. Performance Indices for the Unstable Examplea IAE nominal

SFSP FSP SFSP FSP SFSP FSP

+30% −30%

TV

SR

LDR

SR

LDR

201 201 313 294 229 226

111 119 140 121 114 119

16.2 16.2 22.5 29.3 20.0 39.0

2.7 2.4 5.0 5.0 2.2 4.0

a

IAE, integrated absolute error; TV, total variation of the input; SR, set-point response; and LDR, load disturbance rejection.

Figure 10. System response with −30% dead-time error.

the cases of the ±60% dead-time estimation errors, the performance indices were not computed because the FSP had an unstable response. 4.2. Integrative Process. To show that the SFSP can handle integrative processes, we consider the process model presented in ref 23 0.1 P(s) = e − 8s s(s + 1)(0.5s + 1)(0.1s + 1) (27) which is approximated by the following FOPDT model 0.1 −9.6s Pn(s) = e s

(28)

To design the controllers, the FOPDT model in eq 28 was discretized with a zero-order hold and a sampling time of Ts = 0.1 s. The FSP and SFSP were tuned considering a dead-time uncertainty of ±1 s and the difference between P(s) and Pn(s) (eqs 27 and 28). Thus, for the FSP, the control parameters are z − 0.9714 C(z) = 11.4247 (29) z−1

Figure 11. System response with +60% dead-time error.

Figure 12 shows the closed-loop response for the case with the dead-time estimation error of −60%. As can be seen, the proposed SFSP has a stable closed-loop response, and the FSP is unstable. (The FSP simulations were cut to obtain a clear figure.) Finally, two performance criteria index were computed for both the set-point response (SR) and the load disturbance rejection (LDR): (i) the integrated absolute error (IAE) and (ii) the total variation of the input (TV).27 The results are presented in Table 1. Note the following points: (i) In the nominal case, the performances of the FSP and the SFSP are similar. (ii) In the case of the +30% dead-time estimation error, the FSP had a slightly better IAE; however, it had a worse TV for the SR. (iii) In the case of the −30% dead-time estimation error, the IAEs are similar for the two controllers; nevertheless, the proposed SFSP has a TV almost one-half that of the FSP. In

F(z) = 0.5

z − 0.9428 z − 0.9714

Fr(z) = 0.0102

z 2(z − 0.9969) (z − 0.995)(z − 0.92)2

(30)

(31)

and for the SFSP, they are

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F (z ) = 1

(32)

C(z) = 5.7

(33)


Industrial & Engineering Chemistry Research Fr(z) = 0.0553

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z(z − 0.9964) (z − 0.985)2

(34) jω

Figure 13 shows the values of the robustness index Ir(e ) for both the FSP and the SFSP. In addition, the plant modeling

Figure 15. System system response if the real-plant dead time were 9 s.

Regarding to input disturbance, the SFSP rejected it faster than the FSP. Figure 16 shows the results for the third case, for which the errors in the static gain and dominant time constant of P(s)

Figure 13. Robustness index: Integrative plant.

error δP (ejω) is depicted. Note that, as in the unstable case, the robustness index of the SFSP is larger than that of the FSP at the frequencies where the robust stability condition is closer to being violated. To compare the performance between the FSP and the SFSP, three simulations were performed: first for the nominal case, second using the “real-plant” P(s) with 9 s of dead time, and third using the real-plant P(s) with some modeling errors that were not considered in the analysis. P(s) has 10 s of dead time and +10% error in the static gain and in the dominant time constant. Figure 14 shows the simulation results for the nominal case. An input disturbance was applied at 30 s, and a measurement Figure 16. Response if the real dead time were 10 s, with errors in the static gain and dominant time constant of 10%.

were +10% and the dead time was 10. Note that the SFSP followed the reference and rejected the input disturbance whereas the FSP was unstable. The integrated absolute error (IAE) and the total variation of the input (TV) were computed27 for both the set-point response (SR) and the load disturbance rejection (LDR). The results are presented in Table 2. Observe that (i) in the nominal case, the performances of both the FSP and SFSP were almost the same and (ii) in the case of L = 9 s, the two controllers had similar IAEs, but the proposed SFSP had a better TV. The performance indices were not computed for the other case because the FSP was unstable.

Figure 14. Nominal system response with disturbance and noise.

Table 2. Performance Indices for the Integrative Examplea IAE

noise was applied at 120 s. The noise was generated by means of a Simulink band-limited white noise with Ts = 0.1, noise power = 0.01, and seed = 0. As can be observed, the SFSP rejected the input disturbance faster than the FSP, and the noise filtering behaviors were similar. Figure 15 shows the results for the second case, for which the plant delay was L = 9 s (see eq 27). An input disturbance was applied at 70 s. The FSP and SFSP had similar outputs for setpoint tracking, although the FSP input was more oscillatory.

nominal L=9s

SFSP FSP SFSP FSP

TV

SR

LDR

SR

LDR

10.7 10.7 15 14.8

12.1 14.5 13.4 15.2

5.7 5.7 8.3 13.8

1.1 1.1 1.5 2.2

a

IAE, integrated absolute error; TV, total variation of the input; SR, set-point response; and LDR, load disturbance rejection.

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5. EXPERIMENTAL CASE STUDY In this section, the SFSP algorithm is applied to control the temperature of a neonatal intensive care unit (NICU), shown in Figure 17, that belongs to the Electrical Engineering

Figure 18. Experimental results of SFSP when controlling the temperature of a neonatal incubator.

6. CONCLUSIONS This article presents a new and simple design of the FSP called the SFSP that was specially designed for stable, integrative, and unstable FOPDT models. The algorithm uses a simple parameter to define the set-point following and two others to obtain the desired robustness and noise filtering. This approach is simpler than those presented in refs 21 and 25, and the transfer functions of the SFSP are of lower order. The proposed SFSP presented good results in both simulated and practical experiments. Two simulation cases were used to evaluate the robustness of the proposed SFSP. The SFSP presented a better robustness stability condition, a faster disturbance rejection, and a less oscillatory control signal even in the presence of dead-time error. These features are of interest for practical applications. To evaluate the application of the SFSP algorithm to a real system, the problem of temperature control in a neonatal incubator was considered. The experiments showed responses with no overshoot and almost no oscillations of the control signal. Because of its simplicity and good performance, the SFSP presented in this work has great potential to be implemented in commercial neonatal intensive care units.

Figure 17. Picture of the neonatal intensive care unit.

Department of the Federal University of Ceará (Fortaleza, Brazil). A NICU is a device consisting of a rigid boxlike enclosure in which a newborn infant can be kept in a controlled environment for medical care. The incubator basically includes an ac-powered heater, a fan to circulate the warmed air, and transducers for relative humidity and temperature. Fresh air is driven by the fan toward the heating element, and then the warmed air goes into the NICU. The control signal is measured in percentages in the range from 0% to 100%. The process variable is the temperature (in degrees Celsius) inside the NICU, which is the most important variable to be controlled. The model obtained using some step tests in the range of operation of the process and an offline least-squares identification method is given by P(z) =

0.0018263 −30 z z − 0.99337

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AUTHOR INFORMATION

Corresponding Author

where the sampling time was Ts = 0.4 min. To avoid overheating next to the heater, the primary controller was tuned to have a safe closed-loop settling time of 1 h. Thus, the closed-loop pole was set to zc = 0.98, and a filter with α = 0.93 was used to increase the robustness of the system and to attenuate the effects of measurement noise. Figure 18 illustrates the experimental results obtained with the SFSP algorithm. Initially, the reference temperature (shown in dotted lines) was set to 32 °C. At t = 120 min, the reference temperature was changed to 36 °C. These reference values are commonly used in NICUs. It was observed that, for the two reference step changes, the overshoot was 0 °C. Note that the overshoot was as expected, because, in section 3, it was designed to be zero. This is very important because it represents more comfort for the infant. To test the controller robustness, a disturbance was applied at t = 260 min by opening two port holes of the NICU for 5 min. The port holes allow nurses and caretakers to handle the newborn without risk of contamination. Observe that there was an undershoot of 0.7 °C, but the temperature returned to the set point in 10 min.

*E-mail: [email protected]. Tel.: +55-85-33669575. Fax: +55-85-33669574. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Brazilian funding agencies CNPq and FUNCAP is gratefully acknowledged. The authors also thank the editor and anonymous reviewers for their suggestions and comments.

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REFERENCES

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Simple Tuning Rules for Dead-Time Compensation of Stable

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