Closed-Loop Automatic Tuning Technique for an Event-Based PI

Article pubs.acs.org/IECR

Closed-Loop Automatic Tuning Technique for an Event-Based PI Controller Manuel Beschi,† Sebastián Dormido,‡ José Sánchez,‡ and Antonio Visioli*,§ †

Istituto di Tecnologie Industriali e Automazione, National Research Council, Milan, Italy Departamento de Informática y Automática, UNED, Madrid, Spain § Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia, Brescia, Italy ‡

ABSTRACT: A methodology for the closed-loop automatic tuning of an event-based PI controller is proposed in this article. In particular, the technique consists of performing two relay feedback experiments with a symmetric send-on-delta (possibly roughly tuned) PI controller already in place. The evaluation of the process variable allows for the determination of the parameters of a first-order-plus-dead-time process transfer function that is eventually employed to tune the controller parameters. Simulation and experimental results demonstrate the effectiveness of the methodology.

1. INTRODUCTION There has recently been significant interest, from both academic and industrial researchers, in event-based proportional−integral−derivative (PID) controllers, because of their capability to reduce the number of communications between control agents (namely, sensors, controllers, and actuators) in industrial plants. In fact, a message is sent from one control agent to another only when an event occurs, in contrast to the standard time-driven case where messages are sent at periodic time intervals regardless of the process operating conditions, that is, of whether the process variable is in steady state or a transient is occurring. The reduction of the number of transmissions is indeed of main concern especially when wireless sensors and actuators are used, because reducing the transmissions of these devices minimizes their power consumption, thereby increasing their battery lifetimes and reducing maintenance costs. An additional advantage is the reduction of the risk of lost data and stochastic time delays.1−3 Different event-based PID control methodologies have been proposed in the literature; see, for example, refs 4−14. (Note that the derivative action is almost always not employed, as its physical meaning is lost with a varying and possibly long time interval between two events.) It is recognized that the design of event-based PID controllers is generally more complex than that of standard time-driven PID controllers, as asynchronous sampling does not allow the application of standard design tools. Further, in general, there are more parameters to tune in an event-based PID controller than in a standard PID controller, as the logical conditions that determine the occurrence of an event have to be defined. However, despite the significant effort in providing effective event-based PI control methodologies, to date, no automatic tuning methodologies have been proposed in the literature for this kind of controller. In particular, there is a lack of eventbased procedures for the estimation of process parameters so that tuning rules (possibly devised specifically for event-based controllers) can then be applied. In this article, which is an extension of ref 15, we propose a closed-loop automatic tuning procedure for a symmetric send-on-delta (SSOD) PI controller © XXXX American Chemical Society

that was proposed in ref 16. The control system exploits a modification of send-on-delta (SOD) sampling17 and has the nice feature that the value of the sampled signal is independent of the initial conditions of the system.18 Necessary and sufficient conditions on the controller parameters for the existence of equilibrium points without limit cycles were determined for first-order-plus-dead-time (FOPDT) processes.16 Further, the threshold parameter Δ does not influence the stability properties of the system and can be tuned to handle the tradeoff between the system precision and the number of events.16 Indeed, one of the main features of the SSOD-PI controller is that the tuning of the proportional and integral gains can be done by employing tuning rules devised for standard PI controllers.19 In this context, an automatic tuning strategy should be based on the estimation of an FOPDT process model (which is capable, in general, of satisfactorily modeling self-regulating processes). This is achieved in this work by suitably extending the well-known relay feedback methodology.20,21 The technique consists of performing two consecutive experiments with the (possibly roughly tuned) PI controller in place. Then, the process parameters are computed based on the integrals of the resulting process variables, and a tuning rule is eventually employed. Once a process model is estimated, tuning rules already available in the literature, such as the simple internal model control (SIMC)22 or approximate M-constrained integral gain optimization (AMIGO)23 tuning rules, can be selected depending on the required performance, that is, taking into account the robustness and control effort issues. It is worth noting that, although the use of the integrals of the signal makes the method inherently robust to measurement noise, an analysis of the noise effect was also performed (see section 5). This article is organized as follows: In section 2, the SSOD control architecture is reviewed. The automatic tuning Received: March 18, 2015 Revised: May 6, 2015 Accepted: June 1, 2015

A

DOI: 10.1021/acs.iecr.5b01024 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

relay with hysteresis. The sensor unit (SU) is located in a device, whereas the control unit (CU) and the actuator unit (AU) are placed in another machine (for example, the communication between the two components could be wireless). Thus, the controller computes the control action at a predefined constant sampling rate by taking into account the last received sampled error (even if its input is constant between two consecutive events). This architecture is very interesting because, in a networked control system, the sensor unit can be powered using a battery (so that a reduction in communications can increase the battery life). Conversely, the actuator unit normally requires an external power supply, so a reduction of power consumption is not a critical issue for this control agent. Denote the input signal to the sampling block as v(t) and the sampled output signal as v*(t) [note that, in the SSOD-PI case, v(t) = e(t) and v*(t) = e*(t)], where v*(t) can assume only values that are integer multiples of a predefined threshold Δ multiplied by a gain β > 0, that is, v*(t) = jΔβ with j ∈ . The sampled signal changes its value to the upper quantization level when the input signal v(t) increases by more than Δ and to the lower quantization level when v(t) decreases by more than Δ. This behavior can be mathematically described as

methodology is proposed in section 3. Illustrative simulation results are presented in section 4, and the role of the measurement noise is analyzed in section 5. Experimental results obtained with a laboratory-scale apparatus are presented and discussed in section 6. Finally, conclusions are drawn in section 6.

2. SSOD CONTROL ARCHITECTURE The SSOD-PI control scheme already proposed in ref 16 and shown in Figure 1 is considered in this work. It can be

Figure 1. Control scheme of the SSOD-PI control system. The dashed arrows indicate data sent by the communication medium.

considered as a special case of the send-on-delta sampling method, which can, in turn, be seen as a generalization of a

⎧ v (t ) > (i + 1) ∧ v*(t −) = iΔβ ⎪(i + 1)Δβ if Δ ⎪ ⎪ v (t ) v*(t ) = ssod[v(t ); Δ, β ] = ⎨ iΔβ if ∈ [(i − 1), (i + 1)] ∧ v*(t −) = iΔβ Δ ⎪ ⎪ ⎪(i − 1)Δβ if v(t ) < (i − 1) ∧ v*(t −) = iΔβ ⎩ Δ

(1)

Controller C is a (discretized version of a) continuous-time PI controller, namely ⎛ 1 ⎞ C(s) = K p⎜1 + ⎟ Tsi ⎠ ⎝

The proposed automatic tuning methodology consists of performing two experiments with the (possibly roughly tuned) PI controller already in place and with a modified SSOD sampler. Then, the process parameters are computed based on the integrals of the resulting process variables (this is actually the information attached to the event message), and eventually, a given tuning rule is employed to determine the PI parameters. Details are given hereafter. To excite the system during the autotuning experiment, the CU discards the values of sampled error e* (i.e., the output of the SSOD block) different from −Δ and Δ. In this way, the relationship between e(t) and e*(t) is equal to a relay with hysteresis map, denoted as hystrel(·)

(2)

where Kp is the proportional gain and Ti is the integral time constant. Note that, because the controller is implemented in a single machine, it is possible to implement a standard antiwindup technique without additional effort. It is worth stressing that another control scheme, called PISSOD, in which the controller is placed before the SSOD block in the control loop [namely, v(t) = u(t) and u*(t) = v*(t)], was also proposed in ref 16. In that case, the sensor and control units are placed in the same device, whereas the actuator is located in another physical entity and holds the last received control action value until the next exchange of data. Thus, because the sensor and control unit have information on the process variable and on the control variable, standard automatic tuning techniques can be implemented.

e*(t ) = hystrel[e(t ); Δ, β ] ⎧ Δβ if e(t ) > Δ and e*(t −) = −Δβ ⎪ = ⎨−Δβ if e(t ) < Δ and e*(t −) = Δβ ⎪ ⎪ e*(t −) otherwise ⎩

(3)

Remark 1.The parameter β can be considered as part of the controller gain; therefore, it is possible to consider β = 1 without loss of generality. The proposed technique is based on the approximation of the process as an FOPDT transfer function (see eq 4), which is well-known to be capable of accurately modeling many overdamped self-regulating industrial plants. Thus, we consider the transfer function

3. AUTOMATIC TUNING PROCEDURE In the SSOD-PI control system, no unit has enough information to use standard autotuning techniques. For this reason, it is necessary to modify the sensor unit to send an additional indication to the control unit, which is attached to the event message. B


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Industrial & Engineering Chemistry Research P(s) =

K e−sL τs + 1

⎧ e At x̃ ̃ + F(t )̃ if t ̃ ∈ [0, T̃ /2) ⎪ 0 x(̃ t )̃ = ⎨ ⎪−e A(t −̃ T̃ /2)x ̃ − F(t ̃ − T̃ /2) if t ̃ ∈ [T̃ /2, T̃ ) ⎩ 0

(4)

where K is the process gain (which is assumed to be positive without loss of generality), τ > 0 is the time constant, and L ≥ 0 is the apparent dead time. A system in which the FOPDT process is controlled by the relay with hysteresis in series with a PI controller can be rewritten in state-space form as ⎧ 1 K ⎪ x1̇ (t ) = − x1(t ) + u(t ) τ τ ⎪ ⎪ 1 x ̇ (t ) = e*(t ) ⎪ ⎪ 2 Ti ⎨ ⎪ u(t ) = K p[e*(t ) + x 2(t )] ⎪ ⎪ y(t ) = x1(t − L) ⎪ ⎪ e*(t ) = hystrel[r − y(t ), Δ, β ] ⎩

(9)

where F(t)̃ is the zero-state unit-step response of the open-loop system described by matrices A and B, defined as ⎡ ⎤ t̃ ⎢ ⎥ ̃ F (t ) = ⎢ 1 − ρ ̃ (1 − e−ρt )⎥ ⎢⎣ ρ ⎥⎦

and x̃0 is the initial value of the state, which can be calculated by virtue of the symmetry and the continuity of the limit cycle [i.e., x̃(T̃ /2−) = x̃(T̃ /2+) = −x̃0] as ̃

x0̃ = −(I + e AT /2)−1F(T̃ /2) ⎡ ⎤ −T̃ /4 ⎢ ⎥ = ⎢ρ − 1 ⎥ 1 + ρT̃ /2 −1 ) − 1]⎥ ⎢⎣ ρ [2(e ⎦

(5)

To reduce the problem complexity, equation system 5 was normalized through the following variable changes

To estimate the process transfer function, two time instants are defined: tã is the first time instant when Cx̃(tã ) = 0, and tb̃ is the first time instant when Cx̃(tb̃ ) = 1, as shown in Figure 2. [Note that Cx̃(t)̃ , i.e., ỹ(t)̃ , is the sum of a ramp and an exponential term, as the state matrix A has a null eigenvalue and a negative eigenvalue.]

y(t ) − r y ̃ (t ) = , Δ

e(t ) e (̃ t ) = = −y ̃(t ), Δ TT e*(t ) L e *̃ (t ) = , x∼(̇ t )̃ = i x(̇ t ), l = , Ti Δ Δ ⎡ r ⎤⎞ T T⎛ κ = KK p , ρ = i , x(̃ t ) = ⎜x(t ) − ⎢ −1 ⎥⎟ ⎣ κ r ⎦⎠ τ Δ⎝

t̃ =

t , Ti

(10)

(6)

where the matrix T is given by ⎡0 1 ⎤ T=⎢ ⎣ κ −1⎥⎦

Thus, we obtain ⎧ x∼(̇ t )̃ = Ax(̃ t )̃ + Be *̃ (t )̃ ⎪ ⎪ ⎨ y ̃(t )̃ = Cx(̃ t ̃ − l) ⎪ ⎪ e *̃ (t )̃ = hystrel[−y ̃(t ), ̃ 1, 1] ⎩

Figure 2. Evolution of ẽ*(t)̃ (dash-dotted line), ỹ(t)̃ (dashed line), and Cx̃(t)̃ (solid line) during an oscillation. (7)

Because the normalized time delay l is unknown, the time instants tã and tb̃ are also unknown. However, from ỹ(t)̃ , it is possible to measure the time interval t2̃ between tã and tb̃ and the time interval t1̃ between tb̃ and tã + T̃ /2. Moreover, it is possible to relate these quantities to the period T̃ and the time delay l by the equations

where ⎡0 0 ⎤ A=⎢ ⎥ ⎣ 0 −ρ ⎦

⎡ 1 ⎤ B=⎢ ⎥ ⎣ ρ − 1⎦

tb̃ + l = T̃ /2 t1̃ = T̃ /2 + tã − tb̃

C = [κ κ ]

t 2̃ = tb̃ − tã

⎧ if t ̃ ∈ [0, T̃ /2) ⎪ 1 e *̃ (t )̃ = ⎨ ⎪ ⎩−1 if t ̃ ∈ [T̃ /2, T̃ )

(11)

The area of Cx̃(t)̃ defined on the time interval t1̃ is denoted as T̃ /2+t ̃ t̃ I1̃ = |C∫ tb̃ ax̃(t)̃ dt|̃ , whereas I2̃ = |C∫ tbã x(t)̃ dt|̃ is the integral ̃ = I1̃ + I2̃ = defined on the time interval t2̃ , and I12 ̃ ̃ |C∫ Ttã /2+tax̃(t)̃ dt|̃ . The integral I2̃ can be calculated by finding the quantities t̃ t̃ ∫ tbã x̃1(t)̃ dt ̃ and ∫ tbã x̃2(t)̃ dt ̃ separately. The former can be easily found by noting that

System 7 surely admits a symmetrical limit cycle with a period T̃ , because ẽ(t)̃ cannot assume the null value and hystrel(·) is symmetrical with respect to the origin.24 The trends of ẽ*(t)̃ and x̃(t)̃ are

(8) C


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Industrial & Engineering Chemistry Research ⎧ 1 − T̃ + t ̃ if t ̃ ∈ [0, T̃ /2) ⎪ ⎪ 4 x1̃ (t )̃ = ⎨ ⎪ 3 T̃ − t ̃ if t ̃ ∈ [T̃ /2, T̃ ) ⎪ ⎩ 4

′̃ γ + (2t 2′̃ 2 − 4t 2′̃ + T̃ ′ [(4t 2″̃ − 2t 2″̃ 2 − T̃ ″ + T̃ ″t 2″̃ )I12 ″̃ ]l 2 + [(4 − 4t 2″̃ + T̃ ″)I12 ′̃ γ + 4(t 2′̃ − 4 − T̃ ′) − T̃ ′t 2′̃ )I12 ″̃ ]l + (2I12 ″̃ − 2I12 ′̃ γ ) = 0 I12

and therefore tb̃

∫t ̃

κ=

x1̃ (t )̃ dt ̃ =

a

1 (tb̃ − tã )(2tb̃ + 2tã − T̃ ) 4

C[x(̃ tb̃ ) − x(̃ tã )] = C

∫t ̃

x∼(̇ t )̃ dt ̃ = CA

a

+ CB

∫t ̃

tb̃

∫t ̃

tb̃

a

x(̃ t )̃ dt ̃

K = K p−1κ

a

v (̃ t )̃ dt ̃ = 1

τ = ρ−1Ti L = lTi

1 x 2̃ (t )̃ dt ̃ = (tb̃ − tã ) − κρ

I2̃ = κt 2̃ (1 − t 2̃ /2 − l + T̃ /4) − ρ−1

̃ as In the same way, it is possible to calculate the quantity I12 κ I12̃ = (4l − T̃ + 4t 2̃ − 4lt 2̃ + Tt̃ 2̃ − 2l 2 − 2t 2̃ 2 + lT̃ ) 2

Kp =

(12)

Kp =

⎡⎛ t̃ ⎞ T̃ I1̃ = ρ−1 + κ ⎢⎜1 − l + − 2 ⎟t 2̃ + (2 − l)l 4 2⎠ ⎣⎝

Cx(̃ t )̃ dt ̃ =

1 TiΔ

ι

[Cx(t ) − r ] dt

(20)

6.7Lτ 2 τ + 2Lτ + 10L2 2

(21)

In summary, the autotuning procedure consists of the following steps: (1) Choose an initial set of PI parameters {Kp, Ti} that guarantees system stability. (2) Start an experiment by discarding the values of sampled error e* (i.e., the output of the SSOD block) that are different from −Δ and Δ. (3) When the limit cycle is stabilized (typically, after three limit cycles25), calculate the values of I′1, I′2, I′12, t′̃ , and T̃ ′. (4) Normalize the quantities I1′̃ = [1/(TiΔ)]I1′ , I2′̃ = [1/ (TiΔ)]I2′ , I12 ′̃ = [1/(TiΔ)]I12 ′ , t2′̃ = t2′ /Ti, and T̃ ′ = T′/Ti. (5) Set the proportional gain equal to γKp. (6) Start a new experiment by discarding the values of sampled error e* (i.e., the output of the SSOD block) that are different from −Δ and Δ. (7) When the new limit cycles is stabilized, calculate the values of I1″, I2″, I12 ″ , t2″, and T″. (8) Normalize the quantities I1″̃ = [1/(TiΔ)]I1″, I2″̃ = [1/ (TiΔ)]I2″, I12 ″̃ = [1/(TiΔ)]I12 ″ , t2″̃ = t2″/Ti, and T̃ ″ = T″/Ti. (9) Determine the process parameters by means of eqs 16−19. (10) Set the controller parameters by using the desired tuning rules to meet a given control specification (considering set-point following, load disturbance rejection, robustness, etc.).19

(13)

∫ς

Ti = min{τ , 8L}

0.15 ⎡ Lτ ⎤ τ + ⎢0.35 − ⎥ K ⎣ (L + τ )2 ⎦ KL

Ti = 0.35L +

̃ are related to the Remark 2. The integrals I1̃ , I2̃ , and I12 denormalized integrals I1, I2, and I12 by the equation ι̃

0.5τ , KL

whereas the AMIGO tuning rules are

̃ depends only on l and κ. It is important to note that I12 ̃ The integral I1̃ can be calculated as the difference between I12 ̃ and I2, giving

∫ς ̃

(19)

and then a suitable tuning rule can be employed for the selection of the PI parameters. In ref 19, we showed that, for example, the well-known SIMC22 and AMIGO23 tuning rules, although conceived for the time-driven case, retain their nice properties in the event-based framework as well. It is worth recalling that the SIMC tuning rules are

Thus, from eq 11, I2̃ is equal to

T̃ ⎤ + (l − 1) ⎥ 2⎦

(18)

Finally, the process parameters can be trivially found from eq 6 as follows

which gives

∫t ̃

4l + 4t 2′̃ − 4lt 2′̃ − 2l + T̃′(l + t 2′̃ − 1) − 2t 2′̃ 2

ρ−1 = κt 2′̃ (1 − t 2′̃ /2 − l + T͠ ′/4) − I2′̃

a

tb̃

′̃ 2I12 2

(17)

The latter can be found by integrating Cẋ̃(t)̃ = CAx̃(t)̃ + CBũ(t)̃ on the time interval t2̃ tb̃

(16)

(14)

where Tiς = ς̃ and Tiι = ι̃. To find the process parameters, two limit cycles with different proportional gains of the PI controller have to be obtained. In the second one, the proportional gain is modified by multiplying it by a quantity γ ≠ 1. From a practical point of view, a value of γ < 1 is recommended to preserve the system stability (for example, a convenient default value of γ = 0.5 can be selected). The quantities I1′̃ , I2′̃ , I12 ′̃ , t′̃ , and T̃ ′ represents the values obtained in the first case, whereas I1″̃ , I2″̃ , ̃ , t″̃ , and T̃ ″ are the values obtained in the second oscillation. I″12 Thus, we have κ ′̃ = (4l − T̃′ + 4t 2′̃ − 4lt 2′̃ + T̃ ′t 2′̃ − 2l 2 − 2t 2′̃ 2 + lT̃ ′) I12 2 γκ ″̃ = (4l − T̃ ″ + 4t 2″̃ − 4lt 2″̃ T̃″t 2″̃ − 2l 2 − 2t 2″̃ 2 + lT̃ ″) I12 2 (15)

The quantities l and κ can be therefore be found by solving the following equations, discarding the inadmissible solutions D


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Industrial & Engineering Chemistry Research Remark 3. It is worth stressing that the selection of Δ does not influence the stability properties of the system26 and can be done to handle the tradeoff between the system precision and the number of events.19 Indeed, the value of Δ does not influence the results of the automatic tuning procedure.

Table 1. Maximum Sensitivity Ms Obtained with AMIGO rules and the Proposed Plant Estimation Method P(s) 1

4. SIMULATION RESULTS To illustrate its capabilities, the proposed algorithm was applied to an FOPDT model with K = 82, τ = 22, and L = 3, initially controlled by an SSOD-PI controller with Kp = 0.05, Ti = 7, and Δ = 0.2. For the sake of clarity, measurement noise was not considered (this is done in the next section). The estimated parameters were found to be Kest = 81.93, τest = 21.98, and Lest = 3 and were used to obtain the new controller parameters by applying the AMIGO tuning rules in eqs 21. These rules were selected to obtain a high level of robustness, that is, a low value of the maximum sensitivity. The results obtained were Kp = 0.0237 and Ti = 15.14. Figure 3 shows the obtained results,

2 3* 4 5 6* 7 8 9 10

Pa(s)

Ms

Ms,a

1 e−s 0.1s + 1 1 −s e s+1 1 e−s 100s + 1

1 e−s 0.1s + 1 1 −s e s+1 1 e−0.999s 100s + 1

1.461

1.461

1.249

1.249

1.399

1.398

1 e−s 0.01s 2 + 0.2s + 1 1 e−s s 2 + 2s + 1

0.999 e−1.08s 0.124s + 1

1.468

1.47

1.01 e−1.56s 1.46s + 1

1.257

1.257

1 e−s 10000s 2 + 200s + 1 1 0.01s 2 + 1.01s + 1

1.01 e−59.8s 147s + 1

1.184

1.217

1 −0.00995s e s+1

1.259

1.398

1 0.5s 2 + 1.5s + 1 1 (s + 1)2

1.01 e−0.358s 1.18s + 1

1.203

1.233

1.03 e−0.517s 1.58s + 1

1.209

1.228

1 (s + 1)8

1 e−5.13s 2.89s + 1

1.352

1.348

1 e−0.106s 1.01s + 1

1.24

1.313

1.01 e−1.14s 1.42s + 1

1.228

1.227

1.02 e−1.3s 1.88s + 1

1.219

1.217

1 e−2.94s 1.26s + 1

1.374

1.403

11

1 3

(s + 1) ∑ j = 1 (1 + 0.1s) j 12

1 3

(s + 1) ∑ j = 1 (1 + 0.7s) j 13 14

− 0.1s + 1 s 3 + 3s 2 + 3s + 1 − 1.2s + 1 s 3 + 3s 2 + 3s + 1

quite similar, and therefore, the approximate FOPDT model is sufficiently accurate to tune the controller effectively. It is worth noting at this point that, to improve the accuracy of the model, it is advisible to repeat the estimation procedure if the retuned controlled system has a bandwidth that is significantly different from the limit-cycle frequency. In fact, Pa(s) is an accurate approximation of P(s) only in a frequency range around the limit-cycle frequency, as can be seen in Figure 4, where the process P(s) = 1/[(s + 1)∑3j=1(1 + 0.7s)j] is controlled by a PI controller that was tuned using the proposed method and the AMIGO rules. Finally, to analyze the effects of the noise on the parameter estimation, an FOPDT process with K = 1, τ = 1.8, and L = 0.8 was estimated with the proposed procedure (using initial SSOD-PI parameters of Kp = 1, Ti = 1, and Δ = 1). The considered noise was uniformly distributed on a band NB. In Figure 5, the relationships between the ratio NB/Δ and the parameter-estimation percentage errors are shown. It can be seen that a ratio NB/Δ of less than 25% causes estimation errors that are less than 20%.

Figure 3. Autotuning procedure applied to an FOPDT model (K = 82, τ = 22, and L = 3). Initial control system, thick solid line; first step of the autotuning procedure, thin dashed line; second step of the autotuning procedure, thick dash-dotted line; AMIGO-retuned control system, thin solid line; set-point signal, gray solid line. Top, process variable; bottom, control variable.

where it can be noted that the performance obtained by the SSOD-PI control system is consistent with that expected by using the AMIGO tuning rules, thus confirming the effectiveness of the overall methodology. The proposed autotuning method was then applied to the test batch proposed in ref 23 (which includes different process models including lag-dominant and delay-dominant ones) to analyze the accuracy of the approximate FOPDT model. In particular, an approximate model Pa(s) was calculated for each process P(s) of the batch list; then, the PI controller was retuned using the AMIGO rules,23 and the maximum sensitivities of the actual open loop, C(s) P(s) (denoted as Ms) and of modeled open loop C(s) Pa(s) (denoted as Ms,a) were calculated. The initial controller was chosen equal to C(s) = (s + 1)/s for all processes with the exceptions of processes indicated by an asterisk in Table 1, for which it was chosen equal to C(s) = (10s + 1)/(10s). In all cases, the parameters Δ and γ were set equal to Δ = 1 and γ = 0.5. In Table 1, where results related to some of the considered processes are reported, it can be seen that the values of Ms and Ms,a are

5. COPING WITH MEASUREMENT NOISE As stated in the previous section, the presence of noise in the process variable signal introduces errors in the estimation of the process parameters. This distortion is due to the “early″ crossing of the threshold because of the additive contribution of the noise signal. As shown in Figure 6, this implies that the E


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Figure 4. Comparison between the actual (solid lines) and modeled (dashed lines) control systems for the illustrative example of section 4: (a) process-variable unit-step response, (b) magnitude Bode plot, (c) Nyquist plot, (d) control-variable unit-step response, (e) phase Bode plot, and (f) Bode plot of the sensitivity function.

affected only by the ratio of the integrals, not by their magnitudes. Remark 4. The integrals should also take into account the contribution due to the early crossing of the null threshold, dentied as I3̃ in Figure 6, but this contribution is negligible when np is much smaller than Δ and this calculus requires the knowledge of the system parameters, implying the use of a more complicated iterative approach.

relay switches from one state to another earlier than expected, thus resulting in a bias in the determined values of the areas and eventually in the estimated values of the parameters. Even though a rigorous study of this phenomenon could be ̃ , t′̃ , T̃ ′, I″1̃ , I″2̃ , I″12 ̃ , very difficult (because the quantities I′1̃ , I′2̃ , I′12 T̃ ″, and T̃ ″ become non-Gaussian stochastic signals), it is possible to perform a practically meaningful analysis by approximating this early crossing with a deformation of the threshold values. Actually, the events can be assumed to occur when the noise-free signal crosses values equal to Δ − np and −Δ + np, where np can be set equal to 80% of the peak noise value for uniformly distributed noise signal or 2σ for white noise with a standard deviation equal to σ. With this choice, there is a 10% of probability of an early crossing for a single sample, but because the sampling rate is normally much faster than the system dynamics, the probability for early crossing of the threshold with at least one sample in a limited time interval is quite high. Because of the modification of the threshold values, it is necessary to rewrite eq 14 taking into account the modified threshold value as follows

∫ς ̃

ι̃

Cx(̃ t )̃ dt ̃ =

1 TiΔ(1 − n p)

∫ς

6. EXPERIMENTAL RESULTS To test the proposed autotuning procedure in a practical application, a laboratory-scale setup was employed to implement a level-control task (see Figure 8). In particular, a tank is filled with water by means of a pump whose speed is set by a dc voltage (the manipulated variable), in the range of 0−10 V (normalized to the range of 0−1 in the following figures), through a pulse-width-modulation (PWM) circuit. The tank is fitted with an outlet at the base to let the water return to the reservoir. The measurement of the level (in centimeters) of the water is given by a pressure sensor that provides an output signal between 0 V (empty tank) and 10 V (full tank). Because of the small apparent dead time, an additional time delay of 9 s was added by software at the plant output. Even though the process is nonlinear (as the flow rate out of the tank depends on the square root of the level), an FOPDT model of the system was obtained by applying a least-squares procedure to the response of a series of open-loop steps around the operating point selected equal to 10 cm, resulting in the expression

ι

[Cx(t ) − r ] dt (22)

Hence, steps 4 and 8 of the autotuning procedure (see section 3) are modified by considering I′1̃ = {1/[TiΔ(1 − np)]}I′1, I′2̃ = ′̃ = {1/[TiΔ(1 − np)]}I12 ′ ,I1″̃ = {1/ {1/[TiΔ(1 − np)]}I2′ , I12 ̃ ″̃ = {1/ [TiΔ(1 − np)]}I1″, I2″ = {1/[TiΔ(1 − np)]}I2″, and I12 [TiΔ(1 − np)]}I″12. The modified approach is able to improve the effectiveness of the estimation of parameters K and T, as shown in Figure 7 for the same system as considered in Figure 5, whereas the estimation of L does not change because eq 16 is

P(s) = F

64.86 −9s e 22.3s + 1

(23) DOI: 10.1021/acs.iecr.5b01024 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 5. Parameter-estimation percentage errors caused by a uniform random noise without noise compensation. Dashed lines indicate the mean values and confidence intervals of the estimated values.

Figure 7. Parameter-estimation percentage errors caused by a uniform random noise: (●) with noise compensation, (◊) without noise compensation. Dashed lines indicate the mean values and confidence intervals of the estimated values.

Figure 6. Evolution of ẽ*(t)̃ (dash-dotted line), ỹ(t)̃ (dashed line), modified crossing value (gray dashed line), Cx̃(t)̃ (solid line), ỹ(t)̃ ± np, and Cx̃(t)̃ ± np (gray dash-dotted line) during an oscillation.

Note that this model is given only to evaluate the accuracy of the result, and was not employed in the automatic tuning procedure. The system was initially controlled by an SSOD-PI controller, which was tuned with Kp = 0.0366 cm−1 and Ti = 16 s. The parameter Δ = 0.3 cm was chosen to be larger than the noise band of the sensor and equal to 1% of the full scale of the level, which is 30 cm. During the experiment, two set-point step changes were applied, the first from 10 to 15 cm and the second from 15 to 10 cm. Then, the autotuning procedure was started. The obtained estimated parameters are K = 61.1906 cm, τ =

Figure 8. Sketch of the setup used for the experiments.

21.5964 s, and L = 8.6939 s. The SSOD-PI controller was tuned with the AMIGO tuning rules (Kp = 0.0084 cm−1 and Ti = 19.2695 s), and finally, two set-point step changes were applied to the retuned controlled system. G


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case as well, the obtained results are consistent with the SIMC tuning behavior, namely, a faster response and a minor robustness with respect to the AMIGO case.

7. CONCLUSIONS In this article we have presented a closed-loop automatic tuning procedure for a symmetrical send-on-delta event-based PI controller. The technique is based on a suitable relay-feedbackbased experiment that can be performed during normal process operations because it does not perturb them significantly. Other nice features of the method, as shown by simulation and experimental results, are robustness to the measurement noise and to the initial PI tuning. This makes the overall methodology suitable for implementation in industry. Future work will focus on extending the methodology to other eventbased PI controllers such as the PID plus controller,12 which is available in commercial distributed control systems.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 9. Autotuning procedure applied to the laboratory tank system. Initial control system, thick solid line; first step of the autotuning procedure, thin dashed line; second step of the autotuning procedure, thick dash-dotted line; AMIGO-retuned control system, thin solid line. Top, process variable; bottom, control variable.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by National Plan Projects DPI201127818-C02-02 and DPI2012-31303 of the Spanish Ministry of Economy and Competitiveness and FEDER funds.

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REFERENCES

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Figure 10. Autotuning procedure applied to the laboratory tank system. Initial control system, thick solid line; first step of the autotuning procedure, thin dashed line; second step of the autotuning procedure, thick dash-dotted line; SIMC-retuned control system, thin solid line; set-point signal, gray solid line. Top, process variable; bottom, control variable.

In Figure 9, the results of the autotuning procedure are shown. Note that, although the initial controller was badly tuned, the method allowed the system to find appropriate values of the FOPDT parameters and, therefore, appropriate values of the SSOD-PI controller. The same experiment was then repeated using the SIMC tuning rules in the retuning phase. The obtained parameters are Kp = 0.0155 cm−1 and Ti = 17.3794 s, and the results of the autotuning procedure are shown in Figure 10. Note that, in this H


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Closed-Loop Automatic Tuning Technique for an Event-Based PI

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