An Automatic Tuning Procedure for an Event-Based PI Controller
52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy
An Automatic Tuning Procedure for an Event-based PI Controller Manuel Beschi, Sebasti´an Dormido, Jos´e Sanchez, Antonio Visioli
Abstract— In this paper we propose an automatic tuning procedure for an event-based PI control scheme. In particular, the well-known relay feedback methodology for a standard PI controller is extended to a symmetrical send-on-delta PI controller. In this context, the overall event-based procedure consists of two relay feedback experiments (with a PI controller, possibly roughly tuned, already in place) in order to estimate the parameters of a first-order-plus-dead-time transfer function. Then, based on the process parameters, a tuning rule can be suitably applied. Simulation results demonstrate the effectiveness of the methodology.
I. I NTRODUCTION Event-based Proportional-Integral-Derivative (PID) controllers have been recently the subject of many investigations from academic and industrial researchers because of the need of addressing new technological issues, in particular, the introduction of wireless sensors and actuators in industrial plants [1], [20], [15], [17], [18], [8]. In this context it is essential to minimize the power consumption of the devices in order to increase the battery life time and for this reason, in addition to minimize the risk of lost data and stochastic time delays [13], [11], [14], the communications should be reduced as much as possible by keeping an acceptable level of performance. The rationale of event-based control is to send a communication between the control agents just when an event occurs, on the contrary of the standard time-driven control where the information is sent in any case at periodic time intervals. Thus, a logical condition has to be defined in order for an event to be detected. In the context of event-based PID controllers, one of the most employed condition is the so-called send-on-delta (SOD) sampling (also known as deadband sampling [21] or level crossing sampling [10]) where a variable is sent when its value (or some function of it) crosses predefined quantization levels [16]. A modification of the SOD technique, called Symmetric Send-On-Delta (SSOD), has been proposed and applied to a PI controller in [4], where its advantages with respect to the standard SOD approach have been highlighted (among them, it is worth stressing that the SSOD parameter ∆ does not influence the stability properties of the system but only M. Beschi is with the Dipartimento di Ingegneria dell’Informazione, University of Brescia, Brescia, Italy [email protected] S. Dormido and J. Sanchez are with the Departamento de Informatica y Automatica, UNED, Madrid, Spain {sdormido,jsanchez}@dia.uned.es A. Visioli is with the Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia, Brescia, Italy
[email protected] 978-1-4673-5716-6/13/$31.00 ©2013 IEEE
the precision and the number of events). In particular, two control algorithms, named SSOD-PI and PI-SSOD have been proposed (depending where the triggering function is applied in the control loop) and necessary and sufficient conditions on the controller parameters for the existence of equilibrium points without limit cycles have been determined for first-order-plus-dead-time (FOPDT) processes. Then, the tuning of the controller parameters (which is in general a more challenging task with respect to the time-driven case because the control is asynchronous and there are more parameters to select) has been addressed in [5], [6]. It has been highlighted that tuning rules devised for standard time-driven PI controllers [3], [12] can be effectively applied also to SSOD-PI and PI-SSOD controllers. In any case, it is recognized that the automatic tuning feature is very desirable in industry and it should be available also for event-based PI controllers. In this paper we propose therefore an autotuning procedure for the SSOD-PI controller (standard techniques can be employed for the PI-SSOD controlled system, as it will be clarified in Section III). In particular, for the reason explained above, we will focus on the technique for the estimation of the parameters of a first-order-plus-dead-time (FOPDT) system which is known to model satisfactorily a wide range of self-regulating overdamped processes. For this purpose, an extension of the well-known (closed-loop) relay feedback methodology [3], [22] is presented. It has been specifically devised for the event-based control architecture and it consists of performing two 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, thus making the overall approach inherently robust to measurement noise. Once the process parameters are estimated, a suitable tuning rule (such as the SIMC [19] or AMIGO [9] one) can be employed, depending on the required performance (i.e., by taking into account the robustness and control effort issues).
II. SSOD C ONTROL A RCHITECTURE A. Generalities In event-based control strategies, the controller can be divided into four logical blocks: the sensor unit, the control unit, the actuator unit and a governor (for short SU, CU, AU and G, respectively), as shown in Figure 1. The units and their tasks can be described as follows: • The Sensor Unit is composed of the sensor and its onboard intelligence. Its task is to measure the process
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Governor
G
r
D SU
SU
CU
+
AU
e
SSOD
CU and AU e*
ZOH
−
Plant
C(s)
u +
+
y P(s)
Fig. 2. Control scheme of the SSOD-PI controlled system. The dashed arrows indicate data sending via the communication medium.
Fig. 1. Scheme of a generic event based control strategies. The dashed arrows indicate the possibility of an event-triggered data transmission.
output and to calculate the error between the measured signal and a constant set-point value received from the governor. • The Control Unit implements the control algorithm, which determines the control action by taking into account the last received sampled error and sends it to the Actuator Unit. • The Actuator Unit receives the control action signal from the Control Unit and applies it to the actuator. • The Governor, which, in practice, can be implemented together with one of the previous two blocks, receives the desired set-point value from a user interface or from a higher hierarchical controller, and sends it to the Sensor Unit. These blocks can be implemented in a unique machine or in two or more physical entities. In this last case, the data has to be sent from one to each others on a network. It is clear that the communication between two entities implies more efforts than the data exchanging into a single machine, especially when they are battery-powered. For this reason, it is recommended to use event-triggered data exchanging for all the signals that are sent between two machines and normal time-driven sampling for the data which are elaborated by an unique machine. In this paper, we consider two special cases where two of the three control agents (namely, SU, CU and AU) are placed in a physical entity and the other one is located in another machine. We do not address the case where SU and AU are in the same machine, because in this case it is sufficient to implement also the CU task in this machine in order to obtain a standard controller. The remaining cases are: SU separated from CU and AU (which are in the same machine) and AU separated from SU and CU (which are in the same machine). In both situations, only an eventtriggered data exchanging in the control loop is required. The communications between the governor and the other units (considering only the transmissions concerning the control aspects) are done only when the set-point signal changes. B. Symmetric send-on-delta triggering As explained before, we consider two different architectures, which present, as common characteristic, the presence of only two event-triggered data exchanging. The first one is the set-point value sending, which is not a complex task from a control point of view, the second one is implemented with
the symmetryc send-on-delta (SSOD) sampling technique (see [4]). This can be seen as a special case of the send-on-delta sampling method (see [11], [20]), which can be seen also as a generalization of a relay with hysteresis. Denote as v(t) the input signal to the sampling block and as v ∗ (t) the sampled output signal, which can assume only values multiple of a predefined threshold ∆ multiplied by a gain β > 0, namely v ∗ (t) = j∆β with j ∈ Z. The sampled signal changes its value to the upper quantization level when the input signal v(t) increases more than ∆, or to the lower quantization level when v(t) decreases more than ∆. This behavior can be mathematically described as: v∗ (t) = ssod(v(t); ∆, β) = v(t) ∗ − (i + 1)∆β if ∆ > (i + 1) ∧ v (t ) = i∆β v(t) i∆β if ∆ ∈ [(i − 1), (i + 1)] ∧ v ∗ (t− ) = i∆β ∗ − (i − 1)∆β if v(t) ∆ < (i − 1) ∧ v (t ) = i∆β (1)
C. PI controller
The controller used in this work is a (discretized version of) continuous-time PI controller, namely: 1 (2) C(s) = Kp 1 + Ti s where Kp is the proportional gain and Ti is the integral time constant. The derivative action is not employed (as it often happens in industrial settings), because its implementation is very critical with a variable, and possibly long, sampling period. Remark 1. Because the controller is implemented in a single machine, it is possible to implement a standard anti-windup technique without additional efforts. D. SSOD-PI controller In the first architecture, shown in Figure 2, the SU is located in a machine while CU and AU are placed in another physical entity (for example, the communication between the two components could be wireless). We call this architecture SSOD-PI controller, because the SSOD block is placed before the PI controller in the control loop. Note that the controller computes the control action at a regular sampling rate by taking into account the last received sampled error. This architecture is very interesting because, in a networked control system, the sensor unit can be powered using a battery (thus, a reduction of the communications can increase
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self-regulating industrial plants. Thus, we consider the following transfer function
Governor r + e −
D SU and CU u C(s) SSOD
AU u*
+ ZOH
+
K e−sL (4) τs + 1 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. The system where the FOPDT process is controlled by the relay with hysteresis can be rewritten in the state-space form as follows x˙ 1 (t) = − τ1 x1 (t) + K τ u(t) 1 ∗ x ˙ (t) = e (t) 2 Ti (5) u(t) = Kp (e∗ (t) + x2 (t)) y(t) = x1 (t − L) ∗ e (t) = hystrel(r − y(t), ∆, β) P (s) =
y P(s)
Fig. 3. Control scheme of the PI-SSOD controlled system. The dashed arrows indicate data sending via communication medium.
the battery life). Conversely, the AU normally requires an external power supply, therefore the power consumption reduction is not a critical issue in this control agent. E. PI-SSOD controller In the second architecture, shown in Figure 3, the SU and the CU are located in the same machine, while the AU is placed in another entity. This solution is called for short PISSOD controller because the controller is before the SSOD block in the control loop. Note that the AU holds the last received control action value until the next data exchanging. III. T HE AUTOMATIC TUNING PROCEDURE In the PI-SSOD controlled system, it is possible to implement standard automatic tuning techniques because the sensor and control units (which are located in the same physical entity) have information on the process variable and on the control variable. On the contrary, in the SSODPI controlled system there is not a unit which has enough information to use standard methods. For this reason, it is necessary to modify the sensor unit in order to send an additional indication to the control unit, which is attached to the event message. The proposed automatic tuning technique 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. In order to excite the system during the autotuning experiment, the CU discards the values of sampled error e∗ (namely, the output of the SSOD block) different to −∆ and ∆. In this way, the relationship between e(t) and e∗ (t) is equal to a relay with hysteresis map denoted as hystrel(·), namely e∗ (t) = hystrel(e(t); ∆, β) if e(t) > ∆ and e∗ (t− ) = −∆β ∆β (3) = −∆β if e(t) < ∆ and e∗ (t− ) = −∆β . ∗ − e (t ) if otherwise
Remark 2. The parameter β can be considered as a 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 a FOPDT transfer function (4), which is wellknown to be capable to model accurately many overdamped
In order to reduce the problem complexity, the equation system (5) is normalized with the following variable change: r y(t) − r t r˜ = , y˜(t) = , t˜ = , Ti ∆ ∆ ∗ e(t) e (t) iT x(t), ˙ e˜(t) = , e˜∗ (t) = , x ˜˙ (t˜) = T∆ ∆ ∆ L l= , κ = KKp , ρ = Tτi , Ti T r x ˜(t) = x(t) − κ−1 r ∆
(6)
where the matrix T is equal to: 0 1 T= κ −1 Thus, we obtain ˜˙ (t˜) = A˜ x(t˜) + B˜ e∗ (t˜) x y˜(t˜) = C˜ x(t˜) ∗ ˜ e˜ (t) = hystrel(−˜ y (t˜), 1, 1)
where:
A=
0 0
B=
1 ρ−1
κ
C=
0 −ρ
κ
(7)
System (7) surely admits a symmetrical limit cycle with a period T˜, because e˜(t˜) cannot assume the null value and hystrel(·) is symmetrical with respect to the origin [7]. The trends of e˜∗ (t˜) and x ˜(t˜) are 1 if t˜ ∈ [0, T˜/2) ∗ ˜ e˜ (t) = (8) −1 if t˜ ∈ [T˜/2, T˜) ( ˜ eAt x˜0 + F (t˜) if t˜ ∈ [0, T˜/2) x ˜(t˜) = ˜ ˜ −eA(t−T /2) x ˜0 − F (t˜ − T˜/2) if t˜ ∈ [T˜/2, T˜) (9) where F (t˜) is the unit step response, defined as follows: # " t˜ F (t˜) = 1−ρ 1 − e−ρt˜ ρ
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˜ + CB˜ ˜ on the time interval t˜2 CA˜ x(t) u(t) Z t˜b x ˜˙ (t˜)dt˜ = C x˜(t˜b ) − x˜(t˜a ) = C
2 1 I2
0 Cx ˜(t˜)
I1
y˜(t˜)
v˜∗ (t˜)
−1
CA
−2 0
t˜a
t˜b
t˜a
l t˜2
Z
t˜1 t˜b T˜/2 Normalized time t˜
x ˜(t˜)dt˜ + CB
Z
t˜a t˜b
t˜a
v˜(t˜)dt˜ = 1
and it results t˜a + T˜/2 t˜b + T˜/2
T˜
Fig. 4. Evolution of v˜∗ (t˜) (dash-dot line), y˜(t˜) (dashed line) and C˜ x(t˜) (solid line) during an oscillation.
Z
t˜b
t˜a
1 . x ˜2 (t˜)dt˜ = t˜b − t˜a − κρ
Thus, by using (11), I2 is equal to:
I2 = κt˜2 (1 − t˜2 /2 − l + T˜/4) − ρ−1 and x˜0 is the initial value of the state, which can be calculated by virtue of the symmetry of the limit cycle (namely, x˜(T˜/2) = −˜ x0 ) as: x ˜0
−1 ˜ = − I + eAT /2 F (T˜/2) ˜ −T /4 −1 = ρ−1 ˜ 2 e1+ρT /2 −1 ρ
(10)
In order to estimate the process transfer function, two time instants are defined: t˜a is the first time instant when C˜ x(t˜a ) = 0 and t˜b is the first time instant when C˜ x(t˜b ) = 1, as shown in Figure 4. Because the time delay l is unknown, also the time instants t˜a and t˜b are unknown. However, from y˜(t˜) it is possible to measure the time interval t˜2 between t˜a and t˜b and the time interval t˜1 between t˜b and t˜a + T˜/2. Moreover, it is possible to relate these quantities with the period T˜ and with the time delay l by the following equations: t˜b + l = T˜/2 t˜1 = T˜/2 + t˜a − t˜b
(11)
t˜2 = t˜b − t˜a The area of C˜ x(t˜) defined on the time interval t˜1 is deR T˜ /2+t˜a R t˜ noted as I1 = |C t˜b x ˜(t˜)dt˜|, while I2 = |C t˜ab x(t˜)dt˜| is the integral defined on the time interval t˜2 and I12 = R T˜/2+t˜a I1 + I2 = |C t˜a x˜(t˜)dt˜|. The integral I2 can be calculated by finding the quantities R t˜ R t˜b x ˜ (t˜)dt˜ and t˜ab x˜2 (t˜)dt˜ separately. The first one can be t˜a 1 easily found by noting that 1 − 4 T˜ + t˜ if t˜ ∈ [0, T˜/2) ˜ x ˜1 (t) = 3 ˜ ˜ ˜ ˜ ˜ 4 T − t if t ∈ [T /2, T )
In the same way, it is possible to calculate the quantity I12 , which results to be equal to κ I12 = 4l − T˜ + 4t˜2 − 4lt˜2 + T˜t˜2 − 2l2 − 2t˜22 + lT˜ . 2 (12) It is important to note that I12 depends only on l and κ. The integral I1 can be calculated as the difference between I12 and I2 , and it results: ˜ ˜ ˜ I1 = ρ−1 + κ 1 − l + T4 − t22 t˜2 + (2 − l)l + (l − 1) T2 (13) Remark 3. The integrals I1 , I2 , and I12 are related to the denormalized integrals by the following equation: Z ˜ι Z ι 1 C˜ x(t˜)dt˜ = (Cx(t) − r) dt Ti ∆ ς ς˜ where Ti ς = ς˜ and Ti ι = ˜ι. In order to find the process parameters, two limit cycles with different proportional gains of the PI have to be obtained. In the second one, the proportional gain is modified by multiplying it by a quantity γ 6= 1. From a practical point of view, a value γ < 1 is recommended in order to surely preserve the system stability (for example, a convenient ′ default value γ = 0.5 can be selected). The quantities I1 , ′ ′ ′ ′ I2 , I12 , t˜ , and T˜ represents the obtained values in the first ′′ ′′ ′′ ′′ ′′ case, while I1 , I2 , I12 , t˜ , and T˜ are the obtained values in the second oscillation. Thus, we have ′ ′ ′ ′ ′ ′ ′ ′ κ 4l − T˜ + 4t˜2 − 4lt˜2 + T˜ t˜2 − 2l2 − 2t˜22 + lT˜ I12 = 2 ′′ ′′ ′′ ′′ ′′ ′′ γκ I12 = 2 4(l + t˜2 − lt˜2 ) + T˜ (l + t˜2 − 1) − 2l2 − 2t˜2 2 (14) The quantity l and κ can be therefore found by solving the following equations, discarding the inadmissible solutions: ′′
[(4t˜2 −2t˜2 2 −T˜
′′
′′
′′
′
[(4−4t˜2 +T˜
and therefore Z t˜b 1 ˜ ˜ ˜ x ˜1 (t˜)dt˜ = tb − ta 2tb + 2t˜a − T˜ , 4 t˜a
′′
′′ ′′ ′ ′ ′ ′ ′ ′ ′′ 2 +T˜ t˜2 )I˜12 γ+(2t˜22 −4t˜2 +T˜ −T˜ t˜2 )I˜12 ] l + ′
′
′′
)I˜12 γ+4(t˜2 −4−T˜ )I˜12 ] l
′′
′
+ [2I˜12 −2I˜12 γ ] = 0
2I˜12 κ= ′ ′ ′ ′ 4l + 4t˜2 − 4lt˜2 − 2l2 + T˜ ′ (l + t˜2 − 1) − 2t˜22
(15)
′
˜ = while the second one can be found by integrating Cx ˜˙ (t) 7440
ρ−1 = κt˜2 (1 − t˜2 /2 − l + T˜′ /4) − I2 ′
′
′
(16) (17)
0.5τ Kp = Ti = min{τ, 8L} KL while the AMIGO ones are Lτ τ Kp = 0.15 K + 0.35 − (L+τ )2 KL Ti = 0.35L +
2
6.7Lτ τ 2 +2Lτ +10L2
(19)
(s + 1)
j=1 (1
+ 0.7s)j
10
120
240 Time
360
480
120
240 Time
360
480
1 0.8 0.6 0.4
0 0
In order to clarify better the proposed algorithm, it has been applied to a FOPDT model with K = 82, τ = 22, and L = 3, initially controlled by a SSOD-PI controller with Kp = 0.05, Ti = 7, and ∆ = 0.2. The estimated parameters result equal to Kest = 81.93, τest = 21.98, and Lest = 3, which are used to obtain the new controller parameters by applying the AMIGO tuning rule (20). It results Kp = 0.0237 and Ti = 15.14. Figure 5 shows the obtained results, it is possible to note that the performance obtained by the SSOD-PI control system is consistent with the one expected by using the AMIGO tuning rule, thus confirming the effectiveness of the overall methodology. The proposed autotuning method has then been applied to a high-order process with transfer function 1
15
5 0
(20)
IV. S IMULATION R ESULTS
P3
20
0.2
Summarizing, the autotuning procedure consists of the following steps: • choose an initial PI parameters which guarantee the system stability; • start an experiment by discarding the values of sampled error e∗ (namely, the output of the SSOD block) that are different from −∆ and ∆; • when the limit cycle is stabilized (typically, after three ′ ′ ′ ′ limit cycles ([2]), calculate the values of I1 , I2 , I12 , t˜ , ′ and T˜ . Afterwards, set the proportional gain equal to γKp ; • when the limit cycles is stabilized, calculate the values ′′ ′′ ′′ ′′ ′′ of I1 , I2 , I12 , t˜ , and T˜ ; • determine the process parameters by means of (15)-(16) and (18); • set the controller parameters by using the desired tuning rules.
P (s) =
Process variable
and then a suitable tuning rule can be employed for the selection of the PI parameters. In [5] we have shown that, for example, the well-known SIMC [19] and AMIGO [9] tuning rules, although conceived for the time-driven case, keep their nice properties also in the event-based framework. It is worth recalling that the SIMC tuning rules are
25
Control variable
Finally, the process parameters can be trivially found from (6) as follows: K = Kp−1 κ (18) τ = ρ−1 Ti L = lTi
Fig. 5. Autotuning procedure applied to a FOPDT model (K = 82, τ = 22, and L = 3). Initial control system: solid thick line. First step of the autotuning procedure: dashed thin line. Second step of the autotuning procedure: dash-dot thick line. AMIGO retuned control system: solid thin line. Top: process variable. Bottom: control variable.
in order to analyze the accuracy of the approximated FOPDT model. In particular, an approximated model Pa (s) = 1 −1.14s has been calculated (the initial SSOD-PI 1.42s+1 e parameters have been selected as Kp = 1, Ti = 1 and ∆ = 1), then the PI controller has been retuned by using the AMIGO rules [9]. In Figure 6, it is possible to evaluate the step and frequency responses of process P (s) and its approximated FOPDT model. Note that Pa (s) is an accurate approximation of P (s) only in a frequency range around the limit cycle frequency. For this reason, in order to improve the accuracy of the model, it is advisable to repeat the estimation procedure if the retuned controlled system has a bandwidth which is significantly different from the limit cycle frequency. Finally, in order to analyze the effect of the noise on the parameters estimation, a FOPDT process with K = 2, τ = 3, and L = 2 is estimated with the proposed procedure (the initial SSOD-PI parameters have been selected as Kp = 1, Ti = 1 and ∆ = 1). The considered noise is uniformly distributed on a band N B. In Figure 7, the relationship between the ratio N∆B and the parameters estimation percentage errors are shown. It is possible to note that a ratio N∆B less than 25% causes estimations error that are less than 20%. V. C ONCLUSIONS In this paper we have presented an automatic tuning procedure for a symmetrical send-on-delta event-based PI controller. The technique is based on a suitable closed-loop experiment which can be performed during the normal process operations because it does not perturb them significantly. Other nice features of the method, as shown by simulation results, are the robustness to the measurement noise and to the initial PI tuning. This makes the overall methodology suitable to implement in industry.
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VI. ACKNOWLEDGEMENTS This work has been funded by the National Plan Project DPI2011-27818-C02-02 of the Spanish Ministry of Economy and Competitiveness and FEDER funds.
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Fig. 6. Comparison between the actual control system (solid line) and the modelled one (dashed line). First plot (from the top-left corner): process variable unit step response. Second plot: magnitude Bode plot. Third plot: Nyquist plot. Fourth plot: control variable unit step response. Fifth plot: phase Bode plot. Sixth plot: Bode plot of the sensitivity function.
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40 30 20 10 0 −10 0
Fig. 7. Parameters estimation percentages error caused by a uniform random noise.
˚ en. A simple event-based PID controller. In Proceedings of [1] K.E. Arz` 14th World Congress of IFAC, Bejining, China, 1999. ˚ om. Oscillations in systems with relay feedback. In Adaptive [2] K.J. Astr¨ Control, Filtering, and Signal Processing, volume 74. Springer-Verlag, January 1995. ˚ om and T. H¨agglund. Advanced PID Control. ISA Press, [3] K.J. Astr¨ Research Triangle Park, USA, 2006. [4] M. Beschi, S. Dormido, J. Sanchez, and A. Visioli. Characterization of symmetric send-on-delta PI controllers. Journal of Process Control, 22:1930–1945, 2012. [5] M. Beschi, S. Dormido, J. Sanchez, and A. Visioli. Tuning rules for event-based SSOD-PI controllers. In Proceedings 20th Mediterranean Conference on Control and Automation, pages 1073–1078, Barcelona, E, 2012. [6] M. Beschi, S. Dormido, J. Sanchez, and A. Visioli. Tuning of symmetric send-on-delta PI controllers. IET Control Theory and Applications, submitted, 2013. [7] M. Beschi, A. Visioli, S.Dormido, and J. Sanchez. On the presence of equilibrium points in PI control systems with send-on-delta sampling. In Proceedings of 50th IEEE International Conference on Decision and Control and European Control Conference, Orlando, USA, 2011. [8] T. Blevins. PID advances in industrial control. In Preprints IFAC Conference on Advances in PID Control, Brescia, Italy, 2012. ˚ om and T. H¨agglund. Revisiting the Ziegler-Nichols tuning [9] K.J. Astr¨ rules for PI control. Asian Journal of Control, 4(4):364–380, December 2002. [10] E. Kofman and J. Braslavsky. Level crossing sampling in feedback stabilization under data rate constraints. In Proceedings of 45th IEEE International Conference on Decision and Control, 2006. [11] M. Miskowicz. Send-on-delta: An event-based data reporting strategy. Sensors, 6:49–63, 2006. [12] A. O’Dwyer. Handbook of PI and PID Tuning Rules. Imperial College Press, London, UK, 2006. [13] P. Otanez, J. Moyne, and D. Tilbury. Using deadbands to reduce communication in networked control systems. In Proceedings of American Control Conference, Anchorage, USA, 2002. [14] V. Pawlowskyy, J. L. Guzm´an, F. Rodr´ıguez, M. Berenguel, J. Sanchez, and S. Dormido. Event-based control and wireless sensor network for greenhouse diurnal temperature control: A simulated case study. In Proc. of 13th IEEE International Conference on Emerging Technologies and Factory Automation (EFTA-2008), Hamburg, Germany, 2008. [15] M. Rabi and K. H. Johansson. Event-triggered strategies for industrial control over wireless networks. In Proceedings of 4th Annual International Conference on Wireless Internet, Maui, Hawaii, USA, 2008. [16] J. S´anchez, M.A. Guarnes, S. Dormido, and A. Visioli. Comparative study of event-based control strategies: An experimental approach on a simple tank. In Proceedings European Control Conference, Budapest, Hungary, 2009. [17] J. Sanchez, A. Visioli, and S. Dormido. A two-degree-of-freedom PI controller based on events. Journal of Process Control, 21:639–651, 2011. [18] J. Sanchez, A. Visioli, and S. Dormido. PID Control in the Third Millennium (R. Vilanova and A. Visioli (eds.)), chapter Event-based PID control, pages 495–526. Springer, 2012. [19] S. Skogestad. Probably the best simple PID tuning rules in the world. In AIChE Annual meeting, Reno, NV, USA, 2001. [20] V. Vasyutynskyy and K. Kabitzsh. Implementation of PID controller with send-on-delta sampling. In International Conference Control, 2006. [21] V. Vasyutynskyy and K. Kabitzsh. Towards comparison of deadband sampling types. In IEEE International Symposium on Industrial Electronics (ISIE 2007), June 2007. [22] C.-C. Yu. Autotuning of PID Controllers: A Relay Feedback Approach. Springer, London, UK, 1999.
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An Automatic Tuning Procedure for an Event-based PI Controller Manuel Beschi, Sebasti´an Dormido, Jos´e Sanchez, Antonio Visioli
Abstract— In this paper we propose an automatic tuning procedure for an event-based PI control scheme. In particular, the well-known relay feedback methodology for a standard PI controller is extended to a symmetrical send-on-delta PI controller. In this context, the overall event-based procedure consists of two relay feedback experiments (with a PI controller, possibly roughly tuned, already in place) in order to estimate the parameters of a first-order-plus-dead-time transfer function. Then, based on the process parameters, a tuning rule can be suitably applied. Simulation results demonstrate the effectiveness of the methodology.
I. I NTRODUCTION Event-based Proportional-Integral-Derivative (PID) controllers have been recently the subject of many investigations from academic and industrial researchers because of the need of addressing new technological issues, in particular, the introduction of wireless sensors and actuators in industrial plants [1], [20], [15], [17], [18], [8]. In this context it is essential to minimize the power consumption of the devices in order to increase the battery life time and for this reason, in addition to minimize the risk of lost data and stochastic time delays [13], [11], [14], the communications should be reduced as much as possible by keeping an acceptable level of performance. The rationale of event-based control is to send a communication between the control agents just when an event occurs, on the contrary of the standard time-driven control where the information is sent in any case at periodic time intervals. Thus, a logical condition has to be defined in order for an event to be detected. In the context of event-based PID controllers, one of the most employed condition is the so-called send-on-delta (SOD) sampling (also known as deadband sampling [21] or level crossing sampling [10]) where a variable is sent when its value (or some function of it) crosses predefined quantization levels [16]. A modification of the SOD technique, called Symmetric Send-On-Delta (SSOD), has been proposed and applied to a PI controller in [4], where its advantages with respect to the standard SOD approach have been highlighted (among them, it is worth stressing that the SSOD parameter ∆ does not influence the stability properties of the system but only M. Beschi is with the Dipartimento di Ingegneria dell’Informazione, University of Brescia, Brescia, Italy [email protected] S. Dormido and J. Sanchez are with the Departamento de Informatica y Automatica, UNED, Madrid, Spain {sdormido,jsanchez}@dia.uned.es A. Visioli is with the Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia, Brescia, Italy
[email protected] 978-1-4673-5716-6/13/$31.00 ©2013 IEEE
the precision and the number of events). In particular, two control algorithms, named SSOD-PI and PI-SSOD have been proposed (depending where the triggering function is applied in the control loop) and necessary and sufficient conditions on the controller parameters for the existence of equilibrium points without limit cycles have been determined for first-order-plus-dead-time (FOPDT) processes. Then, the tuning of the controller parameters (which is in general a more challenging task with respect to the time-driven case because the control is asynchronous and there are more parameters to select) has been addressed in [5], [6]. It has been highlighted that tuning rules devised for standard time-driven PI controllers [3], [12] can be effectively applied also to SSOD-PI and PI-SSOD controllers. In any case, it is recognized that the automatic tuning feature is very desirable in industry and it should be available also for event-based PI controllers. In this paper we propose therefore an autotuning procedure for the SSOD-PI controller (standard techniques can be employed for the PI-SSOD controlled system, as it will be clarified in Section III). In particular, for the reason explained above, we will focus on the technique for the estimation of the parameters of a first-order-plus-dead-time (FOPDT) system which is known to model satisfactorily a wide range of self-regulating overdamped processes. For this purpose, an extension of the well-known (closed-loop) relay feedback methodology [3], [22] is presented. It has been specifically devised for the event-based control architecture and it consists of performing two 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, thus making the overall approach inherently robust to measurement noise. Once the process parameters are estimated, a suitable tuning rule (such as the SIMC [19] or AMIGO [9] one) can be employed, depending on the required performance (i.e., by taking into account the robustness and control effort issues).
II. SSOD C ONTROL A RCHITECTURE A. Generalities In event-based control strategies, the controller can be divided into four logical blocks: the sensor unit, the control unit, the actuator unit and a governor (for short SU, CU, AU and G, respectively), as shown in Figure 1. The units and their tasks can be described as follows: • The Sensor Unit is composed of the sensor and its onboard intelligence. Its task is to measure the process
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Governor
G
r
D SU
SU
CU
+
AU
e
SSOD
CU and AU e*
ZOH
−
Plant
C(s)
u +
+
y P(s)
Fig. 2. Control scheme of the SSOD-PI controlled system. The dashed arrows indicate data sending via the communication medium.
Fig. 1. Scheme of a generic event based control strategies. The dashed arrows indicate the possibility of an event-triggered data transmission.
output and to calculate the error between the measured signal and a constant set-point value received from the governor. • The Control Unit implements the control algorithm, which determines the control action by taking into account the last received sampled error and sends it to the Actuator Unit. • The Actuator Unit receives the control action signal from the Control Unit and applies it to the actuator. • The Governor, which, in practice, can be implemented together with one of the previous two blocks, receives the desired set-point value from a user interface or from a higher hierarchical controller, and sends it to the Sensor Unit. These blocks can be implemented in a unique machine or in two or more physical entities. In this last case, the data has to be sent from one to each others on a network. It is clear that the communication between two entities implies more efforts than the data exchanging into a single machine, especially when they are battery-powered. For this reason, it is recommended to use event-triggered data exchanging for all the signals that are sent between two machines and normal time-driven sampling for the data which are elaborated by an unique machine. In this paper, we consider two special cases where two of the three control agents (namely, SU, CU and AU) are placed in a physical entity and the other one is located in another machine. We do not address the case where SU and AU are in the same machine, because in this case it is sufficient to implement also the CU task in this machine in order to obtain a standard controller. The remaining cases are: SU separated from CU and AU (which are in the same machine) and AU separated from SU and CU (which are in the same machine). In both situations, only an eventtriggered data exchanging in the control loop is required. The communications between the governor and the other units (considering only the transmissions concerning the control aspects) are done only when the set-point signal changes. B. Symmetric send-on-delta triggering As explained before, we consider two different architectures, which present, as common characteristic, the presence of only two event-triggered data exchanging. The first one is the set-point value sending, which is not a complex task from a control point of view, the second one is implemented with
the symmetryc send-on-delta (SSOD) sampling technique (see [4]). This can be seen as a special case of the send-on-delta sampling method (see [11], [20]), which can be seen also as a generalization of a relay with hysteresis. Denote as v(t) the input signal to the sampling block and as v ∗ (t) the sampled output signal, which can assume only values multiple of a predefined threshold ∆ multiplied by a gain β > 0, namely v ∗ (t) = j∆β with j ∈ Z. The sampled signal changes its value to the upper quantization level when the input signal v(t) increases more than ∆, or to the lower quantization level when v(t) decreases more than ∆. This behavior can be mathematically described as: v∗ (t) = ssod(v(t); ∆, β) = v(t) ∗ − (i + 1)∆β if ∆ > (i + 1) ∧ v (t ) = i∆β v(t) i∆β if ∆ ∈ [(i − 1), (i + 1)] ∧ v ∗ (t− ) = i∆β ∗ − (i − 1)∆β if v(t) ∆ < (i − 1) ∧ v (t ) = i∆β (1)
C. PI controller
The controller used in this work is a (discretized version of) continuous-time PI controller, namely: 1 (2) C(s) = Kp 1 + Ti s where Kp is the proportional gain and Ti is the integral time constant. The derivative action is not employed (as it often happens in industrial settings), because its implementation is very critical with a variable, and possibly long, sampling period. Remark 1. Because the controller is implemented in a single machine, it is possible to implement a standard anti-windup technique without additional efforts. D. SSOD-PI controller In the first architecture, shown in Figure 2, the SU is located in a machine while CU and AU are placed in another physical entity (for example, the communication between the two components could be wireless). We call this architecture SSOD-PI controller, because the SSOD block is placed before the PI controller in the control loop. Note that the controller computes the control action at a regular sampling rate by taking into account the last received sampled error. This architecture is very interesting because, in a networked control system, the sensor unit can be powered using a battery (thus, a reduction of the communications can increase
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self-regulating industrial plants. Thus, we consider the following transfer function
Governor r + e −
D SU and CU u C(s) SSOD
AU u*
+ ZOH
+
K e−sL (4) τs + 1 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. The system where the FOPDT process is controlled by the relay with hysteresis can be rewritten in the state-space form as follows x˙ 1 (t) = − τ1 x1 (t) + K τ u(t) 1 ∗ x ˙ (t) = e (t) 2 Ti (5) u(t) = Kp (e∗ (t) + x2 (t)) y(t) = x1 (t − L) ∗ e (t) = hystrel(r − y(t), ∆, β) P (s) =
y P(s)
Fig. 3. Control scheme of the PI-SSOD controlled system. The dashed arrows indicate data sending via communication medium.
the battery life). Conversely, the AU normally requires an external power supply, therefore the power consumption reduction is not a critical issue in this control agent. E. PI-SSOD controller In the second architecture, shown in Figure 3, the SU and the CU are located in the same machine, while the AU is placed in another entity. This solution is called for short PISSOD controller because the controller is before the SSOD block in the control loop. Note that the AU holds the last received control action value until the next data exchanging. III. T HE AUTOMATIC TUNING PROCEDURE In the PI-SSOD controlled system, it is possible to implement standard automatic tuning techniques because the sensor and control units (which are located in the same physical entity) have information on the process variable and on the control variable. On the contrary, in the SSODPI controlled system there is not a unit which has enough information to use standard methods. For this reason, it is necessary to modify the sensor unit in order to send an additional indication to the control unit, which is attached to the event message. The proposed automatic tuning technique 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. In order to excite the system during the autotuning experiment, the CU discards the values of sampled error e∗ (namely, the output of the SSOD block) different to −∆ and ∆. In this way, the relationship between e(t) and e∗ (t) is equal to a relay with hysteresis map denoted as hystrel(·), namely e∗ (t) = hystrel(e(t); ∆, β) if e(t) > ∆ and e∗ (t− ) = −∆β ∆β (3) = −∆β if e(t) < ∆ and e∗ (t− ) = −∆β . ∗ − e (t ) if otherwise
Remark 2. The parameter β can be considered as a 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 a FOPDT transfer function (4), which is wellknown to be capable to model accurately many overdamped
In order to reduce the problem complexity, the equation system (5) is normalized with the following variable change: r y(t) − r t r˜ = , y˜(t) = , t˜ = , Ti ∆ ∆ ∗ e(t) e (t) iT x(t), ˙ e˜(t) = , e˜∗ (t) = , x ˜˙ (t˜) = T∆ ∆ ∆ L l= , κ = KKp , ρ = Tτi , Ti T r x ˜(t) = x(t) − κ−1 r ∆
(6)
where the matrix T is equal to: 0 1 T= κ −1 Thus, we obtain ˜˙ (t˜) = A˜ x(t˜) + B˜ e∗ (t˜) x y˜(t˜) = C˜ x(t˜) ∗ ˜ e˜ (t) = hystrel(−˜ y (t˜), 1, 1)
where:
A=
0 0
B=
1 ρ−1
κ
C=
0 −ρ
κ
(7)
System (7) surely admits a symmetrical limit cycle with a period T˜, because e˜(t˜) cannot assume the null value and hystrel(·) is symmetrical with respect to the origin [7]. The trends of e˜∗ (t˜) and x ˜(t˜) are 1 if t˜ ∈ [0, T˜/2) ∗ ˜ e˜ (t) = (8) −1 if t˜ ∈ [T˜/2, T˜) ( ˜ eAt x˜0 + F (t˜) if t˜ ∈ [0, T˜/2) x ˜(t˜) = ˜ ˜ −eA(t−T /2) x ˜0 − F (t˜ − T˜/2) if t˜ ∈ [T˜/2, T˜) (9) where F (t˜) is the unit step response, defined as follows: # " t˜ F (t˜) = 1−ρ 1 − e−ρt˜ ρ
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˜ + CB˜ ˜ on the time interval t˜2 CA˜ x(t) u(t) Z t˜b x ˜˙ (t˜)dt˜ = C x˜(t˜b ) − x˜(t˜a ) = C
2 1 I2
0 Cx ˜(t˜)
I1
y˜(t˜)
v˜∗ (t˜)
−1
CA
−2 0
t˜a
t˜b
t˜a
l t˜2
Z
t˜1 t˜b T˜/2 Normalized time t˜
x ˜(t˜)dt˜ + CB
Z
t˜a t˜b
t˜a
v˜(t˜)dt˜ = 1
and it results t˜a + T˜/2 t˜b + T˜/2
T˜
Fig. 4. Evolution of v˜∗ (t˜) (dash-dot line), y˜(t˜) (dashed line) and C˜ x(t˜) (solid line) during an oscillation.
Z
t˜b
t˜a
1 . x ˜2 (t˜)dt˜ = t˜b − t˜a − κρ
Thus, by using (11), I2 is equal to:
I2 = κt˜2 (1 − t˜2 /2 − l + T˜/4) − ρ−1 and x˜0 is the initial value of the state, which can be calculated by virtue of the symmetry of the limit cycle (namely, x˜(T˜/2) = −˜ x0 ) as: x ˜0
−1 ˜ = − I + eAT /2 F (T˜/2) ˜ −T /4 −1 = ρ−1 ˜ 2 e1+ρT /2 −1 ρ
(10)
In order to estimate the process transfer function, two time instants are defined: t˜a is the first time instant when C˜ x(t˜a ) = 0 and t˜b is the first time instant when C˜ x(t˜b ) = 1, as shown in Figure 4. Because the time delay l is unknown, also the time instants t˜a and t˜b are unknown. However, from y˜(t˜) it is possible to measure the time interval t˜2 between t˜a and t˜b and the time interval t˜1 between t˜b and t˜a + T˜/2. Moreover, it is possible to relate these quantities with the period T˜ and with the time delay l by the following equations: t˜b + l = T˜/2 t˜1 = T˜/2 + t˜a − t˜b
(11)
t˜2 = t˜b − t˜a The area of C˜ x(t˜) defined on the time interval t˜1 is deR T˜ /2+t˜a R t˜ noted as I1 = |C t˜b x ˜(t˜)dt˜|, while I2 = |C t˜ab x(t˜)dt˜| is the integral defined on the time interval t˜2 and I12 = R T˜/2+t˜a I1 + I2 = |C t˜a x˜(t˜)dt˜|. The integral I2 can be calculated by finding the quantities R t˜ R t˜b x ˜ (t˜)dt˜ and t˜ab x˜2 (t˜)dt˜ separately. The first one can be t˜a 1 easily found by noting that 1 − 4 T˜ + t˜ if t˜ ∈ [0, T˜/2) ˜ x ˜1 (t) = 3 ˜ ˜ ˜ ˜ ˜ 4 T − t if t ∈ [T /2, T )
In the same way, it is possible to calculate the quantity I12 , which results to be equal to κ I12 = 4l − T˜ + 4t˜2 − 4lt˜2 + T˜t˜2 − 2l2 − 2t˜22 + lT˜ . 2 (12) It is important to note that I12 depends only on l and κ. The integral I1 can be calculated as the difference between I12 and I2 , and it results: ˜ ˜ ˜ I1 = ρ−1 + κ 1 − l + T4 − t22 t˜2 + (2 − l)l + (l − 1) T2 (13) Remark 3. The integrals I1 , I2 , and I12 are related to the denormalized integrals by the following equation: Z ˜ι Z ι 1 C˜ x(t˜)dt˜ = (Cx(t) − r) dt Ti ∆ ς ς˜ where Ti ς = ς˜ and Ti ι = ˜ι. In order to find the process parameters, two limit cycles with different proportional gains of the PI have to be obtained. In the second one, the proportional gain is modified by multiplying it by a quantity γ 6= 1. From a practical point of view, a value γ < 1 is recommended in order to surely preserve the system stability (for example, a convenient ′ default value γ = 0.5 can be selected). The quantities I1 , ′ ′ ′ ′ I2 , I12 , t˜ , and T˜ represents the obtained values in the first ′′ ′′ ′′ ′′ ′′ case, while I1 , I2 , I12 , t˜ , and T˜ are the obtained values in the second oscillation. Thus, we have ′ ′ ′ ′ ′ ′ ′ ′ κ 4l − T˜ + 4t˜2 − 4lt˜2 + T˜ t˜2 − 2l2 − 2t˜22 + lT˜ I12 = 2 ′′ ′′ ′′ ′′ ′′ ′′ γκ I12 = 2 4(l + t˜2 − lt˜2 ) + T˜ (l + t˜2 − 1) − 2l2 − 2t˜2 2 (14) The quantity l and κ can be therefore found by solving the following equations, discarding the inadmissible solutions: ′′
[(4t˜2 −2t˜2 2 −T˜
′′
′′
′′
′
[(4−4t˜2 +T˜
and therefore Z t˜b 1 ˜ ˜ ˜ x ˜1 (t˜)dt˜ = tb − ta 2tb + 2t˜a − T˜ , 4 t˜a
′′
′′ ′′ ′ ′ ′ ′ ′ ′ ′′ 2 +T˜ t˜2 )I˜12 γ+(2t˜22 −4t˜2 +T˜ −T˜ t˜2 )I˜12 ] l + ′
′
′′
)I˜12 γ+4(t˜2 −4−T˜ )I˜12 ] l
′′
′
+ [2I˜12 −2I˜12 γ ] = 0
2I˜12 κ= ′ ′ ′ ′ 4l + 4t˜2 − 4lt˜2 − 2l2 + T˜ ′ (l + t˜2 − 1) − 2t˜22
(15)
′
˜ = while the second one can be found by integrating Cx ˜˙ (t) 7440
ρ−1 = κt˜2 (1 − t˜2 /2 − l + T˜′ /4) − I2 ′
′
′
(16) (17)
0.5τ Kp = Ti = min{τ, 8L} KL while the AMIGO ones are Lτ τ Kp = 0.15 K + 0.35 − (L+τ )2 KL Ti = 0.35L +
2
6.7Lτ τ 2 +2Lτ +10L2
(19)
(s + 1)
j=1 (1
+ 0.7s)j
10
120
240 Time
360
480
120
240 Time
360
480
1 0.8 0.6 0.4
0 0
In order to clarify better the proposed algorithm, it has been applied to a FOPDT model with K = 82, τ = 22, and L = 3, initially controlled by a SSOD-PI controller with Kp = 0.05, Ti = 7, and ∆ = 0.2. The estimated parameters result equal to Kest = 81.93, τest = 21.98, and Lest = 3, which are used to obtain the new controller parameters by applying the AMIGO tuning rule (20). It results Kp = 0.0237 and Ti = 15.14. Figure 5 shows the obtained results, it is possible to note that the performance obtained by the SSOD-PI control system is consistent with the one expected by using the AMIGO tuning rule, thus confirming the effectiveness of the overall methodology. The proposed autotuning method has then been applied to a high-order process with transfer function 1
15
5 0
(20)
IV. S IMULATION R ESULTS
P3
20
0.2
Summarizing, the autotuning procedure consists of the following steps: • choose an initial PI parameters which guarantee the system stability; • start an experiment by discarding the values of sampled error e∗ (namely, the output of the SSOD block) that are different from −∆ and ∆; • when the limit cycle is stabilized (typically, after three ′ ′ ′ ′ limit cycles ([2]), calculate the values of I1 , I2 , I12 , t˜ , ′ and T˜ . Afterwards, set the proportional gain equal to γKp ; • when the limit cycles is stabilized, calculate the values ′′ ′′ ′′ ′′ ′′ of I1 , I2 , I12 , t˜ , and T˜ ; • determine the process parameters by means of (15)-(16) and (18); • set the controller parameters by using the desired tuning rules.
P (s) =
Process variable
and then a suitable tuning rule can be employed for the selection of the PI parameters. In [5] we have shown that, for example, the well-known SIMC [19] and AMIGO [9] tuning rules, although conceived for the time-driven case, keep their nice properties also in the event-based framework. It is worth recalling that the SIMC tuning rules are
25
Control variable
Finally, the process parameters can be trivially found from (6) as follows: K = Kp−1 κ (18) τ = ρ−1 Ti L = lTi
Fig. 5. Autotuning procedure applied to a FOPDT model (K = 82, τ = 22, and L = 3). Initial control system: solid thick line. First step of the autotuning procedure: dashed thin line. Second step of the autotuning procedure: dash-dot thick line. AMIGO retuned control system: solid thin line. Top: process variable. Bottom: control variable.
in order to analyze the accuracy of the approximated FOPDT model. In particular, an approximated model Pa (s) = 1 −1.14s has been calculated (the initial SSOD-PI 1.42s+1 e parameters have been selected as Kp = 1, Ti = 1 and ∆ = 1), then the PI controller has been retuned by using the AMIGO rules [9]. In Figure 6, it is possible to evaluate the step and frequency responses of process P (s) and its approximated FOPDT model. Note that Pa (s) is an accurate approximation of P (s) only in a frequency range around the limit cycle frequency. For this reason, in order to improve the accuracy of the model, it is advisable to repeat the estimation procedure if the retuned controlled system has a bandwidth which is significantly different from the limit cycle frequency. Finally, in order to analyze the effect of the noise on the parameters estimation, a FOPDT process with K = 2, τ = 3, and L = 2 is estimated with the proposed procedure (the initial SSOD-PI parameters have been selected as Kp = 1, Ti = 1 and ∆ = 1). The considered noise is uniformly distributed on a band N B. In Figure 7, the relationship between the ratio N∆B and the parameters estimation percentage errors are shown. It is possible to note that a ratio N∆B less than 25% causes estimations error that are less than 20%. V. C ONCLUSIONS In this paper we have presented an automatic tuning procedure for a symmetrical send-on-delta event-based PI controller. The technique is based on a suitable closed-loop experiment which can be performed during the normal process operations because it does not perturb them significantly. Other nice features of the method, as shown by simulation results, are the robustness to the measurement noise and to the initial PI tuning. This makes the overall methodology suitable to implement in industry.
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0
0
VI. ACKNOWLEDGEMENTS This work has been funded by the National Plan Project DPI2011-27818-C02-02 of the Spanish Ministry of Economy and Competitiveness and FEDER funds.
0.6
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0.9
15 −90 Phase (deg)
0.7 0.6 0.5 0.4
Sensitivity function (dB)
0.8 Control variable
−0.5 0 Real axis
−180
−270
0.3
10 5 0 −5 −10 −15
0.2 0
−360 10
20
30
Time
−2
−1
0
1
10 10 10 10 Angular frequency (rad/s)
−20
−2
−1
0
1
10 10 10 10 Angular frequency (rad/s)
Time delay L estimation error (%)
Time constant τ estimation error (%)
Process gain K estimation error (%)
Fig. 6. Comparison between the actual control system (solid line) and the modelled one (dashed line). First plot (from the top-left corner): process variable unit step response. Second plot: magnitude Bode plot. Third plot: Nyquist plot. Fourth plot: control variable unit step response. Fifth plot: phase Bode plot. Sixth plot: Bode plot of the sensitivity function.
5 0 −5 −10 −15 −20 −25 −30 0
0.25
0.5 Ratio NB ∆
0.75
1
0.25
0.5 Ratio NB ∆
0.75
1
0.25
0.5 Ratio NB ∆
0.75
1
10 0 −10 −20 −30 −40 −50 −60 0
40 30 20 10 0 −10 0
Fig. 7. Parameters estimation percentages error caused by a uniform random noise.
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