Model Reference PI Controller Tuning for Second Order Inverse
Response and Dead Time Processes
J.A. Martinez†, O. Arrieta†, R. Vilanova?, J.D. Rojas†, L. Marin†, M. Barbu?,§
†Instituto de Investigaciones en Ingeniería, Facultad de Ingeniería,
Universidad de Costa Rica, 11501-2060 San José, Costa Rica.
?Departament de Telecomunicació i d’Enginyeria de Sistemes
Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.
§Department of Automatic Control and Electrical Engineering,
"Dunarea de Jos" University of Galati, 800008 Galati, Romania.
Abstract—In this paper, a One-Degree-of-Freedom PI con- (2DoF) PID controller can be implemented, as it includes a
troller is optimized using the model reference tuning approach set-point weight factor β that can improve the servo-control
for a Second Order Inverse Response and Dead Time Process performance when the controller is designed first to optimize
operating as a servo control. In addition, a graphic user interface
tool that computes the PI optimized controller parameters is pre- the regulatory-control behavior.
sented, also showing the response of the control system operating
as a servo-control (the optimized one) and the associated response In order to obtain the PID parameters (tuning), currently
for the regulatory-control case. there are many techniques in the literature [8]. Most of them
take into account a design parameter, such as the settling time,
I. INTRODUCTION the percent overshoot, the rising time and other performances
Nowadays the most used and implemented feedback indicators such as the integral indexes. But other aspects of
controller in the industry is the PID (Proportional - Integral the control-loop performance such as the system robustness,
Derivative) controller, as it provides a satisfactory performance the control output evolution or the controller fragility must be
in most cases [1]. The design of these type of controllers is also taken into account when tuning the controller, in order
usually done to obtain a good disturbance rejection, as the set- to guarantee an adequate performance of the control-loop
point in most industrial applications is not modified frequently. in many circumstances [1]. When these specifications have
to be achieved simultaneously, it can become a challenging
When designing the controller two main approaches can issue in both 1DoF and 2DoF PID controllers. In this case,
be followed, designing the controller when the set-point optimization techniques have to be used in order to find
tracking is the main objective is known as servo-control, and the PID parameters that fulfill two or more specifications
when a good disturbance rejection is needed, the controller simultaneously and, in some cases, when the control-loop
can be tuned as a regulatory-control [2], [3]. To guarantee a have to operate in both servo and regulatory control.
stable and successful system operation, the controller must
be tuned according to these operation modes and according In this paper, a One-Degree-of-Freedom (1DoF) PI
to controlled process [1]. For this objective, several tuning controller is optimized for servo-control, taking into account
methods exist in the literature, providing the optimal PID the robustness of the control-loop defined with the maximum
controller parameters according to the industrial process [4]. sensibility function Ms and the performance of the system,
defined via a similar model reference approach proposed
Although in industrial processes the emphasis needed in by [9]. The novelty of this proposed approach lies in the
most cases is the regulatory-control, neglecting the set-point extension of the current Model Reference Robust Tuning
tracking can be counterproductive in some cases. In this case, (MoReRT) [10] to a Second Order Inverse Response and
when a good performance is needed in both operating modes, Dead Time Process (previous one was without dead-time).
a well-known problem occurs, as there is a compromise Even the addition of the parameter for the delay can appear
between the performance of a servo and a regulatory control simple, it gives a lot of more information about the optimal
as the usual one degree of freedom (1DoF) PID controller points that is not trivial to manage. This work is a preliminary
can only be optimized to one mode of operation, and this stage that must be continued with the tuning of 1DoF PI
will degrade the performance of the other operating mode controller for regulatory-control and 2DoF PI controllers.
[5]–[7]. To solve this issue, a Two-Degrees-of-Freedom
The rest of the paper is organized as follows: the
O. Arrieta email: Orlando.Arrieta@ucr.ac.cr control system framework is presented in Section II; the
978-1-5090-1314-2/16/$31.00 ©c 2016 IEEE controller design methodology is described in Section III;
P (s)
Myd(s) = (5)
1 + C(s)P (s)
In the formulation of the model reference robust approach,
a reference model has to be stated, in order to use it in the
optimization [9]. First it is necessary to define the model
Figure 1. Closed-loop control system for the process that in this case is a Second Order Inverse
Response and Dead Time shown in (6), where K is the
gain, T the main time constant, a the ratio of the two
the optimization results and the robustness obtained for the main poles time constants (0.1 ≤ a ≤ 1.0), b the relative
inverse response model, along with the developed interface position of the right-half plane zero and L the dead-time. The
are shown in Section IV; and finally some conclusions and parameter set of this model is denoted by θp = {K,T, a, b, L}.
future work are outlined in Section V.
K(−bTs+ 1)e−Ls
II. PROBLEM FORMULATION P (s) = , (6)(Ts+ 1)(aTs+ 1)
The structure of an ideal PID (Proportional-Integral- The reference model required for the process model (6)
Derivative) controller has three parameters that are the is the result of the modification performed to the equations
integral time Ti, the derivative time Td and the controller proposed in [10] by adding the corresponding dead-time,
gain Kc. having as result the target transfer function shown in (7) for
the desired control system response to a set-point change,
In the present work the PI controller is utilized, thus Td and in (8) for the desired control system response to a
is zero. The output expression for the 1DoF PI controller is load-disturbance change.
defined through equation (1) in the time domain, being e(t)
the error signal.
{ ∫ } t (−bTs+ 1)e
−Lss
yr(s) = r(s), (7)(τcTs+ 1)(aτcTs+ 1)
1 t
u(t) = Kc e(t) + e(τ)dτ , (1)
Ti 0 t (Ti/Kc)(−bTs+ 1)e−Lssyd(s) = d(s), (8)
One of the advantages of using a PI controller is the (τcTs+ 1)
2(aτcTs+ 1)
elimination of the steady-state error. The controller equation In (7) and (8), the dimensionless design parameter τc
in the frequency domain is obtained by taking the Laplace represents the relative velocity time constant of closed-loop
Transform of (1) as shown in (2). system. The global control system output target yt can be
{ } defined as
1
u(s) = Kc 1 + e(s), (2)
Tis y
t(s) =Myr(s)r(s) +Myd(s)d(s) = y
t(s) + ytr d(s), (9)
The closed-loop control system is shown in Fig. 1. In this, A comparison of closed-loop system output (3) with de-
P (s) is the controlled process, assumed as a second order sired output of the reference model (9) is performed via
inverse response and dead time model and C(s) is the 1DoF optimization, to guarantee that the difference between them
PI controller. In this system, r(s) is the set-point, u(s) the is minimum. In this case the optimization will provide the
controller output signal, d(s) the load-disturbance and y(s) optimal normalized PI controller parameters, as described in
the process controlled variable (system output). the next section.
The output of the closed loop system y(s) is given by III. CONTROLLER DESIGN AND TUNING
As was discussed on Section II, the general reference
(3) model is described by (9), however, as this paper focuses onlyy(s) =Myr(s)r(s) +Myd(s)d(s) = yr(s) + yd(s) (first stage) on the servo-control operation, only (7) will be
where M (s) and M (s) are the servo-control (set-point used in the optimization, that will compare (7) to the responseyr yd
tracking) and regulatory-control (disturbance rejection) closed- (4) (in the time domain), by means of a Cost Functional that
loop transfer functions, that are given by computes the difference between the responses of both models.
C(s)P (s) According to the work presented in [10], the expression that
Myr(s) = (4)
1 + C(s)P (s) represents the cost functional for the servo-control as
∫ ∞ [ ]
t 2Jr = yr(τc, θp, t)− yr(θc, θp, t) dt, (10)
0
where ytr is the step response of the servo-control closed-loop
target transfer function, and yr is the step response of the
servo-control closed-loop transfer function. By optimizing
(11) the optimal PI controller parameters θc = {Kc, Ti} are
obtained.
{ ( )}
JoT = min Jr τc, θc, θp , (11)
θc
The control system robustness Ms is defined with the
sensibility function on (12).
1
Ms = max |S(jω)| = max , (12)
ω ω |1 + C(jω)P (jω)|
Figure 3. Controlled process normalized gain con a = 0.5 y b = 2
The design looks to accomplish the robustness levels
defined at the values of Ms ∈ {1.4, 1.6, 1.8, 2.0}].
IV. OPTIMIZATION RESULTS AND DEVELOPED INTERFACE
The optimal parameters can be achieving performing the
optimization problem using MATLAB ©R . In Figs. 2 and 3 are presented the variation of the controller
parameters, from the results provided by the optimization
The comparison between the real a reference model has to procedure, for the all cases of robustness values.
be close to zero, but that is just an ideal case, so this value
has a tolerance of 1x10−6, so the process will optimized the In addition, taking into account that one of the goals of
controllers parameters until the cost functional is 1x10−6 or this approach is to satisfy a certain robustness level (from the
less. value of Ms), in Figs. 4, 5 and 6 are shown some cases of
the resulting values that can be achieved with the proposal,
The resulting parameters from the optimization are κ being these very close to the selected ones as desired values.p
(the normalized controller gain) and τ (the normalized Therefore, the accomplishment of the robustness for thei
controller integral time). The results of the optimizations of control system is good enough.
these parameters have been collected for a specific case and
showing in the Figs. 2 and 3.
Figure 4. Accomplishment for the target robustness for the control system
with a = 0.5
Figure 2. Normalized Integral Time with a = 0.5 y b = 2 The corresponding closed-loop responses to a step change
in the set-point for the model (6) with K = 2.0, T = 1.0,
Taking into account that to achieve the PI controller
parameters, it is necessary to variate the five parameters the
model (6), the generated information of the optimal point
must be huge and difficult to fit in a tuning equation rules.
Therefore, it was developed a graphic interface tool in order
to easy compute the controller parameters.
Fig. 8 shows the main screen of the interface, where the
user should enter the information of the model, to achieve
the PI parameters.
As an example in Fig. 9 there is entered a model with
K = 3.0, T = 5.0, a = 0.5, b = 1.2 and L = 2.0, computing
the controller parameters as Kc = 0.116 and Ti = 6.779.
Also in Figs. 10 and 11 are presented the control system
responses for servo-control and regulatory-control operation,
Figure 5. Accomplishment for the target robustness for the control system respectively.
with a = 0.7
Figure 8. Main screen of the graphic interface tool
Figure 6. Accomplishment for the target robustness for the control system
with a = 1.0
Figure 9. Practical example of the use of the interface tool
a = 0.5, b = 1.3 and L = 0.6, are shown in Fig. 7, for servo
control operation.
Figure 10. Servo-control response
Figure 7. Control System Responses in Servo-Control
[6] O. Arrieta and R. Vilanova, “Simple PID tuning rules with guaranteed
Ms robustness achievement,” in 18th IFAC World Congress, August 28-
September 2, Milano-Italy, 2011.
[7] ——, “Simple Servo/Regulation Proportional-Integral-Derivative (PID)
Tuning Rules for Arbitrary Ms-Based Robustness Achievement,” Indus-
trial & Engineering Chemistry Research, 2012, dOI: 10.1021/ie201655c.
[8] A. O’ Dwyer, Handbook of PI and PID Controller Tuning Rules.
McGraw-Hill, Londres, Inglaterra., 2006.
[9] V. Alfaro Ruiz, Model-reference robust tuning of 2DoF PI controllers
for first- and second-order plus dead-time controlled processes. Journal
of Process Control, p 358-374., 2012.
[10] V. Alfaro and R. Vilanova, Two-Degree-of-Freedom Proportional In-
Figure 11. Regulatory-control response tegral Control of Inverse Response Second-Order Processes. 16th
Intenational Conference on System Theory, Control and Computing,
2012.
V. CONCLUSIONS AND FUTURE WORK
As an extension of the model reference tuning presented in
[9], [10], it was proposed a design for second order inverse
response models with also a dead-time, to achieve the 1DoF
PI controller parameters for servo-control operation. The
tuning guarantees the accomplishment of a certain desired
robustness value into the option Ms ∈ {1.4, 1.6, 1.8, 2.0}.
Taking into account the amount of information of the
optimal tuning points, resulting from the optimization
procedure, a graphical user interface tool was developed
with the aim to facilitate the computation of the controller
parameters.
This study is just a stage of a more general work, that is
in progress, and must include the develop of designs for also
1DoF PI controller in regulatory-control mode and for 2DoF
PI controllers. Then it could be looking for tunings for PID
controllers. In addition, the graphical interface tool should
be improved including more quantitative measures for the
performance and robustness of the control system.
ACKNOWLEDGMENTS
The financial support from the University of Costa Rica,
under the grants 731-B4-213 and 322-B4-218, is greatly ap-
preciated. Also, this work has received financial support from
the Spanish CICYT program under grant DPI2013-47825-C3-
1-R.
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