Using Simulated Annealing for Optimal Tuning of a
PID Controller for Time-Delay Systems. An Application
to a High-Performance Drilling Process
Rodolfo E. Haber1,2,*, Rodolfo Haber-Haber3, Raúl M. del Toro1, and José R. Alique1
1
Instituto de Automática Industrial (CSIC)
km. 22,800 N-III, La Poveda. 28500
Madrid, Spain
{rhaber,jralique}@iai.csic.es
2
Escuela Politécnica Superior
Ciudad Universitaria de Cantoblanco
Calle Francisco Tomás y Valiente, 11
28049 – Madrid, Spain
Rodolfo.Haber@uam.es
3
Departamento de Control Automático
FIE. Universidad de Oriente
Ave. Américas s/n. 90400
Santiago de Cuba, Cuba
rhaber@fie.uo.edu.cu
Abstract. This paper shows a strategy based on simulated annealing for the
optimal tuning of a PID controller to deal with time-varying delay. The main
goal is to minimize the integral time absolute error (ITAE) performance index
and the overshoot for a drilling-force control system. The proposed strategy is
compared with other classic tuning rules (the Ziegler-Nichols and Cohen-Coon
tuning formulas). Other tuning laws derived from genetic algorithms and the
Simplex search algorithm for unconstrained optimization are also included in
the comparative study. The results demonstrate that simulated annealing
provides an optimal tuning of the PID controller, which means better transient
response (less overshoot) and less ITAE than with other methods.
Keywords: simulated annealing, time-delay systems, high-performance drilling.
1 Introduction
Simulated Annealing (SA) is a probabilistic hill-climbing technique that is based on
the annealing/cooling process of metals. It is basically a free-gradient method based
on a simple criterion that searches the problem space by piecewise perturbations of
the estimates of the parameters that are being optimized. The Metropolis algorithm is
a well-known method used to accept/reject the perturbed configuration [1]. SA has
*
Corresponding author.
F. Sandoval et al. (Eds.): IWANN 2007, LNCS 4507, pp. 1155–1162, 2007.
© Springer-Verlag Berlin Heidelberg 2007
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R.E. Haber et al.
been used widely as an optimization technique in many fields, from reinforced
composite structures to submarine maneuvering systems [2,3].
In this paper, simulated annealing is used for the optimal tuning of the parameters
of a PID controller for a network-based control system. The main goal is to derive
controller parameters that minimize the integral time absolute error (ITAE)
performance index. The SA algorithm starts with a set of initial proportional, integral
and derivative gain parameters KPID= [Pi Ii Di] and evaluates the ITAE performance
index. The KPID controller parameters are then perturbed to generate another set,
KPIDnew, and the ITAE performance index is again evaluated. The
acceptance/rejection criterion is based on the Metropolis algorithm. This procedure is
repeated according to the annealing schedule.
A PID controller is selected for a network-based application because PID
controllers are easy and intuitive to tune as well as used extensively in the industry
[4]. Nevertheless, the crucial issue is how to tune a PID controller to deal with the
time delay and provide adequate closed-loop performance. The unsolved problem is
that the network induces a varying time delay into the control loop, and that delay has
to be taken into account in control system design and optimization.
A high-performance drilling process is selected as the case study in improving
efficiency in a production environment through a cutting-force control system. This
paper addresses the design and implementation of a PID controller for cutting-force
regulation in a network-based application. The major issue to be dealt with is the
design and implementation of a PID using the computerized numerical control (CNC)
machine tool’s own resources and a fieldbus. The control algorithm is connected to
the process through a multipoint interface (MPI) bus, a proprietary programming
interface port for peer-to-peer communications that resembles the PROFIBUS
protocol. The output (i.e., force) signal is measured from a dynamometer, and the
control signal (i.e., feed command) is transmitted through the MPI, so a networkinduced delay is unavoidable.
This paper is organized as follows: Section II presents the model of a highperformance drilling (HPD) process; Section III describes the design of a PID
controller to optimize the high-performance drilling process; Section IV addresses the
implementation of the PID and connection to the CNC machine tool through a
fieldbus; Section V reviews the experimental results and explores some of the
comparative studies; finally, Section VI contains a number of conclusions.
2 Dynamic Model of a High-Performance Drilling Process
The modeling of a high-performance drilling process includes the modeling of the
feed drive system, the spindle system and the cutting process. In this paper, the
overall plant model is obtained by experimental identification using different stepshaped disturbances in the command feed. The drilling force, F, is proportional to the
machining feed, and the corresponding gain varies according to the workpiece and
drill diameter.
Using SA for Optimal Tuning of a PID Controller for Time-Delay Systems
1157
The overall system of the feed drive, cutting process and dynamometric platform
was modeled as a third-order system, and the experimental identification procedure
yielded the transfer function as:
GP ( s ) =
F (s)
f (s)
=
1958
s + 17.89 ⋅ s + 103.3 ⋅ s + 190.8
3
(1)
2
where s is the Laplace operator, f is the command feed, and F is the cutting force.
The model does have certain limits in representing the complexity and uncertainty
of the drilling process. However, it provides a rough description of the process
behavior that is essential for designing a network-based PID control system.
3 Network-Based PID Control of Cutting Force
This section presents the design of a PID controller to regulate cutting force in a
network-based application. PID controllers are widely used in industry to deal mainly
with first- and second-order dynamic systems [5]. Additionally, they are used for
high-order dynamic systems with dominant second-order behavior. The main
difficulty is that a PID controller does not explicitly take into account the varying
time delay.
The PID controller in continuous time is given by:
⎛
1
f (t ) = K p ⋅ ⎜ e(t ) +
⎜
Ti
⎝
t
∫ e(τ )dτ + T
d
0
de(t ) ⎞⎟
dt ⎟
⎠
(2)
where e(t ) = Fr − F (t ) is the error signal, Fr is the setpoint, and F(t) is the output of
the controlled process (i.e., the high-performance drilling process).
G PID (s ) =
where
G PID (s )
P = Kp, I =
is
the
f (s )
I
= P+ + D⋅s
E (s )
s
transfer
function
(3)
of
the
controller
and
Kp
, D = K p ⋅ Td are the proportional, integral and derivative gains,
Ti
respectively. In this paper, the discrete implementation chosen for Eq. (3) is given by:
f [k ] = f p [k ] + f i [k ] + f d [k ]
f p [k ] = K p ⋅ e[k ]
⎛K ⋅h ⎞
f i [k ] = f i [k − 1] + ⎜ p
⋅ e[k ]
Ti ⎟⎠
⎝
K p ⋅ Td ⋅ N
Td
f d [k ] =
f d [k − 1] +
⋅ (e[k ] − e[k − 1])
Td + N ⋅ h
Td + N ⋅ h
(4)
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R.E. Haber et al.
where N=10 is the filter coefficient for the discrete-time derivative and h=0.069s is
the sample.
Once the control structure is selected, the key issue is how to set the controller
K
parameters ( K p , p , D = K p ⋅ Td ). The following table summarizes some
Ti
methods for tuning PID controller parameters.
Table 1. Tuning rules for time-varying systems
Method/Parameters
Ziegler-Nichols
frequency domain
Cohen-Coon (a firstorder system with dead
time
L
L
)
a K1 ,W
Tc
L Tc
Visioli [6] (process
with a dead time L and
gain K1)
Ti
0.5 ˜ Tu
Kp
0.6 ˜ K u
W ·
§
1.35 / a ˜ ¨ 1 0.18 ˜
¸
1W ¹
©
1.37
L˜
K1 ˜ L
2.5 2 ˜W
1 0.39 ˜W
1.49 ˜ L
Td
0.125 ˜ Tu
L˜
0.37 0.37 ˜W
1 0.81 ˜W
0.59 ˜ L
In accordance with Table 1, the dead-time, T1, and the rise time, T2, are measured
from experimental results. Furthermore, K1 ïis the step amplitude, Tc is the time
constant, and L is the time delay. In the frequency response method, the loop is closed
and a pure proportional controller is used. The gain is increased to the ultimate gain,
Ku, when the system exhibits a steady oscillation, which is used to measure the
oscillation period Tu
3.1 Network-Based PID Control Using a Fieldbus
PROFIBUS is a widely used fieldbus that operates via a master-slave relationship
among devices connected to the network. Each master is assigned a set of slaves
which it regularly polls on a periodic basis. Access to the network is regulated by a
token moving among the masters. Distributed control systems based on PROFIBUS
are affected by jitter due to the retransmission of data with slaves and the
asynchronous activities performed by the masters.
Fig. 1. Network-based PID control system architecture for a high-performance drilling process
Using SA for Optimal Tuning of a PID Controller for Time-Delay Systems
1159
Multipoint interface (MPI) is a programming interface for the Siemens SIMATIC
S7 series that resembles the PROFIBUS protocol. The MPI physical interface is
identical to the PROFIBUS RS485 standard. The transmission speed can be increased
up to 12MB with the use of MPI. The control system architecture for a machine tool
on the basis of an MPI network is shown in Figure 1.
As the control signal (command feed) is transmitted through MPI, some amount of
network-induced delay is unavoidable. Figure 2 shows the cutting force’s step
response to command feed in a high-performance drilling process. The maximum
delay estimated from experiments is 0.4s, including both dead-time process and
network-induced delay.
Fig. 2. Drilling force response to command feed in the high-performance drilling process using
a network-based environment
4 Simulated Annealing for Optimal Tuning of a PID Controller
SA simulates the annealing process as it searches for a solution [7]. A random
perturbation is generated on the design variables (i.e., P, I, and D) and obtains the
change in the objective function (i.e., ITAE performance index). These perturbations
depend on a temperature index, T, and the rate at which it is lowered ( α = [0.8,0.95] ).
The temperature index decreases with each iteration of the algorithm, thus reducing
the size of the perturbations as the search progresses. Each set of PID controller
parameters obtained by this method is substituted into the controller equations, and
the performance of the resulting solution is evaluated through simulation of the
system in the time domain. The ITAE performance index is evaluated through
comparison of the simulated responses with the desired responses and quantified by
calculating the ITAE performance index (usually termed “the energy” in this method).
If the performance index is lower than the previous best performance index, then the
new parameters replace the previous parameters. Otherwise, the new PID controller
parameters are not immediately discarded. Instead, the cost value is subjected to a
probability evaluation where the probability, P, of the new parameters’ cost,
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R.E. Haber et al.
(ITAEnew), relative to the previous best cost, (ITAEprev), is calculated using
Boltzmann’s equation:
(ITAE PREV − ITAE NEW )
P=e
T
(5)
This probability, P, is then compared with a threshold number, n (a value between
0 and 1 with a uniform distribution). If P>n, then the new PID parameters are
accepted as if ITAEnew < ITAEprev, and they are rejected if P<n. This mechanism
avoids premature convergence at a local minimum and thus moves towards the
overall global minimum. After this stage, the temperature index is reduced according
to the annealing schedule using a reduction constant α = [0.8,0.95] . The whole process
is repeated until either the cost has reached an acceptable minimum level or the
temperature value has become too small to perturb the parameters.
5 Simulation Results
In order to carry out simulation studies, the control system is implemented in
Matlab/Simulink (see Fig. 3). First, the PID controller is adjusted using the ZieglerNichols and Cohen-Coon tuning laws, considering a maximum delay L=0.4s and a
sampling frequency Ts=0.069s. Ku=0.181 and Tu=1.6s are obtained by simulation. The
PID controller parameters are obtained by applying the tuning rules given in Table 1.
Likewise, the PID controller is adjusted using tuning rules proposed by Visioli [6].
This tuning rule is derived from genetic algorithms.
Fig. 3. PID control scheme implemented in Matlab/Simulink
In order to perform the comparison, the optimal tuning is carried out using the
proposed simulated-annealing strategy described in Section 4. The SA algorithm
parameters were: initial solution = [0.1086, 0.1483, 0.02183], which corresponds to
the results of the Ziegler-Nichols methods, reduction constant α = 0.9 , initial
temperature T0= 2000, number of Markov chains = 150, and percentage of
acceptance= 90.
Using SA for Optimal Tuning of a PID Controller for Time-Delay Systems
1161
Fig. 4. a) Behavior of force signal for the cases analyzed, (b) control signal
Table 2. Results of applying different tuning rules and the corresponding performance indices
PID Controller Parameters
Tuning Method
P
I
D
Ziegler-Nichols
0.1086
0.1348
0.0219
Cohen-Coon
0.0910
0.1449
0.0097
Visioli [6]
0.2617
0.1705
0.0584
Simulated
Annealing
0.0265
0.000132
0.0072
Overshoot
(%)
Min
Max
Mean
13.2
23.0
16.7
17.2
26.9
22.1
7.6
21.9
15.1
2.1
7.4
3.0
ITAE
Min
Max
Mean
505.3
770.4
614.4
474.0
831.4
595.0
557.7
2656.5
1483.2
372.9
582.9
462.1
The time-varying delay L in the network-based application is simulated assuming a
random delay between 0 and 0.4s, and 100 simulation tests are run for each set of
controller parameters. The maximum, minimum and mean value of the ITAE
performance index and the overshoot are thus calculated. Table 2 shows the PID
controller parameters and performance indices. Figure 4 shows the behavior of the
drilling-force signal and feed control signal for the cases analyzed, considering
L=0.4s. The best performance indices (ITAE and overshoot) are obtained for the PID
controller tuned by simulated annealing.
6 Conclusions
This paper introduces a simulated-annealing strategy for the optimal tuning of a PID
controller to deal with a time-varying delay system. The PID control system adjusts
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the command feed as needed to regulate the drilling force through a multipoint
interface network using the computational resources of computerized numerical
control. Different tuning rules are applied and evaluated through simulations
considering a time-varying delay between 0 and 0.4 seconds. The controller
parameters obtained on the basis of simulated annealing provide better transient
response and a better ITAE performance index than other methods’ parameters for the
worst case analyzed.
Further studies to improve the force control system will be conducted, as well as
the analysis of other intelligent control strategies.
Acknowledgments. This work was supported by the Spanish Ministry of Education
and Science’s project DPI2005-04298 COREMAV.
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