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nt. J. Automation and Control, Vol. 12, No. 4, 2018
Copyright © 2018 Inderscience Enterprises Ltd.
PSO based parameter estimation and PID controller
tuning for 2-DOF nonlinear twin rotor MIMO system
Roshni Maiti*, Kaushik Das Sharma and
Gautam Sarkar
Department of Applied Physics,
University of Calcutta,
Kolkata, India
Email: roshni.maiti@gmail.com
Email: kdsaphy@caluniv.ac.in
Email: gautamgs2010@yahoo.in
*Corresponding author
Abstract: The present paper proposes a control methodology for a nonlinear
multi-input multi-output (MIMO) system that combines a stochastic
optimisation and proportional-integral-differential (PID) control scheme. This
methodology is demonstrated through a laboratory scale helicopter
setup commonly known as the twin rotor MIMO system (TRMS). The
objective is to design a stochastically optimal/near optimal control law that can
simultaneously stabilise the TRMS with considerable cross-couplings and
provide satisfactory tracking performance to reach a desired position. The
proposed control methodology utilises two PID controllers independently
employed for the two rotors of TRMS. A PSO based parameter estimation
technique has been utilised in this paper to estimate the parameters of the
nonlinear TRMS laboratory setup. The nonlinear TRMS model with estimated
parameters is employed to tune the PID gains by means of PSO in an offline
manner and then implemented in real-life experimentations. The proposed
realisation of PID control strategy is implemented for both simulation and
real-life experimentations and their results demonstrate the usefulness of the
proposed methodology.
Keywords: multi-input multi-output system; MIMO; twin rotor MIMO system;
TRMS; proportional-integral-derivative controller; PID; parameter estimation.
Reference to this paper should be made as follows: Maiti, R., Sharma, K.D.
and Sarkar, G. (2018) ‘PSO based parameter estimation and PID controller
tuning for 2-DOF nonlinear twin rotor MIMO system’, Int. J. Automation and
Control, Vol. 12, No. 4, pp.582–609.
Biographical notes: Roshni Maiti is a PhD scholar in the Electrical
Engineering Section, Department of Applied Physics, University of Calcutta,
Kolkata, India. She received her BTech and MTech degrees in Electrical
Engineering from the West Bengal University Technology, Kolkata and
University of Calcutta, Kolkata in 2012 and 2014, respectively. Her research
interests include adaptive control, system identification, stochastic optimisation
techniques, etc.
PSO based parameter estimation and PID controller tunin
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583
Kaushik Das Sharma received his BSc (Physics Hons.), BTech (Electrical
Engg.) and MTech (Electrical Engg.) degrees from the University of Calcutta,
Kolkata, India, in 1998, 2001 and 2004, respectively and PhD (Engg.) degree
from the Jadavpur University, Kolkata, India in 2012. Presently he is an
Associate Professor in Electrical Engineering Section, Department of Applied
Physics, University of Calcutta, Kolkata, India. His research interests include
fuzzy control system design, stochastic optimisation applications, signal
processing, etc. He has published more than 35 research articles in international
and national journals or conferences. He is a senior member of IEEE (USA),
member of IET (UK) and life member of The Indian Science Congress
Association (Engineering Section).
Gautam Sarkar received his BTech, MTech and PhD degrees from the
University of Calcutta, Kolkata, India, in 1975, 1977 and 1991, respectively.
He has retired as LD Chair Professor in Electrical Engineering Section,
Department of Applied Physics, University of Calcutta, Kolkata, India. His
research interests include control system design, smart grid technologies, etc.
He has published more than 50 research articles in international and national
journals or conferences.
1 Introduction
Automatic control theory, principally the concept of feedback, has played an important
role in the advancement of automation (Mayr, 1970). A journey of few centuries
ultimately culminated from classical control theory to modern control approaches. The
controllers in classical control theory are being designed mainly for the linear time
invariant systems (Chen, 1993). The need of the advanced control strategies arose when
accurate control laws were required to be developed for detailed mathematical modelling
of the systems. All most all the practical systems in real-life are of nonlinear in type. The
control laws for nonlinear systems necessitated adopting modern approach, e.g., state
variable theory (Azar and Vaidyanathan, 2015a, 2015b, 2015c, 2016; Azar and Zhu,
2015; Billel et al., 2017, 2016; Boulkroune et al., 2016a, 2016b; Ghoudelbourk et al.,
2016; Azar and Serrano, 2014, 2015a, 2015b, 2015c, 2015d, 2016a, 2016b, 2017; Azar
et al., 2017a, 2017b, 2017c, 2017d; Azar, 2010, 2012; Mekki et al., 2015; Vaidyanathan
and Azar, 2015a, 2015b, 2015c, 2015d, 2016a, 2016b, 2016c, 2016d, 2016e, 2016f,
2016g; Vaidyanathan et al., 2017a, 2017b, 2017c, 2015a, 2015b, 2015c; Zhu and Azar,
2015; Grassi et al., 2017; Ouannas et al., 2016a, 2016b, 2017a, 2017b, 2017c, 2017d,
2017e, 2017f, 2017g, 2017h, 2017i, 2017j; Singh et al., 2017; Wang et al., 2017; Soliman
et al., 2017; Tolba et al., 2017; Mohler, 1991; Ogata, 1967), which evolved as a general
solution for the controller design problem. For the systems, which can be well described
with a linear second order mathematical model, there are a number of conventional
procedures for designing the controllers (Azar and Vaidyanathan, 2015a, 2015b, 2015c,
2016; Azar and Zhu, 2015; Chen, 1993), while for the systems modeled with higher order
linear models, pole-placement design technique (Devison, 1968), or the frequency
domain design procedures can be utilised (Azar and Serrano, 2014). Many of the current
584 R. Maiti et al.
control applications involve complex, time delay (Azar et al., 2015c), nonlinear single-
input single-output (SISO), multi-input single-output (MISO), or multi-input multi-output
(MIMO) models, for which controller(s) is(are) required to deliver high end performance
specifications in the presence of plant uncertainty, input uncertainty (including noise or
disturbances), and actuator constraints. Since classical control theories are not able to
deal with such problems, several modern methodologies emerged in the past four decades
that systematically address the fundamental limitations and capabilities of linear state
feedback controllers, for such challenging systems, both in continuous time case (Azar
and Vaidyanathan, 2016) and in digital implementations (Ellis, 2004). Examples of such
methodologies are the adaptive control (Azar and Serrano, 2015c, 2015d), H∞ based
robust control (Astrom and Wittenmark, 1989; Devison, 1968), sliding mode control
(Azar and Serrano, 2015c; Billel et al., 2016; Mekki et al., 2015), optimal control (Zak,
2003), etc., in conjunction with different intelligent techniques, such as neural networks
(Zak, 2003), fuzzy logic (Boulkroune et al., 2016a, 2016b; Azar, 2010, 2012),
evolutionary computations (Azar and Vaidyanathan, 2015b), etc. In some recent
controller design strategies, intelligent control techniques have been utilised individually
or in combinations to enhance the performance of the designed controllers.
In this paper, a control problem, involving a popular experimental benchmark model
called the twin rotor MIMO system (TRMS) (TRMS 33-220 user manual, 1998) whose
behaviour is much resemblance to that of a practical helicopter, has been investigated.
The TRMS model can freely rotate both in the horizontal and vertical axes responding to
yaw and pitch moments, respectively over a beam, with a pivot and a counter balance
attached to it. At each end of the beam there are two rotors, driven by DC motors,
commonly known as main and tail rotors. The main rotor produce a lifting force allowing
the beam to raise vertically, manipulating the pitch angle, and the tail rotor is used to
control the beam to turn left or right, manipulating the yaw angle. Both of the motors
produce aerodynamic forces through the rotor blades and also provide the coupling effect
between the rotors. Therefore, TRMS can be considered as a higher order nonlinear
system with prominent cross-couplings between the main rotor and the tail rotor (Ahmad
et al., 2000; Rahideh and Shaheed, 2007; TRMS advanced teaching manual, 1998) and
posing a challenge to the control engineering researchers. The main control objective is to
make the beam of the TRMS to track the predefined trajectory or to reach desired
position accurately and as quickly as possible by manipulating the speed of the DC
motors connected to each rotor simultaneously.
There are a very few work where TRMS is run in real-time environment to give
proper justification of simulation case studies. The design of ideal control law for
positioning the TRMS beam to a predefined position predominantly in real-time
environment depends on the accurate mathematical modelling of the TRMS setup and
accurate controller gains. In spite of the very simple construction and operation of
proportional-integral-differential (PID) controller, the problem is to select the proper
gains for keeping the error minimum. The Ziegler–Nichols method of designing PID
controller is not able to tackle the nonlinearity and significant cross-coupling present in
the TRMS model (Biswas et al., 2014). Thus, the motivation of this present paper is to
estimate the parameters of the TRMS model and design a control law for accurate
trajectory tracking utilising stochastic optimisation (PSO) based PID control scheme.
Particle swarm optimisation (PSO) was first introduced by Kennedy and Eberhart (1995)
and is one of the modern heuristic algorithms (Eberhart and Shi, 1998) that can explore
PSO based parameter estimation and PID controller tunin
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585
the solution space globally and automatically tune the PID controller to get an
optimal/near optimal solution for the TRMS to follow the desired trajectory:
• This work utilises stochastic optimisation, such as PSO algorithm to estimate the
model parameters at the outset.
• It is also presents a methodology that combines PSO and PID control scheme to
achieve the tracking control for the TRMS. Proposed PSO based PID tuning is first
realised in an offline manner using simulation on that estimated model of TRMS.
• Then the proposed design of PID controller has been implemented in real-time with
the optimal gains of PID controllers, obtained from simulation, to execute the
trajectory tracking of TRMS experimental setup.
• Results demonstrate the effectiveness of proposed optimisation method in
tuning the PID controllers for different operational modes of TRMS, such as
1-degree-of-freedom (DOF) pitch control, 1-DOF yaw control, and 2-DOF MIMO
control with different reference trajectories. This work employed PSO as the
representative stochastic global optimisation algorithm for designing the PID
controllers for TRMS. The rationale behind this is that PSO is a relatively simple
algorithm among stochastic optimisation algorithms and thus the computational
burden is much lesser than other such algorithms (Sharma et al., 2009a, 2009b).
This paper is oriented as follows. Related works are highlighted in Section 2. TRMS
experimental setup and its parts are described in Section 3. Working principle and
mathematical model are also briefly described Section 3. The proposed PSO based
parameter estimation of TRMS is elaborated in Section 4. Section 5 describes the
proposed PSO based tuning of PID controllers. The results for both in simulation and
real-time study are presented in Sections 6 and 7 concludes the paper.
2 Related work
To estimate the accurate nonlinear effects, effective cross-coupling, etc., of the TRMS
model several approaches of modelling have been proposed by many researchers (Ahmad
et al., 2000). In Ahmad et al. (2000), Ahmed et al. (2009), Dutka et al. (2003), Gabriel,
(2008), Nejjari et al. (2012), Rahideh and Shaheed (2007) and Witczak et al. (2013) a
nonlinear modelling using radial basis function networks has been proposed whereas two
models based on Newtonian and Lagrangian approaches are presented and then compare
them in a rigorous way (Rahideh and Shaheed, 2007). The nonlinearity present in TRMS
model was handled by utilising the state space representation into the linear time-varying
representation (Ahmed et al., 2009; Dutka et al., 2003; Gabriel, 2008; Witczak et al.,
2013). In Sahu and Dash (2011) sliding mode technique based model identification has
been proposed. A quasi-linear parameter varying modelling using a state transformation
and nonlinear identification of a twin rotor system has been found in Nejjari et al. (2012).
Along with different types of modelling many control schemes were also presented by
many researchers (Biswas et al., 2014; Chelihi and Chemachema, 2014; Ge and Wang,
2004; Hametner et al., 2013; Juang et al., 2008; Lopez-Martinez and Rubio, 2003;
Lopez-Martinez et al., 2003; Tao et al., 2010) with varied domain of control law
formulations, e.g., conventional PID control based (Biswas et al., 2014; Hametner et al.,
586 R. Maiti et al.
2013; Juang et al., 2008), nonlinear predictive control (Dutka et al., 2003), model
referenced adaptive control (Chelihi and Chemachema, 2014), robust controller using H∞
control (Lopez-Martinez et al., 2003), fuzzy controller based (Tao et al., 2010),
controllers using artificial neural network (Ge and Wang, 2004), sliding mode control
techniques (Azar and Serrano, 2015b; Daikh and Khelfi, 2015; Nayak et al., 2017) and
many others (Juang et al., 2011; Mekki et al., 2015). Although, there are many options,
explored to design the control law for TRMS, the conventional PID controllers are simple
but accurate enough to cater the desired control objective. So many design techniques of
PID controllers were presented in literature (Biswas et al., 2014; Dutka et al., 2003;
Gabriel, 2008; Hametner et al., 2013; Juang et al., 2008; Witczak et al., 2013).
3 Twin rotor MIMO system
The TRMS experimental setup consists of mechanical and electrical units. Mechanical
part consists of two rotors, main rotor (for pitch control) responsible for vertical
movement and tail rotor (for yaw control) responsible for horizontal movement. They are
pivoted on a vertical beam over which they can freely rotate together with a counter
balance. Input signals are given to the DC motors, attached with two rotors, produces
their rotation. In practical helicopter system angle of attack is changed to regulate the
speed of the rotors. But here angle of attack is fixed due to its two-dimensional nature.
Therefore, the DC motor input is changed to modify rotors speed. Two tachometers,
mounted on these two rotors are used to measure the velocity of the DC motors and one
position sensor is pivoted on the beam to sense rotors position. Electrical unit transfers
the measured signal from the tachometers and the sensor to a computer and allows the
control signal to the TRMS through an input-output port. The schematic diagram of
TRMS is shown in Figure 1 (TRMS 33-220 user manual, 1998; TRMS advanced
teaching manual, 1998).
Figure 1 The schematic diagram of the TRMS
In this present work the TRMS model is controlled in three different modes such as:
PSO based parameter estimation and PID controller tunin
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587
a 1-DOF pitch rotor control, where cross coupling dynamics is ignored and only the
main rotor is controlled
b 1-DOF yaw rotor control, where cross coupling dynamics is again ignored and only
the tail rotor is controlled
c 2-DOF rotor control, where both the rotors are controlled simultaneously and all
coupling effects are included here.
In all modes of operations the control objective is set the beam of the TRMS according to
the desired or reference trajectory by controlling the pitch angle and yaw angle with
minimum possible transient error.
Figure 2 Block diagram representation of the TRMS model
3.1 TRMS model
TRMS is a highly nonlinear system with its higher aerodynamics and significant
cross-coupling in between its two rotors. It consists of several elastic parts like rotor,
engine and control surfaces (Juang et al., 2008). The nonlinear aerodynamic force and
gravitational force act on flexible structure of it and make it difficult for realistic analysis.
Therefore to control this system a representative model has to be introduced that is
capable of giving same dynamic response that of the real one. In Figure 2, a
representative model of TRMS is shown where two linear transfer functions 1.1
1.2 1
s
+ and
0.8
1
s
+ describe two dc motors attached with main rotor and tail rotor respectively.
Nonlinear static characteristics of motors with propellers which produce the required
aerodynamic forces are given by 2
111 1
Gau b
=
×+ and 2
222 2
,Gau b=× + where u1 and
u2 are voltage inputs. The TRMS beam has to compensate aerodynamic thrust produced
588 R. Maiti et al.
by two of its propellers. Therefore, interactions between two rotors represented by two
transfer functions G3 and G4 and are given by:
31 1
2
cos
0.0326 sin 2 sin
2
ψgy
fg
dψd
GBGψK
dt dt
dψMψ
dt
φ
φ
⎛⎞⎛ ⎞
=− × − × × ×
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
⎛⎞
⎛⎞
+××−×
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(1)
41
d
GB dt
ψ
φ
=× (2)
where ψ = pitch angle,
φ
= yaw angle.
In G3 first term comprises of the angular velocity corresponding to pitch, whereas,
the second and third terms corresponds to yaw angular velocity. Third term also
introduces the nonlinearity produces by yaw angular velocity.
4 PSO and parameter estimation
4.1 PSO algorithm
PSO is a stochastic, population-based evolutionary algorithm used for solving different
kinds of problems. It is a kind of swarm intelligence that is based on socio-psychological
principles and provides insights into social behaviour, as well as contributing to
engineering applications, in particular. The PSO algorithm was first developed in 1995 by
Kennedy and Eberhart and Eberhart and Shi 1998. In the PSO computational algorithm,
population dynamics simulates bio-inspired behaviour, i.e., a ‘bird flock’s’ behaviour
which involves sharing of information and allows particles to take profit from the
discoveries and previous experience of all other particles during the search of food. Each
particle in PSO has a randomised velocity vector ()v associated to it, which moves
through the search space. Each particle in PSO keeps track of its coordinates in the
solution space, which are associated with the best solution (fitness) it has achieved so far.
This value is called p (personal best position vector). Another best value that is tracked
by the global version of the particle swarm optimiser is the overall best value (fitness). Its
location, called
g
(global best position vector), is obtained among all the particles in the
population. The past best position of the particle itself and the best overall position in the
entire swarm are employed to obtain new position vector ()
s
for the particle in quest to
minimise (or maximise) the fitness (Cai and Simon, 2013; Sharma et al., 2009a, 2009b).
At each time step ‘t’ the velocity of the particle is updated and the particle is moved to a
new position. This new position is calculated as the sum of the previous position and the
new velocity:
11ttt
s
sv
++
=+ (3)
The update of the velocity from the previous velocity to the new velocity is determined
as:
PSO based parameter estimation and PID controller tunin
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589
()
(
)
11 2 2
1.. .
tt t t
tt
vωvrcps rcg s
+=+ −+ −
(4)
The PSO concept consists, in each time step, of changing the velocity of each particle
flying towards its personal best and global best location with an inertia weight ω, which
controls the magnitude of the old velocity in the calculation of new velocity. The velocity
is weighted by random terms, with separate random numbers (r1 and r2) being generated
for velocities towards personal best and global best locations respectively. c1 and c2 are
the social parameters used for the learning purpose in the optimisation technique (Sharma
et al., 2009b).
4.1.1 PSO algorithm
The outline of the PSO (global version) algorithm is given as follows:
1 Initialise the jth particle’s position and velocity randomly throughout the solution
space using the following formula:
()
1
0 min max min
j
ss rs s=+ − (5)
()
2
min max min
0Δ
jsrss
vt
+−
= (6)
where r1, r2 are the random numbers between 0 and 1, min
s
is the vector of lower
bounds and max
s
is the vector of upper bounds and Δt is the time step.
2 Calculate the fitness value of the process with the candidate solution.
3 Search the fitness value for each particle till now, in all iterations, to get its own best
fitness value (pBest) in history and get the corresponding position vector ().p
4 Determine the particle in the solution space with the best fitness value (gBest) till
now, among all the particles, and denote its position as .
g
5 For each particle calculate the velocity according to the following formula
(Das Sharma et al., 2009b):
()
(
)
11 22
1ΔΔ
jj j
tt
jj t
t
tt
cr p s cr g s
vωvtt
+
−−
=+ + (7)
where ωt is the inertia weight of the particle at iteration t, as introduced by Eberhart
and Shi (Witczak et al., 2013), and is given by:
()( )
max
max
start end
tend
iter t ωω
ωω
iter
−∗ −
=+
(8)
where itermax is the maximum number of iterations of PSO. Normally ωt is reduced
linearly from ωstart to ωend throughout the PSO generation. A typical combination is
ωstart = 0.9 and ωend = 0.4.
590 R. Maiti et al.
c1 and c2 in (7) are the ‘trust’ parameters and are usually set to two. Update each
particle’s position vector according to the following equation:
11
Δ
jjj
ttt
s
sv t
++
=+ (9)
where 1
j
t
s
+ represents the position of particle ‘j’ at (t + 1)th iteration and 1
j
t
v+
represents the corresponding velocity vector. A unit time step (Δt) is taken
throughout the present work.
6 The stopping criteria can be set in two ways, either by choosing the itermax or by
using a minimum error criteria, i.e., continue the PSO adaptation until a maximum
pre-specified error is attained by the controller.
Each particle’s velocity, in each dimension, is clamped to a maximum velocity max
v
as:
max max
max max
if , then
if , then
jj
gg
jj
gg
vv vv
vv vv
>=
<− =−
where max
vis specified by the designer according to the nature of the system.
4.1.2 PSO based parameter estimation technique
A system identification problem can be viewed as the construction of a mathematical
model for a dynamic system using the knowledge and/or observations of that system
(Sharma et al., 2009a, 2009b; Rodrigo et al., 2014). The two most important elements for
system identification are:
1 the selection of a feasible model structure
2 the estimation of model parameters with the minimum error.
In this present work, the model structure is derived from the basic voltage and torque
equations of two DC motors and their coupling effects in operation of TRMS and the
parameters of this model, as shown in Figure 2, are identified by using PSO.
The unknown parameters of the derived TRMS model are: static characteristics
parameter of main rotor (a1), static characteristics parameter of main rotor (b1), gravity
momentum (Mg), friction momentum function parameter (B1ψ), gyroscopic momentum
parameter (KGy), moment of inertia of vertical rotor (I1), static characteristics parameter
of tail rotor (a2), static characteristics parameter of tail rotor (b2), friction momentum
function parameter (B1φ), and moment of inertia of horizontal rotor (I2).
To perform the parameter estimation task, the open-loop test data of the input
reference signals of 1.5 * u(t) for pitch position and –0.4 * u(t) for yaw position and its
corresponding real-time output from the TRMS are acquired for 100 sec., where each
input-output data samples are measured at an interval of 0.1 sec. In PSO based parameter
estimation technique (Sharma et al., 2009b; Cai and Simon, 2013), a particle’s position in
solution space is chosen as a vector containing all required parameter, as mentioned
earlier, to obtain the model of the TRMS. This particle vector
Z
is formed as (Sharma
et al., 2009a, 2009b):
PSO based parameter estimation and PID controller tunin
g
591
[
]
11 1 1 2 2 1 2
|| | | || | | |
gψGy φ
Z
abM B K Ia b B I= (10)
The unknown parameters are estimated utilising the PSO algorithm. In this case the PSO
algorithm determines the best vector comprising the unknown parameters as shown in
(10) for which discrepancy between the model outputs and the actual experimental output
data, for the same inputs, aggregated over the entire set of input-output real experimental
data, are found minimum. The integral absolute error (IAE) between the actual output
and model output are calculated and equated as the fitness function for the PSO
algorithm. The IAE, in this case, can be defined as:
[]
0
IAE ( ) ( ) Δ
PST
A
Mc
n
Yn Y n t
=
=−
∑
where PST = plant simulation time, ∆tc = step size or sampling time, YA = actual output
and YM = model output. The results obtained, after ten test cases carried out, for the
estimated parameters of the TRMS are tabulated in Table 1 along with the TRMS model
data as in TRMS 33-220 user manual (1998). Figures 3(a) and 3(b) show the comparative
plots of the open-loop responses, for the pitch angle and for the yaw angle respectively,
of actual real-time system and model with estimated parameters.
Figure 3 (a) Response for the pitch angle and (b) response for the yaw angle in open-loop
configuration for parameter estimation (see online version for colours)
020 40 60 80 100
0
0.5
1
Time (Sec.)
Amplitude
Model Response
Actual Res ponse
(a)
020 40 60 80 100
-1.5
-1
-0.5
0
Time (Sec.)
Amplitude
Model Response
Actu al Res ponse
(b)
Table 1 Estimated parameters of the TRMS using PSO based parameter estimation technique
Parameters a1 b
1 M
g B
1ψ K
Gy I
1 a
2 b
2 B
1φ I
2
Supplied 0.0135 0.0924 0.3200 0.0060 0.0500 0.0680 0.0200 0.0900 0.1000 0.0200
Estimated 0.0077 0.0949 0.3174 0.0089 0.0450 0.0800 0.0289 0.0500 0.1355 0.0360
592 R. Maiti et al.
5 PSO and PID controller tuning
5.1 Digital implementation of PID controller
In this new era of modern control applications still the most popular type of controller
used in industry is PID controller. A basic PID control action consists of proportional
error signal supplemented with derivative and integral of the error signal. Thus, the PID
control law can be written in time constant form as (Biswas et al., 2014; Cai and Simon,
2013; Chelihi and Chemachema, 2014; Eberhart and Shi, 1998; Ge and Wang, 2004;
Hametner et al., 2013; Juang et al., 2008, 2011; Kennedy and Eberhart, 1995;
Lopez-Martinez et al., 2003; Slotine et al., 1987; Tao et al., 2010):
0
1()
() () ( )
t
d
i
de t
ut K et eτdτT
Tdt
⎛⎞
⎜⎟
=+ +
⎜⎟
⎝⎠
∫ (11)
where K is a constant, e(t) is error between the reference input and actual plant output, Ti
is integral time constant and Td is derivative time constant.
In digital implementation of PID controller all the desirable features of a controller
(e.g., anti-integral wind-up, auto/manual modes of operation with bump-less transfer,
etc.) may be easily incorporated while maintaining the high accuracy and precision and
there will be no drift problem associated to a digital system. In a processor based digital
controller, rapid switching from one algorithm to another (e.g., a P controller to a PID
controller) and automatic tuning of controller parameters are possible (Juang et al., 2008;
Hametner et al., 2013; Chelihi and Chemachema, 2014; Lopez-Martinez et al., 2003; Tao
et al., 2010; Ge and Wang, 2004; Juang et al., 2011; Kennedy and Eberhart, 1995; Cai
and Simon, 2013).
Now from (11) in general form of PID controller is:
0
1t
Pd
i
de
uKe edtT
Tdt
⎛⎞
⎜⎟
=+ +
⎜⎟
⎝⎠
∫ (12)
where
,,
pi dd
i
K
K
KK K KT
T
== =
From backward difference algorithm:
1nn
tnh
de e e
dt h
−
=
−
⎛⎞
≈⎜⎟
⎝⎠
(13)
and using rectangular integration:
11
(1)
nh
nn n n
nh
I
IedtIhe
−−
−
=+ =+
∫ (14)
Now, using (13) and (14) in (12) for nth instant, the digitally realised PID control action is
as:
PSO based parameter estimation and PID controller tunin
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593
1nnn
nPn d
i
Iee
uKe T
Th
−
⎛−⎞
⎛⎞
=++
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(15)
and similarly for (n –1)th instant:
112
11
nnn
nPn d
i
Iee
uKe T
Th
−−−
−−
⎛−⎞
⎛⎞
=++
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(16)
Now, using (15) and (16), we get the incremental form of PID control action as:
()
1
11 21
2
nn d
nn Pnn nn n
i
II T
uu Kee ee e
Th
−
−− −−
−
⎛⎞
−= −+ + +−
⎜⎟
⎝⎠
(17)
Using (14), again we get:
()
11 21
2
d
nn Pnn n nn n
i
hT
uu Kee e ee e
Th
−− −−
⎛⎞
−= −+ + +−
⎜⎟
⎝⎠
(18)
121
2
11
dd Pd
nP nP n n n
i
hT T KT
uK eK e e u
Th h h
−
−−
⎛⎞⎛⎞⎛⎞
=++ − + + +
⎜⎟⎜⎟
⎜⎟
⎝⎠⎝⎠
⎝⎠ (19)
011221nnn nn
uaeae ae u
−−−
=+ + + (20)
where
01dd
PPi
i
hT K
aK KhK
Th h
⎛⎞⎛ ⎞
=++=++
⎜⎟
⎜⎟
⎝⎠
⎝⎠ (21)
1
22
1
dd
PP
TK
aK K
hh
⎛⎞⎛ ⎞
=− + =− +
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
(22)
2
Pd d
KT K
ahh
⎛⎞⎛⎞
==
⎜⎟⎜⎟
⎝⎠⎝⎠
(23)
The block diagram representation of digital implementation of PID control signal is
shown in Figure 4.
Figure 4 Realisation of (20) for PID control action
594 R. Maiti et al.
The advantage of the incremental algorithm in realisation of PID control action is that the
most of the computations are done using increments only. It is only in the final stage
where the increments are added to compute the actual control action considering the
required precision applicable for that particular case.
5.2 PSO based tuning of PID controller
To design a PID controller the values of the proportional, integral and derivative gains
has to be determined. The PSO based tuning methodology can optimise the values of the
controller gains of PID control action and can determine the optimum controller
structure.
In case of 1-DOF control for the PSO based controller design, a particle’s position in
solution space is a vector containing all required information to construct a PID
controller, e.g.,
1 proportional gain (Kp)
2 integral gain (Ki)
3 derivative gain (Kd).
This particle vector is formed as (Sharma et al., 2009a, 2009b):
[
]
||
pid
Z
KKK= (24)
To design a PID controller utilising PSO based algorithm first the population of particles
in the swarm is chosen. Then each candidate control law is formulated from equation (19)
by putting the values of Ti and Td in it from equation (12) yields:
121
2
11
id d d
nP nP n n n
PP P
hK K K K
uK eK e e u
KhK hK h
−
−−
⎛⎞⎛⎞⎛⎞
=++ − + + +
⎜⎟
⎜⎟⎜⎟
⎝⎠
⎝⎠⎝⎠ (25)
The TRMS is then simulated with this control signal by using Runge-Kutta 4th order
method with fixed time step and sampling time Δtc = 0.01sec for PST times to get the
error between the reference signal and output of the plant en = rn – yn (n = 1:PST). The
candidate controller simulation (CCS) algorithm, as shown in Figure 5, is implemented
for each on the candidate controller to calculate the fitness function, the IAE, which can
be defined as:
0
IAE ( )Δ
PST
c
n
en t
=
=∑
where PST = plant simulation time and ∆tc = step size or sampling time.
In case of 2-DOF control two separate PID controllers for pitch and yaw control is
required. The particle’s position in solution space is formed with:
1 proportional gain for pitch control ()
P
P
K
2 integral gain for pitch control ()
P
i
K
3 derivative gain for pitch control ()
P
d
K
P
SO based parameter estimation and PID controller tunin
g
595
4 proportional gain for yaw control ()
y
P
K
5 integral gain yaw control ()
y
i
K
6 derivative gain yaw control ().
y
d
K
This particle vector is formed as (Sharma et al., 2009a, 2009b):
||||
yyy
PP i
PP
dd
Z KK KKK
⎡⎤
=⎣⎦
(26)
Like previous one here also at first the population of particles in the swarm is chosen.
Then each candidate control law is formulated likewise the equation (25) for both pitch
and yaw control, such as:
121
2
11
ppp
p
pp pp p p p
iddd
np np nnn
pp p
pp p
KKK
hK
uK eK e e u
h
KhK hK
−
−−
⎛⎞
⎛⎞ ⎛⎞
=++ − ++ +
⎜⎟
⎜⎟ ⎜⎟
⎜⎟⎜⎟
⎝⎠
⎝⎠⎝⎠ (27)
121
2
11
yyy
y
yy yy y y y
iddd
np np nnn
yy y
pp p
KKK
hK
uK eK e e u
h
KhK hK
−
−−
⎛⎞⎛⎞⎛⎞
=++ − + + +
⎜⎟⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠
⎝⎠⎝⎠ (28)
Figure 5 Flowchart representation of CCS algorithm
The TRMS is then simulated with this control signal by using Runge-Kutta 4th order
method with fixed time step and sampling time ∆tc = 0.01sec for (PST) times to get the
error between the reference signal given to pitch rotor and output of the pitch rotor
ppp
nn n
ery=− (n =1:PST) and reference signal given to yaw rotor and output of the yaw
rotor .
yyy
nn n
ery=− The CCS algorithm, as shown in Figure 5, is implemented for each on
the candidate controller to calculate the fitness function, the IAE, which can be defined
as:
596 R. Maiti et al.
()
0
0.5 Δ
PST
py
nnc
n
I
AE e e t
=
=× +
∑
where PST = plant simulation time and ∆tc = step size or sampling time.
Figure 6 Flowchart representation of PSO based PID controller tuning algorithm
Then, according to the value of the fitness function (IAE) of each particle in each
iteration, the pBest and gBest candidate controllers are calculated and the particles
position and velocity in the search space is updated by the global version PSO method.
The algorithmic flow of the task will be same as described in Section 4.1. The PSO
algorithm will stop searching the solution space when the number of iterations specified
by the designer is reached or a pre specified error is attained by the controller. The
flowchart representation of PSO based PID controller tuning is shown in Figure 6.
This process runs ten times each to get the best candidate solution vector and by using
those values the experimental setup is run for different operating modes with different
types of input reference trajectories.
6 Simulation and experimental results
The effectiveness of the PID control scheme, as described in this paper, is evaluated by
considering both simulation case studies and real-life experiments on TRMS. A fixed
P
SO based parameter estimation and PID controller tunin
g
597
step 4th order Runge-Kutta method of 0.01 sec. sampling time is used to simulate the
TRMS model. The IAE between the reference signal and the corresponding output signal
is taken as the measure to compute the usefulness of the proposed PID control strategy.
6.1 Simulation results
Simulation study of TRMS has been done for three modes of control such as 1-DOF
pitch, 1-DOF yaw and 2-DOF control. Two types of input reference signal, namely
variable step and sinusoidal wave, are utilised to design the PID controller for the TRMS
in simulation. The objective of the design strategy is to track the reference signal with
minimum possible transient error. The nature of the reference signals used in this study
for main rotor (pitch control) and tail rotor (yaw control) over the period of 100 sec. are
given as:
1 reference signal to the main rotor:
a variable step signal:
(0.2 0.5 0.1 0.3)* ( )ut+++
b sinusoidal signal:
0.2 ( ) 0.2 * sin(2* * 0.04* )ut pi t
+
2 reference signal to the yaw rotor:
a variable step signal:
(0.3 0.1 0.2 0.6)* ( )ut−++
b sinusoidal signal:
0.2* ( ) 0.4*sin(2* *0.05* ) ut pi t−+
Figure 7 Evaluation period response and control signal for variable step trajectory for 1-DOF
pitch control in simulation environment (see online version for colours)
020 40 60 80 100
0
0.5
1
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-200
0
200
Time(second)
Amplitud
e
Control Input
598 R. Maiti et al.
Figure 8 Evaluation period response and control signal for pure sine wave trajectory for 1-DOF
pitch control in simulation environment (see online version for colours)
020 40 60 80 100
-0.5
0
0.5
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-100
0
100
Time(second)
Amplitude
Control Input
Figure 9 Evaluation period response and control signal for variable step trajectory for 1-DOF
yaw control in simulation environment (see online version for colours)
020 40 60 80 100
-1
0
1
Time(second)
Amplitude
Referenc e Input
System Response
020 40 60 80 100
-500
0
500
Time(second)
Amplitud
e
Control Input
To perform the simulation, a population size of ten particles for PSO algorithm is chosen.
For each simulation, 100 iterations of new particle optimisations are performed. As the
PSO based PID gain tuning methodology is a stochastic way of searching the solution
space for the PID gains, for each mode of TRMS operation, the simulation is performed
for ten times and a best value among them with its average value and standard deviation
are calculated and tabulated in Table 2. It can be observed from Table 2 that the values of
IAE for 2-DOF control configuration are somewhat higher than the other 1-DOF
configurations and the uncertainty in the result is also higher in that case, as depicted by
the standard deviation value. The 2-DOF configuration is quite difficult to control as the
cross-coupling effect between the main rotor and the tail rotor is in full play and that
creates the uneven thrust over the individual rotor for the other one. Responses for the
best IAE value and corresponding control input signal for different modes of TRMS
operation as well as for two different input reference signals are shown in Figures 7 to 12.
During the simulation study the controllers are trained, i.e., the PID gains are optimised
P
SO based parameter estimation and PID controller tunin
g
599
and then these trained controllers are utilised to control the system in real-time. From the
figures it is also revealed that the control objectives are fulfilled and the designed
controllers are good enough to provide satisfactory transient responses.
Figure 10 Evaluation period response and control signal for pure sine wave trajectory for
1-DOF yaw control in simulation environment (see online version for colours)
020 40 60 80 100
-1
0
1
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-200
0
200
Time(second)
Amplitude
Control Input
Table 2 Performance evaluation of designed PSO based PID controllers in simulation case
study
Input signal TRMS mode of operation Best IAE Average IAE Std. dev. IAE
Variable
step
1-DOF pitch 5.6597 5.8531 0.0125
1-DOF yaw 3.6915 3.8512 0.1590
2-DOF 4.9331 6.0451 0.8921
Sinusoidal
wave
1-DOF pitch 5.6253 5.7853 0.0151
1-DOF yaw 4.2380 4.3901 0.0733
2-DOF 5.9553 6.6521 0.4301
Figure 11 Evaluation period response and control signal for variable step trajectory for 2-DOF
control in simulation environment (see online version for colours)
050 100
0
0.5
1
Time(second)
Amplitude
Refe rence Input
System Response
050 100
-200
0
200
Time(second)
Amplitude
Control In put
050 100
-1
0
1
Time(second)
Amplitude
Refe rence Input
System Response
050 100
-500
0
500
Time(second)
Amplitude
Control In put
600 R. Maiti et al.
Figure 12 Evaluation period response and control signal for pure sine wave trajectory for
2-DOF control in real-time environment (see online version for colours)
050 100
-0.5
0
0.5
Time(second)
Amplitude
Refe rence I nput
System Response
050 100
-50
0
50
100
Time(second)
Amplitude
Control I nput
050 100
-1
-0.5
0
0.5
Time(second)
Amplitude
Refe rence Input
System Response
050 100
-200
-100
0
100
Time(second)
Amplitude
Control I nput
Table 3 Performance evaluation of designed PSO based PID controllers in real-time case
study
Input signal TRMS mode of operation IAE
Variable step 1-DOF Pitch 5.5312
1-DOF Yaw 7.7265
2-DOF 6.8993
Sinusoidal wave 1-DOF Pitch 5.5903
1-DOF Yaw 6.4802
2-DOF 7.1995
Figure 13 Evaluation period response and control signal for variable step trajectory for 1-DOF
pitch control in real-time environment (see online version for colours)
020 40 60 80 100
0
0.5
1
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-200
0
200
Time(second)
Amplitude
Control Input
P
SO based parameter estimation and PID controller tunin
g
601
Figure 14 Evaluation period response and control signal for pure sine wave trajectory for
1-DOF pitch control in real-time environment (see online version for colours)
020 40 60 80 100
-0.5
0
0.5
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-100
0
100
Time(second)
Amplitude
Control Input
Figure 15 Evaluation period response and control signal for variable step trajectory for 1-DOF
yaw control in real-time environment (see online version for colours)
020 40 60 80 100
-1
0
1
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-500
0
500
Time(second)
Amplitude
Control Input
6.2 Experimental results
The trained controllers after simulation study are employed to control the TRMS
experimental model for different modes of operation as well as for different input
reference signals in real-time. The best available trained PID controller, decided by the
best IAE values, for each control strategy is taken for real-time operation of TRMS
experimental setup. From each TRMS mode of operation IAE values are tabulated in
Table 3. Responses for the real-time operation of TRMS and the corresponding control
signal, i.e., TRMS system input, are shown in Figures 13 to 18 as in case of simulation
case study. Table 3 depicts, in a similar fashion to that of simulation study, that the
602 R. Maiti et al.
2-DOF control configurations are producing more transient errors compared to 1-DOF
configurations in real-time experimentation also. From the figures it can also be seem that
the transient responses for 2-DOF control mode are fair enough to achieve a satisfactory
performance out of a practical case study. Again from Figures 17 and 18 these can be
seen that the performance of yaw control is quite challenging rather that the pitch control
in 2-DOF coupled condition.
Figure 16 Evaluation period response and control signal for pure sine wave trajectory for
1-DOF yaw control in real-time environment (see online version for colours)
020 40 60 80 100
-1
0
1
Time(second)
Amplitude
Reference Input
System Response
020 40 60 80 100
-200
0
200
Time(second)
Amplitude
Control Input
Figure 17 Evaluation period response and control signal for variable step trajectory for 2-DOF
control in real-time environment (see online version for colours)
050 100
-0.5
0
0.5
1
Time(second)
Amplitude
Refe rence Input
System Response
050 100
-200
0
200
Time(second)
Amplitude
Control In put
050 100
-0.5
0
0.5
1
Time(second)
Amplitude
Refe rence Input
System Response
050 100
-500
0
500
Time(second)
Amplitude
Control In put
P
SO based parameter estimation and PID controller tunin
g
603
Figure 18 Evaluation period response and control signal for pure sine wave trajectory for
2-DOF control in real-time environment (see online version for colours)
050 100
-0.5
0
0.5
Time(second)
Amplitud
e
Refe rence Input
System Response
050 100
-50
0
50
100
Time(second)
Amplitude
Control I nput
050 100
-1
0
1
Time(se cond)
Amplitude
Refe rence Input
System Response
050 100
-200
-100
0
100
Time(se cond)
Amplitude
Control I nput
7 Conclusions
In this paper a design methodology of digital implementation of PID controller for
MIMO system utilising PSO, a global stochastic optimisation method, is proposed. The
proposed design scheme is implemented for simulation and real-time case studies for the
benchmark TRMS model. The parameters of the TRMS experimental setup are estimated
by a PSO based methodology. The proposed PID design scheme for TRMS model is then
employed to train the PID controller and the trained controllers are tested for different
mode of its operation with different reference trajectories. The proposed method
successfully demonstrated that it can simultaneously provide good tracking performance
with high degree of automation in the design process. The authors intend to design other
control schemes, such as state feedback control, robust controllers, etc., and compare
their performances in the near future.
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ISBN: 978-3-319-12882-5.
