International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 5, May 2014)
Performance Analysis of Speed Control of Direct Current (DC)
Motor using Traditional Tuning Controller
Vivek Shrivastva1, Rameshwar Singh2
1
M.Tech Control System Deptt. of Electrical Engg. NITM Gwalior
2
Assistance Prof. Deptt. Of Electrical Engg. NITM Gwalior
Several approaches have been documented in literatures
for determining the PID parameters of such controllers
which is first found by Ziegler- Nichols tuning
[3,4],Proportional-Integral-Derivative “PID” controller,
due to its simplicity, stability, and robustness, is a type of
controller that is most widely applied [11], The fuzzy logic,
unlike conventional logic system, is able to model
inaccurate or imprecise models. The fuzzy logic approach
offers a simpler, quicker and more reliable solution that is
clear advantages over conventional techniques [6], Genetic
algorithm [7], and PSO [8]. Linear quadratic regulator
design technique is well known in modern optimal control
theory and has been widely used in many applications. It
has a very nice robustness property. This attractive property
appeals to the practicing engineers. Thus, the linear
quadratic regulator theory has received considerable
attention since 1950s. The liner quadratic regulator
technique seeks to find the optimal controller that
minimizes a given cost function (performance index). This
cost function is parameterized by two matrices, Q and R,
that weight the state vector and the system input
respectively. These weighting matrices regulate the
penalties on the excursion of state variables and control
signal. One practical method is to Q and R to be diagonal
matrix. The value of the elements in Q and R is related to
its contribution to the cost function. To find the control
law, Algebraic Riccati Equation (ARE) is first solved, and
an optimal feedback gain matrix, which will lead to optimal
results evaluating from the defined cost function is
obtained [9-10].
In this paper to achieve accurate control performance of
speed control of dc motor by using of Linear Quadratic
Regulator (LQR) technique presented. In this paper
compared with PID-ZN controller, PID-MZN controller
and PI-PSO controller. And minimizing of overshoot,
minimizing of rise time and minimizing setting time.
Abstract— The control of the speed of a DC motor is an
important issue and has been studied since the early decades
in the last century. This paper presents a comparison of time
response specification between different controller and Linear
Quadratic Regulator (LQR) for a speed control of a DC
motor. The goal is to determine which control strategy
delivers better performance with respect to DC motor’s speed.
Performance of these controllers has been verified through
simulation using MATLAB/SIMULINK software package.
According to the simulation results the Comparing with PIDZN controller, PID-MZN controller and PI-PSO controller,
the tuning method was more efficient in improving the step
response characteristics such as, reducing the rise time,
settling time and maximum overshoot in speed control of DC
motor. Linear quadratic regulator method gives the better
performance and superiority of liner quadratic regulator
method over other controller.
Keywords— PID-ZN, PID-MZN, PID –PSO controller, DC
motor, Linear quadratic regulator.
I. INTRODUCTION
Dc motors are controllable over a wide range with stable
and linear characteristics. Therefore, they are the most
common choice in the industries for both constant speed
and constant load operation [1].The field of electrical
energy will be divided into three areas: Electronics, Power
and Control. Electronics basically deals with the study of
semiconductor devices and circuits at lower power. Power
involves generation, transmission and distribution of
electrical energy. The electric motors are perhaps the most
widely used energy converters in the modern machine tools
and robots. These motors require automatic control of their
main parameters such as speed, position, acceleration etc.
In this paper to control the speed of DC motor, their
simplicity, ease of applications such as reliability and
favourable cost have long been a backbone of industrial
applications and it will have a long tradition of use as
adjustable speed machines and a wide range of options
have evolved for this purpose. In these applications, the
motor should be precisely controlled to give the desired
performance [2]. The past decades witnessed many
advancing improvements keeping in mind the requirement
of the end users.
II. SYSTEM MODELING OF DC MOTOR
DC motors are most suitable for wide range speed
control and are therefore used in many adjustable speed
drives.
119
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 5, May 2014)
DC motor shown in Figure 1 is the one of most common
motors which used in industrial motion control systems and
Fig 2 shows a Direct Current (DC) Motor Model. Ra is the
armature resistance and La is the armature inductance of dc
motor. Rf is the field resistance and Lf is the inductance of
the field winding, ia Armature current, J Moment of inertia
of motor, KT Torque factor constant, Kb Back emf constant
B Viscous Frictional coefficient.
[ ]
[
]
[
]
[
[
][ ]
] [ ] …………….... (4)
Obtaining the transfer function of the motor using the
state space model by formula G(s)= C (s I - A)-1 B + D
[14] in the equation (3) and (4) and obtain the equation 5 .
Fig 2 show the dc motor armature control system.
Fig 1 DC motor [12]
Fig. 3- Block Diagram of DC Motor
Fig 2 Direct Current (DC) Motor Model [13]
A linear model of a simple DC motor consists of an
electrical equation and mechanical equation. Using
Kirchhoff‟s Voltage Law (KVL) and Newton‟s second law,
the following equation is obtained:
III. LINEAR-QUADRATIC REGULATOR (LQR) OPTIMAL
CONTROL
LQR is a method in modern control theory that used
state-space approach to analyse such a system. Using state
space methods it is relatively simple to work with MultiInput Multi-Output system. Linear quadratic regulator
design technique is well known in modern optimal control
theory and has been widely used in many applications,
Linear-Quadratic Regulator (LQR) optimal control
problems have been widely investigated in the literature.
Assuming the above equations, the steady state
representation can be obtained as:
120
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 5, May 2014)
Fig 4 Block diagram of DC motor control system used
by LQR Controller. The performance measure is a
quadratic function composed of state vector and control
input. If the linear time-invariant system is controllable, the
optimal control law will be obtained via solving the
algebraic Ricci equation optimal control. The function of
Linear Quadratic Regulator (LQR) is to minimize the
deviation of the speed of the motor. The speed of the motor
is specifying that will be the input voltage of the motor and
the output will be compare with the input.
In general, the system model can be written in state
space equation as follows:
The diagonal-off elements of these matrices are zero for
simplicity. If diagonal matrices are selected, the quadratic
performance index is simply a weighted integral of the
squared error of the states and inputs. The term in the
brackets in equation (8) above are called quadratic forms
and are quite common in matrix algebra. Also, the
performance index will always be a scalar quantity,
whatever the size of Q and R matrices .The conventional
linear quadratic regulator problem is to find the optimal
control input law u* that minimizes the performance index
under the constraints of Q and R matrices. The closed loop
of dc motor with linear quadratic regulator show in the fig
1, The closed loop optimal control law is defined as:
A is the state matrix of order n×n B is the control matrix
of order n×m. Also, the pair (A, B) is assumed to be such
that the system is controllable. The linear quadratic
regulator controller design is a method of reducing the
performance index to a minimize value. The minimization
of it is just the means to the end of achieving acceptable
performance of the system. For the design of a linear
quadratic regulator controller, the performance index (J) is
given by:
Fig 4 Block diagram of DC motor control system used by LQR
Controller.
∫
Where K is the optimal feedback gain matrix, and
determines the proper placement of closed loop poles to
minimize the performance index in equation (8). The
feedback gain matrix K depends on the matrices A, B, Q,
and R. There are two main equations which have to be
calculated to achieve the feedback gain matrix K. Where P
is a symmetric and positive definite matrix obtained by
solution of the ARE is
Defined as:
Where Q is symmetric positive semi-definite state
weighting matrix of order
and R is symmetric
positive definite control weighting matrix of order
The choice of the element Q and R allows the relative
weighting of individual state variables and individual
control inputs as well as relative weighting state vector and
control vector against each other. The weighting matrices Q
and R are important components of an LQR optimization
process. The compositions of Q and R elements have great
influences of system performance. The designer is free to
choose the matrices Q and R, but the selection of matrices
Q and R is normally based on an iterative procedure using
experience and physical understanding of the problems
involved. Commonly, a trial and error method has been
used to construct the matrices Q and R elements. This
method is very simple and very familiar in linear quadratic
regulator application. However, it takes long time to choose
the best values for matrices Q and R. The number of
matrices Q and R elements are dependent on the number of
state variable (n) and the number of input variable (m),
respectively.
–
(10)
Then the feedback gain matrix K is given by:
K=
Substituting the above equation (9) into Equation (7)
gives:
… (12)
If the Eigen values of the matrix (A-BK) have negative
real parts, such a positive definite solution always exits [9,
10, 14].
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Table 2
Show Best Two Results Of Dc Motor Using LQR Controller
IV. RESULTS AND DISCUSSION
The performance and tuning using the Linear-Quadratic
Regulator (LQR) controller Optimal Control has been
compared with several controllers such as PID-ZN
controller, PID-MZN controller and PI-PSO controller and
its two different rating of dc motor as described below in
the table 1and table 4 [5] . Simulations were carried out
using MATLAB 7.0.1 on a Pentium IV processor, 2.8 GHz.
with 1 GB RAM.
Test Case I: in this test system the data of dc motor
show in the table 1 and transfer function of dc motor
equation 6 and 13 used as a system and find out the
response of the system applying the step function as an
input. and the tuning of different point such as the LQR
parameter Q and R:Q=[0.0000006;0.2] , R=[0.000005] and
Q=[0.000006;0.7 ] ,R=[0.000005] used in the motor
1.achieve The best results show in the table 2 and fig 5 and
6.and compare LQR controller than the several controller
such as PID-ZN controller, PID-MZN controller and PIPSO controller for different parameter such as rise time,
setting time and overshoot, using the LQR tuning controller
Q=[0.00004;0.2] , R=[0.000000006] for a dc motor 1and
gives the better rise time (0.0466),setting time (0.0833),
and overshoot(0.00) show in the fig 7and table 3.
Table 1.
Dc motor parameters motor 1[5]
Fig 5 Fig System Response with LQR controller
Show the transfer functions for the dc motor 1
…..(13)
Fig 6 System Response with LQR controller
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 5, May 2014)
Table 3
Comparative analysis of various tuning controller for motor 1
The best results show in the table 5 and fig 8 and 9.and
compare LQR controller than the several controller such as
PID-ZN controller, PID-MZN controller and PI-PSO
controller for different parameter such as rise time, setting
time and overshoot, using the LQR tuning controller
Q=[.000006 0;0 0.3];R=[0.00000000083]; for a dc motor 2
and gives the better rise time (0.0601),setting time
(0.109), and overshoot(0.00) show in the fig 10 and
table 6.
Table 4.
Dc motor parameters motor 4 [5]
Show the transfer functions for the dc motor 2
….
Table 5
Show Best Two Results Of Dc Motor Using Lqr Controller
Fig 7 System Response with LQR controller
Test Case II: for test system 2 the data of dc motor
show in the table 4 and transfer function of dc motor
equation 14 used as a system and find out the response of
the system applying the step function as an input. and the
tuning of different point such as the Linear Quadratic
Regulator (LQR) parameter Q and R : Q=[.00000006 0;0
0.3],R=[0.00000008],Q=[.00000020;00.3];R=[0.00000004]
;used in the motor 2.achieve
123
(14)
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Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 5, May 2014)
Fig 8 System Response with LQR controller
Fig 10 System Response with LQR controller
V. CONCLUSION
In this paper the speed of a DC motor is controlled using
Linear-Quadratic Regulator (LQR). The simulation results
are obtained using MATLAB/SIMULINK. LQR response
is compared with that of PID-ZN, PID-MZN,PI-PSO
controller. The results show that the overshoot, settling
time, peak time and control performance has been
improved greatly by using Linear-Quadratic Regulator
(LQR) controller. The LQR controller has more
advantages, such as higher flexibility, control, better
performance compared with ZN, PID-MZN,PI-PSO
controller. Hence, the comparison results showed that the
Linear-Quadratic Regulator (LQR) controller was the best
controller.
Fig 9 System Response with LQR controller
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Table 6
Comparative analysis of various tuning controller for motor 1
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