Design of the LQG controller using operational
amplifiers for motion control systems has done based on
a new design procedure. We propose to add two extra
steps into the traditional design procedures. By
implementing the controller in different domains, the
performances of each domain are compared. With the
same reference input, if the design steps are implemented
correctly, the performances of the controlled system in sdomain, also in equivalent analog electronic circuits, and
in experimental results will be almost the same. Both
simulation and experimental results confirm the precise,
featured by the proposed procedure.
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Design of LQG Controller Using Operational
Amplifiers for Motion Control Systems
Nguyen Duy Cuong, Nguyen Van Lanh, and Dang Van Huyen
Faculty of International Training, Thai nguyen University of Technology, Thai nguyen City, Viet nam
Email: {nguyenduycuong, lanhnguyen, dtk1051020639}@tnut.edu.vn
Abstract—This paper proposes additional steps to the
traditional design procedures for continuous control systems
using operational amplifiers. Designing an analog Linear
Quadratic Gaussian (LQG) controller is selected as a case
study. The controller in the s-domain is firstly designed
based on the mathematical model of the plant to be
controlled, and then simulated and adjusted. Next, the plant
and the controller in the s-domain will be converted to
equivalent corresponding continuous electronic circuits
using operational amplifiers and continued for simulating.
The main purpose of these proposed additional steps is to
confirm that converting the controller from s-domain to
corresponding analog electronic circuits using operational
amplifiers is correct or not. After that, the controller will be
implemented and applied to the real setup. For a good
design, the simulation results of the resulting controlled
system in s-domain, also in equivalent analog electronic
circuits, and experimental results in the real setup are
almost the same.
Index Terms—linear quadratic gaussian (LQG), linear
quadratic regulator (LQR), linear quadratic estimator
(LQE), motion control systems, operational amplifiers (Op-
Amp)
I. INTRODUCTION
In analog control systems, the controllers use
continuous devices and circuits. In digital control systems,
the controllers use digital devices and circuits. The choice
between analog or digital control systems depends on the
application requirements. The most important advantage
of analog control systems overs digital control systems is
that in the analog control systems, any change in either
reference inputs or system disturbances is immediately
sensed, and the controllers adjust their outputs
accordingly [1]. However, the analog controllers are
recommended for use in the non-sophisticated systems. In
practice, most analog control systems have relied on
operational amplifiers as essential building blocks [2].
Operational Amplifiers are among the most widely
used electronic devices today. An operational amplifier is
a DC-coupled high-gain electronic voltage amplifier with
a differential input and, usually, a single-ended output [2].
The instrumentation amplifiers using operational
amplifiers provide great benefits to the designers. The
Manuscript received December 16, 2013; revised February 2, 2014.
mathematical operations such as inversion, addition,
subtraction, integration, differentiation, and
multiplication can be performed by using operational
amplifiers [2]. Such that many practical continuous
control systems can be constructed using operational
amplifiers. Electronic circuits using operational
amplifiers can be used to compare to most physical
systems such that analog electronic simulation was
effectively used in the research and development of
electro-mechanical systems.
Model-based design is a mathematical and visual
method of addressing problems associated with designing
complex control, signal processing and communication
systems [3]. In model-based design of control systems,
development is constructed in these five main steps:
Modeling the system;
Analyzing and synthesizing a controller for the
system;
Simulating the resulting controlled system;
Validating the simulation results;
Implementing the controller.
The mathematical model is used to identify dynamic
characteristics of the system model. A controller can then
be synthesized based on these characteristics. The time
response of the dynamic system is investigated through
offline simulation and real-time simulation
It is clear that when Step 4 has done the resulting
controlled system is in s-domain. However, in Step 5 the
analog controller is implemented using operational
amplifiers. The problem is that if correctly follow the
above procedure, we cannot confirm converting the
controller in s-domain to equivalent analog electronic
circuits using operational amplifiers is correct or not.
In this paper we propose additional steps between Step
4 and Step 5. First of all the controller and also the plant
are converted to analog electronic circuits using
operational amplifiers. Next, the equivalent continuous
electronic circuits are simulated. This step will be shown
in Section IV. The same inputs, the response outputs in
Step 3 and in the proposed steps are almost the same.
Such that after doing with the proposed steps we can
retest the final resulting continuous controller.
This paper is organized as follows: First, the dynamic
characteristic of the setup is analyzed in Section II. In
Section III, design of LQG controller is shown. Analog
LQG controller using operational amplifiers is introduced
in Section IV. Simulation and experimental tests are
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
157©2015 Engineering and Technology Publishing
doi: 10.12720/joace.3.2.157-163
performed in Section V. Finally, conclusions are given in
Section VI.
II. MATHEMATICAL MODEL OF THE SETUP
The setup (see Fig. 1) is designed for the purpose of
testing the results of the controller for linear and non-
linear systems. It consists of a slider which can move
back and forth over a rail. A DC motor, rail and slider are
fixed on a frame. The parameters of this setup are shown
in Table I [4].
Figure 1. The configuration of the setup
The mechanical part of the setup is designed
mimicking printer technology. For this process, a
computer based control system has been implemented
with software generated by MATLAB. This setup is also
suitable for the analog electronic circuits based control
systems.
Figure 2. Second order model of the setup
TABLE I. PLANT PARAMETERS OF THE SETUP
Elements Parameters Labels Values
Motor-Gain Motor constant 8.5 N/A
Motor-Inertia Inertia of the motor 2e-5 kg
Load Mass of the slider 0.35 kg
Belt-Flex Spring constant 80 kN/m
Damping in belt 1 Ns/m
Damper Viscous friction 8 Ns/m
Coulomb friction 0.75 N
The Damper component represents a viscous and
Coulomb friction. Coulomb friction always opposes
relative motion and is simply modeled as
( ̇) (1)
where is the Coulomb parameter of the Damper
element, ̇ is the velocity of the load. Viscous friction is
proportional to the velocity. It is normally described as
̇ (2)
where is the viscous parameter of the Damper element.
The mathematical expression for the combination of
viscous and Coulomb friction is
̇ ( ̇) (3)
If the non-linear Coulomb friction part is disregarded,
the model only contains linear components. In this case
we get a linear process model. A second order
approximation model is obtained with a state space
description as given in (4) [4].
[
̇
̇
] [
] [
] [
] (4)
[ ] [
] [ ]
where is the velocity of the load; is the position of
the load; and is applied force on the process. When we
mention the nonlinear friction term of the Damper
element then:
[
̇
̇
] [
] [
] [
( )
] [
] (5)
The second order model of the setup is given in Fig. 2.
III. DESIGN OF LQG CONTROLLER
A. Linear Quadratic Regulator
In the theory of optimal control, the Linear Quadratic
Regulator (LQR) is a method of designing state feedback
control laws for linear systems that minimize a given
quadratic cost function [5], [6].
Figure 3. Principle of state feedback
In the so called Linear Quadratic Regulator, the term
“Linear” refers to the system dynamics which are
described by a set of linear differential equations and the
term “Quadratic” refers to the performance index which
is described by a quadratic functional. The aim of the
LQR algorithm is finding an appropriate state-feedback
controller. The design procedure is implemented by
choosing the appropriate positive semi-definite weighting
matrix and positive definite weighting matrix . The
advantage of the control algorithm is that it provides a
robust system by guaranteeing stability margins.
An LQR however requires access to system state
variables. A state feedback system is depicted in Fig. 3
[7]. The internal states of the system are fed back to the
controller, which converts these signals into the control
signal for the process. In order to implement the
deterministic LQR, it is necessary to measure all the
states of the system. This can be implemented by means
of sensors in the system. However, these sensors have
noise associated with them, which means that the
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
158©2015 Engineering and Technology Publishing
measured states of the system are not clean. That is,
controller designs based on LQR theory fail to be robust
to measurement noise. In addition, it may be difficult or
too expensive to measure all states.
Figure 4. Clean state feedback of the process is obtained by using
State Variable filter
State Variable Filters (SVFs) can be used to make a
complete state feedback (see Fig. 4) [7]. When the noise
spectrum is principally located outside the band pass of
the filter, measurement noise can be suppressed by
properly choosing of the filter. For example, in the
experimental setup, information about the position is
measured with a lot of noise at any time instant. The
SVFs remove the effects of the noise and produce a good
estimate of the positions and velocities
However, SVFs cause phase lags. The phase lags can
be reduced by means of increasing the omega of the SVF.
In practice, the choice of the omega is a compromise
between the phase lag and the sensitivity for noise.
B. Linear Quadratic Estimator
Another way to estimate the internal state of the
system is by using a Linear Quadratic Estimator (LQE)
(see Fig. 5). In control theory, the LQE is most
commonly referred to as a Kalman filter or an Observer
[8], [9].
Figure 5. Principle of an observer
The Kalman filter is a recursive estimator. This implies
that to compute the estimate for the current state, the
estimated state from the previous time step and the
current measurement are required. The Kalman filter is
implemented with two distinct phases: i - The prediction
phase, the estimate from the previous step is used to
create an estimate of the current state; ii - The update
phase uses measurement information from the current
step to refine this prediction to arrive at a new estimate. A
Kalman filter is based on a mathematical model of a
process. It is driven by the control signals to the process
and the measured signals. When we use Kalman filters or
observers disturbances at the input of the process are
mostly referred to as “system noise” as in Fig. 5.
Its output is an estimate of the states of the system
including the signals that cannot be measured directly.
The Kalman filter provides an optimal estimate of the
states of the system in the presence of measurement noise
and system noise. In order to obtain optimality the
following conditions must be satisfied [10], [11]: i -
Structure and parameters of process and model must be
identical; ii - Measurement and system noise have
average zero and known variance. The LQE design
determines the optimal steady-state filter gain based on
linear parameters of the process, the system noise
covariance and the measurement noise covariance .
The states of the model will follow the states of the plant,
depending on the choice of and .
Figure 6. LQG explanation
C. Linear Quadratic Gaussian
Linear Quadratic Gaussian (LQG) is simply the
combination of a Linear Quadratic Regulator (LQR) and
a Linear Quadratic Estimator (LQE) [8]. This means that
LQG is a method of designing state feedback control laws
for linear systems with additive Gaussian noise that
minimizes a given quadratic cost function. The control
configuration is shown in Fig. 6. The design of the LQR
and LQE can be carried out separately. LQG enables us
to optimize the system performance and to reduce
measurement noise. The LQE yields the estimated states
of the process. The LQR calculates the optimal gain
vector and then calculates the control signal.
Figure 7. Addition of integrator to LQG
However, in state feedback controller designs
reduction of the tracking error is not automatically
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
159©2015 Engineering and Technology Publishing
realized. In motion control systems, Coulomb friction is
the major non-linearity, which causes a static error. This
problem can be solved, by introducing an additional
integral action to the LQG control structure. The
difference between process and model is integrated,
instead of the error between reference and process output.
Adding the integral term to the LQG control structure
leads to the system indicated in Fig. 7.
LQG design
We consider the LQG design based on the 2nd order
mathematical model of the plant to be controlled [4], [7].
Continuous LQR design
We consider a continuous-time linear plant described
by
{
̇
(6)
With a performance index defined as
∫ (
)
(7)
In (6) and (7) and are continuous state matrices
of the plant to be controlled, denotes the state of the
plant, is the tracking error, is the control signal,
and are matrices in the optimization criterion ( is
positive semi-definite weighting matrix and is positive
definite weighting). The optimal state feedback controller
will be achieved by choosing a feedback vector
(8)
in which is found by solving the continuous time
algebraic Riccati equation
(9)
The output of the state feedback controller is
̂ (10)
where
̂ [ ̂ ̂ ]
, (11)
̂ and ̂ denote the states of the estimator (see Fig. 8).
The following parameters are used in the simulation:
[
] [
]; [
];
[
]; . (12)
These values results in the following feedback
controller gains
[ ] [ ] (13)
Continuous LQE design
The feedback matrix yielding optimal estimation
of the process states is computed as
(14)
where is the solution of the following matrix Riccati
equation
(15)
in which and are continuous state matrices of the
plant to be controlled, is the system noise covariance,
and the sensor noise covariance. The following
settings were used:
[
] [
]; [ ];
[
]; . (16)
These values results in the following feedback
controller gains
[
] [
] (17)
The results lead to the control structure shown in Fig. 8.
Figure 8. The setup with order continuous LQG controller
Figure 9. Control signal is insensitive for measurement noise
With the LQG regulator, noise on the measurements of
the process has almost no influence on the system. This is
illustrated in Fig. 9; the real position state (second line)
and the position state error (fourth line) are corrupted by
measurement noise, whereas, the estimated position state
(third line) and the control signal (lowest line) are almost
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
160©2015 Engineering and Technology Publishing
clean. The integral action is used to compensate the effect
of the process disturbances (see Fig. 7 and Fig. 8). The
gain of the integrator can be tuned manually, or it can be
included in the solution of the Riccati equation [8]. By
comparing two simulation results as indicated in Fig. 10,
it is observed that when the integral action is used with
the tracking error is decreased.
Figure 10. Tracking error without (first line) and with (second line)
integral action
The estimator and feed-back controller may be
designed independently. It enables us to compromise
between regulation performance and control effort, and to
take into account process and measurement noise.
However, it is not always obvious to find the relative
weights between state variables and control variables.
Most real world control problems involve nonlinear
models while the LQG control theory is limited to linear
models. Even for linear plants, the mathematical models
of the plants are subject to uncertainties that may arise
from un-modeled dynamics, and parameter variations.
These uncertainties are not explicitly taken into account
in the LQG design.
IV. DESIGN OF CONTROLLED SYSTEM USING
OPERATIONAL AMPLIFIERS
A. Design of LQG Controller Using Operational
Amplifiers
Inverting and non-inverting amplifiers, weighted-sum
adders, integrators, and differentiators are used to design
analog controllers. From analog LQR and LQE blocks in
s-domain as shown in Fig. 8 we can design corresponding
analog electronic LQR and LQE circuits using
operational amplifiers that indicated in Fig. 11 and Fig.
12, respectively. The controller and observer parameters
with some explanations are listed in Table II and Table
III.
TABLE II. ANALOG LQR PARAMETERS
Symbols Parameters Notes
Sum 1
Calculate the tracking error
Proportional gain
Derivative gain
Sum 2
Calculate the control signal
Figure 11. Analog LQR using operational amplifiers
Figure 12. Analog LQE using operational amplifiers
In this step the LQG using operational amplifiers is
obtained. However we do not sure that converting the
controller in s-domain to corresponding analog electronic
circuits using operational amplifiers is correct or not.
B. Design of the Plant Using Operational Amplifiers
The proposed step is shown in this section. From
analog plant in s-domain is shown in Fig. 8 we can design
corresponding analog electronic plant using operational
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
161©2015 Engineering and Technology Publishing
amplifiers that shown in Fig. 13. The plant parameters
with some explanations are listed in Table IV.
Figure 13. The plant model using operational amplifiers
TABLE III. ANALOG LQE PARAMETERS
Symbols Parameters Notes
Sum 4
Calculate the prediction
error
̂
Inv
Create suitable sign for
estimated slider position
,
Optimal steady-state
filter gain
,
Optimal steady-state
filter gain
Sum 5
;
Weighted-sum adder
Int 2 ,
Create estimated slider
velocity ̂
Model parameter
Model parameter
Sum 6
Weighted-sum adder
Int 3
Create estimated slider
position ̂
TABLE IV. PLANT PARAMETERS
Symbols Parameters Notes
Plant parameter
Sum 7
Create slider velocity
Int 4 Integrator
Plant parameter
Int 5 Integrator
dc/m Invertor
V. SIMULATION AND EXPERIMENTAL TESTS
In this section the resulting LQG controller in s-
domain is firstly simulated using the Matlab Simulik.
Next the equivalent analog electronic LQG circuit is
performed and simulated using the Multisim software.
The simulation results in this step confirm converting
LQG in s-domain to equivalent electronic circuits is
correct or not. Finally, the analog electronic LQG circuit
is implemented and tested in the real setup.
We address the problem relating to the precision
control of the DC motor which requires the needs of
mechanical transmission from the rotary to linear motion.
In order to show implementation of the resulting
controlled systems, the following numerical values of the
input reference are fixed for all simulation and tests:
stroke [m], period [s].
Figure 14. Simulation results in s-domain using Matlab Simulink
software
Figure 15. Simulation results for equivalent electronic circuits using
Multisim software
Figure 16. Experimental results in the real setup
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
162©2015 Engineering and Technology Publishing
As we expected with the same input reference the
simulation results of the resulting controlled system in s-
domain (see Fig. 14), also in equivalent analog electronic
circuits (see Fig. 15), and experimental results in the real
setup (see Fig. 16) are almost the same.
VI. DISCUSSION
In order to design an analog controller using
operational amplifiers, we propose a design procedure
that has 6 following main steps:
Modeling the system in s-domain;
Analyzing and synthesizing the control system in
s-domain;
Simulating the controlled system in s-domain and
validating the simulation results;
Converting both the controller and the plant in s-
domain to equivalent electronic circuits;
Simulating the resulting control system in step 4
and checking the simulation results;
Implementing the real controller and testing on-
line in the real setup.
However, the above design procedure is recommended
for use in the non-sophisticated systems.
VII. CONCLUSIONS
Design of the LQG controller using operational
amplifiers for motion control systems has done based on
a new design procedure. We propose to add two extra
steps into the traditional design procedures. By
implementing the controller in different domains, the
performances of each domain are compared. With the
same reference input, if the design steps are implemented
correctly, the performances of the controlled system in s-
domain, also in equivalent analog electronic circuits, and
in experimental results will be almost the same. Both
simulation and experimental results confirm the precise,
featured by the proposed procedure.
REFERENCES
[1] Wikibooks. Control systems/Digital and analog. [Online].
Available:
and Analog
[2] WIKIPEDIA. Operational amplifier. [Online]. Available:
[3] WIKIPEDIA. Model-based design. [Online]. Available:
design
[4] N. D. Cuong, “Advanced controllers for electromechanical motion
systems,” Ph.D. thesis, University of Twente, Enschede, The
Netherlands, 2008.
[5] WIKIPEDIA. Linear-quadratic-Gaussian control. [Online].
Available:
control
[6] J. V. Amerongen and T. J. A. De Vries, Digital Control
Engineering; University of Twente, The Netherlands, May 2005.
[7] N. D. Cuong, “Application of LQG combined with MRAS-based
LFFC to electromechanical motion systems,” in Proc. 3rd IFAC
International Conference on Intelligent Control and Automation
Science, 2013, pp. 263-268.
[8] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems -
Theory and Design, Third Edition, Prentice Hall Information and
System sciences Series, Prentice Hall, Upper Saddle River, 1997.
[9] R. Burkan, “Modelling of bound estimation laws and robust
controllers for robustness to parametric uncertainty for control of
robot manipulators,” Journal of Intelligent and Robotic Systems,
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[10] D. Simon, “Kalman filtering with state constraints-A survey of
linear and nonlinear algorithm,” Control Theory and Application,
IET, vol. 4, no. 8, pp. 1303-1318, 2010.
[11] L. Lessard, “Decentralized LQG control of systems with a
broadcast architecture,” in Proc. IEEE Conference on Decision
and Control, 2012, pp. 6241-6246.
Nguyen Duy Cuong received the M.S. degree in
electrical engineering from the Thainguyen
University of technology, Thainguyen city,
Vietnam, in 2001, the Ph.D. degree from the
University of Twente, Enschede city, the
Netherlands, in 2008. He is currently a lecturer
with Electronics Faculty, Thainguyen University
of Technology, Thainguyen City, Vietnam.
His current research interests include real-time
control, linear, parameter-varying systems, and applications in the
industry.
Dr. Nguyen Duy Cuong has held visiting positions with the University
at Buffalo - the State University of New York (USA) in 2009.
Nguyen Van Lanh earned his Bachelor degree
in 2011 at the Nguyen University of Technology,
Thai nguyen city, Vietnam. He has been working
as a lecturer in Thai Nguyen University of
Technology since 2012. His research areas are
electrical, electronic and automatic control
engineering.
Bs. Nguyen Van Lanh has held visiting positions
with the Oklahoma State University (USA) in
2013.
Dang Van Huyen is a fourth-year student,
majoring in electrical engineering in Faculty of
International Training, Thai Nguyen University
of Technology. He is interested in electrical,
electronic and automatic control engineering.
Journal of Automation and Control Engineering Vol. 3, No. 2, April 2015
163©2015 Engineering and Technology Publishing
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