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Volume 2, Number 3, March 2012 (Serial Number 9)
Journal of Mechanics
and
Automation
Engineering
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Publication Information:
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Journal of Mechanics
Engineering andAutomation
Volume 2, Number 3, March 2012 (Serial Number 9)
ContentsTechniques and Methods
137 An Adaptive Dominant Type Hybrid Adaptive and Learning Controller for Geometrically
Constrained Robot Manipulators
Munadi, Tomohide Naniwa and Yoshiaki Taniai
149 Basic Aspects of Defining Mechanical-Technological Solutions for the Production of Biogas from
Liquid Manure
Nataa Soldat and Mirjana Radii
154 Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method
Vjacheslav Pshikhopov and Mikhail Medvedev
163 Cycle Analysis of Internally Reformed MCFC/SOFC-Gas Turbine Combined System
Abdullatif Musa and Abdussalam Elansari
169 Design, Development and Testing of a Wireless Multi-Sensors Network System
Chelakara Subramanian, Jean-Paul Pinelli, Ivica Kostanic and Gabriel Lapilli
184 Distributed Robot Control System Based on the Real-Time Linux Platform
Goran Ferenc, Zoran Dimi, Maja Lutovac, Vladimir Kvrgiand Vojkan Cvijanovi
190 Hybrid Algorithms for Multiobjective Optimization of Mechanical and Hydromechanical Systems
Valeriy D. Sulimov and Pavel M. Shkapov
Investigation and Analysis
197 Preventive Maintenance of Passengers Cars Driving in the Territory of the Republic of Kosovo
Xhemajl Mehmeti, Naser Lajqi, Bashkim Baxhaku, Shpetim Lajqi and Hajredin Tytyri
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mechanism against a force applied to it. Next, as the
name implies, the hybrid position/force control can be
used to track positional trajectories and force
trajectories in different subspaces simultaneously by
using two feedback loops for direct separate control of
position and force.
Refs. [3-4] have proposed the approach of
impedance control and hybrid position/force control,
in which they assumed that the dynamical parameters
of robot manipulator are known precisely such as
moment of inertia, mass of link, and joint friction. In
fact, it is difficult to measure the exact value of
dynamical parameters due to the uncertainty ofdynamical parameters of robot manipulator, hence, the
identification and estimation techniques [5-6] are
proposed. Furthermore, when the end-effector touches
with an object, the model-based adaptive control
methods [7-8] are proposed to track the desired
position trajectory and contact force trajectory by
estimating the unknown dynamical parameter of robot
manipulator accurately. It is also reinforced by the
experimental results in Ref. [9].
Meanwhile, the learning control is used to
compensate for the repeated errors for robot
manipulators to perform a given task repeatedly. The
learning control is a concept for controlling uncertain
dynamical system in an iterative manner or repetitive
manner. In Refs. [10-11], several learning control laws
have been discussed in which the learning control
does not require exact knowledge of the dynamics of
robot manipulator. So far, most researches on the
learning control have been focused on the problem ofperiodic trajectory tracking. Furthermore, relating to
the implementation of the hybrid position/force
control, there were several approaches which have
been proposed to describe a constraint surface for
force control [12-15]. In Ref. [16], the principle of
orthogonalization for position and force control was
presented. It introduces a projection matrix that
projects error vectors to the tangent plane of the
constraint surface in joint space.
In this paper, we are motivated to extend a simpler
of hybrid adaptive and learning control (HALC) for
hybrid position/force control that can achieve both
desired position and contact force trajectories
accurately when the robot manipulator is limited in
motion for keeping in touch with the smooth
constraint surface. The hybrid position/force approach
is developed based on the HALC law in Ref. [17] that
consists of the model-based adaptive control to cope
the unknown dynamical parameters, the learning
control to handle an assigned task repeatedly and also
the proportional-derivative control to stabilize the
closed-loop system and ensure the error convergence,in which the adaptive control input becomes dominant
than other inputs. Domination of adaptive control
input gives the advantage that the hybrid
position/force control could adjust the feed-forward
motion control input immediately. Whereas, if the
learning control input becomes dominant, the
proposed controller will need much time to relearn the
learning control input when the desired trajectory is
changed during motion. Furthermore, a Lyapunov-like
method is presented to prove the stability of the
proposed hybrid position/force control in which
asymptotic convergence of position and force errors to
zero is guaranteed. The effectiveness of the proposed
method is evaluated by computer simulation with a
model of two-link robot manipulator.
The paper is organized as follows: The dynamics of
robot manipulators and a constraint surface for the
end-effector are described in section 2. Section 3 then
presents the proposed controller for geometricallyconstrained robot manipulator. The stability analysis is
explained in section 4. Section 5 reports the
simulations results and section 6 concludes the paper.
2. Description of Constrained Robotic
System
2.1 Dynamics of Robot Manipulators
In this section, we consider a constraint surface of
robot system in which an end-effector ofnserial-link
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robot manipulator is required to move on a smooth
constraint surface, as shown in Fig. 1. Furthermore, let
, , denote a position vector of a
contact point of end-effector in a fixed task space
coordinates such as Cartesian coordinates and let also denote the joint angle position of robotmanipulator. The forward kinematic relation of and is expressed by a vector function as follows: (1)where . is generally a nonlineartransformation describing the relation between the
joint space and task space. Then, the velocity isgiven as follows:
(2)where is the manipulators Jacobianmatrix of with respect to . Therefore, theequation of motion for the constraint robot
manipulator is expressed in following form by
neglecting joint friction:
12 , (3)where
, are the joint angle velocity
and acceleration vector, respectively. represents inertia matrix, which is symmetric andpositive definite, , represents askew-symmetric matrix came from Coriolis and
Centrifugal force that is expressed by
, (4)And also represents the gravitational
force vector, represents the control inputvector generated by independent torque sources at
each joint, and represents the contactforce vector.2.2 Constraint Surface
Further, we consider a constraint surface for the
end-effector of robot manipulator, which moves in
touch with an object, and it can be defined in the
algebraic term as follows:
0
(5)
where : is a given scalar function.
Fig. 1 The constrained robot manipulator.
And taking the derivative of Eq. (5) with respect to
time gives the following expression:
0
(6)
It is assumed that the normal vector of the
constraint surface / is not a zero vector, so weobtain // (7)where /// is a unit normalvector in task space and is anormal vector of 0 in joint space suchthat
0 (8)
It means we can obtain the derivative of Eq. (8)
with respect to time as follows: 0 (9)Hence, using Eq. (7), the dynamics of robot
manipulator in Eq. (3) can be represented as follows:
12 , (10)in which represents the magnitude of contactforce, and
(11)3. Adaptive Dominant Type HALC
3.1 Definition of Error Signals and Regressor Matrix
A HALC is designed to make the robot
manipulators follow a periodic desired position
trajectory and a desired contact force trajectoryin accurately during an interval of finite duration 0, , in which denotes a period of desired
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trajectory. Furthermore, in order to define the
proposed controller, we make the standard assumption
with regard to Eq. (10) that all of the terms on
left-hand side of the dynamics of robot manipulators
are bounded if , and are boundedand uniformly continuous.
Later, we can define the joint angular error and integrated contact force error as follows: , (12)where is also assumed uniformly continuous.In this proposed controller, we define a reference
nominal velocity which is denoted as follows: (13)where and is a positive scalar gain, and denotes a projection matrix. The role ofthe projection matrix is to project the joint space
vectors onto a tangent plane of the surface 0 at point . It is defined by thefollowing form: (14)in which
is the pseudo inverse matrix of
defined as follows: (15)As long as the end-effector of robot manipulator
touches a constraint surface, it holds 0 atpoint , therefore we can confirm the followingproperties: , 0 (16)
Next, we consider the control input adopted from
Eq. (10). The dynamics of robot manipulators can be
rearranged in terms of unknown dynamical parameters which will be rewritten in this following expression:,,, (17)where ,,, is the nonlinear functionmatrix known as the regressor matrix that consists of
known functions of joint position, velocity and
acceleration, while represents vector ofunknown dynamical parameters such as mass,
moments of inertia, and distance from joint to center
of mass of each link. Moreover, based on the periodic
desired trajectory, we can denote the desired regressor
matrix as follows:
, , ,
, (18)And also, we present another type of regressormatrix which can be regarded as the residual regressor
matrix based on the reference nominal velocity in Eq.
(13) as the following expression:,, , , (19)Based on Eq. (18) and Eq. (19), we can express a
correlation regressor matrix between
and
, and
it is defined by as follows: ,, , , , , (20)Substituting Eq. (18), Eq. (20) into Eq. (10), in
which ,, , is linear in and ,we obtain the another form of dynamics of robot
manipulator that can be formulated by using inthe following expression: , (21)
is another error signal called as a
filtering tracking error. It is defined as the difference
between current velocity and nominal referencevelocity which is defined as follows: (22)where the right-hand side of the above equation
corresponds with following expression: (23)Note that this filtering tracking error
plays
important role to design the proposed controller and to
prove the stability of the proposed controller.
3.2 Design of Proposed Controller
Certainly, the proposed controller is designed to be
capable of guaranteeing the convergence of position
and force tracking errors when time towards infinity.
Hence, we propose an adaptive dominant type HALC
for position/force control of the constrained robot
manipulator. To illustrate the detailed design of
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proposed controller, we express the control input in
the right-hand side of Eq. (10) by following form:
(24)where , , and are the MBACinput, the RLC input and the proportional-derivative
(PD) control input, respectively. Now let us to
describe the MBAC input based on the residual
regressor matrix in Eq. (19), which can be resulted
according to following law: ,, , (25)where is an estimation of unknowndynamical parameter vector
at time
,and
. is updated on-line accordingto the following adaptive update rule: 0 ,, , (26)which implies ,, , (27)where is adaptive gain selected as a symmetricpositive definite matrix.
As the second component of control input, the RLC
input is resulted based on recalling the filtering
tracking error and storing error to update the controlinput for next period. The RLC input is declared as an
original learning law by adding a forgetting factor in
the following expression: (28)where is a forgetting factor selected as a positivescalar value which satisfies 0 1. We have tonotice for defining a forgetting factor in the RLC law,
because will make the RLC input approach tozero and also make the MBAC input to be dominant.
This strategy will make the adaptive control input be
greater compared with other inputs when the control
input of the proposed controller achieves the actual
position and force trajectories converging to the
desired position and force trajectories. Next, is alearning gain selected as a symmetric positive
definite matrix. In this approach, the RLC input is
initialized as 0 for 0, , also satisfies
0 0.
For the PD control input, it is resulted by the
filtering tracking error multiplied by its gain and is
defined as follows:
(29)where is a PD gain selected as a symmetricpositive definite matrix. Absolutely, utilization of this
PD feedback in the proposed controller is to stabilize
the closed-loop system and ensure the error
convergence.
Finally, we combine the dynamics of robot
manipulator in Eq. (21) with the proposed control
input law in Eq. (24), and obtain the closed-loop
system of robot manipulators expressed in following
compact form: , (30)4. Stability Analysis
Stability is a fundamental issue in analysis and
design of control system. In this section, we will prove
the stability of proposed controller, so the actual angle
position and contact force trajectories of the robot
manipulator converge to their desired position and
force trajectories as .4.1 Lyapunov Function Candidate
We use the Lyapunov-like method to prove
asymptotic stability of the proposed controller. Now, a
Lyapunov function candidate is defined inthe following lower bounded function:
0 (31)where (32)12
(33)
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We show that the above equations satisfy 0since , and are positive definite, andalso0 1.
Having determined the Lyapunov function
candidate to be positive definite, then we compute its
time derivative. By substituting Eq. (32) and Eq. (33)
into Eq. (31), and differentiating with respect totime, we have
(34)
Utilizing the closed-loop system of robot
manipulators in Eq. (30) and differentiating byconsidering the properties of robot manipulators in
which , is skew-symmetric, , 0 , we can simplify as follows:
(35)where
(36) (37)
(38)
Next, substituting Eq. (20), Eq. (25) and Eq. (27)
into Eq. (36) yield another form of expressedas follows:
(39)
While based on the RLC law in Eq. (28), inEq. (37) can be expressed in following result:
12
12
12
(40)
And in Eq. (37) can be asserted as follows:
(41)
Meanwhile, after substituting Eq. (22) and Eq. (16)
into Eq. (38), we have :
(42)
Finally, by substituting, and into in Eq. (35), we can represent the derivative of theLyapunov function candidate as follows:
12
(43)Since and are positive definite, and is
positive, we can select the control gains in above
based on the following sufficient condition:
0, and 0 (44)
and it means that 0 . This implies theboundedness of of and . In addition,based on Eq. (32) and Eq. (33),
and
are
also bounded.
4.2 Uniform Boundedness of Joint Angular Error
In this section, we will prove the uniform
boundedness of when the constraint surface 0 is smooth enough. According todefinition of in Eq. (23) and in Eq. (14),we can denote
1
2
(45)
where
(46) is a projection matrix into the
complementary subspace of the contact force vector.
Next, we consider the fact of Eq. (8), we have
(47)
and the inner product of and in Eq. (45)
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can be rewritten as follows:12
(48)
since (49)Furthermore, the assumption of the constraint
surface is smooth, it will make becomingnearly perpendicular to the unit normal when tracks approaching to in same direction,in which it satisfies
. This
means that there is a constant scalar value satisfying which implies (50)At the same time and in the same way, we can also
declare a constant scalar value which is used tosatisfy the following assumption: (51)when . To note in above equation that
depends on the radius of curvature of the
constraint surface at point , in which theconstant will be smaller as long as the radius ofcurvature increases.
Thus, we submit Eq. (50) and Eq. (51) into Eq. (48),
and the result is divided by , then we willobtain following equation: (52)where (53)
According to Eq. (31),
is non-increasing in
and that consists of and areorthogonal to each other is bounded, so we can
express more precisely by following equation: 0 000 (54)where and denote the minimum andmaximum eigenvalues of matrix over all .Later, we can define
(55)
From Eq. (54), it follows that for any 0, 0 1 0 0 0 0 (56)
where
; 0 00Further, substituting Eq. (56) into Eq. (52), it yields
0 0 (57)provided that 2. Based on Eq. (56) whichdefines , and the fact 1, and referring Eq. (56),Eq. (57) can be represented as follows:
0 0 0
(58)
Next, if is large enough satisfying 0 0 (59)
and 0 satiesfies0 (60)then it follows from Eq. (58) that (61)
Eq. (61) shows the uniformly boundedness of joint
angular error
.
4.3 Convergence of Filtering Tracking Error and
Contact Force Error
For showing convergence of and , wecan assume that the second order differential of the
constraint surface 0 is continuous andbounded in and thus , / , , and/ is uniformly continuous in . Next, thedynamics of robot manipulator in Eq. (30) can be
rewritten as follows:
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, (62)Since
is used in the dynamics of robot
manipulator, it can be described in the following
expression: , (63)Now, we can rewrite the dynamics of manipulator
as follows: (64)where
12 , 12 , , , , (65) indicates the inertia matrix calculated on thebasis of the estimation of unknown dynamical
parameter vector , and represents theremaining bounded terms of
and
. Since
is bounded, based on Eq. (22), is alsobounded together with and . Then, thereference nominal velocity in Eq. (13) can bederived with respect to time, and it yields (66)where (67)
Hence, are also bounded.Next, substituting Eq. (66) into Eq. (64) yields:
(68)Multiplying Eq. (68) by and then it
is continued by substituting , we have
(69)
For the left-hand side of eq. (69), it can be rewritten
as follows:
(70)Since 0 is small enough as a positive constant
value, is bounded, and the inside of the squarebracket of Eq. (70) is also bounded and positive
definite. Whereas in Eq. (66) shows theboundedness of , and the boundedness of follows next equation:
(71)Further, both and are uniformlycontinuous referring to Eq. (69), and belong to the since is bounded, and certainly it implieslim 0 ; lim 0 (72)
This condition is identical form to in Eq. (23)converging to zero:lim lim 0
(73)
5. Computer Simulation Results
In this section, the simulation results are presented
to illustrate the performance of the proposed controller.
We consider a model of two-link robot manipulator
with two revolute joints of joint variables , , linklengths 0.4 m, masses 0.2 kg, distance between the joint to the center ofmass 0.2 m , and moment inertias 0.0107 kg m. Fig. 2 shows the model oftwo-link robot manipulator in which the end-effector
is required to move along on a constraint surface in
the Cartesian coordinates.
For the dynamics of robot manipulator, according
Eq. (10), it can be written in detail as follows:
(74)for 2
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Fig. 2 A model of two-link constrained robot manipulator.12 , 0 (75)and
(76)where sin, sin, sin , cos , cos, cos
and
9.81 m/s.
Furthermore, for the regressor matrix ,,, ,the unknown dynamical parameters is defined by
(77)
Meanwhile, when the length of links is denoted by and with reference to the geometry given in Fig.2, the position of the end-point is given as follows:
(78)For the Jacobian matrix which maps fromjoint space to Cartesian space for two-link robot
manipulator is given by
(79)In this simulation, the geometric constraint of
end-point is described by following expression: (80)in which we specify
0.275 . Furthermore, the
robot manipulator is requested to track the periodic
desired position trajectory , in which byconsidering the relation between the constraint surface
and for , we have to define and it yields as follows:
2 2 (81)Both desired position trajectories are shown in Fig.
3. For , the desired force trajectory is given as aconstant value, 2 [N]. And according to Eq. (44), the
control gains
, , and
are selected as:
(a)
(b)
Fig. 3 The periodic desired trajectory for eachperiod at joint (a) 1 and (b) 2.
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0.5, 0.01,0.01,0.01,0.01,0.01 , 3.0,3.0, and 2.9,2.9respectively. For other gains, we define
3.0,3.0, 8.0, and 3.0.The simulation is started by initialing all initialcondition of estimation of unknown dynamical
parameters as 0, 0 0 . Later, the simulationresults of the proposed controller are shown in Figs.
4-7. Fig. 4a shows position tracking error resulted at
joint 1 and Fig. 4b is error resulted at joint 2,
respectively. At joint 1, the position tracking error is
reduced each period, especially the error shrinks
drastically after the second repetition at 4 [s] from0.045 to 0.006 [rad]. And also at joint 2, It is
decreased from 0.0330 to 0.0055 [s]. It can be seen
that the tracking performance was considerably
improved after the second repetition and it converges
asymptotically to 0.
(a)
(b)Fig. 4 The position tracking error
at joint (a) 1
and (b) 2.
(a)
(b)
Fig. 5 The velocity tracking error from repetitionof trajectory at joint (a) 1 and (b) 2.Figs. 5a-5b show the velocity tracking performance
improvement for the two joint. At the initial repetition,
the maximum velocity errors were about 0.18 and
0.15 [rad/s], respectively. But after second repetition,
the maximum values were reduced to 0.050 and 0.008
[rad/s]. Meanwhile, Fig. 6a shows the force tracking
performance of the controller and the force tracking
error converges asymptotically, but it is very slowly.
And Fig. 6b shows the estimation of unknown
dynamical parameters during the execution of
prescribed desired trajectory. Based on Fig. 6b for 0 s, all estimated value of is 0, then , and increase leading up to relatively fixed valueof , and , in which we obtain variousestimation of dynamical parameters 0.03 , 0.022 and 0.015. Meanwhile, and
decrease and will tend to -0.015 and -0.042,
respectively.
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(a)
(b)
Fig. 6 The force error trajectory (a) and theestimation of unknown dynamical parameter (b).
Furthermore, the motion control inputs are shown
in Fig. 7, in which the MBAC input is shown by solid
line and the RLC input by dashed line. Based on Fig.
7a which shows the MBAC input and RLC input at
joint 1, the initial MBAC input is comparable with
RLC input in early period, but for next period, the
MBAC input increases and becomes dominant toachieve desired trajectory be compared with other
inputs. This condition describes the meaning of
domination of adaptive input in the proposed
controller. Whereas, the RLC input has the highest
torque in the second period, and then it decreases
according to the learning updated law.
6. Conclusions
In this paper, we have studied the hybrid
position/force control problem with the geometric
(a)
(b)
Fig. 7 The required torque profile for MBAC input and
RLC input at joint (a) 1 and (b) 2.
constraint of end-effector of robot manipulator. This
control method is a simpler combination of
model-based adaptive control that estimates the
unknown dynamical parameters, a repetitive learning
control that uses the input torque profile obtained
from the previous repetition, and a traditional PD
control. The proposed controller incorporates both
adaptive and learning capabilities, therefore, it can
provide an incrementally improved tracking
performance of position error by increasing the
number of repetitive tasks. The position/force tracking
errors have been proven to converge asymptotically
via Lyapunov-like stability analysis. The numerical
simulations have validated the effectiveness of the
proposed controller by showing the position and
velocity tracking errors decrease with the increase of
the repetition number.
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operation of robots by learning, J. Robot. Syst. 1 (2)
(1984) 440-447.
[11] M. Aicardi, G. Cannata, G. Gasalino, Hybrid learning
control for constrained manipulators, Advanced Robotics
6 (1992) 69-94.
[12] M. Vukobratovic, Y. Ekalo, New approach to control of
robotic manipulators interacting with dynamic
environment, Robotica 14 (1) (1996) 31-39.
[13] C.C. Cheah, S. Kawamura, S. Arimoto, Stability of
hybrid position and force control for periodic manipulator
with kinematics and dynamics uncertainties, Automatica
39 (2002) 847-855.
[14] C.S. Chiu, K.Y. Lian, T.C. Wu, Robust adaptive
motion/force tracking control design for uncertain
constrained robot manipulators, Automatica 40 (2004)
2111-2119.
[15] Y. Karayiannidis, G. Rovithakis, Z. Doulgeri,
Force/position tracking for a robotic manipulator in
compliant contact with a surface using neuro-adaptive
control, Automatica 43 (2007) 1281-1288.
[16] S. Arimoto, Y.H. Liu, T. Naniwa, Principle of
orthogonalization for hybrid control of robot arms, in:
Proceedings of 12th IFAC World Congress, 1993, pp.
507-512.
[17] Munadi, T. Naniwa, An adaptive controller dominant
type hybrid adaptive and learning controller for trajectory
tracking of robot manipulators, Advanced Robotics 26
(2012) 45-61.
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Journal of Mechanics Engineering and Automation 2 (2012) 149-153
Basic Aspects of Defining Mechanical-Technological
Solutions for the Production of Biogas from Liquid
Manure
Nataa Soldat1and Mirjana Radii
2
1. Department of Mechanical Engineering, Faculty of Mechanical Engineering, University of Belgrade, Belgrade 11000, Serbia
2. Department of Mechanical Engineering, Technical College of Applied Sciences in Zrenjanin, Zrenjanin 23000, Serbia
Received: January 03, 2012 / Accepted: January 30, 2012 / Published: March 25, 2012.
Abstract: Defining mechanical-technological solutions for the production of biogas requires defining of single devices and their
functional integration into a whole. This paper defines mechanical and technological solutions for the production of biogas from
liquid manure, the choice of material, the method, the possibility of adaptation of existing devices and the removal of hydrogen
sulphide. The research and the results thereof are based on years of research done on existing facilities for the production of biogas.
The results show that the volume of the digester increases proportionally to the daily influx of fertilizer. Besides that, it should be
noted that all the insulating materials must be coated with a watertight material (thin aluminum foil) in order to prevent the alteration
of thermal insulating material.
Key words: Liquid manure, biogas, facilities.
1. Introduction
The production of biogas from liquid manure is
carried out using anaerobic fermentation in a facility
called reactor, where this process is carried out in
multiple phases. For each phase a functional and
dedicated group of devices is used.
This paper shows the research that was carried out
with regard to finding mechanical-technological
solutions for the production of biogas fro liquid manure,
the selection of a facility for anaerobic fermentation,the method for their construction, and also the
possibility of adjusting existing devices and purifying
biogas.
The objective of this paper is to identify the most
suitable reactors for the production of biogas
fromliquid manure by studying the mentioned
Mirjana Radii, Ph.D., research field: manufacturing ofbiomaterial.
Corresponding author:Nataa Soldat, M.Sc., research field:
materials. E-mail: [email protected].
mechanical-technological solutions, along with thepossibility of adjusting devices and purifying biogas.
The paper consists of the following sections:
Section 2 proposes mechanical-technological solution
for the production of biogas; section 3 introduces
digester volume; section 4 is the insulation of digester;
section 5 is choosing an anaerobic fermentation
facility and section 6 talks about the possibility of
reactor adjustment and purification of biogas.
2. Mechanical-Technological Solution for theProduction of Biogas
Defining a mechanical and technological solution
for the production of biogas from liquid manure is
based on defining the single devices and putting them
together into one functional unit.
The production of biogas comprises a number of
phases [1]:
Preparation of the substrate;
Fermentation;
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Capturing the gas;
Using biogas.
To every single phase a functional and dedicated
group of devices is assigned.
Every biogas production plant consists of the
following:
Raw material storage and processing facility;
Digester;
Overfermented substrate disposal system;
Biogas storing system.
The basic-centre part of a biogas facility is the
digester, i.e., biogas reactor for anaerobic digestion, to
which other components of the biogas facility, i.e.,biogas processing equipment is connected.
A group of devices for maceration and separation of
coarse particles is used for the process of collecting
and preparation of the substrate. Actually,
polypropylene meshes and centrifugal fecal sledge
pumps, usually with two buckets are used for this.
The main phase of the biogas production process is
the anaerobic fermentation which is a multistage
biochemical process used on a number of different
types of organic substances. The technological process
of anaerobic fermentation depends on a number of
conditions, which, when met, produce a result
showing a high degradation level for organic matter
together with an acceptable quality and amount of
biogas.
Some of them can be monitored and in that way the
production of biogas can be controlled. That refers
mostly to: pH level (acidity), temperature, retention
time, filling level and toxicity.
3. Digester Volume
In almost all types of digesters, all three phases of
anaerobic digestion are carried out simultaneously
within the same volume-digester. That also is the
reason why the substrate hydraulically is kept for such
a long time in the digester. This time ranges between
12 and 20 days (sometimes 30). The degradation levelof organic matter is from 45 to 65%. The volume of
the digester increases proportionally to the daily influx
of fertilizer. This function is shown in Tables 1-2.
4. Insulation of Digester
As most digesters for biogas production work in a
mesophile temperature range, it is necessary to
thermally insulate the digesters well, to minimize the
loss of heat. Insulating materials can be natural and
synthetic. Synthetic materials (polystyrol and
polyurethane) are used more frequently, because they
are easier to shape and process, and they also have
better thermal insulating characteristics and they are
also cheaper.
Table 1 Required digester volume depending on the content of dry material in liquid fertilizer.
Content of dry material (%) 8 7 6
Liquid fertilizer mass (kg) 76,240 87,129 101,650
Usable digester volume (m3) 1,530 1,742 2,033
Index 100 114 133
Organic load of digester (kg SM/m3/day) 3.98 3.50 3.00
Specific production of biogas (Nm/m3/day) 2.14 1.85 1.58
Table 2 Required digester volume depending on the content of dry material in liquid fertilizer.
Content of dry material (%) 5 4 3.5
Liquid fertilizer mass (kg) 121,980 152,475 174,254
Usable digester volume (m3) 2,433 3,044 3,485
Index 159 199 278
Organic load of digester (kg SM/m3/day) 2.50 2.00 1.75
Specific production of biogas (Nm/m3/day) 1.34 1.06 0.92
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All insulating materials must be coated with
watertight material (thin aluminium foil) in order to
prevent the alteration of thermal insulating material.The technical process of biogas production is
economical if the heat loss does not exceed 15 to 40%.
Tables 3-4 show the thickness of the digesters
thermal insulation.
5. Choosing an Anaerobic Fermentation
Facility
The biogas production facility consists of a
complex installation incorporating a large number of
parts. The appearance of the facility dependa on the
type and amount of raw material that will be used for
the production of biogas.
The facility where the anaerobic fermentation takes
place is called reactor. Its solution is a basic condition
for a good performance of the whole installation.
Globally, as well as in our country, there are a number
of reactors that are different in structure and the
material they are made of [2-3].
Reactors most often are discontinuous and usually
there are at least two or three in series. Continuous
reactors are rarer. Both reactor types are suppliedthrough a hydraulic sealing system, which, for now, is
a practical and simple solution.
As already mentioned, during the process of
anaerobic fermentation, the matter is biologically
degraded. Besides the absence of oxygen, this process
also needs constant temperature. The degradation of
matter is most efficient on a temperature of 15C
(psychrophile), 35C (mesophile) and 55
C
(termophile process). The mesophile process is used
most frequently, and in the summer season also the
termophile process [2].
Reactors are classified by shape, size, type, by the
material they are made of, the mixing system or
substrate heating used. Depending on the material they
are made of reactors are from: steel, concrete or
plastics. Very rarely they are made of stainless steel.
Fig. 1 shows a vertical reactor made of concrete,
one that is most widely used [3].
Table 3 Required thickness of the digesters thermal insulation.
Container diameterMinimum production
of biogas in m3/dayMineral wool Epoxide resin
1.6 7.1 125 109
1.8 10.1 111 97
2.0 13.8 100 88
2.2 18.5 91 80
2.4 24.0 83 73
2.8 36.0 73 64
3.0 47.0 67 59
3.5 75.0 58 51
Table 4 Required thickness of the digesters thermal insulation.
Container diameterMinimum productionof biogas in m3/day
Polystyrene plates Dry chips
1.6 7.1 105 250
1.8 10.1 93 230
2.0 13.8 84 210
2.2 18.5 76 190
2.4 24.0 70 170
2.8 36.0 61 150
3.0 47.0 56 135
3.5 75.0 49 120
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152
Fig. 1 Vertical reactor made of concrete.
Due to the corrosion of steel and the porosity of
concrete, it is necessary to plastify the first two types.Due to its chemical stability, easy use and low
plastification cost and unsaturated polyester is used as
a basic reinforcing material. In order to achieve
mechanical reinforcement, the polyester layer is
reinforced with glass fiber, and to enhance the shock
strength, a bitumen component is added to polyester in
form of a 50% bitumen stirene or toluene solution. An
ortho-or isonaphtal polyester resin is used for this.
Such reinforced protection containing approx. 30% of
alcohol-free glass fiber, draws us towards the idea that
most of the vital device parts, such as the ractor and
the bell, can entirely be made of polyester laminate [2].
Due to gas permeability and water absorption, and
also the inhibition of enzymate reactions, especially
saponification under the influence of different types of
estherases, the hydrophility of the laminate is reduced
by adding 3% of a five percent solution pf paraffin in
styrene. Gas permeability is reduced by foliar
multilayer lamination.
Mechanical and other properties of laminate areshown in Table 5.
As for the overfermented substrate disposal system,
it is pumped out of the digester and through the piping it
reaches the tanks where it is stored. The tanks are located
near the digester where it is stored for a limited time (few
days). The digested material can be stored in concrete
facilities which are covered with natural or artificial
floating layers or membranes or even in lagoons.
Fig. 2 shows a storage tank covered by a
membrane.
It is possible that a part of the methane and
nutritious matter is lost during storing and handling of
overfermented substrate. It has been empirically
shown that up to 20% of total biogas production
occurs in the tanks. To avoid methane emissions and
to collect the additionally produced gas, tanks always
have to be covered with a gas-non-permeable
membrane in order to collect the gas.
Fig. 2 Storage tank covered by a membrane.
Table 5 Mechanical and other properties of laminate.
Tensile strength 100-140 N/mm2
Bending strength 120-160 N/mm2
E-module from bending 6000-8000 N/mm2
Pressure strength 240-300 N/mm2
Shock strength 70-90 N/mm2
Max. Stretch till break 2%
Content of glass 30%
Density 1450 kg/m3
Coefficient of thermic conductivity 0.20 W/mK
Resistance to temperature change -40-+120 C
Linear expansion coefficient 3 10-5K-1
Absorption of water (24 h on 20
C) max 0.2 %
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Biogas can also be stored within the gas space of
the digester itself (with a mobile or a fixed canopy)
and in biogas tanks which can be as follows: Wet, low-pressure tanks (gas bell over substrate
or water;
Plastic foil gas cap;
Low-pressure dry tank;
Pressurized dry tank.
The pressure on low-pressure tanks is never greater
than 30 kPa. U Empty space is left in the digester (at
least 2-8% of the volume on large, and about a third
on small digesters) that is intended for the reception
and storage of biogas. If that space is small or if the
production and consumption of biogas on a farm are
not balanced, then it is necessary to build a special
biogas tank.
The storage of biogas in special tanks is expensive.
Building costs for tanks of up to 200 m3capacity are
20 to 30% of the digesters price.
6. Possibility of Reactor Adjustment and
Purification of Biogas
The adjustment of both steel and concrete reactors
is easy. On concrete facilities, non-permeability of gas
is also achieved using a penetration layer and foliar
lamination.
Purification of biogas, i.e., removing hydrogen
sulfate and water prevents the installations from
corroding, and the removal of CO2 increases the
caloric capacity of biogas.
The level and method for the purification of biogas
depends on the method of use, purpose and other
factors. When biogas is compressed it has to be dried.
Usually it is dried through absorption, i.e., using
agents that are binding water, as are calcium
hydroxide or calcium chloride.
The deficit of this procedure is that also a part of
CO2, CaO and CaCl2 is absorbed, increasing the
consumption of lime. When biogas passes through the
granular layer, the granules of CaO and CaCl2adhereto each other, binding water. In that way the absorbent
cloddes and prevents the flow of biogas.
7. Conclusions
Studying mechanical-technological solutions for the
production of biogas from liquid manure, we can
conclude that due to their characteristics, concrete,
steel and plastic reactors are most frequently used.
Due to the corrosion of steel reactors and the
porosity of concrete reactors, they need to be
plastified. Polyester is used as a basic reinforcement
material, due to its chemical stability, easy use and
low plastification cost, in form of a polyester coating,
the reinforcement is achieved through glass fiber and
the shock strength is enhanced by adding a bitumen
component in shape of a 50% bitumen-styrene or
toluene solution.
The adjustment of steel or concrete reactors is
carried out in using a penetration layer and foliarlamination for concrete facilities, achieving gas
non-permeability at the same time.
Purification of biogas, i.e., the removal of hydrogen
sulfate and water prevents the installations from
corroding, and the removal of CO2 increases the
caloric capacity of biogas.
References
[1] M. ulbi, Biogas (dobijanje, korenje i gradnja
ureaja), Novinsko-izdavaka radna organizacijaTehnika knjiga, Beograd, 1986.
[2] M. Radii, Proizvodnja i primena biogasa, Technical
College of Appplied Sciences in Zrenjanin, Zrenjanin.
2006.
[3] N. Soldat, Biogas-mogunosti proizvodnje i primene
(diplomski rad), Faculty of Mechanical Engineering,
University of Belgrade, Beograd, 2010.
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Journal of Mechanics Engineering and Automation 2 (2012) 154-162
Block Design of Robust Control for a Class of Dynamic
Systems by Direct Lyapunov Method
Vjacheslav Pshikhopov and Mikhail Medvedev
Department of Mechanical Engineering and Mechatronics, Technological Institute of Southern Federal University, Taganrog
347928, Russian Federation
Received: January 31, 2012 / Accepted: February 15, 2012 / Published: March 25, 2012.
Abstract: In this paper, a new method of a robust control design for a class of nonlinear multilinked dynamic systems is developed.
The method is advanced block control theory, which allow to provide compensation of object nonlinearity to closed-loop system with
sliding regime. This paper suggests approach, which allows to eliminate disadvantage of all sliding regime systems, that is unstability
in small. It is made by means of substitution of relay control by similar continuous control. The considered class of systems is called
block systems. Block form allows to describe a big class of controlled objects. In this work new controllability estimation approach
for nonlinear multilinked systems is suggested on the base of optimal control ability conditions check. The design procedure is based
on the direct Lyapunov method. It provides closed-loop system asymptotic stability. Control design for a single block system is based
on quadratic function of Lyapunov. For a general block system step-by-step design procedure is developed. Suggested synthesis
method provides closed-loop system stability characteristic robustness to right side equations of object in an area, defined by
limitations for control actions. Suggested approach also considers limitations for object variable states. The results of theoretical
analysis, solvability conditions of the control design equations, and robust control algorithms are presented. Theoretic results are
implemented on experimental robotic mini-airship and wheeled vehicle.
Key words:Nonlinear system, robust control, block system.
1. Introduction
Design of adaptive control systems is of significant
interest nowadays. This interest is attracted by unique
capability of adaptive control systems [1-5]: capability
to operate under condition of uncertain parameters,
uncertain mathematical model, and unmeasured
disturbances. Nowadays adaptation of systems are
based on searchless direct and indirect adaptivecontrol, robust control, search adaptive control,
invariant control, relay control, fuzzy logic control,
and neural network control.
In Refs. [6-7], a new method of the relay robust
control systems design was proposed. In this paper a
block method of the robust control by a class of
nonlinear control systems is developed. The method
Corresponding author: Mikhail Medvedev, professor,
D.Sc., research fields: automatic control, robotics, adaptive
control, estimation. E-mail: [email protected].
takes into account state variables, control actions
limits. In addition the function of Lyapunov for the
designed control systems is constructed. Moreover the
matrix controllability conditions are presented via
scalar inequalities.
In Refs. [6-7], a relay control is designed. Therefore
the designed closed-loop systems are unstable in small
neighborhood of steady-state point. In this paper, a
continues approximation of relay control is applied.
Parameters of the approximation are defined by direct
Lyapunov method.
The paper is organized as follows: Section 2
contains description of synthesis method for object,
consisting of one block, and also stability conditions
of these objects. Suggested approach is generalized for
system class, composed of a few sequentially
connected blocks in section 3. Suggested approach is
advanced in section 4 according to limitations of
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method 155
object variables states. Similarities of suggested
synthesis method and time optimal control are
discussed in section 5. Robust control systems
synthesis examples for test objects, for wheeled
mobile robot and for airship-based robotized complex
are described in section 6. Implementation of robots
using algorithms, synthesized with suggested in the
paper algorithm is described in section 7.
2. Robust Control Design
Suppose that a control object is
( ) ( )uxBxfx += (1)
where xa vector of state variables; ua vector of
control action; ( ) ( )( )Tif xxf = , ni ,1= afunctional vector of the state variables;
( ) ( )( )Ti xbxB = , ni ,1= a mn functionalmatrix.
Let the control action u be bounded:
max
jj uu , mj ,1= (2)
where
max
pu positive constants.Let the control ( )Tmuuu 21=u be a function
of the state variables x . We have to design the
control vector u such that the closed-loop system is
stable, and robust.
Now we introduce the following auxiliary vectors:
x= , ( )Tmuuu maxmax2max1max ,,=u (3)
Using the vectors , andmax
u we introduce the
next theorem.Theorem 1: Suppose the control vector for object
(1), (2) is
( )( )xBuu Tsign= max (4)and the next inequalities
( ) ( )xuxb ii f>max , ni ,1= (5)
are satisfied; then function of Lyapunov for
closed-loop system (1)-(4) is
T
V2
1=
(6)
Theorem 1 can be proved by direct calculation of
function (6) time derivative. We have
( ) ( ) ( )( )( )xxBuxBxfxxx
TT
TT
sign
V
===max
(7)
If (5) is satisfied, then the time derivative (7) is a
negative definite function. The theorem is proved.
From theorem 1, it follows that control (4) ensures
stability of the closed-loop system (1)-(4) if (5) is
satisfied. The assumptions of theorem 1 does not
include continuity of the ( )xf . But the functionalvector ( )xf is bounded. Note that control (4) doesnot depend from the vector ( )xf .
Inequalities (5) are equivalent to the controllability
condition presented in Ref. [8].
If nm then we have the following theorem.Theorem 2: Suppose (5) is satisfied, and nm ,
then the state commonness condition [9] is satisfied:
)nD GGG ,...,, 21= , nrankD = (8)( )xBG =1
( ) ( )( )
( ) ( )
j 1
j
j 1
= +
+
GG f x B x u
x
f x B xu G
x x ,
nj ,2=
Using (5) we get
( ) 0max >uxbi , ni ,1= (9)
Since (9) is performed it follows that
( ) 0max uxbi , ni ,1= (10)
Writing (10) in a vector form, we obtain
( ) 0uxB max (11)
where 0a zero vector.
From (11) it follows that
( )( ) nrank =xB (12)
If (12) is performed, then (8) is satisfied. Theorem 2
is proved.
Theorem 2 defines sufficient conditions of
controllability of the system (1)-(2). Thus the
sufficient condition of controllability (8) can be
substituted by scalar inequalities (5).
Moreover condition (12) is necessary condition of
the system (1), (2), (4) stability. Function (7) must be
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method156
continues. Therefore we have
( ) ( ) ( )( )( )
( )
( ) ( )( )
max
0
0
max
0
lim
lim
lim 0.
+
+
+
= +
=
T T
x
T
x
T T
x
sign
sign
x f x B x u B x x
x f x
x B x u B x x
( ) ( ) ( )( )( )( )
( ) ( )( )
max
0
0
max
0
lim
lim
lim 0.
= +
=
T T
x
T
x
T T
x
sign
sign
x f x B x u B x x
x f x
x B x u B x x
But the last equations are satisfied if (12) is
performed.
3. Block Robust Control Design
In general case controls can directly impact just on
part of state space variables. Therefore the block
control design is developed in this section.
Now we introduce the next definition.
Definition 1: System (1) is a block system if it can
be presented in the next form:
) ( ) ( )uxBxfxxxfx +== + kkiii ,,..., 11 (13)1,1 = ki
where
( )Til
iii
ixxx 21=x , ( )
Tkxxxx ...,,,
21= ;
( ) ( ) ( )( )Tiiliiii ff 1111111 ,...,,...,,..., i+++ = xxxxxxf
,
( ) ( ) ( ) ( )( )Tkk xfxfxfxf ,...,, 2211=;
( ) ( )( )xxB ijb= n x m-matrix, ka positive integer.
A block system is a special case of a controllable
Jordan form [10].If 2k , then the vector u impacts to the vectork
x . The vectork
x impacts to the vector1k
x ,etc.
In general case a vectorix is a fictitious control if
ix impacts to
1ix .
If system (1) is a block system, then the control
design block procedure is used.
To design the control vector ( )kxxuu ,...,1= weintroduce the next auxiliary vectors:
kk
x = ,kiii
xH += , 1,1 = ki (14)
whereiH matrices of weighting factors.
We introduce the next theorem using (14).
Theorem 3: Suppose for block system (2), (13) the
next conditions are satisfied:
( )
=
=
n
i
iTsign
1
maxxBuu (15)
( ) kii
ij ff +>max
uxb (16)
Then Lyapunovs function of system (2), (13)-(15)
is
( )=
=k
i
iTiV
12
1 (17)
Theorem 3 can be proved by direct calculation offunction (17) time derivative. We have
( ) ( ) ( ) ( )
( )( )
k kiT max T i
i 1 i 1
k 1iT i k kT k
i 1
V sign= =
=
=
+ + +
B x u B x
f f f
(18)
Under condition (16) of the theorem we have
function (18), which is a negative definite function.
The theorem is proved.
Any arbitrary system can be presented by (13) via
corresponding denotations of state variables.
Expression (14) transforms system (13) to the single
block system (1). It is clear, that transformation (14) is
nonsingular ifi
H is nonsingular.
4. Block Robust Control Design under State
Variables Bounded
Let the vector ( )Tiliii ixxx 21=x , ki ,2= be
bounded by constant. Then we havemax
j
i
j xx , ki ,1= , ilj ,1= (19)
wheremax
jx are positive constants.
Assume system (13) is presented in the next form:
( )
( )
1 1,...,i i i i i
k k k
+= +
= +
x f x x B x
x f x B u (20)
1,1 = ki
Now we shall give the following theorem.
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method 157
Theorem 4: Suppose for block system (20), (2), (19)
the next conditions are satisfied:
iii
xf < ,1
max
+
T
,111 +> ii
TiiiiTiqBBqBB
,
ki ,2= (24)
Then Lyapunov function of system (20), (22) is
kTkV 5.0= (25)
Theorem 4 can be proved by direct calculation of
function (25) time derivative. In small neighborhood
of the 0=i we have
( ) iiiiii BqBq ~tanh (26)From (22), (25)-(26) we obtain
(
( )( ))(
( )( ))
k k max k 1T k 1
Tk 1 k 1max k 2T k 2 k 1
k k max kT k k
k max k 1T k 1
k 1 k 1max k 2T k 2 k 2
V
...
...
= +
+ +
+
+ +
x x B q
x x B q x
f B u B q
x B q
x x B q x
(27)
Under the conditions of (23)-(24), we have time
derivative (27) is a negative definite function.
In a tail region of the 0=i we have
( )( )0
tanh 1
dt
d iii Bq (28)
From (25) and (28) we get
kTkkkTksignV BuBf = max (29)
Time derivative (29) is a negative definite function
under condition (21). The theorem is proved.
If theorem 4 is satisfied, then control algorithm (22)
ensures stability of the closed-loop system. According
to conditions (21) the functionsi
f shall be bounded
both by the sectorsii
x , as well as by the
constants1
max
+ iij xb , and maxub k
j . According to
(24) the sectorii x encloses the sector
11 ii x .
Thus control (22) in a tail region is close to
bang-bang control. In a small neighborhood of
steady-state mode the control (22) is close to linear
quadratic regulator.
5. Interconnection of the Proposed Method
and the Pontryagin Principle of Maximum
Let H be a function of Pontryagin. Then the
interconnection between the Lyapunov method and
the Pontryagin principle of maximum is
VH = (30)where V is determined by (6).
If a function of Pontryagin is (30), then control (4)
satisfies to the Pontryagin principle of maximum [6].
In Ref. [11], a method of time suboptimal controls
for nonlinear systems was proposed. Let consider
system (1)-(2). Suppose the ( )xB be a symmetricalpositive definite matrix. Let the dimension of the x
is equal to the dimension of u : mn= . The purposeof control is given by (3). According to Ref. [11], we
have to satisfy the next equation:
0T =+ (31)
where Ta positive definite function.
Combining (1), (3), and (31) we get
( ) ( )[ ] 0xuxBxfT =++ (32)
From (32) we obtain
( )( ) ( )( ) ( )[ ]xfxBxxTBu 11 = (33)According to Ref. [11] to get a time suboptimal
control we have to calculate
( )( ) ( )( ) ( )xfxBxxTBuTU
11
0lim
max
= sat (34)
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method158
wheremaxU
satthe saturation function.
Expression (34) is the time suboptimal control
algorithm. Calculating limit in (34), we get
( )( ) ( )( ) ( )
( )( )[ ] ( )( )xxBuxxTB
xfxBxxTBu
TU
TU
signsat
sat
=
==
max
1
0
11
0
lim
lim
max
max
(35)
It is clear that control (35) is equal to control (4).
6. Examples
Example 1: The example is to demonstrate the
proposed design method as a time optimal systems
method design.
Suppose that a control object is
( ) ( )u
dt
tdxx
dt
tdx== 22
1 , (36)
maxUu (37)It is known that the time optimal closed-loop
control for system (36)-(37) is
( )( )2221max 5.0 xsignxxsignUu += (38)Applying the method proposed in this paper we get
( )( )( )11max222max tanhtanh xqxxqUu += (39)Multiplying the right side of (38) by the right side
of (39) we obtain that the region of controls (38), and
(39) coincidence is
max22 xx < (40)2
21 5.0 xx > (41)In Fig. 1, there are both phase-plane portraits of
system (36)-(38), as well as system (36)-(37) and (39).
It is clear that in the region (40)-(41) the phase path ofsystem (36)-(38) and the phase path of system
(36)-(37) and (39) are same.
Parameters of Fig. 1 modeling results are:
5.0,2 max2max == xU , 1021 ==qq .
Tr1 is the time optimal control system path. Tr2 is
the robust control system path. It is clear path Tr2 is
close to path Tr1 in the area bounded by (40)-(41).
Example 2: Consider a wheeled vehicle that
described by the next system:
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
x1
x2
Tr1
Tr2
Fig. 1 Phase portraits of time optimal, and robust control
system.
( )( ) ( )
( )( ) ( )
( ),2
,
22122
12111
lr
rl
rl
b
r
dt
d
dt
tdx
dt
tdx
=
+=
+=
(42)
( )
( ),
,
2221212
2121111
ububddt
td
ububddt
td
r
r
l
l
++=
++=
(43)
where 21,xx external coordinates of the vehicle;
angle of orientation of the vehicle; rl , thewheel rotation speeds; rthe wheel radius; aa
kinematic factor; id , ijb , 2,1=i ,2,1=j constants; 21,uu control actions.
Functions ( )11 , ( ) 21 , ( ) 12 , ( )22 are
( ) ( )( )11 0.5 cos sinr a = + ( ) ( )( )12 0.5 cos sinr a =
( ) ( )( )2 1 0.5 sin cosr a =
( ) ( )( )12 0.5 sin cosr a = + Let bounds be given by
maxmax , rl (44)
2max21max1 , uuuu (45)
Eqs. (42) describe a kinematics of the vehicle. Eqs.
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method 159
(43) describe a dynamics of the vehicle.
The initial state ( ) ( )0,0 21 xx of the vehiclebelongs to some area . Let the purpose state of the
vehicle is given by 0,0 21 == xx . The orientationof platform is arbitrary.
From section 4 of this paper we get
( )( )( )( )tanh
,tanh
222121
2
2max2
212111
2
1max1
bbquu
bbquu
=
= (46)
( ) ( )( )( )
( ) ( )( )( )
1 l
1
max 1 11 2 21
2 r
1max 1 12 2 22
tanh q x x 0
tanh q x x 0
=
=
=
=
(47)
Let the Lyapunov function is given by
( )22215.0 +=V (48)Differentiating the Lyapunov function (48) we obtain
( )( )
( )( )
( )( )
( )( )
2
11 1max 1 11 2 21
2
12 2max 1 12 2 22 1 l 1
2
21 1max 1 11 2 21
2
22 2max 1 11 2 21 2 r 2
V b u tanh q b b
b u tanh q b b d
b u tanh q b b
b u tanh q b b d
=
+ +
+
(49)
Function (49) is a negative definite function if
,
,
2max222max121
1max212max111
r
l
dubub
dubub
>+
>+ (50)
( ) ( )
( ) ( ) .0
,0
max2212
max2111
>+
>+ (51)
Necessary conditions for solution existence of
(50)-(51) are
22221
1211 = bb
bbrang (52)
( ) ( )( ) ( )
22221
1211 =
rang (53)
Easy to prove that (52)-(53) are sufficient
conditions of the vehicle controllability.
There are modeling results of system (42)-(47) in
Figs. 2-4. Parameters of Figs. 2-4 modeling results are
102max1max ==uu , 10max = , 2.0=r , 1=a ,( )15.005.0J , ( )35.01d , ( )35.02d ,
12211 == bb , 02112 == bb , 1021 ==qq .
Fig. 2 Path of the vehicle.
0 5 10 15 20-15
-10
-5
0
5
10
15
omegar
omegal
Fig. 3 Wheels speeds of the vehicle.
0 5 10 15 20-10
-8
-6
-4
-2
0
2
4
6
8
10
t,c
u1 u2
Fig. 4 The control action of the vehicle.
Example 3: Consider an airship that described by
the next system:
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method160
,
,
du FFxM
Ry
+=
=
(54)
where x is a speed vector of the airship; y is a
coordinate vector of the airship;uF is a control
action vector;d
F is a nonlinear uncertain functional
vector; M is matrix of masses as well as moments
of inertia; R is functional matrix of kinematical
connections.
Model (1) is described detailed in Ref. [1].
The aim of this example is to design the vectoru
F
such that airship (54) is stable in an area of the
undisturbed motion0
x ,0
y .
According to (14) we have
Eyxxxx +==21 , (55)
where E is a nonsingular matrix.
Let the function of Lyapunov be given by
( ) ( )225.0 xx TV = (56)
Differentiating (56) in time we get
( ) ( ) ( )( )11222 xREFFMxxx ++== duTT
V (57)
According to section 3 of this paper, and (57) the
robust control is
( )( )EyxMFF += Tuu sign 1max (58)
wheremax
uF vector of the control action bounds.
It is clear that (57) is a negative definite function if
11max1ERxFMFM +>
du (59)
There are modeling results of system (54), (58) in
Figs. 5-6. The purpose of the control system ismovement along the straight line with speed about
5 m/s.
7. Hardware Implementation of the Control
System
Results of research are implemented in prototype of
airship-based autonomous mobile robot Sterkh,
shown in Fig. 7. These results also are implemented in
the control system of medium airship.
0 0.5 1 1.5 2 2.5 32
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
t,c
Vx
Vy
Fig. 5 The airship speed.
7 8 9 10 11 12 20
308
0
2
4
6
8
1
2
4
Zg
Xg
Yg
Fig. 6 The airship path.
Fig. 7 Autonomous mobile robot Sterkh based on
mini-airship.
Volume of the medium airship is 2 000 m3. Length
of the medium airship is 40 m. The movement of
airship is controlled by two engine installed in pylons.
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method 161
-3 -2 -1 0 1 2
x 104
-2
0
2
4
x 104
0
200
400
600
800
Fig. 8 Path of the mobile robot on base of medium airship.
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110000
5
10
15
20
25
15 15 25 15 2 5 20 15 15 15 15 15 7151515151515
Fig. 9 Speed of the airship.
Fig. 10 Mobile robot Skif.
If speed of the airship exceeds 10 m/s (the inverse
speed), then aerodynamic control surfaces are used.
The airship control system is implemented as two
separate units. The first unit is a calculation block.
The second unit is a power electronic block. The
calculation block is based on computer PC/104
CMD58886PX1400HR-BRG512 (Rtd Embedded
Technologies). Computer allows to calculate
non-linear dynamic complex controls for the airship.
Navigation system of the airship consists of satellite
navigation system, inertial navigation system, laser
rangefinder, and video camera. In addition the airship
is equipped by sensor of air temperature, humidity,
sensor of atmospheric pressure, and wind sensor.
Actuators are servo-drives with local control systems.
Experimental results of the robust control systems
of the autonomous medium airship are shown in Figs.
8-9.
There are both paths of the autonomous airship as
well as the path speed. The deviation closed-loop
system is about deviation is 28 m2for path and 1.5 m
2
for speed. Causes of the deviation are control errors,
navigation errors, and actuators errors.In addition results of research are implemented in
the wheeled mobile robot Skif, shown in Fig. 10.
8. Conclusions
New design methods of robust control systems
based on the direct Lyapunov method are developed in
this report. Function of Lyapunov is defined for the
block systems. The minimum of the Lyapunov
function is ensured by the robust relay control.
It was proved that the state commonness condition
is satisfied if the control actions are above or equal to
the disturbances.
Further the transformation of the block system to
the single block system was found.
References
[1] V.K. Pshikhopov, M.Y. Medvedev, M.Y. Sirotenko, V.A.
Kostjukov, Control system design for robotic airship, in:
Proceedings of the 9-th IFAC Symposium on Robot
Control, Gifu, Japan, September 9-12, 2009, pp. 123-128.
[2]
J.D. Landan, Adaptive Control: the Model Reference
Adaptive Control, New York, Dekker, 1980.
[3] K.J. Astrm, V. Borrison, L. Ljung, B. Wittenmark,
Theory and applications of self-tuning regulators,
Automatica 13 (1977) 457-476.
[4] S.D. Zemlyakov, Some problems of analytical synthesis
in model reference control systems by the direct method
of Lyapunov: Theory of self tuning adaptive control
systems, in: Proc. of 1965 IFAC Symposium on Adaptive
Control, Teddington, England, 1965, pp. 145-152.
[5]
V.Y. Rutkouvsky, V.M. Sukhanov, V.M. Glumov, S.J.
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Block Design of Robust Control for a Class of Dynamic Systems by Direct Lyapunov Method162
Dodds, Nonexciting control by orientation of flexible
space vehicle, in: Proc. of the 14-th IFAC Symposium on
Automatic Control in Aerospace, Seoul, Seoul National
University, 1998, pp. 339-344.
[6]
M.Y. Medvedev, Design of suboptimal controls for
nonlinear multi-linked dynamical systems, Mechatronics,
Automatics, and Control 12 (2009) 2-8.
[7]
V.K. Pshikhopov, M.Y. Medvedev, Block design of
robust systems with bounded controls and state variables,
Mechatronics, Automatics, and Control 1 (2011) 2-8.
[8] E.S. Pyatnickiy, Controllability of Lagrange systems with
limited controls, Automation and Remote Control 12
(1996) 29-37.
[9] V.A. Oleinikov, N.C. Zotov, A.N. Prishvin, The basis of
optimal and extremal control, oscow, High School,
1969.
[10]
A.R. Gaiduk, Design of nonlinear systems on base of
controllable Jordan form, Automation and Remote
Control 7 (2006) 3-13.
[11] V.K. Pshikhopov, Time optimal path control of
electromechanical robot manipulator, Electromechanics 1
(2007) 51-57.
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Journal of Mechanics Engineering and Automation 2 (2012) 163-168
Cycle Analysis of Internally Reformed MCFC/SOFC-Gas
Turbine Combined System
Abdullatif Musa1and Abdussalam Elansari
2
1. Marine Engineering Department, Faculty of Engineering, Tripoli University, Tripoli 21821, Libya
2. Renewable Energy Authority of Libya, Tripoli 21821, Libya
Received: January 25, 2012 / Accepted: February 06, 2012 / Published: March 25, 2012.
Abstract: The use of high temperature fuel cells for distributed power generation presents advantages related to high electrical
efficiency that can be achieved through hybrid systems and low emissions. The intermediate temperature solid oxide fuel cell
(IT-SOFC) and molten carbonate fuel cell (MCFC) performances are calculated using numerical models which are built in Aspen
customer modeler for the internally reformed (IR) fuel cells. These models are integrated in Aspen PlusTM. In this paper, a new
combined cycle is proposed. This combined cycle consists of two-staged of MCFC and IT-SOFC. The combined and single-staged
MCFC cycles are simulated in order to evaluate and compare their performances. Moreover, the effects of important parameters such
as operating temperature, and cell pressure on the system performance are evaluated. The simulations results indicate that the net
efficiency of MCFC/IT-SOFC combined cycle is 64.6% under standard operation conditions. On the other hand, the net efficiency of
single-staged MCFC cycle is 51.6%. In other words, the cycle with two-staged MCFC and IT-SOFC gives much better net efficiency
than the cycle with single-staged MCFC.
Key words:Fuel cells, SOFC (solid oxide fuel cell), MCFC (molten carbonate fuel cell), gas turbine, cycle analysis.
1. Introduction
Currently, the annual global population growth rate
is about 2% while rising even more sharply in many
countries. Consequently, the global energy services
demand is expected to increase dramatically, with
primary energy doubling or tripling over the next five
decades. Therefore, there is a need for renewable and
environmentally benign power production in order to
resolve the seemingly inevitable energy crisis [1].
Fuel cells are electrochemical energy conversion
devices which typically run on hydrogen or methane
or methanol and produce electricity, heat and benign
emissions (water and, in the case of methane and
methanol, CO2). The fuel cells used for stationary
energy production are typically high temperature fuel
cells (HTFCs) such as solid oxide fuel cell (SOFC)
and molten carbonate fuel cell (MCFC).
Corresponding author: Abdullatif Musa, Ph.D., research
field: fuel cells. E-mail: [email protected].
MCFC technology was established about 30 years
ago and has been developed considerably fast in the
USA, Korea, Japan and Europe during the last 10
years. In the literature there have been several studies
published on MCFC and SOFC systems and their
analyses [1-8].
Using serially connect fuel cells has emerged as a
new and highly efficient source of power source. In
Ref. [2] Araki and co-authors analysed a power
generation system consisting of two-stages externallyreformed SOFCs with serial connection of low and
high temperature SOFCs. They showed that the power
generation efficiency of the two-staged SOFCs is 50.3%
and the total efficiency of power generation with gas
turbine is 56.1% under standard operating conditions.
In previously paper [3], Two types of combined cycles
is investigated: a combined cycle consisting of a
two-staged combination of IT-SOFC and HT-SOFC
and another consisting of two stages of IT-SOFC. The
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Cycle Analysis of Internally Reformed MCFC/SOFC-Gas Turbine Combined System164
simulation results show that a combined cycle of
two-staged IT-SOFC can give 65.5% under standard
operational conditions. Furthermore, by optimizing
the heat recovery and the gas turbine use, the
efficiency can go up to 68.3%.
In this paper, a new combined cycle is proposed and
investigated. Thermodynamic models for internally
reformed IT-SOFC and MCFC are developed. These
fuel cells are combined in different ways in order
construct single-staged and two-staged fuel cell system
combining different cell types. The aim of the paper is
to find the best configuration for a single-staged or a
two-staged combined system. Therefore, theperformance of two types of cycles is analysed:
combined cycle consisting of two-staged of MCFC and
IT-SOFC and single-staged cycle with MCFC.
2. Cycles Description
The SOFC and MCFC cells currently in operation
are fuelled with natural gas. The high temperature
inside the cells stack allows for reforming the methane
directly inside the cell if the steam is provided at the
inlet. The heat necessary for this reforming reaction is
delivered by the electrochemical reaction in the cell.
Fuel is provided at atmospheric conditions. The fuel is
pure methane (CH4). In the cycles part of the anode
gasses is recycled, as the anode gasses contain steam
needed in the reforming reaction. This is a way of
avoiding a steam generator in the cycles. The
characteristics of the systems are given in Table 1.
2.1 Combined Cycle Configuration
Fig.1 shows a cycle diagram of the combined cycle
consisting of an MCFC and IT-SOFC. In this cycle,
the anode flow of the MCFC and IT-SOFC is in
parallel connected. Methane is admitted into the heat
exchanger H/E2 to preheat the methane. The
preheated methane is split into two parts. Part of the
preheated methane is mixed with the recycling anode
gases; the mixture is supplied to the anode side of
MCFC stack. The remaining part of the preheated
methane is mixed with the part of recycling anode
gases and then enters into the anode side in the
IT-SOFC stack. The compressors (C1 and C2) are
used to compensate the pressure drop through the
stacks. In both stacks the remaining part of anode
gases is recycled to the combustor. The combustor exitgas which contains a major part of air, and CO2is split
into two parts. The first part is the cathode inlet gas of
the MCFC stack. The remaining part of the combustor
exit gas and cathode outlet gas of MCFC are mixed,
the mixture is sent to a gas turbine and heat exchanger
(H/E2) respectively. The cathode gases of the
IT-SOFC stack is recycled to the combustor. The
compressed air from the compressor (AC) is supplied
to the heat exchanger (H/E1) and then enters into the
cathode side of the IT-SOFC stack.
2.2 Single-Staged Cycle Configuration
The single-staged MCFC cycle is similar to the
combined cycle (Fig. 1), except that there is no
IT-SOFC stack. The combustor exit gas is split into
two parts. The first part of the combustor exit gas and
the compressed air from the compressor (AC) are
mixed. This mixture is sent to the heat exchanger
(H/E1) and then enters into the cathode side of theMCFC stack. The remaining part of the combustor
exit gas is mixed with part of the cathode outlet gas,
Table 1 Setting parameters of the cycles.
Setting parameter Value Setting parameter Value
Current density 0.250 Acm-2 Steam-to-carbon ratio 2
Active cell area 250 m Pressure drop in combustor 0.2 bar
Total fuel utilization rate 85% Pressure drop in SOFC 0.01 bar
Compressors isentropic efficiency 80% Pressure drop in heat exchangers 0.02 bar
Gas turbine and pump isentropic efficiencies 85% Gas turbine and compressor mechanical efficiencies 98%
Fuel recirculation rate 55% Pressure drop in MCFC 0.05 bar
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Cycle Analysis of Internally Reformed MCFC/SOFC-Gas Turbine Combined System 165
Fig. 1 Configuration of the MCFC/IT-SOFC combined cycle (AC: air compressor; FC: fuel compressor; GT: gas turbine;
H/E: heat exchanger).
the mixture is sent to a gas turbine and heat exchanger
(H/E2) for recovering energy. The remaining part of
the cathode outlet gas is recycled to the combustor. In
this cycle part of the preheated methane is bypassed to
the combustor.
3. Water Gas Shift and Methane Reforming
Reactions
In the models, the chemical reactions are assumed
to be in equilibrium. This means that the reactions
occur instantaneously and reach the equilibrium
condition spontaneously at each position.
For SOFC model the electrochemical reaction is
implemented: + 222
1 2 OeO
cathode (1)
++ eOHOH 222
2 anode (2)
OHOH 2221
2 + overallreaction (3)
For MCFC model the electrochemical reaction is
implemented: ++ 2322
12 2 COeOCO
cathode (4)
+++ eCOOHCOH 2222
32 anode (5)
OHOH 2221
2 + overallreaction (6)
The IT-SOFC and MCFC operate at a temperature