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Volume 2, Number 3, March 2012 (Serial Number 9)

Journal of Mechanics

and

Automation

Engineering

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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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An Adapt ive Dominant Type Hybr id Adapt ive and Learning Control ler fo r Geometrical lyConstrained Robot Manipulators

138

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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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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Basic Aspects of Defining Mechanical-Technologi cal Solutionsfor the Production of Biogas from Liquid Manure

150

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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151

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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153

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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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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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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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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(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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,

,

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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-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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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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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

DDAVID PUBLISHING


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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

open architecture robot control system

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