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Articles2

Computer aided methods for stability analysis of 2D linear systems described by the first Fornasini-Marchesini modelMikolaj Buslowicz, Andrzej Ruszewski DOI 10.14313/JAMRIS_2-2014/12

An adequate mathematical model of four-rotor flying robot in the context of control simulationsStanislaw Gardecki, Wojciech Giernacki, Jaroslaw Goslinski, Andrzej Kasinski DOI 10.14313/JAMRIS_2-2014/13

Mathematical modeling and computer aided planning of communal sewage networksLucyna Bogdan, Grazyna Petriczek, Jan StudzinskiDOI 10.14313/JAMRIS_2-2014/14

Failures location within water-supply systems by means of neural networks Izabela Rojek, Jan StudzinskiDOI 10.14313/JAMRIS_2-2014/15

Outside the box: an alternative data analytics frameworkPlamen AngelovDOI 10.14313/JAMRIS_2-2014/16

Arm Manipulator Position Control Based On Multi-Input Multi-Output PID StrategyFatima Zahra Baghli, Larbi El Bakkali, Yassine Lakhal, Abdelfatah Nasri, Brahim GasbaouiDOI 10.14313/JAMRIS_2-2014/17

A new heuristic possibilistic clustering algorithm for feature selectionJanusz Kacprzyk, Jan W. Owsinski, Dimitri A. ViattcheninDOI 10.14313/JAMRIS_2-2014/18

Extracting fuzzy classifications rules from three-way dataJanusz Kacprzyk, Jan W. Owsinski, Dimitri A. ViattcheninDOI 10.14313/JAMRIS_2-2014/19

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Journal of automation, mobile robotics & intelligent systems

Volume 8, n 2, 2014 Doi: 10.14313/Jamris_2-2014

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contents

Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 8, N 2 2014

C A M S A 2D L SD F F-M M




Submied: 2nd July 2013; accepted: 31th July 2013

Andrzej Ruszewski, Mikoaj Busowicz

DOI: 10.14313/JAMRIS_2-2014/12Abstract:Computer aided methods for invesgaon of the asymp-toc stability of 2D discrete linear systems described bythe rst Fornasini-Marchesinimodel are given. Themeth-ods require computaon of eigenvalues of complex ma-trices or values of complex funcons. Eecveness of thestability tests are demonstrated on numerical examples.Keywords: 2D system, linear, discrete, stability, compu-taonal method

1. IntroduconThere are several models of 2D discrete linear sys-

tem [9, 11, 12]. The most popular is the Fornasini-Marchesini model introduced in [9].

The problem of asymptotic stability of linear 2Dsystems has considerable attention since about 40years. For the stability analysis of these systems var-iousmethods can be applied: analytical (similar to theSchur stability test of 1D systems) [1], based on Lya-punov stability theory [21, 22], based on LMI [8, 13,23, 24], based on spectral radius [10, 17, 20, 25, 26],frequency domainmethods [23] or algebraic methodsfor positive systems [12, 13, 14, 15, 16, 19]. The analyt-ical methods require symbolic computations whereasthe methods based on Lyapunov stability theory, LMIor spectral radius give only suficient but not neces-sary conditions for stability of standard systems.

The main purpose of this paper is to present newfrequency domain necessary and suficient conditionsfor investigation of asymptotic stability of the irstFornasini-Marchesinimodel of 2D standard linear sys-tems.

The following notation will be used: - the set ofnon-negative integers; - the set of realma-trices; {} -th eigenvalue of .

2. Problem FormulaonConsider the state equation of the irst Fornasini-

Marchesini model of 2D linear system [9, 11, 12]

( + 1, + 1) = (, ) + ( + 1, )+(, + 1) + (, ), , ,

(1)

where (, ) , (, ) and , , , .

The boundary conditions for (1) are as follows

(, 0) = , (0, ) = , , . (2)

The characteristic matrix of the model (1) has theform

(, ) = , (3)where and are complex variables.

The characteristic function(, ) = det(, )= det[ ]

(4)

of the model (1) is a polynomial in two independentvariables and , of the form

(, ) =

, = 1. (5)

The model (1) is called asymptotically stable(Schur stable) if for (, ) 0 and bounded bound-ary conditions (2) the condition (, ) 0 holds for, .

From [1, 11] we have the following theorem.Theorem 1. The model (1) is asymptotically stable ifand only if(, ) 0, || 1 || 1. (6)The polynomial (5) satisfying the condition (6) is

called discrete stable or Schur stable. Several algebraicmethods for asymptotic stability checking of such bi-variate polynomials were given in [1].

Computational method for investigation of asymp-totic stability of the Fornasini-Marchesini model (1)has been given in [2]. This method requires computa-tion of eigenvalue-loci of complex matrices.

The main purpose of this paper is to present newcomputationalmethods for checking the condition (6)of asymptotic stability of the model (1) which do notrequire a priori knowledge of the characteristic bivari-ate polynomial (5).

3. Soluon of the ProblemTheorem 2. The model (1) is asymptotically stable ifand only if the following two conditions hold:( , ) 0, || 1, [0, 2], = 1, (7)

(, ) 0, || 1, [0, 2]. (8)

Proof. From [1, 2] it follows that (6) is equivalent tothe conditions

(, ) 0, || = 1, || 1, (9)

3


(, ) 0, || 1, || = 1. (10)It is easy to see that conditions (9) and (10) can be

written in the forms (7) and (8), respectively. Lemma 1. If the model (1) is asymptotically stablethen

|()| < 1, = 1, 2, ..., , (11)and

|()| < 1, = 1, 2, ..., . (12)Proof. From (1) for 0, 0 and 0 one ob-tains the homogeneous state equation of the discrete-time linear system

( + 1, + 1) = ( + 1, ). (13)

The system (13) is asymptotically stable if andonlyif the condition (11) holds, i.e. the matrix is Schurstable (is a Schur matrix).

Similarly, substitution of 0, 0 and 0 in (1) gives the homogeneous state equationof discrete-time linear system

( + 1, + 1) = (, + 1), (14)

which is asymptotically stable if and only if the condi-tion (12) holds, i.e. the matrix is Schur stable (is aSchur matrix).

Asymptotic stability of the model (1) with anyixed triple of matrices , and means that thecondition (6) holds for this triple. In particular, asymp-totic stability of the systemwith 0 and 0 (or 0 and 0) is equivalent to satisfaction of thecondition (6) for 0 and 0 (or 0 and 0). Hence, the conditions (11) and (12) are nec-essary for asymptotic stability of the model (1).

To show that the conditions (11) and (12) are notsuficient, we consider the scalar system (1)with =0, = 0.5 ((11) and (12) hold) and = 1.In this case the characteristic equation has the form0.51 = 0.Fromthis equationwehave that if,for example, = 0 then = 2 and if = 0.5 then = 2.5. This means that there exist such values ofroots of the characteristic equation which do not sat-isfy the condition (6) and the system is unstable.

Using the rules for computing the determinant ofblock matrices, we obtain that the characteristic ma-trix (3) of the model (1) can be computed from one ofthe following equivalent formulae

(, ) = [ ][ ()], (15)

(, ) = [ ][ ()], (16)where

() = ( )( + ), (17)

() = ( )( + ). (18)Using (4) and (15), (16) we can write

(, ) = det[ ] det[ ()], (19)

(, ) = det[ ] det[ ()]. (20)

From (15) for = we have

( , ) = [ ][ ()], (21)

where

() = ( )( + ). (22)

Lemma 2. Let the necessary condition (12) be satis-ied. The condition (7) holds if and only if all eigenval-ues of the complex matrix (22) have absolute valuesless than one for all = [0, 2].Proof. From (21) we have

( , ) = det[ ] det[ ()].(23)

If (12) holds then the matrix is non-singular for all . Hence, from (23) it follows thatthe condition (7) is satisied if and only if

det[ ()] 0, || 1, . (24)

Satisfaction of (24) means that all eigenvalues ofthe complexmatrix (22)have absolute values less thanone for all .

Eigenvalue-loci of () for [0, ] and for [, 2] are symmetric respect to the real axisof the complex plane. Therefore, we can equivalentlyconsider in (24) the interval = [0, ] instead of theinterval = [0, 2].

From (16) for = we have

(, ) = [ ][ ()] (25)

and

(, ) = det[ ] det[ ()],(26)

where

() = ( )( + ). (27)

Lemma 3. Let the necessary condition (11) be satis-ied. The condition (8) holds if and only if all eigenval-ues of the complex matrix (27) have absolute valuesless than one for all = [0, 2].Proof. If (11) holds then thematrix is non-singular for all . From (26) we have that the con-dition (8) is satisied if and only if

det[ ()] 0, || 1, , (28)

i.e. all eigenvalues of thematrix (27) have absolute val-ues less than one for all .

Similarly as in Lemma 2, we can equivalently con-sider the interval = [0, ] instead of the interval = [0, 2].

The conditions of Lemmas 2 and 3 can be writtenin the following forms

|{()}| < 1, , = 1, 2, ..., (29)

4


and

|{()}| < 1, , = 1, 2, ..., , (30)

respectively.Theorem 3. The model (1) is asymptotically stable ifand only if the conditions (11), (12), (29) and (30) aresatisied.Proof. The proof follows directly from Theorem 2 andLemmas 1, 2 and 3.

Computational methods for checking the con-ditions (29) and (30) for the Fornasini-Marchesinimodel (1), based on the eigenvalues-loci of the matri-ces (22) and (27), are given in [2].

It is easy to see that the conditions (29) and (30)can be written in the forms: () > 0 for all and() > 0 for all ,where

() = 1 max,...,

|{()}|, (31)

() = 1 max,...,

|{()}|. (32)

Hence, from Theorem 3 one obtains the followinglemma.Lemma 4. Let the necessary conditions (11), (12)hold. Themodel (1) is asymptotically stable if and onlyif () > 0 for all and () > 0 for all orequivalently, the conditions

= min () > 0, = min () > 0, (33)

are satisied.Example 1. Consider the model (1) with thematrices

= 0.3 0.1 0.40.4 0.1 00 0.3 0.2

,

= 0.1 0.2 00 0.4 0.30.1 0.3 0.1

,

= 0.3 0.1 0.20 0.2 0.10.3 0.2 0.4

.

(34)

Computing eigenvalues of and we obtain- eigenvalues of : -0.1233; 0.1577; 0.5656.- eigenvalues of : 0.1166; 0.2343; 0.5491.

Hence, the necessary conditions (11) and (12)hold, i.e. the matrices and are Schur stable.

Plots of the functions () ( ) and () ( ) are shown in Figure 1. By o are denoted the end-points of the plots. The ranges = [0, 2] and =[0, 2] were digitized with the steps = 0.01 and = 0.01.

From Figure 1 and also from the fact that =0.3012 > 0 and = 0.2737 > 0 it follows that theconditions of Lemma 4 are satisied and the model isasymptotically stable.

Checking the conditions of Theorem 3 and Lemma4 require computation of eigenvalues of the matrices(22) and (27). This may be inconvenient with respect

0 1 2 3 4 5 6 70.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

y,

,

1

2

Fig. 1. Plots of the funcons (31) (curve 1) and (32)(curve 2) for = [0, 2]

to computational problems, particularly in the case ofill conditioned matrices. Therefore, we present a newmethod for investigation of asymptotic stability of themodel (1) which require computation only determi-nants of some matrices, not eigenvalues.

Consider the polynomial

( , ) = det( ()), (35)

where the matrix () is deined by (22). From theclassical Mikhailov theorem (see for example [18]) itfollows that the condition (24) holds if and only if forany ixed plot of ( , ) for encir-cles in the positive direction times the origin of thecomplex plane.

Direct application of the Mikhailov theorem tochecking the condition (24) is not practically reliablefor a large values of . Therefore, we introduce the ra-tional function

( , ) =( , )()

, , (36)

instead of ( , ), where () is any Schurstable polynomial of degree .Lemma 5. The condition (29) holds if and only if forall ixed plot of the function (36) does not en-circle or cross the origin of the complex plane.Proof. If the reference polynomial () is Schurstable then from the Argument Principle we have

arg

() = . (37)

From (36) it follows that for any ixed

arg ( , ) = arg( , ) arg().

(38)

The matrix (22) for any ixed is Schur stableif and only if

arg[,]

( , ) = arg[,]

() = ,

5


which holds if and only if arg ( , ) = 0.Taking into account all , we obtain that the

above holds if and only if for all ixed plot of(36) as a function of does not encircle or crossthe origin of the complex plane.

The reference polynomial () can be chosenin the form

(1, ) = det( (1)), (39)

where (1) = ( )( + ), which we getfrom (35) and (22) substituting = 0. Schur stabil-ity of (39) is necessary for Schur stability of complexpolynomial (35) for all .

If() = (1, ), then

( , ) =( , )(1, )

, . (40)

Plot of (40) as a function of (with any ixed ) is a closed curve. It beginswith = 0 and endswith = 2 in the point (1, ) = 1.

Now, we consider the complex polynomial

(, ) = det( ()), (41)

where the matrix () is deined by (27).Let () be any Schur stable polynomial of de-

gree .Proceeding similarly as in the case of Lemma 5, we

obtain the following lemma.Lemma 6. The condition (30) holds if and only if forall ixed plot of the function

( , ) =( , )()

, , (42)

does not encircle or cross the origin of the complexplane, where ( , ) has the form (41) for = .

The reference polynomial () can be chosenin the form

(, 1) = det( (1)), (43)

where (1) = ( )( + ). Schur stabilityof (43) is necessary for Schur stability of the complexpolynomial (41) for all .

If() = (, 1), then

( , ) =( , )( , 1)

, . (44)

Plot of (44) as a function of with the ixed is a closed curve. It begins with = 0 and endswith = 2 in the point ( , 1) = 1.

From Theorem 3 and Lemmas 5 and 6 we have thefollowing theorem.Theorem 4. Assume that the necessary conditions(11) and (12) are satisied and the polynomials (39)and (43) are Schur stable. The model (1) is asymptot-ically stable if and only if the following two conditionshold:

1) plots of the function (40) do not encircle or crossthe origin of the complex plane for all ixed ,

2) plots of the function (44) do not encircle or crossthe origin of the complex plane for all ixed .Applying computational method given in Theorem

4we can take into consideration the following remark.Remark. Refer to point 1) of Theorem 4, one shouldset any ixed , determined with appropriatelysmall step, and drawplots of the function (40) sep-arately digitizing the range with a suficiently smallstep . For point 2) of Theorem 4 one should set anyixed , determined with appropriately small step, and draw plots of the function (44) separately dig-itizing the range with a suficiently small step .Plots should be smooth especially near the origin ofthe complexplane so that the important parts havenotbeen neglected.Example 2.Using Theorem 4 check asymptotic stabil-ity of the model (1) with the matrices (34).

In Example 1 it has been shown that the necessaryconditions (11) and (12) hold.

Computing eigenvalues of the matrices (1) =()(+) and (1) = ()(+)we obtain respectively:

= 0.4201 + 0.2872, = 0.4201 0.2872,

= 0.6204,(45)

= 0.4762 + 0.2152, = 0.4762 0.2152,

= 0.5703.(46)

Moduli of all eigenvalues (45) and (46) are lessthan one and the reference polynomials (39) and (43)are Schur stable.

Plots of (40) and (44) are shown in Figures 2 and3,respectively. The ranges = [0, 2] and = [0, 2]for all plots was digitized with the steps = 0.01and = 0.01.

0 0.5 1 1.5 2 2.5 31.5

1

0.5

0

0.5

1

1.5

Real Axis

Imag

inar

y Ax

is

Fig. 2. Plots of (40)

6


0 0.5 1 1.5 2 2.5 3 3.52

1.5

1

0.5

0

0.5

1

1.5

2

Real Axis

Imag

inar

y Ax

is

Fig. 3. Plots of (44)

From Figures 2 and 3 is follows that the plots donot encircle the origin of the complex plane for all and .Thismeans, according to Theorem4, thatthe model (1), (34) is Schur stable.

Now we consider the 1st order Fornasini-Marchesini model described by the equation

( + 1, + 1) = (, ) + ( + 1, )+(, + 1) + (, ),

(47)

where , , and are real coeficients.For the system (47) the necessary conditions (11)

and (12) take the forms

|| < 1, || < 1. (48)

The matrix (22) for the system has the form

() = +

. (49)

It is easy to check that plot of (49) for =[0, 2] is a circlewith the center at real axis. This circlecrosses real axis in points

= () = + 1

, = () = 1 +

.

Hence, the irst condition (33) holds if and only if

= 1 max {|| , ||} > 0. (50)

Similarly, we can show that the second condition(33) holds if and only if

= 1 max {|| , ||} > 0, (51)

where

= () = + 1

, = () = 1 +

.

From the above andTheorem3wehave the follow-ing condition.Lemma 7. The 1st order Fornasini-Marchesini model(47) is asymptotically stable if and only if the condi-tions (48) and (50), (51) are satisied.

4. Concluding RemarksSimple necessary conditions (Lemma 1) and two

computational methods for investigation of asymp-totic stability of the irst Fornasini-Marchesini model(1) of 2D discrete linear systems have been given.

The irst method (Theorem 3, Lemma 4) requirecomputation of eigenvalues of complex matrices (22)and (27). Similarmethods have been applied in [7, 23]to asymptotic stability analysis of the Roesser modelof 2D systems and in [3] for the Fornasini-Marchesiniand the Roesser type models of 2D continuous-discrete linear systems.

The second method (Theorem 4) require compu-tation of values of functions (40) and (44) and there-fore is simpler from the computation point of view.Similar methods have been applied in [3], [4], [5] and[6], respectively, to asymptotic stability analysis of 2Dcontinuous-discrete linear systems described by theirst and the second Fornasini-Marchesini type mod-els and the Roesser type model.

The proposed methods can be applied to thestability checking of the second Fornasini-Marchesinimodel describedby the state equation (1)with 0.

AcknowledgmentThis work was supported by the Ministry of Scienceand High Education of Poland under the grant no.S/WE/1/2011.

AUTHORSAndrzej Ruszewski Bialystok University of Tech-nology, Faculty of Electrical Engineering,Wiejska 45D,15-351 Biaystok, Poland, e-mail: [email protected],www: pb.edu.pl.Mikoaj Busowicz Bialystok University of Technol-ogy, Faculty of Electrical Engineering, Wiejska 45D,15-351Biaystok, Poland, e-mail: [email protected],www: pb.edu.pl.Corresponding author

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[21] W.-S. Lu, On a Lyapunov approach tostability analysis of 2-D digital ilters,IEEE Trans. Circuits Syst. I, Fundam. The-ory Appl., vol. 45, 1994, pp. 665669. DOI:http://dx.doi.org/10.1109/81.329727

[22] T. Ooba, On stability analysis of 2-D systemsbased on 2-D Lyapunov matrix inequalities, IEEETrans. Circuit Syst. I, Fundam. Theory Appl., vol. 47,2000, pp. 12631265.

[23] W. Paszke, E. Rogers, P. Rapisarda, K. Gakowski,A. Kummert, New frequency domain basedstability tests for 2D linear systems, Proc.of 17 Int. Conf. Methods and Models in Au-tomation and Robotics, 2012 (CD-ROM). DOI:http://dx.doi.org/10.1109/MMAR.2012.6347922

[24] M. Twardy, An LMI approach to checking sta-bility of 2D positive systems, Bull. Pol. Acad. Sci.,Tech. Sci., vol. 55, no. 4, 2007, pp. 385395.

[25] X. Xiao, R. Unbehauen, New stability test al-gorithm for two-dimensional digital ilters, IEEETrans. Circuits Syst. I, Fundam.TheoryAppl.,vol. 45,no. 7, 1998, pp. 739741.

[26] S.-F. Yang, C. Hwang, s, IEEE Trans. Circuits Syst.I, Fundam. Theory Appl., vol. 47, no. 7, 2000, pp.11201123.

8


A A M M F- F R C C S



A A M M F- F R C C SSubmied: 23rd April 2013; accepted: 26th July 2013

Stanislaw Gardecki, Wojciech Giernacki, Jaroslaw Goslinski, Andrzej Kasinski

DOI: 10.14313/JAMRIS_2-2014/13Abstract:In this paper a model of the dynamics of four-rotor y-ing robot is described in details. Control design must bepreceded by the modeling and subsequent analysis ofthe robot behavior in simulator. It is therefore necessaryto develop the mathemacal model as accurate as it ispossible. The paper contains a detailed derivaon of themathemacal model in the context of physics laws aect-ing the quadrocopter. The novelty of presented notaonis an extenon of Coriolis forces in linear acceleraon andthe gyroscopic eect on angular acceleraon. In the vali-daon phase, the mathemacal model was veried withthe use of proposed control algorithms. Simulaon stud-ies have demonstrated the adequacy of aMATLABmodelto properly reect the real quadrocopter dynamics. Thiswould allow for its use in the simulator and aerwardsto implement and verify of control laws on the real four-rotor ying robot.Keywords:unmannedaerial vehicle (UAV), quadrocopter,modeling, mathemacal model

1. IntroduconUnderstanding and control at the acceptable level

of physical processes is possible due to proper mod-eling. By bringing together all the essential processfeatures against the background of non-essential onesand the use of mathematics as well as computer, itis possible to build simulators to evaluate the morecomplex system properties such as dynamics. In themodeling phase of engineering work, in addition tothe added value which is a new model and a possi-bility of acquiring a new knowledge through experi-menting with it, modeling allows also for large sav-ings on experiments costs. Studies on the model arerepeatable and time-saving. They allow also for theintroduction of certain extra conditions to be simu-lated. Finally, the results of simulation studies, whichare normally available before the experiment, maybe more easily established, described, and archived.To develop eficient and rapid control algorithms forunmanned lying vehicles having complex structure,good quality mathematical models, which reliably re-lect the dynamics of these vehicles, are needed. Inthis article authors focused their attention on selected,four-rotor lying robot - called quadrocopter, whichfor the correct operation requires an appropriate con-trol strategy. Naturally, proper control may be devel-oped by experimental tests - on the real object, orby the use of simulator - which is a dedicated com-

puter program. The irst approach (physical model-ing) is expensive and it is associated with tests car-ried out directly on the system hardware. The secondway to its use requires an availability of themathemat-ical model. There are already many dynamics modelsfocused on the four-rotor lying robot. They differ inthe complexity and the level of the adopted simpli-ications. It is important that quadrocopter is a non-linear dynamical object with multiple inputs and out-puts. It is inherently unstable and its parameters arenon-stationary in time. Of course, it would be bene-icial to know all the necessary physical parametersof the quadrocopters elements a priori [2], but with-out a wind tunnel experiment - it is impossible to di-rectly evaluate quadrocopters geometry, dynamics ofits propellers, or other relevant aerodynamic parame-ters. The particular literature studies [4] let to drawthe following conclusion: To overcome the inlexi-bility of the complex models, the National Aeronau-tics and Space Administration (NASA) has developed aso-calledMinimum-Complexity Helicopter SimulationMathModel (MCHSMM) [3],which is amathmodel de-pending only on the basic data sources with the inten-tion of low cost real-time simulation possibilities (...).One of the additional beneits from a MCHSMM, is thepotential for a more clear understanding of the heli-copter and its dynamics in general. On the basis ofMCHSMM, in further works, e.g. [4] there was a striveto develop methods for robust and optimal control ofa robot based on this model. However, in the opinionof authors of this paper, too far-reaching simpliica-tions narrow the spectrum of control methods used.While the robust control in its idea compensatesmod-eling errors and guarantees stability [11], such a con-trol is not optimal, which is not without the signii-cance from the perspective of energy expenditure (asrobot batteries allow up to tens of minutes of light).On the other hand most often optimal control is as-sociated with considerable amount of computing [8],which makes it dificult to perform the relevant real-time calculations in the case of robots on-board con-trol unit. Thus it would be preferable to use the adap-tive control methods [8], among which eficient meth-ods of low computational complexity are available.Therefore, authors postulated a compromise, i.e. tocreate a mathematical model of the certain simpliiedrobot structure (solid body), but taking into accountall relevant physical phenomena, which the robot issubjected to, andwhich are often not included in othercommonly used models, such as introduction of rela-tionship describing the linear accelerationdue to Cori-

9


olis and angular acceleration due to the gyroscopic ef-fect. In this way, this article is a supplement to previ-ous approaches [10], [6], contributing to more faith-fully relect the dynamics of the robot.

2. Quadrocopter Physical ObjectQuadrocopters differ in technical details, but have

one common feature: the design of a system is basedon four rotors powered by DC motors and suppliedby battery (Fig. 1). The layout of the quadrocopter isdiscussed in details in [5]. The use of such a struc-ture reduces the instrumentation costs by eliminatinga variable rotor blades compared to the classic solu-tion used in helicopters. In this way an increased sta-bility and a relatively large capacity of the lying robotcan be achieved.The robots avionics consists of the followingon-boardsensors: inertial measurement unit (IMU) - used forthe stabilization of the robots position and orienta-tion, an altimeter for the height stabilization, a ther-mometer and a power consumption meter [5]. TheIMUcontains three gyroscopes and triaxial accelerom-eter. This sensor communicates with on board proces-sor via the SPI bus. Gyroscopes are used to determinethe orientation of the robot in space (relative to thebase). The accelerometers are used to reset the gyrodrift.In quadrocopters, currently powertrain consisting ofa speed controller, brushless motor and propeller, aremostly used. Their task is to generate a lift force andappropriate power steering. These are essential fea-tures from the perspective of robot motion control.The propeller used in each unit is responsible for thegeneration of the lift force and it is rotating in one di-rection, which is due to a constant pitch of the blade.For this reason, the controller is used only tomodulateits speed, and it is not responsible for the direction ofthe propeller rotation.

Fig. 1. Quadrocopter Hornet

3. Mathemacal Model Reference FramesDuring mathematical modeling (for the general

quadrocopters geometric model form Figure 4), thephysical parameters are described in three differentreference systems (or with respect to these systems),i.e. the Earth system (), the robot system (local-)and auxiliary system (). The relationship between

Fig. 2. Quadrocopters geometric model

Fig. 3. Reference systems

, and are shown in Figure 3. System is avirtual system whose aim is to adjust the orientationof the global system to the local system. In thenotation of the mathematical model all physical quan-tities except the Euler angles are expressed in the localcoordinate system . Euler angles are the angles be-tween the and (or ) frame.

4. Mathemacal Model, Equaons4.1. Euler angles

Euler angles deine the orientation of the robotsframe ( system) relative to the global system - theEarth [9]. During the following calculations wewill apply the standard rotationmatrices (transforma-tions): matrix of rotation in the axis about angle :

() = 1 0 00 0

, (1)

matrix of rotation in the axis about angle :

() = 0 0 1 0

0 (2)

and matrix of rotation in the axis about angle :

10


() = 0 00 0 1

. (3)

While determining the transformationmatrix, thatallows to translate the system to , authors fol-low the principle of rotation sequence 3-2-1 [7]. Thissequence assumes that during rotations the followingsteps are performed:- Rotation Yaw; about angle in the axis of the localcoordinate system ()

- Rotation Pitch; about angle in the axis of the newlocal coordinate system ()

- Rotation Roll; about angle in the axis of the newlocal coordinate system ()

Under this assumption and taking into account thematrices: (1), (2), (3), the total transformation matrixform the to the systemmay be written:

() = () () () = (4)

+ +

, (5)

where and stands for and respectively.In the next step, knowledge about the transforma-

tion from to the ( ()) is required, thereforethe inverse of the matrix () must be calculated.However, this is an orthonormal matrix, so its inverseis equal to the transpose of this matrix:

() = (()) = (()) (6)

Hence: () = (7)

+ +

(8)

In the notation above aswell as in (5) vector consistsof Euler angles:

=

(9)

In the further part of this work, the vector of Euler an-gles will be used for the purpose of deining the trans-formation matrices.4.2. Angular velocies

In the irst point of the description of the math-ematical model (Section 4.1), the model assumptionand a description of Eulers notation (Chapter 3) havebeen presented. Euler angles also constitutes the basefor the notation of angular velocities of the quadroro-tor. The Euler rates (vector ) are projections of therobots angular velocities (in the local coordinate sys-tem ) onto the axes of the coordinate system.This is an alternative notation in relation to the no-tation in the system (vector ). This calculation isnecessary, because measurements are performed by

gyroscopes, which refer to the robots frame. Thus, inorder to calculate the frequency of Eulers speed (Eu-ler rates) projection of the system () on axes ofthe coordinate systemmust bemade. Algorithm fortheEuler rates calculationhas beendeinedby [9]. Theirst step is the theoretical calculation of the angularvelocity in the local coordinate system:

=

= (10)

00 + ()

00 + ()()

00

, (11)

which may be written in a matrix form:

= , (12)

where:

= 1 0 0 0

(13)

The next step is to calculate the inverse of the matrix:

= () = 1 0 0 )

(14)

Finally, having already matrix, which depends onthe angular speed of the local coordinate system, Eulerrates may be deined:

= (15)

4.3. Angular acceleraonOne of the simpliications is the assumption of

robots structure perfect stiffness. On the basis of thissimpliication, the description of angular accelerationof the robot may be determined in the same way asfor the rigid body in space. Mathematical descriptionfor this effect is possible thanks to Euler equation ofa rigid body motion [1]. The general form of the Eulerequation may be deined as:

+ = , (16)

where: - angular momentum vector - vector of input torques - vector of angular velocity

For a rigid body is also true that:

= , (17)

where is the matrix of inertia moments of an object[7]:

= 0 00 00 0

(18)

11


Substituting (17) to (16) (assuming the constancy ofinertia moment) following equation may be obtained:

= + ( ) (19)This equation, however, due to the gyroscopic effectcaused by the motors and rotors, must be modiied.Gyroscopic effect is described by the basic equation ofthe gyroscope:

= , (20)

where: vector of angular momentum of rotors vector of input torques precession angular velocity vector

The effect of gyroscopic precession motion mani-fests itself on the object, which is equipped with a ro-tating mass. It is therefore true that:

= , (21)

where: is the angular velocity vector of the robot inthe local coordinate system .Angular momentum vector is formed by multiplyingthe moment of inertia of the rotor () by the rotorsangular velocity about the axis where the rotation oc-curs ():

= =

= (22)

00

( + + + ) (23)

It must be noted that the rotor angular velocity vector() contains zeros for the components in the axesof , and the sum of the rotor angular velocity (where varies in the range of < 1, 4 >) in -axis.This is because rotors axes are parallel to the axisin the system, which is valid in this description ofgyroscopic effect.

Knowing the and , equation (20) may beexpanded:

=

00

( + + + ) =

(24)

( + + + )( + + + )

0 (25)

With the inal form of gyroscopic torques vector, Eu-lers equation for the object (16) may be upgraded:

= + ( ) + (26)

In this equation the resulting quantity is the angularacceleration, so after transformations it may be writ-ten:

= ( ( ) ), (27)

which, after total multiplication of all components,leads to the form:

= ( ( )) ( ( )) ( ( ))

+ (28)

+( + + + ))( + + + ))

0 (29)

4.4. Linear acceleraonSimilar to the case of angular accelerations, linear

accelerations are also calculated in the local coordi-nate system BF. This is because all forces and torquesoperate in the local coordinate system - directly on therobot.The general form of the equation for linear accelera-tion may be written in the form of the second law ofdynamics:

= , (30)

where: - total mass of the robot, - accelerations vector of the robot, - active force vector in the local coordinate system, - gravitational forces vector in local coordinate sys-tem, - Coriolis forces vector due to rotations of the en-tire robot, - Coriolis forces vector due to rotors rotations ofthe robot,As may be seen above, this equation consists of a con-straint, which is the active force decreased by the grav-itational force and the Coriolis force.The irst of Coriolis forces results from the rotation ofthe entire robot:

= 2 = 2

, (31)

while the second Coriolis force is due to the rotationof rotors:

= 2 = (32)

2 ( + + + ) ( + + + )

0 , (33)

where mass of the entire object is equal to , whilethe mass of propellers and rotors is equal to .

In equation (30) the force of gravity should alsobe explained - it is a projection of force from thesystem to the system :

= ()

00 =

(34)

12


Having already all components, the equation (30)maybe expanded to its extended inal form:

= 1

+ (35)

2

+ (36)

2 ( + + + ) ( + + + )

0 (37)

5. Forces and Forcing TorquesIn the mathematical transformations derived

above, two forcing vectors were used, namely theforces vector and the torques vector . However,these physical quantities were not yet speciied. Todo so, the lift force of the quadrocopter must bedetermined. This force is equal to the product of thesum of squared rotors angular velocities and the gainfactor .

= ( + + +) (38)Because this force acts only in the axis direction

Fig. 4. Coordinate systems on the robot

(Figure 4) of the local system, therefore it may bewritten as:

=

= 00

( + + + ) (39)

Another issue is the generation of moments of forces,which arise only in the case of unbalanced rotationspeeds of rotors. That is when the:

( + + +) 0 (40)The equation of generated moments may be writtenas:

=

= ( ) ( )

( + ) , (41)

where is equal to arm force (length of the quadro-copter arm), while refers to the reaction torque gaincoeficient in the axis direction.

6. The Complete ModelHaving the successive elements of model and

forces, one may write a complete model of a lying ob-ject. The irst part refers to a linear acceleration,whichis deined in the local coordinate system:

=

= (42)

+ 2( )2( )2( )

+ (43)

+2 ( ( + + + ))

2 ( ( + + + ))0+ (44)

+00

( + + + )

(45)

The second part are Euler rates, which will be used inorder to determine changes of robots local systemorientation ralative to the auxiliary system :

=

= (46)

+ +

+

(47)

The last part refers directly to angular accelerations:

=

= (48)

(

)

(

)

(

+ )

(49)

(( )) (( )) (( ))

(50)

(( + + + )) (( + + + ))

0 (51)

In this model, both the vector as well as the vector describe the relationships which exist in the local co-ordinate system. Euler rates vector describes therelationship between systems: and .

13


7. The Results of Simulaon TestsThe mathematical model of the quadrocopter, de-

scribed above, was implemented in the MATLAB envi-ronment for testing and to assess the impact of the dy-namics on the behavior of real four-rotor lying robot.Simulation tests allowed to verify the light trajec-tory of the robot at various control signal levels. Themost important advantage in this case was the abil-ity to analyze the object stability to the different exter-nal disturbances, which in the work with a real robotequipped with several sensors to record light, in gen-eral might easily expose him to crash. Implementationand simulation tests were preceded by the identiica-tion of physical quantities characterizing parametersof the robot from the Figure 2 in the general model(42) - (51). According to thenotation (42) - (51), trans-parency of the model is guaranteed by the knowledgeof: a mass of the propeller-rotor (), total mass ofthe robot (),moment of inertia in X, Y, Z axes (respec-tively , , , ), aswell as rotor coeficients andand the arm length . For the purpose of further simu-lations, appropriatemeasurementswere done andnu-merical values of coeficients were obtained:- total mass of the robot: = 2.3 []- mass of the rotor: = 0.01 []- arm length of the robot: = 0.2825 []- moment of inertia in theX axis: = 0.0250 []- moment of inertia in theY axis: = 0.0230 []- moment of inertia in the Z axis: = 0.0475 []- moment of inertia of the rotor: = 0.000065 [ ]

- coeficient: = 0.01149- coeficient: = 0.001For the quadrocopter mathematical model with pa-rameters given above, a number of simulations wasundertaken. During analysis and results veriication,particular attention was paid to Euler angles. Theyprovide themost important information about the be-havior of the orientation while maneuvering in the air.This information is key and essential knowledge forthe development of an eficient algorithm of quadro-copter control. For tests purposes of implementedmathematical robots model, authors have proposedthe use of controllers due to clear, intuitive opera-tion and simplicity of parameters tuning (with heuris-ticmethod). Each controller has been dedicated to oneof the Euler angles. For proper settings of the con-trollers parameters (selected by use of Swarm algo-rithm), graphs shown in Figures 5, 6, 7, 8 and 9, havebeen generated.

In Figures 5, 6 and 7 Euler angles change overtime is presented for the given, external disturbance.The controller response is directly visible on coursesshown in Figure 8 and 9 - it is noted that every changeof angular velocity begins and ends at the speed of21 radians/s. This value corresponds to the gener-ated thrust, which provides the compensation of grav-ity forces by the drivetrain of a quadrocopter. Figures:8 and 9 purposely grouped rotors angular velocities

Fig. 5. Euler angle during the simulated ight



Fig. 8. Angular velocies of the robots rotors (pair ofrotors no.2 and no.4) associated with the Euler angles

pairs as: 1-3 and 2-4. It should be emphasized thatthis grouping, results from quadrocopters architec-ture - each pair of rotors is associated precisely withone axis in the system coordinates of the robot (,

14


Fig. 9. Angular velocies of the robots rotors (pair ofrotors no.1 and no.3) associated with the Euler angles

local coordinate system). The next part of the experi-ment was to ind the differences that may result froman incomplete physical model. For this purpose, au-thors removed the gyroscopic effect from the equation(28). This effect refers only to the axis (angle) andis due to the rotation of the entire robot:

= ( ) (52)The results are shown in Figures: 10, 11 and 12.

Fig. 10. Euler angle with and without (WGE) thegyroscopic eect


Itmaybenoted that the gyroscopic effect is a resultof unbalanced of inertia movements about for axes and (52). As it turns out, this is a very important ef-fect, which in dynamic situations may strongly inlu-ence the orientation of the robot. The omission of thiseffect is therefore unacceptable.


8. SummaryThe ways how to assess the eficiency and valida-

tion of the mathematical model in science and engi-neering aremultiple, but physical forces impose some-how the use of certain measures of assessment. Inthe case of such a complex object as four-rotor lyingrobot, a reasonable way to assess the adequacy of itsmathematical model, is to compare responses of both:real object and model - for the forces set in determin-istic way. It should be remembered that certain fac-tors can not be measured and predicted in a real sys-tem as they have a random character. Also, the samemeasurement method may be unreliable, having con-sequences on the dynamics of the lying robot undercontrol. Data acquisition system with sensors or on-board vision system which might be used for exper-imentation should have to be quick and precise. Suchinstrumentation increases signiicantly the costs of ex-periment in the modeling phase. Besides, not all be-haviors of quadrocopter can be checked on the realobject - as the risk of robot damaging is high (for ex-ample a rapid set value in control signal). Therefore,in searching for a compromise, it was decided to usethe mathemathical analysis of Euler angles while con-sidering the orientation of robot model in the contextof the gravity force. The obtained results conirm thatthe mathematical model adequately describes the be-havior of a simulated robot, and allows for its imple-mentation in the simulator (computer program) re-lecting the physical aspects of the environment. Theobtained simulation tool enables a better understand-ing of robot navigation aspects, which can not be di-rectly tested on a the real robot. Tests in the simula-tor will be the next step undertaken by the authors inorder to develop and propose further eficient controlalgorithms of quadrocopter subjected to the externaldisturbances (wind, turbulence, air temperature vari-ation).

AUTHORSStanislaw Gardecki Poznan University of Technol-ogy, Institute of Control and Information Engineer-ing, ul. Piotrowo 3A, 60-965 Poznan, Poland, e-mail:[email protected] Giernacki Poznan University of Technol-

15


ogy, Institute of Control and Information Engineering,ul. Piotrowo 3A, 60-965 Poznan, Poland, e-mail: [email protected] Goslinski Poznan University of Technol-ogy, Institute of Control and Information Engineer-ing, ul. Piotrowo 3A, 60-965 Poznan, Poland, e-mail:[email protected] Kasinski Poznan University of Technol-ogy, Institute of Control and Information Engineering,ul. Piotrowo 3A, 60-965 Poznan, Poland, e-mail: [email protected] author

REFERENCES[1] Wie B., Space Vehicle Dynamics and Control,

The American Institute of Aeronautics andAstronautics - Educational Series, 1998. DOI:http://dx.doi.org/10.2514/4.860119

[2] Ohanian O.J., Ducted Fan Aerodynamics and Mod-eling,withApplications of Steady and Synthetic JetFlow Control, Virginia Polytechnic Institute, 2011.

[3] Hefley R.K., Mnich M.A., Minimum-ComplexityHelicopter SimulationMathModel,National Aero-nautics and Space Administration, 2003.

[4] Hald U.B.,Autonomous Helicopter -Modelling andControl, Aalborg University, 2005.

[5] Gardecki S., Kasinski A., Testing and selection ofelectrical actuators for multi-rotor lying robot,PAK, 2012.

[6] Erginer B., Altug, E., Modeling and PD Controlof a Quadrotor VTOL Veh. In: Proceedings of the2007 IEEE Intelligent Vehicles Symposium, 2007.

[7] Craig J.J., Introduction to Robotics Mechanics andControl, Addison-Wesley, 1989.

[8] Banka St., Multivariable Control Systems: A Poly-nomial Approach, ZUT University Publishing,2007.

[9] Bak T., Modeling of Mechanical Systems, Lecturenote Aalborg University, 2002.

[10] Azzam A., Wang X., Quad Rotor ArialRobot Dynamic Modeling and Conigura-tion Stabilization. In: 2nd InternationalAsia Conference on Informatics in Con-trol, Automation and Robotics, 2010.DOI:http://dx.doi.org/10.1109/CAR.2010.5456804

[11] Albertos P., Sala A., Multivariable Control Sys-tems: An Engineering Approach, Springer-VerlagLondon, 2004. DOI: http://dx.doi.org/10.1016/j.automatica.2005.04.003

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17

Mathematical Modeling and Computer Aided Planing of Communal Sewage Networks

Lucyna Bogdan, Grayna Petriczek, Jan Studziski

Submitted: 20th March; accepted 30th August 2013

DOI: 10.14313/JAMRIS_2-2014/14

Abstract:In the paper the basic questions connected with modeling of wastewater networks are presented. Methods of modeling basic sewage parameters and appropriate calculation al-gorithms are described. The problem concerns the gravita-tional networks divided by nodes into branches and sectors. The nodes are the points of connection of several network segments or branches or the points of changing network parameters as well as of location of sewage inflows to the network. The presented algorithms for networks hydraulic calculation concern sanitary or combined sewage nets. It is assumed that the segments parameters such as shape, ca-nal dimension, bottom slope or roughness are constant. Be-cause of these assumptions all relations considered concern the steady state conditions for the network. The calculation of flow velocities and the filling heights in the segments of the wastewater net are carried out for the known slopes and diameters of the canals.

Keywords: mathematical modeling of sewage network, hydraulic parameter of canal

1. Characteristic of Sewage SystemsTaking into account a design and the operating processes we can distinguish the following sorts of sewage: housekeeping (sanitary) sewage, industrial sewage, rain wastewater, drainage sewage and ground water. The following sewage systems can be marked out depending on the kind of wastewater dump:a) combined sewage systemb) separated sewage systemc) semi-separated sewage system.In an universal sewage system (combined system) all kinds of the wastewater are led using the common canals. At the present time the separated sewage sys-tems are mostly used and there are two separated sewage nets to notice:a) a sewage net, used for the housekeeping sewage and for the industrial sewageb) a rainwater net, used for carrying out the rain-wastewater.The semi-separated sewage system is a system enclosing two kinds of nets: the housekeeping net and the rainwater one. In this system the sewage net can receive a part of the rain run-off.In this paper the following basic assumptions are made:

Only housekeeping or combined sewage nets are considered, divided into branches and segments by nodes. The nodes are the points of connection of several network segments or branches or the points of changing of network parameters as well as of location of sewage inflows to the network (sink basins, rain inlets, connecting basins). In the connecting nodes the flow balance equations and the condition of levels consistence are satisfied. It is assumed that the segments parameters such as shape, canal dimension, bottom slope or roughness are constant. Because of these assumptions all relations concern the steady state problem. The nets considered are of gravitational type.2. Basic ProblemsDesigning and analysis of sewage networks are connected with the following tasks:1. Making hydraulic analysis of the network for known section crosses and for known canal

slopes. In this case the calculation of filling heights of the canals as well as the calculation of flow velocities depending on the sewage flow rates must be done. These calculations are done for the respective net segments using the earlier received flow values.2. Designing of new segments of the network. It concerns the case when the new segments of the network must be added to the existing ones. In this situation diameters and canals slopes must be chosen for the new canals. It is assumed that the sewage inflows are known.

3. Basic Hydraulic Dependences in Sewage NetworkAccording to Manning formula the flow velocity of sewage depends on hydraulic radius R and radius R depends on the filling height H. The Manning formula for velocity v has the form:

2132 JRv n1 = (1)where: R hydraulic radius, J canal slope, n rough-ness coefficient, v flow velocity.The relations presented in the following concern the canals with circular section. From Manning for-mula and taking into account canal geometry one can obtain the following relation:


Articles18

for H 0.5d: ( )pi= sin8d4dA 22 (3a) ( )12arccos2 dH = (3b) pi += 5.0sin8d4dR (3c)where: A cross-section area, H filling height, r ra-dius of circular canal, central angle, d canal in-side diameter.From the above expressions one can see that for circular canals the cross-section area A and the hy-draulic radius R depend on the canal filling height H and as a result the sewage flow velocity depends also on the canal filling height H when canal slope J and diameter d are given.We define for the following the canal filling degree in form of relation H/d. In Figures 1 and 2 the rela-tions between A and H/d and between R and H/d for different diameters values d are shown. The figures show that section area A increases monotonically with growing canal filling degree H/d. For greater diameter values the increase of section area is faster and its values are greater. The greatest value of A is in the case of total canal filling and it equals d2/4. Hydraulic radius R increases from zero and achieves its maximum for the filling ratio of 81.3% and then it decreases to the value equal to half of the canal height. For the total filling and for the half canal filling the value of radius is d/4. For greater diameters d also

the hydraulic radius grows but the shape of the curves does not depend on d. The sewage velocity depends on the canal parameters like diameter, canal slope and roughness coefficient and on the canal filling de-gree (Fig. 3).The sections of the surface from Fig. 3 using planes J=const. are presented in Fig. 4. It shows that the func-tion describing velocity v depending on filling degree H/d has the shape similar to the function describing hydraulic radius R. The sewage velocity increases from zero and achieves its maximum for the filling degree of 81.3% and then it decreases to the value equal to half of the canal height. Greater diameters d

Fig. 1. Dependences between the cross-section area and canal filling degree for different diameter values

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

H/d

A

d=0.2d=0.3d=0.4d=0.5d=0.6

Fig. 2. Dependences between the hydraulic radius and canal filling degree for different diameter values

0

0.02

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R

d=0.2

d=0.3

d=0.4

d=0.5

d=0.6

0

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0.95

0.5%

8.0%

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1

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3

4

5

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7

8

V

H/d

J

Fig. 3. Dependences between flow velocity v, canal fill-ing degree H/d and canal slope J for roughness coef-ficient n=0.013 and for canal diameter d=0.6

Fig. 4. Dependences between flow velocity v and canal filling degree H/d for roughness coefficient n=0.013 and for canal slope J=5% by different diameter values d

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

H/d

v

d=0.2d=0.3d=0.4d=0.5d=0.6


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increase only velocities v but the shape of the curves presented does not depend on d.The section of the surface from Fig. 3 using plane H/d=const is shown in Fig.5. It shows that the flow ve-locity increases monotonically with the growing canal slope for the given filling degree.Fig. 6 shows the relation between flow velocity v and canal filling degree H/d for different slope val-ues J. One can see from Fig. 6 that greater values of canal slope increase only velocity values and they do not influence the shape of the curves presented.4. Algorithm for the Calculation of Waste-

water Networks 4.1 The algorithm for Calculation of Canal Filling

Heights and Flow Velocities The algorithm presented requires the following data for its calculation: type of the network housekeeping sewage net or combined sewage net structure of the network numbers of segments and nodes and type of nodes maximal sewage inflow into the network and the corresponding node number slows of canal bottoms and the canal dimensions.The task of the algorithm is to determine the fol lowing values for given values of rate inflows Qi: filling heights in each wastewater network segment, flow velocity for each network segment.

The calculation scheme presented below is for the canals with circular section. The algorithm consists of the following steps:Step 1. Entering the network structure and input data, i.e. number of nodes NW, number of segments NO, set of nodes W={j=1,..,NW}, set of segments U={i=1,..,NO}, set of diameters {di}, set of slopes for segments Ji, i=1,,NO, roughness coefficients ni.Step 2. Calculating the inflow rates for network input nodes; they are calculated depending on the kind of sewage. For the housekeeping and industrial sewages the maximal hour inflow Q for given network segment can be calculated according to the relation: [1], [4], [7], [8] = hmaxrhmax N MqQ 24 (4)where: M number of residents for the given segment of the net, q

r average wastewater amount for aver-age housekeeping unit, Nhmax rate of irregularity for twenty four hours.For the rain the wastewater inflow can be ex-pressed as follows: [1], [4], [7], [8]

Q=qd F (5)where: Q rain wastewater inflow caused by infiltration [dm3l/s], F area of drainage basin for the canal segment considered [ha], ratio between the rain wastewater amount passing into canals and the rain wastewater amount coming from the whole area given, rate of delay between the rain time and the time of infiltration result, dq rain intensity.Step 3. For given rate inflows Qi in segments i=1,., NO one can determine the following values: filling heights Hi, hydraulic radius values Ri and flow velocities vi.1. From the Manning formula and taking into account the canal geometry one can obtain the following relations with

dHx = :

For H/d 0.5 0Q)x(F1 = (6a) ( )( )32 3

5

)x(

)x(sin)x()x(F1

111

= (6b)

( )x21arccos2)x(1 = (6c)For H/d > 0.5 0Q)x(F2 = (7a)

Fig. 5. Dependences between flow velocity v and canal slope J for given canal filling degree H/d

0123456789

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%J

v

H/d=0.75

Fig. 6. Dependences between flow velocity v and canal filling degree H/d for roughness coefficient n= 0.013 and for diameter value d=0.6 for different canal slopes J

0123456789

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

H/d

v

J=10%J=8%J=6%J=4%J=1%J=0,5%


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( )( )( )32 35)x(5.0 )x(sin5.0)x(5.02)x(F 2 222 pi +pi= (7b) ( )1x2arccos2)x(2 = (7c) 213538 J415.0 )d(n1 = (8)where: H filling height, j central angle, d inside canal diameter, J canal slope, n roughness coeffi-cient, Q rate inflow, H/d - canal filling degree.

The parameter in (8) depends on canal diameter d and on canal slope J and for the fixed diameter val-ues and canal slopes it is constant.Solving equations (6a)(7b) we obtain canal filling degree H/d as a function of flow rate Q.2. For canal filling degree H/d calculated above the hydraulic radius Ri should be determined according to the formula:For H/d 0.5: pi +pi= 5.0 )sin(5.05.04dR (10a) ( )12arccos2 dH = (10b)

3. The flow velocity should be calculated from: 2132 JRv n1 = (11)

Knowing the network geometry, i.e. slopes, shapes and diameters of canals as well as the wastewater inflows Qi, one can calculate filling heights and flow velocities for each network canal. The calculations are carried out for each network segment beginning from the farthest one and going step by step to the nearest segment regarding the wastewater treatment plant.Step 4. The equations of flow balances 0Q

ijj =

and the conditions of surface levels equality are calcu-lated in each network node.

Step 5. The whole network will be calculated once again with the wastewater inflows changed. Under

assumption of constant sewage flows in the network segments the sewage system simulation can be ex-ecuted for a sequence of time steps, for a couple of hours or days; by such the calculation the change of the wastewater inflows occurring with the time must be considered.There is to notice that the parameters analyzed in the algorithm, i.e. filling heights, hydraulic radius values and flow velocities depend on the wastewater inflows and by the rain wastewaters there is important to take into account their changes and to repeat the simulation runs according to their frequency. The algorithm presented can be considered as a part of the complex model for calculation sewage networks also under unsteady state conditions.4.2 Analysis of Equations (6a)-(6c) and (7a)-(7b)Equations (6a)(7b) for calculating the canal fill-ing degree are nonlinear and to solve them the stan-dard numerical methods for solving nonlinear alge-braic equations can be applied. In order to determine the equation roots some conditions for parameter and sewage flow Q must be fulfilled that will be dis-cussed in the following.Function F(x)=F1(x)+F2(x) is continuous in values range (0; 1>. For x=1, i.e. for the full canal filling, there is F= 2p and for x=0.5 we get F= p. In values range (0; 0.8> function F(x) is growing monotone. In values range (0.8; 1>. function reaches its maximum Fmax = 6.7588 for x = 0.9381. It is diminishing in values range (0.9381; 1>. This analysis has been done for d=0.6, J=1% and n=0.013. For fixed network parameters like canal diam-eter d and canal slope J, equation F(x)Q=0 has solu-tions depending on sewage flow Q (Fig. 7).

Fig. 7. Diagrams of function F(x)Q for different val-ues of Q in values range (0; >

-0.4

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1H/d

*F-Q

Q=0.1 Q=0.2 Q=pi*

Equation F(x)Q=0 has the following roots:1. For x(0; 0.5> there is only one root and the fol-lowing inequality must be fulfilled: 0< Q p. This inequality defines a values range for sewage flows Q for fixed canal diameters d and canal slopes J.2. For x(0.5; 1> equation F(x)Q=0 has the fol-lowing roots: one root for x(0.5; 1) and p < Q < 2p two roots for x(0.5; 1> and 2p Q <

6.7586936, whereas for Q=2p there are x1=1 and x2=0.81963.


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The results of the discussion are shown in Fig-ures 8 and in Fig. 9 the case with two roots of equation F(x)Q=0 is presented with Q=2p and Q=0.63 (Q < 6.7588). For the fixed network parameters such as a canal diameter d and canal slope J the above relations let to

decide what are the solutions for the given flow Q and whether the value of Q is not greater than the upper limit 6.7586936, what means the lack of solutions. In such the case a change of one or of both of the fixed network parameters d and J must be considered.The result of the above relations is that the flow value Q depends on the parameter . The parameter depends on the canal diameter d and on the canal slope J. The equation describing the dependence of canal filling on the flow in the range ( 0; 2p) has one solution in this range and that is why this range is relevant. In Fig. 10 the relation between the solution of equation F(x)Q=0 and flow Q for d=0.6, J=2%, n=0.013 and 00.225 kg/m2 for communal and industrial wastewater, specific gravity of sew-age kg/m3, R hydraulic radius.The canal slope shall be calculated for 60%70% of the canal filling height.

Fig. 8. Diagrams of function F(x)Q for < Q < 2

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*F-Q

Q=0.45 Q=0.58

`

Fig. 9. Diagrams of function F(x)Q for 2 Q


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Step 2. Solution of the following equation: 0Qd 38 = (13) ( ) 2135 J41n = pi

If a solution d of the equation exists, then in-equality 0Qd 38 > is valid for all values > dd .If canal slope J has been calculated from relations (12a)(12b) and now a value d greater than d will be taken into account, then one shall pass to Step 1 and the canal slope must be calculated again.If a solution of equation (13) does not exists then one shall return to Step 1, change the value J and solve once again equation (13).In Fig. 11 the relations between the solution of equation (13) (concerning canal diameter d) and ca-nal flow Q for different canal slopes J are shown.5. Computational Example The considered algorithm has been tested on an ex-emplary housekeeping network consisting of 17 nodes connected by 16 segments. The net has got 9 input nodes (W6, W7, W8, W10, W11, W14, W15, W16, W17) and 1 output node W1. Other nodes constitute the connections between different segments of the network. [11]

The arrows in Fig. 12 show the sewage flow direc-tion. The sewage flow rates values for the input nodes are given. The flow rates in the connection nodes should be calculated according to the balance equa-tion. For the respective segments the values of diam-eters d and canal slopes J are given.For such a structure of the net the fillings H/d and the velocities of flows v in respective segments are calculated. The conclusion is that for these values of sewage rate flows and for the given values of geo-metric parameters (diameters and canal slopes), the heights of filling are lower than the half of canal di-ameters. So there is a possibility of increasing of the input flows in some sewage nodes. The calculations results are shown in Table 1.Table 1. The results of hydraulic computations for the exemplary net shown in Fig. 12Upper node Lower node Segment. D [m] Q [dm3/s] J H/d v [m/s]W6 W5 1 0.2 0.53 5 10.72% 0.,309W7 W5 2 0.2 0.31 5 8.09% 0.259W5 W4 3 0.2 1.14 5 15.08% 0.383W10 W9 4 0.2 0.36 6 8.32% 0.289W11 W9 5 0.2 1.13 9 13.03% 0.469W9 W4 6 0.2 2.13 5 20.48% 0.460W4 W3 7 0.2 3.91 5 27.78% 0.549W8 W3 8 0.2 0.11 5 4.98% 0.189W3 W2 9 0.2 4.12 5 28.53% 0.557W14 W13 10 0.2 0.11 5 4.98% 0.189W15 W13 11 0.2 0.32 5 8.22% 0.261W13 W12 12 0.2 0.66 5 11.59% 0.325W16 W12 13 0.2 0.24 5 7.18% 0.24W12 W2 14 0.2 2.76 5 23.29% 0.497W17 W1 15 0.2 6.33 5 35.70% 0.629W2 W1 16 0.2 7.61 5 39.42% 0.661

Fig. 11. Relations between canal diameter d and canal flow Q for different canal slopes J

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Q

d

J=J self-purification J=0.5% J=1% J=1/d


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ACKNOWLEDGEMENTSThe paper is a result of the research project No N N519 6521 40 financed by the Polish National Center of Science NCN.AUTHORSLucyna Bogdan, Grayna Petriczek, Jan Studziski Systems Research Institute Polish Academy of Sci-ences, Newelska 6, 01447 Warsaw, Poland, email: [email protected], [email protected], [email protected]*Corresponding authorREFERENCES[1] Biedugnis S., Metody informatyczne

w wodocigach i kanalizacji, Oficyna Wydawnicza Politechniki Warszawskiej, Warsaw 1998, in Polish.[2] Bogdan L., Petriczek G., Zagadnienia modelowania sieci kanalizacyjnej dla potrzeb zarzdzania przedsibiorstwem wodocigowym, series: Studia i Materiay Polskiego Stowazrzyszenia Zarzdzania Wiedz, vol. 22, 2009, pp. 3242, in Polish.

[3] Baszczyk W., Stamatello H., Blaszczyk P., Kanalizacja. Sieci i pompownie. vol. 1, publ.: Arkady, Warsaw 1983, in Polish.[4] Chudzicki J., Sosnowski S., Instalacje kanalizacyjne, publ.: Seidel-Przywecki, Warsaw 2004 in Polish.

[5] Jaromin K., Jlilati A., Borkowski T., Widomski M., adg G., Rodzaje materiau i sposoby eksploatacji a wspczynniki szorstkoci w przewodach kanalizacji grawitacyjnej. In: Proceedings of ECOpole, vol. 2, no. 2, 2008, in Polish.[6] Karnowski J.M., Warunki transportu wleczonych czci mineralnych w przewodach koowych o dowolnym nachyleniu. In: Materiay Konferencji Naukowo-Technicznej PZITS, Pozna 1973, in Polish.

[7] Kwietniewski M., NowakowskaBaszczyk A., Obliczenia hydrauliczne kanaw ciekowych na podstawie krytycznych nate stycznych. Wodocigi i Kanalizacja, 13, 1981, in Polish.

[8] Niedzielski W., Charakter przepywu w sieci kanalizacji deszczowej, Ochrona rodowiska, no. 434/34, 1984, pp. 2021, in Polish.[9] Puchalska E., Sowiski N., Wymiarowanie

kanaw ciekowych metod krytycznych napre stycznych, Ochrona rodowiska, no. 34, 1984, pp. 5362, in Polish.

[10] Serek M., Zastosowanie mikrokomputerw do obliczania sieci kanalizacji deszczowej, Ochrona rodowiska, no. 488/12, 1986, pp. 2728, in Polish.

[11] Sualec A., Sie kanalizacji ciekowej obliczenia hydrauliczne. Raport Badawczy IBS PAN, Warsaw 2010, in Polish.[12] Wartalski J., Komputerowe metody projektowania i analizy hydraulicznej sieciowych ukadw kanalizacyjnych, Ochrona rodowiska, no. 434/34, 1984, pp. 2021, in Polish.[13] Wartalski A., Wartalski J., Projektowanie hydrauliczne rurocigw z tworzyw sztucznych. Ochrona rodowiska, no. 1/76, 2000, pp. 1924, in Polish.

[14] WILO Polska Producent pomp i urzdze sanitarnych, Podstawy odprowadzania i pompowania ciekw. Oferta handlowa, in Polish.

Fig.12. Structure of the sewage net used computational example

W6 W5 W7 W8 W4 W3 W9 W10 W16 W2 W11 W12 W17 W13 W15 W14 W1


24

Failures Location within Water Supply Systemsby Means of Neural Networks

Izabela Rojek, Jan Studziski

Submitted: 20th March 2013; accepted 30th August 2013

DOI 10.14313/JAMRIS_2-2014/15

Abstract:In the article the neural networks used for failures loca-tion for water supply networks are presented. To do this a hydraulic model of the water net, as well as an appro-priate developed monitoring system have to be used. The current applications of monitoring systems installed in the waterworks do not realize their possibilities. The monitor-ing systems provided as autonomic programs to collect and record the information about flows and pressures of water in source pumping stations, in the pump stations bringing up the water pressure inside the water net and in the pipes of water supply network give a general knowledge about state of its work, but if they would be used as elements of IT systems supporting the water network management, they could help to solve the tasks concerning detection and lo-calization of water leaks. The models of failures location in water nets described in the paper are created by means of neural networks in the form of MLP nets.

Keywords: water-supply networks, network hydraulic model, detection and location of water leakages, neural networks

1. IntroductionThe main goals of a municipal water network are the supply of water to the water net users and correct operation of the water net assuring an appropriate water pressure in the water net end nodes, efficient removing of the failures, as well as planning and ex-ecuting of activities concerning conservation, mod-ernization and extension of the network whereas the water supplied and distributed under the water net users has to be of a suitable quality and sufficient quantity [1]. The operation and control of a water net-work is a difficult and complex process. The problem of detection and localization of hidden leakages in the water network is one of the most important water net management tasks. This is because of the water losses caused by the water net damages; and the resulted water losses can reach sometimes even 30% of the total water production what has essentially and nega-tively financial results of waterworks. Therefore, the fast location and elimination of water leaks and (espe-cially of these hidden ones) can bring the measurable economic advantages both for the waterworks and for the water net end users.The different stages of the whole process of elimi-nation of water leaks can be defined in the follow-ing way:

failures detection a failure case can be determined by observation of a bigger water tribute, but the failure location cannot be defined;

failures location the failure place in the water net can be determined by means of some suitable algorithms and with the use of a monitoring system, the water net hydraulic model and particularly also of neuronal networks; failures counteraction using the failures historical data, development of models to forecast the water net emergency and the subsequent planning of network revitalization, the rate of the water net unreliability can be essentially reduced.2. The Algorithm for Water Net Failures

LocalizationDifferent approaches and computational algo-rithms to aid detection and location of water leaks in water networks have been already presented in the past and current literature [1, 2, 3, 4]. In every case a water network hydraulic model and a monitoring system installed on the water net are the basic tools for making the calculations. An appropriate comput-er infrastructure exploited on the water network is needed for practical realization of these algorithms. A monitoring system, a calibrated hydraulic model of the water net, as well as a GIS system for generat-ing the water net numerical map should be included as key components into this infrastructure. Such the extended computer infrastructure permits not only to detect and locate the water net failures but also to manage the network executing the tasks like wa-ter net control, water quality analysis and improve-ment, water net optimization and design, etc. [5, 6]. This means that the high developed ICT tools are use-ful and indispensable for water network management making it easy, right and optimal.In the following an algorithm to detect and locate the water leaks in municipal water networks is described. It uses the neuronal nets to create a classifier identify-ing and situating the water leaks arising in the water net. The algorithm consists of the following steps:1. Determination of a ranking list of sensitive points in the water net, using an algorithm for planning the monitoring systems.2. Choice of a suitable number of the most sensitive measuring points for the monitoring system to be installed on the water network.3. Development of a hydraulic model of the investigated water net and its calibration using the data from the monitoring system installed.


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4. Determination of standard distributions of pressure and flow values using the data from the measuring points of the monitoring system; these distributions are calculated for standard loads of the whole water network and of its end nodes.5. Simulation of leakage events in subsequent nodes of the water net by means of the hydraulic model and recording of pressure and flow values measured in the measuring points of the monitoring system.6. On the base of the failures data recorded, creation of the water leaks classifiers in form of neural networks and choice of the best classifier regarding the criterion of largest sensibility.7. On line measuring the water flow and pressure values in the water net using the monitoring system and comparison of the current data with these standard ones.8. In case of an essential difference between the standard and current data recorded by the monitoring system, use of the classifier to find out the water net node in which the water leak possibly happened.

2.1. Determination of Sensitive Points in the Water Net InvestigatedIn order to find out the best location of sensors for the measuring points of the monitoring system to be installed on the water net the so-called sensitive

points of the water net have to be determined. There are in state to collect the information concerning the changes in the water network not only in the points where they are installed but also from the remote surroundings. These sensitive points one can name as characteristic points of the water net in contrary to the so-called dead points in which only the local changes of the water network can be noticed. The usual prac-tice while developing monitoring systems consists in extension of number of the monitoring points what stays in opposition to the procedure shown above. A suitable choice of a comparatively small number of characteristic points in the water net can be equiva-lent regarding the quality and quantity of the informa-tion collected with larger number of points situated in less sensitive places of the network. To determine the sensitive points of the water net the following formu-las [7] can be used:

( ) ( )

= =

m m km m m km

k m k mpm qm

km kmk m k m

p / p L q / q LS S

L L where: k node with the water leak simulated, m measurement point considered, p water pressure, q water flow, Dpm and Dqm differences in measure-ments for standard and emergency states of opera-tion of the water net, L distance between the points k and m.The correct measurement points are these ones with the highest sensitivity values. In Fig. 1 the circu-lation of information while planning the monitoring system is shown. The data collected from the water

net by the monitoring system are recorded in the data base, which is mostly the branch data base of a GIS system and then there are used by a hydraulic model to calculate the sensitivity values of the water net nodes. One can see that to calculate these values a monitoring system installed on the water net, as well as a calibrated water net hydraulic model are needed what is not the case at the beginning of the procedure. Because of that the procedure is realized iteratively in the following steps: the first step means a calibration of the hydraulic model using data got from a measure-ment experiment performed at the water net; the sec-ond step means the sensitive points calculation using the hydraulic model calibrated and then the installa-tion of the monitoring system in the selected measur-ing points; the third step if realized means mostly the recalibration of the hydraulic model with use of the monitoring system already installed.2.2. Development of a Hydraulic Model of the

Water NetTo find out the sensitive points of the water net for planning the monitoring system the hydraulic model of the real Polish water net has been used. The hydrau-lic calculations can be made only with the hydraulic graph of the water net, which is topologically correct, that is compact and without any un-continuities. Hy-draulic graphs can be generated and exported to the hydraulic models by GIS systems and such the mech-anism is shown in Fig. 2. Such the operation makes

Fig. 1. Structure of the procedure for planning the monitoring system

Fig. 2. Export of a water net hydraulic graph from a GIS system to the water net hydraulic model


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the calculation with the hydraulic model much easier and faster as if the hydraulic graph would be designed using the software interface of the hydraulic model. After the data export from GIS system is already com-pleted then the calculation with the hydraulic model can be executed quite apart from GIS (Fig. 3).2.3. Simulation of Leakage Events in The

Subsequent Water Net NodesIn the research presented two cases of investiga-tion have been realized: for the monitoring systems consisted of 10 and of 20 monitoring points located on the water network in its most sensitive nodes. The execution of hydraulic calculations of the water net for its standard load without any water leaks, simula-tion of leakages in the subsequent water net nodes us-ing the hydraulic model and recording of flow values from the monitoring points for both cases of the mon-itoring systems and for both cases of the water net op-eration, in standard and in failures modes, leads to the preparation of learning files for the neural networks. In Fig. 4 one can see the data file got from the hy-draulic model with the flow values calculated for 10 measuring points of monitoring system for the stan-dard

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