Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Advanced D-Partitioning Analyi and it Comparion with the haritonov Theorem Aement amen M. Yanev Profeor, Department of Electrical Engineering, Faculty of Engineering and Technology Univerity of Botwana Gaborone, Botwana yanevkm@yahoo.com; yanevkm@mopipi.ub.bw Abtract Thi paper contribute for further advancement of the D-Partitioning analyi applied to ytem with multivariable parameter. It alo explore the effect of imultaneou ytem uncertaintie by determining graphically region of tability in the pace of the ytem parameter. The interaction between the varying parameter will alo bring a new light in the graphical olution of the problem of tability. Coniderable advantage of the uggeted advanced D-partitioning analyi tool are illutrated comparing it with the haritonov theorem aement. The advanced D-partitioning i beneficial for further development of control theory in the area of ytem tability analyi. eyword parameter; D-Partitioning; tability; haritonov theorem; analyi; I. INTRODUCTION Following ome initial idea of Neimark [], [], [] the D-partitioning method wa better clarified and further advanced by the author in previou publihed work [], [5], [6], [7]. It enable a quick and convenient determination of the region of tability in cae of variation of ytem parameter. The method of the D-partitioning i a powerful tool for ytem analyi. It can be eaily implemented and ha a coniderable of advantage compared to other tability analyi method. Baically, it ha the advantage of a clear graphical diplay of the variation of each parameter and it effect on the ytem tability. MATLAB oftware package can be employed for automatically plotting of the region of tability. The objective of thi reearch i to demontrate the application of the developed by the author advanced D-partitioning method for cae of imultaneou variation of two ytem parameter and it advantage compared to other well-known method, a the haritonov Theorem aement [8], [9], []. II. ADVANCED D-PARTITIONING IN CASE OF TWO SIMULTANEOUSLY VARIABLE PARAMETERS To implement the method of the D-partitioning, a general characteritic equation i preented in the format: n n n G ( ) a a... a () A characteritic equation of a hypothetical third order unity feedback ytem can be preented a follow: G ) ( T )( T )( T ) () ( It i uggeted that imultaneouly two of the ytem parameter are variable: T = (time-contant), = (gain) () The initial objective i to determine the region of variation of thee two parameter, for which the ytem will be table. Equation () are ubtituted in (), from where: G( ) [ T T ( T T T T ) ( T ] T ) () By ubtituting = j, the equation () could be preented in the detailed form: P( j) [ T T ( j) Q( j) R( j) T T ( j) Where ( T ( T P( j) P ( ) jp ( ) T )( j) T ) j j] (5) Q( j) Q ( ) jq ( ) (6) R( j) R ( ) jr ( ) Then the equation () can be preented by a et of two equation: P ( ) Q ( ) R ( ) P ( ) Q ( ) R ( ) (7) www.jmet.org JMESTN56 8
Conidering equation (7), the variable parameter can be determined a: T T T T ( T T ) ( T T T T ) The determinant of equation (8) i: (8) T T (9) The determinant become = at a pecific frequency = that can be found out from equation (8) a: () T T If =, both ytem parameter are approaching infinity: ( ), ( ), () Thi implie that the main D-Partitioning curve ha an interruption, or a breakdown, at a frequency =. It conit of two part, the firt one i plotted within the frequency range < <, while the econd one i obtained for < <. For a better clarification, firt the function () and () are plotted, a hown in Fig. (a) and Fig. (b). Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 a n =, at = ; a o =, at = () The pecial line are determined by comparing the equation () and () identifying the coefficient a n and a o and equalizing them to zero: T T =, + = or () = = Cont. = = Cont. The region of tability are determined by the D- Partitioning curve, defined by equation (8) and the pecial line, defined by equation (). The D- Partitioning region could be determined by plotting the main D-Partitioning curve, together with the pecial line on the (, ) plane. The locked region between thee part of the curve, correponding to realitic phyically realized ytem parameter and the pecial line are identified a the region of tability. The realitic table region are alo alway located on the left-hand ide of the D- Partitioning curve, following the frequency increment. Finally, by combining the curve (), () from Figure (a) and Figure (b) and the pecial line, the D-Partitioning i obtained in the (,) plane a een in Figure. D() min D() Region of Intability Special Line ( = ) D () = (T +T ) ω Region of Stability (,,) Plane ω=ω D () = (T +T ) (a) (b) Fig. : The graphical preentation of () and () howing the interruption of the curve at a frequency = The region of the D-partitioning alo depend on two traight line in the (,) plane, conidered a pecial line. The pecial line are plotted for the two border frequencie = and =. The equation of the pecial line are obtained from equation () by: Special Line ( = ) Fig.. Advanced D-Partitioning, Defining the Region of Stability and the Region of Intability Conidering equation (8), the variable parameter and are conidered a even function. It follow that each one of the parameter and ha overtracing value within the frequency region, a een from equation (): ( ) ( ) ( ) ( ) () Then, if in the plain (, ), the D-Partitioning curve i plotted following the frequency increment from to, the ret part of the curve, plotted for frequency increment from to + i over-tracing the already plotted curve in revere order. www.jmet.org JMESTN56 9
Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Taking into account that = T i a time-contant and it can adopt only poitive value, practically only the table region D () hould be conidered. The region of tability D () i locked within the left-hand ide of the D-Partitioning curve, correponding to frequency rie from = to and the pecial line =. Since the gain = may alo adopt only poitive value, the realitic border of the table region D () hould be conidered = =. Further, the concluion i that for mall value of the gain < min, the ytem i table for any value of the timecontant = T. A an example, a ytem coniting of an armature-controlled dc motor and a type-driving mechanim i uggeted to illutrate the application of the advanced D-Partitioning analyi in cae of variable gain and variable time-contant. Two of the motor time-contant T =.5 ec and T =.8 ec are known and contant value. The variation of the ambient temperature may caue the change of the ytem gain, while the variation of the load caue change of the mechanim time-contant T. The tranfer function of the open loop ytem i preented a: G o ( ) ( T )( T )( T ) ( T )(.5)(.8).T (.T.) ( T.) (5) Then, the characteritic equation of the unity feedback control ytem i determined a follow: G( ).T (.T.) ( T.) (6) By ubtituting = j and T = T, equation (6) i modified to: (.T.) ² j(.t ². T) (7) Since the gain may obtain only real value, the imaginary term of equation (7) i et to zero, then:. T ² (8).T The reult of (8) i ubtituted into the real part of equation (7), from where:.t ².69T.5..5T.5 (9).T T The D-partitioning curve = f (T) define the border between the region of tability D() and intability D() for the cae of imultaneou variation of the two ytem parameter. The D-partitioning curve = f (T), a preented in Fig., i plotted with the aid of the following MATLAB code: >> T = :.:5; >> =.5.*T+.5+../T = Column through Inf 7.55.75 9.5 8.775 8.5 8.7 8.57 8.5 8.59 Column through 8.775 8.988 9.8 9.5 9.76 9.9667.75.57.797.8 Column through.75.669.9659.65.5667.87.75.85.789.98 Column through.8.79 5. 5.9 5.657 5.97 6.86 6.6 6.97 7. Column through 5 7.55 7.867 8.85 8.5 8.85 9.89 9.576 9.7766.958.5 Column 5.75 >> plot(t,) Each point of the D-partitioning curve repreent alo the marginal value of the two imultaneouly variable parameter, which i a unique advancement and an innovation in the theory of control ytem tability analyi. Fig. : Advanced D-Partitioning in Term of Two Variable Parameter Initially, the illutration of the ytem performance in cae of variation of the time-contant T can be done when the gain et to =. Then, if the time-contant www.jmet.org JMESTN56
i within the range < T <.5 ec and T >.5 ec the ytem i table. But for the ame value of the gain =, the ytem become untable if the timecontant i in the range.5 ec < T <.5 ec. The ytem performance can alo be invetigated for any other value of the variable gain, like =, =, etc. It i obviou that if i varied, thi affect the value of T at which the ytem may become untable. Higher value of, enlarge the range of T at which the ytem will fall into intability. If < 8.7, a limit determined with the aid of a MATLAB procedure, the ytem i table for any value of the T. It i obviou that the ytem performance and tability depend on the interaction between the two imultaneouly varying parameter. III. COMPARIZON OF THE ADVANCED D-PARTITIONING WITH THE HARITONOV S THEOREM ASSESSMENT The well-known and popular haritonov' theorem aement can be ued in the cae where the coefficient are only known to be within pecified range [8], []. It provide a tet of tability for a ocalled interval polynomial. An interval polynomial i the family of all polynomial: n P( ) a a a... an ) The interval polynomial () i the characteritic equation of a control ytem with variable parameter, where each of it coefficient a i can take any value in the pecified interval a i [a i, a + i ], or a i a i a + i. The notation a i repreent the lower limit of the variable coefficient, while a + i repreent the upper limit of the variable coefficient. An interval polynomial characterized by equation () i table (i.e. all member of the family are table) if and only if the four o-called haritonov polynomial repreented in equation () are table [8], []. P ( ) a P ( ) a a a a a a a...... P ( ) a a a a... () P ( ) a a a a... It i obviou that while there i imilarity in the four haritonov polynomial, at the ame time, there i a pecific arrangement of the lower limit and upper limit coefficient at each one of thee polynomial. What i alo extraordinary about haritonov' reult i that although in principle an infinite number of polynomial are teted for tability, in fact only four polynomial need to be teted. Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Further, each of the four haritonov polynomial i teted for tability with the aid of the well-known Routh-Hurwitz tability criterion. The reult are placed in table for final aement of the ytem tability decribed by the haritonov interval polynomial. The objective of thi dicuion i the uggeted in thi reearch advanced D-Partitioning analyi, to be compared with the haritonov aement when the variation of the ytem uncertain parameter i defined within pecific limit [8], []. To validate thi comparion, the control ytem of the armature-controlled dc motor with a type-driving mechanim i conidered once again. In that cae, the characteritic equation (6) of the ytem can be preented a an interval polynomial, while the variable gain and the variable time-contant T are defined within pecific limit. To demontrate the application of the haritonov Theorem aement of the ytem tability, two cae are preented a follow: Cae : The original characteritic equation (6) i modified to the interval polynomial, hown in equation (), now being a family of all polynomial: P ( ) ( T (.T.).).T [8,], or 8 () T [,], or ec T ec The interval polynomial P () i table, (i.e. all member of the family are table) if and only if the four o-called haritonov polynomial are table: (.) (..) (.) 8 (.). (..) (..) 8 (.) (..)... () After the calculation, the haritonov polynomial are preented in the proper tate for aement: www.jmet.org JMESTN56
7.5 5.75.5.75.5.75.5.5.75 ().5 5.75.5 Taking into account any of the third order equation of (), repreenting the haritonov polynomial, in order to apply the Routh-Hurwitz tability criterion [], [] the following table i created: TABLE I ARRAY OF THE ROUTH-HURWITZ STABILITY TEST where (CASE OF A THIRD ORDER SYSTEM k i () a n a n- a n- a n- b c ( a a ) ( a a ) n n n n b (5) ( a a ) ( b c a n n ) ( a b n n ) (6) The Routh-Hurwitz tability criterion i applied to all thee polynomial k i (),where (i =,,, ). TABLE II RESULTS FROM THE FOUR HARITONOV POLYNOMIALS (CASE ) k () k () k () k () 5.75.. 5.75.5 7.5.75.75.75.5.5.5 -.7.6..6 7.5.75.5.5 In thi particular cae, the firt column of the Routh array for the three polynomial k (), k () and k () are all poitive (that i, there i no change of ign in the firt column). But the polynomial k () ha change of ign in the firt column of the Routh array. Thi mean that the cloed-loop ytem will be untable for the given et of coefficient variation. Cae : Although the polynomial k () in Table II have change of ign in the firt column and one of the component of the Routh array i negative, it value i Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 cloe to zero. Thi mean that the cloed-loop ytem i cloe to the tate of margin of tability. If the et of parameter variation i changed, the cloed-loop ytem may become table. Thi i demontrated with the following change of the parameter variation limit in the characteritic interval polynomial, a hown in equation (): P ( ) ( T (.T.).).T [,6], or 6 (7) T [,], or ec T ec Similarly, the interval polynomial P () i table if and only if the four o-called haritonov polynomial are alo table: 6 (.) (..) 6 (.) (..) (.) (..) (.) (..).... (8) After further calculation, all of the haritonov polynomial are in the proper tate for aement: 8.75.875.75.75.5.5 (9).5.5.5 6.5.875.75 Again the Routh-Hurwitz tability criterion i applied to all thee polynomial k i (),(i =,,, ) taking into account Table I. TABLE III RESULTS FROM THE FOUR HARITONOV POLYNOMIALS (CASE ) k () k () k () k () 8 6 8 6 www.jmet.org JMESTN56
Since the firt column of each haritonov Polynomial, hown in Table III, contain no change in ign and all it component are poitive, the concluion i that all of the root of each k i (), (i =,,, ) polynomial have negative real part. Therefore the cloed-loop control ytem i table for all coefficient value in the pecific range. That i, the feedback control ytem i guaranteed aymptotically table. The haritonov aement can be ueful for determining ytem tability in the cae of variation of large number of the ytem parameter when defined within pecific limit. At the ame time the haritonov aement method ha ubtantial diadvantage. It i hort of determination of the parameter marginal value of tability, alo the reult are achieved after coniderable calculation and there i lack of any graphical diplay viualizing thee reult. The haritonov aement i alo applicable only for a prearranged and pecified et of ytem parameter variation. The major diadvantage of the haritonov method i that the haritonov polynomial deal with the coefficient variation of the haritonov characteritic interval polynomial, rather than directly with the ytem parameter variation. The variation of the ytem parameter remain in a hidden mode. Thee variation cannot be directly oberved from the four haritonov polynomial. Alternatively, the advanced D-Partitioning analyi, preented in thi reearch, ha coniderable advantage, compared with the haritonov theorem aement. The advanced D-Partitioning analyi doe not need a pecified et of limit of parameter variation. It i applicable generally and can deliver reult repreenting the exact marginal value of the multivariable parameter. The D-Partitioning analyi reult are obtained eaily with the aid of the interactive MATLAB procedure. The D-Partitioning curve in term of the two variable parameter i plotted by the imple MATLAB code, a already demontrated. The clear graphical diplay of the region of tability and intability i another ignificant advantage of the D- partitioning. A graphical evaluation between the two method of analyi, a een from Fig., i validating the coniderable advantage of the achieved advanced D- Partitioning analyi compared with the haritonov aement. Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 Fig. : Advanced D-Partitioning Analyi compared with the haritonov Aement in Term of Two Simultaneouly Variable Parameter (Stability Aement for Cae and Cae ) The graphical reult of the advanced D-Partitioning analyi i illutrating immediately the region of tability D() and the region of intability D() that can be ued for the complete general aement of the cloed-loop ytem tability. It i obviou from Fig. that for Cae, when the ytem gain i within the limit [8,], and imultaneouly the ytem time-contant i within the limit T [,], thee parameter limit are entering the region of intability D() and the ytem will become aymptotically untable. For cae, when the two variable parameter are within the limit [,6] and T [,], thee parameter limit are entirely within the region of tability D() and therefore the feedback control ytem will be guaranteed aymptotically table. Thi ditinctive phenomenon i demontrating the coniderable advantage of the D-Partitioning analyi in comparion with the haritonov aement. By applying the D-Partitioning analyi and implementing a imple interactive MATLAB procedure, the ytem aymptotic tability can be promptly determined and it can be graphically demontrated, avoiding the ignificant calculation needed for the haritonov theorem aement. III. Concluion Contribution of thi reearch i the application of the further advancement of the D-partitioning tability analyi, accomplihed by applying the method in cae of multivariable ytem parameter. The advancement of the D-partitioning tability analyi, developed by the author, proved to be a unique method that introduce a clear graphical diplay of the ytem parameter variation and their interaction. A a reult, in cae of two imultaneouly variable parameter, region of tability and intability are determined in the parameter plane. www.jmet.org JMESTN56
Each point on the D-Partitioning curve repreent the marginal value of the two imultaneouly variable parameter, being a unique property of the advanced D-Partitioning tability analyi that i not offered by the haritonov aement or any other known tability analyi method. Alo, by applying the D-Partitioning analyi, the ytem tability can be aeed immediately for any imultaneou variation of the two variable parameter without the need of determining the haritonov polynomial and calculating the value of the Routh array column. Thi reearch i worth achieving it, not only becaue it advance knowledge. It ha a ubtantial practical apect a well, ince it can be ued for analyi of a lot of indutrial control ytem that have uncertain or variable parameter due to variou ambient condition. REFERENCES [] Neimark Y., Robut tability and D- partition, Automation and Remote Control Vol. 5(Iue 7),99, pp. 957 965, 99. [] Neimark Y., D-partition and Robut Stability, Computational Mathematic and Modeling, Vol. 9(Iue ), pp. 6-66, 6. [] Neimark Y., Determination of the value of parameter for which an automatic ytem i table, Automatic and Tele-mechanic, Vol. 9, pp.9-, 98. [] Yanev.M., Application of the Method of the D-Partitioning for Stability of Control Sytem with Variable Parameter, Botwana Journal of Technology, Vol. 5(Iue ): 6-, 5. [5] Yanev.M, Anderon G., Maupe S., Multivariable Sytem' Parameter Interaction and Robut Control Deign, International Review of Automatic Control (IREACO), Napoli, Italy, Vol.,N., ISSN: 97-659, pp. 8-9,. Journal of Multidiciplinary Engineering Science and Technology (JMEST) ISSN: 59- Vol. Iue, January - 5 [6] Yanev.M., Analyi of Sytem with Variable Parameter and Robut Control Deign, Proceeding of the 6th IASTED International Conference on Modeling, Simulation and Optimization, pp. 75-8, 6. [7] Yanev.M, Anderon G., Obok Opok A, Achieving Robutne in Control Sytem with Variable Time-Contant, Int. Journal of Energy Sytem, Computer and Control (IJESCC), Vol., No., pp. -. 9. [8] haritonov theorem. Retrieved October,, http://en.wikipedia.org/wiki/haritonov'_theorem [9] haritonov V. L., Aymptotic Stability of an Equilibrium Poition of a Family of Sytem of Differential Equation, Journal of Differential equation,, pp. 86-88, 978. [] Barmih, B.R., A generalization of haritonov' four-polynomial concept for robut tability problem with linearly dependent coefficient perturbation, IEEE Tranaction Volume: Iue:, 989. [] Chapellat, H., A generalization of haritonov' theorem; Robut tability of interval plant, Automatic Control, IEEE Tranaction Volume: Iue:, 989. [] Peteren, I.R., haritonov polynomial theory, Encyclopedia of Mathematic. Retrieved October,, http://www.encyclopediaofmath.org/index.php?title =haritonov_polynomial_theory&oldid=985 [] Routh Hurwitz tability criterion. Retrieved October,, http://en.wikipedia.org/wiki/routh%e%8%9hu rwitz_tability_criterion [] Control Sytem/Routh-Hurwitz Criterion. Retrieved October,, http://en.wikibook.org/wiki/control_sytem/rout h-hurwitz_criterion www.jmet.org JMESTN56