0Parametric Robust StabilityCsar Elizondo-GonzlezFacultad de Ingeniera Mecnica y ElctricaUniversidad Autnoma de Nuevo LenMxico1. IntroductionRobust stability of LTI systems with parametric uncertainty is a very interesting topic to study,industrialworldiscontainedinparametricuncertainty. Inindustrialreality, thereisnotaparticularsystem to analyze, there is afamily of systems tobe analyzedbecausethevaluesofphysicalparametersarenotknown, weknowonlythelowerandupperboundsofeachparameter involved in the process, this is known as Parametric Uncertainty (Ackermann et al.,1993; Barmish, 1994; Bhattacharyya et al., 1995). Theset of parameters involved in a systemmakes a Parametric Vector, the set of all vectors that can exists such that each parameter is keptwithin its lower and upper bounds is called a Parametric Uncertainty Box.The system we are studying is now composed of an innite number of systems, each systemcorrespondstoaparametervectorcontainedintheparametricuncertaintybox. Soastotest the stability of the LTI system with parametric uncertainty we have to prove thatall theinnite number of systems are stable, this is called Parametric Robust Stability. The parametricrobust stability problem is considerably more complicated than determine the stability of anLTI system with xed parameters. The stability of a LTI system can be analyzed in differentways,thischapterwill beanalyzedby meansof its characteristicpolynomial,in thecaseofparametric uncertainty now exists a set with an innite number of characteristic polynomials,this is known as a Family of Polynomials, and we have to test the stability of the whole family.The parametric robust stability problem in LTI systems with parametric uncertainty is solvedinthischapterbymeansoftwotools,therstisarecentstabilitycriterion forLTIsystems(Elizondo, 2001B) and thesecond is themathematicaltool Sign Decomposition (Elizondo,1999). Therecent stabilitycriterion mapstheprametric robuststabilityproblem toarobustpositivityproblemof multivariablepolynomic functions, signdecompositionsolves thisproblem in necessary and sufcient conditions.By means of the recent stability criterion(Elizondo, 2001B) is possible to analyze thecharacteristicpolynomialanddeterminethenumberofunstablerootsontherightsideinthe complex plane.This criterion is similar to the Routh criterion although without using thetraditional division of the Routh criterion. This small difference makes a big advantage whenit is analized the robust stability in LTI systems with parametric uncertainty, the elements ofthe rst column of the table (Elizondo, 2001B) they are multivariable polynomic functions andthese must be positive for stability conditions. Robust positivity of a multivariable polynomialfunction is more easier to prove that in the case of quotients of this class of functions, therefore,therecentcriterion(Elizondo, 2001B)iseasiertousethanRouthcriterion. Thereareother1www.intechopen.com2 Will-be-set-by-IN-TECHcriterionswhoseitselementsaremultivariablepolynomicfunctions, suchastheHurwitzcriterionandLienard-Chipart criterion(Gantmacher, 1990), but bothuseahugeamountof mathematical operations incomparisonwiththerecentlystablishedstabilitycriterionElizondo et al. (2005). Whenindustrial casesareanalyzed, thedifferenceof mathematicaloperations is paramount, if the recently stability criterion takes several hours to determine therobust stability, the other criterions take several days. For these reasons the recently stabilitycriterion is used in this chapter instead of other criterions.Sign Decomposition (Elizondo, 1999) also called by some authors as Sign denite Decompositionisamathematical tool abletodetermineinnecessaryandsufcient conditionstherobustpositivityofmultivariablepolynomicfunctionsbymeansofextremepointsanalysis. SignDecompositionbegunasincipient orthogonal ideasof theauthorinhisPhDresearch. Itwas not easy to develop this tool as thus it happens in orthogonal works with respect to thecontemporary research line,theorthogonalideas normally are notwell seen. This is averydifcult situation on any research work, there may be many opinions, but we must accept thatthe world keeps working by the aligned but it changes by the orthogonals.In LTI systems with parametric uncertainty applications, the multivariable polynomicfunctionstobeanalyzeddependonboundedphysicalparametersandsomeboundscouldbe negative. Sosigndecompositionbegins withacoordinates transformationfromthephysical parameters to a set of mathematical parameters such thatall the vectors of the newparameters are contained in a positive convex cone; in other words, all the newparameters arenon-negatives. In this way, the multivariable polynomic function is made by non-decreasingterms, some of them are precededby a positive sign and some by a negative sign. Grouping allthe positive terms and grouping all the negative terms, then factorizing the negative sign anddening a positive part and a negative part of the function we obtain two non-decreasingfunctions. Now the function can be expressed as the positive part minus the negative part. Itis obvious thatboth parts are independent functions, so they can be taken as a basis in withagraphicalrepresentationusingtwoaxis, theaxisofthenegativepartandtheaxisofthepositivepart. Now, supposethat wehaveaparticularvectorcontainedintheparametricuncertaintybox, thenevaluatingthenegativepart andthepositivepart apoint onthenegativepart, positive part plane is obtained,this point represents thefunction evaluatedin the particular vector in . The forty ve degree line crossing at the origin on the negativepart,positive partplanerepresents thesetof functionswith zero value,apointabovethisline represents a function with positive value and a point below this line represents a functionwith negative value.The decomposition of the function in its negative and positive parts may look very simple andnon-transcendentbuttakingintoaccountthatthenegativeandpositivepartsaremadebythe addition of non-decreasing terms, then the negative and positive parts are nondecreasingfunctions in a vector space, this implies that the positive part and the negative part are bonded.So, geometrically,anypointrepresentingthefunctionevaluatedatanyparametervectoriscontainedinarectangleonthenegativepart, positivepartplaneandifthelowestrightvertex is above the forty ve degree line then the function is robust positive, obtaining in thisway the basis of the rectangle theorem.By means of this theorem upper and lower boundsof themultivariablepolynomicfunctionintheparametricuncertaintyboxareobtained.Signdecomposition containsasetofdenitions, propositions,facts, lemmas,theoremsandcorollaries, sign decomposition can be applied to several disciplines; in the case of LTI systemswithparametricuncertainty, thismathematicaltoolcanbeappliedtorobustcontrollability,4 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 3obsevability or stability analysis. In this chapter sign decomposition is applied to parametricrobust stability.In this chapter the following topics are studied: recent stability criterion, linear time invariantsystemswithparametricuncertainty, briefdescriptionofsigndecompositionandnallyasolutionfortheparametricrobuststabilityproblem. All demonstrationsofthecriterions,theorems, corollaries, lemmas, etc, will be omittedbecause theyare results previouslypublished.2. A recent stability criterion for LTI systemsThe studyof stabilityof the LTI systems begunapproximatelyone anda half centuryago with three important criterions: Hermite in 1856 (Ackermann et al., 1993), 1854(Bhattacharyya et al., 1995);Routh in 1875(Ackermann et al., 1993), 1877(Gantmacher, 1990)andHurwitzin1895(Gantmacher,1990). Routh, usingSturmstheoremandCauchyIndextheoryofarealrational function, setupatheoremtodeterminethenumberkofrootsofpolynomial with real coefcients on the right half plane of the complex numbers.Theorem1. (Routh) (Gantmacher, 1990) The number of roots of the real polynomial p(s) =c0+ c1s + c2s2+ + cnsninthe right half of the complex plane is equal to the number ofvariationsofsignin therstcolumn oftheRouthstablewithcoefcients: ai,j=(ai1,1ai2,j+1ai2,1ai1,j+1)/ai1,1i 3, ai,j= cn+1i2(j1) i 2Thereareseveral results relatedtotheRouthcriterion,forexample(Fuller,1977;Meinsma,1995), buttheyarenotappropriatetouseintheparametricuncertaintycaseandtheyusemore mathematical calculations than the Routh criterion.Inthischapterarecentcriterion,anarrangesimilar totheRouhttable, itispresented. Thestability in this recent criterion depends on the positivity of a sign column. The recent criterionhas two advantages: 1) the numerical operations are reduced with respect to above mentionedcriterions; 2) the coefcients are multivariable polynomic functions in the case of parametricuncertaintyandrobustpositivityiseasiertotest thanRouthcriterion. Thecriterionisasdescribed below.Theorem 2. (Elizondo, 2001B) Given a polynomial p(s) = c0 +c1s +c2s2+ +cn1sn1+cnsnwith real coefcients, the number of roots on the right half of the complex plane is equal to the numberof variations of sign in the sign column on the follow arrange.1cncn2cn4 2cn1cn3cn5 3e3,1e3,2 .........ei,j= (ei1,1ei2,j+1ei2,1ei1,j+1), 3 i n +1ei,j= cn+1i2(j1) i 2i= Sign(ei,1) i 2, i= Sign(ei,1)(i+1m)/2j=1Sign(em+2(j1),1) i 3Theprocedureforcalculatingtheelements(ei,j)issimilartotheRouthtablebut withoutusingthedivision. Ontheother hand, the calculationof anelement iis moreeasierthanitlooksmathematical expression. Wecangetthesigni, multiplyingthesignofthe5 Parametric Robust Stabilitywww.intechopen.com4 Will-be-set-by-IN-TECHelement(ei,1)bythesignoftheimmediatesuperiorelement(ei1,1)andthenjumpinginpairs. Forexample6=Sign(e6,1)Sign(e5,1)Sign(e3,1)Sign(e1,1). Also1=Sign(cn)and2=Sign(cn2). Soalsoitisnotnecessarytocalculatethelastelement(en+1,1), onlyitssignisnecessarytocalculate. Eachrowof(ei,j)elementsisobtainedbymeansof(ei1,j)and (ei2,j) elements previously calculated and in Hurwitz criterion a principal minor is notcalculatedfromprevious, thentheElizondo-GonzlezcriterionismoreadvantageousthanHurwitz criterion as shown in table (1)Remark 3. a) Given the relation of the above criterion with the Routh criterion, the cases in that oneelement ei,j is equal to zero or all the elements of a row are zero, they are treated as so as it is done in theRouth criterion. b) The last element en+1,1is not necessary to calculate, but it is necessaryto obtainonly its signMathematical operations in polynomials n degreedegree Hurwitz C. Elizondon +o +o 3 4 1 2 14 9 2 5 25 66 18 9 46 193 45 14 67 780 145 20 9Table 1. A comparison of stability criterions.2.1 ExamplesExample 1. Given the polynomial p(s) = s5+2s4+1s3+5s2+2s +2 by means of criterion 2determine the number of roots in the right half of the complex plane and compare the resultswith the Routh criterion.Applying2criterion weobtainthelefttable. Asanexampleof theprocedure toobtaintheelementsei,jandi, wehave: e3,1=2 1 1 5, e3,2=2 2 1 2, 6=Sign(+) Sign(56) Sign(3) Sign(1), 5= Sign(56) Sign(19) Sign(2).Elizondo-Gonzlez 20011= +1 1 22= +2 5 23= 3 24= +1965= +566= ++Routh1 122 521.5 16.333321.4737+Table 2. Example 1. Comparison of stability criterions.Theleft arrangementshows twosign changesincolumn so thepolynomial hastworootsontherighthalfofthecomplexplane. BymeansofRouthcriterionisobtainedtherighttable, it shows too two sign changes in the rst column which is the same previous result. Aninterestingobservation (see table(2))is thattheleft tablepresents aminussign in thethirdrow of the column and the right table presents a minus sign in the same third row but in therst column.6 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 5Example 2. Given the polynomialp(s)=s5+ 2s4+ 2s3+ 2s2+ s + 3 by means of criterion 2determine the number of roots in the right half of the complex plane and compare the resultswith the Routh criterion.Elizondo-Gonzlez 20011= +1 2 12= +2 2 33= +2 14= +6 65= 186= +Routh1 2 12 2 31 0.53 31.5+Table 3. Example 2. Comparison of stability criterions.It is easy to see by means of two criterions that the polynomial has two roots on the right halfof the complex plane in accordance to the table (3).Example 3. Given the polynomial p(s) = s5+1s4+2s3+2s2+2s +1 by means of criterion 2determine the number of roots in the right half of the complex plane.Whenwetrytomakethetablebymeans of Elizondo-Gonzlez2001criterionor Routhcriterion, it is truncated because e3,1= 011222121301Table 4. Example 3. Presence of a zero in the rst column of elements.Since the element e3,1 is equal zero (see table (4)) then this element is replaced by by an > 0,thus obtaining the following arrangement.11 2221 213 142 1 52 1 26(2 1 2)Table 5. Example 3. Solution of the problem of zero in the rst column.Applying the limit 0 in table (5) is obtained the table (6).1= +1 222= +1 213= + 14= 15= +16= +Table 6. Example 3. Final result to the solution of the problem of zero in the rst column.Fromthetable(6)iseasytoseethatthepolynomialhastworootsontherighthalfofthecomplex plane.7 Parametric Robust Stabilitywww.intechopen.com6 Will-be-set-by-IN-TECHExample 4. Given the polynomial p(s) = s5+1s4+2s3+2s2+1s +1 by means of criterion 2determine the number of roots in the right half of the complex plane. Applying this criterionwe get as following.11212121300Table 7. Example 4. A row equal zero.The table (7) generated, it shows the third rowequal zero. Then obtaining the derivative of thepolynomial corresponding to the immediately superior row p(s) = s4+2s2+1 is obtainedp(s) = 4s3+4s. Now the coefcients of this polynomial replace the zeros of the third rowandthe procedure continues, obtaining in this way the follow arrangement.1= +1 212= +1 213= +4 44= +4 45= +6= +4Table 8. Example 4. Solution to the problem of a row equal zero.We can see in table (8) that there is no sign change in sigma column, then there are not rootsin the right half complex plane.3. Linear time invariant systems with parametric uncertainty3.1 Parametric uncertaintyAll physical systems are dependent on parameters qi and in the physical world does not knowthe value of the parameters, only know the lower qiand upper q+ibounds of each parameter,so that qi qi q+i, this expression is also written as qi [qi, q+i].Forexampleifwehaveseveral electrical resistanceswithcolorcodeof 1,000ohm, ifonemeasures one of them, the measurement can be: 938, 1,024, or a value close to 1,000 ohm butitis rather difcult thatitis exactly 1,000ohm. By meansof tolerance code can bededucedthat theresistancewill begreaterthan900andlessthan1,100ohm. Anotherexampleisthemassofacommercialaircraft,itcanywithfewpassengers andlittlebaggageorwithwith many passengers and much baggage, then the mass of the plane is not known until thelastpassenger toberegistered, butnotwhentheplanewasdesigned, howevertheplaneisdesigned to y from a minimum mass to a maximum mass.The set ofparameters involved in a system makes a Parametric Vectorq=[q1, q2, , q

]T,q

andthesetofallthepossibleparametervectorsthatmayexistmakesaParametricUncertaintyBoxQ={q = [q1, q2, , q

]Tqi[qi, q+i]i}. In thecaseofqi>0ithenQ = {q = [q1, q2, , q

]Tqi> 0, qi [qi, q+i] i} and Q is contained in a positive convexcone P, Q P

.For the study of cases involving parametric uncertaintyis necessary to dene the minimumand maximumvertices of the parametric uncertainty box, so the minimumvminand8 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 7maximum vmaxEuclidean vertices of Q are dened as so as__vmin__2=minqQq2, vmax

2=maxqQq2.3.2 Parametric robust stability in LTI systemsIn the LTI systems with parametric uncertainty, the characteristic polynomial has coefcientsdependent on physical parameters, p(s, q)=c0(q) + c1(q)s + c2(q)s2+ cn(q)sn; so Routhcriterionis verydifcult touse because it is necessaryto test the robust positivityofrational functions dependent onphysical parameters. Bymeans of Hurwitzcriterionispossible tosolve theproblem of parametric robust stability by means of robust positivity ofprincipal minors of a matrix dependent on physical parameters, this procedure uses a lot ofmathematicalcalculations. The robust positivity of rationalfunction dependent on physicalparameterscanbeconsideredassoasaverymuchdifcultproblemsinceonlytherobustpositive test of multivariable polynomic function is very difcult problem (Ackermann et al.,1993)(page93). SotheparametricrobuststabilityprobleminLTIsystemswithparametricuncertaintyinthegeneralcaseisnotaneasyproblemtosolve, howeverinthischapterispresented a solution.The characteristic polynomials are classied according to its coefcient of maximumcomplexity; from thesimplest structurecoefcienttothemostcomplexare: Interval, Afne,Multilinear and Polynomic. For example, the coefcients: ci(q)=qi, ci(q)=2q1 +3q2 + 5q3 +q4, ci(q)=5q1q2 + 2q2q4 + 5q3 + q4, ci(q)=2q31q2 + 2q22q54 + q3, correspond to classication:Interval,Afne, Multilinear and Polynomic respectively. The number of polynomialsp(s, q)thatcan exist is innitesince thenumberof vectors thatexist is innite,thecollection of allpolynomials that exist is a Family of Polynomials P(s, Q) = {p(s, q)|q Q}.The families of polynomials interval andanare convex sets andthese families havesubsettingtest. Thisconcept, subsettingtest,meansthatafamilyofpolynomialsisrobustlystable if and only if all polynomials contained in the subsetting test are stable.Kharitonovin(Kharitonov, 1978), bymeansof histheoremdemonstratesthat afamilyofinterval polynomialsisrobuststableifandonlyifasetoffourpolynomialsarestable. In(Bartlett et al., 1988) bymeansof their edgetheorem, demonstratedthat afamilyof anpolynomials isrobustly stableifandonlyif allthepolynomials corresponding totheedgesoftheparametricuncertaintyboxare stable. Themultilinearanpolynomic families arenotconvexset andtheydonot havesubsettingtest. Soparametricrobust stabilityof thesefamilies can not be resolved by tools based on convexity.In (Elizondo, 1999) was presented asolution for parametric robust stability of any kind of family: Interval, Afne, Multilinear orPolynomic. The solution is based on sign decomposition, and by means of this tool can alsosolve the problem of robust controllability or robust observability.3.3 Robuststability mappedto robust positivityThe parametric robust stability problem of LTI systems can be mapped to a problem of robustpositivity of polynomial functions for at least three ways.Therst twoare: theHurwitzandLienard-Chipart criterions, theother is therecentlystabilitycriterion (2). ByHurwitzorLienard-Chipartcriterionscandothemappingbutasexplained these require making a lot of mathematical calculations. The criterion (2) requiresmuch less mathematical calculations that the criterions mentioned as was shown in table (1),(Elizondo et al., 2005)9 Parametric Robust Stabilitywww.intechopen.com8 Will-be-set-by-IN-TECH4. Brief description of sign decompositionIn different areas of sciences the fundamental problem can be mapped to a problem of robustpositivity ofmultivariablepolynomic functions. For example thenosingularity of amatrixcan be analyzed by mean of the robust positivity of its determinant, so it is very useful to haveamathematicaltool thatsolves theproblem ofrobust positivity ofmultivariable polynomicfunctions. Practically there are three tools for this purpose:Interval Arithmetic (Moor, 1966);Bernstein Polynomials (Zettler, et all 1998) and Sign Decomposition ((Elizondo, 1999)) whosecompleteversionisdevelopedin(Elizondo, 1999)anditspartialversionsarepresentedin(Elizondo, 2000; 2001A;B; 2002A;B),forsimplicityonlywillbementioned(Elizondo, 1999).Interval arithmeticisverydifculttousebecauseitrequiresmuchmorecalculationsthanother methods. Whenrobust positivityisanalyzedinaverysimplefunction, Bernsteinpolynomials have advantages over sign decomposition, but when the function is not simple,sign decomposition has advantages over Bernstein polynomials (Graziano et al., 2004). Thereare several works using sign decomposition instead of Bernstein polynomials, some of themare: (Bhattacharyya et al., 2009; Guerrero, 2006; Keel et al., 2008; 2009; Keel, 2011; Knap et al.,2010; 2011)4.1 Denition of sign decompositionThefollowingis abrief descriptionof the morerelevant results of SignDecomposition(Elizondo, 1999). By means of this tool it is possible to determine, in necessary and sufcientconditions, the robust positivityof a multivariable polynomic functiondependingon parameters, employing extreme points analysis.Since mathematically exist the possibility that a parameter qi has negative value , then this toolbegins by a coordinates transformation from qito qisuch thatthe new parameters will bepositive qi> 0, then an uncertainty box Q = {q = [q1, q2, , q

]Tqi> 0, qi [qi, q+i]} ismakes, in other words, Q is in a positive convex cone P, Q P

with minimum vminandmaximum vmaxEuclidean vertices.The transformation is very easy as shown in the equation(1)qi= qi+ qi qi q+i qi(q+iqi) (1)From here on we will assume that if necessary, the transformation was made and work withparameters qi> 0.Under this consideration will continue with the rest of this topic.Denition4. (Elizondo,1999)Let f :

beacontinuousfunctionandletQP

beabox. Itissaidthat f (q)hasSignDecomposition inQifthereexisttwoboundedcontinuousnondecreasingandnonnegativefunctions fn() 0, fp() 0, suchthat f (q) =fp(q) fn(q)qQ. InthiswaytherearedenedthePositivePart fp(q)andNegativePart fn(q)ofthefunction.Negative Part is only a name since Negative Part and Positive Part are nonnegative.4.2 ( fn,fp) representationIs obvious that for the general case, fn() andfp() are independent functions then they makea basis in 2with graphical representation in the ( fn(), fp()) plane in accordance with gure( 1).10 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 9Ifwetakeaparticular vectorqQandevaluatedthefn(q)andfp(q)parts,weobtainthecoordinates ( fn(q), fp(q)) of the function in the ( fn, fp) plane. The 45oline is the set of pointswhere the function is equal zero becausefp(q) =fn(q) sof (q) =fp(q) fn(q) = 0 . If a pointis above the 45oline means that fp(q)>fn(q) thenf (q)>0. If a point is below the 45olinemeans that fp(q) 0 qQ; d) if the upper left vertex ( fn(vmin), fp(vmax)) is belowthe 45oline thenf (q) < 0 q Q. In accordance with gure ( 3 ).The above result seems to be very useful, we can say that the rectangle is the house wherethemultivariablefunctionlivesin2. Wecanknowtherobust positivityof afunctionanalyzingonlyonepoint. Itisimportanttonotethatthisisonlysufcientconditions, thelower right vertex can be below the 45oline and the function could be robustly positive or notbe. But if the lower right vertex is above the 45oline then the function is robustly positive.For example, the functionf (q) = 4 q2 + q1q3 +8q21q29q33q1q22 such that qQ P3,Q={q = [q1, q2, q3]Tqi[0, 1]}, hassign decomposition,itsminimumandmaximumEuclideanverticesaremin=[0, 0]T, max=[1, 1]T, theirpositiveandnegativepartsare:fp(q) = 4 + q1q3 +8q21q2, fn(q) = q2 +9q33q1q22. Then the lower bound isfp(vmin) fn(vmax),fp(vmin) =4 + (0)(0) + 8(0)(0)=4, fn(vmax) =1 + 9(1)(1)(1)=10, thelowerboundis4 10 =9. Thefunctioncouldberobustlypositive, butfornowwedonotknow, Itisnecessary see more signs of decomposition items.Remark 6. Should be noted three important concepts:The graph of the function does not "lls" the whole rectangle, but it is contained in.12 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 11The graph of the function always "touches " the rectangle in lower left vertice and upper right vertice.The graph of the function is not necessarily convex.Fig. 3. Rectangle theorem4.4 The polygontheoremForthepurpose ofimproving theresults shownuptothispoint, thefollowingpropositionis necessary. In some cases it is necessary to analyze thefunction in a box contained inQ, Q. The box has Euclidean Vertices minand max. So, a vector in is expressed as so asq = min+ , where is a vector in , with origins in min.Proposition7. (Elizondo,1999)Let f :

beacontinuous functioninQP

,letjQbeaboxwithitsverticesset{i}withminimumandmaximumEuclideanverticesmin,max, let ={| i[0, maxi], maxi=maximini}P

be a box with its vertices set{i} with minimumand maximum Euclidean vertices 0, max= maxmin, and let q ja vectorsuch that q = min+ where . Then the functionf (q) is expressed by its: linear, nonlinear andindependent parts, in its minimum expression for all q j.f (q) =fmin+fL() +fN() | q jfminIndependent Part =f (min)fL() Linear Part =f (q)|min fN() Nonlinear Part =f (min+ ) fminfL() f (q)|min = f (q)q1min1 + f (q)q2min2 + + f (q)q

min

Must be noted that fmin=f (min). On other hand, it is clear that we can use the concepts ofpositive part andnegative part in the aboveproposition, So, fp(q) fn(q)=fminpfminn+fLp() fLn() +fNp() fNn()obtainingthefollowingequations(3)wheretherelationbetween and q can be appreciated in the gure (4).fp(q) =fminp+fLp() +fNp()fn(q) =fminn+fLn() +fNn()(3)13 Parametric Robust Stabilitywww.intechopen.com12 Will-be-set-by-IN-TECHFig. 4. Gamma boxTheorem8. Polygon Theorem (Elizondo, 1999). Let f :

be acontinuous function withsigndecompositioninQ, let q, , jandinaccordancewiththeproposition(7). Then, a)thelower and upper bounds of the functionf (q) are: LowerBound=fmin+fLmin fNn(max) andUpperBound=fmin+fLmax +fNp(max)qQ,b)theboundsofincisea, arecontainedintheinterval denedbytheboundsof therectangletheorem3. fp(min) fn(max) LowerBoundUpperBoundfp(max) fn(min),c)Thegraphical representationofthefunctionf (q)q inthe( fn, fp)planeiscontainedinthepolygondenedbytheintersectionof therectangle of the rectangle theorem (5) and the space between the two 45olines separated from the originby the Lower Bound and Upper Bound in accordance with gure (5).Fig. 5. Bounding of the functionThe symbolic expression of the nonlinear part used in the above theorem is not necessary toobtain, because we will use only its numerical value. So, from the equations (3), the nonlinearparts are obtained as so as equations ( 4).14 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 13fNp() =fp(q) fminpfLp()fNn() =fn(q) fminnfLn()fLp() =fp(q)min fLn() =fn(q)|min (4)As an illustration of this theme, by means of rectangle theorem and polygon, we will analyzethelowerboundof afunctioninagammabox. Considerthefunctioncorrespondingtothegure( 2), f (q) = 4 q2+ q1q3+ 8q21q2 9q33q1q22suchthat q Q P 3,Q={q = [q1, q2, q3]Tqi[0, 1]}. Supposethatthefunctionisanalyzedintoagammabox Q, with Euclidean vertices min= [0.2 0.2 0.2 ]Tand max= [0.85 0.85 0.85 ]T.InaccordancewiththeRectangleTheorem(3) thelowerboundis fp(vmin) fn(vmax) =0.1403. ApplyingthePolygon Theorem (8)thelower boundis fmin+fLmin fNn(max),soit isnecessarytoobtaineachof theseexpressions, theresultsareasfollows: fmin=f (min) =3.9034, fLmin=0.4457, fNn(max) =3.3825. Thelast valueisobtainedofequations (4), thus the lower bound is 0.0752.By means of the Rectangle Theorem is obtainedf (q) >0.1403 q, following the Polygon Theorem is obtainedf (q) > 0.0752 q, sothe function is robustly positive in the box.4.5 The box partition theoremBy means of Rectangle Theorem(3) and Polygon Theorem(8) are obtained sufcientconditionsofrobustpositivity,sotoobtainnecessaryandsufcientconditionsisnecessaryto obtain new results.Whenitisnot possibletoknowwhetherthefunctionispositiveornot inQ=[q1 , q+1 ][q2 , q+2 ] [q

, q+

]. In this case it is possible to divide each variable [qi, q+i] in k parts,generating knew intervals: [qi, q1i ], [q1i, q2i ], , [qji, qj+1i], [qk1i, q+i], let [i, +i] be aknew interval,giving cause to the generation of k

new boxesi=[1, +1] [2 , +2] [

, +

]withmin,maximinimumandmaximumEuclidean vertices ofiandQ = ii. Through these concepts, the following theorem is obtained.Theorem 9. Box Partition Theorem (Elizondo, 1999). Let f :

be a continuous functionwith sign decomposition in Q such that Q P

is a box with minimumand maximum Euclideanvertices vmin, vmax. Then the functionf (q) is positive (negative) in Q if and only if a boxes set exists,such that Q = jjand Lower Bound c > 0 for each jbox (Upper Bound c < 0 for each onejbox).Thistheoremcanbeappliedintwoways, oneofthemwecallAnalyticalPartitionandthe other one Constant Partition. In analytical partition, the box where the function has anegative lower bound is subdivided iteratively. In the case of the function is robustly positiveis also obtained information about where the function is close to losing positivity. By meansof constant partition is only obtained information on whether the function is robustly positiveor not.To illustrate both procedures, we analyze the robust positivity of the function (Elizondo, 1999)f (q)=_4 + q1 +8q21q2__q2 +9q1q22_, such thatQ={q = [q1, q2]Tqi[0, 1] i} . Therobustpositivity isanalyzedbymeansoftherectangle theorembecauseitis more easier to15 Parametric Robust Stabilitywww.intechopen.com14 Will-be-set-by-IN-TECHapply, althoughitmustbesaidthattheboundsofthepolygon theoremarebetterthantherectangle theorem.Analytical Partition (Elizondo, 1999). In the subgure 1 of gure (6) shows that the functionisrobustlypositiveinboxes1and3but notintheboxes2and4. Soitisnecessaryapplyiterativelythepartitionboxtotheboxeswherethefunctionis notrobustpositive,inthis way is obtained the subgure 2 of gure (6). Since there is a set of boxes such thatQ=

jj| f (q) > 0 j, then the function is robustly positive in Q. The graphs were made to showthe procedure in visual way, but for more than two dimensions, using software we can get thecoordinates and dimensions of sub boxes where the function is close to losing positivity.(a) Subgure 1 (b) Subgure 2Fig. 6. Partition boxConstant Partition(Elizondo, 1999). Inthis procedurethedomainof eachoneof the parametersisdivideinkequal parts(notnecessarilyequal), inthisway, itisgeneratedaboxes set ofk

sub boxesisuch thatQ= jj. The robust positivity of eachibox can beanalyzed by a computer programso that the computer give us the nal result about the robustpositivity of the function.Another way is through a software which plot a (blue) mark in the ( fn, fp) plane in each( fn(min), fp(min))and( fn(max), fp(max))coordinates corresponding totheminimumand maximum vertices of each ibox, and plot too a + (red) mark corresponding to the lowerbound of each ibox, as can be appreciated in gure (7) that it was obtained with k = 13.If a (blue) mark is below the 45oline, means that there is at least one vector for which thefunction is negative and therefore the function is not robustly positive. If all the (blue) marksare above the 45oline, and a + (red) mark is below the 45oline means that it is necessary toincrease the k number of partitions up to all the + (red) and (blue) marks are above the 45oline. If this is achieved then the function is robustly positive, as shown in gure (7).Inthegure (7)wecan see thatitis difcult tosee thatall +(red) marks are abovethe45oline, then with purpose to resolve this difculty is proposed the following representation.16 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 15Fig. 7. Function in ( fn, fp) plane4.6 (, ) RepresentationInsomecasesassoasgure (7)itisnoteasy todetermine ingraphicwaywhetherapointclose to the 45oline is over this line or not. So in (Elizondo, 1999) the (, ) representation wasdeveloped, (q) =fp(q) +fn(q), (q) =fp(q) fn(q) , it is similar to rotated 45othe axis withrespect to ( fn, fp) representation implying some graphical and algebraic advantages over thenegative and positive representation.Denition 10. (Elizondo, 1999) Letfn(q) andfp(q) be the negative and positive parts of a continuousfunction f (q)withsigndecompositioninQ. Let Tbethe linear transformationdescribedbelowsuchthatT1exists, thenitiscalledarepresentationofthefunctionf (q),in(, )coordinates,tothelineartransformation((q), (q)) =T( fn(q), fp(q))andtheinversetransformationof an((q), (q)) representation is a ( fn(q), fp(q)) representation of the functionf (q).T=_1 111_T1=12_111 1__(q)(q)_ = T_fn(q)fp(q)_ _fn(q)fp(q)_ = T1_(q)(q)_(q) =fp(q) +fn(q) fp(q) =12((q) + (q))(q) =fp(q) fn(q) fn(q) =12((q) (q))With the purpose to showthe advantages of the (, ) representation, by means oftherectangletheoremweanalyzethesamefunctionintheprevioussubsection f (q) =_4 + q1 +8q21q2__q2 +9q1q22_ applying k = 13.We can see in the gure (8) beta axis scale ispositive implying that all the bounds are positives and consequently the function is robustlypositive.The functionf (q) = 4 q2 + q1q3 +8q21q29q33q1q22 corresponding to the gure (2) is shownin the gure (9) in (, ) representation. We can see that beta axis scale is positive implyingthe function is robustly positive.The original idea to develop the representation ( , ) (Elizondo, 1999) was to solve a visualgeometric problem, but this representation has interesting algebraic properties on continuousfunctions f (q), g(q), h(q) withsigndecompositioninQandfor all u(q) nondecreasingfunction in Q, (Elizondo, 1999) as the following:17 Parametric Robust Stabilitywww.intechopen.com16 Will-be-set-by-IN-TECHFig. 8. Function in (, ) representationFig. 9. Function in (, ) representationa)(q)is anon-decreasing andnon-negativefunctioninQ; b)(q)(q); c)(q)=f (q)f (q), q Q; d) the((q) + u(q), (q) + u(q))isa, representationof f (q); e) the((q) + u(q), (q))representationisreducedtoitsminimumexpression((q), (q)); f)Additionf (q) + g(q) : (q) =f (q) + g(q), (q) =f (q) + g(q); g)Subtraction f (q) g(q):(q)=f (q) + g(q),(q)=f (q) g(q);h) Product f (q)g(q), (q)=f (q)g(q),(q)=f (q)g(q);i) the(, )representation of g(q) is as follows: (g(q), g(q));j) iff (q)=g(q) + h(q)thenthealphaanbetapartsof f (q) g(q)arereduced toitsminimumexpression as follows (q) = f (q) g(q), (q) =f (q) g(q).Computationally the (, ) representation is better than ( fn, fp) because if the computer doesnot generate the negative scale in the axis it is implying that all marks are positives.This18 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 17isanuseful andinterestingproperty, but aboveall propertiestherearethreeoutstandingproperties, itwouldbeveryuseful iftheywerefullledincomplexnumbers, theyareasfollows:Addition f (q) + g(q) (q) = f (q) + g(q) (q) =f (q) + g(q)Subtractionf (q) g(q) (q) = f (q) + g(q) (q) =f (q) g(q)Product f (q)g(q) (q) = f (q)g(q) (q) =f (q)g(q)(5)Most be noted that the alpha component of subtraction is correct with (q)=f (q) + g(q),it is an addition of alphas. It is also important to highlight the simplicity with which madethe addition, subtraction and product in alpha beta representation.4.7 Sign decompositionof the determinantSign decompositionof the determinant was developedin(Elizondo, 1999) and it waspresentedan application in (Elizondo, 2001A; 2002B), by simplicity only will mention(Elizondo, 1999). Inparametricrobuststabilityisnot veryuseful thesigndecompositionof the determinant, but it is a part of sign decomposition. We can analyze robust stability bymeans of the Hurwitz criterion means the robust positivity of determinants, but it is so mucheasierbymeansofcriterion (2), see table(1). Takingaccountthatthereader couldwork inother areas where the nonsingularity of a matrix dependent in parameters is important, thensign decomposition of the determinant is included in this chapter.4.7.1 The (, ) representation of the determinantIn order to achieve the procedure to determine the robust positivity in necessary and sufcientconditions of a determinant with real coefcients depending onparameters qi, the followingfact ispresented. Bymeansof the(, )properties(5) isobtainedthefollowingfact, inthedevelopmentofthedeterminantappearsthealphapartandbetapart, asshowninthefollowing fact.Fact 1. (Elizondo, 1999) Let M(q) be a (2 2) matrix with elements mi,j(q) with representation(i,j(q) , i,j(q)). Then the (, ) representation of the determinant of the matrix M(q) is:(det(M(q)))= (1,1(q)2,2(q) + 2,1(q)1,2(q))(det(M(q)))= (1,1(q)2,2(q) 2,1(q)1,2(q)).Denition 11. (Elizondo, 1999) Let M(q)=_mi,j(q)_ be a matrix with elements mi,j(q) with(i,j(q) , i,j(q)) representation. Then the matrixM(q)=_i,j(q)_ will be called the alpha part ofthe matrixM(q), and the determinant det(M(q))=|M(q)|=|M(q)|will be called the alphapart of the determinant |M(q)| , which is similar to the usual determinant changing all the subtractionsby additions including the sign rule of Cramer. In a similar way, the matrixM(q)=_i,j(q)_ willbe called the beta part of the matrixM(q), and the determinant det(M(q))=|M(q)|=M(q)will be called the beta part of the determinant |M(q)| .Most be noted that: a)i,j(q)=mi,j(q), then, M(q) =M(q) and det(M(q))=det(M(q)),b)Inaccordancewiththeabovefact, fora(2 2)matrix, the(, )representationofthedeterminant of thematrixM(q)is_det(M(q)), det(M(q))_. Inthefollowinglemmaageneralization of the last expression for a (n n) matrix is established.19 Parametric Robust Stabilitywww.intechopen.com18 Will-be-set-by-IN-TECHLemma12. (Elizondo, 1999) Let M(q) be a(n n) matrixwithelements mi,j(q) withrepresentation(i,j(q), i,j(q)). Thenthe(, )representationof thedeterminantof thematrixM(q) is_det(M(q)), det(M(q))_. In accordance with denition (11)4.7.2 Linear, nonlinearand independentparts of the determinantWhen the positivity of the determinant of a matrix with elements mi,j(q) is analyzed via signdecomposition, it is normally necessary to use the box partition and polygon theorems. Then,the independent, linear and nonlinear parts of the determinant need to be obtained. These areobtained in the following theorem.Theorem 13. (Elizondo, 1999) (Sign Decomposition of the Determinant Theorem) Let qQ|q = min+ be according to the proposition (7 ). Let M(q) nnbe a matrix with elements mi,j(q)withsigndecompositioninQwithrepresentation(mini,j+ i,j,L() + i,j,N(), mini,j+ i,j,L() +i,j,N()), then the (, ) representation of the determinant of the matrix M(q) is as follows:(q)=min+ L() + N(),(q)=min+ L() + N()min=det__mini,j__, min= det__mini,j__L(q)=k=nk=1det_(k)_mini,j_+ [I (k)]_i,j,L()_L(q)=k=nk=1det_(k)_mini,j_+ [I (k)]_i,j,L()_(k)=_i,j(k) |1,1(k)=|sign(1 k)|2,2(k)=|sign(2 k)|...n,n(k)=|sign(n k)|i,j(k)=0 i =jN() = (q) minL(), N() =(q) min L()4.7.3 Example(Elizondo, 1999; 2001A). The Frazer and Duncan Theorem is presented in (Ackermann et al.,1993)intheboundarycrossingversionasfollows. LetP(s, Q) ={p(s, q) | q QP

}beafamilyof polynomialsof invariant degreewithparametricuncertaintyandrealcontinuouscoefcients, thenthefamilyP(s, Q)isrobust stableif andonlyif: 1) astablepolynomial p(s, q) P(s, Q) exists, 2) det (H(q)) = 0 for all q Q.(Ackermann et al., 1993)Giventhefamilyofinvariantdegreepolynomialswithparametricuncertaintydescribedby: p(s, q) = c0+ c1s + c2s2+ c3s3+ c4s4, withreal continuouscoefcients:c0(q) = 3, c1(q) = 2, c2(q) = 0.25 +2q1 +2q2, c3(q) = 0.5(q1 + q2), c4(q) = q1q2,suchthat qi[1, 5]. DeterminetherobuststabilityofthefamilybymeansoftheFrazerandDuncantheorem applying in graphicalwaythesign decomposition of thedeterminanttheorem (13).20 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 19TheHurwitzmatrixH(q)isobtained, itisprovedthatthepolynomial p(s, q)isstablefor q=[1 1]Tand that the determinant of the Hurwitz matrixH( q) is positive. Having the rstconditionoftheFrazerandDuncantheoremsatised, andprovingthatthedeterminantisrobust positive in Q, the second condition of the Frazer and Duncan theorem will be satisedtoo.H(q) =__c3(q)c1(q) 0 0c4(q)c2(q)c0(q) 00 c3(q)c1(q) 00 c4(q)c2(q)c0(q)__Therobustpositivityofthedeterminantproblemissolvedbymeansof: theboxpartitiontheorem9, thepolygontheorem8in(, )representationandthesigndecompositionofthedeterminanttheorem(13). Takingthepartitionin9equalpartsineachoneofthetwovariablesqiandapplying sign decomposition in constantpartition way,thefunction valuesin minimum and maximum vertices and lower bound + are plotted for each ibox, asit appears in the gure (10). All lower bound marks + are above the alpha axis, then all ofbounds are positive, therefore the determinant of the Hurwitz matrixH(q) is robust positiveimplying that the polynomials family is robust stable.Fig. 10. Positivity of the determinant5. A solution for the parametric robuststability problem5.1 Problem identicationIncontrol area,therobust stability ofLTI systems with parametricuncertaintyproblem hasbeen studied in different interesting ways. The problem can be divided in two parts. One ofthem is that it is not possible to be obtained roots of a polynomial by analytical means for thegeneral case. The second is thatwe havenowa family of polynomials to study instead of asingle polynomial.Sincetoobtainrootsof polynomialsforthegeneral caseisadifcult problem. Thentheextraction of roots of polynomials went mapped rstly to a position of roots problem in thecomplex plane, Routh never tried to extract the roots, his work begun studying the position ofthe roots. This problemwas subjected to a second mapping, it was transferredto mathematicalproblemsofsmallerlevel forexampletoapositivityproblem, asitisthecaseof: Routh,Hurwitz, Lienard-Chipart and Elizondo-Gonzlez 2001 criterions.21 Parametric Robust Stabilitywww.intechopen.com20 Will-be-set-by-IN-TECHThe objective in this chapter is to study the stability of a family of polynomials with invariantdegree (the reader can see poles and zeros cancellation cases) and real continuous coefcientsdependent on parameters with uncertainty. The essence of the problem is that we have nowasetofrootsinthethecomplexesplane, andforstabilityconditionallofthemmustbeinthe left half of the complex plane for asymptotic stability. How to obtain that the set of rootsremains in the left side of the complex plane?Awellknownsolutionis: a)thefamilyP(s, Q)hasatleastoneelement p(s, q)stableandb)| H(q)|=0qQ. TheexplanationisbecausethedeterminantofaHurwitzmatrixiszero when the polynomial has roots in the imaginary axis, so if there is a qQ vector suchthat p(s, q)is stablethenitsroots are atthelefthalfofthecomplex plane. On otherhand,ifavectorqslidesintoQstartingfromqimpliesthatthecoefcientsci(q)willchangeincontinuous way and the roots of p(s, q) will slides too on the complex plane. But if | H(q)|=0 qQ, it means that does not exist a vector q for whichp(s, q) has roots in the imaginaryaxis, implying that the displacement of the roots never cross the imaginary axis. This solutionis very difcult to use because to test the robust positivity of a determinant in the general caseis a very difcult problem (Ackermann et al., 1993)(page 93).Anothersolutionwasthroughthesubsettingtest, theideaworkedwellinconvexfamiliesasinterval (Kharitonov, 1978)andafne(Bartlett et al., 1988), butitwasnotinnonconvexfamilies as multilinear and polynomic.Then it can be concluded that the solution for robust stability of LTI systems with parametricuncertaintyproblemforthegeneralcase: interval, afne, multilinear, polinomic, cannotbesustained in convexity properties nor subsetting test.5.2 A proposedsolutionIn (Elizondo, 1999) it was developed a solution for the general case of robust stability of LTIsystemswithparametricuncertaintywithoutconcerningtheconvexityofthefamilies, thesolution consists of two parts.Apart of the solution was the development of a stability criterion, operating withmultivariablepolynomicfunctions inparametric uncertaintycase, simpler thanHurwitzandLienard-Chipartcriterions(Elizondo et al., 2005). Thementionedcriterionissimilartocriterion(Elizondo, 2001B) but without thecolumn, thereforeit doesnot determinethenumberofunstableroots, itonlydetermineswhetherthepolynomialisstableornot. Theamount of mathematical operations required in this criterion is equal to the one of (Elizondo,2001B) but theyare muchless that the requiredones inHurwitz andLienard-Chipartcriterions (Elizondo et al., 2005).The other part of the solution was the development of a mathematical tool capable of solvingrobustpositivityproblems ofmultivariablepolynomic functionsinnecessary andsufcientconditions by means of extreme point analysis.The mathematical tool developed in (Elizondo,1999) was Sign Decomposition.Then, the solution proposed for robust stability in LTI systems with parametric uncertainty inthe general case is supported in two results: the stability criterion for LTI systems (Elizondo,2001B)andsigndecomposition(Elizondo, 1999). Givenapolynomial p(s, q) =cn(q)sn+cn1(q)sn1+ + c0(q) with real coefcients, where qQP, Q={[q1q2 q

]T|qi[0, 1] i}. Theprocedureeasiertouseisbymeansofthepartitionboxtheorem(9)inthemodality Constant Partition, its application could be of the following way.a) Take the equations of the coefcients ci(q) and decompose them into positive and negativeparts ci p(q) and cin(q). In symbolic way.22 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 21b) By means of thepositive and negativeparts,to obtainthe componentsin alphaand betarepresentation. i= ci p(q) + cin(q), i= ci p(q) cin(q).c) To make a table in accordance to the criterion (2).d) Bymeans of therectangletheorem(5) or polygontheorem(8), toanalyzetherobustpositivity in Q of the coefcients cn(q) and cn1(q). In case of negative bound in a coefcient,include its graph in the following software.e) To make a software to develop the table in accordance to the partition box theorem and tograph the wished ei,1 element.Remark 14. The sigma column in the criterion (2) is not necessary calculate for robust stability5.3 ExampleGivenaLTI systemwithparametric uncertaintyQ = {[q1q2q3]T|qi [0, 1] i}, itscharacteristic polynomial of invariant degreeis p(s, q) =c4(q)s4+ c3(q)s3+ +c2(q)s2++c1(q)s + c0(q). To analyze the robust stability of the system.a) Positive and negative parts cpi(q) and cni(q).c0(q) = 2 + q1q2q33q2q3c1(q) = 5 + q1q32q2q3c2(q) = 10 +4q1q3q1q22q32c3(q) = 5 + q22q1q22c4(q) = 3 + q1q32q2q3c0p(q) = 2 + q1q2q33c1p(q) = 5 + q1q32c2p(q) = 10 +4q1q3c3p(q) = 5 + q22c4p(q) = 3 + q1q32c0n(q) = q2q3c1n(q) = q2q3c2n(q) = q1q22 + q32c3n(q) = q1q22c4n(q) = q2q3b) The alpha and beta representation of the coefcients is as follows.i= cpi(q) + cni(q),0= cp0(q) + cn0(q)1= cp1(q) + cn1(q)2= cp2(q) + cn2(q)3= cp3(q) + cn3(q)4= cp4(q) + cn4(q)i= cpi(q) cni(q)0= cp0(q) cn0(q)1= cp1(q) cn1(q)2= cp2(q) cn2(q)3= cp3(q) cn3(q)4= cp4(q) cn4(q)c) To make a table in accordance to the criterion (2).1(4, 4) (2, 2) (0, 0)2(3, 3) (1, 1)33,1= c3c2 + c4c1, 3,1= c3c2c4c13,2= c3c0, 3,2= c3c044,1= 3,1c1 + c33,2, 4,1=3,1c1 c33,25Check robust positivity of 4,1and 3,2d) The lower bound of c4(q) and c3(q) are as follows.For c4(q) is LB c4= c4p_[0 0 0]T_c4n_[1 1 1]T_ = 3 + (0)(0)3(1)(1) = 2.For c3(q) is LB c3= c3p_[0 0 0]T_c3n_[1 1 1]T_ = 5 + (0)2(1)(1)2= 4.23 Parametric Robust Stabilitywww.intechopen.com22 Will-be-set-by-IN-TECHThen c4(q) and c3(q) are robustly positives in Qe) Bymeansof softwareapplying8partitionsthegraphse3,1, e3,2, e4,1wereobtainedasfollowing.Fig. 11. Element e31 in (, ) representationFig. 12. Element e32 in (, ) representation24 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comParametric Robust Stability 23Fig. 13. Element e41 in (, ) representationSince c4(q), c3(q), e31(q), e32(q), e41(q) are robustly positive, then the system is robustly stable.6. ReferencesAckermann, J. & Bartlett, A. (1993). Robust Control Systems with Uncertain Physical Parameters,Springer, ISBN 978-0387198439.Barmish, B.R. (1990). NewTools for Robustness of Linear Systems, Prentice Hall, ISBN978-0023060557.Bartlett, A.C.; Hollot, C.V. &Lin, H. (1988). Root locations of an entire polytope ofpolynomials: Itsufcestochecktheedges. Mathematicsof Control SignalsSystems,Vol. 1, No. 1, 61-71, DOI: 10.1007/BF02551236.Bhattacharyya,S.P.; Chapellat, H. & Keel, L.H. (1995). Robust Control the Parametric Approach,Prentice Hall, ISBN 0-13-781576-X, NJ, USA.Bhattacharyya, S.P.;Keel,L.H. & Datta, A. (2009).Linear Control Theory: Structure, Robustnessand Optimization, CRC Press,ISBN 978-0-8493-4063-5, Boca Raton.Elizondo-Gonzlez, C. (1999). Estabilidad y Controlabidad Robusta de Sistemas Lineales conIncertidumbreMultilineal. ProgramaDoctoral delaFacultaddeIngenieraMecnicayElctrica de la Universidad Autnoma de Nuevo Len.Elizondo-Gonzlez, C. (2000). NecessaryandSufcientConditionsforRobustPositivityofPolynomic Functions Via Sign Decomposition, 3 rd IFAC Symposium on Robust ControlDesignROCOND2000, pp. 14-17,ISBN-13:9780080432496,Prague Czech Republic,April, 2000.Elizondo-GonzlezC. (2001).RobustPositivity of theDeterminantVia Sign Decomposition,The5thWorld Multi-Conference on SystemicsCybernetics andInformatics SCI 2001, pp.14-17, ISBN 980-07-7545-5, Orlando, Florida, USA, July, 2001.Elizondo-GonzlezC. (2001). NewStabilityCriteriononSpaceCoefcients, ConferencesonDecisionandControl IEEE, SBN0-7803-7063-5, Orlando, Florida, USA. Diciembre,2001.25 Parametric Robust Stabilitywww.intechopen.com24 Will-be-set-by-IN-TECHElizondo-Gonzlez, C. (2002). An Application of Recent Reslts on Parametric Robust Stability,X Congreso Latinoamericano de Control Autmtico, Guadalajara Jalisco, 2002.Elizondo-Gonzlez, C. (2002). AnApplicatonof SignDecompositionof theDeterminantonParametric Robust Stability, XCongreso Latinoamericano de Control Autmtico,Guadalajara Jalisco, 2002.Elizondo-Gonzlez, C. & Alcorta-Garca, E. (2005). Anlisis de cotas de races de polinomioscaractersticos y nuevo criterio de estabilidad, Congreso Nacional 2005 de la AsociacindeMxicodeControlAutomtico, ISBN970-32-2974-3Cuernavaca, Morelos,Mxico,Octubre, 2005.Fuller, A.T. (1977). On Redundance in Stability Criteria, Internatinal Journal Control, Vol. 26, No.2, pp. 207-224.Gantmacher, F.R. (1990). TheTheoryof Matrices, AmericanMathematical Society, ISBN-10:0821813935, ISBN-13:978-0821813935.Guerrero, J.; Romero, G.; Mendez, A.; Dominguez, R.; Panduro, M. & Perez, I. (2006). LectureNotes in Control and Information Sciences, Robust Absolute Stability Using PolynomialPositivity and Sign Decomposition, Vol. 341, No. 1, pp. 423-430, 2006. ISSN: 0170-8643.Graziano-Torres, R.; Elizondo-Gonzlez, C. (2010). Herramientas para el Anlisis deEstabilidad Robusta de Sistemas LTI con Incertidumbre Paramtrica, CongresoNacional2004delaAsociacindeMxicodeControlAutomtico, ISBN:970-32-2137-8,Mxico D.F., Octubre, 2004.Keel, L.H. &Bhattacharyya, S.P. (2008). FixedOrderMultivariableControllerSynthesis: ANewAlgorithm, Proceedingsof the47thConferenceonDecisionandControl, Cancun,Mexico, December, 2008.Keel, L.H. &Bhattacharyya, S.P. (2009). FixedOrderMultivariableDiscrete-TimeControl,Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference,Shangai, P.R. China, December, 2009.Keel, L.H. &Bhattacharyya, S.P. (2011). Robust StabilityviaSign-DeniteDecomposition,Journal of IEEETransactions onAutomatic Control, Vol. 56, No. 140 155, ISSN:0018-9286, Jan -2011.Kharitonov, V. (1978). On a Generalization of a Stability Criterion. Seria Fizico-matematicheskaia,Vol.1, pp. 53-57, Izvestiia Akademii Nauk Kazakhskoi SSR.Knap, M.J.; Keel, L.H. &Bhattacharyya, S.P. (2010). Robust stabilityof complexsystemswith applications to performance attainment problems, American Control Conference.,Marriot Waterfront, Baltimore, MD, USA, July, 2010.Knap, M.J.; Keel, L.H. & Bhattacharyya, S.P. (2010). Robust Hurwitz stability via sign-denitedecomposition , Linear Algebra and its Applications., Volume 434, Issue 7, pp 1663-1676,1 April 2011.Meinsma G. (1995). Elementary Proof of the Routh-Hurwitz Test Systems & Control Letters 25,pp. 237-242.Moore, Ramon E.(1966). Robust Control the Parametric Approach, Prentice Hall, NJ, USA.Zettler M.; Garloff J. (1998). Robustness Analysis of Polynomials with Polynomial ParameterDependency Using Bernstein Expansion, IEEE Transactions on Automatic Control vol.43 pages 1017-1049, 1998.26 Recent Advances in Robust Control Theory and Applications in Robotics and Electromechanicswww.intechopen.comRecent Advances in Robust Control - Theory and Applications inRobotics and ElectromechanicsEdited by Dr. Andreas MuellerISBN 978-953-307-421-4Hard cover, 396 pagesPublisher InTechPublished online 21, November, 2011Published in print edition November, 2011InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.comInTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China Phone: +86-21-62489820 Fax: +86-21-62489821Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and\mu design methods followed by research on parametric robustness, initially motivated by Kharitonov'stheorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes ofRecent Advances in Robust Control give a selective overview of recent theoretical developments and presentselected application examples. The volumes comprise 39 contributions covering various theoretical aspects aswell as different application areas. The first volume covers selected problems in the theory of robust controland its application to robotic and electromechanical systems. The second volume is dedicated to special topicsin robust control and problem specific solutions. Recent Advances in Robust Control will be a valuablereference for those interested in the recent theoretical advances and for researchers working in the broad fieldof robotics and mechatronics.How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:Cesar Elizondo-Gonzalez (2011). Parametric Robust Stability, Recent Advances in Robust Control - Theoryand Applications in Robotics and Electromechanics, Dr. Andreas Mueller (Ed.), ISBN: 978-953-307-421-4,InTech, Available from: http://www.intechopen.com/books/recent-advances-in-robust-control-theory-and-applications-in-robotics-and-electromechanics/parametric-robust-stability