1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 10Lecture 10RouthRouth--Hurwitz stability criterionHurwitz stability criterion2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••NyquistNyquistDesign specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratories3Stability summary (review)Stability summary (review)(BIBO, asymptotically) stable(BIBO, asymptotically) stableififRe(sRe(sii)<0 for all i.)<0 for all i.marginally stablemarginally stableififRe(sRe(sii)<=0 for all i, and)<=0 for all i, andsimple root for simple root for Re(sRe(sii)=0)=0unstableunstableififit is neither stable nor it is neither stable nor marginally stable.marginally stable.Let Let ssiibe be polespolesof of rational G. Then, G is rational G. Then, G is ……4RouthRouth--Hurwitz criterionHurwitz criterionThis is for LTI systems with a This is for LTI systems with a polynomialpolynomialdenominator (without sin, denominator (without sin, coscos, exponential etc.), exponential etc.)It determines if all the roots of a polynomial It determines if all the roots of a polynomial lie in the open LHP (left halflie in the open LHP (left half--plane),plane),or equivalently, have negative real parts.or equivalently, have negative real parts.It also determines the number of roots of a It also determines the number of roots of a polynomial in the open RHP (right halfpolynomial in the open RHP (right half--plane).plane).It does It does NOTNOTexplicitly compute the roots.explicitly compute the roots.No proof is provided in any control textbook.No proof is provided in any control textbook.5Polynomial and an assumptionPolynomial and an assumptionConsider a polynomialConsider a polynomialAssumeAssumeIf this assumption does not hold, Q can be factored asIf this assumption does not hold, Q can be factored aswherewhereThe following method applies to the polynomialThe following method applies to the polynomial6RouthRoutharrayarrayFrom the given From the given polynomialpolynomial7RouthRoutharray array (How to compute the third row)(How to compute the third row)8RouthRoutharray array (How to compute the fourth row)(How to compute the fourth row)9RouthRouth--Hurwitz criterionHurwitz criterionThe number of roots The number of roots in the open right halfin the open right half--plane plane is equal to is equal to the number of sign changesthe number of sign changesin the in the first columnfirst columnof of RouthRoutharray.array.10Example 1Example 1RouthRoutharrayarrayTwo sign changesTwo sign changesin the first columnin the first columnTwo roots in RHPTwo roots in RHP11Example 2Example 2RouthRoutharrayarrayIf 0 appears in the first column of a If 0 appears in the first column of a nonzero row in nonzero row in RouthRoutharray, replace it array, replace it with a small positive number. In this with a small positive number. In this case, Q has some roots in RHP.case, Q has some roots in RHP.Two sign changesTwo sign changesin the first columnin the first columnTwo roots Two roots in RHPin RHP12Example 3Example 3RouthRoutharrayarrayIf zero row appears in If zero row appears in RouthRoutharray, Q array, Q has roots either on the imaginary axis has roots either on the imaginary axis or in RHP.or in RHP.No sign changes No sign changes in the first columnin the first columnNo roots No roots in RHPin RHPButButsome some roots are on roots are on imagimag. axis.. axis.Take derivativeTake derivativeof an of an auxiliary polynomialauxiliary polynomial(which is a factor of (which is a factor of Q(sQ(s))))13Example 4Example 4RouthRoutharrayarrayNo sign changes No sign changes in the first columnin the first columnFind the range of K Find the range of K s.ts.t. . Q(sQ(s) has all roots in the left ) has all roots in the left half plane. (Here, K is a design parameter.)half plane. (Here, K is a design parameter.)14Simple & important criteria for stabilitySimple & important criteria for stability11ststorder polynomialorder polynomial22ndndorder polynomialorder polynomialHigher order polynomialHigher order polynomial15ExamplesExamplesAll roots in open LHP?All roots in open LHP?Yes / NoYes / NoYes / NoYes / NoYes / NoYes / NoYes / NoYes / NoYes / NoYes / No16Summary and ExercisesSummary and ExercisesRouthRouth--Hurwitz stability criterionHurwitz stability criterionRouthRoutharrayarrayRouthRouth--Hurwitz criterion is applicable to only Hurwitz criterion is applicable to only polynomials (so, it is not possible to deal with polynomials (so, it is not possible to deal with exponential, sin, exponential, sin, coscosetc.).etc.).Next,Next,RouthRouth--Hurwitz criterion in control examplesHurwitz criterion in control examplesExercisesExercisesRead Section 6.Read Section 6.Do Examples and Problems 6Do Examples and Problems
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