控制工程基础 南理工 HPWANG-FCE9

Fundamentals of Control Engineering Ch9 Stability in the Frequency DomainHaoping WANG E-mail: Hp.wang@Automation SchoolNanjing University of Science and Technology Fundamentals of Control Engineering -hp.wang@j g y gyy q yCh9 Stability in the Frequency Domain IntroductionMapping contours in the s-planeNyquist CriterionRelative Stability and Nyquist CriterionTime-domain performance criteria specified in the Time-domain performance criteria specified in thefrequency domainStability of control System with time delaysFundamentals of Control Engineering -hp.wang@Introduction The frequency response of a system represents the sinusoidal steady-state response of a system and provides sufficient information for the state response of a system and provides sufficient information for the determination of the relative stability of the system.A f d i t bilit it i ll d N i t t bilit it i A frequency domain stability criterion called Nyquist stability criterion was developed by H. Nyquist in 1932.The Nyquist stability criterion is based on a theorem in the theory of the function of a complex variable due to Cauchy .The characteristic equation of the system isThe characteristic equation of the system is F(s) = 1 + L(s)where L(s) is a rational function of s .()To ensure stability, we must ascertain that all the zeros of F (s) lie in the left-hand s-plane. Nyquist thus proposed a mapping of the right-hand s-plane into the F(s)plane Fundamentals of Control Engineering -hp.wang@ plane into the F(s)-plane.y q yCh9 Stability in the Frequency Domain IntroductionMapping contours in the s-planeNyquist CriterionRelative Stability and Nyquist CriterionTime-domain performance criteria specified in the Time-domain performance criteria specified in thefrequency domainStability of control System with time delaysFundamentals of Control Engineering -hp.wang@A contours map is a contours or trajectory in one plane mapped or translated into another plane by a relation F(s)translated into another plane by a relation F(s).Since s is a complex variable, the function F(s) is itselfcomplex it can be defined as F(s)=u +j s j σω=+complex, it can be defined as F(s) = u + jv Exe 1. Let us consider a function F(s) = 2s + 1d t i th l h i Fi ()A di t th and a contours in the s-plane, as shown in Fig. (a). According to the relation ()2()1u jv F s j σω+==++we have21u σ=+⎧⎨The mapped contours in the F(s)-plane is shown in Fig 1. (b).2v ω=⎩Fundamentals of Control Engineering -hp.wang@21u σ=+⎧Fi 1M i t b F()2+12v ω⎨=⎩Fundamentals of Control Engineering -hp.wang@ Fig 1. Mapping a square contours by F(s) = 2s + 1.We assume clockwise traversal of a contour to be positive and the area enclosed within the contour to be on the right. and the area enclosed within the contour to be on the right Exe 2. Let us consider a rational function of sF(s) =s/(s + 2)Solution 2:Fundamentals of Control Engineering -hp.wang@Solution 2:Fig2. (a) square contour in the s-plane (b)and the resulting contour in the F(s)-planeFundamentals of Control Engineering -hp.wang@If the function F(s)is the characteristic equation, then we haven∏+()K s sCauchy’s theorem or Principle of the argument: If a contour Tin thess-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour T F in the F(s)-plane encircles the origin of the F(s)-plane N = Z − P times in the clockwise direction.Fundamentals of Control Engineering -hp.wang@Exe 3. Let us consider the functionSolution 3: then we haveNow considering the vectors as shown for a specific contour T s in Fig.(a).Fundamentals of Control Engineering -hp.wang@Fi3E l ti f th t l f Fundamentals of Control Engineering -hp.wang@Fig 3: Evaluation of the net angle of T F.We can extend this case to a general case: If Z zeros and P poles are enclosed within T s , then the net angle would be equal to and, respectively.2z Z φπ=2p P φπ=If Z zeros and P poles are encircled as T s is traversed, then the net change angle of F(s)change angle of F(s) is22()F z p N Z P φφφππ=−⇒=−and the net number of encirclements of the origin of the F (s)-plane is −N Z P=Fundamentals of Control Engineering -hp.wang@Exe 4. Reflecting exampleFig 4Example of Cauchy’s theorem with three zeros and one pole within Completes two clockwise encirclements ofthe origin in the F(s)-planeFundamentals of Control Engineering -hp.wang@ Fig 4. Example of Cauchy s theorem with three zeros and one pole within T sExe 5. Reflecting exampleFig 5Example of Cauchy’s theorem with one pole within Completes one counterclockwise encirclementsof the origin in the F(s)-planeFundamentals of Control Engineering -hp.wang@ Fig 5. Example of Cauchy s theorem with one pole within T sy q yCh9 Stability in the Frequency Domain IntroductionMapping contours in the s-planeNyquist CriterionRelative Stability and Nyquist CriterionTime-domain performance criteria specified in the Time-domain performance criteria specified in thefrequency domainStability of control System with time delaysFundamentals of Control Engineering -hp.wang@Considering the characteristic equation, then we haven. The zeros of F(s) are the characteristic roots of the system Exe 6.Fundamentals of Control Engineering -hp.wang@Nyquist stability criterion(1) Choose the contour T s in the s-plane that enclosed the entire right-hand (1)Ch h i h l h l d h i i h h d s-plane, and we determine whether any zeros of F(s) lie within T s by utilizing Cauchy's theorem.(2) Plot the T F in the F(s) plane and determine the number of encirclementsgof the origin N;Then the number of zeros of F(s) within the T s contour (and therefore, the unstable zeros of F(s)) isZ N P=+Thus, if P= 0, as is usually the case, we find that the number of unstable Th if0i ll th fi d th t th b f t bl roots of the system is equal to N, the number of encirclements of the origin of the F(s)-plane.Fundamentals of Control Engineering -hp.wang@The Nyquist contour that encloses the entire right-hand s-plane is shown inFig 7.Fig 7.The Nyquist contour includes two parts:(1) The first part is the jw-axis from −infinityto +infinity ;y (2) The second part is the right semicircularpath with infinity radius r.Fig 7 Nyquist contour is shown as the heavy line.Fundamentals of Control Engineering -hp.wang@ g yqu st co tou s s o as t e ea y eThe Nyquist criterion is concerned with the mapping of the characteristic equationF (s) = 1 + L(s)(s)(s)and the number of encirclements of the origin .We may define the function :F’(s)=F(s)−1=L(s)We may define the function : F (s) F(s) 1 L(s)Then the mapping of T s in the s-plane will be through the function F’(s) = L(s) into the L(s)-plane.The number of clockwise encirclements of the origin of theF()l b th b f l k i i l t f th 1i t i th F(s)-plane becomes the number of clockwise encirclements of the −1 point in the L(s)-plane.case 1. A feedback system is stable if and only if the contour T L in the L(s)-plane does not encircle the (−1, 0) point when the number of poles of L(s) in the right-hand s-plane is zero (P = 0).case 2. A feedback system is stable if and only if, for the contour T L the number of counterclockwise encirclements of the (−1, 0) point is equal to the number of poles of L(s) with positive real parts.Fundamentals of Control Engineering -hp.wang@ 0Z PN P N =+=⇒=−Exe 8. Considering a system with 2 real poles K 121,1/10,100K ττ===Solution 8:12()(1)(1)L s s s ττ=++(1)+jw-axis -> solidcolored line;(2) The −jw-axis ->dashed colored line;(3) The semicirclewith r -> infinity ismapped into thei iFig 8Nyquist contour and mapping for GH(s)origin.Fundamentals of Control Engineering -hp.wang@ Fig 8. Nyquist contour and mapping for GH(s).Exe 9. Considering a system with a pole at origin()KGH s=s sτ+(1)g yq pp g()Fig 9. Nyquist contour and mapping for GH(s).Fundamentals of Control Engineering -hp.wang@Solution 9:()1)K L s s s τ=+()(1) The Portion from to . In this case, the contour T s is mapped by the functionL()|L(j )0ω+=ω=+∞L(s)|s=jw = L(jw)(2) The Portion from . In this case, the contour T s is to 0ωω−=−∞=plane In this case the mapping is represented by plane. In this case, the mapping is represented byFundamentals of Control Engineering -hp.wang@(4) The Origin of the s-plane . In Fig. 9, the small semicircular detour at the origin isdenoted as and to vary from -90°to +90°The mapping j φdenoted as and to vary from 90to +90. The mapping for GH(s) is,0s e εε=→φThus, the angle of the contour in the GH(s)-plane changes from +90°at to -90°at , passing through 0 at .0ω=0ω−=0ω+=Then using the Nyquist criterion, we obtain Z = N + P = 0 + 0 = 0, Thus the system is stableFundamentals of Control Engineering -hp.wang@There are three general conclusions from this example:g p(i)It is sufficient to construct the contour T L for the frequency rangein order to investigate the stability.0 ω+<<+∞(ii) The magnitude of L(s) as s = r.exp(j Ԅ) and r to infinity will normallyapproach zero or a constant.(iii)The contour T L for the small semicircular detour at the originalways starts at and ends at in the clockwise0ω+=0ω−=direction with infinity radius.The net angle change of the contour is , where v is the typev πg g yp number of L (s).Fundamentals of Control Engineering -hp.wang@Exe 10. system with 3 poles K L ==y p 12()()(1)(1)s GH s s s s ττ++Fundamentals of Control Engineering -hp.wang@Solution 10: The point where the GH(s)-locus intersects the real axis can be foundby setting the imaginary part of GH(jw)=u+jv equal to zero The transfer found by setting the imaginary part of GH(jw) u jv equal to zero. The transfer function GH(jw) can be decomposed asThen we haveThus,v=0when or .The magnitudeFundamentals of Control Engineering -hp.wang@ Thus, v 0 when or .The magnitudeof the real part of GH(jw) is thenTherefore, the system is stable whenConsider the case where , so thatUsing the further above relation, when K < 2, the system is stable.Fundamentals of Control Engineering -hp.wang@Fig: Nyquist diagrams forFundamentals of Control Engineering -hp.wang@Exe 11 (System with two poles at the origin). Consider a single-loop control system loop control system 2()1)K L s s s τ=+()According to Fig, we haveZ =N +P =2+0=2Z N + P 2 + 0 2Thus the system is unstable.Fundamentals of Control Engineering -hp.wang@Exe 12 (System with a pole in the right-hand s-plane). Consider a single-loop control system K 1()()(1)L s GH s s s ==−Fundamentals of Control Engineering -hp.wang@ Z=N+P=2, the system unstable Fig.12The Nyquist CriterionExe 13. Consider a open-loop control systemK1 (1 + K 2 s ) GH ( s ) = s ( s − 1)The GH(jw)-locus intersects the u-axis at a point u= −K1K2 (i) If −K1K2 < −1, the contour TGH encircles the −1 point once in a counterclockwise t l k i di direction, ti and d therefore N = −1. Then Z = N + P = −1 + 1 = 0, the system is stable. (ii) If −K1K2 > −1,the contour TGH encircles the −1 point once in a clockwise direction, and therefore N = +1 +1. Then Z = N + P = 1 + 1 = 2, the system is unstable.Fundamentals of Control Engineering - hp.wang@31The Nyquist CriterionExe. 14 System with a zero in the right hand s-planeGH ( s ) = K ( s − 2) ( s + 1) )2unstableFundamentals of Control Engineering - hp.wang@32Ch9 Stability y in the Frequency q y Domain„ „ „ „ „Introduction Mapping contours in the s-plane Nyquist Criterion Relative Stability and Nyquist Criterion Time-domain performance criteria specified in the frequency domain„The stability of control System with time delaysFundamentals of Control Engineering - hp.wang@33Relative Stability and Nyquist Criterion• Absolute stability and relative stability th system the t i is. - In time-domain, the relative stability is measured by maximum overshoot and damping ratio. - In frequency frequency-domain domain, the relative stability is measured by resonance peak and how close the Nyquist plot of L(jw) is to the (-1,j0) ( 1 j0) point• For a stable system, relative stability describes how stableFundamentals of Control Engineering - hp.wang@34Relative Stability and Nyquist CriterionExe 15. Consider the open-loop transfer function a part of Nyquist plot of this system is shown in the following figureK GH ( s ) = s(τ 1s + 1)(τ 2 s + 1)The relative Th l ti stability t bilit of f th the curves are Lk1 > LK2 > LK3Fundamentals of Control Engineering - hp.wang@Fig. Polar plot for GH(jw) for three vales of gain.35Relative Stability and Nyquist Criterion‹This measure of relative stability is called the gain margin, which can be defined as follows:L( jωg ) at the The Gain margin is defined as the reciprocal of the gain frequency at which the phase angle reaches –1800 . The gain margin can be defined mathematically asGM =Or1 L( jωg )j ImL(jw)-planeωgL( jωg )1 GM = 20 lg = −20 lg L( jωg ) dB L( jωg )w e e is where st the e frequency eque cy at which w c the t e phase p ase of o L(jw) becomes -180°and called the phase crossover frequency.∠L( jωg ) = −180Dω=∞0R l Realω→0Fundamentals of Control Engineering - hp.wang@36Relative Stability and Nyquist CriterionGM = −20log10 L( jω ) dBWhen L( jω ) < 1 (stable), log10 L( jω ) < 0 ⇒ GM > 0L( jω ) ↓ (closer to the origin) ⇒ GM ↑ (more stable) L( jω ) ↑ (closer to -1) ⇒ GM ↓ (less stable)When L( jω ) = 1 (marginally stable), log10 L( jω ) > 0 ⇒ GM=0When L( jω ) > 1 (unstable), log10 L( jω ) > 0 ⇒ GM<0Gain G i margin i represents t the th amount t of f gain i i in d decibels ib l (dB) that can be added to the loop before the closed-loop system becomes unstable. unstableFundamentals of Control Engineering - hp.wang@37Relative Stability and Nyquist CriterionGain margin alone is inadequate to indicate relative stability when h system t parameters t other th th the l loop gain i are subject bj t t to variation. j Im With the same gain margin, system represented by plot A is more stable than plot B B. −1∠L( jωc )PM L( jωc ) = 10R ReBA38Fundamentals of Control Engineering - hp.wang@Relative Stability and Nyquist Criterion‹ The phase margin is defined as the phase angle through which the L( jω ) must be rotated so that the unity magnitude L( jωc ) = 1 point will pass through the critical point (-1, j0) in the L( jω ) plane. The p phase margin g can be defined mathematically y asPM = 1800 + ∠GH ( jωc )Where the frequency ωc at which GH ( jωc ) = 1 called the gain crossover frequency.‹ Nyquist CriterionFor minimum-phase system, If G M > 0, PM > 1 the system is stable. If the G M < 0, PM < 1 the system is unstable. G M , PM e ect t the e degree deg ee of o relative e at ve stability. stab ty. reflectFundamentals of Control Engineering - hp.wang@39Relative Stability and Nyquist CriterionExe 14.The Bode plot ofL( jω ) =1 jω ( jω + 1)(0.2 jω + 1)is shown in following figure, whose PM is 430 , GM is 15dBFundamentals of Control Engineering - hp.wang@40Exe 15. Comparison of relativestability : consider the following y gtwo systems:11()(1)(0.21)GH j j j j ωωωω=++221()(1)GH j j j ωωω=+01143,20lg 15 dB;20l 57dB PM GM ==02220,20lg 5.7 dB PM GM ==Fundamentals of Control Engineering -hp.wang@Advantages of Nyquist plot:-By Nyquist plot of the loop transfer function, the closed-loop stability can be easily determined with reference to the critical point (-1,j0).iti l i t(1j0)y y p-It can analyze systems with either minimum phase or nonminimum phase loop transfer function.Disadvantages of Nyquist plot:-By Nyquist plot only, it is not convenient to carry out controller design.Fundamentals of Control Engineering -hp.wang@y q yCh9 Stability in the Frequency Domain IntroductionMapping contours in the s-planeNyquist CriterionRelative Stability and Nyquist CriterionTime-domain performance criteria specified in the Time-domain performance criteria specified in thefrequency domainStability of control System with time delaysFundamentals of Control Engineering -hp.wang@in the frequency domainin the frequency domain•Open or Closed-loop frequency responsee yqu s c e o a d e p ase a g de a e de ed o eThe Nyquist criterion and the phase margin index are defined for theloop transfer functionThe maximum magnitude of the closed-loop frequency response can be related to the damping ratio of a second-order system ofy yFor unity feedback systems, Equation becomesFundamentals of Control Engineering -hp.wang@in the frequency domain in the frequency domain•Constant M circles()c G G j u jv ω=+221/2221/2()()1()[(1)]c c G G j u v M G G j u v ωω+==+++M 222222()()11M u v M M−+=−−Constant M circles.Fundamentals of Control Engineering -hp.wang@in the frequency domainin the frequency domainThe maximum magnitude of the closed-loop frequency response, M pw, is the value of the M circle that is tangenti th l f th M i l th t i t t to the L(jw) locus.Fundamentals of Control Engineering -hp.wang@in the frequency domainin the frequency domainNichols chart•Constant M circlesConstant N circles•Constant N•Nichols chartlog-magnitude-phase diagramFundamentals of Control Engineering -hp.wang@y q yCh9 Stability in the Frequency Domain IntroductionMapping contours in the s-planeNyquist CriterionRelative Stability and Nyquist CriterionTime-domain performance criteria specified in the Time-domain performance criteria specified in thefrequency domainStability of control System with time delaysFundamentals of Control Engineering -hp.wang@Stability of control System with time delays y y yMany control systems have a time delay within the closed loop of the system that affects the stability of the system system that affects the stability of the system.A time delay: is the time interval between the start of an event at i t i t d it lti ti t th i t i th one point in a system and its resulting action at another point in the system.Fortunately, the Nyquist criterion can be utilized to determine the effect of the time delay on the relative stability of the feedback system . A pure time delay, without attenuation, is represented by the transfer function y ywhere T is the delay time.The Nyquist criterion remains valid for a system where T is the delay time. The Nyquist criterion remains valid for a system with a time delay because the factor e -sT does not introduce any additional poles or zeros within the contour. The factor adds a phase shift to the frequency response without altering the magnitude curve Fundamentals of Control Engineering -hp.wang@ frequency response without altering the magnitude curve.Stability of control System with time delays y y yExe 16. Steel Rolling mill control system: the motor adjusts the separation of the rolls so that the thickness error is minimized. If the steel is traveling of the rolls so that the thickness error is minimized.If the steel is traveling at a velocity v, then the time delay between the roll adjustment and the measurement isT=d/v.Fundamentals of Control Engineering -hp.wang@。