ME 3600 Cntl Systems Fequency Dmain Analysis The fequency espnse f a system is defined as the steady-state espnse f the system t a sinusidal (hamnic) input. F linea systems, the esulting utput is itself hamnic; it diffes fm the input in amplitude and phase nly. This may be illustated in a blck diagam as fllws: Linea System Hee, epesents the multiplicatin fact f the magnitude, and epesents the elative phase shift between the input and the utput. If 1, the system amplifies the input, and if 1, the system attenuates the input. One cmmn way t pesent the fequency espnse f a linea system is using a Bde diagam. The Bde diagam f a typical secnd de system is shwn in the diagam belw. Using the magnitude plt, the esnant fequency ( ) the esnant magnitude ( M ), the bandwidth (BW), and the ate f decay f the system can be identified. The bandwidth (BW) is defined as the fequency at which the system is 3dB dwn fm its lw fequency value. Nte that 3dB epesents an amplitude f 0.71, s the system epnds at 71% f its lw fequency value. Resnant Magnitude Bandwidth (BW) (@ -3dB) Rate f Decay Resnant Fequency, Kamman ME 3600 page: 1/7
If a system is secnd de and has a damping ati 0.707, the esnant fequency and esnant magnitude may be estimated using the equatins n 1 and M 1 1 (1) Nte: Equatins (1) can be used f any pai f cmplex ples (in highe de systems) that ae sufficiently islated (in fequency) fm the ples and zes. In this case, M epesents the ise in magnitude fm the pe-esnance value. These chaacteistics in the fequency-dmain celate with behavi f the system in the time-dmain. The esnant magnitude M gives an indicatin f the elative stability. Lage values f M ae indicative f lw damping, suggesting scillaty espnse with lage veshts. Systems with lage bandwidths have faste espnse than systems with small bandwidths; hweve, they may be me sensitive t nise. Sensitivity t nise is detemined by a cmbinatin f the bandwidth and the ate f decay f the magnitude at high fequencies. Minimum Phase Systems The lp tansfe functin GH () s f a minimum phase system has n zes ples in the ight half f the s-plane. If a system has ples zes in the ight half plane, it is efeed t as a nn-minimum phase system. R(s) + - Y(s) G(s) H(s) Clsed Lp System If clsed-lp system is a minimum phase system, then the stability f the system can be detemined by examining the Bde diagam f the lp tansfe functin GH () s. If the system is a nn-minimum phase system, a Nyquist diagam may be used t detemine stability. Bde diagams ae usually pefeed ve Nyquist diagams f minimum phase systems, because it is easie t measue the gain and phase magins n a Bde diagam. It is als easie t see hw the Bde diagam changes shape as ples and zes ae added t ( emved fm) the system. The Nyquist diagam and the Nyquist stability citein ae pesented belw. Kamman ME 3600 page: /7
Gain and Phase Magins and the Bde Diagam Gain and phase magins f a minimum phase system ae detemined by pltting the Bde diagam f the lp tansfe functin GH () s. F example, cnside the Bde diagam f the lp tansfe functin 100( s ) () ( s 5)( s 10)( s s ) Phase magin (PM) is the additinal phase lag equied t make the phase angle 180 (deg) at the fequency whee the magnitude f the system csses the ze-db line. Gain magin (GM) is the additinal magnitude equied t make the magnitude ze db when the phase angle is 180 (deg). In the case shwn belw, the phase magin is PM 68 (deg) (measued at. (ad/s)), and the gain magin is GM 17.4 db (measued at 7.15 (ad/s)). Gain Magin (GM>0) Phase Magin (PM>0) Gain and Phase Magins and Stability The gain and phase magins detemine the stability f minimum phase systems. A minimum phase system is stable if bth magins ae psitive, and unstable if they ae negative. Systems with a highe degee f stability have lage magins and less stable systems have smalle magins. The Bde diagam abve epesents a stable clsed-lp system. Kamman ME 3600 page: 3/7
MATLAB Cmmands f Bde Diagams and Gain and Phase Magins A set f MATLAB cmmands t display the Bde diagam and the gain and phase magins f the tansfe functin f Equatin () ae shwn belw. The figue shws the display esulting fm the magin cmmand. >> num=100*[1,]; >> den=cnv([1,5],cnv([1,10],[1,,])); >> sys=tf(num,den) Tansfe functin: 100 s + 00 ----------------------------------- s^4 + 17 s^3 + 8 s^ + 130 s + 100 >> bde(sys); gid; >> magin(sys) Nyquist Diagam A Nyquist diagam is a plt f the eal-pat f GH ( j ) vesus the imaginay-pat f GH ( j ). (Recall that the Bde diagam is a plt f the magnitude f GH ( j ) vesus and a plt f the phase f GH ( j ) vesus.) F example, the Nyquist diagam f the lp tansfe functin ( s 10) ( s )( s 3)( s s ) (3) is shwn in the diagam belw. Nte that GH ( j ) 0.833 j0, and the aw heads shw 0 the diectins f inceasing fequency. S, the pat f the diagam that is belw the eal axis is f the fequency ange 0, and the pat abve the eal axis (the cmplex cnjugate f the pat belw) is f the ange 0. Nte als that the plt fms a clsed cuve. Kamman ME 3600 page: 4/7
Befe stating the Nyquist stability citein, a few definitins ae helpful. N = the numbe f times the Nyquist plt encicles the pint 1 j0 in a clckwise sense ( 1 j0 is dented n the diagam with a ed "+".) Z = the numbe f zes f 1 GH ( s) in the ight half f the s-plane P = the numbe f ples f 1 GH ( s) in the ight half f the s-plane (Nte that the ples f 1 GH ( s) ae als the ples f GH () s ) Nyquist Stability Citein F a clsed lp system t be stable the Nyquist citein states that Z 0 and N Z P P. This means that the Nyquist plt must encicle the pint 1 j0 "P" times in the cunteclckwise diectin f the clsed-lp system t be stable. Nte that if the lp tansfe functin GH () s is f minimum phase, then P 0 and N 0. Hence, the Nyquist plt f a stable, minimum-phase system des nt encicle the pint 1 j0. The Nyquist diagam abve indicates the clsed lp system is stable. If the clsed-lp system is unstable, the Nyquist citein states that the numbe f ples f the clsed-lp system (als the numbe f zes f 1 GH ( s) ) in the ight half plane is equal t Z N P. If the system is minimum phase, the numbe f ples in the ight half plane is equal t the numbe f clckwise enciclements f the pint 1 j0. Kamman ME 3600 page: 5/7
As an example f this last statement, cnside the Nyquist plt f the lp tansfe functin in Eq. (4) as shwn in the figue belw. 10 ( s 10) ( s )( s 3)( s s ) (4) In this case, the Nyquist plt encicles the pint 1 j0 (shwn as a ed "+") twice in a clckwise diectin. This means that the clsed-lp system has tw ples in the ight half plane, indicating the system is unstable. Unstable system with tw clckwise enciclements f the pint. Cmpaisn f Results fm a Rt Lcus Diagam, a Bde Diagam and a Nyquist Diagam Nw we cmpae the esults btained fm t lcus, Bde, and Nyquist diagams f a single system. Cnside the clsed-lp system whse lp tansfe functin is K ( s 10) ( s )( s 3)( s s ) (5) The fist diagam belw shws the t lcus diagam f the paamete K indicating the system is unstable when K 4.1. The next tw diagams ae the Bde and Nyquist diagams f the system with K 10. It is clea fm these diagams that the system is unstable f that value f K. Kamman ME 3600 page: 6/7
Rt Lcus Diagam System is unstable f Bde Diagam Veifies the system is unstable f. Nyquist Diagam Veifies the system is unstable f. (Plt encicles twice in a clckwise diectin, indicating tw clsed-lp ples in the ight half s-plane.) Kamman ME 3600 page: 7/7