ME 4710 Mtin and Cntl 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: Asin( t) Asin( t ) 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. Kamman ME 4710: page 1/5
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. Kamman ME 4710: page /5
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 GH () s 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 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 4710: page 3/5
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. MATLAB Cmmands >> 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 >> magin(sys) Cmpaisn f Results fm Rt Lcus and Bde Diagams Nw we cmpae the esults btained fm t lcus and Bde diagams f a single system. Cnside the clsed-lp system whse lp tansfe functin is GH () s K ( s10) ( s )( s 3)( s s ) Hee, the RL diagam f the paamete K indicates the system is unstable when K 4.14. Kamman ME 4710: page 4/5
Bde diagams f the lp tansfe functins f K 1 and K 10 ae shwn belw. The gain and phase magins ae bth psitive (indicating a stable system) f K 1 and bth ae negative (indicating an unstable system) f K 10. Using eithe diagam, we can calculate K max the gain equied t make the system maginally stable. 1.3 0lg ( x) 1.3 x 10 0 4.1 K 1 x 4.1 10 max 7.69 0lg ( x) 7.69 x 10 0 0.413 K 10 x 4.13 10 max Kamman ME 4710: page 5/5