ME 3600 Control Systems Frequency Domain Analysis

Transcription

1 ME 3600 Contol Systems Fequency Domain Analysis The fequency esponse of a system is defined as the steady-state esponse of the system to a sinusoidal (hamonic) input. Fo linea systems, the esulting steady-state output is itself hamonic; it diffes fom the input in amplitude and phase only. Linea System Tansient Response Steady-state Response Hee, epesents the multiplication facto fo the magnitude, and epesents the elative phase shift between the input and the output. If 1, the system amplifies the input, and if 1, the system attenuates the input. One common way to epesent the fequency esponse of a linea system is using a Bode diagam. The Bode diagam of a typical second ode system is shown in the diagam below. Using the magnitude plot, the esonant fequency ( ) the esonant magnitude ( M ), the bandwidth (BW), and the ate of decay of the system afte esonance can be identified. Bandwidth (BW) (@ db) Rate of Decay Resonant Fequency, Kamman Contol Systems page: 1/8

2 The Bode diagam shown above is fo a second-ode system whose natual fequency is n 1 (ad/s) and whose damping atio is 0.3. The esonant fequency is defined as the fequency at which the magnitude is maximum. The magnitude at this fequency is M. In this case, MATLAB indicates the esonant fequency and esonant magnitude ae (ad/s) and 0log( M) 4.85 (db). The bandwidth (BW) of the system is defined as the fequency at which the system is 3 db down fom its constant low-fequency value. Note that 3 db epesents an amplitude multiplie of 0.71, so the system eponds at 71% of its low fequency value. The decay ate afte esonance fo a second-ode system is 40(dB/decade). If a system is second-ode and has a damping atio esonant magnitude may be estimated using the equations n 1 and M 1 1, the esonant fequency and (1) Eqs. (1) can be used fo any pai of complex poles (in highe-ode systems) that ae sufficiently isolated (in fequency) fom othe poles and zeos. In these cases, magnitude fom the pe-esonance value. M epesents the ise in These chaacteistics in the fequency-domain coelate with behavio of the system in the time-domain. The esonant magnitude values of M gives an indication of the elative stability. Lage M ae indicative of low damping, suggesting oscillatoy esponse with potentially lage oveshoots. Systems with lage bandwidths have faste esponse than systems with small bandwidths; howeve, they may be moe noise sensitive. Sensitivity to noise is detemined by a combination of the bandwidth and the ate of decay of the magnitude at high fequencies. Minimum Phase Systems The loop tansfe function GH () s of a minimum phase system has no zeos o poles in the ight-half of the s-plane. If a system has poles o zeos in the ight-half plane, it is efeed to as a non-minimum phase system. Simple Closed Loop System R(s) + G(s) H(s) Y(s) Kamman Contol Systems page: /8

3 If a closed-loop system is a minimum phase system, then the stability of the system can be detemined by examining the Bode diagam of the loop tansfe function GH () s. If the system is a non-minimum phase system, a Nyquist diagam can be used to detemine stability. Bode diagams ae usually pefeed ove Nyquist diagams fo minimum phase systems, because it is easie to measue the gain and phase magins on a Bode diagam. It is also easie to see how the Bode diagam changes shape as poles and zeos ae added to (o emoved fom) the system. The Nyquist diagam and the Nyquist stability citeion ae pesented below. Gain and Phase Magins and the Bode Diagam Gain and phase magins of a minimum phase system ae detemined by plotting the Bode diagam of the loop tansfe function GH () s. To illustate this pocess, conside the Bode diagam of the loop tansfe function GH () s 100( s ) ( s 5)( s 10)( s s ) () Phase magin (PM) is the additional phase lag equied to make the phase angle 180 (deg) at the fequency whee the magnitude of the system cosses the zeo-db line. Gain magin (GM) is the additional magnitude equied to make the magnitude zeo db when the phase angle is 180 (deg). In the case shown below, the phase magin is PM 68 (deg) (measued at. (ad/s)), and the gain magin is Gain and Phase Magins and Stability GM 17.4 db (measued at 7.15 (ad/s)). The gain and phase magins detemine the stability of minimum phase systems. A minimum phase system is stable if both magins ae positive, and unstable if they ae negative. Systems with a highe degee of stability have lage magins and less stable systems have smalle magins. The Bode diagam below epesents a stable closed-loop system. Kamman Contol Systems page: 3/8

4 Gain Magin (GM>0) Phase Magin (PM>0) MATLAB Commands fo Bode Diagams and Gain and Phase Magins A set of MATLAB commands to display the Bode diagam and the gain and phase magins fo the tansfe function of Eq. () ae shown below. The figue shows the display esulting fom the magin command. Note that the convolution function conv is used to build the tansfe function fom its component pats. >> num = 100*[1,]; >> den = conv([1,5],conv([1,10],[1,,])); >> sys = tf(num, den) Tansfe function: 100 s s^ s^3 + 8 s^ s >> bode(sys); gid; >> magin(sys) Kamman Contol Systems page: 4/8

5 Nyquist Diagam A Nyquist diagam is a plot of the eal-pat of GH ( j ) vesus the imaginay-pat of GH ( j ). (Recall that the Bode diagam is a plot of the magnitude of GH ( j ) vesus and a plot of the phase of GH ( j ) vesus.) The Nyquist diagam fo the loop tansfe function GH () s ( s 10) ( s )( s 3)( s s ) (3) is shown in the diagam below. Note that GH ( j ) j0, and the aow heads show 0 the diections of inceasing fequency. The pat of the diagam that is below the eal axis is fo the fequency ange 0, and the pat above the eal axis (the complex conjugate of the pat below) is fo the ange 0. Note also that the plot foms a closed cuve. Befoe stating the Nyquist stability citeion, a few definitions ae helpful. N = numbe of times the Nyquist plot encicles the point 1 j0 in a clockwise sense ( 1 j0 is denoted on the diagam with a ed + ) Z = numbe of zeos of 1 GH ( s) in the ight-half of the s-plane P = numbe of poles of 1 GH ( s) in the ight-half of the s-plane (Note that the poles of 1 GH ( s) ae also the poles of GH () s ) Kamman Contol Systems page: 5/8

6 Nyquist Stability Citeion Fo a closed loop system to be stable the Nyquist citeion states that Z 0 and N Z P P. This means that the Nyquist plot must encicle the point 1 j0 P times in the counteclockwise diection fo the closed-loop system to be stable. Note that if the loop tansfe function GH () s is of minimum phase, then P 0 and N 0. Hence, the Nyquist plot fo a stable, minimum-phase system does not encicle the point 1 j0. The Nyquist diagam above indicates the closed loop system is stable. If the closed-loop system is unstable, the Nyquist citeion states that the numbe of poles of the closed-loop system (also the numbe of zeos of 1 GH ( s) ) in the ight-half plane is equal to Z N P. If the system is minimum phase, the numbe of poles in the ight-half plane is equal to the numbe of clockwise enciclements of the point 1 j0. To illustate this last statement, conside the Nyquist plot fo the loop tansfe function in Eq. (4) as shown in the figue below. GH () s 10 ( s 10) ( s )( s 3)( s s ) (4) The Nyquist plot encicles the point 1 j0 twice in a clockwise diection. This means that the closed-loop system has two poles in the ight-half plane, indicating the system is unstable. Unstable system with two clockwise enciclements of the point. Kamman Contol Systems page: 6/8

7 Compaison of Results fom a Root Locus Diagam, a Bode Diagam, and a Nyquist Diagam To compae the esults obtained fom oot locus, Bode, and Nyquist diagams fo a single system. Conside the closed-loop system whose loop tansfe function is GH () s K ( s 10) ( s )( s 3)( s s ) (5) The fist diagam below shows the oot locus diagam fo the paamete K indicating the system is unstable when K 4.1. The next two diagams ae the Bode and Nyquist diagams fo the system with K 10. It is clea fom these diagams that the system is unstable fo K 10. Root Locus Diagam System is unstable fo Bode Diagam Veifies the system is unstable fo. Kamman Contol Systems page: 7/8

8 Nyquist Diagam Veifies the system is unstable fo. (Plot encicles twice in a clockwise diection, indicating two closed-loop poles in the ight half s-plane.) Kamman Contol Systems page: 8/8