Signal Processing and Linear Systems-B.P.Lathi copy

T he a symptotic value for w 100 is 90 iv t he zero a

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Unformatted text preview: 4) w hen K = 24. T he s ame information is p resented in polar form in t he c orresponding Nyquist plot in Fig. 7.lOc. For example, a t w = 1, IH(jw)1 = 2.6 a nd L.H (jw) = - 130.6°. We plot a point a t a d istance 2.6 u nits a nd a t a n angle - 130.6° from t he h orizontal axis. This point is labeled as w = 1 for identification (see Fig. 7.lOc). We p lot such points for several frequencies in t he r ange from w = 0 t o 0 0 a nd d raw a smooth curve through them t o o btain t he N yquist plot. T he same information is presented in Cartesian form in Nichols plot. For instance, a t w = 1, t he log magnitude is 20 log 2.6 = 8.3 dB, a nd t he p hase a t w = 1 is - 130.6°. We p lot a point a t c oordinates x = 8.3, Y = - 130.6° a nd label this point as w = 1 for identification. We do this for several values of w from 0 t o 0 0 a nd d raw a curve through these points t o o btain Nichols plot. Using Bode or Nyquist (or Nichols) plots of t he open-loop transfer function, we c an readily investigate stability aspect of t he c orresponding closed-loop system as discussed below. 7.3-1 Relative Stability: Gain and Phase margins For t he s ystem in Fig. 7.10a, t he c haracteristic equation is 1 + K G(8)H (8) = 0 a nd t he c haracteristic roots are t he r oots of K G (8) H (8) = - 1. T he s ystem becomes unstable when t he r oot loci cross over t o RHP. T he crossing occurs on t he i maginary axis where 8 = jw (see Fig. 6.43). Hence, a t t he verge of instability (marginal stability) K G(jw)H(jw) T he t ime-domain method of control system design discussed in Sec. 6.7 works only when t he t ransfer function of t he p lant (the system t o be controlled) is known and is a r ational function (ratio of two polynomials in 8). T he i nput-output description of practical s ystems is often unknown a nd is more likely to b e nonrational. A system containing a n ideal time delay (dead time) is a n example of a nonrational system. In such cases, we c an determine t he frequency response of t he open-loop system empirically a nd use this d ata t o design t he (closed-loop) system. In this section we shall discuss feedback system design procedure based on frequency response description o f a system. However, t he frequency response design method is n ot as convenient a s t he t ime-domain design method from the viewpoint of t he t ransient a nd t he s teady-state error specifications. Consequently, t he t ime-domain method in = - 1 = 1 e±j1f T hus, a t t he verge of instability, t he m agnitude a nd angle of t he o pen loop gain K G(jw)H(jw) a re I KG(jw)H(jw)1 = 1, K G(8) T(8) = 1 + K G(8)H(8) 491 a nd L.G(jw)H(jw) = ±7r T hus, on t he verge of instability, t he open-loop transfer function has unity gain a nd p hase of ±7r. I n order t o u nderstand t he significance of these conditions, let us consider t he s ystem in Fig. 7.lOa, with open-loop transfer function K / 8 (s+2)(8+4). T he Bode plot for this transfer function (for K = 24) is d epicted in Fig. 7.lOb. T he r oot locus for this system is illustrated in Fig....
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