33 ASYMPTOTICTRA KING 33 which violates(a)-■ Finally,let us recall for lat er use the Nyquist stability crit erion-It can be derived from Theo- rem2 and the principle of the argument-Begin with the curve D in the compler plane It starts at the origin,goes up the imaginary aris,turns int o the right halfplane following a semicircle of infinite radius,and comes up the negative imagiary aris to the origin again: D As a point s makes me circuit around this curve,the point P(s)C(s)F(s)traces out a curve called the Nyquist plot of PCF-If PCF has a pole on the imaginary aris,then D mst hawe a small indentation to avoid it- NyquiSt CriteriOn Construct the Nyquist plot of PCF,indenting to the left around poles on the imaginary aris,Let n denote the total number cf poles of P,C,and F in Res >0 Then the feedback system is internally stable i,the Nyquist plot does not pass through the point 1 and encircles it exactly n times counterclockibise 3.3 Asym ptotic Tracking In this section we look at a typical performance specification,perfect asymptotic tracking of a reference signal-Both time domain and frequency domain occur,so the notation distinction is required- For the remainder of this chapter we specialize to the unity feedback case,F=1,so the block diagram is as in Figure 34-Here e is the tracking error;with h =d=0,e equals the reference input (ideal response),r,minus the plant output (actual resp anse),y- We wish to study this systems capability of tracking certain test inputs asympt otically as time tends to infinity-The two test inputs are the step r()= ft≥0 0 ift(0 and the ramp ct-ft≥0 r(t)= 0-ft(0 (c is a nonzero real number)-As an application of the former think of the temperature-control thermost at in aroom when you change the setting on the thermost at (step input),you would likeASYMPTOTIC TRACKING  which violates a ￾ Finally let us recall for later use the Nyquist stability criterion It can be derived from Theo rem  and the principle of the argument Begin with the curve D in the complex plane It starts at the origin goes up the imaginary axis turns into the right halfplane following a semicircle of innite radius and comes up the negative imaginary axis to the origin again D As a point s makes one circuit around this curve the point P sCsF s traces out a curve called the Nyquist plot of PCF If PCF has a pole on the imaginary axis then D must have a small indentation to avoid it Nyquist Criterion Construct the Nyquist plot of PCF  indenting to the left around poles on the imaginary axis Let n denote the total number of poles of P  C and F in Res   Then the feedback system is internal ly stable i the Nyquist plot does not pass through the point and encircles it exactly n times counterclockwise ￾￾ Asymptotic Tracking In this section we look at a typical performance specication perfect asymptotic tracking of a reference signal Both time domain and frequency domain occur so the notation distinction is required For the remainder of this chapter we specialize to the unityfeedback case F   so the block diagram is as in Figure  Here e is the tracking error with n  d   e equals the reference input ideal response r minus the plant output actual response y We wish to study this systems capability of tracking certain test inputs asymptotically as time tends to innity The two test inputs are the step rt   c if t    if t   and the ramp rt   ct if t    if t   c is a nonzero real number As an application of the former think of the temperaturecontrol thermostat in a room when you change the setting on the thermostat step input you would like