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Combination of Eqs. (32.9) to (32.12) yields 1 3dr  d2c +2;i;+e = 2 dt2
E >O E <O
E >O E CO (32.13) Equation (32.13) can be rewritten in phase notation as de dt =& (32.14) d6 (3& + 2c + 2) c >O dt= (38 + 2e  2) Et<0 I Equation (32.14) breaks up into two regions, the region for which E > 0 will be referred to as R, and that far which e < 0 as L. The critical point for R occurs at c = 1 g = 0 and that for L at E = 1 6 = 0 Note that each critical point is outside the region to which it pertains. In region R, the isocline equation is 2 + 2t I 36 = SR t , METHODS
OF PHASEPLANE ANALYSIS
or (32.15) The corresponding isocline equation in L is (32.16) The isoclines in R, which is the right half of the e e plane, radiate from the R critical point ( 1,O) and have slopes  ~/(SR + 3). The isoclines in L radiate from the critical point (1 ,O) and have slopes ~/(SL + 3). These isoclines are indicated in Fig. 32.7. Note that, in this figure, the E scale has been expanded by a factor of. 10 to magnify the behavior near the origin. A typical trajectory has been constructed, using the method of isoclines. When the trajectory crosses from one region to the other on the k axis, the applicable isoclines also change. It can be seen from Fig. 32.7 that the trajectory approaches the origin, Since the trajectories must be vertical as they cross the E axis, the final state is a limit cycle of zero amplitude and infinite frequency about the origin. In other words, the relay alternately opens and closes at very high frequency, a condition known as chattering. Solemimidve
Solen$valve
s=21 s=11 t FIGURE 327 Phaseplane trajectory for onoff control of system of Fig. 32.4.
NONLINEARCONTROL
Physically, this condition will never be realized because the dynamics of the solenoid valve and therelay itself would become important. Instead, the final condition will be a limit cycle of high, rather than infinite, frequency and low, rather than zero, amplitude. However, the basic idealization which has led us to this suspect conclusion is in the behavior of the relay. True relays have inputoutput characteristics more similar to that shown in Fig. 32.8. There is a dead band around the set point, of width 2~6, over which the relay is insensitive to changes in the error signal. Anyone who has made fine adjustments in the setting of a home thermostat has observed this behavior. Consider as an example the case for ~6 = 0.01. The effect of this dead zone is to change the dividing line between R and L to that shown in Fig. 32.9. The new dividing line has the equation: 0.01 k >O 0.01 k <O 1 Now, as shown in Fig. 32.9, all trajectories approach a limit cycle, for which the error amplitude is approximately 0.03, The frequency is finite and is obtained by computing the time around the limit cycle. Although we have not presented here the graphical methods for determining this time, it can always be calculated by noting from the first of Eqs. (32.14) that E = (32.17) Thus, time around the limit cycle can be computed by graphical evaluation of the integral in Eq. (32.17). The only difficulty is near the E axis, where i goes to zero. To circumvent this, we may use the second of Eqs. (32.14)

