By D. Atherton
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Additional info for An Introduction to Nonlinearity in Control Systems
Because the DF solution is approximate, the actual measured frequency of oscillation will differ from this value by an amount which will be smaller the closer the oscillation is to a sinusoid. 41%. A symmetrical square wave has a harmonic content which is 1/nth that of the fundamental for n odd. 21% of the fundamental. 9) gives the amplitude of the assumed sinusoidal limit cycle a as 5h/r . 20) gives 2 N (a) = 2 1/2 4h (a - D ) 2 a r - j 4h2 D from which a r 2 2 1/2 C (a) = - 1 = - r 6(a - D ) + jD @ .
3) over the interval - r/2 to r/2 and assuming that the input amplitude a is greater than d + D , gives # h cos idi = (2h/r) (sin b + sin a) b a1 = (2/r) -a where a = cos- (d - D) /a and b = cos- (d + D) /a , and 1 b1 = (2/r) # b -a h sin idi = (2h/r)` 1 (d + D) (d - D) j = 4hD/ar . 4. 13) one obtains # x (h/2) p (x) dx + # xhp (x) dx d+D a1 = (4/a) d-D a d+D 2 2 1/2 2 2 1/2 = 2h "6 a - (d + D) @ + 6 a - ^d - Dh @ , ar # (h/2) dx = 4hD/ar = (Area of nonlinearity loop) /ar. d+D b1 = (4/ar) d-D The DFs for other relay characteristics can easily be found from this result or worked out directly.
Thus, 3 a limit cycle solution, which is independent of n , with frequency 1 rad/s and amplitudes of 2 for x1 and x2 , is predicted. The result is therefore only reasonably accurate for small values of n , where the phase plane plot is almost elliptical. 1 is considered with n(x) a relay with dead zone and G (s) = 2/s (s + 1) 2 . 20) with D = 0 and is given by 2 2 1/2 N (a) = 4h (a - d ) 2 /a r for a > d, which is real because the nonlinearity is single valued. 2. 2 Relay with dead zone and its DF It shows that N (a) starts at zero, when a = d , increases to a maximum, with a value of 2h/rd at a = d 2 , then decreases toward zero for larger inputs.