An Introduction to Nonlinearity in Control Systems by D. Atherton

By D. Atherton

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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.