By D. Atherton
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Extra info for An Introduction to Nonlinearity in Control Systems
27) -a Thus for the input c + a cos i the output is cNc (c, a) + Na (c, a) a cos i . The important point about the SBDF model is the interaction between the signals caused by the nonlinearity, so that the gain to both signals is affected by a change in one of them. 26) can be plotted against γ to give a new nonlinear characteristic. This is often called a modified nonlinearity and denoted by n (a, c) rather than a0 . The slope of the modified nonlinearity at any input value γ gives the gain g (a, c) to any small unrelated signal in the presence of a and γ.
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. The latter is very straightforward when the relay has no dead zone as its output is a square wave of amplitude ! h , which has a fundamental component of amplitude 4h/r , either in phase with the input for D = 0 , or lagging when Δ is finite.
15) 2 where x10 and x20 are the initial values of x1 and x2 . 15) describes a parabola, which for the special case of K = 0 has the solution x2 = x20 , it is easy to calculate the system’s response from any initial condition (x10, x20) in the phase plane. 7. 5x2 = 1 is reached at C. 5, meets the same switching boundary again at D. The next motion is again at constant velocity until the point E is reached on the first switching boundary. 5, brings the trajectory straight back to the switching boundary.