By D. Atherton
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Extra info for An Introduction to Nonlinearity in Control Systems
16) An Introduction to Nonlinearity in Control Systems The Phase Plane Method It is easily shown from symmetry considerations, when the above lines are drawn on a phase plane, that a limit cycle exists for the system with no damping with a maximum value of x2 = D/m . 5 . 51 and -4 are shown converging to the limit cycle from within and without. 5. 16 shows a Simulink diagram for a position control system with no viscous damping and with nonlinear effects due to torque saturation and Coulomb friction.
Many simulation languages have special routines for taking care of the discontinuous sliding motion, but good results can usually be obtained by using a fixed step length integration algorithm or by incorporating a small hysteresis in the relay. 13 Initial condition response for system with When the velocity feedback signal saturates, that is, when | mx2 | > h , the input signal to the relay is - x1 ! h . 12, although the equations describing the motion between the boundaries remain unaltered. 5.
Dependent on various factors the loops may continue to maintain both frequencies or lock to a limit cycle with two related, say 1 to 3, dominant frequencies. In a single loop configuration the assumption of sinusoidal signals into all the nonlinear elements is usually not justifiable. 1 prior to Gc(s) then it may be reasonable to assume this has a sinusoidal input but since most controller transfer functions Gc(s) are not low pass filters the signal into the nonlinearity before the plant may be quite highly distorted.