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Extra resources for An Introduction to Nonlinearity in Control Systems
Slotine JJE & Li W 1991, Applied Nonlinear Control, Prentice-Hall International, New Jersey. 11 Appendix -Tables of Describing Functions Two tables of DFs, or SDFs, are given below, one for single valued nonlinearities and the other for double valued nonlinearities. Extensive use is made in the tables of the describing function for saturation, Ns (d/a) , defined in eqn. 19). 7, and for other inputs. 1 Introduction The problem of determining whether a linear feedback system is stable is simply one of determination of the roots of its characteristic equation.
9 then the gain when the complex roots cross the axis will be twice that for the real root crossing the axis. 9 is presumably due to errors in the DF and the small range of attraction of the limit cycle, which is a closed curve in three space. 6 More than one Nonlinear Element When the closed loop system has more than one nonlinear element it is possible to write down nonlinear algebraic equations based on the DF method to give any possible limit cycle solutions. The difficulty with this approach, however, is that satisfactory accuracy can only be obtained if, when a limit cycle occurs, it is reasonable to assume the inputs at all the nonlinear elements are nearly sinusoidal.
Thus a steady-state error of unit magnitude will result from any motion. Finally a word of warning that difficulties may arise in simulating systems with sliding, since if a variable step length integration algorithm is used it may reduce the step length to zero and stop in the sliding motion. 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.