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17. Compare these results to those in Fig. 12. 22 M. Borre and H. 755 Fig. 17 Simulation showing limit cycles excited from rigid body initial conditions noted on ROA map in Fig. 100 constant dead-zone Fig. 3 Fig. 1 Parametric Study The following parametric study shows how the limit cycle frequencies and their ROA change as the relay hysteresis is varied. All other parameters are held constant per the values shown in Fig. 10. 18 shows the values of hysteresis that were used, and Fig. 19 shows the limit cycle frequencies found using Step 2 of the method, for intermediate values of hysteresis.

3) This approximation is characterized in the following section. • An approximation of max{JCM(a), a ∈ I } is computed using optimization methods which are appropriate for dealing with the mentioned restriction (non-availability of an algebraic expression). Precisely, Genetic Algorithms, a Kiefer–Wolfowitz scheme, and some Neural Network-based techniques have been employed. 1 Size of Domain of Attraction The size of a domain of attraction is considered as the performance index J to be maximized.

Assuming f2 < 0, we get that for f1 < À 0. 2 the origin is asymptotically stable, and the system has two additional (unstable) equilibria at ( À 5, 0) and (5, 0). Therefore, the analysis focuses on the region ðf 1 ; f 2 Þ 2 ðÀ1; À0:2Þ ÂðÀ1; 0Þ. When applying the different optimization approaches described in this chapter, the results are quite sensitive to the initial conditions (seeds for the genetic scheme, and initial conditions for Kiefer–Wolfowitz and NN schemes) since the domain CM-estimate reaches a maximum in a parameter region including ðf 1 ; f 2 Þ 2 ½À1:8; À0:6 Â ½À2:0; À1:0 as it can be seen at Fig.

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### A Course in Real Analysis by J. McDonald, N. Weiss

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