Wireless Ad Hoc and Sensor Networks

76 Wireless Ad Hoc and Sensor NetworksSastry, S. and Bodson, M., Adaptive Control, Prentice Hall, Englewood Cliffs, NJ,1989.Slotine, J.-J.E. and Li, W., Applied Nonlinear Control, Prentice Hall, NJ, 1991.Von Bertalanffy, L., General System Theory, Braziller, New York, 1998.Whitehead, A.N., Science and the Modern World, Lowell Lectures (1925), Macmillan,New York, 1953.ProblemsSection 2.1Problem 2.1.1: Simulation of compound interest system. Simulatethe system of Example 2.1.3 and plot the state with respect to time.Problem 2.1.2: Genetics. Many congenital diseases can be explainedas a result of both genes at a single location being the samerecessive gene (Luenberger 1979). Under some assumptions,the frequency of the recessive gene at generation k is given by therecursionxk ( )xk ( + 1)=1 + xk ( )Simulate in MATLAB using x( 0) = 75.Observe that x(k) convergesto zero, but very slowly. This explains why deadly genetic diseasescan remain active for a very long time. Simulate the systemstarting for a small negative value of x( 0)and observe that it tendsaway from zero.Problem 2.1.3: Discrete-time system. Simulate the systemx2( k)x1( k+ 1)=1 + x ( k)2x1( k)x2( k+ 1)=1 + x ( k)using MATLAB. Plot the phase-plane plot.2