CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2004Homework Set #3 Issued: 11 Oct 04Due: 18 Oct 04Note: In the upper left hand corner of the first page of your homework set, pleaseput the class you are taking (CDS 101, CDS 110) and the number of hours that youspent on this homework set (including reading).The following exercise should be completed only by students taking CDS 101:1. List two different fluid instabilities and give a brief (1-2 sentence) description of the physicalmechanism. Identify (in words or a simple sketch) the equilibrium flow about which pertur-bations are unstable, and if the process involves internal feedback in an obvious way, statehow.All students should complete the following problems:2. For each of the following systems, locate the equilibrium points for the system and indicatewhether each is asymptotically stable, stable (but not asymptotically stable), or unstable.To determine stability, you can either use a phase portrait (if appropriate) or simulate thesystem using multiple nearby initial conditions to how the state evolves. (Note: if you knowhow to check stability through the linearization, you can also use this approach.)(a) Nonlinear spring mass. Consider a nonlinear spring mass system,m¨x = −k(x − ax3) − b ˙x,where m = 1000 kg is the mass, k = 250 kg/sec2is the nominal spring constant, a = 0.01represents the nonlinear “softening” of the spring, and b = 100 kg/sec is the dampingcoefficient. Note that this is very similar to the spring mass system we have studied inclass, except for the nonlinearity.(b) Predator prey ODE. Use the ODE model described in class,˙x1= brx1− ax1x2˙x2= bx1x2− dfx2,with the parameters br= 0.7, df= 0.5, a = 0.007, b = 0.0005.(c) Congestion control of the Internet. A simple model for congestion control between Ncomputers connected by a router is given by the differential equation˙xi= −bx2i2+ (bmax− b)˙b =NXi=1xi− cwhere xi∈ R, i = 1, N are the transmission rates for the sources of data, b ∈ R is thecurrent buffer size of the router, bmax> 0 is the maximum buffer size, and c > 0 is thecapacity of the link connecting the router to the computers. The ˙xiequation representsthe control law that the individual computers use to determine how fast to send dataacross the network (this version is motivated by a protocol called “Reno”) and the˙bequation represents the rate at which the buffer on the router fills up. Consider the casewhere N = 2 (so that we have three states, x1, x2, and b), and take bmax= 1 Mb andc = 2 Mb/sec.3. (MATLAB/SIMULINK) Consider the cruise control system from Homework Set #1, problem1. Set the gains of the system to their default values (Ki= 100, Kp= 500).(a) Using hw1cruise.mdl from the course home page, plot the step response of the system(from 55 mph to 65 mph) and measure the rise time, overshoot, settling time, and steadystate error.(b) Modify the block diagram to allow a sinusoidal reference signal superimposed on top ofa commanded reference (so that you get something that oscillates around the nominalspeed of 55 m/s). Plot the response of the system to a commanded reference speed thatvaries sinusoidally between 50 m/s and 60 m/s at a frequency of 1 Hz (about 6 rad/sec).Measure the relative amplitude and phase of the velocity with respect to the commandedinput. Your answer should be the ratio of the output amplitude to the input amplitude(after subtracting off the means) and the number of radians of phase “lead” or “lag”between the sinusoids.(c) In most real-life systems, inputs magnitudes are limited by the capabilities of the actu-ator. A modified version of the cruise controller with input saturation is available fromthe lecture homepage, with the file name hw1cruise_sat.mdl (you can see the satura-tion by clicking into the vehicle block). Using this model, show that if we increase theamplitude of the desired oscillations sufficiently high, that the response of the system isno longer a pure sinusoid at the desired frequency.Only CDS 110a students need to complete the following additional problems:4. Consider a second order system of the form¨y + 2ζωn˙y + ω2ny = u(t)with initial conditions y(0) = y0, ˙y(0) = ˙y0. The constant ωn> 0 is called the naturalfrequency and ζ > 0 (zeta) the damping ratio.(a) Compute the homogenous solution to this equation (u(t) = 0) with initial conditiony0= 0, ˙y0= 1. This is the “impulse response” for this system. Plot the impulseresponse as a function of time for ωn= 1, ζ = 0.5.(b) Compute the response of the system to a sinusoidal input u(t) = A sin(ωt). Your resultshould be analytical (a formula) and you should make sure to keep the effects of theinitial conditions. Now assuming that the initial conditions have died out (i.e., ignoringthe homogeneous part of the solution), plot the “frequency response” of the system. Youranswer should be in the form of two plots: the relative amplitude and the relative phaseof the output compared to the input, both as a function of frequency. Use a logarithmicscale for the frequency and amplitude, and a linear scale for the phase. (This type ofplot is called a “Bode plot”).2Note: you can find this solution worked out in many textbooks. You are encouraged tolook for the solution, but make sure that you provide a derivation of your results andthat you understand them. (Pretend that this might be the type of thing you were askedon a closed book section of the midterm.)(c) Suppose that we now implement a feedback control law of the formu(t) = k1(y − v(t)) + k2˙y,which is intended to allow us to track a new input v(t) (just like the cruise controlexample). Compute the frequency response of the closed loop and show that we canset the closed loop natural frequency ω0nand damping ratio ζ0to arbitrary values byadjusting the gains k1and k2. Give formulas for the gains in terms of the desired ω0nand ζ0.(d) Optional:1Use the results from this problem to design a cruise control law for thesystem in problem #2 of last week’s homework that has a settling time of 1 second andno overshoot.5. For each of the systems in the table below, defined in more detail in Problem 2, determine ifthere exists a Lyapunov function of the given form that proves that the indicated equilibriumpoint is asymptotically stable. The parameter γ should be taken as a free (scalar) parameterand used as needed to satisfy the conditions of the Lyapunov
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