CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2004Homework Set #2 Issued: 4 Oct 04Due: 11 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).All students should complete the following problems:1. Consider the following block diagram of the flight control system of a fly:Using the paper “Vision as a Compensatory Mechanism for Disturbance Rejection in UpwindFlight” by Reiser et al. (available via the course web page), identify the state, input, outputs,and dynamics for each block in the diagram. You may give you answer in words, but beprecise as possible. (Hint: not all of the blocks are “dynamic”; some are static maps withinputs and outputs, but no state.)2. (MATLAB/SIMULINK) In this problem you will build a model of a vehicle in SIMULINKand control the vehicle using feedback control. The vehicle will consist of a body (chassis +wheels) and a drive train (engine + transmission). Assume that the vehicle dynamics are ofthe formm ˙v = −bv + Fengine+ Fhillwhere m = 1000 kg is the mass of the vehicle, b = 50 N sec/m is the viscous dampingcoefficient, and Fengineand Fhillrepresent the forces on the vehicle due to the engine andthe terrain, respectively. We can implement this in SIMULINK as a two input, one outputsystem, written in state space form as˙xv=£−b/m¤xv+£1/m 1/m¤uvyv= xv(1)where xv= v is the vehicle state, uv=£FengineFhill¤Tis the vehicle input (two dimensional),and yv= v is the vehicle output (velocity). You should make this into a single SIMULINKblock using the “State Space” block (under Simulink→Continuous→State Space in theSimulink Library Browser). You may also want to use the Mux block (under Signals &Systems).We will model the engine dynamics as a “first order lag”. Let τ represent the engine torqueand assume the engine has the following dynamics:˙τ = −aτ + ueye= Kτ(2)where a = 0.2 is the lag coefficient, K = 5 is the conversion factor between engine torque andforce applied to the vehicle (representing the transmission) and ueis the accelerator input(which we will assume has the proper units). You should also create a SIMULINK block forthis subsystem.Finally, we include the effects of a hill. The hill simply exerts a force on the car that is basedon the angle of the hill:Fhill= −mg sin(θ) (3)where g = 9.8 kg m/sec2and θ = π/18 is the angle of the hill (10 degrees).(a) Plot the output of the open loop vehicle model (1) for a step input of Fengine= 500Newtons (assume Fhill= 0). What is the rise time (0 to 95% of the final value)?(b) Plot the output of the open loop engine model (2) for a step input of ue= 100 Nm.What is the rise time?(c) In the homework from last week, you built a simple cruise controller. Replace thevehicle/engine model in that system with your vehicle and engine models and plot theresponse for the default gains (Ki= 50, Kp= 1000). Make sure to set your simulationtime to be sufficiently long.(d) Now include the effect of a hill on your system. You should model the system so thatthe car is initially on a flat surface and then encounters the hill at T = 100 seconds.Plot the response of the system and compute the rise time.Note: if you are having trouble figuring out how to create these blocks in SIMULINK, take alook at “hw1cruise.mdl” from last week’s homework and see if you can modify it appropriately.It has all of the subsystems you will need (except for the Mux block). You may also find thefollowing web-based tutorial helpful:http://www.engin.umich.edu/group/ctm/examples/cruise/cc.html(ignore the sections on transfer functions; we will get to these later in the class).Only CDS 110a students need to complete the following additional problems:3. Consider the following discrete time systemz[k + 1] = Az[k] + Bu[k]y[k + 1] = Cz[k + 1]2wherez =·z1z2¸A =·a11a120 a22¸B =·01¸C =£1 0¤In this problem, we will explore some of the properties of this discrete time system as afunction of the parameters, the initial conditions, and the inputs.(a) Assume that the off diagonal element a12= 0 and that there is no input, u = 0. Writea closed form expression for the output of the system from a nonzero initial conditionz[0] = (z1[0], z2[0]) and give conditions on a11and a22under which the output getssmaller as k gets larger.(b) Now assume that a126= 0 and write a closed form expression for the response of thesystem from a nonzero initial conditions. Given a condition on the elements of A underwhich the output gets smaller as k gets larger.(c) Write a MATLAB program to plot the output of the system in response to a unit stepinput, u[k] = 1, k ≥ 0. Plot the response of your system with z[0] = 0 and A given byA =·0.5 10 0.25¸4. In this problem we will look at how to play with fire without getting burned. The system wewant to consider is a finger being moved back and forth across a flame, as shown below:xf= 1finger(axial view)flamexf= 0The description of the system is as follows:• The temperature of a finger is regulated by an internal feedback mechanism. To firstorder, we will say that heat is convected away by blood flow, at a rateFb= αb(Tf− Tb)where Tfis the temperature of the fingertip, Tbis the temperature of the blood, and αbis the convection coefficient (the F signifies the heat flux).• A flame gives off heat into the ambient air, and we assume a steady-state temperaturefield around the flame. The ambient air far from the flame is at 25 degrees Celsius.• The flame is fixed at xF= 1, and fingertip begins at a position xf= 0, where theambient air is precisely at the same temperature as the blood.3• Suppose that the temperature of the air varies exponentially with distance from theflame, soTa(x) = 25 + (TF− 25)µTb− 25TF− 25¶(x−1)2where TFis the flame temperature.• Heat convects into the finger from the ambient air at a rateFa= αa(Ta− Tf).• The dynamics of the fingertip temperature is given bycf˙Tf= −Fb+ Fawhere cfis the fingertip thermal capacity.• The fingertip is rapidly passed into and out of the flame, according toxf(t) = sin(ωt).Using the MATLAB ode45 function (or something similar), build a model for the system andsolve the following:(a) Assume that the finger moves sinusoidally in and out of the flame at frequency ω = 1rad/sec. Plot the temperature of the finger as
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