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ECE 549 -Fall 2005 Midterm Do each question. Exam is open book. You can consult with other students but make sure to write up on your own. 1. Write a differential equation that corresponds to the diagram:
- (^25) Ans: dv/dt = -2, dx/dt = 5v Does the model have an equilibrium state? No, the model doesn’t have an equilibrium state as dv/dt is a constant (and hence it can’t be equal to zero which is a condition for the system to be in equilibrium). 2. For the differential equation: 2
dx v dt dv v x x dt
a) What is the equilibrium state? Ans: The equilibrium state is given by dx/dt = 0 and dv/dt = 0 => x=1, v = 1. b) Draw an integrator diagram as in 1) that corresponds to the differential equation Ans: The diagram (using SIMULINK) is as follows:
c) Is the model stable at the equilibrium state? (The answer can be obtained either by simulating the model in appropriate package or linearizing the model around the equilibrium similar to the discussion of Lotka-Volterra). Ans: When simulated, the model in part (b) gives the following output: which clearly shows that the system is unstable around the equilibrium state.
Analysis:
x v Ans: A well-defined differential equation always arrives at the same trajectory from a given initial state. It is not possible to have a limit cycle as shown because by definition, no matter what initial state the model is started in, it eventually always winds up in a cycle. As shown in the figure, it cannot off shoot if it displays limit cycle behavior. b) Discuss why, in a differential equation that exhibits chaos, behavior similar to that in a) can be observed. Ans: For a differential equation that exhibits chaotic behavior, it is possible to diverge from the cycle as shown in the figure. This is because, as the trajectory comes closer to an initial state the large gain can cause it diverges from that earlier portion to follow a completely new trajectory. Even if it tries to fall into a limit cycle, this sensitivity to initial states causes this divergence. 4. (This question does not require simulation to answer – but you are welcome to implement it in an appropriate simulation package to check your answers/or obtain insight.) During fall and spring, every day has a high temperature of 80F and a low of 50F. In a house, the thermostat is set to turn on the heater if the inside temperature falls below 68F and to turn on the air conditioner (cooler) if the temperature rises above 78F. The inside temperature is modeled by: inside ( (^) outside inside ) Heat Cool dT R T T sw H sw C dt
where (^) R is a factor depending on the insulation in the walls, (^) H is the rate at which the inside temperature Tinside^ can rise when the heater is on, C is the rate at which the inside temperature can fall when the cooler is on; swHeat^ and swCool^ are 1 when the heater and cooler are on, respectively, and 0 otherwise. a) If the outside temperature Toutside^ ^80 F and remains there for awhile. Indicate whether each of the following is true or false:
i) the inside temperature, Tinside^ will stay constant during this period This is False. If the inside temperature is below 78F will start to rise because Toutside is 80 F ; if it is above 78F then the AC will cause it to start to move toward 78F. ii) the inside temperature, Tinside^ will approach 80 F^ at an exponential rate True. Tinside will approach 80 F at an exponential rate (but may not get there, see next question) a) If the outside temperature Toutside^ ^0 F and remains there for awhile. Indicate whether each of the following is true or false: i) the inside temperature, Tinside^ will rise to reach 68F This is False. If Tinside^ is initially above 68F then heat exchange with the outside will cause it drop towards 0F. If Tinside is below 68F then it will rise toward 68F but it may not reach 68 F, it depends on the ratio H/R (see part ii). ii) the inside temperature, Tinside^ will approach H / R at an exponential rate This is True. Consider ( ) with 0, 68 , 1, 0 we have:
So for equilibrium we have 0 * The solutioin is: ( ) inside outside inside Heat Cool outside inside Heat Cool inside inside inside eq inside inside dT R T T sw H sw C dt T T F sw sw dT R T H dt R T H H T R T t
Rt inside
H H
T e R R b) to implement this model, indicate whether each of the following is true or false: i) we need level crossing detectors to represent the operation of the thermostat This is True. ii) we need to reset the inside temperature integrator whenever a level is crossed This is False, because we only need to change the derivatives (by adding/removing heating/cooling) whenever a level is crossed. We need not reset the inside temperature integrator.
