Sample Midterm Solutions - East Carolina University CSDI 8104

Document Content and Description Below

5. Consider a rotating robot where you can control the angular velocity around a fixed axis. (a) Model this as a system where the input is angular velocity θ˙ and the output is angle θ. Give your ... model as an equation relating the input and output as functions of time. Solution: ∀ t ∈ R, θ(t) = θ(0) +Zt 0 θ˙(τ)dτ, where θ(0) is the initial position. (b) Is this model BIBO stable? Solution: The model is not BIBO stable. For example, the input θ˙(t) = u(t) is bounded but yields an unbounded output. (c) Design a proportional controller to set the robot onto a desired angle. That is, assume that the initial angle is θ(0) = 0, and let the desired angle be ψ(t) = au(t), where u is the unit step function. Find the actual angle as a function of time and the proportional controller feedback gain K. What is your output at t = 0? What does it approach as t gets large? Solution: A proportional controller has the same structure as the helicopter controller: Just as with the helicopter controller, we can solve the integral equation to get θ(t) = au(t)(1−e −Kt). The output at zero is θ(0) = 0, as expected. As t gets large, the output approaches a. Lee & Seshia, Introduction to Embedded Systems, Solutions 11 3 Discrete Dynamics — Exercises 1. Consider an event counter that is a simplified version of the counter in Section 3.1. It has an icon like this: This actor starts with state i and upon arrival of an event at the input, increments the state and sends the new value to the output. Thus, e is a pure signal, and c has the form c : R → {absent} ∪ N, assuming i ∈ N. Suppose you are to use such an event counter in a weather station to count the number of times that a temperature rises above some threshold. Your task in this exercise is to generate a reasonable input signal e for the event counter. You will create several versions. For all versions, you will design a state machine whose input is a signal τ : R → {absent} ∪Z that gives the current temperature (in degrees centigrade) once per hour. The output e : R → {absent,present} will be a pure signal that goes to an event counter. (a) For the first version, your state machine should simply produce a present output whenever the input is present and greater than 38 degrees. Otherwise, the output should be absent. Solution: This state machine does not require more than one state: /output (b) For the second version, your state machine should have hysteresis. Specifically, it should produce a present output the first time the input is greater than 38 degrees, and subsequently, it should produce a present output anytime the input is greater than 38 degrees but has dropped below 36 degrees since the last time a present output was produced. 17 SOLUTIONS Solution: (c) For the third version, your state machine should implement the same hysteresis as in part (b), but also produce a present output at most once per day. Solution: Note that this problem statement is ambiguous. What is meant by “at most once per day?” Is it OK to produce a present output at 11 PM and again at 1 AM? Or does it mean that at least 24 hours should elapse between present outputs? Either would be correct, given the problem statement. Here is a solution under the second interpretation: Note that with this solution, if the temperature stays high for several days, there will nonetheless be no present output for those several days. Is this likely to be what we intended? The problem appears to ask for this behavior, but it is probably not the behavior we want. 18 Lee & Seshia, Introduction to Embedded Systems, Solutions SOLUTIONS 3. Consider the following state machine: Determine whether the following statement is true or false, and give a supporting argument: The output will eventually be a constant 0, or it will eventually be a constant 1. That is, for some n ∈ N, after the n-th reaction, either the output will be 0 in every subsequent reaction, or it will be 1 in every subsequent reaction. Note that Chapter 13 gives mechanisms for making such statements precise and for reasoning about them. Solution: TRUE. In an infinite execution, if the transition from A to B is ever taken, then after that point, the output will always be 0. If that transition is never taken, then the output will be a constant 1 for the entire execution. This too is allowed behavior for the state machine. 20 Lee & Seshia, Introduction to Embedded Systems, Solutions 3. DISCRETE DYNAMICS 7. Consider the state machine in Figure 3.2. State whether each of the following is a behavior for this machine. In each of the following, the ellipsis “··· ” means that the last symbol is repeated forever. Also, for readability, absent is denoted by the shorthand a and present by the shorthand p. (a) x = (p, p, p, p, p,···), y = (0,1,1,0,0,···) (b) x = (p, p, p, p, p,···), y = (0,1,1,0,a,···) (c) x = (a, p,a, p,a,···), y = (a,1,a,0,a,···) (d) x = (p, p, p, p, p,···), y = (0,0,a,a,a,···) (e) x = (p, p, p, p, p,···), y = (0,a,0,a,a,···) Solution: (a) no (b) yes (c) no (d) yes (e) no [Show More]

Last updated: 1 year ago

Preview 1 out of 22 pages