Continuous and Modal Behaviour

  1. For the model from the following figure:

    1. What is the mathematical expression that the model defines?
    2. Does the system have feedback?
  2. (from Lee Shia's book)  Consider a rotating robot where you can control the angular velocity around a fixed axis. Model this as a system where the input is angular velocity \( w \) and the output is angle θ. Give your model as an equation relating the input and output as functions of time.
  3. (from Lee Shia's book) A DC motor produces a torque that is proportional to the current through the windings of the motor. Neglecting friction, the net torque on the motor, therefore, is this torque minus the torque applied by whatever load is connected to the motor. Newton’s second law (the rotational version) gives

    \( k_{T} \cdot i(t) − T(t) = I\frac{\partial}{\partial t}ω(t), \)

    where \( k_{T} \) is the motor torque constant, i(t) is the current at time t, \( T(t) \) is the torque applied by the load at time t, \( I \) is the moment of inertia of the motor, and \(  ω(t) \) is the angular velocity of the motor.
    1.  Assuming the motor is initially at rest, rewrite the equation as an integral equation.
    2. Assuming that both \( T \) and \( i \) are inputs and \( ω\) is an output, construct an actor model (a block diagram) that models this motor. You should use only primitive actors such as integrators and basic arithmetic actors such as scale and adder.
  4. Consider the control of a robot's motor, for moving forward and backward. When the button for going forward is being pressed, the motor should rotate forward. When the button for moving backwards is being pressed the motor should rotate backwards. If neither button is pressed the motor should stop. The robot must first stop before changing direction. The robot has a sensor on the front and another on the back to detect objects. If the object detect is closer than 10 cm the robot stops.
    1. Draw the state diagram for this system.
    2. Identify the states, inputs, outputs, initial state and update function.
  5. (from Lee Shia's book) Consider a variant of the thermostat described in the class. In this variant, there is only one temperature threshold, and to avoid chattering the thermostat simply leaves the heat on or off for at least a fixed amount of time. In the initial state, if the temperature is less than or equal to 20 degrees Celsius, it turns the heater on, and leaves it on for at least 30 seconds. After that, if the temperature is greater than 20 degrees, it turns the heater off and leaves it off for at least 2 minutes. It turns it on again only if the temperature is less than or equal to 20 degrees.
    1. Design an FSM that behaves as described, assuming it reacts exactly once every 30 seconds.
    2. How many possible states does your thermostat have? Is this the smallest number of states possible?
    3. Does this model thermostat have the time-scale invariance property (if we change the time scale for sampling is the output the same, scale-wise)?
  6. (from Lee Shia's book)  Consider the deterministic finite-state machine in the following figure that models a simple traffic light.
    1.  Formally write down the description of this FSM as a 5-tuple: (States,Inputs,Outputs,update,initialState).
    2.  Give an execution trace of this FSM of length 4 assuming the input tick is present on each reaction.
    3.  Now consider merging the red and yellow states into a single stop state. Transitions that pointed into or out of those states are now directed into or out of the new stop state. Other transitions and the inputs and outputs stay the same. The new stop state is the new initial state. Is the resulting state machine deterministic? Why or why not? If it is deterministic, give a prefix of the trace of length 4. If it is nondeterministic, draw the computation tree up to depth 4.
Última alteração: terça, 16 de março de 2021 às 16:53