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					EGR 326                                Homework 4                        Due Tuesday Feb 23
                                                                        4pm to Ford Hall 352

1: HW 1 revisited
    Returning to HW 1, from the first week (hopefully this will be fun) … revisit one of the
    systems for which you developed a first pass dynamic model. That was before we had
    talked about storage elements, across and through variables, feedback…
        Hand in: Redefine and redraw the one system of your choice from the two you
        developed the first week (or a new one; but not a circuit or mass-spring system), using
        the concepts of state variables, state equations, output equations, and feedback. Hand
        in a system definition including a purpose, state variable definition, state and output
        equations in matrix format, a brief state on expected system behavior, and a simulation
    A good approach: To the extent that it makes sense with your system
    a. Assign state variables as: the dynamic variables associated with storage elements in
       your system.
            i. For many of those first-week examples, the system elements may be storing
               information, people, chemical compounds, etc, rather than energy.
           ii. Think in terms of ‘stocks’ and ‘flows.’ What is flowing and what is being
               stored in your system? In many cases these will be analogous to the across and
               through variables from our electrical and mechanical examples.
    b. For the diagram
            i. Assign one integrator/delay block for each state variable, and draw the
               structure as in the other homework simulation diagrams (these represent the
               ‘stocks’ or the quantity being stored)
           ii. Connect the dynamic variable (state variable) blocks to show the ‘flows’
               between the variables. Assign generic gain blocks along these connections,
               with parameter values that you may or may not (probably not) know.
         iii. Use summing nodes, keeping track of where there is positive and negative
               feedback and feedforward flows.
          iv. Identify the input(s) and the output(s)
    c. For the equations
            i. You should be able to write the equations from your diagram, and/or iterate
               between the diagram and the equations until you are satisifed

2: State space model comparisons
    Synthesize what you have learned so far about state space models.
           a. Think about the different models we have used, identifying each model in
               terms of how the state variables are obtained.
           b. Think about the characteristics of each type of model.
           c. Review class examples, notes, and section 3.4 in the text.
           d. Hand in: Compare and contrast the types of state space models by creating an
               easy-to-read table or matrix that could be organized with…
                   ii. The types of state space models listed down the left side for the row
                  iii. The characteristics you will use to compare and contrast the types of
                       state space models listed along the top as column headings.
                  iv. The matrix cells filled in with your comparing and contrasting.
                   v. By all means, collaborate on this.

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EGR 326                                 Homework 4                        Due Tuesday Feb 23
                                                                         4pm to Ford Hall 352

3: Vehicle suspension system

A vehicle suspension system is an interesting dynamic problem (that shows up in many text
books and online tutorials), and an interesting control problem. When the suspension system
is designed, a ¼ vehicle model (e.g., one of the four wheels) is used to simplify the problem to
a one dimensional mass-spring-damper system.


Defining/simplifying the system:
    The stiffness of the tire is modeled by a spring
    The tire, axle and other tire-axle parts are modeled as a mass
    The suspension + shock absorber system is modeled as a second spring and a damper
    The vehicle is modeled as a second mass
More definition for the model
   Define the coordinate system so ‘y’ is positive pointing up
          o So, use ‘z’ perhaps for the output variable, to avoid confusion
   The lower end of the tire follows the road surface, described by y0(t)
   Selecting and defining the input to this system is likely to take some thought and
Task: Develop a state space model of this system, including a purpose for the model, a clear
definition of the state variables, the state equations, the output equations, and a simulation
diagram. Put this all into Simulink and run your model from a Matlab script.
        a. Build the model in detail, with an integrator for each energy storage element/state
            variable (rather than using the state space block).
        b. Try different inputs until you find one you like. A good input is one that:
                     i. has physical meaning (which y0(t) does, as does Fin = k1y0(t)), and
                    ii. gives understandable and hopefully expected output
        c. Identify the output(s) you want and obtain them from the model using the ‘To
            Workspace’ blocks, along with ‘scope’ blocks if you want them too.
        d. Define your model parameters in the Matlab script (i.e., the ‘.m’ file) – which is to
            say, use the variable names in the Simulink model and assign values to these
            variables in the Matlab script.
        e. Plot your output from the Matlab script – including well labeled plots with a title,
            axis labels and a legend.
        f. Hand in: your model development, the model including the Simulink diagram and
            the beautifully commented Matlab script (with the input you used explained briefly
            in comments in the script or in a text box in the diagram), some plots and a brief
            discussion (up to one paragraph) on the behavior of the system as revealed through
            your plot(s).

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EGR 326                                  Homework 4                       Due Tuesday Feb 23
                                                                         4pm to Ford Hall 352

         g. We will plan to return to this model when we get into control, because the
            objective of developing a model of a bouncy bus is to allow us to build a controller
            to make it less bouncy.

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