ECE 590, Fall 1996, Due 10/17/96

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Shown below is a block diagram for a particular continuous-time system.
The reference input is R(s), and the signals of interest are the error E(s),
the feedback F(s), the output Y(s), and the control U(s). The purpose of the
project is to investigate the ability of various operational substitution
methods to obtain stable and accurate simulations of this system.

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Assume that the reference input R(s) is a unit step function, and that
all initial conditions in the system are zero. Using a time increment of 0.1
seconds, obtain time-domain plots of e(t), f(t), u(t), and y(t) for the
continuous-time system. The MATLAB functions "step" or "lsim" would be
useful for this, along with "series" and "feedback". The plots obtained in
this step will serve as the basis of comparison for the operational
substitution methods. The simulation should be run long enough until the
system reaches steady state.

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Using a timestep of 0.1 seconds and a unit step input, obtain a block-by-block
simulation for all the signals e(nT), f(nT), u(nT), and y(nT) for each of the
following substitution methods: Euler, Tustin, Halijak, Boxer-Thaler.
Block-by-block simulation means that difference equations are to be developed
for each block which relate its output signal to the output of the previous
block in the loop. The accuracy of each simulation is determined by the
closeness of the signals to those generated from the continuous-time system
directly.

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For the Euler and Tustin methods only, develop the closed-loop transfer
function from R(z) to Y(z) using the substitution results from part 2. With
the same timestep and input as before, simulate the system with each of these
two methods.

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Develop the closed-loop transfer function from R(s) to Y(s) of the
continuous-time system. For the Tustin and Halijak methods only, develop
the discrete-time transfer function of the closed-loop system. With the
same timestep and input as before, simulate the system with each of these
two methods.

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Document your activities in a written report. The factors to be
considered should include at least the following: (a) discussion and
comparison of the accuracy and stability of the various methods; (b) design
ease, programming ease, and computational burden of the methods; (c)
comparison of block-by-block simulation with closed-loop simulation; (d)
comparison of associative with non-associative methods; and (e) your opinion
of the effects of the timestep value on the accuracy and stability of the
simulation.

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