Problem 1

Consider an asymptotically stable first-order system: y = ay+bu. The coefficients a and b are

real-valued and, furthermore, a < 0 due to asymptotic stability. You have available the test data

from a "sine wave" experiment (the dependent variable has converged its steady-state sinusoidal

response in the measurement). The results of the test is shown below:

Sine wave test results

Input and output

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

0.1

0.2

- Input sinusoid

-Output sinudoid

0.3

Time (second)

0.4

0.5

0.6

Use this information to estimate a and b. Hints: first determine the frequency (in rad/s) of the input

sinusoid (the output sinusoid has the same frequency); the magnitude and phase of the frequency

response function can be determined at this frequency from the relative magnitudes and phases.