Linear Continuous Systems: Summary

1.   Linearity á convolution:

á á á á á á á á á á á á á á á á á á á á á á á   

2.   Hence in the frequency domain we have a filter:

á á á á á á á á á á á á á á á á á á á á á á á   

3.   á is a general eigenfunction of the system á hence we can define the Laplace Transform:
á á á á á á á á á á   

4.   Inverse Laplace Transform most easily done by looking up a table.

5.   Represent a function graphically by means of a pole-zero plot. This gives an easy route to visualizing the frequency response (behaviour when á ). Magnitude is the product of the distances from a point on the axis to each of the zeros, divided by the product of the distances to the poles.

6.   Hence when á is close to the imaginary part of a pole, we have a resonance.á 

7.   A linear system is stable if it has no poles in the Right Half-Plane

8.   Test with Routh-Hurwitz criteria:

a.    All coefficients must have the same sign.

b.   For a cubic,  

9.   Alternatively test with Nyquist criterion: locus of GH must not encircle the point (-1,0).