Control Systems

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By G. S. Virk (auth.)

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The other two transformations will always give stable digital realisations for stable continuous systems. It is also important to note that, since Tustin's rule maps the stable region of the s-plane exactly into the stable region of the z-plane, it gives rise to better approximations as seen in chapter 2. However this does not imply perfect results because the entire jw-axis of the s-plane is squeezed into the 271" length of the unit circle in the z-plane. Clearly a distortion takes place in the mapping in spite of the agreement of the stability regions.

5 that the forward rectangular rule can give an unstable transformation for a stable continuous transfer function (as illustrated by the example considered in chapter 2). The other two transformations will always give stable digital realisations for stable continuous systems. It is also important to note that, since Tustin's rule maps the stable region of the s-plane exactly into the stable region of the z-plane, it gives rise to better approximations as seen in chapter 2. However this does not imply perfect results because the entire jw-axis of the s-plane is squeezed into the 271" length of the unit circle in the z-plane.

To assist in this transformation there are conversion tables (see, for example Kuo [70]). Now, since the w-plane is similar to the s-plane, it is possible to apply the standard Routh method to the characteristic equation in w for assessing system stability. 1, to which the Routh stability result can be applied. This states that the system has no unstable roots if all the numbers in the first column have the same sign, and also that none of the numbers vanishes. In our example there are two sign changes, and so there are two unstable poles.

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