The stability of sampled data systems

PROCESS CONTROL 7.17.5. The Stability of Sampled Data Systems 

It is shown in Section 7.10.1 that a continuous system is unstable if any root of the associated characteristic equation (i.e. any pole of the system transfer function) lies in the right half of the complex s-plane (Fig. 7.93a). If this root is s, then i, can be expressed in terms of its real and imaginary parts, i.e.  

Putting z = exp(j, 2T) to obtain the corresponding z-transform (i.e. to map the root on to the complex z-plane) gives  

If st lies in the right half of the 5-plane, then 0, 0 and z I. The latter condition applies to the region outside the unit circle (i.e. outside z = 1) in the z-plane (Fig. 7.93b). Thus the roots of the z-transformed characteristic equation must all lie within the unit circle in the z-plane for the corresponding system to be stable. 

In order to determine the number of roots of the z-transformed characteristic equation that lie outside the unit circle, a procedure analogous to the Routh-Hurwitz approach for continuous systems (Section 7.10.2) can be used. The Routh-Hurwitz criterion cannot be applied directly to the characteristic equation f(z) = 0. However, by mapping the interior of the unit circle in the z-piane on to the left half of a new complex variable -plane, the Routh-Hurwitz criterion can be applied as for continuous systems to the corresponding characteristic equation in terms of the new variable 4,). This mapping can be achieved using the bilinear transformation07  

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The examples below illustrate the use of the bilinear transformation to analyze the stability of sampled-data systems. We can use all the classical methods that we are used to employing in the s plane. The price that we pay is the additional algebra to convert to ID from z. 

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