Characteristic equation: sampled-data system

Consider the characteristic equation of a sampled-data system... 

The stability of a sampled-data system is determined by the location of the roots of a characteristic equation that is a polynomial in the complex variable z. This characteristic equation is the denominator of the system transfer function set equal to zero. The roots of this polynomial (the poles of the system transfer function) are plotted in the z plane. The ordinate is the imaginary part of z, and the abscissa is the real part of z. 

A sampled-data system is stable if all the roots of its characteristic equation (the poles of its transfer function) lie inside the unit circle in the z plane. 

It is sometimes convenient to write the closedloop characteristic equation of a general sampled-data system as... 

With sampled-data systems, root locus plots can be made in the z plane in almost exactly the same way. Controller gain is varied from zero to infinity, and the roots of the closedloop characteristic equation 1 - - = 0 are plotted. When... 

Using a proportional sampled-data controller, the closedloop characteristic equation for this system was... 

It can be shown that there is a Nyquist criterion for sampled data systems which is equivalent to that for continuous systems (see Section 7.10.5) and equation 7.131 can be applied in its comparable r-transformed form(42). In practice it is generally sufficient to ascertain whether the polar plot of G(z) in the complex z-plane encircles the (-1,0) point (as with continuous systems in the j-plane) where 1 + G(r) = 0 is the system z-transformed characteristic equation. The polar plot is constructed from... 

The closedloop characteristic equation for the second-order system with a proportional sampled-data controller is... 

Sampled-data control systems can be designed in the frequency domain by using the same techniques that we employed for continuous systems. The Nyquist stability criterion is applied to the appropriate closedloop characteristic equation to find the number of zeros outside the unit circle. 

The two constants K and n (n a 1) are characteristic of a given adsorption system. The Freundlich isotherm is derivable from certain rather artificial assumptions [e.g.. Ref. (10)], but it is essentially an empirical fitting function. Equation (3-7) commonly provides a good match for experimental adsorption data over a limited range in sample concentration. Systems which exhibit an extended linear isotherm region (and these are today of greatest interest in adsorption chromatography) are less well described by the Freundlich isotherm. 

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