Characteristic equation: sampled-data

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... 

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. 

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... 

Equation (11) represents the time-discrete dynamic equivalent of the steady-state balance equations (2). The dynamic balance equations (11) present some characteristic properties of the sampled-data input and output relationship, that are not present in the corresponding steady-state equations 1) There are as many equations as the number of outputs 2) Each equation contain only one output 3) Each equation contain, except for special cases, all the inputs variables. 

As can be seen from Table 1, the estimated coefficients b[0] are not equal to zero for different samples, whereas the estimated coefficients b[l] are close to 1 within confidence interval. That means that coefficients b[0] estimated for different points of the territory are generalized relative characteristics of elements abundance at the chosen sampling points. Statistical analysis has confirmed that hypotheses Hi and H2 are true with 95% confidence level for the data obtained by any of the analytical groups involved. This conclusion allowed us to verify hypothesis H3 considering that the estimated average variances of the correlation equation (1) are homogeneous for all snow samples in each analytical group. Hypothesis H3... 

The basic features of equations of state are not complicated when they are expressed as PV/RT versus density. Figure 3 is a sample plot for methanol. These curves are characteristic of all fluids, and equations of state only differ in their ability to accurately predict these curves. The actual curves are relatively simple and they change only slightly from one material to another for this reason, simple equations of state such as the Redlich-Kwong equation have been about as successful as the BWR equation. The simplicity of the actual curves is often hidden because the data are not usually plotted as PV/RT versus density. More often the data are plotted as PV/RT versus pressure shown in Figure 4, or pressure versus volume shown in Figure 5. Both of these plots obscure the real simplicity shown in Figure 3. 

Based on current knowledge of the process and its disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed initial parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the controller accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. 

Table I summarizes the experimentally determined resistivity characteristics for the nine formulations investigated. On the basis of Equation 6 and these data, a high degree of spray charge-ability by the electrostatic induction process could be predicted for all the pesticide samples tested. For these particular pesticide formulations, laboratory spray tests confirmed excellent droplet charging to greater than 10 mC/kg. Similar electrical resistivity measurements will serve as a suitable predictor of the chargeability of other formulations of interest in the electrostatic pesticide-spraying process.

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