Characteristic Loci Plots

The multivariable Nyquist plots discussed above give one curve. These curves can be quite complex, particularly with high-order systems and with multiple deadtlmes. Loops can appear in the curves, making it difficult sometimes to see if the (— 1,0) point is being encircled. 

A different kind of plot, called a characteristic loci plot, is sometimes easier to understand. The method is as follows  

Calculate the eigenvalues of. If the system is N x N (JV controlled variables and N manipulated variables) there will be N complex eigenvalues. The IMSL subroutine EIGCC is used in the program given in Table 16.2. 

Plot these N eigenvalues as frequency is varied from 0 to oo. 

ChAracterisric loci plots for Wood and Berry colomn 

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Characteristic loci plots for the Wood and Berry column are shown in Fig. 16.3. They show that the empirical controllers settings give a stable closedloop system, but the ZN settings do not since the eigenvalue goes through the (- 1,0) point... 

The idea of a root locus plot is simple—if we have a computer. We pick one design parameter, say, the proportional gain Kc, and write a small program to calculate the roots of the characteristic polynomial for each chosen value of as in 0, 1, 2, 3,., 100,..., etc. The results (the values of the roots) can be tabulated or better yet, plotted on the complex plane. Even though the idea of plotting a root locus sounds so simple, it is one of the most powerful techniques in controller design and analysis when there is no time delay. 

To determine the shape of a root locus plot, we need other rules to determine the locations of the so-called breakaway and break-in points, the corresponding angles of departure and arrival, and the angle of the asymptotes if the loci approach infinity. They all arise from the analysis of the characteristic equation. These features, including item 4 above, are explained in our Web Support pages. With MATLAB, our need for them is minimal. 

To begin with, this is a second order system with no positive zeros and so stability is not an issue. Theoretically speaking, we could have derived and proved all results with the simple second order characteristic equation, but we take the easy way out with root locus plots. 

From the characteristic polynomial, it is probable that we ll get either overdamped or underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that no matter what, both Zq and p0 should be larger than the value of l/xp in Gp. That s how we may "steer" the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.)... 

Here are some useful suggestions regarding root locus plots of control systems. In the following exercises, we consider only the simple unity feedback closed-loop characteristic equation ... 

Remember also that for gains greater than the ultimate gain, the root locus plot showed two roots of the closedloop characteristic equation in the... 

With continuous systems we made root locus plots in the s plane. Controller gain was varied from zero to infinity, and the roots of the closedloop characteristic equation were plotted. Time constants, damping coefficients, and stability could be easily determined from the positions of the roots in the s plane. The limit of stability was the imaginary axis. Lines of constant closedloop damping coefficient were radial straight lines from the origin. The closedloop time constant was the reciprocal of the distance from the origin. 

These parameters are varied to achieve some desired performance criteria. In the z-plane root locus plots, the specifications of closedloop time constant and damping coefficient are usually used. The roots of the closedloop characteristic equation 1 -I- are modified by changing. ... 

Figure 3.2 gives root locus plots for the two designs at reactor temperature of 350 K with the measurement lags included. You may remember that a root locus plot is a plot of the roots of the closedloop characteristic equation as a function of the controller gain Kc. The plots start (Kc = 0) at the poles of the openloop transfer function and end (Kc —> oo) at its zeros. 

Representing the roots of the characteristic equation in the complex domain offers a simple way to perform a stability analysis. The system is stable if and only if all the poles are located in the open left-half-plane (LHP). If there is at least one pole in the right-half-plane (RHP), the system is unstable. The representation is similar wiA the well-known root-locus plot used to evaluate the stability of a closed-loop system. 

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 + HGm z) (z) = 0 are plotted. When the roots lie inside the unit circle, the system is closedloop stable. When the roots lie outside the unit circle, the system is closedloop unstable. 

For a given value of lu, equation (6.9) represents a point in complex space P(lu). When LU is varied from zero to infinity, a locus will be generated in the complex space. This locus, shown in Figure 6.2, is in effect a polar plot, and is sometimes called a harmonic response diagram. An important feature of such a diagram is that its shape is uniquely related to the dynamic characteristics of the system. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

As with the continuous systems described in Chapter 5, the root locus of a discrete system is a plot of the locus of the roots of the characteristic equation... 

Root locus is a graphical representation of the roots of the closed-loop characteristic polynomial (i.e., the closed-loop poles) as a chosen parameter is varied. Only the roots are plotted. The values of the parameter are not shown explicitly. The analysis most commonly uses the proportional gain as the parameter. The value of the proportional gain is varied from 0 to infinity, or in practice, just "large enough." Now, we need a simple example to get this idea across. 

Admittance-plane plots are presented in Figure 16.6 for the series and parallel circuit arrangements shown in Figure 4.3(a). The data are presented as a locus of points, where each data point corresponds to a different measurement frequency. As discussed for the impedance-plane representation (Figure 16.1), the admittance-plane format obscures the frequency dependence. This disadvantage can be mitigated somewhat by labeling some characteristic frequencies. 

Thus a detector positioned at a scattering angle 0 will perform a scan in time whose locus is a parabola in (0,[Pg.112]

Figure 3.8 shows the conductivity Wessel diagram. Also here the characteristic frequency is >100 Hz, and in contrast to the permittivity plot with a complete semicircle locus, there is a strong deviation with the a" diverging proportional to... 

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