Plane Root Locus Plots

Lines of constant damping coefficient C in the s plane are radial lines from the origin as sketched in Fig. 19.2. 

Using Eq. (19.15), the real part of s, cc, can be expressed in terms of the imaginary part of s, CO, and the damping coefficient 

These lines of constant damping coefficient can be mapped into the z plane. The z variable along a line of constant damping coefficient is 

Lines of constant damping coefficient in the z plane can be generated by picking a value of C and varying a in Eq. (19.17) from 0 to oojl. See Sec. 19.4 for a discussion of the required range of tu. 

Notice the very significant result that the damping coefficient is less than one on the negative real axis. This means that in sampled-data systems a real root can give underdamped response. This can never happen in a continuous system the roots must be complex to give underdamped response. 

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Figure 19.7h,c compare z-plane and log-z-plane root locus plots for first-order and second-order systems. For the first-order system, the single root moves to minus infinity in the log-z plane as z goes to zero. Then the root comes back... 

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. ... 

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. 

In the simplest scenario, we can think of the equation as a unity feedback system with only a proportional controller (i.e., k = Kc) and G(s) as the process function. We are interested in finding the roots for different values of the parameter k. We can either tabulate the results or we can plot the solutions s in the complex plane—the result is the root-locus plot. 

There is one root and there will be only one curve in the s plane. Figure 10.6 gives the root locus plot. The curve starts ats = — 1/rg when X, = 0. The dosedloop root moves out along the negative real axis as K, is increased. 

Note that the effect of adding a zero or a lead is to pull the root locus toward a more stable region of the s plane. The root locus starts at the poles of the open-loop transfer function. As the gain goes to infinity the two paths of the root locus go to minus infinity and to the zero of the transfer function at s = -2. We will find that this is true in general the root locus plot ends at the zeros of the openloop transfer function. 

When a system has poles that are widely different in value, it is difficult to plot them all on a root locus plot using conventional rectangular coordinates in the s plane. U is sometimes more convenient to make the root locus plots in the log s plane. Instead of using the conventional axis Re s and Im s, an ordinate of the arg s and an abscissa of the log s arc used, since the natural logarithm of a complex number is defined ... 

A process has a positive pole located at (-1-1,0) in the s plane (with time in minutes). The process steadystate gain is 2. An addition lag of 20 seconds exists in the control loop. Sketch root locus plots and calculate controller gains which give a dosedloop damping coeHicient of0.707 when... 

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. 

Figure 19.4h shows the root locus plot in the z plane. There are now three loci. The ultimate gain can occur on either the real-root path (at z = — 1) or on the complex-conjugate roots path. We can solve for these two values of controller gain and see which is smaller. 

Instead of making root locus plots in the z plane, it is sometimes convenient to make them in the log-z plane. In the z plane, the ordinate is the imaginary part of z and the abscissa is the real part of z. In the log-z plane, the ordinate is the... 

The bilinear transformation is another change of variables. We convert from the z variable into the lU variable. The transformation maps the unit circle in the z plane into the left half of the ID plane. This mapping converts the stability region back to the familiar LHP region. The Routh criterion can then be used. Root locus plots can be made in the 11 plane with the system going closedloop unstable when the loci cross over into the RHP. 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

Sampled-data controllers can be designed in the same way continuous controllers are designed. Root locus plots in the z plane or frequency-response plots are made with various types of >(z) s (different orders of M and N and different values of the a, and 6, coefficients). This is the same as using different combinations of lead-lag elements in continuous systems. 

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. 

Make a root locus plot for the process considered in Problem 15.7 in the z plane. 

If a digital proportional controller and zero-order hold are used, derive HGm(z)-Sketch a root locus plot in the i plane and find the ultimate gain and ultimate frequency. 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

From Eq. (7.174), we see that the ideal terms obtained with infinite GB have been corrupted by additional terms caused by finite GB. To see the effects graphically, we select several values of Q and then factor the polynomial in Eq. (7.175) for many values of GB in order to draw loci. The polynomial in Eq. (7.175) is fifth order, but three of the roots are in the far left-half normalized s plane. The dominant poles are a pair of complex poles that correspond to the ideal poles in Eq. (7.172) but are shifted because of finite GB. Figure 7.117 shows the family of loci generated, one locus for each value of Q selected. Only the lod of the dominant second quadrant pole are plotted. The other dominant pole is the conjugate. 

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