Root locus curve

We will discuss how to solve for the roots of this cubic equation in the next section. The root locus curves are sketched in Fig. 10.7c, There are three curves because there are three roots. The root locus plot starts at the three openloop poles of the transfer function —1, —2,and... 

Root locus curves for a. three-CSTR system. 

Figure 10. l,e,/ gives root locus curves for three values of derivative time Tp with integral time t/ constant at 1.32. There are now five loci. They start at the poles of... 

Root locus curves for openloop unstable processes (positive poles). 

The root locus curves are shown in Fig. 11.10c. The loci start at the poles of the openloop transfer function . s = — 1 and 5 = — Since the loci must end at the zeros of the openloop transfer function (.t — + ) the curves swing over into the RHP. Therefore the system is closedloop unstable for gains greater than 2. 

The complete Nyquist plot is shown in Fig. 13.4o for several values of gain X,. Notice that the curves will never encircle the (—1,0) point, even as the gain is made infinitely large. This says that this second-order system can never be closedloop unstable. This is exactly what our root locus curves showed in Chap. 10. 

Definition / 8.4.2 Construction of Root Locus Curves 8.5 Conclusion Problems... 

Fig. 5.6. Root locus curve (real and imaginary axis) for active damping via proof mass actuators in the secondary mirror support truss (Ci percentage of critical damping)...

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. 

The effect of adding a lag or a pole is to pull the root locus plot toward the unstable region. The two curves that s ait ats=— Jand.s=—1 become complex conjugates and curve off into the RHP. Therefore this third-order system is closed-loop unstable if is greater than = 20. This was the same result that we obtained in Example 10.5,... 

It is useful to compare these values with those found for conventional control = 19.8 and oi. = 1.61. We can see that cascade control results in higher controller gain and smaller dosedloop time constant (the reciprocal of the frequency). Figure 11.26 gives a root locus plot for the primary controller with the secondary controller gain set at Two of the loci start at the complex poles s = rj which Come from the dosedloop secondary loop. The other curve... 

The entire curve is given in Fig. 19.10d. It crosses the negative real axis at = —0.087A,.. So the ultimate gain is = 1/0.087 = 11.6, which is the same result we obtained from a root locus analysis. 

As long as Kc satisfies inequality (15.2), we have two distinct real and negative roots. Therefore, the root locus is given by two distinct curves which emanate from points A and B and remain on the real axis. Furthermore, the two curves move toward each other and meet at point C (Figure 15.5). At this point, Kc has the value given by eq. (15.3) and we have a double root. 

For larger values of Kc satisfying inequality (15.4), we have again two distinct curves of the root locus because we have distinct, complex conjugate roots. Since the real part of the complex roots is constant (see eq. 15.5), the two branches of the root locus are perpendicular to the real axis and extend to infinity as Kc - oo. 

Figure 8.8 gives the root locus plot for a proportional controller. The three curves start at - 10 on the real axis (only one complex root is shown). Two of the loci go off at 60 angles and cross the imaginary axis at 17.3 (the ultimate frequency) vdien the gain is 6 (the ultimate gain). For a closedloop damping coefficient of 0.3 (the radial line), the closed-loop time constant is about 0.085 hours. 

The full curves in the /c-/r parameter plane, expressed by (4.46) and (4.47), are shown in Fig. 4.3. There are two closed loops emerging from the origin. The outer loop, given by the upper root in (4.46) and the lower root with 0 < 6SS < 1, also touches the k axis at (fi = 0, k = 1). Outside this locus the stationary state is a stable node inside this loop it is a focus (we will discuss stability within this region in the next subsection). The maximum in this curve occurs for k = (3 + y/S) exp[ — (3 — /5)] 1.787. For larger values of the dimensionless reaction rate constant, e.g. for high ambient temperatures, no damped oscillatory states will be found. 

A root of A(C) exists near r 36.7 s. The value of A(C) also appears to approach zero as the value of t approaches large values. This suggests that the equilibrium CSTR point also acts as a critical CSTR point to the Van de Vusse system. The curve in Figure 7.4 implies that there are exactly two concentrations that lie on the AR boundary. The remaining concentrations do not lie on the boundary and are thus not associated with an optimal reactor structure from Chapter 6. PFR trajectories initiated from the CSTR locus not associated with the two critical CSTR solutions therefore also do not form part of the manifold of PFR trajectories... 

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