Nichols chart

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. 

Figure 6.27 (see also Appendix 1, fig627.m) shows the Nichols chart for K = 4 (controller gain K = 1). These are the settings shown in the Bode diagram in Figure 6.23(a), curve (i), and (b), where... 

Fig. 6.28 Nichols chart showing best flatband response (curve (a)) and response with /Wp=3dB (curve (b)).

Fig. 6.33 Nichols chart for uncompensated laser guided missile.

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

The Nichols chart for the uncompensated and compensated system (curve (a)) is shown in Figure 6.34 (see also Appendix, fig634.m). From Figure 6.34, curve (a)... 

Plot the results between 0.1 and 2.0 rad/s on a Nichols Chart and determine... 

Plot these results on a Nichols Chart and adjust the compensator gain A lSO that the system achieves the required performance specification. 

This tutorial shows how MATLAB can be used to construct all the classical frequency domain plots, i.e. Bode gain and phase diagrams, Nyquist diagrams and Nichols charts. Control system design problems from Chapter 6 are used as examples. 

Script file fig627.m produces the Nichols chart for Example 6.4 when K = 4, as illustrated in Figure 6.27. The command ngrid produces the closed-loop magnitude and phase contours and axis provides user-defined axes. Some versions of MATLAB appear to have problems with the nichols command. 

Figure 6.34 is generated usmgfig634.m and shows the Nichols Chart for the uncompensated system. Curve (a) is when the compensator gain K =, and curve (b) is when K = 0.537 (a gain reduction of 5.4dB). 

Nichols Chart for Case Study Example 6.6 %Lead Compensator Design One, Figure 6.34 clf... 

Nichols chart Nichols chart is a frequency parametric plot of open-loop function magnitude vs. phase angle. The closed-loop magnitude and phase angle are overlaid as contours. 

With gain or phase margin, calculate proportional gain. Can also estimate the peak amplitude ratio, and assess the degree of oscillation. The peak amplitude ratio for a chosen proportional gain. Nichols chart is usually constructed for unity feedback loops only. 

A Nichols chart is a graph that shows what the closedloop log modulus and closedloop phase angle d, are for any given openloop log modulus Lq and openloop phase angle 0q. See Fig. 13.11a. The graph is a completely general one... 

Thus for any arbitrary system with the given opcnloop parameters 0(, and Lq, Eqs. (13 34) and (13.35) give the closedloop parameters 0 and L. The Nichols chart is a plot of these relationships. 

To use a Nichols chart, we first construct the openloop G, B Bode plots. Then we drawn an openloop Nichols plot of Finally we sketch this... 

Figure lillb is a Nichols chart with two G B curves plotted on it. They are from the three-CSTR system with a proportional controller. 

The lines of constant closedloop log modulus L, are part of the Nichols chart. If we are designing a closedloop system for an L specification, we merely have to adjust the controller type and settings so that the openloop B curve is tangent to the desired line on the Nichols chart. For example, the G B curve in Fig. 13,11b with X, = 20 is just tangent to the +2 dB line of the Nichols chart. The value of frequency at the point of tangency, 1.1 radians per minute, is the closedloop resonant frequency aif. The peak in the log modulus plot is clearly seen in the closedloop curves given in Fig. 13.12. 

Nichols chart with a three-CSTR system openloop Gmho plotted. 

Nichols chart with the B curve for this system. A gain of 20 makes the open-loop curve tangent to the +2-dB L, curve on the Nichols chart. 

Let us design a PI controller for a +2-dB L specification. For proportional controllers, all we have to do is find the value of that makes the Gjh B curve on a Nichols chart tangent to the -t-2-dB line. For a PI controller there are two parameters to find. Design procedures and guides have been developed over the years for finding the values of Tj. The procedure has the following steps ... 

Move the curve vertically on the Nichols chart until it is tangent to the -f2-dB line. Read off the resonant frequency m,. (Figure 13.lift shows w, 1.1 radians per minute.)... 

Both the openloop and the closedloop frequency response curves can be easily generated on a digital computer by using the complex variables and functions in FORTRAN discussed in Chapter 10 or by using MATLAB software. The frequency response curves for the closedloop servo transfer function can also be fairly easily found graphically by using a Nichols chart. This chart was developed many years ago, before computers were available, and was widely used because it greatly facilitated the conversion of openloop frequency response to closedloop frequency response. 

The graphical procedure for using a Niehols ehart is first to construet the Open-loop G Gc Bode plots. Then we draw an openloop Niehols plot of GM(ih))Gc icj)-Finally we sketch this openloop curve of Lq versus 0q onto a Nichols chart. At each point on this curve (which corresponds to a eertain value of frequency), the values of the elosedldop log modulus Lc ean be read off... 

Maximum closedloop log modulus. We have already designed (in Section 11.2.3) a proportional controller that gave of +2 dB. Figure 11.116 gives a Nichols chart with the G Gq curve for this system. A gain of 20 makes the open-loop GmGc curve tangent to the +2-dB Lc curve on the Nichols chart. 

Nichols plots are generated (Fig. 11.25). The ngrid command draws the Nichols chart curves of closedloop log modulus L,.. 

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