Closed-loop frequency response

5 Relationship between open-loop and closed-loop frequency response 

Bandwidth (wb) This is the frequeney at whieh the elosed-loop modulus M has fallen by 3dB below its zero frequeney value. (This is not true for systems that do not operate down to de levels.) 

Peak frequency (cup) This is the frequeney that the peak modulus oeeurs. Note that 

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Second-order system closed-loop frequency response... 

The M and N circles can be superimposed on a Nyquist diagram (called a Hall chart) to directly obtain closed-loop frequency response information. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Fig. 6.25 Closed-loop frequency response from Nyquist diagram using the direct construction method.

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. 

Fig. 6.29 Closed-loop frequency response showing best flatband response (curve (b)) and response with...

Fig. 6.36 Closed-loop frequency response for lead compensator one.

Fig. 6.39 Closed-loop frequency response for both lead compensator designs.

Running script file fig629.m will produce the closed-loop frequency response gain diagrams shown in Figure 6.29 for Example 6.4 when K = 3.8 and 3.2 (value of K for best flatband response). 

The command cloop is used to find the closed-loop transfer function. The command max is used to find the maximum value of 20 logio (mag), i.e. Mp and the frequency at which it occurs i.e. tUp = uj k). A while loop is used to find the —3 dB point and hence bandwidth = ca (n). Thus, in addition to plotting the closed-loop frequency response gain diagrams,/ gd29.7 will print in the command window ... 

A generally used set of criteria for good control is that the controlled variable in response to a unit step change in set point (a) overshoot by not more than 20 per cent of the step and (b) damp out with a subsidence ratio of about one-third. This behavior is approximated by many systems if the closed-loop frequency response and the corresponding open-loop frequency response have certain simple characteristics. Since the closed-loop frequency response characteristics can be determined readily from the open-loop frequency response, the latter characteristics of simple control systems can be used as a convenient basis for design. 

To verify the predictions of the control structure selection method, controllers were designed for all four structures, and the closed-loop behaviors were simulated. The controllers were obtained by applying the frequency response approximation method developed by Engell and Muller [9, 42]. In this approach, the frequency response of a controller that achieves the specified performance exactly is approximated by a low-order, possible structured controller. The optimization is performed with respect to the approximation of the desired closed-loop frequency response, not by approximation of the controller response. As the desired performance (specified closed-loop frequency response) for the frequency response approximation, the results of the attainable performance calculation as listed in Table 6 were used. 

Oiur objective is to find the PID controller parameters such that the actual closed-loop frequency response is in some sense close to the desired closed-loop fi quency response Gr- y jw). However, the direct approach to this problem leads to a nonlinear optimization problem. Instead, we choose to work with the equivalent open-loop transfer function because, in this case, the problem becomes linear in the controller parameters, enabling us to consider a linear least squares approach to solving this problem. 

Figure 6.8 Comparison of the desired and actual closed-loop frequency responses using low frequency region (solid desired dash-dotted actual)...

Assuming that the sampling instants correspond to to = 0, ti = At,. .., tjv-i = N — l)At, where N = we know from Equation (5.4) that the following relationship exists between the closed-loop step response and the closed-loop frequency response... 

Step 3 Calculate the frequency response G jw) and Gr- uijw) at wi = and W2 = to form the desired closed-loop frequency response... 

Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)... 

J.5 Closed-Loop Frequency Response and Sensitivity Functions J.5.1 Sensitivity Functions J.5.2 Bandwidth... 

J.5 CLOSED-LOOP FREQUENCY RESPONSE AND SENSITIVITY FUNCTIONS... 

The previous sections have demonstrated that open-loop frequency response characteristics play a key role in the analysis of closed-loop stability and robustness. However, the closed-loop frequency response characteristics also provide important information, as discussed in this section. 

Next, we consider the desired closed-loop frequency response characteristics for disturbances. Ideally, we would like to have the closed-loop amplitude ratio 5(7 a)) be zero for all frequencies. However, this ideal situation is physically impossible for feedback control, and thus a more realistic goal is to minimize 5(7(o) over as wide a frequency range as possible. According to the guideline, the controller should be designed so that 1.2 [Pg.587]

The closed-loop frequency response can be calculated analytically from the open-loop frequency response. Again, consider Fig. J.l, and assume that the sensor dynamics are negligible so that G s) = From Fig. J.l... 

The Nichols chart in Fig. J.15 provides a graphical display of the closed-loop frequency response characteristics for set-point changes when G is) = Kjn Contours of constant AR l and ql are shown on a plot of ARol vs. OL- In a typical Nichols chart application, ARq and ql are calculated from Gol s) and plotted on the Nichols chart as a series of points. Then AR l and CL are obtained by interpolation. For example, if ARol 1 and ql = -100° at a certain frequency, then interpolation of Fig. J.15 gives ARcl = 0.76 and CL = 50° for the same frequency. The Nichols chart can be generated in the MATLAB Control Toolbox by a single command, nichols. 

Frequency response techniques are powerful tools for the design and analysis of feedback control systems. The Bode and Nyquist stability criteria provide exact stability results for a wide variety of control problems, including processes with time delays. They also provide convenient measures of relative stability, such as gain and phase margins. Closed-loop frequency response concepts such as sensitivity functions and bandwidth can be used to characterize closed-loop performance. 

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