< Day Day Up > |
4.4 Bode Design
The Bode design method, also called the frequency response method, is probably the most common control system design approach (other than PID control) used in industry [8]. It is popular because this approach results in good controller designs even when there is significant uncertainty in the plant model.
Bode design is based on the use of open-loop frequency response plots of the plant plus controller. The open-loop system configuration from which this data is generated appears in Figure 4.10. The only difference between Figure 4.9 and Figure 4.10 is that the feedback path and the summing junction have been removed.
Figure 4.10: Open-loop system configuration used in Bode design.
The MATLAB Control System Toolbox provides limited support for the Bode design approach. You can view Bode diagrams within the SISO Design Tool and use those diagrams in the design process. You can also view the gain margin and phase margin (defined in Section 4.4.1) of your design as you make changes. The Bode diagrams provide a useful alternate view of the system when working with a root locus design.
The default behavior of the SISO Design Tool is to display both a root locus plot and a set of Bode diagrams. Figure 4.11 shows the resulting dialog when the following commands are issued. Note that the only difference from the method previously used to start the SISO Design Tool is that the rlocus argument to the sisotool() command has been omitted.
Figure 4.11: SISO Design Tool displaying root locus and Bode views.
>> tfplant = tf(100, [1 1.8 100])
>> sisotool(tfplant)
The Bode diagrams on the right side of the SISO Design Tool display the magnitude and phase of the open-loop frequency response on the upper and lower plots, respectively. The horizontal axis frequency range is automatically selected on the basis of the plant model characteristics. The magnitude plot is in units of decibels, and the phase plot is in degrees.
Together, the Bode diagrams indicate the steady-state amplitude and phase of an output sine wave relative to an input sine wave across the frequency range shown on the horizontal axis.
4.4.1 Phase Margin and Gain Margin
In a stable system, all closed-loop poles are in the left half of the complex plane. The point of neutral stability occurs when one or more of the poles are located directly on the imaginary axis. Instability results when any closed-loop pole moves into the right half of the plane.
It is valuable to understand "how far" a design is from instability. This notion is captured in the concept of stability margin. Two measures of the stability margin are directly available from the open-loop Bode diagrams: gain margin and phase margin.
In a neutrally stable system, the open-loop gain is unity (0dB) at the frequency where the open-loop response is 180° out of phase from the input. If the gain at that frequency is less than 0dB, the system is stable. A gain greater than 0dB at that frequency produces an unstable system.
On the Bode diagrams, the frequency of interest for this test is where the phase plot has a value of -180°. For a stable system, the gain curve passes downward through 0dB at a frequency where the phase plot has a value greater than -180°. The phase margin is defined as the number of degrees the phase curve is above -180° at the frequency where the gain plot passes through 0dB.
The SISO Design Tool automatically determines the phase margin, displays it on the plot, and gives its value in degrees. In Figure 4.11, the phase margin is indicated by the text "P.M.: 14.6 deg." The phase margin is shown on the phase plot as a small circle with a line drawn downward to the -180° value to indicate its magnitude. The frequency indicated by "Freq: 14 rad/sec" below the phase margin is the frequency at which the gain plot passes through 0dB.
The gain margin is defined as the negative of the open-loop gain (in dB) at the frequency where the phase is -180°. A stable system has a gain less than 0dB at that frequency, so the negative of that value will result in a positive gain margin. The gain margin indicates the amount by which the compensator gain can be increased before the system becomes neutrally stable.
In Figure 4.11, the phase never reaches -180°, no matter how large the controller gain becomes, so the gain margin for this system is infinite. This is indicated by the text "G.M.: Inf." The frequency indicated below the gain margin is the frequency at which the phase is -180° if it exists or "Inf" if no such frequency exists. The line of text below that indicates whether the system is stable or unstable. If the gain margin is finite, it will be drawn on the Bode gain plot in a manner similar to that of the phase margin.
Phase margin is used more frequently than gain margin as a specification and performance metric for control system design. Gain margin is not easily translated into a meaningful measure of system performance. The following two rules guide the application of a phase margin to control system design.
The damping ratio is approximately 0.01 multiplied by the phase margin in degrees.
For good transient response characteristics, the phase margin should normally be at least 60°. A value of less than 45° will tend to produce unacceptable overshoot and oscillation in the response.
After repeating the steps of the example in Section 4.3 that place the pole location constraints in the Root Locus Editor, add poles and zeros to the compensator, and adjust the compensator gain, the resulting compensator design appears in Figure 4.12. The phase margin is 69.5° and the gain margin is 20.6dB.
Figure 4.12: SISO Design Tool displaying root locus and Bode diagram views.
No comments:
Post a Comment