How to Measure Percent Overshoot Using Siso Design Tool

Design Constraints

When designing compensators, it is common to have design specifications that call for specific settling times, damping ratios, and other characteristics. The SISO Design Tool provides design constraints that can help make the task of meeting design specifications easier. The New Constraint window, which allows you to create design constraints, automatically changes to reflect which constraints are available for the view in which you are working. Select Design Constraints and then New to open the New Constraint window, which is shown below.

Since each view has a different set of constraint types, click on the following links to go to the appropriate descriptions:

• Root locus
• Open-loop Bode diagram and prefilter Bode diagram (same)
• Nichols plot

Design Constraints for the Root Locus

For the root locus, you have the following constraint types:

• Settling Time
• Percent Overshoot
• Damping Ratio
• Natural Frequency

Use the Constraint Type menu to select a design constraint. In each case, to specify the constraint, enter the value in the Constraint Parameters panel. You can select any or all of them, or have more than one of each.

Settling Time. If you specify a settling time in the continuous-time root locus, a vertical line appears on the root locus plot at the pole locations associated with the value provided (using a first-order approximation). In the discrete-time case, the constraint is a curved line.

Percent Overshoot. Specifying percent overshoot in the continuous-time root locus causes two rays, starting at the root locus origin, to appear. These rays are the locus of poles associated with the percent value (using a second-order approximation). In the discrete-time case, In the discrete-time case, the constraint appears as two curves originating at (1,0) and meeting on the real axis in the left-hand plane.

Note that the percent overshoot (p.o.) constraint can be expressed in terms of the damping ratio, as in this equation.

where is the damping ratio.

Damping Ratio. Specifying a damping ratio in the continuous-time root locus causes two rays, starting at the root locus origin, to appear. These rays are the locus of poles associated with the damping ratio. In the discrete-time case, the constraint appears as curved lines originating at (1,0) and meeting on the real axis in the left-hand plane.

Natural Frequency. If you specify a natural frequency, a semicircle centered around the root locus origin appears. The radius equals the natural frequency.

Example: Adding Damping Ratio Constraints

This example add a damping ratio of 0.707 inequality constraint. First, type

• load ltiexamples sisotool(sys_dc)          

at the MATLAB prompt. This opens the SISO Design Tool with the DC motor example imported.

From the root locus right-click menu, select Design Constraints and then New to open the New Constraint window. To add the constraint, select Damping Ratio as the constraint type. The default damping ratio is 0.707. The SISO Design Tool should now look similar to this figure.

Damping Ratio Constraints in the Root Locus

The two rays centered at (0,0) represent the damping ratio constraint. The dark edge is the region boundary, and the shaded area outlines the exclusion region. This figure explains what this means for this constraint.

You can, for example, use this design constraint to ensure that the closed-loop poles, represented by the red squares, have some minimum damping. Try adjusting the gain until the damping ratio of the closed-loop poles is 0.7.

Design Constraints for Open-Loop and Prefilter Bode Diagrams

For both the open-loop and prefilter Bode diagrams, you have the following options:

• Upper Gain Limit
• Lower Gain Limit

Specifying any of these constraint types causes lines to appear in the Bode magnitude curve. To specify an upper or lower gain limit, enter the frequency range, the magnitude limit, and/or the slope in decibels per decade, in the appropriate fields of the Constraint Parameters panel. You can have as many gain limit constraints as you like in your Bode magnitude plots.

Upper Gain Limit. You can specify an upper gain limit, which appears as a straight line on the Bode magnitude curve. You must select frequency limits, the upper gain limit in decibels, and the slope in dB/decade.

Lower Gain Limit. Specify the lower gain limit in the same fashion as the upper gain limit.

Example: Adding Upper Gain Limits

This example shows you how to add two upper gain limit constraints to the open-loop Bode diagram. First, type

• load ltiexamples sisotool('bode',Gservo)          

at the MATLAB prompt. This opens the SISO Design Tool with the servomechanism model loaded. Use the right-click menu to add a grid.

First, add an upper gain limit constraint of 0 dB from 10 rad/sec to 100 rad/sec. This figure shows the New Constraint editor with the correct parameters.

Your SISO Design Tool should now look like this.

Now, to constraint the roll off, open the New Constraint editor and add an upper gain limit from 100 rad/sec to 1000 rad/sec with a slope of -20 db/decade. This figure shows the result.

With these constraints in place, you can see how much you can increase the compensator gain and still meet design specifications.

Note that you can change the constraints by moving them with your mouse. See Editing Constraints for more information.

Design Constraints for Open-Loop Nichols Plots

For open-loop Nichols plots, you have the following design constraint options:

• Phase Margin
• Gain Margin
• Closed-Loop Peak Gain

Specifying any of these constraint types causes lines or curves to appear in the Nichols plot. In each case, to specify the constraint, enter the value in the Constraint Parameters panel. You can select any or all of them, or have more than one of each.

Phase Margin. Specify a minimum phase amount at a given location. For example, you can require a minimum of 30 degrees at the -180 degree crossover. The phase margin specified should be a number greater than 0. The location must be a -180 plus a multiple of 360 degrees. If you enter an invalid location point, the closed valid location is selected.

Gain Margin. Specify a gain margin at a given location. For example, you can require a minimum of 20 dB at the -180 degree crossover. The location must be -180 plus a multiple of 360 degrees. If you enter an invalid location point, the closed valid location is selected.

Closed-Loop Peak Gain. Specify a peak closed-loop gain at a given location. The specified value can be positive or negative in dB. The constraint follows the curves of the Nichols plot grid, so it is recommended that you have the grid on when using this feature.

Example: Adding a Closed-Loop Peak Gain Constraint

This example shows how to add a closed-loop peak gain constraint to the Nichols plot. First, type

• load ltiexamples sisotool('nichols',Gservo)          

This opens the SISO Design Tool with Gservo imported as the plant. Use the right-click menu to add a grid, as this figure shows.

To add closed-loop peak gain of 1 dB at -180 degrees, open the New Constraint editor and select Closed-Loop Peak Gain from the pull-down menu. Set the peak gain field to 1 dB. The figure shows the resulting design constraint; use Zoom X-Y to zoom in on the plot for clarity.

As long as the curve is outside of the grey region, the closed-loop gain is guaranteed to be less than 1 dB. Note that this is equivalent, up to second order, to specifying the peak overshoot in the time domain. In this case, a 1 dB closed-loop peak gain corresponds to an overshoot of 15%.

Editing Constraints

To edit an existing constraint, left-click on the constraint itself to select it. Two black squares appear on the constraint when it is selected, and your mouse cursor turns into a large black cross (+). In general, there are two ways to adjust a constraint:

• Click on the constraint and drag it. This does not change the shape of the constraint. That is, the adjustment is strictly a translation of the constraint.
• Grab a black square and drag it. In this case, you can rotate, expand, and/or contract the constraint.

For example, in Bode diagrams you can move an upper gain limit by clicking on it and moving it anywhere in the plot region. As long as you haven't grabbed a black square, the length and slope of the gain limit will not change as you move the line. On the other hand, you can change the slope of the upper gain limit by grabbing one of the black squares and rotating the line. In all cases, the Status panel at the bottom of the SISO Design Tool displays the constraint values as they change.

This figure shows the process of editing an upper gain limit in the open-loop Bode diagram.

An alternative way to adjust a constraint is to select Design Constraints and then Edit from the right-click menu. The Edit Constraints window opens.

To adjust a constraint, select the constraint by clicking on it and change the values in the fields of the Constraint parameters panel. If you have additional constraints in, for example, the Bode diagram, you can edit them directly from this window by selecting Open-Loop Bode from the Editor menu.

Deleting Constraints

To delete a constraint, place your cursor directly over the constraint itself. You cursor changes into a large `x'. Right-click to open a menu containing Edit and Delete. Select Delete from the menu list; this eliminates the constraints. You can also delete constraints by left-clicking on the constraint and then pressing the BackSpace or Delete key on your keyboard.

Finally, you can delete constraints by selecting Undo Add Constraint from the Edit menu, or pressing Ctrl+Z if adding constraints was the last action you took.

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How to Measure Percent Overshoot Using Siso Design Tool

Source: http://www.ece.northwestern.edu/local-apps/matlabhelp/toolbox/control/toolviewer/sisocs17.html

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