In Root Locus Editor: Keep the mouse pointer on a closed-loop pole (squares) on the root locus. The arrow cursor changes to a hand. Hold down the left mouse button and drag the closed-loop pole along the root locus. Bode plot and the closed-loop response in the LTI Viewer will immediately change to reflect the gain change. The value of the gain will be displayed in the
section of the
SISO Design Tool window.
In Open-Loop Bode Editor: Keep the mouse pointer anywhere on the Bode magnitude curve. The arrow cursor changes to a hand. Hold down the left mouse button and shift the curve up or down. Root locus and closed-loop response in the LTI Viewer will immediately change to reflect the gain change.
In the Current Compensator Window: Type the desired gain value in the
box in the
SISO Design Tool window.
With gain changes, you can read the gain and phase margins and gain crossover and phase crossover frequencies at the bottom of the Bode magnitude and phase plots. Also, at the bottom of the Bode magnitude plot, you are told whether or not the closed-loop system is stable.
Fig. M7.20 Root locus and Bode plots of G in SISO Design Tool window
Fig. M7.21 LTI Viewer for SISO Design Tool window
5. Design constraints: Design constraints may be added to your plots. These constraints may be selected by right-clicking a respective plot and selecting Design Constraints. To put new constraints, choose New… and to edit existing constraints, choose Edit… . For example, Fig.M7.22 shows the selection of design constraint: damping ratio=0.5. On pressing OK, indicators appear identifying portions of the root locus where the damping ratio is less than 0.5(shaded gray), equal to 0.5(damping line), and greater than 0.5( Fig.M7.23 ). Note the change made in axes limits in this figure with respect to Fig.M7.20 using the Property Editor (the description of the Property Editor is given in the next step).
Constraints may also be edited on the plots. Two black squares appear on the constraint. You can drag these with your mouse anywhere in the plot region. Point the mouse at the boundary of the constraint. When it changes to four-pointed arrow, you can drag the boundary to a new position. The values of the constraints are displayed in the Status Bar at the bottom of the plots.
Fig. M7.22 Adding design constraints
Fig.M7.23 Adding design constraints
6. Properties: Right-clicking in a plot's window and selecting
Property Editor window. From this window, some of the properties of the plot such as axes labels and limits can be controlled. Try exploring various options available with root locus and Bode plot properties editor.
7. Add poles, zeros, and compensators: Poles and zeros may be added from the
SISO Design Tool
window toolbar shown in Fig. M7.23. Let your mouse pointer rest on the button for a few seconds to see the functionality of the button in the form of screen tips.
Add real pole; Add real zero;
Add complex pole; Add complex zero; Delete pole/zero; .....
functions are available.
SISO Design Tool toolbar
and select the desired real/complex pole/zero compensator. Move the mouse on the plots; your cursor shows that a compensator was selected.. Place the cursor arrow to the point on the root locus or Bode plot where you want to add the compensator, and click. The compensator will be updated in the
section of the
SISO Design Tool
window. Compensator addition will be reflected immediately in the root locus, Bode plots, and
LTI Viewer for SISO Design Tool
SISO Tool Preferences…
menu to change the way the compensator is represented.
8. Editing compensators and prefilters: The pole and zero values of the compensators and prefilters can be edited in several ways. The most convenient is to click on C or F blocks in the block diagram representation in top-right corner of the SISO Design Tool window (Fig. M7.23).This operation will open prefilter or compensator editor window shown in Fig. M7.24. Desired real/complex zero/pole locations can be edited here. The same windows can also be opened by following Compensators Edit C or F from the SISO Design Tool window.
In control systems design, we use compensators of the form
that alter the roots of the characteristic equation of the closed-loop system. However, the closed-loop transfer function, M(s), will contain the zero of D(s) as a zero of M(s).This zero may significantly affect the response of the system M(s). We may use a prefilter F(s) to reduce the effect of this zero on M(s). For example, if we select
we cancel the effect of the zero without changing the dc gain.
Prefilters may be employed with lead/PD compensation.
Fig. M7.24 Editing compensators and prefilters
Consider a plant with the following transfer function:
We will design a cascade controller using SISO Design Tool in interactive mode, so that the unity feedback closed-loop system meets the following criteria (refer Kuo and Golnaraghi(2003)):
Steady-state error due to unit-ramp input 0.000443
Maximum overshoot 5 percent
Rise time 0.002 sec
Settling time sec
The first step is to import the model into
SISO Design Tool. System transfer functions can be imported in SISO Design Tool by clicking on
File and then going to
Import… Before executing this sequence, create transfer function G in MATLAB Command Window:
den=[1 361.2 0];
In order to examine the system performance, we start by using a proportional controller. The system root locus can be obtained by clicking on
View in the main menu and then selecting
Root Locus only. Fig. M7.25 shows the root locus of the system. The plot is for K =1(by default).
In order to see the poles and zeros of
H , go to the
View menu and select
System Data, or alternatively, double click the block
H in the top-right corner of the block diagram in the
SISO Design Tool window. The
System Data window is shown in Fig.M7.26.
You may obtain the closed-loop system poles by selecting
Closed-Loop Poles from the
View menu. Closed-loop poles are given in Fig.M7.27.
In order to see the closed-loop system time response to a unit-step input, select the
Response to Step Command in the
Analysis main menu. With specific selections made in
Characteristics submenus, we generate Fig. M7.28, which shows the unit step response of the closed-loop system with unity gain controller, i.e. , K= 1.
As a first step to design a controller, we use the built-in design criteria option within the
SISO Design Tool to establish the desired closed-loop poles regions on the root locus. To add the design constraints, use the
Edit menu and choose the
Root Locus option. Select
New to enter the design constraints. The
Design Constraints option allows the user to investigate the effect of the following:
- Settling time
- Percent overshoot
- Damping ratio
- Natural frequency
We will use the settling time and the percent overshoot as primary constraints. After designing a controller based on these constraints, we will determine whether the system complies with the rise time constraint or not. Figure M7.29 shows the addition of settling time constraint. Peak overshoot constraint is added on similar lines.
Figure M7.30 shows the desired closed-loop system pole locations on the root locus after inclusion of the design constraints (Note that the scale has been modified by following
Right click --> Properties --> Limits.
Left-clicking anywhere inside plot will remove the black squares on the zeta lines). Obviously, closed-loop poles of the system for K = 1 are not in the desired area. Note the definition of the desired area: the vertical gray bar signifies the boundary for that portion of the root locus where the settling time requirement is not met; and the gray bars on damping lines signify the boundary for that portion of the root locus where the peak overshoot requirement is not met.
Changing K will affect the pole locations. In the
represents the controller transfer function. Fig.M7.30 corresponds to
C(s) = K =
. Hence, if
is increased, the effective value of K increases, forcing the closed-loop poles to move together on the real axis, and then ultimately to move apart to become complex. See Fig. M7.31 wherein K= 16.8 gives closed-loop poles on the damping lines. However, the settling time requirement is not met.
The closed-loop poles of the system must lie to the left of the boundary imposed by the settling time(Fig.M7.31). Obviously, it is impossible to use the proportional controller (for any value of K ) to move the poles of the closed-loop system farther to the left-hand plane. However, a PD controller C(s)= K (1+ sTD ) may be used to accomplish this task. The zero s = - 1/ TD of the compensator has to be placed far into the left half plane to move the root-locus plot to the left. We have tried various values:
s = - 1/0.0005; - 1/0.001; - 1/0.0015; - 1/0.002,...
The value s =
-1/0.00095 gives satisfactory results.
To add a zero to the controller, click the
C block in the block diagram in top-right corner of Fig. M7.31. Fig.M7.32 shows the
Edit Compensator window and how the PD controller is added:
Figure M7.33 shows the plot for s= - 1/0.00095. Note that the closed-loop poles have been dragged to the desired locations. The value of K that achieves the desired dominent closed-loop poles is 282. This value of K forces the closed-loop poles to the desired region. The system closed-loop poles are shown in Fig. M7.34.
The step response of the controlled system in Fig. M7.35 shows that the system has now complied with all design criteria.