792 Chapter 10 / Control Systems Design in State SpaceFrequency (rad/sec)Bode Diagrams of Closed-Loop Systems− 3000− 100− 200100− 150− 100Phase (deg); Magnitude (dB)50− 50010 −^1100101102System 1System 2System 1
System 2Figure 10–34
Bode diagrams of
closed-loop system 1
(shown in Figure
10–29) and closed-
loop system 2 (shown
in Figure 10–31).Note that, in these two systems, the rise time and settling time are determined primari-
ly by the desired closed-loop poles for pole placement. (See Figures 10–32 and 10–33.)
The Bode diagrams of closed-loop system 1 (shown in Figure 10–29) and closed-
loop system 2 (shown in Figure 10–31) are shown in Figure 10–34. From this figure, we
find that the bandwidth of system 1 is 5 radsec and that of system 2 is 1.3 radsec.
Summary of State-Space Design Method
1.The state-space design method based on the pole-placement-combined-with-
observer approach is very powerful. It is a time-domain method. The desired closed-
loop poles can be arbitrarily placed, provided the plant is completely state
controllable.
2.If not all state variables can be measured, an observer must be incorporated to
estimate the unmeasurable state variables.
3.In designing a system using the pole-placement approach, several different sets of
desired closed-loop poles need be considered, the response characteristics
compared, and the best one chosen.
4.The bandwidth of the observer controller is generally large, because we choose
observer poles far to the left in the splane. A large bandwidth passes high-
frequency noises and causes the noise problem.
5.Adding an observer to the system generally reduces the stability margin. In some
cases, an observer controller may have zero(s) in the right-half splane, which
means that the controller may be stable but of nonminimum phase. In other cases,
the controller may have pole(s) in the right-half splane—that is, the controller is
unstable. Then the designed system may become conditionally stable.
6.When the system is designed by the pole-placement-with-observer approach, it is
advisable to check the stability margins (phase margin and gain margin), using a
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