This example considers the design of a second-order system that will satisfy
certain time-domain specifications. This will be done by comparing the given
form for the open-loop (or closed-loop) transfer function with the corresponding
"standard" form for second-order systems. The given form and standard
form are:
The specifications for the system will be on the percent overshoot and the
settling time of the closed-loop system. The required values are:
Percent Overshoot = PO = 15%
Settling Time = Ts = 2 seconds
The goal is to determine the values of the gain K and the open-loop pole location p so that the specifications will be satisfied. Since there is a specified overshoot, the second-order system is underdamped.
The values for overshoot and settling time are related to the damping ratio
and undamped natural frequency given in the standard form for the second-order
system. The following relationships exist between the system parameters and the
specifications:
The two equations shown above can be solved to provide unique solutions for the
two parameters. First, the overshoot equation will be solved for the damping
ratio, and then the settling time equation can be solved for the undamped
natural frequency.
By comparing the given and standard forms of the transfer functions in either
open-loop or closed-loop, the necessary relations between the standard
parameters and the gain and open-loop pole location in the original system can
be obtained. These relations and the resulting values are:
Thus, the required transfer function for the system is
The step response for the closed-loop
system using these values should satisfy the specifications. The figure
shows that the overshoot specification is satisfied exactly and the settling
time is essentially satisfied also. The settling time shown in the figure was
obtained by searching the data array storing output values for the earliest time
when the output stays within 2% of the final value. Thus, the time shown for
settling time is a function of the resolution in the data and time arrays.
The ramp response is seen to have a
slope equal to the slope of the reference input signal (dashed curve).
Therefore, the two curves are parallel in steady-state. The steady-state error
in the ramp response depends on both the damping ratio and undamped natural
frequency according to
The open-loop poles (at s = 0 and -4) and the closed-loop
poles can be graphically displayed. The real part of the closed-loop poles
is midway between the two open-loop poles and is determined by the settling time
specification. This is always the case for the underdamped condition. The relation between the imaginary part and real part of the
closed-loop pole is determined by the percent overshoot specification through
the damping ratio. This can be seen from
where beta is the angle of the radial line from the origin passing through p1, and p1 is the closed-loop pole in the second quadrant of the s-plane. Thus, a settling time specification establishes a vertical line in the s-plane on which the closed-loop poles must lie, and an overshoot specification establishes a relationship between the real part and imaginary part of the closed-loop pole. The correct location for the closed-loop pole is that location that satisfies both of the specifications.
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Lastest revision on
Wednesday, June 7, 2006 12:08 PM
