` `
This example considers the relationship between the locations of the closed-loop
poles for the standard second-order system and various time-domain
specifications that might be imposed on the system's closed-loop step response.
The open-loop and closed-loop transfer functions for the standard second-order
system are:

` `We will
only consider the underdamped case in this example, so the damping ratio is less
than one. The specifications for the system's step response that are often used
are the percent overshoot and the
settling time. The specified values for percent overshoot are:

and for the settling time (in seconds), the specifications are

There will also be a specification on the frequency of oscillation in the step response. This specification is:

The goal is to determine the region in the *s*-plane where
the closed-loop poles must lie in order to satisfy the specifications. Since the
system is underdamped, the closed-loop poles are complex conjugates, so only one
of the poles needs to be considered.

`
`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:

` `
First we will deal with the settling time specifications. The relationship
between settling time and the closed-loop poles is

where *p*_{1} is one of the closed-loop poles (we
will assume that it is the one with positive imaginary part). Therefore, a
settling time specification imposes a limit on the real part of the complex
conjugate closed-loop poles. This is equivalent to requiring the closed-loop
poles to lie on a specified vertical line in the *s*-plane. Since there is
an inverse relation between the settling time and the real part of the pole,
increasing the settling time reduces the absolute value of the real part of the
closed-loop pole, and vice versa. If an upper limit is imposed on settling time,
this means that the closed-loop pole has to lie on or to the left of a vertical
line located in accordance with the above equation. If a lower limit is imposed
on settling time, this means that the closed-loop pole has to lie on or to the
right of a specified vertical line. For our example, the constraints on the real
part of the closed-loop poles are

Therefore, the closed-loop poles must lie in the vertical strip given by the real part of the poles between -2 and -8. No constraint is imposed on the imaginary part of the poles or on the relationship between real and imaginary parts of the pole by the settling time specification.

` `
The percent overshoot specification will be tackled next. Percent overshoot is
only a function of the damping ratio. The angle that the closed-loop pole makes
relative to the negative real axis is also a function only of the damping ratio.
For a given percent overshoot, the damping ratio and angle can be computed from

where the angle beta is measured in a clockwise direction from the negative real axis. Small overshoots require large damping ratios which correspond to small angles. Large overshoots allow small damping ratios and have large angles. For this example, the values are

For each allowed value of overshoot, constraints on the damping ratio and angle are imposed. The closed-loop pole must lie on a radial line drawn from the origin at the specified angle, so a relationship between the real and imaginary parts of the closed-loop pole is established. For a maximum overshoot, a minimum damping ratio is specified and this corresponds to a maximum angle. Therefore, for a maximum overshoot, the closed-loop pole must lie on or below the radial line at the required angle. For a minimum overshoot, the pole must lie on or above the corresponding radial line. Therefore, if both minimum and maximum overshoots are specified, the closed-loop pole must lie in the cone-shaped section between the two lines. For this example, the constraints on the damping ratio and angle are

The
relationship between the real and imaginary parts of the closed-loop pole is
given by the tangent of the angle beta. For this example, the real and imaginary
parts of *p*_{1} must satisfy the following expression.

` `
The last specification in this example is the upper bound on the frequency of
oscillation in the step response. The relationship between this parameter and
the closed-loop pole *p*_{1} is

Thus,
the specification on the damped natural frequency is directly a specification on
the imaginary part of the closed-loop pole. An upper bound on that parameter
means that the closed-loop pole must lie on or below the corresponding
horizontal line at the specified value for the imaginary part of *p*_{1}.

` `
The final, overall requirements on the locations of the closed-loop pole *p*_{1}
can be determined by considering all of the constraints together. These are:

The maximum settling time means that the closed-loop pole must lie on or to the left of a vertical line at

s= -2;The minimum settling time means that the closed-loop pole must lie on or to the right of a vertical line at

s= -8;The maximum overshoot means that the closed-loop pole must lie on or below a radial line at an angle of 66.2 deg, which means that the slope of a line from the origin to

p_{1}must be less than or equal to 2.2673;The minimum overshoot means that the closed-loop pole must lie on or above a radial line at an angle of 46.4 deg, which means that the slope of a line from the origin to

p_{1}must be greater than or equal to 1.0501;The maximum frequency of oscillation means that the closed-loop pole must lie on or below a horizontal line at

s=j12.6.

These constraints can be displayed graphically to show the
acceptable
locations for the closed-loop pole *p*_{1}. The dotted area in
the intersection of the 5 constraint curves are the acceptable locations for
that pole. Any interior point or a point on the boundary is a valid pole
location. The other pole will be located at its complex conjugate.

As an
example, choose the pole location to be *p*_{1} = -5+*j*7.5.
This is in the acceptable region. The settling time should be 4/5 = 0.8 seconds
which is acceptable. The frequency of oscillation will be 7.5 r/s which is
acceptable. The angle of the radial line through that pole is atan2(7.5, 5) =
56.31 degrees, so the damping ratio is cos(56.31) = 0.5547. This yields an
overshoot of 12.3% which is acceptable. The undamped natural frequency is 9.0139
r/s. Since all specifications are satisfied, the choice of a closed-loop
pole at *s* = -5 + *j*7.5 is a valid choice. The
graph
of the closed-loop step response shows the time-domain results of our
choice.

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*Lastest revision on
Wednesday, June 7, 2006 12:09 PM
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