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 p1 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 p1 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 p1 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 p1.
The final, overall requirements on the locations of the closed-loop pole p1 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 p1 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 p1 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 p1. 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 p1 = -5+j7.5. 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 + j7.5 is a valid choice. The graph of the closed-loop step response shows the time-domain results of our choice.
Click the icon to return to the Dr. Beale's home page
Lastest revision on
Wednesday, June 7, 2006 12:09 PM