ECE 421

Second-Order System Example #2



This example examines the effects that adding either a pole or a zero to the open-loop system has on the step response of the standard second-order system. The open-loop and closed-loop transfer functions of the standard second-order system are shown below, and the step response for damping ratio = 0.5 and undamped natural frequency = 4 r/s is shown. The closed-loop poles are located at s = -2 +/- j3.46.
Step Response: zeta = 0.5, wn = 4 r/s

The standard second-order system has no zeros in the transfer function. If the system is modified to include a zero in it, the overshoot and settling time are greatly affected by the location of that zero. Assume that the open-loop system is modified as shown below by the addition of a zero. The gain of the transfer function is also modified by the 1/z term so that the velocity error constant is not changed from its original value as the zero is moved. This normalizes the problem that is being studied. The values of zeta and wn will be unchanged.

Six different values will be used for the location of the zero. Each of them places the zero in the left-half of the s-plane. The locations of the zero are: s = -0.2, -0.5, -1, -2, -5, -10. The two open-loop poles are fixed at s = 0 and s = -4. The step responses for the closed-loop system with the different values for z are shown in the next figure. They should be compared with that of the standard second-order system. The locations of the closed-loop poles are given in the following table. Compare the real value of the pole closest to the jw axis with the settling time for the corresponding step response. Also note that for the first four values of z (0.2, 0.5, 1, 2), both closed-loop poles are real and there is no overshoot in the step response. This is similar to the overdamped case for the standard second-order system. For the last two values of z (5, 10), the closed-loop poles are complex conjugates and there is some overshoot in the step reponse. For z=5, the effective damping ratio of the complex poles is 0.9, which would correspond to an overshoot of 0.15%. For z=10, the effective damping ratio of the complex poles is 0.7, which would correspond to an overshoot of 4.6%. Although the actual overshoots are larger than that in each case, the concept of relating the time domain step response characteristics with the effective damping ratio and natural frequency still gives a good "ball park" figure. The rise time increases as the zero is moved further to the left in the s-plane. When the zero is close to the jw axis, one of the closed-loop poles is also close to the jw axis, and the settling time is longer than in the standard case.
Step Response with Added Zero
Table of Closed-Loop Pole Locations with Added Zero

The standard second-order system has two poles in the transfer function. If the system is modified to include a third pole in it, the overshoot and settling time are again greatly affected by the location of that pole, particularly in terms of whether it is to the left or the right of the original open-loop pole at s = -4. Assume that the open-loop system is modified as shown below by the addition of a third pole. The gain of the transfer function is also modified by the p term so that the velocity error constant is again unchanged from its original value.

Five different values will be used for the location of the extra pole. Each of them places the pole in the left-half of the s-plane. The locations of the pole are: s = -1, -2, -5, -10, -20. The two original open-loop poles are still fixed at s = 0 and s = -4. The step responses for the closed-loop system with the different values for p are shown in the next figure. They should be compared with that of the standard second-order system. The locations of the closed-loop poles are given in the following table. Note that there are now three closed-loop poles. Again, compare the real value of the pole closest to the jw axis (complex in each case) with the settling time for the corresponding step response. Also compare the ratio of imaginary part to real part for the closed-loop poles closest to the jw axis. This ratio is related to the effective damping ratio by zeta_eff = cos(atan(imag(clp)/abs(real(clp)))). Note that the overshoot and settling time decrease as the extra pole is moved to the left. However, even with the extra pole at s = -20, the overhoot and settling time are still larger than the original second-order system.
Step Response with Added Pole
Table of Closed-Loop Pole Locations with Added Pole

The ability to analyze the behavior of higher-order systems and systems with zeros based on second-order approximations is very useful. Although the calculations developed for the standard second-order system will not yield exact results for other system, they can provide an acceptable approximation in many cases. The most accurate calculation will be for settling time; that will be 4 divided by the absolute value of the pole (or real part of complex poles) closest to the jw axis in most cases. This is called the dominant pole (or pair of poles).

MATLAB Code

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Lastest revision on Wednesday, June 7, 2006 12:05 PM