There is still steady-state error, but that can be reduced with a lag controller later. The overshoot and settling time are both smaller. Inspection of the above demonstrates that this is a much better step response. Title( 'Step Response w/ Notch Filter and Loop Gain K= 200') The closed-loop step response can then be generated using the following code. When prompted to select a point, pick a point near one of the crosses in the following root locus. ![]() = rlocfind(C_notch*P) at the command line. If we close this loop (using the rlocfind command to find a specific gain), and inspect the step response, the system response should be much improved. Furthermore, higher damping ratio can be achieved using lower gains ( zeta term will be larger). This means that a larger gain, K, can be employed, while maintaining stability. The complex poles near the imaginary axis have been nearly canceled and more of the root locus is now in the left half plane. Title( 'Root Locus of Plant with a Notch Filter') Given below and run at the command line, we will generate a root locus plot like the one shown below. If we implement this controller employing the code The plant the denominator of the controller introduces two poles at -10. Try the following compensator.Īs you can see, the roots of the numerator of the controller are almost the same as the complex poles of the denominator of Approximate cancellation will give us many of the desirable characteristicsįor the example above, let's try placing the controller zeros slightly to the left of the lightly-damped plant poles (it isĪ good idea to pull the poles to the left instead of to the right). Before getting into the specifics of a notch filter, it should be noted that due to the nature of most systems, exact pole/zero cancellation cannot be obtained nor should it be attempted. Such a controller is called a notch filter. The dominant closed-loop poles of the system can then be placed in a ![]() These zeros can attenuate the effect of these poles. One way to control this system is to design a controller with zeros near the undesirable, lightly-damped poles of the plant. Proportional control is obviously not a good way to control this system. See that the response initially improves slightly, but becomes unstable before a desirable response can be achieved. If you try to increase the gain, you will There is a large overshoot, long settling time, and a large steady-state error. Title( 'Step Response w/ Complex Poles Near the Imaginary Axis') Plane (for a gain of one), you can see that the response is poor. By closing the loop and plotting the step response for this system in the portion of the root locus that is in the left-half ![]() ![]() The portion that is stable will be only lightly damped (small zeta). Title( 'Root Locus w/ Complex Poles Near the Imaginary Axis') Īs you can see, the plot shows that this system is only stable for a small region of the root locus (range of gains K). If you were to generate the root locus of this system using the following code. This will result in an undesirable closed-loop system that is unstable or only lightly damped. That lie close to the imaginary axis in the s-plane. There are many times when the transfer function of a controlled process contains one or more pairs of complex-conjugate poles
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