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1. #HOT TO DRAW PHASE PLANES FOR DIFFERENTIAL EQUATION SYSTEMS HOW TO#
2. #HOT TO DRAW PHASE PLANES FOR DIFFERENTIAL EQUATION SYSTEMS CODE#

In MATLAB, a phase-lead compensator C( s) in frequency response form is implemented using the following code (where a and T are defined).

This can increase the crossover frequency, which will help to decrease the rise time and settling time of the system (but The lead compensator increases the gain of the systemĪt high frequencies (the amount of this gain is equal to a). Is designed by determining a from the amount of phase needed to satisfy the phase margin requirements, and determing T to place the added phase at the new gain-crossover frequency.Īnother effect of the lead compensator can be seen in the magnitude plot. The equation which determines the maximum phase is given below.Īdditional positive phase increases the phase margin and thus increases the stability of the system. The maximum amount of phase is added at the center frequency, which is calculated according to the following Depending on the value of a, the maximum added phase can be up to 90 degrees if you need more than 90 degrees of phase, two lead compensators in seriesĬan be employed. The two corner frequencies are at 1 / aT and 1 / T note the positive phase that is added to the system between these two frequencies. A Bode plot of a phase-lead compensator C( s) has the following form. In frequency response design, the phase-lead compensator adds positive phase to the system over the frequency range 1 /ĪT to 1 / T. Note that this is equivalent to the root locus form repeated below Lead or phase-lead compensator using frequency responseĪ first-order phase-lead compensator can also be designed using a frequency reponse approach. We can interconnect this compensator C( s) with a plant P( s) using the following code. In MATLAB a phase-lead compensator in root locus form is implemented using the following commands (where Kc, z, and p are defined). Of stability and the system's response speed.

The left in the complex plane, and the entire root locus is shifted to the left as well. Thus, the result of a lead compensator is that the asymptotes' intersection is moved further to The net number of zeros and poles will be same (one zero and one pole are added), but the added pole is a larger negative When a lead compensator is added to a system, the value of this intersection will be a larger negative number than it wasīefore. To determine the intersection of the asymptotes along the real axis is the following. How is this accomplished? If you recall finding the asymptotes of the root locus that lead to the zeros at infinity, the equation This results in an improvement in the system's stability and an increase in its response speed. A phase-lead compensator tends to shift the root locus toward to the left in the complex s-plane. Where the magnitude of z0 is less than the magnitude of p0. A lead compensator in root locus form is given by Lead or phase-lead compensator using root locusĪ first-order lead compensator C( s) can be designed using the root locus.

#HOT TO DRAW PHASE PLANES FOR DIFFERENTIAL EQUATION SYSTEMS HOW TO#

The conversions page explains how to convert a state-space model into transfer function form. Lead, lag, and lead/lag compensators are usually designed for a system in transfer function form. One or more lead and lag compensators may be used in various combinations. Reponse of a system a lag compensator can reduce (but not eliminate) the steady-state error. A lead compensator can increase the stability or speed of Lead and lag compensators are used quite extensively in control.