Tham khảo tài liệu 'essentials of control techniques and keyword stats_3', kỹ thuật - công nghệ, điện - điện tử phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Systems with Real Components and Saturating Signals 67 For the phase plane instead of plotting position against time we plot velocity against position. An extra change has been made in 6 . Whenever the drive reaches its limit the color of the plot is changed though this is hard to see in the black-and-white image of Figure . For an explanation of the new plot we will explore how to construct it without the aid of a computer. Firstly let us look at the system without its drive limit. To allow the computer to give us a preview we can remark out the two lines that impose the limit. By putting in front of a line of code it becomes a comment and is not executed. You can do this in the step model window. Without the limit there is no overshoot and the velocity runs off the screen as in Figure . Figure Phase plane with limit. Figure Phase-plane response without a limit. 68 Essentials of Control Techniques and Theory To construct the plot by hand we can rearrange the differential equation to set the acceleration on the left equal to the feedback terms on the right x 5x 6x. If we are to trace the track of the position velocity coordinates around the phase plane it would be helpful to know in which direction they might move. Of particular interest is the slope of their curve at any point dx dx We wish to look at the derivative of the velocity with respect to x not with respect to time as we usually do. These derivatives are closely related however since for the general function f f_d f_ 1 df . dx dx dt x dt So we have Slope x. dx x But Equation gives the acceleration as x 5x 6x so we have Slope 5x 6x 5 6 . x x The first thing we notice is that for all points where position and velocity are in the same proportion the slope is the same. The lines of constant ratio all pass through the origin. On the line x 0 we have slope 5. On the line x x we have slope 11. On the line x x we have slope 1 and so on. We can make up a .
