Procedure for sketching bode plots (mentioned on Oct 5 th notes, Pg. 20) 1. Rewrite the transfer function in proper p form. 2. Separate the transfer function into its constituent parts. 3. Draw the Bode diagram for each part. (you work from a table) 4. Draw the overall Bode diagram by adding up the results from step 3. Good walkthrough (used as class handout for today): http://www.ece.utah.edu/~ee3110/bodeplot.pdf 1
Work through a few examples of this technique from the handout (like this one, from pg. 6)
When in doubt, just plug in numbers and see whether h your magnitude graph is correct. You can plug in actual numerical values of s into H(s) to get estimates for the magnitude plot This doesn t work for the phase plot (would involve calculating complex number values, not worth the effort)
So we learned how to sketch Bode Plots by hand d( (when we could have used a computer program to generate sketches). Why exactly? Can more clearly see how individual terms affect system performance and system output. Easier to understand why different controllers are used to compensate for specific shortcomings and specific terms in the system. 4
Defining stability Interpreting stability from the magnitude and phase plots 5
Stability definition A stable system is a dynamic system with a bounded response to a bounded input. E.g. a sine wave input results in a sine wave output that doesn t grow unbounded until the system finally breaks. Resonance is still acceptable (the output can become very large, as long as it does not become unbounded) 6
From Oct 6 th session, pg. 9: Systems become unstable in negative feedback loops with a negative loop gain. X(s) + G(s) Y(s) X(s) G( ( ) s 1 G( s) F( s) Y(s) F(s) overall transfer function What happens when GF gets to be close to 1? H gets bigger and bigger, becoming an unstable system. H Y X G 1 GF GF can become 1 depending on the gain and phase of the system. 7
Stability analysis for open loop usually has F(s) =1 (unity gain) in the feedback loop. Most of the time (80% of the time?), stability analysis in control theory assumes that the feedback gain is a constant value of 1. This simplifies analysis to just looking at the true open loop gain G(s): X(s) G(s) Y(s) + 1 X(s) G( s) 1 G( s) Y(s) overall transfer function H Y X G 1 G H becomes unstable if G becomes a value of 1 G is called the open loop gain since G is the gain in the system if the feedback kloop is removed. 8
Stability comes from avoiding G(s)= 1 G(s) = 11 when magnitude plot = 1 = 0 db, and phase plot* = 180 0 A stable system is one where the open loop gain is less than 1 when the open loop phase angle is 180 0 An unstable system is one where the open loop gain is greater than 1 when the open loop phase is 180 0 (since this guarantees that at some point, the value G(s) () would be equal to 1**)) Aside: ( * the word phase is from the fact that s, H(s), and G(s) are all really complex number (with real and imaginary components), which can be represented with a magnitude and phase value ) ( **actually, this is only the real part of G(s) which equals 1, since G(s) is a complex # with real and imaginary parts, but really that doesn t matter in this analysis, it still 9 guarantees that the system will become unstable at some point )
Aside: Why analyze the open loop response instead of the closed loop l response directly? X(s) H(s) G( s) 1 G( s) Y(s) Why not analyze this entire expression H(s) instead of just analyzing G(s)? Because G(s) remains bounded and stable even as the entire expression Because G(s) remains bounded and stable even as the entire expression H(s) becomes unstable. We can use Bode plots and straight line asymptotes to examine G(s) even when the system becomes unstable, but we don t have any tools to accurately analyze H(s) as the entire expression becomes unbounded and unstable.
You check stability by looking at the gain and phase margin of the open loop system From Oct 5 th, Pg. 23 Magnitude plot Gain Margin the amount of gain that you could add to the system before reaching a gain of 1 = 0 db. Measured at the phase crossing frequency (phase = +/ 180 0 ) 0db Phase Margin the amount of phase you could add/subtract to the system before reaching +/ 180 0. Measured at the gain crossing frequency (gain = 1 = 0dB) 0 Phase plot W gain_cross W phase_cross W gain_cross = gain crossing (frequency at which gain = 0dB, and phase margin is measured) W phase_cross = phase crossing (frequency at which phase = +/ 180deg, gain margin is measured) 180 Plot from lecture notes, 8B, pg. 12 11
What can you do with ihthe gain and phase margin? Tells you how much proportional gain (k) you can add to the system (to make the system operate faster) without becoming unstable. Tells you how you might select and design a controller to compensate for aspects of the system (e.g. PID, lag/lead compensators) to increase stability or improve performance 12