There are a number of ways to analyse the stability of your system but the best way to determine how stable your system is, is to determine its Gain and Phase margins. In this course we’ll focus on two methods. The first method, which I prefer, uses a Bode Diagram of the open-loop system and in Matlab the margin function will plot the Bode Diagram and calculate the margins for you. The second method uses a Nyquist Plot of the open-loop system, and I think it was devised by Electrical Engineers to punish the world. Matlab has a nyquist function that draws the Nyquist Plot but you have to determine the margins from the plot. Nyquist Plots are preferred when your system has a delay in it.
You might be asking, if our closed-loop poles are in the LHP, aren’t we good? Yes and no. If your system was able to exist under perfect conditions and not change with time, then yes, it would remain stable.
87 Unfortunately, stupid reality gets in the way. The main culprits are model inaccuracies, aging, noise and disturbances. Remember, the poles represent physical properties, and as a system ages, the properties change and so do the locations of the poles. Likewise, noise and disturbances can push a stable pole to the right of the Imaginary Axis and cause the system to go unstable. As a result, the best way to protect your system from going unstable is to insure it has a sufficiently large stability margin.
Figure 5.5 – Comparison Plot of Unit Step Response of Modeled System vs. Actual System using a Phase Lead Controller
5.2.1 Stability Analysis using Bode Diagrams
As discussed in Section 4.1.2, Bode Diagrams can be used to determine the Gain and Phase Margins so I won’t go into further detail other than reiterating that the conservative rule of thumb for stability margins are:
𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 10𝑑𝐵 and 𝑃ℎ𝑎𝑠𝑒 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 60°
88 5.2.1.1 Example Continued - Stability Analysis using Bode Diagrams for System with PD Controller Using Matlab’s margin function for an open-loop system with the PD Controller, the following Bode Diagram was plotted:
Figure 5.6 – Bode Diagram of Example System with a PD Controller
Using this function, Matlab calculates a Gain Margin of infinity and a Phase Margin of 65.5° at 38rad/s. Both values exceed our conservative rule of thumb for Gain and Phase Margins so I’m confident our PD Controller will remain stable regardless of aging, noise and/or disturbances.
5.2.1.2 Example Continued – Stability Analysis using Bode Diagrams for System with Phase Lead Controller
Using Matlab’s margin function for an open-loop system with the PD Controller, the Bode Diagram in Figure 5.7 was plotted.
89 Figure 5.7 – Bode Diagram of Example System with a Phase Lead Controller
Using this function, Matlab calculates a Gain Margin of infinity and a Phase Margin of 69.9° at 71.4rad/s. Both values exceed our conservative rule of thumb for Gain and Phase Margins so I’m confident our Phase Lead Controller will remain stable regardless of aging, noise and/or disturbances.
5.2.1 Stability Analysis using Nyquist Plots
Using Matlab’s nyquist function to draw our Nyquist Plot, we’re left to determine the Gain and Phase margins ourselves. For Nyquist Plots, the Gain Margin is defined as the amount that the Nyquist Plot would have to increase to intersect the (-1, 0) point. On a Nyquist Plot, the distance along the negative real axis is measured between the origin and the intersection of the Nyquist Plot and the negative real axis. This distance is the inverse of the Gain Factor. The Gain Margin, expressed in decibels, is calculated using the following:
𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛(𝑑𝐵) = 20𝑙𝑜𝑔10( 1
90 If the Nyquist Plot intersects the real axis at the origin or on the positive real axis, the Gain Margin is considered infinite.
Similarly, the Phase Margin is defined as the angle that the Nyquist Plot would have to change to move to intersect the (-1, 0) point. On a Nyquist Plot, the angle is measured from the negative real axis about the origin towards the intersection between the Nyquist Plot and the unit circle. If the angle is counter- clockwise direction, it is considered positive. If the Nyquist Plot never intersects the unit circle, the Phase Margin is considered infinite.
As shown below in Figure 5.8, the Sample Nyquist Plot has a Phase Margin of 112°. It also has a Distance of 0.2, and using Equation 5.1, the Gain Margin is:
𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛(𝑑𝐵) = 20𝑙𝑜𝑔10(
1
0.2) = 20𝑙𝑜𝑔10(5) = 15𝑑𝐵
Figure 5.8 – Sample Nyquist Plot
We’ll use the same conservative rule of thumb for Gain and Phase margins that we used stability analysis using Bode Diagrams.
Unit Circle
Phase Margin = 112°
91
6.0 References
[1] Ogata, Katsuhiko, “Modern Control Engineering”, page 159, Prentice Hall, New Jersey, 5th Edition, 2010.
[2] “Avago Download Page”, Accessed June 1st, 2015, http://www.avagotech.com/docs/AV02-1046EN. [3] “Avago Download Page”, Accessed June 1st 2015, http://www.avagotech.com/docs/AV02-0096EN. [4] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas,”Feedback Control of Dynamic Systems”, page 48, Prentice Hall, New Jersey, 6th Edition, 2010.
[5] “Chirp”, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., May 5, 2015, Accessed June 12th, 2015, http://en.wikipedia.org/wiki/Chirp.
[6] Ogata, Katsuhiko, “Modern Control Engineering”, page 161, Prentice Hall, New Jersey, 5th Edition, 2010.
[7] “Extras: System Identification”, Accessed June 13th,
http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Identification.
[8] “Root Locus”, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., January 24, 2015, Accessed June 20th, 2015, http://en.wikipedia.org/wiki/Root_locus.
[9] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, ”Feedback Control of Dynamic Systems”, page 120, Prentice Hall, New Jersey, 6th Edition, 2010.
[10] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, ”Feedback Control of Dynamic Systems”, page 119, Prentice Hall, New Jersey, 6th Edition, 2010.
[11] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, ”Feedback Control of Dynamic Systems”, pages 116-119, Prentice Hall, New Jersey, 6th Edition, 2010.
[12] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, ”Feedback Control of Dynamic Systems”, page 129, Prentice Hall, New Jersey, 6th Edition, 2010.
[13] SYDE 352, Introduction to Control Systems, Course Notes, Package 2 of 2, Winter 2006, Prof. Dan Davison, Dept. of Electrical and Computer Engineering, University of Waterloo.
[14] Balemi, Silvano, “Advanced Control”, June 3, 2011, Accessed August 13, 2015, http://www.dti.supsi.ch/~smt/courses/DigImpl.pdf.