MIT 2 004  Determining Stability Bounds in ClosedLoop Systems (10 pages)
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Determining Stability Bounds in ClosedLoop Systems
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 10
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 Massachusetts Institute of Technology
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 2 004  Dynamics and Control II
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MIT OpenCourseWare http ocw mit edu 2 004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Massachusetts Institute of Technology Department of Mechanical Engineering 2 004 Dynamics and Control II Spring Term 2008 Lecture 261 Reading Nise Chapter 6 Nise Chapter 8 1 Determining Stability Bounds in Closed Loop Systems Consider the closed loop third order system with proportional controller gain K with openloop transfer function K Gf s 3 s 3s2 5s 2 shown below R s P c o n tr o lle r p la n t K s 3 3 s 2 1 C s 5 s 2 The closed loop transfer function is Gcl s N s K 3 2 D s N s s 3s 5s 2 K Let s examine the closed loop stability by using the pzmap function in MATLAB sys tf 1 1 3 5 2 pzmap sys hold on for K 2 2 30 sys tf K 1 3 5 2 K pzmap sys end which superimposes the closed loop pole zero plots for K 0 30 on a single plot 1 c D Rowell 2008 copyright 26 1 Pole Zero Map 3 K 30 2 K increasing K 0 Imaginary Axis 1 K increasing K 30 K 0 0 1 K 0 K increasing 2 3 4 K 30 3 5 3 2 5 2 1 5 Real Axis 1 0 5 0 0 5 1 From the plot we note the following This system always has two complex conjugate poles and a single real pole When K 0 the poles are the open loop poles As K increases the real pole moves deeper into the l h plane and the complex con jugate poles approach and cross the imaginary j axis and the system becomes unstable Close examination of the plot shows that the system becomes unstable at a value of K between K 12 and K 14 We now look at three methods for determining the stability limit of the proportional gain K for this system Example 1 Use the Routh Hurwitz method to nd the range of proportional controller gain K for which the above system will be stable The rst two rows of the Routh array are taken directly from D s s3 s2 1 5 3 2 K 0 0 26 2 and the next two rows are computed as above 1 1 1 1 an an 2 5 b1 K 13 3 2 K a a an 1 3 3 n 1 n 3 1 an 1 1 0 an 4 b2 0 an 1 an 1 an 5 3 3 0 Similarly the
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