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MECH 466 - Lecture7-Routh-HurwitzCriterion-2009W

# Is a factor of qs qs no roots in rhp but some roots

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Unformatted text preview: or of Q(s).) Q(s).) No roots in RHP But some roots are on imag. axis. imag. MECH466 : Automatic Control No sign changes in the first column 9 2008/09 MECH466 : Automatic Control Example 4 Routh array Derivative of auxiliary poly. 4 2 Derivative of auxiliary poly. 0 4 No sign changes in the first column No root in OPEN(!) RHP 2008/09 10 Example 5 Routh array 4 No root in OPEN(!) RHP MECH466 : Automatic Control 11 One sign changes in the first column 2008/09 One root in OPEN(!) RHP MECH466 : Automatic Control 12 3 Example 6 Example 7 Find the range of K s.t. Q(s) has all roots in the left s.t. Q(s) half plane. (Here, K is a design parameter.) Routh array No sign changes in the first column Design K(s) that stabilizes the closed-loop Design K(s) closedsystem for the following cases. K(s) = K (constant, P controller) K(s) K(s) = KP+KI/s (PI (Proportional-Integral) controller) K(s) (Proportional2008/09 MECH466 : Automatic Control 13 2008/09 Example 7: K(s)=K MECH466 : Automatic Control 14 Example 7: K(s)=KP+KI/s Characteristic equation Characteristic Characteristic equation Characteristic Routh array Routh Routh array Routh 2008/09 MECH466 : Automatic Control 15 2008/09 MECH466 : Automatic Control 16 4 Example 7: Range of (KP,KI) Example 7: K(s)=KP+KI/s (cont’d) From Routh array, From Select KP=3 (<9) Select Routh array (cont’d) Routh (cont’ 3.5 3 2.5 2 1.5 1 If we select different KP, the range of KI changes. If changes. 0.5 0 -1 2008/09 0 1 2 3 4 5 6 7 8 9 MECH466 : Automatic Control 17 2008/09 MECH466 : Automatic Control 18 Example 8 Example 7: What happens if KP=KI=3 Auxiliary equation Auxiliary 2 1.8 1.6 1.4 1.2 Oscillation frequency Oscillation 1 0.8 0.6 Period Period Determine the range of K and a that stabilize the Determine closed-loop system. closed- 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Unit step response 2008/09 MECH466 : Automatic Control 19 2008/09 MECH466 : Automatic Control 20 5 Example 8 (cont’d) Example 8 (cont’d) Characteristic equation Characteristic 2008/09 MECH466 : Automatic Control 21 2008/09 Example 8 (cont’d) MECH466 : Automatic Control 22 Summary and Exercises Routh array Routh Examples for Routh-Hurwitz criterion Examples RouthCases when zeros appear in Routh array Cases P controller gain range for stability controller PI controller gain range for stability PI Oscillation frequency Oscillation Nex...
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