Stability. ME 344/144L Prof. R.G. Longoria Dynamic Systems and Controls/Lab. Department of Mechanical Engineering The University of Texas at Austin

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1 Stability The tability of a ytem refer to it ability or tendency to eek a condition of tatic equilibrium after it ha been diturbed. If given a mall perturbation from the equilibrium, it i table if it return. Nonlinear ytem involve a very detailed and careful definition of tability: table in the mall, aymptotically table in the mall, table in the large. For linear ytem, the character of the natural motion doe not depend on their amplitude, o tability analyi i much impler for thee cae. Second order ytem, in particular, are fairly traightforward. The characteritic equation i tudied to undertand how the root vary with the ytem parameter. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

2 Imaginary axi Natural Repone veru -plane poition UNSTABLE Real axi Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

3 Example: nd order Rotational Sytem From Cannon (967) Characteritic equation: + b k J + J Root of characteritic equation: b k b ± j J J J Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

4 Negative Spring A fuel-injection valve deigned to have negative pring characteritic. Force-diplacement curve The retoring force on the ma i the um effect of the mechanical pring force and a Bernoulli force due to preure variation at the valve eat. *Key on a keyboard alo have a ort of untable deign. Thee example are not really imple pring. Flow of water provide the input of energy that feed thi effect. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

5 Inverted Pendulum Intability ɺ ɺɺ θ ɺ θ inθ h ml M o b + mgl ɺɺ θ b g + ɺ ml θ l θ "negative" pring b b ± j g + ml ml l You get one poitive root. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

6 Effect of Damping Thi plot of root how the effect of varying the damping. The cae where the root are on the imaginary axi correpond to pure harmonic ocillation. We ometime ay thi i marginally or neutrally table. When b give root at (f), the ytem i alway untable. Cannon (967) Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

7 Negative Damping Negative damping from aerodynamic force namely lifting Negative damping due to dry friction on a belt Cannon (967) Note, the effective damping, indicated by the SLOPE of the curve, i negative in both cae. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

8 Tire traction force on different urface how that an untable ituation can exit for certain cae Which curve indicate that an untable traction condition might arie? Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

9 Stability Analyi For more complex ytem, it may not be eay to intuitively ae tability, and a mathematical approach i neceary. The tability of a linear cloed-loop ytem can alo be determined from the location of the cloed-loop pole in the plane The Routh tability criterion can help anwer quetion about abolute tability: i the ytem table or not. Other method, like root locu analyi, can tell you omething about relative tability; i.e., a more quantitative aement of the ytem tability baed on pole/zero location. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

10 Routh Stability Criterion Thi criterion allow u to determine the number of pole that lie in the right-half -plane without having to factor the polynomial. Thi applie to tranfer function with finite polynomial of the form G( ) m bm + b + + b + b B( ) n a + a + + a + a A( ) n m m n n Thi i ueful in helping identify the range, for example, that certain parameter can take without the ytem becoming untable. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

11 Routh Stability Criterion n n an + an + + a + a b b c c a a a a a n n n n 3 n a a a a a n n 4 n n 5 n b a a b b n 3 n b a a b b Dynamic Sytem and Control/Lab n 5 n 3 a a a n n n n 4 n an an 3 an 5 Routh Table b b b 3 c c c 3 Criterion: The root have negative real part if and only if the element of the firt column of the Routh Table have the ame ign. The number of ign change i equal to the number of root with poitive real part. The Univerity of Texa at Autin

12 Routh Stability Example 3 K We require that both of thee condition hold for tability, Therefore, 8 K > + K > < K < 8 Dynamic Sytem and Control/Lab K 8 K 3 + K K v. 3 root -: i i.599 -: i i : i -. +.i 7: i i 8: i i 9: i i The Univerity of Texa at Autin

13 Example tability range Since there are no ign change in the firt column, all the root of the characteritic equation have negative real part and the ytem i table. In Matlab: >> c [ 4 8 ] c >> root(c) an i i All root have negative real part, confirming Routh analyi. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

14 Example 3 tability range 4 3 K For tabilty, 6 6 > Therefore, or and K K <, K > < K < 4 3 K 6 6 K 6 6K K Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

15 Problem tability range Conider the characteritic equation, 4 3 K Ue the Routh tability criterion to how that there i no allowable range of K that allow a ytem with thi characteritic equation to be table. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

16 Problem autopilot* Thi i an example of finding a cloed loop TF and correponding tability range a a parameter i varied. A implified form of the open-loop tranfer function of an airplane with an autopilot in the longitudinal mode i G( ) H ( ) K( + ) ( )( ) Show that the range of gain K for tability i, 3.3<K<35.68 *From Ogata (998) Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

17 Wheel himmy intability Cater type wivel wheel himmy Thi figure illutrate a wheel himmy common in the 93. A 3 DOF decription i required to explain. Thi i a elf-excited vibration, but may alo be excited by imbalance of the tire. Stability analyi for thi cae how you hould atify mal > I + ma G Solved by independent front-wheel upenion. Den Hartog, p Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

18 Example: Routh Stability Analyi for Wheel Shimmy 3 amv kl Vkl IG + ma IG + ma IG + ma For tability, you can how that, aml > I + ma and, G Vkl > 3 G ( IG + ma ) G kl I + ma amv Vkl I + ma I + ma kl I G G Vkl amv Vkl + ma Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

19 Example: Cloed-Loop Speed Control of a Ga Engine R( ) + e K + Te τ + C R - Throttle controller GG + G G H G( ) Ignore diturbance for now, H ( ) T t T d τ + m τ ec τ τ m 4ec.5ec Engine Dynamic K τ + G( ) meaurement K K?. C( ) Speed Note, ometime the τ value are eay to meaure and form bai of thi implified model. Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

20 Example: Cloed-Loop Speed Control of a Ga Engine C KK(.5 + ) R ( + )(4 + )(.5 + ) + K K 5 < K < K K KK < K. K < and Dynamic Sytem and Control/Lab K > 5 K K K K See Appendix A for dicuion on thi method. The Univerity of Texa at Autin

21 Example: Cloed-Loop Speed Control of a Ga Engine If you have a linear tranfer function, you can find the SS value by invoking the final value theorem, which effectively i applied by letting,. For the engine peed control, C KK(.5 + ) R ( + )(4 + )(.5 + ) + K K C R SS KK + K K Note there i a teady-tate error. For no diturbance, C e R C R R K K lim R SS + KK The error converge to zero if K Dynamic Sytem and Control/Lab The Univerity of Texa at Autin

22 Example: Cloed-Loop Speed Control of a Ga Engine A imulation can be very helpful in tudying the trend of thi ytem. Here i an example uing Matlab/Simulink. Uing the Routh tability criterion i helpful in narrowing the range over which we can et the gain, K (abolute tability). Once we know thi value, we can eaily conduct imulation to tune the ytem further. R C K 4 Dynamic Sytem and Control/Lab The Univerity of Texa at Autin