(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)

Transcription

1 Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion of leads o a d cb ad Therefore, we will simply consider he second-order linear sysem given by a b 4

2 To obain he phase porrai of he linear sysem, we firs solve for he ime hisory where he consans and are he soluions of he characerisic for e e for e e equaion s s b as s 4 4 b a a b a a For linear sysems described by 4, here is only one singular poin assuming b, namely he origin. However, he rajecories in he viciniy of his singular poin can display quie differen characerisics, depending on he values of a and b.

3 s as b s s. and are boh real and have he same sign posiive or negaive. and are boh real and have opposie signs. and are comple conjugae wih non-zero real pars 4. and are comple conjugae wih real pars equal o zero

4 Conceps of Sabiliy I is imporan o poin ou he qualiaive difference beween insabiliy and he inuiive noion of blowing up all rajecories close o origin move furher and furher away o infiniy. In linear sysems, insabiliy is equivalen o blowing up, because unsable poles always lead o eponenial growh of he sysem saes. However, for nonlinear sysems, blowing up is only one way of insabiliy. Eample: an der Pol Oscillaor Unsable origin of he an der Pol Oscillaor 4

5 Lyapunov s Linearizaion Mehod and Local Sabiliy A nonlinear sysem should behave similarly o is linearized approimaion for small range moion. Because all physical sysems are inherenly nonlinear, Lyapunov s linearizaion mehod serves as he fundamenal jusificaion of using linear conrol echniques in pracice. f Consider he nonlinear auonomous sysem and assume ha f is coninuously differeniable. Taylor epansion around gives f f fh. o.. Noe ha f and f... sands for higher-order erms in h o. Le A denoe he Jacobian mari of f wih respec o a A f A Then he sysem is called he linearizaion or linear approimaion of he original nonlinear sysem a he equilibrium poin. 5

6 6 The basic philosophy of Lyapunov s sabiliy heory is ha if he oal energy of a mechanical or elecrical sysem is coninuously dissipaed, hen he sysem, wheher linear or nonlinear, mus evenually sele down o an equilibrium poin. Consider he nonlinear mass-damper-spring sysem given below. 5 b m b : nonlinear spring : nonlinear dissipaion or damping If he mass is pulled away from he naural lengh of he spring by a large disance, and hen released, will he resuling moion be sable? m m m b Linearizing he sae space equaion, m Therefore, he linear approimaion is only marginally sable.

7 Le us consider he energy sored in he sysem: 4 m d m 4 Comparison beween he mechanical energy and he sabiliy conceps zero energy corresponds o he equilibrium poin, asympoic sabiliy implies he convergence of mechanical energy o zero insabiliy is relaed o he growh of mechanical energy Differeniaing he energy equaion 6 m b b 7 Equaion 7 implies ha he energy of he sysem is coninuously dissipaed by he damper unil he mass seles down, i.e., unil 7

8 Definiion A scalar coninuous funcion is said o be locally posiive definie if and, in a ball > If and he above propery holds over he whole sae space, hen is said o be globally posiive definie. B R Definiion If, in a ball B, he funcion is posiive definie and R has coninuous parial derivaives and if is ime derivaive along any sae rajecory of he sysem is negaive semi-definie, i.e., hen is said o be a Lyapunov funcion for he sysem. Typical shape of a posiive definie funcion Conour curves of a posiive definie funcion 8

9 Eamples Local Sabiliy A simple pendulum wih viscous damping is described by θ θ sin θ Consider he following scalar funcion and is ime derivaive θ cosθ θ sinθ θθ θ One can easily chec ha only if θ and θ, herefore posiive definie, bu is only negaive semi-definie. In his case, is he oal energy of he pendulum and is precisely he power dissipaed in he pendulum. One can conclude ha he origin is a sable equilibrium, bu canno draw conclusions on he asympoic sabiliy of he sysem, because is only negaive semi-definie. 9

10 Eamples Asympoic Sabiliy Le us consider he nonlinear sysem described by Consider he following scalar funcion and is ime derivaive Thus, is locally negaive definie in he -dimensional ball, i.e., in he region defined by. Therefore, he origin is asympoically sable., B <

11 Eamples Global Asympoic Sabiliy Consider he nonlinear sysem described by Consider he following scalar funcion and is ime derivaive is a posiive definie funcion and is negaive definie. Therefore, he origin is a globally asympoically sable equilibrium poin. Noe ha he globalness of his sabiliy resul also implies ha he origin is he only equilibrium poin of he sysem.