3_phase_plane - 3 Phase-Plane Analysis Key points Phase plane analysis is limited to second-order systems For second order systems solution trajectories

3_phase_plane - 3 Phase-Plane Analysis Key points Phase...

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3 Phase-Plane Analysis Phase plane analysis is a technique for the analysis of the qualitative behavior of second- order systems. It provides physical insights. Reference: Graham and McRuer, Analysis of Nonlinear Control Systems, Dover Press, 1971. Consider the second-order system described by the following equations: ) , ( 2 1 1 x x p x = & ) , ( 2 1 2 x x q x = & x 1 and x 2 are states of the system p and q are nonlinear functions of the states Key points Phase plane analysis is limited to second-order systems. For second order systems, solution trajectories can be represented by curves in the plane, which allows for visualization of the qualitative behavior of the system. In particular, it is interesting to consider the behavior of systems around equilibrium points.
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phase plane = plane having x 1 and x 2 as coordinates get “rid” of time ) , ( ) , ( 2 1 2 1 1 2 x x p x x q dx dx = We look for equilibrium points of the system (also called singular points), i.e. points at which: 0 ) , ( 2 1 = e e x x p 0 ) , ( 2 1 = e e x x q Example: Find the equilibrium point(s) of the system described by the following equation: 3 2 ) ( x a x x & & & + = Start by putting the system in the standard form by setting 2 1 , x x x x = = & : 3 2 2 1 2 2 1 ) ( x a x x x x + = = & & Looking at the slope of the phase plane trajectory: 2 3 2 2 1 1 2 ) ( x x a x dx dx + = This yields the following equilibrium point: 0 2 1 = = e e x a x Investigate the linear behaviour about a singular point : 1 1 1 x x x e δ + = 2 2 2 x x x e δ + =
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2 2 1 1 1 x x p x x p x e e δ δ δ + = & 2 2 1 1 2 x x q x x q x e e δ δ δ + = & Set e e e e x q d x q c x p b x p a 2 1 2 1 , , , = = = = Then 2 1 1 x b x a x δ δ δ + = & 2 1 2 x d x c x δ δ δ + = & This is the general form of a second-order linear system. Next, we obtain the characteristic equation: 0 ) )( ( = bc d a λ λ This equation admits the roots: 2 ) ( 4 ) ( 2 2 , 1 bc ad ad d a ± + = λ This yields the following possible cases : λ 1 , λ 2 real and negative Stable node λ 1 , λ 2 real and positive Unstable node λ 1 , λ 2 real and opposite signs Saddle point λ 1 , λ 2 complex and negative real parts Stable focus λ 1 , λ 2 complex and positive real parts Unstable focus λ 1 , λ 2 complex and zero real parts Center
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Stability (Lyapunov’s First Method) Consider the system described by the equation: ) ( x f x = & Write x as : x x x e δ + = Then ) , ( . ) , ( . x x h x A x x x h x x f x e e e δ δ δ δ δ δ + = + = & & Lyapunov proved that the eigenvalues of A indicate “local” stability of the nonlinear system about the equilibrium point if: a) 0 ) , ( lim 0 = x x x h e x δ δ δ (“The linear terms dominate”) b) There are no eigenvalues with zero real part. Example: Consider the equation: 3 x ax x + = & If x is small enough, then 3 x ax >> Thought question: What if a = 0?
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Example: Simplified satellite control problem Built in the 1960s.
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  • Spring '08
  • HEDRICK
  • Nonlinear system, Stability theory, phase plane, Sliding mode control, Nonlinear control

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