10 MRAS Lyapunov Method - SYS635 Adaptive Systems KaCC...

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SYS635 Adaptive Systems KaCC MRAS Lyapunov Method 1 10/29/05 Lyapunov Stability Method ± MIT gradient rule uses simple a 1 st order logic (slope & step size) as a guide in searching for a minimum. With appropriate step size, it can converge to equilibrium, which can be a local of global minimum. Extension of topics: ± Extend search using 2 nd order logic, (slope of slope) . ± How can we adapt step size using decisions ± Lyapunov Method uses a stability criterion to guide a search to converge to an equilibrium that can be a local or global minimum. Lyapunov theory for Time-invariant System Consider an n-th order homogeneous nonlinear vector differential equation given by () n d dt = ∈ℜ x fx x Definition: A point for which ( ) ss = xf x 0 is called a singular point . E.g., (1 ) dx x x dt =− has two singular points at 0 and x = 1 x = . Definition: The vector function () is assumed to be locally Lipschitz , that is () () 0 LL <− > fx fy x y (Explain what this means. Sketch an example for this condition. In plain English, the Lipschitz condition means that the rate d d x is limited by L .) Note: x denotes a norm of x, which can be defined from any of the p -norm definition 1 1 pp p n p xx == + + L Definition: A singular point s x is said to be stable if for any hyperspherical region S R of radius R centered at s x , there exist a hyperspherical region S r of radius r R also centered at s x , in which any motion x (t) of the system beginning in S r remains in S R ever after. S r S R x 1
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SYS635 Adaptive Systems KaCC MRAS Lyapunov Method 2 10/29/05 Rephrase: The solution s = x0 is stable if given an 0( ) 0 ε δε >∃ > such that all solution with (0) δ < x has the property () 0 tt <∀≤ < x , where denotes a norm operation. A singular point is asymptotically stable if is stable and all trajectory (motion) x(t) tends toward s x s as time goes to infinity. An asymptotically stable singular point is also called an equilibrium point . A successful bungee jump: A accidental broken bungee is an unstable scenario: : Mass-Damper-Spring-Air Friction Example (Bungee jump) x t time t [sec] x t x s x t Start with a jump up Mass-Damper-Spring-Air Friction Example (Bungee jump) x t time t [sec] x t x s x t Start with a jump up Aaaaiiiieeee! Oh sh_t! I’m too heavy! (See the Jackass! movie…now showing)
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SYS635 Adaptive Systems KaCC MRAS Lyapunov Method 3 10/29/05 Lyapunov stability criterion : If the origin is a singular point ( ) () = f0 0 , then it is stable if a Lyapunov function V ( x ( t )) can be found with the following properties: a. ( ( )) 0, values of Vt >∀ xx 0 b. 0, dV x dt ≤∀ Furthermore, if dV dt is never zero except at the origin, then the origin is asymptotically stable . Application of Lyapunov Stability Criterion to a Linear System. Let’s first see how the LSC can be applied to a linear system, which we already have a good idea on.
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10 MRAS Lyapunov Method - SYS635 Adaptive Systems KaCC...

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