{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

10 MRAS Lyapunov Method

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 14

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online