Contents
What is sufficient condition of Lyapunov stability?
The sufficient and necessary condition for global exponential stability of the zero solution of system (4) is that the zero solution of system (4) on partial variable m ~ or p ~ is globally exponentially stable.
How is Lyapunov function determined?
A Lyapunov function is defined by the following properties: it is zero at x=0, is positive definite for x≠0, and has a negative semidefinite derivative with respect to time, V˙.
How do you determine if a system is Lyapunov stable?
1. If V (x, t) is locally positive definite and ˙V (x, t) ≤ 0 locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov). 2.
What is Lyapunov stability function?
In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability.
What is the basic concept of Lyapunov stability analysis?
Lyapunov stability theory was come out of Lyapunov, a Russian mathematician in 1892, and came from his doctoral dissertation. The control of the trajectory of the space transports is based on the Lyapunov stability theory. The Lyapunov stability theory is used to describe the stability of a dynamic system (Fig. 1.2).
How do you determine if a system is asymptotically stable?
A time-invariant system is asymptotically stable if all the eigenvalues of the system matrix A have negative real parts. If a system is asymptotically stable, it is also BIBO stable.
What is a strict Lyapunov function?
A strict Lyapunov function is a proper positive definite function whose time derivative is negative along all solutions of the system outside the equilibrium. Strict Lyapunov functions also allow us to robustify controls, e.g., to prove robustness in the key sense of input-to-state stability (or ISS).
How do you know if asymptotically stable?
If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. If a solution does not have either of these properties, it is called unstable.
Is Lyapunov function unique?
The discrete-time Lyapunov equation has a unique solution P, for any Q = QT , if and only if λi(A)λj(A) = 1, for i, j = 1,…,n. (AT )tQAt is the unique solution of the Lyapunov equation AT PA − P + Q = 0.
How do you know if an equilibrium point is stable?
The stability of equilibrium points is determined by the general theorems on stability. So, if the real eigenvalues (or real parts of complex eigenvalues) are negative, then the equilibrium point is asymptotically stable. Examples of such equilibrium positions are stable node and stable focus.
What is the stability of nonlinear system?
Roughly speaking, stability means that the system out- puts and its internal signals are bounded within admissi- ble limits (the so-called bounded-input/bounded-output stability) or, sometimes more strictly, the system outputs tend to an equilibrium state of interest (the so-called as- ymptotic stability).