Lyapunov stability analysis

Lyapunov stability analysis in analog and digital control system.

1. LYAPUNOV STABILITY ANALYSIS There are two Lyapunov methods for stability analysis. The first method usually requires the analytical solution of the differential equation. Its an indirect method. In the second method , it is not necessary to solve the differential equation. Instead, a Lyapunov function is constructed to check the motion stability. Therefore, it is called as direct method. Lyapunov direct method is the most effective method for studying non linear and time-varying systems and is a basic method for stability analysis.

2. SECOND METHOD OF LYAPUNOV STABILITY The second method is based on a the fact that if the system has an asymptotically stable equilibrium state, then the stored energy of the system displaced within a domain of attraction decays with increasing time until it finally assumes its minimum value at the equilibrium state.

3. LYAPUNOV STABILITY ANALYSIS Lyapunov stability is used to described the stability of a dynamic system. Lyapunov stability for linear time invariant system, X˙=Ax is Lyapunov stable if no eigenvalues of a are in the right half of the complex plane. Lyapunov introduced the lyapunov function, But before discussing the lyapunov function, it is necessary to define the definiteness of scalar function.

4. DEFINITENESS OF SCALAR FUNCTION Positive definite of function- V(x 1 ,x 2 ,x 3 … x n ) >0 for any non zero value of x 1 ,x 2 ,x 3 … x n Negative definite of function- V( x 1 ,x 2 ,x 3 ,… x n ) <0 For any non zero value of x 1 ,x 2 ,x 3 … x n Positive semi definite of function- V(x 1 ,x 2 ,x 3 … x n ) ≥0 For any non zero value of x 1 ,x 2 ,x 3 ,… x n Negative semi definite of function- V(x 1 ,x 2 ,x 3 ,… x n )≤0 For any non zero value of x 1 ,x 2 ,x 3 ,… x n

5. Stability in the sense of Lyapunov An equilibrium state x e of an autonomous system is stable in the sense of Lyapunov, if corresponding to each S( ε ), there is an S( δ ) such that trajectories starting in S( δ ) do not leave S( ε ) as time increases indefinitely.

6. Asymptotic stability An equilibrium state x e of the system is said to be asymptotically stable in the sense of Lyapunov if every solution starting within S( δ ) converges, without leaving S( ε ), to x e as time increases indefinitely.

7. I nstability An equilibrium state x e is said to be unstable if for some real number ε >0 and any real number δ >0, no matter how small, there is always a state x in S( δ ) such that the trajectory starting at this state leaves S( ε ).

8. Stability of continuous-time linear system Consider a linear system described by the state equation- x˙=Ax Where A is n×n real constant matrix. The linear system is asymptotically stable at the origin if, for any given symmetric positive definite matrix Q, there exists a symmetric positive definite matrix P, that satisfies the matrix equation- A’P+PA = -Q We may choose Q=I, the identity matrix.

9. Example- Let us determine the stability of the system described by the following equation x˙ = Ax With Solution- Equation for P- A’P + PA = -I

10. After solving, we get the equations- And solving for P We obtain By using Sylvester’s test, we find that P is positive definite. P1= (23/60)>0 and P2= (17/300)>0 Therefore, the system is consider as asymptotically stable at origin.

11. Stability of discrete-time linear system Consider a linear system described by the state equation- x (k+1) = Fx (k) Where F is n×n real constant matrix. The linear system is asymptotically stable at the origin if, for any given symmetric positive definite matrix Q , there exists a symmetric positive definite matrix P , that satisfies the matrix equation- F’PF – P = -Q Here also we may choose Q=I, the identity matrix.

12. Example Let us determine the stability of the system described by the following equation- x (k+1) = Fx (k) With Solution- Solving the equation for P- F’PF – P = -I

13. After solving we get the equations- Solving for P By using Sylvester’s test, we find that P is negative definite. P1= (-43/60)<0 P2= (-1/49)<0 Therefore, the system is unstable.