LYAPUNOV STABILITY PROBLEM SOLUTION

02/05/2023 Rohit Kumar 1

LYAPUNOV STABILITY PROBLEM SOLUTION

Submitted to:Mrs. Shimi S.LAssistant ProfessorNITTTR Chandigarh

Submitted by:Rohit KumarM.E(R)162519

e

d

02/05/2023 Rohit Kumar 2

STEP TO SOLVE NONLINEAR PROBLEM

Consider nonlinear state space equation Select Lyapunov or energy function V(x) Check V(x) is positive definite or not Determine derivative of V(x) Check is negative definite function or not Check stability area

02/05/2023 Rohit Kumar 3

MATLAB INSTRUCTION

syms assume Jacobian Jacobian matrix jacobian(f , v) lyap Continuous Lyapunov equation solution lyap(A,Q) eig Eigenvalues and eigenvectors eig(A) transpose Transpose vector or matrix transpose(A) det Matrix determinant det(A) disp Display value of variable disp(‘X’)

Create symbolic variables and functions syms x ySet permanent assumption assume(condition)

02/05/2023 Rohit Kumar 4

Q.1 Consider the nonlinear system described by the equations Find the region in the state plane for which the equilibrium state of the systems asymptotically stable.Solution: Select Lyapunov function

+Which is positive definite functionIts derivative is

02/05/2023 Rohit Kumar 5

Case1:

to be negative definite if or If these conditions are fulfil then system remain asymptotically stable.Case2: = 0 if at this point rate of change of energy will become zero. At this point there is no trajectory. But energy is not zero at this point.

02/05/2023 Rohit Kumar 6

Case3: is not negative definite if In this region system will become unstable.

Possibly unstablePossibly unstable

Stable Stable

X1

X2

02/05/2023 Rohit Kumar 7

Select another Lyapunov function

P1>0 and P2>0Which is positive definite functionIts derivative is

+

02/05/2023 Rohit Kumar 8

Case1:If P1=P2

Same conditions as of previous case.Case2:If and is negative definite. Then system will be asymptotically stable.Case2:If and may be negative or positive definite , hence not identified i.e. stability can not be found by Lyapunov function.

02/05/2023 Rohit Kumar 9

MATLAB SOLUTION

02/05/2023 Rohit Kumar 10

02/05/2023 Rohit Kumar 11

02/05/2023 Rohit Kumar 12

OUTPUT

02/05/2023 Rohit Kumar 13

02/05/2023 Rohit Kumar 14

02/05/2023 Rohit Kumar 15

Q.2 Check the stability of the equilibrium state of the system described by

Solution: Select Lyapunov function+Which is positive definite functionIts derivative is

02/05/2023 Rohit Kumar 16

is always negative definite. The rate of change of energy is negative and therefore system energy continuously decreases along the trajectory X(t), t>0. Rate of change of energy can be zero at any axis(X1axis or X2 axis), but not the energy(which is zero at origin). X(t) immediately moves to the point at which the rate of change of energy is negative and the system energy therefore continually decrease from its initial value along the trajectory X(t), t>0 till it reaches a value V=0, at the equilibrium point at origin. as Then system is asymptotically stable in-the-large.

02/05/2023 Rohit Kumar 17

MATLAB SOLUATION

02/05/2023 Rohit Kumar 18

02/05/2023 Rohit Kumar 19

02/05/2023 Rohit Kumar 20

OUTPUT

02/05/2023 Rohit Kumar 21

02/05/2023 Rohit Kumar 22

OUTPUTQ.3 Consider the nonlinear system described by the equations

Using the krasovski for corresponding the Lyapunov function which P as identity ,investigate the stability the stability of the equilibrium state.Solution: Jacobion matrix of f(x) is

J==

02/05/2023 Rohit Kumar 23

Consider P as identityThe matrix

]

Q=

Q=

02/05/2023 Rohit Kumar 24

Q=Q=

always remain positive deficient . System is asymptotically stable at origin.

02/05/2023 Rohit Kumar 25

MATLAB SOLUATION

02/05/2023 Rohit Kumar 26

02/05/2023 Rohit Kumar 27

02/05/2023 Rohit Kumar 28

02/05/2023 Rohit Kumar 29