Wireless Ad Hoc and Sensor Networks

Background 63The subsequent examples show the utility of the Lyapunov approach andmake several points. Among the points of emphasis are those stating that theLyapunov function is intimately related to the energy properties of a system.Example 2.4.1: Local and Global StabilityLOCAL STABILITYConsider the system( )( )x ( k+ 1) = x ( k) x ( k) + x ( k)−21 1 1 2 2 2x ( k+ 1) = x ( k) x ( k) + x ( k)−22 2 1 2 2 2Stability for nonlinear discrete-time systems can be examined by selectingthe quadratic Lyapunov function candidateLxk ( ( )) = x( k) + x( k)1 2 2 2which is a direct realization of an energy function and has first difference∆L( xk ( )) = x( k+ 1) − x( k) + x ( k+ 1) −x ( k)1 2 1 2 2 2 2 2Evaluating this along the system trajectories simply involves substitutingthe state differences from the dynamics to obtain, in this case,( 1 )( − − )2 2 2 1 2 2 2∆L( xk ( )) =− x( k) + x( k) 1 x ( k) x ( k)which is negative as long as|( xk)| = x( k) + x ( k)