Wireless Ad Hoc and Sensor Networks

64 Wireless Ad Hoc and Sensor NetworksGLOBAL STABILITYConsider now the systemx ( k+ 1) = x ( k) x ( k)1 1 2 2x ( k+ 1) = x ( k) x ( k)2 2 1 22where the states satisfy ( x1( k) x2( k)) < 1.Selecting the 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( kx ) ( k)Applying the constraint, the system is globally stable because the statesare restricted.Example 2.4.2: Lyapunov StabilityConsider now the systemx ( k+ 1) = x ( k) −x ( k)1 1 2x ( k+ 1) = 2x ( k) x ( k) −x( k)1 1 2 1 2Selecting the 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 2