Wireless Ad Hoc and Sensor Networks

Background 65Evaluating this along the system trajectories simply involves substitutingthe state differences from the dynamics to obtain, in this case,2∆L( xk ( )) =−x 1 ( k)This is only negative semidefinite (note that ∆L( xk ( )) can be zero, whenx2( k) ≠ 0).Therefore, Lxk ( ( )) is a Lyapunov function, but the system is onlyshown by this method to be SISL — that is, | x1( k)| and | x2( k)|are bothbounded.2.4.2 Controller Design Using Lyapunov TechniquesThough we have presented Lyapunov analysis only for unforced systemsin the form described in Theorem 2.4.1, which have no control input, thesetechniques also provide a powerful set of tools for designing feedbackcontrol systems of the formxk ( + 1) = f( xk ( )) + gxk ( ( )) uk ( )(2.37)Thus, select a Lyapunov function candidatealong the system trajectories to obtainLx ( )> 0 and differentiateTT∆L( x) = Lxk ( ( + 1)) − Lxk ( ( )) = x ( k+ 1) xk ( + 1) −x ( k)xk ( )f x k gxk uk TT= ( ( ( )) + ( ( )) ( )) ( f( x(k)) + gxk ( ( )) uk ( )) −x ( kxk ) ( )(2.38)Then, it is often possible to ensure that ∆L ≤ 0 by appropriate selectionof uk ( ). When this is possible, it generally yields controllers in state-feedbackform, that is, where uk ( ) is a function of the states xk ( ).Practical systems with actuator limits and saturation often contain discontinuousfunctions including the signum function defined for scalarsx ∈R as⎧ 1,x ≥ 0sgn( x)= ⎨⎩⎪ − 1,x < 0(2.39)T nshown in Figure 2.3, and for vectors x= [ x x x n ] ∈R as1 2 ⋯sgn( x) = [sgn( )]x i(2.40)