Wireless Ad Hoc and Sensor Networks

66 Wireless Ad Hoc and Sensor Networkssgn(x)10x−1FIGURE 2.3Signum function.where [ ] denotes a vector z with components z i . The discontinuousnature of such functions often makes it impossible to apply input/outputfeedback linearization where differentiation is required. In some cases,controller design can be carried out for systems containing discontinuitiesusing Lyapunov techniques.z ix k x k x kExample 2.4.3: Controller Design by Lyapunov AnalysisConsider the system( + 1) = ( )sgn( ( ))1 2 1x ( k+ 1) = x ( k) x ( k)+ u( k)2 1 2having an actuator nonlinearity. A control input has to be designed usingfeedback linearization techniques (i.e., cancels all nonlinearities). A stabilizingcontroller can be easily designed using Lyapunov techniques.Select the Lyapunov function candidateand evaluateLxk ( ( )) = x( k) + x( k),1 2 2 2∆L( xk ( )) = x( k+ 1) − x( k) + x ( k+ 1) −x ( k).1 2 1 2 2 2 2 2Substituting the system dynamics in the previous equation results in∆L( xk ( )) = x2 2 ( k)sgn 2 ( x1( k)) − x1 2 2( k) + ( x1( kx ) 2(k) + u( k)) −x 2 ( k)