Wireless Ad Hoc and Sensor Networks

74 Wireless Ad Hoc and Sensor Networkswhere( PQ , )satisfy the Lyapunov equationOne may now use the norm equalities to writewhich is negative as long asTAPA− P=−Q2∆L( xk ( )) ≤−[ σ ( Q)|( xk)| −2|( xk)| σ ( AT 2P) |( d k)| ( P)|( d k)|]minmax−σ maxT2 T 2σmax( APd ) M + σmax( APd ) M + σmin( Q)σ|()| xk ≥σ ( Q)Thus, if the disturbance magnitude bound increases, the norm of thestate will also increase.Example 2.4.6: UUB of Closed-Loop SystemThe UUB extension can be utilized to design stable closed-loop systems.The system described byis excited by an unknown disturbance whose magnitude is bounded sothat |( dk)| < d M . To find a control that stabilizes the system and mitigatesthe effect of disturbances, select the control input asThis helps cancel the sinusoidal nonlinearity and provides a stabilizingterm yielding the closed-loop systemminxk ( + 1) = x 2 ( k) − 10xk ( )sin xk ( ) + dk ( ) + uk ( )uk ( ) =− x 2 ( k) + 10xk ( )sin xk ( ) + kxk ( )vmax( Pd )M 2xk ( + 1) = kxk ( ) + dk ( )Select the Lyapunov function candidatevwhose first difference is given byLxk ( ( )) = x 2 ( k)∆L( xk ( )) = x 2 ( k+ 1)−x 2 ( k)