Wireless Ad Hoc and Sensor Networks

Background 73for some R > 0 such that the ball of radius R is contained in S, then thesystem is UUB, and the norm of the state is bounded to within a neighborhoodof R. In this result, note that ∆L must be strictly less than zerooutside the ball of radius R. If one only has ∆L( xk ( ), k) ≤ 0 for all | x| > R,then nothing may be concluded about the system stability.For systems that satisfy the theorem, there may be some disturbanceeffects that push the state away from the equilibrium. However, if the statebecomes too large, the dynamics tend to pull it back towards the equilibrium.Because of these two opposing effects that balance when | x| ≈ R,thetime histories tend to remain in the vicinity of | x| = R.In effect, the normof the state is effectively or practically bounded by R.The notion of the ball outside which ∆L is negative should not be confusedwith that of domain of attraction — in Example 2.4.1a. It was shownthere that the system is AS provided | x | < 1,defining a domain of attractionof radius one.The subsequent examples show how to use this result. They make thepoint that it can also be used as a control design technique where thecontrol input is selected to guarantee that the conditions of the theoremhold.Example 2.4.5: UUB of Linear Systems with DisturbanceIt is common in practical systems to have unknown disturbances, whichare often bounded by some known amount. Such disturbances result inUUB and require the UUB extension for analysis. Suppose the systemxk ( + 1) = Axk ( ) + dk ( )has A stable, and a disturbance d(k) that is unknown, but bounded so that|( dk)| < d M , with the bound d M known.Select the Lyapunov function candidateTLxk ( ( )) = x ( kPxk ) ( )and evaluateTT∆L( xk ( )) = x ( k+ 1) Pxk ( + 1) −x ( kPxk ) ( )= x T ( k)( A T PA− P) x( k) + 2x T ( k) APdk T ( ) + d T ( kPdk ) ( )=− x T ( k) Qx( k) + 2xT ( k) A T Pd( k) + d T ( k) Pd( k)