Wireless Ad Hoc and Sensor Networks

72 Wireless Ad Hoc and Sensor Networksn3. Global stability: If the equilibrium point is SISL or AS, if S =R and,in addition, the radial unboundedness condition holds,then the stability is global.L(x(k), k) →∞ as |( xk)| →∞,∀k(2.58)4. Uniform stability: If the equilibrium point is SISL or AS, and inaddition Lxk ( ( ), k)is decrescent (e.g., Equation 2.54 holds), thenthe stability is uniform (e.g., independent of k 0 ).The equilibrium point may be both uniformly and globally stable —e.g., if all the conditions of the theorem hold, then one has GUAS.2.4.7 Extensions of Lyapunov Techniques and Bounded StabilityThe Lyapunov results so far presented have allowed the determination ofSISL. If there exists a function such that Lxk ( ( ), k) > 0, ∆L( xk ( ), k)≤ 0, and AS,if there exists a function such that Lxk ( ( ), k) > 0, ∆L( xk ( ), k)< 0. Various extensionsof these results allow one to determine more about the stability propertiesby further examining the deeper structure of the system dynamics.2.4.7.1 UUB Analysis and Controls DesignWe have seen how to demonstrate that a system is SISL or AS usingLyapunov techniques. However, in practical applications, there are oftenunknown disturbances or modeling errors that make even SISL too muchto expect in closed-loop systems. Typical examples are systems of the formxk ( + 1) = f( xk ( ), k) + dk ( )(2.59)with d(k) an unknown but bounded disturbance. A more practical notionof stability is uniform ultimate boundedness (UUB). The next result showsthat UUB is guaranteed if the Lyapunov derivative is negative outsidesome bounded region of R n .THEOREM 2.4.6 (UUB BY LYAPUNOV ANALYSIS)If, for system as described in Equation 2.59, there exists a function Lxk ( , ) withncontinuous partial differences such that for x in a compact set S ⊂RLxk ( ( ), k) is positive definite, Lxk ( ( ), k)> 0∆L( xk ( ), k) < 0 for | x|> R