Wireless Ad Hoc and Sensor Networks

58 Wireless Ad Hoc and Sensor Networkspassivity properties, this design procedure is simplified, and additionalclosed-loop properties such as robustness can be guaranteed. On the otherhand, properties such as stability may not be present in the original openloopsystem but are design requirements for closed-loop performance.Stability along with robustness (discussed in the following subsection)is a performance requirement for closed-loop systems. In other words,though the open-loop stability properties of the original system may notbe satisfactory, it is desirable to design a feedback control system suchthat the closed-loop stability is adequate. We will discuss stability fordiscrete-time systems, but the same definitions also hold for continuoustimesystems with obvious modifications.Consider the dynamical systemxk ( + 1) = f( xk ( ), k)(2.27)nwhere xk ( ) ∈R , which might represent either an uncontrolled openloopsystem, or a closed-loop system after the control input uk ( ) hasbeen specified in terms of the state xk ( ). Let the initial time be k 0 andthe initial condition be xk ( 0) = x0.This system is said to be nonautonomousbecause the time k appears explicitly. If k does not appearexplicitly in f (), ⋅ then the system is autonomous. A primary cause ofexplicit time dependence in control systems is the presence of timedependentdisturbances dk ( ).A state x e is an equilibrium point of the system f( xe, k) = 0, k≥k0.Ifx 0 = x e , so that the system starts out in the equilibrium state, then it willremain there forever. For linear systems, the only possible equilibriumpoint is x e = 0; for nonlinear systems x e may be nonzero. In fact, there mayeven be an equilibrium set, such as a limit cycle.2.3.1 Asymptotic StabilityAn equilibrium point x e is locally asymptotically stable (AS) at k 0 , if therenexists a compact set S ⊂R such that, for every initial condition x0 ∈ S,one has |( xk) −x e | →0 as k →∞. That is, the state xk ( ) converges to x e .nIf S =R , so that xk ( )→ x e for all xk ( 0),then x e is said to be globallyasymptotically stable (GAS) at k 0 . If the conditions hold for all k 0 , thestability is said to be uniform (e.g., UAS, GUAS).Asymptotic stability is a very strong property that is extremely difficultto achieve in closed-loop systems, even using advanced feedback controllerdesign techniques. The primary reason is the presence of unknown butbounded system disturbances. A milder requirement is provided asfollows.