Wireless Ad Hoc and Sensor Networks

60 Wireless Ad Hoc and Sensor NetworksThe term ultimate indicates that the boundedness property holds after antime lapse N. If S =R , the system is said to be globally UUB (GUUB).2.3.4 A Note on Autonomous Systems and Linear SystemsIf the system is autonomous so thatxk ( + 1) = f( xk ( ))(2.28)where f( x( k))is not an explicit function of time, the state trajectory isindependent of the initial time. This means that if an equilibrium pointis stable by any of the three definitions, the stability is automaticallyuniform. Nonuniformity is only a problem with nonautonomous systems.If the system is linear so thatxk ( + 1) = Akxk ( ) ( )(2.29)with A(k) being an n×n matrix, then the only possible equilibrium pointis the origin.For linear time-invariant (LTI) systems, matrix A is time-invariant. Thenthe system poles are given by the roots of the characteristic equation∆( z) = | zI− A| =0,(2.30)where||⋅ is the matrix determinant, and z is the Z transform variable. ForLTI systems, AS corresponds to the requirement that all the system polesstay within the unit disc (i.e., none of them are allowed on the unit disc).SISL corresponds to marginal stability, that is, all the poles are within theunit disc, and those on the unit disc are not repeated.2.4 Nonlinear Stability Analysis and Controls DesignFor LTI systems, it is straightforward to investigate stability by examiningthe locations of the poles in the s-plane. However, for nonlinear or nonautonomous(e.g., time-varying) systems there are no direct techniques.The (direct) Lyapunov approach provides methods for studying thestability of nonlinear systems and shows how to design control systemsfor such complex nonlinear systems. For more information see Lewis,Abdallah, and Dawson (1993), which deals with robot manipulator control,