Wireless Ad Hoc and Sensor Networks

Background 61as well as Jagannathan (2006), Goodwin and Sin (1984), Landau (1979),Sastry and Bodson (1989), and Slotine and Li (1991), which have proofsand many excellent examples in continuous and discrete-time.2.4.1 Lyapunov Analysis for Autonomous SystemsThe autonomous (time-invariant) dynamical systemxk ( + 1) = f( xk ( )),(2.31)nx ∈R , could represent a closed-loop system after the controller has beendesigned. In Section 2.3.1, we defined several types of stability. We shallshow here how to examine stability properties using a generalized energyapproach. An isolated equilibrium point x e can always be brought to theorigin by redefinition of coordinates; therefore, let us assume without lossof generality that the origin is an equilibrium point. First, we give somedefinitions and results. Then, some examples are presented to illustratethe power of the Lyapunov approach.nLet Lx ( ):R →R be a scalar function such that L(0) = 0, and S be acompact subset of R n . Then Lx ( ) is said to be,Locally positive definite, if L(x) > 0 when x ≠ 0, for all x∈ S.(Denotedby Lx ( ) > 0.)Locally positive semidefinite, if L(x) ≥ 0 when x ≠ 0, for all x∈ S.(Denoted by Lx ( ) ≥ 0.)Locally negative definite, if L(x) < 0 when x ≠ 0, for all x∈ S.(Denotedby Lx ( ) < 0.)Locally negative semidefinite, if L(x) ≤ 0 when x ≠ 0, for all x∈ S.(Denoted by Lx ( ) ≤ 0.)An example of a positive definite function is the quadraticTform Lx ( ) = x Px,where P is any matrix that is symmetric and positivedefinite. A definite function is allowed to be zero only when x = 0, asemidefinite function may vanish at points where x ≠ 0. All these definitionsare said to hold globally if S =R .nnA function Lx ( ):R →R with continuous partial differences (or derivatives)is said to be a Lyapunov function for the system described innTheorem 2.4.1, if, for some compact set S ⊂R , one has locally:∆L( x)is negative semidefinite,L(x) is positive definite, L(x) > 0 (2.32)∆L( x)≤ 0(2.33)