Wireless Ad Hoc and Sensor Networks

68 Wireless Ad Hoc and Sensor Networkswhere P is a constant symmetric positive definite matrix. Because P > 0,Tthen xPxis a positive function. This function is a generalized norm,which serves as a system energy function. Then,1 TT∆L( xk ( )) = Lxk ( ( + 1)) − Lxk ( ( )) = [ x ( k+ 1) Pxk ( + 1) − x ( k) Px( k)]2(2.43)1= −2 x Tk A T( )[ PA P ] x ( k )(2.44)For stability, one requires negative semidefiniteness. Thus, there mustexist a symmetric positive semidefinite matrix Q, such thatT∆L( x) =−x ( kQxk ) ( )(2.45)This results in the next theorem.THEOREM 2.4.4 (LYAPUNOV THEOREM FOR LINEAR SYSTEMS)The system discussed in Equation 2.41 is SISL, if there exist matrices P > 0,Q ≥ 0 that satisfy the Lyapunov equationTAPA− P=−Q(2.46)If there exists a solution such that both P and Q are positive definite,the system is AS.It can be shown that this theorem is both necessary and sufficient. Thatis, for LTI systems, if there is no Lyapunov function of the quadratic formdescribed in Equation 2.42, then there is no Lyapunov function. This resultprovides an alternative to examining the eigenvalues of the A matrix.2.4.5 Lyapunov Design of LTI Feedback ControllersThese notions offer a valuable procedure for LTI control system design.Note that the closed-loop system with state feedbackxk ( + 1) = Axk ( ) + Buk ( )(2.47)u=−Kx(2.48)