Wireless Ad Hoc and Sensor Networks

Background 69is SISL if and only if there exist matrices P > 0, Q ≥ 0 that satisfy the closedloopLyapunov equationT( A−BK) P( A−BK)− P=−Q(2.49)If there exists a solution such that both P and Q are positive definite,the system is AS.Now suppose there exist P > 0, Q > 0 that satisfy the Riccati equationPk APk TT( ) = ( + )( I+ BR − 1 −11 BPk ( + 1))A+Q(2.50)Now select the feedback gain asKk R BPk TT( ) =− ( + ( + ) B) − 11 B P( k+1)A(2.51)and the control input asuk ( ) =−Kkxk( ) ( )(2.52)for some matrix R > 0.These equations verify that this selection of the control input guaranteesclosed-loop asymptotic stability.Note that the Riccati equation depends only on known matrices — thesystem (A, B) and two symmetric design matrices Q and R that need tobe selected positive definite. There are many good routines that can findthe solution P to this equation provided that (A, B) is controllable(e.g., MATLAB). Then, a stabilizing gain is given by Equation 2.51. Ifdifferent design matrices Q and R are selected, different closed-loop poleswill result. This approach goes far beyond classical frequency domain orroot locus design techniques in that it allows the determination of stabilizingfeedbacks for complex multivariable systems by simply solving amatrix design equation. For more details on this linear quadratic (LQ)design technique, see Lewis and Syrmos (1995).2.4.6 Lyapunov Analysis for Nonautonomous SystemsWe now consider nonautonomous (time-varying) dynamical systems ofthe formxk ( + ) = f( xk ( ), k),k≥k1 0(2.53)