Wireless Ad Hoc and Sensor Networks

Background 77Section 2.4Problem 2.4.1: Lyapunov stability analysis. Using the Lyapunovstability analysis, examine stability for the following systems. Plottime histories to substantiate your claims.( )a. x ( k+ 1) = x ( k) x ( k) + x ( k)1 1 1 2 2 2( )x ( k+ 1) = x ( k) x ( k) + x ( k)2 2 1 2 2 22b. xk ( + 1) =−x( k) −10xk ( )sin xk ( )Problem 2.4.2: Lyapunov control design. Using Lyapunov techniquesdesign controllers to stabilize the following system. Plottime histories of the states to verify your design. Verify passivityand dissipativity of the systems.a. xk ( + 1) =−x 2 ( k)cos xk ( ) − 10xk ( )sin xk ( ) + uk ( )x1( k+ 1) = x1( k) x2( k)b.x ( k+ 1) = x ( k) − sin x ( k)+ u( k)2 1 2 2Problem 2.4.3: Stability improvement using feedback. The systemxk ( + 1) = Axk ( ) + Buk ( ) + dk ( )has a disturbance dk ( ) that is unknown but bounded so that|( dk)| < d M , with the bounding constant known. In Example 2.4.6,the system with no control input, B = 0, and A stable was shown tobe UUB. Show that by selecting the control input as uk ( ) =−Kxk( ) itis possible to improve the UUB stability properties of the systemby making the bound on |( xk)|smaller. In fact, if feedback is allowed,the initial system matrix A need not be stable as long as ( AB , ) isstabilizable.