Wireless Ad Hoc and Sensor Networks

Background 67Now, select the feedback controlThis yields,uk ( ) =− x( k)sgn ( x( k)) + x ( k) −x ( k) x ( k)2 2 2 1 1 2 1 22∆L( xk ( )) =−x 2 ( k)so that Lxk ( ( )) is rendered a (closed-loop) Lyapunov function. Because∆L( xk ( )) is negative semidefinite, the closed-loop system with this controlleris SISL.It is important to note that by slightly changing the controller, one can alsoshow global asymptotic stability of the closed-loop system. Moreover, notethat this controller has elements of feedback linearization (discussed in Lewis,Jagannathan, and Yesiderek 1999), in that the control input uk ( ) is selected tocancel nonlinearities. However, no difference of the right-hand side of thestate equation is needed in the Lyapunov approach, but the right-hand sidebecomes quadratic, which makes it hard to design controllers and showstability. This will be a problem for the discrete-time systems, and we willbe presenting how to select suitable Lyapunov function candidates for complexsystems when standard adaptive control and NN-based controllers aredeployed. Finally, there are some issues in this example, such as the selectionof the discontinuous control signal, which could cause chattering. In practice,the system dynamics act as a low-pass filter so that the controllers work well.2.4.3 Lyapunov Analysis and Controls Design for Linear SystemsFor general nonlinear systems, it is not always easy to find a Lyapunovfunction. Thus, failure to find a Lyapunov function may be because thesystem is not stable, or because the designer simply lacks insight andexperience. However, in the case of LTI systemsxk ( + 1)= Ax(2.41)Lyapunov analysis is simplified and a Lyapunov function is easy tofind, if one exists.2.4.4 Stability AnalysisSelect as a Lyapunov function candidate, the quadratic formTLxk ( ( )) = 1 x ( kPxk ) ( ),2(2.42)