Wireless Ad Hoc and Sensor Networks

50 Wireless Ad Hoc and Sensor Networks2.1.1 Discrete-Time SystemsIf the time index is an integer k instead of a real number t, the system issaid to be of discrete-time. A general class of discrete-time systems canbe described by the nonlinear ordinary difference equation in discrete-timestate space formxk ( + 1) = f( xk ( ), uk ( )), yk ( ) = hxk ( ( ), uk ( ))(2.1)nwhere xk ( ) ∈R is the internal state vector, u( k)∈R m is the contol input, andpyk ( )∈R is the system output.These equations may be derived directly from an analysis of the dynamicalsystem or process being studied, or they may be sampled or discretizedversions of continuous-time dynamics of a nonlinear system. Today, controllersare implemented in digital form by using embedded hardwaremaking it necessary to have a discrete-time description of the controller.This may be determined by design, based on the discrete-time systemdynamics. Sampling of linear systems is well understood with many designtechniques available. However, sampling of nonlinear systems is not aneasy topic. In fact, the exact discretization of nonlinear continuous dynamicsis based on the Lie derivatives and leads to an infinite series representation(e.g., Kalkkuhl and Hunt 1996). Various approximation and discretizationtechniques use truncated versions of the exact series.2.1.2 Brunovsky Canonical FormTLetting xk ( ) = [ x 1 ( k) … xn( k)] , a special form of nonlinear dynamics isgiven by the class of systems in discrete Brunovsky canonical formx ( k+ 1) = x ( k)1 2x ( k+ 1) = x ( k)2 3⋮(2.2)xn( k+ 1) = f( x( k)) + g( x( k)) u( k)yk ( ) = hxk ( ( ))As seen from Figure 2.1, this is a chain or cascade of unit delay elements z −1 ,that is, a shift register. Each delay element stores information and requiresan initial condition. The measured output yk ( ) can be a general functionof the states as shown, or can have more specialized forms such asyk ( ) hx ( ( k))= 1(2.3)