Wireless Ad Hoc and Sensor Networks

Background 51u(k)x n (k + 1)x n (k)x 2 (k)z −1 z −1 z −1x 1 (k)x(k)h(x(k))g(x(k))f (x(k))x(k)FIGURE 2.1Discrete-time single-input Brunovsky form.The discrete Brunovsky canonical form may equivalently be written asxk ( + 1) = Axk ( ) + bf( xk ( )) + bgxk ( ( )) uk ( )(2.4)where,⎡0 1 0 . . . 0⎤⎢⎥⎢0 0 1 . . . 0⎥⎢⎥⎢ . ⎥A = ⎢⎢.⎥⎥,⎢ . ⎥⎢⎥⎢0 0 . . . 1 0⎥⎣⎢0 0 0 . . . 0⎦⎥⎡0⎤⎢ ⎥⎢0⎥⎢ ⎥⎢ . ⎥b = ⎢⎢.⎥⎥⎢ . ⎥⎢ ⎥⎢0⎥⎣⎢1⎦⎥(2.5)A discrete-time form of the more general version may also be written.It is a system with m parallel chains of delay elements of lengthsn1, n2, …(e.g., m shift registers), each driven by one of the control inputs.Many practical systems occur in the continuous-time Brunovskyform. However, if a system of the continuous Brunovsky form (Lewis,Jagannathan, and Yesilderek 1999) is sampled, the result is not thegeneral form as in Equation 2.2. Under certain conditions, generaldiscrete-time systems of the form having the form of Equation 2.1 canbe converted to discrete Brunovsky canonical form systems (e.g.,Kalkkuhl and Hunt 1996).