Figure 4
Discretization using sampling. (A) This diagram illustrates the use of sampling (S) and first-order-hold (H) to discretize a continuous-time system. The sampling circuit's output is equal to the inputs at the sampling times. The first-order-hold circuits generate a piecewise-constant signal equal to the inputs. If we introduce two copies of these circuits into the loop of Fig. 3B, and group the subsystems as shown here, the effect is to generate a discrete-time nominal model G d as well as a discrete-time uncertainty structure Δ d . (B) The validity of this approximation will depend on the value of the sampling period, h = T/n, chosen. For the Laub & Loomis model, the error for values of n greater than 8 is negligible. A comparison of the responses of the non-linear (solid blue), linearized continuous-time (dashed green) and discrete-time (dotted red) systems when n = 16 is shown, where we have plotted x1 as a function of time for all responses. The latter two have been superimposed onto the nominal limit cycle (x*(t)) computed using the harmonic balance method.