gain for a low pass filter. This method is somewhat heuristic and may not
produce a desirable compensator.
Main disadvantage of designing controller based on an analog prototype is
that the discrete compensator is only an approximation to the designed analog
prototype. The analog prototype sets the upper bound on the effectiveness of
the closed-loop response using the digital compensator.
3.1.2 Fast Dynamic Systems
Fast dynamic system refers to a system with a relatively slow sampling rate.
Let us consider a continuous-time dynamic system described by the following
x = A c x + B c u,
y = C c x,
with zero-order hold (ZOH). Then its discretized model is as follows:
x(k +1) = Ax(k)+Bu(k),
A = exp[A c ,T],B=
exp[A c ,t]dtB c .
Thedesignercanthenuseoneofthemanywelldefined methods for finding
the discrete controller of the discrete-time model of the plant. Available design
methods include shaping of the open loop transfer function, pole placement,
minimizing certain norm or cost function etc , , .
3.1.3 Numerical Search to Find Controller Parameters
Various gains or coeﬃcients of the controller must be found so that the desired
performance criteria are met. To do this, it is a common practice to define a
performance index or cost function C that includes in some way all the design
criteria. The cost function may put different weightage on different design
objectives. The coeﬃcients or parameters of the controller must minimize the
cost function. One possible way of finding the minimum of the cost function
involves computation of the gradient of the cost function. As an alternative,
numerical search methods that do not require gradient information can be uti-
lized to search for the controller parameters directly in the closed-loop control
system . The size of the search space which is supposed to contain the
optimum point and the relative weighting of various terms in the performance
index are up to the designer to decide.
Among those non-gradient based optimization methods, Random Neigh-
borhood Search (RNS) (), Genetic Algorithm (GA) (, ), neural