Hardware Reference

In-Depth Information

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,

(3.3)

with zero-order hold (ZOH). Then its discretized model is as follows:

x(k +1) = Ax(k)+Bu(k),

y(k)=Cx(k),

(3.4)

where

Z

T

A = exp[A
c
,T],B=

exp[A
c
,t]dtB
c
.

0

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 [166], [27], [167].

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 [101]. 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) ([54]), Genetic Algorithm (GA) ([147], [62]), neural