082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

694 Chapter 22 Difference equations and the z transform
The most common type of controller used in industry is the proportional/integral/
derivative (p.i.d.)controller. Itcanbeshown thattheanalogue formofthiscontrolleris
modelled by the equation
m(t) =K p
e(t)+K i
∫ t
0
e(t)dt +K d
de(t)
dt
(22.4)
whereK p
,K i
andK d
are constants. In order to implement a discrete-time (digital) controlleritisnecessarytoconvertthisequationintoanequivalentdifferenceequation.The
approximation for the process of differentiation has already been examined in
Engineering application 22.3, and isgiven by
de(t)
dt
≈
e[n] −e[n −1]
T
Thereareseveralpossiblewaysofapproximatingtheprocessofintegration.Onemethod
is illustrated in Figure 22.14. Here the area under the curve is approximated by a series
of rectangles, each of widthT. If the approximate area under the curve fromt = 0 to
t =nT isdenoted byx[n], then
x[n] =x[n −1] +Te[n] (22.5)
The discrete form ofEquation (22.4) can now be formulated. Itisgiven by
m[n] =K p
e[n] +K i
x[n] +K d
e[n] −e[n −1]
T
(22.6)
Equations (22.5) and (22.6) form a set of equations toimplement a discrete form of the
p.i.d.controlleronadigitalcomputerormicroprocessor.Thesetwoequationsaretermed
coupled difference equations because both are needed to calculate m[n]. In addition,
Equation(22.5)isrecursive.Aflowchartforimplementingtheseequationsisshownin
Figure 22.15.
Initialization
Time to
sample?
YES
NO
Get e[n]
e(t)
Calculate
m[n]
Output m[n]
t
Figure22.14
Approximating the areaunder the curve by a
seriesofrectangles.
Figure22.15
Flow chartforap.i.d. controller.