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Page 277

integ.fig


7.3. Other Popular Controllers 261

×

Im(z)

Re(z)

Figure 7.17: Pole–zero location of the integral mode

Taking the Z-transform of both sides, we obtain

Y (z) = z−1Y (z) +
Ts
2
[
U(z) + z−1U(z)

]
Simplifying this, we obtain

Y (z) =
Ts
2

1 + z−1

1− z−1U(z) (7.38)

If we denote the transfer function of the integrator by Gi, we see that it is given by

Gi(z) =
Ts
2
z + 1
z − 1 . (7.39)

This is known as the trapezoidal approximation or Tustin approximation or bilinear
approximation. The transfer function Gi is low pass as it has a pole at ω = 0 and a
zero at ω = π, see Fig. 7.17. Thus we see that integration, a smoothing operation, is
low pass. We explain the role of the integral mode with an example.

Example 7.4 Evaluate the effect of an integrating controller

Gi(z) =
z + 1
z − 1

when used with a nonoscillating plant given by

G(z) =
z

z − a
where a > 0. The closed loop transfer function becomes

T (z) =
z

z−a
z+1
z−1

1 + z
z−a

z+1
z−1

=
z(z + 1)

2z2 − az + a

The poles are at (a ±

a2 − 8a)/4. For all a < 8, the closed loop system is

oscillatory. When a PI controller of the following form is used

Gc(z) = K
(

1 +
1
τi

z + 1
z − 1

)

Page 278

262 7. Structures and Specifications

the overall transfer function becomes

T (z) =
K
(

1 + 1
τi

z+1
z−1

)
z

z−a

1 +K
(

1 + 1
τi

z+1
z−1

)
z

z−a

The steady state output for a step input is given by limn→∞ y(n), which is equal
to limz→1 T (z) = 1, see Eq. 4.21 on page 83. This shows that there is no steady
state offset.

Because the reciprocal of integration is differentiation, we obtain the transfer
function of discrete time differentiation Gd as the reciprocal of Gi, given in Eq. 7.39.
We obtain

Gd(z) =
1

Gi(z)
=

2
Ts

z − 1
z + 1

(7.40)

This form, however, has a problem: Gd(z) has a pole at z = −1 and hence it produces,
in partial fraction expansion, a term of the form

z

z + 1
↔ (−1)n (7.41)

which results in a wildly oscillating control effort, see Sec. 5.1.1. Because of this,
we consider other ways of calculating the area given in Fig. 7.16. For example, we
approximate the area under the curve with the following backward difference formula:

y(k) = y(k − 1) + Tsu(k) (7.42)
Taking the Z-transform and simplifying it, we obtain

Y (z) = Ts
z

z − 1U(z)

We arrive at the following transfer functions of the integrator and differentiator as

Gi(z) = Ts
z

z − 1
Gd(z) =

1
Ts

z − 1
z

(7.43)

It is also possible to use the forward difference approximation for integration

y(k) = y(k − 1) + Tsu(k − 1) (7.44)
This results in the following transfer functions of the integrator and differentiator as

Gi(z) =
Ts
z − 1

Gd(z) =
z − 1
Ts

(7.45)

It is easy to check that all the definitions of the differentiator given above are high
pass.

Page 554

Index of Matlab Code 543

ma.m, 227
ma pacf.m, 230
mat exp.m, 30
max ex.m, 226
miller.m, 433
motor.m, 371
motor pd.m, 378
mv.m, 429, 430
mv mac1.m, 429
mv nm.m, 431, 431, 432
mv visc.m, 432
myc2d.m, 331, 367, 367, 370, 371, 373,

374, 377, 378, 399, 400

nmp.m, 150
nyquist ex1.m, 296

oe est.m, 233

pacf ex.m, 228
pacf.m, 228, 229, 230, 235
pacf mat.m, 229, 229
pd.m, 378, 378
pend.m, 499, 516
pend model.m, 12, 30, 499
pend ss c.mdl, 499
pid neg.m, 377
plotacf.m, 226, 227, 228, 230, 235
pm 10.m, 429
poladd.m, 376, 397, 400, 428, 430, 432,

458, 481, 524
polmul.m, 369, 370, 372, 377, 430–433,

456–459, 481–483, 524
polsplit2.m, 368, 369, 370
polsplit3.m, 372, 372, 431
pp basic.m, 367, 369, 369, 524
pp im.m, 370, 371, 375, 380, 397
pp im2.m, 372, 376
pp pid, 297
pp pid.m, 375, 377, 377
putin.m, 482, 524
pz.m, 106

recursion ex1.m, 429
recursion.m, 428, 429
respol.m, 106, 106, 107, 108
respol1.m, 106
respol2.m, 107
respol3.m, 107
respol5.m, 108
respol6.m, 108
rlocus ex1.m, 296

selstruc, 217
sigurd his.m, 374
sigurd.m, 373
smith disc.mdl, 386, 397
smith.m, 397
spec ex.m, 481
spec.m, 481, 481
specfac.m, 481, 482, 483
stb disc.mdl, 352, 369, 371, 397, 525
stb disc sat.mdl, 358, 376
sumsq.m, 431

tfvar.m, 430, 431
type 2DOF.m, 375
type test.m, 297

unique ma.m, 228
unstb.m, 369

vande imc.m, 400
vande imc1.m, 399
visc imc1.m, 399

xdync.m, 298, 300, 332, 340, 369, 370,
372, 377, 409, 429–433,
456–460, 483, 524

ZOH1.m, 30
zpowk.m, 369, 369, 370–374, 376, 377,

397, 399, 400, 430, 431, 433,
460, 483

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