Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf

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216

DISCRETE CONTROLLER DESIGN

R(z)

E(z)

Y(z)

z 3− 0.904z2

U(z)

0.095

0.095(z 3− 1)

+

_

z 3 − 0.904z 2

HG(z)

D(z)

Figure 9.4 Block diagram of the system of Example 9.1

For realizability, we can choose k ≥ 3. Choosing k = 3, we obtain

D(z)

=

1 − 0.904z−1

z−3

or

0.095z

−

3

1

−

z

−

3

D(z)

z3 − 0.904z2

.

= 0.095(z3 − 1)

Figure 9.4 shows the system block diagram with the controller, while Figure 9.5 shows the step response of the system. The output response is unity after 3 s (third sample) and stays at this value. It is important to realize that the response is correct only at the sampling instants and the response can have an oscillatory behaviour between the sampling instants.

The control signal applied to the plant is shown in Figure 9.6. Although the dead-beat controller has provided an excellent response, the magnitude of the control signal may not be acceptable, and it may even saturate in practice.

Figure 9.5 Step response of the system

DIGITAL CONTROLLERS

217

Figure 9.6 Control signal

The dead-beat controller is very sensitive to plant characteristics and a small change in the plant may lead to ringing or oscillatory response.

9.1.2 Dahlin Controller

The Dahlin controller is a modification of the dead-beat controller and produces an exponential response which is smoother than that of the dead-beat controller.

The required response of the system in the s-plane can be shown to be

1

e−as

Y (s) =

,

s

1

+

sq

where a and q are chosen to give the required response (see Figure 9.7). If we let a = k T , then the z-transform of the output is

Y (z) =

z−k−1(1 − e−T /q )

(1 − z−1)(1 − e−T /q z−1)

and the required transfer function is

T (z)

Y (z)

z−k−1(1 − e−T /q )

(1 − z−1)

= R(z)

= (1

−

z

−

1)(1

−

−

T /q z

−

1) 1

e

218 DISCRETE CONTROLLER DESIGN

Time

constant q

0

a

q

Figure 9.7 Dahlin controller response

or

T (z)

=

z−k−1(1 − e−T /q )

.

1

−

e

−

T /q z

−

1

Using (9.3), we can find the transfer function of the required controller:

D(z)

=

1

T (z)

1

z−k−1(1 − e−T /q )

.

HG(z) 1 − T (z)

= HG(Z ) 1 − e−T /q z−1 − (1 − e−T /q )z−k−1

An example is given below to illustrate the use of the Dahlin controller.

Example 9.2

The open-loop transfer function of a plant is given by

G(s) =

e−2s

.

1

+

10s

Design a Dahlin digital controller for the system. Assume that T = 1 s.

Solution

The transfer function of the system with a zero-order hold is given by

e

−

s T

2s

HG(z) = Z

1 −

G(s) = (1 − z−1)Z

e−

s

s(1

10s)

or

+

1

10

HG(z) = (1 − z−1)z−2 Z

= (1 − z−1)z−2 Z

1/

.

s(1

10s)

s(s 1/10)

From z-transform tables we obtain

+

+

HG(z)

=

(1

−

z−1)z−2

z(1 − e−0.1)

z−3

(1 − e−0.1)

(z

−

1)(z

−

e

−

0.1) =

1

−

e

−

0.1z

−

1

or

HG(z) =

0.095z−3

.

1 − 0.904z−1

DIGITAL CONTROLLERS

219

Figure 9.8 System response with Dahlin controller

For the controller, if we choose q = 10, then

D(z)

=

1

T (z)

=

1 − 0.904z−1

z−k−1(1 − e−0.1)

or

HG(z) 1 − T (z)

0.095z−3

1 − e−0.1z−1 − (1 − e−0.1)z−k−1

1 − 0.904z−1

D(z)

=

0.095z−k−1

0.095z−3 1 − 0.904z−1 − 0.095z−k−1

For realizability, if we choose k = 2, we obtain

D(z)

=

0.095z3 − 0.0858z2

.

0.095z3 − 0.0858z2 − 0.0090

Figure 9.8 shows the step response of the system. It is clear that the response is exponential as expected.

The response of the controller is shown in Figure 9.9. Although the system response is slower, the controller signal is more acceptable.

9.1.3 Pole-Placement Control – Analytical

The response of a system is determined by the positions of its closed-loop poles. Thus, by placing the poles at the required points we should be able to control the response of a system.

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