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

z−1

rk−1

−b1 a1

z−1

rk−2

−b2 a2

z−1

rk−n

−bn an

Figure 10.1 Canonical direct structure

10.1.2 Direct Noncanonical Structure

Consider Equation (10.2) with b0 = 1:

Cross multiplying and rewriting this equation we obtain

Writing (10.11) in the time domain, we obtain the noncanonical form of the direct realization

n n

Figure 10.2 Block diagram for Example 10.1

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Figure 10.3 Noncanonical direct structure

The block diagram of the noncanonical direct realization is shown in Figure 10.3. This structure has only one adder, but 2n delay elements.

Example 10.2

The transfer function of a digital controller is found to be

1 + 2z−1 + 4z−2 1 + 2z−1 + 5z−2 .

Draw the block diagram of the direct noncanonical realization of this controller.

Solution

With reference to (10.12) and Figure 10.3, we can draw the required block diagram as in Figure 10.4.

10.2 CASCADE REALIZATION

The cascade realization is less sensitive to coefficient sensitivity problems. In this method the transfer function is implemented as a product of first-order and second-order transfer functions. Thus, the controller transfer function is written as

Figure 10.4 Block diagram for Example 10.2

−β

Figure 10.5 Realization of P (z)

and

where m is the smallest integer greater than or equal to n/2.P (z) in (10.13) is the first-order transfer function

In practice, in order to avoid coefficient sensitivity problems second-order transfer function modules of the form given by (10.18) are frequently used and the modules are cascaded to give the required order. The block diagram of Figure 10.6 is sometimes drawn vertically, as shown in Figure 10.7.

−b1

−b2

Figure 10.6 Realization of Q(z)

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Figure 10.7 Figure 10.6 drawn vertically

Example 10.3

The transfer function of a digital controller is given by 3(z + 1)(z + 2)

.

z2 + 0.4z + 0.03

Use two-first order cascaded transfer functions to implement this controller.

Solution

The transfer function can be factorized as

3(z + 1)(z + 2) 3(1 + z−1)(1 + 2z−1) (z + 0.1)(z + 0.3) = (1 + 0.1z−1)(1 + 0.3z−1) .

The required cascaded realization is shown in Figure 10.8.

Example 10.4

The transfer function of a digital controller is given by

(1 + 0.6z−1)(1 + 2z−1 + 4z−2) (1 + 0.4z−1)(1 + 0.1z−1 + 0.3z−2) .

Use a first-order and a second-order cascaded transfer function to implement this controller.

Solution

The transfer function can be implemented as a cascade of a first-order and a second-order function, as shown in Figure 10.9.

Figure 10.8 Cascaded realization for Example 10.3