Series programming of a digital controller

In series (cascade) programming, $G(z)$ is factored into first- and second-order sections. Each section is realized by standard programming, and the sections are connected in cascade.

Step 1: Factor into sections (divide each factor by the highest power of $z$)

$$G(z) = 4\cdot\underbrace{\frac{1 - z^{-1}}{1 + 0.1z^{-1}}}_{G_1(z)}\cdot\underbrace{\frac{1 + 1.2z^{-1} + z^{-2}}{1 - 0.3z^{-1} + 0.8z^{-2}}}_{G_2(z)}$$

Step 2: Difference equations

Section 1: $G_1 = Y_1/X$, with intermediate variable $w_1$:

• $w_1(k) = x(k) - 0.1w_1(k-1)$
• $y_1(k) = w_1(k) - w_1(k-1)$

Section 2: input $y_1$, intermediate $w_2$

• $w_2(k) = y_1(k) + 0.3w_2(k-1) - 0.8w_2(k-2)$
• $y(k) = 4\big[w_2(k) + 1.2w_2(k-1) + w_2(k-2)\big]$

Step 3: Block diagram (two standard-programmed sections in cascade)

Section 1, $G_1(z)$:

x(k)-->(S)------------o--[1]-------->(S)--> y1(k)
        ^             |               ^
        |          [z^-1]             |
        |             | w1(k-1)       |
        +<[-0.1]------o--[-1]---------+

Section 2, $G_2(z)$, followed by the gain 4:

y1(k)-->(S)------------o--[1]-------->(S)->[4]-> y(k)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-1)       |
        (S)<[0.3]------o--[1.2]------>(S)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-2)       |
         +<[-0.8]------o--[1]----------+

The realization uses only $1 + 2 = 3$ delay elements, the minimum for a third-order filter. The overall gain 4 can be placed at either end.

Why series programming

• A coefficient error in one section moves only that section's poles and zeros. The structure is less sensitive to coefficient quantization than direct programming of the whole third-order polynomial.
• The complex poles $z = 0.15 \pm j0.882$ (with $|z| = 0.894$) are kept together in one real-coefficient second-order section.