Series programming of a third-order system

In series programming, the pulse transfer function is split into a cascade of first- and second-order sections. Each section is realized by standard programming.

Step 1: Split into sections (in powers of $z^{-1}$)

$$G(z) = \underbrace{\frac{1 + 3z^{-1}}{1 - 5z^{-1}}}_{G_1(z)}\cdot\underbrace{\frac{1 - 2z^{-1} + 10z^{-2}}{1 - z^{-1} + 8z^{-2}}}_{G_2(z)}$$

The quadratics have complex roots, so they stay as one second-order section with real coefficients.

Step 2: Difference equations

Section 1 (input $x$, output $y_1$):

• $w_1(k) = x(k) + 5w_1(k-1)$
• $y_1(k) = w_1(k) + 3w_1(k-1)$

Section 2 (input $y_1$, output $y$):

• $w_2(k) = y_1(k) + w_2(k-1) - 8w_2(k-2)$
• $y(k) = w_2(k) - 2w_2(k-1) + 10w_2(k-2)$

Step 3: Block diagram

Section 1:

x(k)-->(S)------------o--[1]-------->(S)--> y1(k)
        ^             |               ^
        |          [z^-1]             |
        |             | w1(k-1)       |
        +<[5]---------o--[3]----------+

Section 2:

y1(k)-->(S)------------o--[1]-------->(S)--> y(k)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-1)       |
        (S)<[1]--------o--[-2]------->(S)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-2)       |
         +<[-8]--------o--[10]---------+

x(k) --> [ G1(z) ] --y1(k)--> [ G2(z) ] --> y(k)

A total of 3 delays are used. The order of the sections can be swapped.

Note: The poles are $z = 5$ and $z = 0.5 \pm j2.78$ (with $|z| = 2.83$), all outside the unit circle. So this system is unstable, although it can still be realized.