Response sequence with digital PI controller

Step 1: Pulse transfer functions

PI controller ($K_P = 1$, $K_I = 0.2$):

$$G_D(z) = K_P + \frac{K_I}{1-z^{-1}} = \frac{1.2 - z^{-1}}{1-z^{-1}} = \frac{1.2z - 1}{z-1}$$

ZOH + plant $1/[s(s+1)]$ with $T = 1$:

$$G(z) = (1-z^{-1})\,\mathcal{Z}\left[\frac{1}{s^2(s+1)}\right] = \frac{0.3679z + 0.2642}{(z-1)(z-0.3679)}$$

Step 2: Closed-loop pulse transfer function

$$\begin{aligned} G_DG &= \frac{(1.2z-1)(0.3679z+0.2642)}{(z-1)^2(z-0.3679)} = \frac{0.4415z^2 - 0.0508z - 0.2642}{z^3 - 2.3679z^2 + 1.7358z - 0.3679} \\ \frac{C(z)}{R(z)} &= \frac{0.4415z^2 - 0.0508z - 0.2642}{z^3 - 1.9264z^2 + 1.6850z - 0.6321} \end{aligned}$$

Step 3: Kronecker delta input

For $r(0) = 1$ and $r(k) = 0$ for $k \neq 0$, we have $R(z) = 1$, so $C(z)$ equals the closed-loop pulse transfer function.

The closed-loop poles are the roots of $z^3 - 1.9264z^2 + 1.6850z - 0.6321 = 0$:

$$p_1 = 0.8096,\qquad p_{2,3} = 0.5584 \pm j0.6848 = 0.8836\,\angle \pm 0.8867\ \text{rad}\ (\pm 50.80^\circ)$$

All the poles are inside the unit circle, so the system is stable.

Step 4: Inverse z-transform (residue method)

$$c(k) = \sum \text{Res}\left[C(z)z^{k-1}\right] = \sum_i A_i\,p_i^{\,k-1}, \qquad A_i = \frac{N(p_i)}{D'(p_i)}$$

• $A_1 = -0.0301$ for $p_1 = 0.8096$
• $A_{2,3} = 0.2358 \mp j0.4094 = 0.4724\,\angle \mp 1.0483\ \text{rad}$

The complex pair combines into one real cosine term:

$$c(k) = 0.9448\,(0.8836)^{k-1}\cos\big[0.8867(k-1) - 1.0483\big] - 0.0301\,(0.8096)^{k-1}, \quad k \ge 1$$

and $c(0) = 0$, because the numerator degree is less than the denominator degree.

Step 5: First few values

Check the closed form against long division or the difference equation $c(k) = 1.9264c(k-1) - 1.6850c(k-2) + 0.6321c(k-3) + 0.4415r(k-1) - 0.0508r(k-2) - 0.2642r(k-3)$.

For example, at $k = 1$: $0.9448\cos(-1.0483) - 0.0301 = 0.4716 - 0.0301 = 0.4415$ (here $0.9448\cos 1.0483 = 2\,\text{Re}(A_2) = 0.4716$). ✔

Answer: $c(kT) = 0.9448(0.8836)^{k-1}\cos[0.8867(k-1) - 1.0483] - 0.0301(0.8096)^{k-1}$ for $k = 1, 2, 3, \dots$ (with $c(0) = 0$). This is a damped oscillation that decays to zero.