Error coefficients for k(s+2)/[s²(s²+7s+12)]

Importance of error coefficients

Static error coefficients ($K_p$, $K_v$, $K_a$) measure how well a closed-loop system follows standard test inputs in steady state. Their role in design:

• They give the steady-state error directly: step $e_{ss}=\frac{A}{1+K_p}$, ramp $e_{ss}=\frac{A}{K_v}$, parabola $e_{ss}=\frac{A}{K_a}$.
• They act as design specifications: e.g. "$K_v \ge 20$" fixes the minimum gain needed for a tracking system.
• They show the effect of the type of the system: adding an integrator (higher type) makes the error constant for that input infinite, so the error becomes zero.
• They show the trade-off between accuracy and stability: increasing gain increases $K_p, K_v, K_a$ (better accuracy) but usually reduces relative stability. Lag/PI compensators are designed mainly to raise error constants without disturbing the transient response.

(i) Type and order

$$G(s) = \frac{k(s+2)}{s^2(s^2+7s+12)} = \frac{k(s+2)}{s^2(s+3)(s+4)}$$

• Two poles at the origin: Type 2.
• Highest power of $s$ in the denominator is $s^4$: Order 4.

(ii) Error coefficients

$$\begin{aligned} K_p &= \lim_{s\to0}G(s) = \infty \\ K_v &= \lim_{s\to0}sG(s) = \infty \\ K_a &= \lim_{s\to0}s^2G(s) = \frac{k\times2}{12} = \frac{k}{6} \end{aligned}$$

(iii) Steady-state error for $r(t)=2.5t^2$

Standard parabolic input is $r(t)=\frac{A t^2}{2}$, so $\frac{A}{2}=2.5 \Rightarrow A=5$.

$$e_{ss} = \frac{A}{K_a} = \frac{5}{k/6} = \frac{30}{k}$$

Answer: Type 2, order 4; $K_p=\infty$, $K_v=\infty$, $K_a = k/6$; $e_{ss} = 30/k$.

Note on stability: the error formula is valid only if the closed loop is stable. The characteristic equation is $s^4+7s^3+12s^2+ks+2k=0$. Its Routh array has first column $1,\ 7,\ \frac{84-k}{7},\ \frac{k(k+14)}{k-84},\ 2k$, which always has a sign change for $k>0$. So this particular closed loop is unstable for every positive $k$; $30/k$ is the theoretical value that would apply once the system is stabilised (e.g. by a lead compensator).