Second-Order Dominant-Poles Models in s- and Frequency Domain: 22-Elec-A2 Systems and Control, December 2019, Q8

Question 8: Second-Order Dominant-Poles Models in s- and Frequency Domain (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $G_{cl}(s) = 60(s+50)/[(s+30)(s^2+4s+100)]$, together with two plots. Read from Figure Q8.1: steady-state value 1.0, first peak 1.51 at $t_p = 0.32$ s. Read from Figure Q8.2: low-frequency magnitude 1.0, resonant peak $M_r = 2.5$ at $\omega_r \approx 9.6$ rad/s.

Find. A justification for the dominant-pair reduction and the analytic model $G_{m1}$; the model $G_{m2}$ identified from the step plot; the model $G_{m3}$ identified from the magnitude plot; and a comparison of the three.

Check: figure reads. The values above were read from the printed figures. Graphical reads carry roughly $\pm 0.02$ on the peak amplitude and $\pm 0.02$ s on the peak time, which propagates to about $\pm 2\%$ on $\zeta$ and $\omega_n$ — the level of agreement seen between the three models below is therefore about as good as chart reading permits.

Approach. Factor the denominator to expose the pole pattern and test the separation ratio, then read $\zeta$ and $\omega_n$ three independent ways — analytically from the dominant pair, from overshoot and peak time on the step plot, and from resonant peak and resonant frequency on the magnitude plot — normalising each model to the correct DC gain.

Factor and examine the poles (part 1). The quadratic factor gives $$s = \frac{-4 \pm \sqrt{16-400}}{2} = -2 \pm j9.798$$ so the three closed-loop poles are $-30$ and $-2 \pm j9.798$, with a single finite zero at $-50$. The real pole sits $30/2 = 15$ times further from the imaginary axis than the complex pair, and the zero is further still. Since each mode decays as $e^{\sigma t}$, the mode at $-30$ dies out roughly $e^{28t}$ times faster than the pair — it has effectively vanished within about 0.13 s, while the oscillation persists for some 2 s. Both the far pole and the far zero therefore contribute negligible residues, and a second-order dominant-poles model is appropriate. The usual engineering threshold is a separation of at least five; here it is fifteen.
Write the analytic model $G_{m1}$. The dominant pair alone gives $$\omega_n = |{-2+j9.798}| = \sqrt{4+96} = 10\ \text{rad/s}, \qquad \zeta = \frac{2}{10} = 0.20$$ The model must reproduce the true DC gain, and $G_{cl}(0) = (60)(50)/[(30)(100)] = 1$, so $K_{dc} = 1$ and $$\boxed{\,G_{m1}(s) = \frac{100}{s^2+4s+100}\,}$$
Read the step plot (part 2). With a steady-state value of 1.0 and a first peak of 1.51, $$PO = \frac{1.51-1.0}{1.0}\times100 = 51\% \;\Longrightarrow\; \zeta = \frac{-\ln(0.51)}{\sqrt{\pi^2+\ln^2(0.51)}} = 0.2096$$ The peak time fixes the damped frequency, $t_p = \pi/\omega_d = 0.32$ s, so $$\omega_d = \frac{\pi}{0.32} = 9.817\ \text{rad/s}, \qquad \omega_n = \frac{\omega_d}{\sqrt{1-\zeta^2}} = 10.04\ \text{rad/s}$$ With $K_{dc} = 1$ read directly off the settled value, $$\boxed{\,G_{m2}(s) = \frac{100.8}{s^2+4.208s+100.8}\,}$$
Read the magnitude plot (part 3). The resonant peak of a second-order system relative to its DC gain is $M_r/K_{dc} = 1/(2\zeta\sqrt{1-\zeta^2})$. With $K_{dc} = 1$ and $M_r = 2.5$, solving $$2\zeta\sqrt{1-\zeta^2} = \frac{1}{2.5} = 0.4 \;\Longrightarrow\; \zeta = 0.2043$$ (the smaller of the two roots, the lightly damped one consistent with a tall peak). The resonant frequency then gives $\omega_n$ through $\omega_r = \omega_n\sqrt{1-2\zeta^2}$: $$\omega_n = \frac{9.6}{\sqrt{1-2(0.2043)^2}} = \frac{9.6}{0.9574} = 10.03\ \text{rad/s}$$ so $$\boxed{\,G_{m3}(s) = \frac{100.6}{s^2+4.098s+100.6}\,}$$

Figure 8.1 — Step response of the full third-order Gcl(s) with the second-order model Gm1(s) overlaid. The two are almost indistinguishable, which is the dominant-pair reduction verified directly.

Figure 8.2 — Closed-loop magnitude response. The resonant peak of about 2.55 occurs near 9.6 rad/s, matching the reading taken from Figure Q8.2 of the paper.

Compare the three models (part 4). The agreement is close:

Model	Source	$\zeta$	$\omega_n$ (rad/s)	$K_{dc}$
$G_{m1}$	Analytic, from the pole pair	0.2000	10.00	1
$G_{m2}$	Step response, Fig. Q8.1	0.2096	10.04	1
$G_{m3}$	Magnitude plot, Fig. Q8.2	0.2043	10.03	1

All three natural frequencies agree to better than 0.5% and the damping ratios spread by under 5%, which is well inside chart-reading tolerance. The two graphical models both return a slightly higher damping ratio than the analytic one, and that bias is real rather than random: the neglected pole at $-30$ removes a little of the initial rise and the zero at $-50$ mildly reshapes the peak, so the measured overshoot and resonant peak are marginally smaller than a pure second-order system with $\zeta = 0.20$ would produce, and inverting those relations returns a slightly larger $\zeta$.

$G_{m1}$ is the most accurate. It is derived exactly from the known transfer function, with no measurement uncertainty at all, and the two graphical models simply confirm it to within reading error. $G_{m2}$ and $G_{m3}$ are nonetheless the more valuable techniques in practice: they are the only ones available when the plant transfer function is unknown and all one has is measured data. Of those two, the frequency-domain reading is usually the more reliable because a resonant peak can be measured with a swept sine to high precision, whereas peak time on a transient trace is difficult to pinpoint. The overlays above show both the full third-order response and $G_{m1}$; the curves are almost indistinguishable, confirming the reduction.

Quantity	Result
Closed-loop poles	$-30$, $-2 \pm j9.798$
Separation ratio	15:1 — dominant-pair reduction justified
DC gain	$G_{cl}(0) = 1$
$G_{m1}(s)$ (analytic)	$\dfrac{100}{s^2+4s+100}$; $\zeta = 0.200$, $\omega_n = 10.00$
$G_{m2}(s)$ (step plot)	$\dfrac{100.8}{s^2+4.208s+100.8}$; $\zeta = 0.2096$, $\omega_n = 10.04$
$G_{m3}(s)$ (magnitude plot)	$\dfrac{100.6}{s^2+4.098s+100.6}$; $\zeta = 0.2043$, $\omega_n = 10.03$
Most accurate	$G_{m1}$ — exact, no measurement error

Topic. Reduction of a higher-order closed loop to a second-order dominant-poles model, and the three independent routes to its parameters.

Key relations. $\omega_n = |p|$ and $\zeta = -\text{Re}(p)/|p|$ for a complex pole pair; $PO = 100e^{-\zeta\pi/\sqrt{1-\zeta^2}}$ and $t_p = \pi/\omega_d$ with $\omega_d = \omega_n\sqrt{1-\zeta^2}$; $M_r/K_{dc} = 1/(2\zeta\sqrt{1-\zeta^2})$ and $\omega_r = \omega_n\sqrt{1-2\zeta^2}$ (a resonant peak exists only for $\zeta \lt 0.707$).

Why this works. In a partial-fraction expansion each pole contributes a term weighted by its residue and decaying at its own rate. A pole far to the left both decays quickly and, because the residue involves the distance to the other poles, carries a small weight, so its contribution is doubly suppressed. That is why a separation ratio is the right test, and why the rule of thumb is stated as a ratio rather than an absolute distance. The three parameter routes work because a standard second-order system has exactly two free parameters, so any two independent measurements — two time-domain features, or two frequency-domain features — determine it completely; the agreement between the routes is then a consistency check on the model assumption itself.

Common pitfall. Forgetting to normalise the DC gain is the most common error: writing $G_{m1} = 1/(s^2+4s+100)$ instead of $100/(s^2+4s+100)$ leaves a model whose steady-state value is 0.01 rather than 1. On the frequency side, candidates routinely confuse $\omega_r$ with $\omega_n$; they differ by $\sqrt{1-2\zeta^2}$, which is only 4% here but grows rapidly as damping increases and vanishes entirely at $\zeta = 0.707$, above which there is no peak to read. Finally, the quadratic in $\zeta$ from the resonant-peak relation has two roots — the physically meaningful one for a tall peak is the smaller.

Source. Dorf & Bishop, Modern Control Systems, 13th ed., §5.3 and §8.2 (performance and dominant poles); Nise, Control Systems Engineering, 8th ed., §4.8 and §10.10.

Question 8 of 8: Second-Order Dominant-Poles Models in s- and Frequency Domain

Question 8: Second-Order Dominant-Poles Models in s- and Frequency Domain (20 marks)