M3-T1 — Transfer Functions, Poles and Zeros

Learning objectives

After this lesson, you should be able to:

• Write the transfer function of a linear model from its differential equation.
• Read the poles and zeros of a transfer function and say what each does to the response.
• Place the poles of a second-order loop to meet an overshoot and a settling time, and compute the feedback gains.
• Say what a left-half-plane and a right-half-plane zero do to a step response.

The intuition

From module 2 you have a model of Hopper's pitch axis, $\ddot{\theta} = \lambda^2\theta + b\,\delta$, a linear differential equation. Differential equations are awkward to combine: put a controller and an actuator in series with the plant, and the equation grows with every block. The Laplace transform removes the awkwardness. It replaces the derivative by a multiplication, and a chain of blocks becomes a product.

Exponentials go through unchanged

The key fact is about one family of inputs, the exponentials $e^{st}$. Feed a linear system with constant coefficients an input $e^{st}$ and its steady response is the same exponential scaled by a number, $G(s)\,e^{st}$. That number, as a function of $s$, is the transfer function. Because the derivative of $e^{st}$ is $s\,e^{st}$, you find $G(s)$ by writing $s$ wherever the equation has a derivative. Sines and cosines are exponentials with $s = j\omega$, which is why the same function gives the frequency response of the next lesson.

Poles: how the vehicle moves by itself

The poles are the values of $s$ at which the transfer function becomes infinite. Equivalently, they are the exponents $p$ for which $e^{pt}$ can exist with no input at all: the natural motions of the vehicle. A pole at $p = +0.7$ is a motion that grows as $e^{0.7t}$, which is Hopper's pitch divergence. A pole at $p = -2.8 + 2.86j$ is a decaying oscillation.

Reading the picture gives three rules.

• Real part: decay rate. A pole at $-\sigma$ dies as $e^{-\sigma t}$. The further left, the faster. A pole in the right half-plane grows.
• Imaginary part: oscillation. A pair at $-\sigma \pm j\omega_d$ oscillates at $\omega_d$.
• Angle from the negative real axis: damping. The cosine of that angle is the damping ratio $\zeta$. Poles on the negative real axis have $\zeta = 1$, poles on the imaginary axis have $\zeta = 0$.

Zeros: what the response is built from

A zero is a value of $s$ at which the transfer function vanishes. Zeros do not decide stability, but they shape the response. A zero in the left half-plane makes the output react more strongly to changes in the input, adding overshoot. A zero in the right half-plane makes the response start in the wrong direction before correcting, an effect called undershoot that the worked example shows and that limits what any controller can do (M3-T6: Limits of Performance: Delay, Right-Half-Plane Zeros and the Bode Integral).

Design means choosing poles

This is how Hopper's pitch loop is designed. Choose where the closed-loop poles should be (fast enough, damped enough), and compute the feedback gains that put them there. The loop of this lesson has two poles and two gains, a one-to-one match, so every placement is reachable.

A transfer function turns a differential equation into algebra. Its poles are the natural motions of the system, and a feedback design is a choice of where they go.

Definitions

Laplace variable $s$

A complex number. Replacing $d/dt$ by $s$ in a linear equation with zero initial conditions turns the equation into an algebraic one.

Transfer function $G(s)$

The ratio of the output to the input, as a function of $s$: $G(s) = N(s)/D(s)$, a ratio of two polynomials.

Poles and zeros

The roots of the denominator $D(s)$ (poles) and of the numerator $N(s)$ (zeros). $D(s) = 0$ is the characteristic equation.

Natural frequency $\omega_n$ and damping ratio $\zeta$

The two numbers that describe a pair of complex poles: its distance from the origin and the cosine of its angle from the negative real axis.

DC gain

The value of $G(0)$: the steady output per unit of steady input.

For Hopper's pitch axis, replacing each derivative by $s$ in $\ddot\theta = \lambda^2\theta + b\delta$ gives the plant from gimbal angle to pitch:

A pair of complex poles is described by a natural frequency and a damping ratio. The standard second-order system, and the location of its poles, are:

Its step response has an overshoot and a settling time that depend on the damping and the natural frequency:

Derivation: the pitch plant, the closed loop, and the specifications from the poles

Everything above follows from a few lines. First the transfer function, then the loop, then the formulas for the response.

1. $$\theta(t) = \Theta\,e^{st},\ \delta(t) = \Delta\,e^{st} \;\Rightarrow\; \ddot\theta = s^2\Theta\,e^{st}$$

   Try an exponential input and an exponential output at the same $s$, with constant amplitudes $\Delta$ and $\Theta$. Each derivative brings down a factor $s$.

2. $$s^2\Theta = \lambda^2\Theta + b\Delta \;\Rightarrow\; \frac{\Theta}{\Delta} = \frac{b}{s^2 - \lambda^2}$$

   Substitute into the pitch equation and divide out the common factor $e^{st}$. The ratio of the amplitudes is the transfer function.

3. $$\delta = K_p(\theta_{\text{ref}} - \theta) - K_d\,\dot\theta \;\Rightarrow\; s^2\Theta = \lambda^2\Theta + b\bigl(K_p(\Theta_{\text{ref}} - \Theta) - K_d\,s\,\Theta\bigr)$$

   Close the loop with a controller that pushes on the pitch error and brakes on the pitch rate (a PD law with the derivative taken on the measured rate). Substitute it for the gimbal angle.

4. $$\bigl(s^2 + b K_d\,s + (b K_p - \lambda^2)\bigr)\,\Theta = b K_p\,\Theta_{\text{ref}}$$

   Collect the terms in $\Theta$. The left side is the characteristic polynomial of the loop, and the right side gives the numerator of the transfer function from reference to pitch.

5. $$\omega_n^2 = b K_p - \lambda^2, \qquad 2\zeta\omega_n = b K_d$$

   Match the characteristic polynomial to $s^2 + 2\zeta\omega_n s + \omega_n^2$. Two gains and two unknowns: for any $\omega_n$ and $\zeta$ there is exactly one pair of gains: $K_p = (\omega_n^2 + \lambda^2)/b$ and $K_d = 2\zeta\omega_n/b$.

Where the specifications come from

1. $$\text{envelope} = e^{-\zeta\omega_n t}$$

   The response of a pair of poles at $-\zeta\omega_n \pm j\omega_d$ is a sinusoid under this decaying envelope. The real part of the pole alone sets the decay.

2. $$e^{-\zeta\omega_n t_s} = 0.02 \;\Rightarrow\; t_s = \frac{\ln 50}{\zeta\omega_n} \approx \frac{4}{\zeta\omega_n}$$

   The time at which the envelope has fallen to 2%. For $\zeta\omega_n = 2.8$ it is 1.4 s.

3. $$M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$$

   The first peak of the step response of the standard system occurs half a damped period after the start, at which time the envelope has fallen by exactly this factor. The overshoot depends on the damping alone, so on the angle of the poles and nothing else.

Place the poles, and the response follows: their distance from the imaginary axis sets how fast the response settles, and their angle sets how much it overshoots.

A worked example

Design the pitch loop of Hopper at liftoff ($b = 19.62$, $\lambda = 0.7$) for a natural frequency of 4 rad/s, the pitch-angle bandwidth of M1-T2: Nested Loops and Separation of Timescales, and a damping ratio of 0.7.

1. Open loop. The plant poles are at $\pm 0.7$. The one at +0.7 is the divergence of the pitch axis.
2. Gains. $K_d = 2 \times 0.7 \times 4/19.62 = 0.2854$ and $K_p = (16 + 0.49)/19.62 = 0.8405$. The $\lambda^2$ in $K_p$ is the price of the unstable pole: part of the proportional gain only cancels the divergence.
3. Closed-loop poles. The characteristic polynomial is $s^2 + 5.6\,s + 16$, with roots $-2.8 \pm 2.857j$, the blue crosses of the figure.
4. Predict. The overshoot is $e^{-\pi\cdot 0.7/\sqrt{0.51}} = 4.6\%$ and the settling time $4/(0.7\times 4) = 1.43$ s.
5. Simulate. A step of the pitch reference gives an overshoot of 4.6%, as predicted, a 10 to 90% rise time of 0.53 s and a settling time of 1.49 s (the formula is an approximation, which is why it is 4% off). The steady value is 1.031, not 1: the DC gain $bK_p/(bK_p - \lambda^2)$ is above 1 because the loop spends part of its stiffness holding the unstable axis in place. Module 4 adds integral action to remove it.
6. Zeros. The last three lines show what a zero does to the standard system $s^2 + 2.8s + 4$. With no zero the peak is 1.046. A zero at −2 raises it to 1.112. A zero at +2 gives a peak of 1.057 and a dip to −0.249 first: the output goes the wrong way before it goes the right way.

import math
import gnckit as g

b, lam = 19.62, 0.7                     # Hopper at liftoff: pitch authority (rad/s^2 per rad) and divergence rate (1/s)
wn, zeta = 4.0, 0.7                     # chosen closed-loop natural frequency (rad/s) and damping ratio
plant = g.TF([b], [1, 0, -lam ** 2])    # pitch angle from gimbal angle: theta'' = lam^2 theta + b delta
print('open-loop poles', [round(p.real, 3) for p in plant.poles()], '(one grows like e^(0.7 t))')

Kd = 2 * zeta * wn / b                  # match s^2 + b Kd s + (b Kp - lam^2) to s^2 + 2 zeta wn s + wn^2
Kp = (wn ** 2 + lam ** 2) / b
cl = g.TF([b * Kp], [1, b * Kd, b * Kp - lam ** 2])
poles = [complex(round(p.real, 3), round(p.imag, 3)) for p in cl.poles()]
print(f'Kp = {Kp:.4f}, Kd = {Kd:.4f}, closed-loop poles {poles}')

ts, ys = g.step_response(cl, 6.0, 0.001)
final = ys[-1]
t10 = next(t for t, y in zip(ts, ys) if y >= 0.1 * final)
t90 = next(t for t, y in zip(ts, ys) if y >= 0.9 * final)
settle = max(t for t, y in zip(ts, ys) if abs(y - final) > 0.02 * final)
print(f'DC gain {cl.dcgain():.4f}, overshoot {100 * (max(ys) / final - 1):.1f}% (formula {100 * math.exp(-math.pi * zeta / math.sqrt(1 - zeta ** 2)):.1f}%)')
print(f'10-90% rise time {t90 - t10:.2f} s, 2% settling time {settle:.2f} s (formula 4/(zeta wn) = {4 / (zeta * wn):.2f} s)')

print('effect of a zero on s^2 + 2.8 s + 4 (step response):')
for name, num in (('no zero', [4]), ('zero at -2', [2, 4]), ('zero at +2', [-2, 4])):
    ts, ys = g.step_response(g.TF(num, [1, 2.8, 4]), 8.0, 0.001)
    print(f'  {name:11s} peak {max(ys):.3f}, lowest {min(ys):6.3f}')

open-loop poles [-0.7, 0.7] (one grows like e^(0.7 t))
Kp = 0.8405, Kd = 0.2854, closed-loop poles [(-2.8-2.857j), (-2.8+2.857j)]
DC gain 1.0306, overshoot 4.6% (formula 4.6%)
10-90% rise time 0.53 s, 2% settling time 1.49 s (formula 4/(zeta wn) = 1.43 s)
effect of a zero on s^2 + 2.8 s + 4 (step response):
  no zero     peak 1.046, lowest  0.000
  zero at -2  peak 1.112, lowest  0.000
  zero at +2  peak 1.057, lowest -0.249

In robotics

Pole placement for a second-order loop is the first thing every control engineer does with a new axis, and the specifications are written in the same terms.

• Handling qualities. Aircraft requirements specify the frequency and damping of the short-period mode, for example short-period damping ratios of roughly 0.3 to 1.3 for the best level of flying qualities. These are exactly $\omega_n$ and $\zeta$. Module 10 uses them.
• Attitude loops. Multirotor and spacecraft attitude loops are designed for a damping ratio around 0.7 (a good compromise between speed and overshoot) and a bandwidth set by the actuator and the sensor, as in M1-T2: Nested Loops and Separation of Timescales. UAV M5-T5: Attitude control gives the cascade form of the same design.
• Pointing requirements. A telescope that must settle within a given time and overshoot by no more than a given amount translates directly into a damping ratio and a natural frequency, and from them into gains.
• Zeros in practice. A response that starts in the wrong direction is not exotic. Many aircraft show a right-half-plane zero in the transfer function from elevator to normal acceleration: the tail pushes down first (a loss of lift) before the nose rises. The pilot sees it as a momentary sink, and the autopilot designer as a limit on bandwidth.
• Tools. python-control computes poles, zeros and step responses with poles, zeros and step_response, and gnckit has the same functions so that the worked examples run in the browser. They are checked against each other in the library's tests.

The linear model that this lesson assumes was found in M2-T4: Trim and Linearization. All the formulas here are exact for a pure second-order loop without zeros, like this one. Real loops have extra poles (a servo, a filter) and zeros (a derivative on the error instead of the rate), and the formulas are then approximations that the margins of the next lessons replace.

Common misconceptions

Key takeaways

1. A transfer function is the differential equation with $d/dt$ replaced by $s$. A chain of blocks is a product of transfer functions.
2. Poles are the natural motions: the real part sets the decay rate, the imaginary part the oscillation, and the angle the damping ratio $\zeta$.
3. The overshoot depends only on $\zeta$: $e^{-\pi\zeta/\sqrt{1-\zeta^2}}$, and the 2% settling time is about $4/(\zeta\omega_n)$.
4. For Hopper's pitch loop, $K_p = (\omega_n^2 + \lambda^2)/b$ and $K_d = 2\zeta\omega_n/b$ place the poles; the $\lambda^2$ is the price of the unstable pole.
5. A left-half-plane zero adds overshoot, and a right-half-plane zero makes the response start in the wrong direction.
6. Pole placement is a design for a pure second-order loop. Real loops add dynamics, and the next lessons measure how much margin that leaves.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• K. J. Åström and R. M. Murray, Feedback Systems, 2nd edition, Princeton University Press, 2021 , the chapters on dynamic behaviour and on transfer functions. Free to read as the authors' PDF; the book is copyrighted.
• B. Douglas, “Classical Control Theory” (video series) , the first section on transfer functions and poles. Free to watch.
• J. How and E. Frazzoli, MIT 16.30 Feedback Control Systems (OpenCourseWare) , the lectures on transfer functions and second-order systems. Free; licence CC BY-NC-SA 4.0.
• Python Control Systems Library documentation , to check the numbers of this lesson. Free; BSD licence.
• G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th edition, Pearson, 2019, the chapters on dynamic response and on the root locus. A standard text. Paid book.

Knowledge check

Three questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

Related content

Nothing below is required to finish this lesson.

Other courses on this site treat neighbouring topics from their own angle:

• Lab: Tune a PID Controller
• Lab: Fix the Oscillating Robot
• UAV M5-T6: PID controllers
• UAV M7-T6: End-to-end latency

This course leaves the following to the pages that already teach them:

• What P, I and D do, and tuning them by hand on a step response
• PID cascades for a multirotor