Second Order Dynamics

Second order ODE

\frac{d^2x}{dt^2}+a_1\frac{dx}{dt}+a_2x=b\cdot u(t)

Transfer function

g(s)=\frac{K_{p}\omega_{n}^2}{s^2+2\zeta\omega_{n}s+\omega_{n}^2}

\omega_{n}

Natural frequency

\zeta

Damping ratio

K_{p}

Steady state gain

Overdamped

\zeta>1

Overdamped properties

Two real poles

No oscillation

Looks like two first order systems in series

Overdamped generic transfer function

g(s)=\frac{K_{p}}{(\tau_1s+1)(\tau_2s+1)}

Underdamped

0<\zeta<1

Underdamped properties

Complex conjugate poles

Oscillates

Why must the poles be complex conjugates?

Because the signal itself is real

Underdamped generic transfer function

g(s)=\frac{K_{p}\omega_{n}^2}{s^2+2\zeta\omega_{n}s+\omega_{n}^2}

How are the poles defined for underdamped?

s_{1,2}=-\zeta\omega_{n}+j\omega_{n}\sqrt{1-\zeta^2}

How do we simplify the expression for underdamped poles?

s_{1,2}=-\sigma\pm j\omega_{t}

\sigma

\sigma=\zeta\omega_{n}

\omega_{t}

\omega_{t}=\omega_{n}\sqrt{1-\zeta^2}

Impulse response: cause signal

u(t)=\delta(t)

\overline{u}(s)=1

Relationship between u and x for impulse response

\overline{x}(s)=g(s)\times\overline{u}(s)=g(s)

Method for overdamped impulse response

Find x(s)

Partial fraction expansion

Inverse transform

Impulse response (overdamped)

\overline{x}(s)=\frac{K_{p}}{\left(s\tau_1+1\right)\left(s\tau_2+1\right)}

Partial fraction expansion for overdamped impulse response

x(s)=\left(\frac{K_{p}}{\tau_1-\tau_2}\right)(\frac{1}{\tau_1s+1}-\frac{1}{\tau_2s-1})

Overdamped impulse response: inverse transform gives…

Key features of overdamped impulse response

Two exponentials (fast 1/tau2, slow 1/tau1)

Dominant pole is the slower one (bigger time constant)

Response starts high and decays smoothly

No oscillation

Impulse response (underdamped)

Underdamped impulse response: After partial fractions and inverse transform

x(t)=Ae^{-\sigma t}\sin(\omega_{t}t)

What is A?

Amplitude constant

A=\frac{K_{p}\omega_{n}^2}{\omega_{t}}=\frac{K_{p}\omega_{n}}{\sqrt{1-\zeta^2}}

Underdamped impulse response: Period of oscillation

T_{p}=\frac{2\pi}{\omega_{t}}

How does the amplitude decay for underdamped impulse response?

Ae^{-\sigma t}

Underdamped impulse response: If damping ratio is small, then?

Damped frequency ~ natural frequency

System responds at natural frequency, called ‘ringing’

Oscillation takes a long time to die away

Step response: cause signal

u(t)=H(t)

\overline{u}(s)=\frac{1}{s}

Overdamped step response

\overline{x}(s)=\frac{1}{s}\cdot\frac{K_{p}}{\left(\tau_1s+1\right)\left(\tau_2s+1\right)}

Overdamped step response method

Combine 1/s into transfer function

Partial fraction expansion

Inverse transform

Key features of overdamped step response

Smooth S-shaped rise

No oscillation

Slowest pole dominates longterm

Underdamped step response

x(t)=K_{p}\left(1-\frac{e^{-\sigma t}}{\sqrt{1-\zeta^2}}\sin(\omega_{t}+\phi\right)

Phase shift for underdamped step response?

\phi=\arccos(\zeta)

Key features of underdamped step response

Overshoot (depends on damping ratio)

Oscillates around final value

Oscillations decay wrt the exponential term

Effect of damping

Higher damping = less oscillation and overshoot

Lower damping = more oscillation and overshoot

Physical example of underdamped

Sloshing of liquid in tanks / ships

M_{pt}

Peak overshoot

Depends strongly on damping ratio

Performance curves vs damping ratio

Why are sinusoidal inputs necessary?

Real systems experience oscillating disturbances

Stability of feedback control depends on frequency response

Powerful diagnostic tool

What is the method for frequency response?

Substitute s=j\omega into transfer function

Convert g(j\omega) to magnitude and phase

Read off

Overdamped second order system output

x(t)=A(\omega)\cos\left(\omega t+\phi\left(\omega\right)\right)=\frac{K_{p}}{\sqrt{\omega^2\tau_1^2+1}\sqrt{\omega^2\tau_2^2+1}}

A(\omega)=

\vert g(j\omega)\vert

\phi(\omega)=

angle g(jw)

Overdamped second order frequency response

A(\omega)=\frac{K_{p}}{\sqrt{\left(1+\left(\omega\tau_1^2\right)\right)\left(1+\left(\omega\tau_{2^{}}^2\right)\right)}}

\phi(\omega)=-\left(\arctan\left(\omega\tau_1\right)+\arctan\left(\omega\tau_2\right)\right)

At low w

Underdamped second order frequency response

Algebraic definitions not required, memorise transfer function and read off curves for g(jw)

Underdamped frequency response curves

When does critical damping occur?

When the second order system has a repeated real pole

Damping ratio = 1

Transfer function for critical damping

g(s)=\frac{K}{(1+\tau s)^2}

What is important about partial fraction expansion for critical damping?

Requires both \frac{B}{s+\frac{1}{\tau}} AND \frac{D}{\left(s+\frac{1}{\tau}\right)^2}