Time Response*, ME451

Instructor: Jongeun Choi

* This presentation is created by Jongeun Choi and Gabrial Gomes Zeros and poles of a transfer function

• Let G(s)=N(s)/D(s), then – Zeros of G(s) are the roots of N(s)=0 – Poles of G(s) are the roots of D(s)=0

Im(s)

Re(s) Theorems

• Initial Value Theorem

• Final Value Theorem – If all poles of sX(s) are in the left half plane (LHP), then DC gain or static gain of a stable system

1.4

1.2

0.8

0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 3 DC Gain of a stable transfer function

• DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response • Use final value theorem to compute the steady state of the unit step response Pure integrator

• ODE :

• Impulse response :

• Step response :

• If the initial condition is not zero, then :

Physical meaning of the impulse response First order system R C

• ODE :

• Impulse response :

• Step response :

• DC gain: (Use the final value theorem) First order system

• If the initial condition was not zero, then

Physical meaning of the impulse response Matlab Simulation

• G=tf([0 5],[1 2]); Impulse Response • 5 impulse(G) 4.5 4 3.5 3 2.5 2

1.5Amplitude 1 0.5 0 0 0.5 1 1.5 2 2.5 3 Time (sec)

Step Response • step(G) 2.5 2

1.5

1 • Time constant Amplitude 0.5

0 0 0.5 1 1.5 2 2.5 3 Time (sec) First order system response

System transfer function : First order system response

System transfer function :

Impulse response : First order system response

System transfer function :

Impulse response : First order system response

System transfer function :

Impulse response :

Step response :

Step Response 100 90 80 70 60 50 40 Amplitude 30 20 10 0 0 100 200 300 400 500 600 Time (sec) First order system response

Im(s)

Re(s) First order system response

Im(s)

Unstable

Re(s) First order system response

Im(s)

Unstable

-1 Re(s) First order system response

Im(s)

Unstable

Re(s) -2 First order system response

Im(s)

Unstable faster response slower response

Re(s)

constant First order system – Time specifications. First order system – Time specifications.

Time specs:

Steady state value :

Time constant :

Rise time : Time to go from to

Settling time : First order system – Simple behavior.

No overshoot No oscillations Second order system (mass-spring-damper system)

• ODE :

• Transfer function : Polar vs. Cartesian representations.

Cartesian representation :

… Imaginary part (frequency) … Real part (rate of decay) Polar vs. Cartesian representations.

Cartesian representation :

… Imaginary part (frequency) … Real part (rate of decay)

Polar representation :

… natural frequency … damping ratio Polar vs. Cartesian representations.

Cartesian representation :

… Imaginary part (frequency) … Real part (rate of decay)

Polar representation :

… natural frequency … damping ratio Polar vs. Cartesian representations.

Cartesian representation :

… Imaginary part (frequency) … Real part (rate of decay)

Polar representation :

… natural frequency … damping ratio

Unless overdamped Polar vs. Cartesian representations. System transfer function :

Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped Polar vs. Cartesian representations. System transfer function :

All 4 cases Unless overdamped

Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped Underdamped second order system

• Underdamped

• Two complex poles: Underdamped second order system Impulse response of the second order system Matlab Simulation

• zeta = 0.3; wn=1; • G=tf([wn],[1 2*zeta*wn wn^2]); • impulse(G)

Impulse Response 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Amplitude 0 -0.1 -0.2 -0.3 0 2 4 6 8 10 12 14 16 18 20 Time (sec) Unit step response of undamped systems

• Unit step response :

• DC gain : Unit step response of undamped system Matlab Simulation

• zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]); • step(G)

Step Response 1.4 1.2 1 0.8 0.6

0.4Amplitude 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) Second order system response.

Stable 2nd order system: 2 distinct real poles A pair of repeated real poles A pair of complex poles

Im(s)

Unstable