First and Second Order Systems
Solve a second-order differential equation analytically, describe it and verify using MATLAB numerical integration and symbolics. Assume no initial conditions (but write original with initial conditions).
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Write a first order equation, find the response to a step input, define tau, rise time and settling time.
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Which poles of a first order system generate the natural and forced response? What does this describe at the output?
An input pole generates the form of the forced response. A system pole generates the form of the natural response. In general, poles on the real axis generate an exponential response. The total solution is the sum of both.
What determines the amplitudes of the forced and natural response for a first order system?
Both the poles and the zeros.
Write the general transfer function and analytical solution for a first order system with no zeros present, responding to a step input. (Prove it).
C(s) = a/(s*(s+a)) c(t) = 1-e^-a*t
Describe a second order system in canonical form, what are the roots?
G(s) = wn^2/(s^2+2*zeta*wn*s+wn^2), s(1,2)=-zeta*wn+-wn*sqrt(zeta^2-1)
What is the damping ratio?
It is the ratio of the exponential decay frequency to the natural frequency of the system.
Define rise time for a first order system.
It is the time it takes the waveform to go from 10-90% of its final value: Tr = 2.2/a.
Describe and draw a critically damped system.
Poles are equal and real. Zeta = 1. Fastest transient.
What is the natural frequency?
The frequency a system would oscillate at if there were no damping.
Define the settling time for a first order system.
The time it takes for a response to reach and stay within 2% of its final value: Ts = 4/a.
Name the performance specification for a first order system.
This is called the time constant, also known as tau = 1/a. It is the time it takes for the transient to move 63% towards its final value in response to a step input.
Describe and draw an underdamped system.
Two complex conjugate poles. 0<zeta<1. Oscillates at the damped frequency of oscillation with a decaying envelope.
Describe and draw an undamped system.
Two imaginary, conjugate poles. Oscillates for infinity. Zeta = 0.
Describe and draw an overdamped system.
Two real poles. Zeta>1.
