Chapter 3-Dynamic Response Flashcards
Dynamic Model (from chapter 2)
A mathematical description of the process to be controlled.
Two ways to approach solving the dynamic equations
One is a quick, approximate analysis using linear analysis techniques.
For more precise pictures, we use numerical simulations of non linear equations.
Three domains to study dynamic response
The Laplace transform (s-plane), the frequency response, and the state space.
Laplace transform
Mathematical tool for transforming differential equations into an easier-to-manipulate algebraic form.
Two attributes of linear time-invariant systems (LTIs) form teh basis for almost all analytical techniques applied to these systems
A linear system response obeys the principle of superposition.
The response of an LTI system can be expressed as the convolution fo the input with the unit impulse response of the system.
Principles of superposition
If the system has an input that can be expressed as a sum of signals, then the response of teh system can be expressed as the sum of the individual responses to the respective signals. Applies if and only if the system is linear.
Transfer Function
Relating the output divided by the input with a polynomial in both. Always has to be in terms of S
Poles
In some form they are root numbers of the denominator of a polynomial. This is the important because once you hit the pole numbers the transfer function goes to infinity.
Zero
number that makes in the numerator polynomial of a transfer function zero.
S-plane
It is a complex plane with a real component and an imaginary component.
Frequency Domain
Where s is the input, and s is a complex number.
Stable impulse response
For the first order pole, H(s)=1/(s+sigma) the impulse response is stable when sigma>0 and s<0.
Unstable impulse response
if sigma<0 and s is greater then 0 then it is unstable.
Time constant
A way to explain how the transient section of the function occurs.
First-order transfer function with an impulse response
A system’s transient response is determined by the poles of the transfer function, which in turn are obtained by setting the denominator of the transfer function to zero and solving the resulting expression, known as the characteristic equation.
Damping ratio
How fast the function decays. Written as zeta.
Undamped natural frequency
The funciton will not decay and oscillate together. Shown as Wn
Rise time
The time it takes the system to reach teh vicinity of its new set point. Written as tr
Settling time
The time it takes the system transients to decay. Written as ts
Overshoot
The maximum amount the system overshoots its final valued divided by its final value. Written as Mp. Occurs when the derivative is zero.
Peak time
Is the time it takes the system to reach the maximum overshoot point. Written as Tp. Occurs when the derivative is zero.
Steps for block diagram Allegra
Step 1: Conbine all serial blocks (rule 1)
Step 2: Combine all parallel blocks (rule 2)
Step 3: Close all inner loops (rule 3)
Step 4: Move summing junctions to the left and pick off points to the right (rule 4-7)
Time-domain specifications for step response.
Rise time, settling time, overshoot, peak time.
Finding wn given an s plane
For the value on the s plane s=zeta(wn)+-jwd the magnitude of that vector is omegan.
LTI System for Stability
it is said to be stable if all the roots of the transfer function denominator polynomial have negative real parts (that is, they are all in the left hand s-plane) and is unstable otherwise.
