Exam 2 Flashcards
The zeros of the characteristic equation change as ________ changes
K
Lecture 13: Routh Stability Criterion
Routh Stability Criterion (definition)
A method of obtaining information about the location of the roots of a polynomial without actually solving for the roots
Lecture 13: Routh Stability Criterion
What are the tests for Routh Stability?
- Are all of the coefficients of the characteristic polynomial positive (and present)?
- Are all the elements in the 1st column of the Routh array POSITIVE?
Lecture 13: Routh Stability Criterion
In a Routh Array, ________________ gives the number of poles of the characteristic equation that are on the right side of the S-plane.
The number of sign changes of the elements in the 1st column
Lecture 13: Routh Stability Criterion
Steps to analyzing system using a Routh Array
- Find the transfer function of the closed loop system
- Re-arrange characteristic equation to the form:
1s^n + a_1s^(n-1) + … + a_n-1s^1+a_ns^0 - Apply test 1
- Form Routh Array
- Apply test 2
Lecture 13: Routh Stability Criterion
Two assumptions for this class
- The plant and controller are LTI systems
- Single input - single output (SISO) systems
Lecture 14: Basic Eqn.s of Control
Basic concerns of controls engineers
- Stability
- Tracking
- Regulation
- Sensitivity
Our mission is to create an appropriate control signal to drive the system
Lecture 14: Basic Eqn.s of Control
Types of control
- Tracking: cause the output to follow the reference as closely as possible
- Regulation: keep the error as small as possible
Lecture 15: Conflicting Tradeoffs
What is the purpose of the Root Locus?
Determines
- The values of K at which the system is stable
- How the time response changes as K changes
AKA it is a guide to how poles change as K changes
Lecture 15: Conflicting Tradeoffs
The Root Locus can be thought of as…
A method for inferring the dynamic properties of the closed loop system a K changes
The value of s is a closed-loop pole of a closed loop system if:
- | KG(s)H(s) | = 1
- < KG(s)H(s) = (2k+1)180deg
Lecture 15: Conflicting Tradeoffs
Characteristic polynomial form for using Root Locus
1+KL(s) where L(s) = b(s)/a(s) for a(s)+K*b(s)
Lecture 16: Root Locus
Root Locus Rules
- The n branches of the locus start at the poles of L(s) when K=0, and the m branches end on the zeros of L(s) when K = infinity.
- The loci are on the real axis to the left of an odd number of poles and zeros
- For large s and K, n-m branches of the loci are asymptotic to lines at angles p radiating out from the center point s = alpha on the real axis
p = (180 + 360(l-1)) / (n-m), l = 1, 2, …, n-m
anchor point = alpha = [sum(pi) - sum(zi)] / (n-m)
Lecture 16: Root Locus
For the Root Locus, what does changing K mean?
- The poles change
- Consequently, the step response (i.e. time response) changes
Lecture 16: Root Locus
Integral Controller Expression
Ki/s
Lecture 17: PI Controller
What is the purpose of an integral controller?
Minimize the steady-state tracking error and the steady-state output response to disturbances
AKA: Remove the steady state error
Lecture 17: PI Controller
When should you use a PD controller?
When the root locus does not fit the time domain response you need
Lecture 18: PD Controller
Derivative Controller Expression
Kd*s
Lecture 18: PD Controller
What is the goal of derivative feedback?
3 things
- Improve closed loop system stability
- Speed up the transient response
- Reduce overshoot
Lecture 18: PD Controller
Control Law for PD Controller
Trend of error:
U(t) = Kd*de(t)/dt
Increasing control amplifies error/noise
Lecture 18: PD Controller
An extra zero may….
Increase the overshoot of the step response by a little
Lecture 18: PD Controller
To reduce the steady state error to constant reference and disturbance inputs use a(n) ___________ controller
Integral
To speed up the transient response and reduce overshoot use this type of controller:
Derivative
“anticipatory term”
Derivative
Gain Margin, GM
The change in the open-loop gain required at 180deg phase shift to make the closed loop system unstable
Expressed in dB
Phase Margin, PM
The change in open-loop phase shift required at unity gain to make the closed-loop system unstable
Stability: Typically an open-loop system is stable in closed-loop if…
The open-loop magnitude frequency response has a gain of less than 0dB at the frequency where the phase frequency response is 180
Percent overshoot is reduced by…
Increasing the phase margin
Speed of Response is increased by….
Increasing the bandwidth
Steady-state error is improved by
Increasing the low-frequency magnitude response (even if the high-f response is attenuated)
A _____ PM is required for stability
positive
How to find GM
- Find where phase crosses -180deg
- Follow the frequency up to magnitude plot
- GM = magnitude at that point-0dB; log(w)
