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
