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ME451: Control Systems Course Roadmap

1) This document outlines the key concepts around steady-state error that were covered in Lecture 13 of ME451: Control Systems. It discusses system types, error constants, and how the final value theorem can be used to determine steady-state error for step, ramp, and parabolic inputs. 2) Examples are provided to illustrate how to determine stability, compute error constants, and find the steady-state error for different system types. 3) The summary at the end emphasizes that the system type determines if steady-state error is zero for stable unity feedback systems, and that the final value theorem is the key tool used.

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Vu Nghia
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57 views4 pages

ME451: Control Systems Course Roadmap

1) This document outlines the key concepts around steady-state error that were covered in Lecture 13 of ME451: Control Systems. It discusses system types, error constants, and how the final value theorem can be used to determine steady-state error for step, ramp, and parabolic inputs. 2) Examples are provided to illustrate how to determine stability, compute error constants, and find the steady-state error for different system types. 3) The summary at the end emphasizes that the system type determines if steady-state error is zero for stable unity feedback systems, and that the final value theorem is the key tool used.

Uploaded by

Vu Nghia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Course roadmap

ME451: Control Systems Modeling Analysis Design

Laplace transform Time response


Design specs
• Transient
Lecture 13 Transfer function • Steady state
Root locus
Steady-state error
Models for systems Frequency response
Frequency domain
• electrical • Bode plot
• mechanical
PID & Lead-
Lead-lag
• electromechanical Stability
Dr. Jongeun Choi Block diagrams • Routh-
Routh-Hurwitz
Design examples
Department of Mechanical Engineering Linearization • Nyquist

Michigan State University

(Matlab simulations &) laboratories


1 2

Performance measures (review) Steady-state error: unity feedback


ƒ Transient response (From next lecture) We assume that the
ƒ Peak value CL system is stable!
ƒ Peak time Unity feedback!
ƒ Percent overshoot
Next, we will connect
ƒ Delay time these measures ƒ Suppose that we want output y(t) to track r(t).
ƒ Rise time with s-
s-domain.
ƒ Settling time
ƒ Error
ƒ Steady state response ƒ Steady-state error
ƒ Steady state error (Today’
(Today’s lecture)

Final value theorem


(Suppose CL system is stable!!!)
3 4
Error constants Steady-state error for step r(t)
ƒ Step-error (position-error) constant

ƒ Ramp-error (velocity-error) constant

ƒ Parabolic-error (acceleration-error) constant

Kp

ƒ Kp, Kv, Ka : ability to reduce steady-state error


5 6

Steady-state error for ramp r(t) Steady-state error for parabolic r(t)

Kv Ka
7 8
System type Zero steady-state error
ƒ System type of G is defined as the order ƒ If error constant is infinite, we can achieve zero
(number) of poles of G(s) at s=0. steady-state error. (Accurate tracking)
ƒ Examples ƒ For step r(t)

type 1
ƒ For ramp r(t)

type 2
ƒ For parabolic r(t)
type 3

9 10

Example 1 Example 2
ƒ G(s) of type 2 ƒ G(s) of type 1 G(s)
G(s)

ƒ Characteristic equation ƒ By Routh-Hurwitz criterion, CL is stable iff

ƒ Step r(t)

ƒ CL system is NOT stable for any K. ƒ Ramp r(t)


ƒ e(t) goes to infinity. (Don’t use today’s results if
CL system is not stable!!!) ƒ Parabolic r(t)
11 12
Example 3 A control example
ƒ G(s) of type 2 G(s)

ƒ By Routh-Hurwitz criterion, we can show that CL


system is stable.
ƒ Closed-loop stable?
ƒ Step r(t)
ƒ Compute error constants
ƒ Ramp r(t)
ƒ Compute steady state errors
ƒ Parabolic r(t)

13 14

Summary and Exercises


ƒ Steady-state error
ƒ For unity feedback (STABLE!) systems, the system
type of the forward-
forward-path system determines if the
steady-
steady-state error is zero.
ƒ The key tool is the final value theorem!
theorem!
ƒ Next, time response of 1st-order systems
ƒ Exercises
ƒ Go over the examples in this lecture.

15

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