Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Lecture 13
Steady-state error
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Linearization
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Design
Time response
• Transient
• Steady state
Design specs
Root locus
Frequency response
• Bode plot
Stability
• RouthRouth-Hurwitz
• Nyquist
Frequency domain
PID & LeadLead-lag
Design examples
(Matlab simulations &) laboratories
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Performance measures (review)
ƒ Transient response
2
Steady-state error: unity feedback
(From next lecture)
ƒ Peak value
We assume that the
CL system is stable!
ƒ Peak time
Unity feedback!
ƒ Percent overshoot
ƒ Delay time
ƒ Rise time
Next, we will connect
these measures
with ss-domain.
ƒ Settling time
ƒ Steady state response
ƒ Steady state error
ƒ
ƒ
ƒ
Suppose that we want output y(t) to track r(t).
Error
Steady-state error
(Today’
(Today’s lecture)
Final value theorem
(Suppose CL system is stable!!!)
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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
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Steady-state error for ramp r(t)
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Steady-state error for parabolic r(t)
Kv
Ka
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System type
Zero steady-state error
ƒ System type of G is defined as the order
(number) of poles of G(s) at s=0.
ƒ Examples
ƒ If error constant is infinite, we can achieve zero
steady-state error. (Accurate tracking)
ƒ For step r(t)
type 1
ƒ For ramp r(t)
type 2
ƒ For parabolic r(t)
type 3
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Example 1
ƒ G(s) of type 2
Example 2
ƒ G(s) of type 1
G(s)
G(s)
ƒ By Routh-Hurwitz criterion, CL is stable iff
ƒ Characteristic equation
ƒ Step r(t)
ƒ CL system is NOT stable for any K.
ƒ e(t) goes to infinity. (Don’t use today’s results if
CL system is not stable!!!)
ƒ Ramp r(t)
ƒ Parabolic r(t)
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Example 3
ƒ G(s) of type 2
A control example
G(s)
ƒ By Routh-Hurwitz criterion, we can show that CL
system is stable.
ƒ Step r(t)
ƒ Ramp r(t)
ƒ Closed-loop stable?
ƒ Compute error constants
ƒ Compute steady state errors
ƒ Parabolic r(t)
13
Summary and Exercises
ƒ Steady-state error
ƒ For unity feedback (STABLE!) systems, the system
type of the forwardforward-path system determines if the
steadysteady-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.
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