ece317 : feedback and controlweb.cecs.pdx.edu/~tymerski/ece317/ece317 l9_stability.pdfrouth-hurwitz...
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ECE317 : Feedback and Control
Lecture :Stability
Routh-Hurwitz stability criterion
Dr. Richard Tymerski
Dept. of Electrical and Computer Engineering
Portland State University
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Course roadmap
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Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
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Stability
• Utmost important specification in control design!
• Unstable systems have to be stabilized by feedback.
• Unstable closed-loop systems are useless.
• What if a system is unstable? (“out-of-control”)• It may hit mechanical/electrical “stops” (saturation).
• It may break down or burn out.
• Signals diverge.
• Examples of unstable systems• Tacoma Narrows Bridge collapse in 1940
• SAAB Gripen JAS-39 prototype accident in 1989
• Wind turbine explosion in Denmark in 2008
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Definitions of stability
• BIBO (Bounded-Input-Bounded-Output) stabilityAny bounded input generates a bounded output.
• Asymptotic stability
Any ICs generates y(t) converging to zero.
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BIBO stable
system
r(t) y(t)ICs=0
Asymp. stable
systemr(t)=0
y(t)ICs
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Some terminologies
• Zero: roots of n(s)
• Pole: roots of d(s)
• Characteristic polynomial: d(s)
• Characteristic equation: d(s)=0
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Ex.
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Stability condition in s-domain (Proof omitted, and not required)
• For a system represented by transfer function G(s),
System is BIBO stable
All the poles of G(s) are in the open left half of the complex plane.
System is asymptotically stable
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Re
Im
0
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Idea of stability condition
• Example
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Asym. Stability:
(r(t)=R(s)=0)
BIBO Stability:
(y(0)=0)
Bounded if Re(-a)<0
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Remarks on stability
• For general systems (nonlinear, time-varying), BIBO stability condition and asymptotic stability condition are different.
• For linear time-invariant (LTI) systems (to which we can use Laplace transform and we can obtain transfer functions), these two conditions happen to be the same.
• In this course, since we are interested in only LTI systems, we use simply “stable” to mean both BIBO and asymptotic stability.
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Time-invariant & time-varying
• A system is called time-invariant (time-varying) if system parameters do not (do) change in time.
• Example: Mx’’(t)=f(t) & M(t)x’’(t)=f(t)
• For time-invariant systems:
• This course deals with time-invariant systems.
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SystemTime shift Time shift
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Remarks on stability (cont’d)
• Marginally stable if• G(s) has no pole in the open RHP (Right Half Plane), and
• G(s) has at least one simple pole on jw-axis, and
• G(s) has no multiple pole on jw-axis.
• Unstable if a system is neither stable nor marginally stable.
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Marginally stable NOT marginally stable
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“Marginally stable” in t-domain
• For any bounded input, except only special sinusoidal (bounded) inputs, the output is bounded.• In the example above, the special inputs are in the form of:
• For any nonzero initial condition, the output neither converge to zero nor diverge.
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M=1
x(t)
f(t)K
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Stability summary
Let si be poles of G(s). Then, G(s) is …
• (BIBO, asymptotically) stable if
Re(si)<0 for all i.
• marginally stable if
• Re(si)<=0 for all i, and
• simple pole for Re(si)=0
• unstable if it is neither stable nor marginally stable.
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Mechanical examples
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M
x(t)
f(t)
M
x(t)
f(t)K
M
x(t)
f(t)B
M
x(t)
f(t)
B
K
Poles=
stable?Poles=
stable?
Poles=
stable?
Poles=
stable?
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Examples
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Stable/marginally stable
/unstable
?
?
?
?
???
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Course roadmap
15
Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
![Page 16: ECE317 : Feedback and Controlweb.cecs.pdx.edu/~tymerski/ece317/ECE317 L9_Stability.pdfRouth-Hurwitz criterion •This is for LTI systems with a polynomial denominator (without sin,](https://reader035.vdocuments.net/reader035/viewer/2022070216/611c5cd8c28da34acc719717/html5/thumbnails/16.jpg)
Routh-Hurwitz criterion
• This is for LTI systems with a polynomialdenominator (without sin, cos, exponential etc.)
• It determines if all the roots of a polynomial • lie in the open LHP (left half-plane),
• or equivalently, have negative real parts.
• It also determines the number of roots of a polynomial in the open RHP (right half-plane).
• It does NOT explicitly compute the roots.
• No proof is provided in any control textbook.
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Polynomial and an assumption
• Consider a polynomial
• Assume• If this assumption does not hold, Q can be factored as
where
• The following method applies to the polynomial
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Routh array
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From the given
polynomial
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Routh array (How to compute the third row)
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Routh array (How to compute the fourth row)
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Routh-Hurwitz criterion
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The number of roots
in the open right half-plane
is equal to
the number of sign changes
in the first column of Routh array.
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Example 1
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Routh array
Two sign changes
in the first columnTwo roots in RHP
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Example 2
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Routh array
No sign changes
in the first columnNo roots in RHP
Always same!
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Example 3 (from slide 14)
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Routh array
No sign changes
in the first columnNo roots in RHP
Always same!
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Simple important criteria for stability
• 1st order polynomial
• 2nd order polynomial
• Higher order polynomial
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Examples
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All roots in open LHP?
Yes / No
Yes / No
Yes / No
Yes / No
Yes / No
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Summary
• Stability for LTI systems• (BIBO, asymptotically) stable, marginally stable, unstable
• Stability for G(s) is determined by poles of G(s).
• Routh-Hurwitz stability criterion• to determine stability without explicitly computing the
poles of a system
• Next, examples of Routh-Hurwitz criterion
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