ECE/ME 2646: Linear System Theory (3 Credits, Fall 2016)

Lecture 6: State-Space Realizations and Stability October 4, 2016

Zhi-Hong Mao Associate Professor of ECE and Bioengineering University of Pittsburgh, Pittsburgh, PA 1

Outline of this lecture • Homework 3 (no need to turn in; midterm on October 11) • Realizations • Definitions of stability in various domains • Input-output stability • Internal stability

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Realizations • •

Definitions Conditions for realizability

• Controllable canonical form – Procedure to obtain the controllable canonical form ˆ ( s)  G ˆ ( )  G ˆ ( s ), G sp ˆ ( s ) is a q  p proper rational matrix where G ˆ ˆ ( s) and G ( s ) is the strictly proper part of G sp

1 ˆ ( s )  1 [N( s )]  G [N1s r 1  N 2 s r  2    N r 1s  N r ], sp d (s) s r  1s r 1     r 1s   r ˆ ( s ), where d ( s ) is the least common denominato r of all entries of G sp

it is monic, i.e., with 1 as its leading coefficien t, and Ni are q  p constant matrices

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Realizations • •

Definitions Conditions for realizability

• Controllable canonical form – Procedure to obtain controllable canonical form 1 ˆ ( s )  1 [N( s )]  G [N1s r 1  N 2 s r  2    N r 1s  N r ], sp d (s) s r  1s r 1     r 1s   r ˆ ( s ), where d ( s ) is the least common denominato r of all entries of G

ˆ ( s)  G ˆ ( )  G ˆ ( s ), G sp ˆ ( s ) is a q  p proper rational matrix where G ˆ ( s ) is the strictly proper part of G ˆ ( s) and G

sp

sp

it is monic, i.e., with 1 as its leading coefficien t, and Ni are q  p constant matrices

 1I p   2 I p    r 1I p   r I p   I p    0  0 0  0   Ip   x   0 Ip  0 0  x   0 u                0  Ip 0   0   0 ˆ ()u y  [N1 N 2  N r 1 N r ]x  G

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Realizations • •

Definitions Conditions for realizability

• Controllable canonical form – Procedure to obtain controllable canonical form Example: Find realizations of the following systems

ˆ ( s)  G 1

s2  s  1 (2s  1)( s  2)

 4 s  10  2s  1 ˆ G2 ( s)   1   ( 2 s  1)( s  2)

3  s2   s 1  ( s  2)2 

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Realizations • •

Definitions Conditions for realizability

• Controllable canonical form – Procedure to obtain controllable canonical form Exercise: Find realizations of the following system

 4 s  10    2s  1 ˆ ( s)   G  1    ( 2 s  1)( s  2)  6

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Definitions of stability in various domains • Ecological stability, measure of the probability of a population returning quickly to a previous state, or not going extinct • Social stability, lack of civil unrest in a society • Quotes: “Every time I try to define a perfectly stable person, I am appalled by the dullness of that person.” − J. D. Griffin

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Definitions of stability in various domains • • •

Ecological stability, measure of the probability of a population returning quickly to a previous state, or not going extinct Social stability, lack of civil unrest in a society Quotes: “Every time I try to define a perfectly stable person, I am appalled by the dullness of that person.” − J. D. Griffin

• In control theory, by stability, we usually mean that a stable system remains under control

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Input-output stability • BIBO stable – An input u(t) is said to be bounded if u(t) does not grow to positive or negative infinity or, equivalently, there exists a constant um such that

| u(t ) | um   for all t  0

– A system is said to be BIBO stable (bounded-input bounded-output stable) if every bounded input excites a bounded output. Note that this stability is defined for the zero-state response and is applicable only if the system is initially relaxed

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Input-output stability •

BIBO stable

• Theorems A SISO system with proper rational transfer function is BIBO stable if and only if every pole of the transfer function has a negative real part or, equivalently, lies inside the left-half s-plane. Question: Are the following systems BIBO stable?

1 ( s  2)( s  1) s2 , . , s 2  2 ( s  2) 2  2 s3 10

Input-output stability •

BIBO stable

• Theorems A multivariable system with proper rational transfer-function matrix is BIBO stable if and only if every pole of every element of the transfer-function matrix has a negative real part.

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Internal stability • Definition – The zero-input response of the following system x  Ax  Bu

y  Cx  Du or the equation x  Ax is marginally stable or stable in the sense of Lyapunov if every finite initial state excites a bounded response. It is asymptotically stable if every finite initial state excites a bounded response, which in addition, approaches 0 as t approaches 

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Internal stability • Definition Example: An experiment about rotational stability. Consider a rigid body for which all of the principal moments of inertia are distinct. Let I1 > I2 > I3. Suppose that the body is freely rotating about one of its principal axes. What happens when the body is slightly disturbed? Let the body be initially rotating about principal axis 1, so that = 1e1. If we apply a slight perturbation then the angular velocity becomes  = 1e1 + ue2 + ve3, where u and v are both assumed to be small. By using Euler’s equations and performing linearization, we can get the following state-space equation

( I 3  I1 )1   0  u  I2 u     v    ( I  I )  v     2 1 1 0   I3  

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A little more about rotational stability 14

Internal stability •

Definition

• Theorems If A has distinct eigenvalues, then the equation x  Ax is marginally stable if and only if all eigenvalues of A have zero or negative real parts.

  Axis asymptotically stable if and The equation x only all eigenvalues of A have negative real parts.

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Internal stability •

Definition

• Theorems Question: Does “marginally stable” implies “BIBO stable”? Does “BIBO stable” implies “marginally stable”?

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References • • • • • • • • •

C.-T. Chen. Linear System Theory and Design, 4th Edition, Oxford University Press, 2013. C. L. Phillips and R. D. Harbor. Feedback Control Systems, 4th Edition, Prentice Hall, 2000. http://en.wikipedia.org/wiki/Bullet http://farside.ph.utexas.edu/teaching/336k/lectures/node74.html http://homestudy.ihea.com/aboutfirearms/11bore.htm http://www.answers.com/topic/stability http://www.aviapedia.com/category/fighters/f-22/ http://www.nebraskahistory.org/images/oversite/store/catalog4/Peg-Top.jpg http://www.nothingtoxic.com/uploads/headspinner.wmv

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