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

Lecture 8: Observability and Canonical Decomposition November 1, 2016 Zhi-Hong Mao Associate Professor of ECE and Bioengineering University of Pittsburgh, Pittsburgh, PA 1

Outline of this lecture • An example of controllability analysis • Observability • Canonical decomposition

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An example of controllability analysis • Segway human transportation system

A question concerning controllability: Can we move from one stationary point to another by appropriate application of forces through the wheels? 3

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An example of controllability analysis • Segway human transportation system M: mass of the base m: mass of the system to be balanced J: moment of inertia of the system to be balanced l: distance from the base to the center of mass of the balanced body c: coefficient of viscous friction

( M  m) p  ml cos( )  cp  ml sin( ) 2  F ( J  ml 2 )  ml cos( ) p  mgl sin( ) 4

An example of controllability analysis • Segway human transportation system ( M  m) p  ml cos( )  cp  ml sin( ) 2  F ( J  ml 2 )  ml cos( ) p  mgl sin( )

Question: How to derive a state-space equation from the above equations? How to linearize it?

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An example of controllability analysis • Segway human transportation system ( M  m) p  ml cos( )  cp  ml sin( ) 2  F ( J  ml 2 )  ml cos( ) p  mgl sin( ) x  [ p  p ]' , u  F , y  [ p  ]'

Linearized around the equilibrium [p 0 0 0]’

M t  M  m, J t  J  ml 2 , c  0

0  p  0   d     0 dt  p        0  1 0 0 0 y x 0 1 0 0 

0 0

1 0

m 2l 2 g M t J t  m 2l 2

0

M t mgl M t J t  m 2l 2

0

0 0   1   p  0        Jt  u 0 2 2   p   M t J t  m l       ml 0  2 2  M t Jt  m l   6

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An example of controllability analysis • Segway human transportation system 0  p  0    d    0 dt  p        0 

0 0

1 0

m 2l 2 g M t J t  m 2l 2

0

M t mgl M t J t  m 2l 2

0

0 0   1   p  0        Jt u 0     2 2    p M J  m l t t          ml 0  2 2  M t Jt  m l  

Question: Is it marginally stable? Asymptotically stable?

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An example of controllability analysis • Segway human transportation system 0  p  0   d     0 dt  p        0 

Controllability matrix

0 0

1 0

m 2l 2 g M t J t  m 2l 2

0

M t mgl M t J t  m 2l 2

0

 0    0  C   Jt  2 2  M t Jt  m l  ml  2 2  M t J t  m l

0 0   1   p  0        Jt  u 0 2 2    p M J  m l t t           ml 0    2 2  M t Jt  m l  

 m 3l 3 g  ( M t J t  m 2l 2 ) 2  2 2  Mtm l g ml 0  M t J t  m 2l 2 ( M t J t  m 2l 2 ) 2  3 3  ml g  0 0 ( M t J t  m 2l 2 ) 2   M t m 2l 2 g  0 0 2 2 2 (M t J t  m l )  Jt M t J t  m 2l 2

0

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An example of controllability analysis • Segway human transportation system

Controllability matrix

 0    0  C   Jt  2 2  M t Jt  m l  ml  2 2  M t J t  m l

 m 3l 3 g  ( M t J t  m 2l 2 ) 2   M t m 2l 2 g ml 0  M t J t  m 2l 2 ( M t J t  m 2l 2 ) 2   m3l 3 g  0 0 2 2 2 (M t J t  m l )   M t m 2l 2 g  0 0 2 2 2 (M t J t  m l )  Jt M t J t  m 2l 2

0

Question: Is it controllable?

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An example of controllability analysis • Segway human transportation system

Controllability matrix

det(C ) 

 0    0  C   Jt  2 2  M t Jt  m l  ml  2 2  M t J t  m l

 m 3l 3 g  ( M t J t  m 2l 2 ) 2  2 2  Mtm l g ml 0  M t J t  m 2l 2 ( M t J t  m 2l 2 ) 2   m3l 3 g  0 0 ( M t J t  m 2l 2 ) 2   M t m 2l 2 g  0 0 2 2 2 (M t J t  m l )  Jt M t J t  m 2l 2

m 4l 4 g 2 0 ( M t J t  m 2l 2 ) 4

0

The system is controllable!

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An example of controllability analysis • Segway human transportation system

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Observability • Definition – The concept of observability is dual to that of controllability – Observability studies the possibility of estimating the state from the output (Reminder: Controllability studies the possibility of steering the state from the input)

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Observability • Definition – –

The concept of observability is dual to that of controllability Observability studies the possibility of estimating the state from the output

– Consider an n-dimensional p-input q-output state equation x  Ax  Bu

y  Cx  Du. This state equation is said to be observable if for any unknown initial state x(0), there exists a finite t1 > 0 such that the knowledge of the input u and the output y over [0, t1] suffices to determine uniquely the initial state x(0). Otherwise, equation is said to be unobservable (Reminder: The state equation is said to be controllable if for any initial state x(0) = x0 and any final state x1, there exists an input that transfer x0 to x1 in a finite time) 13

Observability • Definition – – –

The concept of observability is dual to that of controllability Observability studies the possibility of estimating the state from the output and input Observable and unobservable

– Significance of the concept of observability • If a system is observable, then there are no “hidden” dynamics inside it; we can understand everything that is going on through observation (over time) of the inputs and outputs • The problem of observability is of significant practical interest because it will tell if a set of sensors are sufficient for controlling a system

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Observability •

Definition

• Examples of observable and unobservable systems + u 

+

1W

1W

y

+x  1W

1W

_

Question: Is the above system observable (the state variable is x)? 15

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Observability •

Definition

• Examples of observable and unobservable systems

Question: Is the left system observable? (Reminder: Is the right system controllable?)

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Observability •

Definition

• Examples of observable and unobservable systems

Diagnostic techniques in traditional Chinese medicine

Detection of drowsiness 17

Observability • •

Definition Examples of observable and unobservable systems

• Theorems The state equation x  Ax  Bu, y  Cx  Du is observable if and only if the matrix t

Wo (t )   eA ' C' CeA d 0

is nonsingular for any t > 0. Reminder: The pair (A, B) is controllable if and only if the matrix t t Wc (t )   eA BB' eA ' d   eA ( t  ) BB' eA '( t  ) d 0

0

is nonsingular for any t > 0. 18

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Observability • •

Definition Examples of observable and unobservable systems

• Theorems

t

Wo (t )   eA ' C'CeA d 0

Remark: If the state equation is observable, then t1

x(0)  Wo (t1 ) eA 't C' y(t )dt 1

where

0

t

y(t )  y(t )  C eA ( t  ) Bu( )d  Du(t ). 0

Remark: Observability depends only on A and C. Thus observability is a property of the pair (A, C) and is independent of B and D. 19

Observability • •

Definition Examples of observable and unobservable systems

• Theorems Theorem of duality: The pair (A, B) is controllable if and only if the pair (A’, B’) is observable. Question: Can you prove the theorem of duality? Hint:

t

t

0

0

Wo (t )   eA ' C'CeA d , Wc (t )   eA BB' eA ' d .

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Observability • •

Definition Examples of observable and unobservable systems

• Theorems The n-dimensional pair (A, C), where A and C are n by n and q by n matrices respectively, is observable if and only if the nq by n observability matrix C

  CA   O     n 1  CA 

has rank n (full column rank). Question: Can you prove this? 21

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Observability • •

Definition Examples of observable and unobservable systems

• Theorems The n-dimensional pair (A, C) is observable if and only if the matrix C  CA   On  q 1       nq  CA  where the rank of C is q, has rank n or the n by n matrix On'  q1On q1 is nonsingular.

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Observability • •

Definition Examples of observable and unobservable systems

• Theorems The observability property is invariant under any equivalence transformation.

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Canonical Decomposition • Stability, controllability, and observability are preserved under equivalence transform x  Ax  Bu y  Cx  Du

x  Px

x  Ax  Bu y  C x  Du

A  PAP 1, B  PB, C  CP1, D  D C  PC , O  OP 1

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Canonical Decomposition •

Stability, controllability, and observability are preserved under equivalence transform

• Controllable and uncontrollable subspaces Consider the n-dimensional state equation x  Ax  Bu, y  Cx  Du, with the controllability matrix C (i.e. [B AB … An-1B]) of rank n1, which is strictly less than n. We form the n by n matrix P1  [q1 qn1 qn ] where the first n1 columns are any n1 linearly independent columns of C, and the remaining columns can arbitrarily be chosen as long as P is nonsingular. Then the equivalence transformation x  Px will transform the original state equation into  x c  A c A12   xc   Bc   xc           u, y  [ Cc Cc ]   Du,  xc   xc   0 A c   xc   0  where A c is n1 by n1 and A c is (n − n1) by (n − n1), and the n1-dimensional subequation of the derived equation, x c A c xc  Bcu, y  Cc xc  Du, is controllable and has the same transfer matrix as the original equation. 25

Canonical Decomposition •

Stability, controllability, and observability are preserved under equivalence transform

• Controllable and uncontrollable subspaces Example: Reduce the following state equation to a controllable one

1 1 0 0 1 x  0 1 0 x  1 0u, y  [1 1 1]x.     0 1 1  0 1

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Canonical Decomposition • •

Stability, controllability, and observability are preserved under equivalence transform Controllable and uncontrollable subspaces

• Observable and unobservable subspaces Consider the n-dimensional state equation x  Ax  Bu, y  Cx  Du, with the observability matrix O of rank n2, which is strictly less than n. We form the n by n matrix p1      P  p n 2      p   n 

where the first n2 rows are any n2 linearly independent rows of O, and the remaining rows can arbitrarily be chosen as long as P is nonsingular.

Then the equivalence transformation

x  Px will transform the original

state equation into  x o  A o 0   x o   Bo           u,  x o  A 21 A o   x o   Bo  xo  y  [ Co 0]   Du, xo  where A o is n2 by n2 and A o is (n − n2) by (n − n2), and the n2-dimensional subequation of the derived equation, x o  A o xo  Bou, y  Co xo  Du,

is observable and has the same transfer matrix as the original equation. 27

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Canonical Decomposition • •

Stability, controllability, and observability are preserved under equivalence transform Controllable and uncontrollable subspaces

•

Observable and unobservable subspaces

• Kalman decomposition CO

u

CO

y

CO

C CO 28

References • • • • • •

K. J. Astrom and R. M. Murray. Feedback Systems: An Introduction for Scientists and Engineers. Manuscript, 2006. C.-T. Chen. Linear System Theory and Design, 4th Edition, Oxford University Press, 2013. http://images.icnetwork.co.uk/upl/mirror/jun2003/0/6/000C1C14-01A2-1EEB81A280BFB6FA0000.jpg http://www.entershanghai.info/country/Ci_11_set.htm http://www.msichicago.org/scrapbook/scrapbook_exhibits/segway/index.html http://www-static.cc.gatech.edu/ai/robot-lab/segway/sewerruns.wmv

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