Lecture 10-3. Observability and state Estimation

- State variable feedback plays a key role in the state space approach to the linear feedback control system - However, often we do not have direct access to all the state variables - They must be computed from inputs and measurable outputs Consider the discrete-time system

x( k+= 1) Gx ( k ) + Hu ( k ), x (0)= x0 (*1) yk()= Cxk () xk () ∈∈ Rnm , uk () R , yk () = R p

The solution k −1 k kj−−1 x() k= G x0 + ∑ G Hu() j j=0

free response forced-response

k −1 k kj−−1 y() k= Cx () k = CG x0 + C∑ G Hu() j j=0 u( j ) j=−− 0,..., k 1 [0, k 1] : known inputs y() j :measured outputs

The question is how and when the state can be determined from the output measurements.

If x0 is found, we can determine x(k) based on the sol of the state equation Definition (observability) : A system described by

x( k+= 1) Gx () k + ( Hu ()), k x (0)= x0 y() k= Cx () k

is said to be observable if every initial state x0 can be exactly determined from the output measurements, y(j), in finite time steps, 0 ≤ j ≤ k

Once x0 is determined, x(k) can be computed

yu(0) (0) y(0)= Cx (0) = + y(1)= Cx (1) = CG1 x + CHu(0) Oxkk(0) T  0 −−  2 yk( 1) uk( 2) y(2)==++ Cx (2) CG x0 CGHu(0) CHu (1)  k −2 y( k−= 1) CGk−12 x (0) + C∑ Gkj−− Hu() j j=0 C 00    CG CH 00  OTk = = k   k−1  kk −− 23  CG CG H CG H CH  yu(0)  (0)  =  −  Oxk (0)  Tk  yk(−− 1)  uk ( 1)  RHS is known, x(0) is to be determined x(0) can be uniquely determined

NO(k )= {0} if and only if rank() O= n  k RO()= Rn k if x(0)= N ( Ok ) and u= 0, y()j = 0, k = 0,1, , k − 1 by Cayley-Hamilton theorem G k is linear combination of GG 01 ,, n − (for any k) C  = CG rank( Ok ) rank () O OO= =  Hence, k ≥ n , where n  observability matrix NO( )= NO ()  k n−1 CG NO () : unobservable subspace

System is called observable if NO( )= {0}

i.e. Rank () O= n Theorem (observability) : The discrete time system is observable if and only if rank() O= n

Where, O is the observability matrix defined by

C  CG O =   n−1 CG Proof

For a given arbitrary initial state, x0 and u(i)=0

y(0) C  y(1) CG y = =x = Ox   00 n−1 yn(− 1) CG

r1 r2 rn If rank O = n, then there exist n independentOr row vectors m ,m ,…,m of O and from the above equation we can have

ymrr11    =  =   x00 Oxr    ymm rm 

Or is nonsigular and Or-1exists. Thus x0 can be uniquely determined from yr1,…,yrm Continuous-time Observability

Continuous-time system(without sensor noise and disturbance)

x =+=+ Ax Bu, y Cx Bu

The question : How and when the state can be determine from the output measurements?

Let’s look at derivatives of y : y= Cx + Du y=+= Cx Du CAx + CBu + Du y= CA2 x + CABu ++ CBu Du   y(nn−− 1) = CA1 x + CA n −2 Bu + CA n −3 Bu ++... CBu(n− 2) + Du ( n − 1) yC   D 0 ... u     y CA CB D 0 u hence,  = x +          (nn−− 1) 1   nn−−23  (n− 1) y  CA  CA B CA B... CBD u y(nn−− 1) = Ox + Tu( 1) (nn−− 1) ( 1) Rewriting Ox= y − Tu

known

Hence, if NO ()0 k = { } , we can determine x(t) from derivatives of u(t) and y(t) up to order n-1 . = i.e., if NO ()0 k = { } , i.e., rank () O n

then the system is observable x= F( y(nn−− 1) − Tu ( 1) ) F: left inverse of O Definition

(Observability) C-T systems :

If every x0 can be determined by measuring output y in a “finite” time, the system is “completely observable” or “observable”

Theorem : Continuous system is observable if and only if rank() O= n Observability-Controllability

Let (,,,ABCD ) be dual of system (,,,ABCD ) i.e., Original system x =+=+ Ax Bu, y Cx Du

Dual system : all matrices Swap inputs and outputs Reverse directions of signal flow arrows

Observability-Controllability duality

zt()=+=+ AztTT () Cvt () Azt () But () wt()=+=+ BTT zt () Dvt () Czt () Dvt ()

AA=TTT,,, BC = CB = DD = T Observability-Controllability duality

Controllability matrix C  of dual system

C= [] B AB A n−1 B  TTT Tn−1 T = [C AC () A C ] =OT

Similary, OC = T

Thus, system is observable (controllable) if and only if dual system is controllable (observable)

ㅗ ㅗ In fact, N() O= range ( OT )= range () C i.e., unobservable subspace is of controllable subspace of dual system End of 10-3