ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio ad velocity of a automobile. Howeve, it is ofte difficult o eve impossible to measue all state vaiables diectly, e.g., the iteal tempeatues ad pessues of a chemical pocess. It is theefoe desiable to estimate the cuet state fom the kowledge of the histoy of iputs ad outputs of the system ove some fiite iteval. A device which accomplishes estimates of the state is called a obseve, ad this leads to the cocepts of obsevability ad costuctibility, which ae coceed with detemiig whethe a estimate of the state ca be costucted whe iput-output histoies ae kow. O the othe had, if measuemets of u(), y() ae cotamiated by oise, the the best we ca do is to estimate xt () appoximately, by ivokig filtes which achieve the best appoximatio as measued by some objective fuctio, e.g., the Kalma filte. Obsevability: Obsevability is coceed with the ability of detemiig the peset state xt ( fom kowledge of peset ad futue system output yt () ad iput ut (), t [ t, t. ut () yt () xt () =? S t t 2 t t 2 Costuctibility: Costuctibility is coceed with the ability of detemiig the peset state xt () fom kowledge of peset ad past system output yt () ad iput ut (), t [ t, t. Clealy, this is moe iteestig tha obsevability. ut () xt () =? S yt () t t t 2 t t t 2 24- /9
It will be show that obsevability always imply costuctibilty, wheeas costuctibility implies obsevability oly whe the state tasitio matix of the system is osigula. 7. Discete Time Ivaiat Case Coside the system goveed by the diffeetial system x( j+ = Ax( j) + Bu( j), x( ) = x y( j) = Cx( j) + Du( j) (7. whee x () = x is the iitial state. We ae iteested i uiquely detemiig the iitial state x () = x fom peset ad futue system iput ad output (obsevability poblem.) The output of the system is j j j ( i+ i= y( j) = x + Bu( i) + Du( j), j >. (7.2) Rewite (7.2) as follows: j j ( i+ y ( j): = y( j) Bu( i) + Du( j), j > i=. (7.3) y (): = y () Du () Thus, ( ) j y j = x, j >. (7.4) ad this equatio ca be solved by fist computig y ( j), ad ivetig y () C y ( + = x k y ( + k Y k Ok (7.5) o, Y = O x, Y R, O R (7.6) ( k + p ( k + p k k k k Clealy, the system is obsevable if the solutio x of (7.5) is uique, i.e., if it is the oly iitial state that, togethe with the give iput-output histoies ca geeate the obseved output histoy. Now, (7.6) has a uique solutio x N { Ok} = {} θ ak{ O k} =. 24-2/9
Defiitio: Obsevability Matix The liea time-ivaiat system (7. is said to be obsevable if the obsevability opeato i (7.4) which defies a matix O has full colum ak, whee O C : =, (7.7) ak O =. e.g., { } Remak: Note that by the CHT, N { O} = N { O k}, k, but N { O} N { O k}, k <. Theefoe, i geeal oe has to obseve the output i steps. Hece, (7.6) educes to p Y = O x, Y R, O R (7.8) p Fo the system to be obsevable, the cotollability matix must have full colum ak, i.e., it must be oe-to-oe. A state x is called uobsevable if O x = θ, ad N { O } is called the uobsevable space. Examples: x( j+ = Ax( j), x(0) = x y( j) = Cx( j) 0 0, 0 (a) A= C = [ ] C 0 Obsevability matix, ak { } 2 detemied fom 2 output measuemets (hee the system has o iput.) O 2 = = O 2 =, theefoe y(0) C x(0) 0 x(0) y( = x2(0) = x2(0) x (0) y(0) y( y(0) x2 (0) = 0 = y( y( x ca be uiquely 24-3/9
0, 0 (b) A= C = [ ] O C 0 2 = =, ak 2 = 0 O, theefoe the system is ot obsevable. 0 N O 2 = :α α R, such states ae uobsevable. Obsevability matix { } Sice { } 7.2 Cotiuous-Time Case Coside the TV system: xt () = At () xt () + Btut () (), xt ( ) = x yt () = Ctxt () () + Dtut () () (7.9) whee xt () R, ut () R m, yt () R p, ad ABCD,,, have piecewise cotiuous eties. 7.2. Special Case Obsevability Coside fist the special case of estimatig the state xt ( fom obsevig the output o the iteval [ t, t whe ut () = 0. The estictio of the zeo-iput espose y() to [ t, t 2 ], deoted as y () is called the tail of the output. The obseve equatio elates the state to the tail: y () t = C() t Φ(, t t ) x, t [ t, t ] (7.0) 2 Thus, the obsevability opeato, which maps states at t = t ito tails, is the opeato defied by: : R [ t, t ] ( ) 2 2 x () t = C() t Φ(, t t ) x (7. The opeato fom of (7.0) is: x = y (7.2) Defiitio: Complete Obsevability The system ( A(), C() ) is completely obsevable (C.O.) o [ t, t if the obsevability opeato is oe-to-oe, i.e., if x is completely detemied by the tail y. 24-4/9
7.2.2 Geeal Obsevability Defiitio: Complete Obsevability The system ( A(), B(), C(), D() ) is completely obsevable (C.O.) o [ t, t if, give the histoy of u(), y() o [ t, t, x() is completely detemied. The output equatio is: yt () = Ct () Φ (, tt) x + Ct () Φ(, tτ) B( τ) u( τ) dτ t t ( ) = Ct () Φ (, tt x + Ct () u () t + Dtut () () y ( t) (tail) yu () t (foced) (7.3) The geeal obseve equatio is y = y y = x (7.4) : u which elates states ad tails, as i the special case. Notes: (a) The effect of past ( t < t ) iput o futue ( t > t ) output ca be summaized by eithe xt ( o the tail afte t. (b) The state acts as a "tag" o the tail by uiquely detemiig it. The covese is ot geeally tue as a tail may have bee poduced by may states xt ( fo a system that is ot completely obsevable. (c) The poblem of C.O., e.g., give u(), y() o [ t, t, fid x() ca be educed to the poblem of fidig the iitial state xt ( fom peset ad futue iput ad output because xs () =Φ(, st x, s [ t,] t. Moeove, the costuctibility poblem of estimatig the cuet state xt (), t [ t, t fom peset ad past iput ad output yields the solutio of the C.O. poblem via xs () =Φ (,)(), tsxt s [ t,] t. Clealy, costuctibility ad C.O. ae equivalet fo cotiuous-time TV systems, but ot ecessaily fo discete-time systems because the state tasitio matix is ot always ivetible. (d) The C.O. coditio that is oe-to-oe is equivalet to the statemet that { } = { θ} { } { θ} N. If N, the the dimesio of the state space is highe tha is eeded to detemie futue behavio, ad thee is edudacy. The coditio { } = { θ} N suggests the followig equivalet defiitio of complete obsevability. Defiitio: Uobsevable State 24-5/9
I a liea system, the state x is uobsevable at time t if the zeo-iput espose (tail) of the system is zeo fo t t. That is, a state is uobsevable if it is ozeo ad it poduces a zeo tail. Defiitio: Uobsevable Subspace t The uobsevable subspace at time t, R, is Defiitio: Complete Obsevability (alteative) The system (7.9) is completely obsevable at O { } t R : = R : is uobsevable at. (7.5) O x x t t t if = { θ} R. O 7.3 Obsevability ad the Stuctue of the Obsevability Opeato The discete-time case is staightfowad as Equatio (7.8) is a fiite-dimesioal vecto equatio i which is a tall matix, as depicted below. y x D { } = R N{ } θ R{ } C{ } = l [, k] θ 2 N{ } R { } 24-6/9
The obsevability equatio does ot have a solutio if y R{ }, i.e., whe the measued output is oisy. I this case, we ca set up the poblem of miimizig y x ove the possible iitial state vectos (moe o this late.) The cotiuous-time case has the followig stuctue diagam: D { } = R N{ } θ C{ } = [ t, t ] 2 2 N{ } θ R{ } R { } The above emak holds i the cotiuous-time case as well: The obsevability equatio does ot have a solutio if y R{ }, but y x ca be miimized. 7.4 The Optimal Obseve Poblem i 2 Suppose the obseved tail y outside the { } + v is cotamiated by a oise v. The, the obseved tail may be R, i which case the obsevability equatio x = y + v has o solutio. Defiitio: Obseve Poblem Suppose the system [ A(), B(), C(), D() ] is C.O. o [, ] ( ) 2 t, t 2 t t 2. Fid x to miimize the om x y + v, (7.6) i.e., we ae iteested i computig x which poduces the tail x that best appoximates y + v. Note that, it is the iput-output behavio athe tha the state that fially mattes to us, ad theefoe we measue the quality of appoximatio by how fa the tail x is fom the obseved y + v (athe tha ty to miimize the distace betwee the tue ad estimated states i the state-space R.) Theoem: 24-7/9
Suppose the system [ A(), B(), C(), D() ] is C.O. o [ t, t ] o the iteval [, ] Poof: 2 mi ( ) 2 t, t 2 x y + v x R 2. The the miimizatio poblem, (7.7) t t has the uique solutio ( x ) = ( ) ( y + v) By the stuctue diagam, [ t, t ] Thus, y 2 2 opt ca be decomposed ito a sum of two othogoal subspaces: [ t, t ] = ( ) ( ) R N. 2 2 + v ca be decomposed ito compoets lyig i these same subspaces By the Pojectio Theoem y ( R) N R ( ) N ( ) y + v= y + v + v, v R, v N (7.8) + vr is the uique vecto i R( ) closest to y + v. Sice y + vr R( ), thee exists ( x ) R such that ( x ) opt = y opt + vr be two solutios sice is assumed to be oe-to-oe, ad hece ( x ) is uique. opt The,. Thee caot ( x ) = ( y + v ) opt R ( R) ( y v) = y + v + v = + N, (7.9) ad sice is ivetible (assumig that the system is C.O.), we have: ( x ) = ( ) ( y + v). (7.20) opt Remaks: (a) The expessio ( ) defies the pseudoivese of ay oe-to-oe opeato. 24-8/9
(b) The oise compoet v N ( ) R ( ) N is filteed out by the obseve, but the compoet v R caot be distiguished fom the tue tail ad emais as a eo i the estimate of the tue tail. 7.4. Evaluatio of the Pseudoivese Recall that: : R 2[ t, t. (7.2 x () t = C() t Φ(, t t ) x ( ) The adjoit : 2[ t, t R is obtaied as: t2 = Φ t y: ( τ, t ) C ( τ) y( τ) dτ (7.22) Defiitio: Backwad Obsevability Gammia The backwad obsevability Gammia is defied as: t2 2 = = Φ Φ t Nt (, t): ( τ, t) C( τ) C( τ) ( τ, t) dτ (7.23) The optimal obseve (o ecostuctio) law is give by: t2 ( ) = (, ) Φ (, ) ( )( ( ) + ( )) x N t t τ t C τ y τ v τ dτ. (7.24) opt 2 t whee y (): t = y() t yu () t (7.25) measued output zeo-state espose 24-9/9