《随机预算与调节》（英文版）Lecture 9 Last time: Linearized error propagation

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture g Last time: Linearized error propagation trajectory surface Integrate the errors at deployment to find the error at the surface e=ee See s E SE,S Or p can be integrated from: d=Fd, whereΦ(0)=1 文=f(x) F where F is the linearized system matrix. But this requires the full a(same number of equations as finite differencing) I, =time when the nominal trajectory impacts e(Ln)=Φ(tn)8 e(n)=旦=Φ,旦 where a, is the upper 3 rows of a(n) Covariance matrix E2=d,EΦ Page 1 of 8

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 9 Last time: Linearized error propagation s 1 e Se = Integrate the errors at deployment to find the error at the surface. 1 1 1 T s ss T T T E ee See S SE S = = = Or Φ can be integrated from: , where (0) ( ) F I x fx df F dx Φ= Φ Φ = = = & & where F is the linearized system matrix. But this requires the full Φ (same number of equations as finite differencing). nt = time when the nominal trajectory impacts. 1 2 1 () () ( ) n n rn r et t e et e e = Φ = =Φ where Φr is the upper 3 rows of ( ) n Φ t . Covariance matrix: 2 1 T E E =Φ Φ r r

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde e= Fe E(1)=e(l)e(t) E()=(n)e()y+e()() =Fe(n)e(1)+e()e()F You can integrate this differential equation to t, from E(O)=E. This requires the full6×6 Ematrix E eeee ee ee E,= upper left 3 x3 partition of E(t,) perturbed trajectory e (t) For small times around t e(n=e(t,)+v(,(t-L) e(,)+(v,(t,)+e()(t-L) =e(t,)+v((t-t) 1e(1)=1g+1(n)(t-Ln)=0

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 () () () () () () () () () () () () () () T T T T TT T e Fe Et etet Et etet etet Fe t e t e t e t F FE t E t F = = = + = + = + & & & & You can integrate this differential equation to tn from 1 E E (0) = . This requires the full 6 6 × E matrix. 2 ( ) upper left 3 3 partition of ( ) T T rr rv n T T vr vv n ee ee E t ee ee E Et ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = × For small times around tn, ( ) ( ) ( )( ) ( ) ( ( ) ( ))( ) ( ) ( )( ) n nn n nn vn n n nn n et et vt t t et v t e t t t et v t t t =+ − =+ + − =+ − 2 2 1 ( ) 1 1 ( )( ) 0 1 ( ) 1 T TT v v v nn n T v i n T v n et e v t t t e t t v = + −= − =−

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde e,= position error at impact I'y e"projection matrix" V 1 altitude, H range, R rack, T (into pag e 3=[rle, COS R= R R=[11y L=unit vectors along the jth axis of the 2 frame expressed in the coordinates of the 3 frame e=re RER

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 3 2 2 2 position error at impact 1 1 1 "projection matrix" 1 T v n T v n T n v T v n e e e v v v I e v = = − ⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦ [ ] 3 2 123 [ ] cos ... ... ... ... ... ... ... ... cos 111 ij ij ij e Re R R R θ θ ′ = ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = = 1j = unit vectors along the jth axis of the 2 frame expressed in the coordinates of the 3 frame. 3 2 3 2 T e Re E RE R ′ = ′ =

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde er(t=e+v, cosy(t-t,) =已 en()=eH. -v, siny(t-t,)=0 (1-L)= sIn =eg. cot yeH The transformation which relates rhterrors at the nominal end time to r and t errors when h=o e4 Pe If the e defined earlier, based on integration of perturbed trajectories, measured in R, Tcoordinates, then the sensitivity matrix defined at that point is equivalent to e

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 3 3 3 ( ) cos ( ) ( ) ( ) sin ( ) R Rn n T T H Hn n et e v tt et e et e v tt γ γ =+ − = =− − Impact: 3 3 3 3 3 3 3 ( ) sin ( ) 0 1 ( ) sin cos ( ) sin cot ( ) Hi H n i n in H n n Ri R H n R H Ti T et e v t t tt e v v et e e v e e et e γ γ γ γ γ = − −= − = = + = + = The transformation which relates R,H,T errors at the nominal end time to R and T errors when H=0 is: 3 3 3 4 3 3 ( ) ( ) cot 1 0 cot 01 0 R i T i R H T e t e e t e e e e Pe γ γ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ + = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = ≡′ ′ ⎢ ⎥ ⎣ ⎦ If the s e defined earlier, based on integration of perturbed trajectories, is measured in R,T coordinates, then the sensitivity matrix defined at that point is equivalent to s 1 r e Se S PR = = Φ

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde E4= PE3 ITR T grr」LpoR0r0 If all the original error sources are assumed normal, R and T will have a joint binormal distribution since they are derived from the error sources by linear operations only. This joint probability density function is f(r,)= where o,, Or and p can be identified from E4. Recall that we are co ng unbiased errors Contour of constant probability density function is r=xcos 8- yin 6 ne Get (6)x2+()xy+()y2=c Coefficient of x, y equals zero for principal axes tan 20=LpGROT Use a 4 quadrant tan.function. ge 5 of 8

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 4 3 2 2 2 2 2 2 T R RT R R T RT T R T T E PE P R RT TR T σ µ σ ρσ σ µ σ ρσ σ σ = ′ ⎡ ⎤ ⎡ ⎤⎡ ⎤ == = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ If all the original error sources are assumed normal, R and T will have a joint binormal distribution since they are derived from the error sources by linear operations only. This joint probability density function is ( ) ( ) 2 2 2 2 2 1 2 1 , 2 1 R RT T r rt t R T f rt e ρ σ σσ σ ρ πσ σ ρ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ − + ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ − − ⎣ ⎦ = − where , σ R T σ and ρ can be identified from E4 . Recall that we are considering unbiased errors. Contour of constant probability density function is 2 2 2 2 R RT T r rt t ρ c σ σσ σ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ − += ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ cos sin sin cos rx y tx y θ θ θ θ = − = + Get: { 2 22 0 () () () θθ θ x xy y c = ++= Coefficient of x, y equals zero for principal axes. 22 22 2 2 tan 2 R T RT RT RT ρσσ µ θ σ σ σσ = = − − Use a 4 quadrant tan-1 function

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Once 8 is found, can plug into pdf expression, get o and o h=√(a2-a)+(2pog1)2 +h) (a+G2-h) V =CO sin May want to choose c to achieve a certain probability of lying in that contour In principal coordinates, the probability of a point inside a"co ellipse is People often choose c to find what is called the circular probable error(CPe

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 8 Once θ is found, can plug into pdf expression, get σ x and σ y . 2 22 2 2 22 2 22 ( ) (2 ) 1 ( ) 2 1 ( ) 2 R T RT x RT y RT h h h σ σ ρσ σ σ σσ σ σσ = −+ = ++ = +− cos sin i xi i yi x c y c σ φ σ φ = = May want to choose c to achieve a certain probability of lying in that contour. In principal coordinates, the probability of a point inside a “ cσ ” ellipse is 2 2 1 c P e− = − People often choose c to find what is called the circular probable error (CPE)

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Choosing P=0.5, c=1.177 CPE=0.588(0x+,) This approximation is good to an ellipticity of around 3 Random processes A random process is an ensemble of functions of time which occur at random x(t) x(t) In most instances we have to imagine a non-countable infinity of possible functions in the ensemble There is also a probability law which determines the chances of selecting the different members of the ensemble We generally characterize random processes only partially One important descriptor- the first order distribution This is the classical description of random processes. We will also give the state space description later Page 7 of 8

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 1 ( ) 2 σ = + σ σ x y Choosing P=0.5, c=1.177 0.588( ) CPE = + σ x y σ This approximation is good to an ellipticity of around 3. Random Processes A random process is an ensemble of functions of time which occur at random. In most instances we have to imagine a non-countable infinity of possible functions in the ensemble. There is also a probability law which determines the chances of selecting the different members of the ensemble. We generally characterize random processes only partially. One important descriptor – the first order distribution. This is the classical description of random processes. We will also give the state space description later

16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde x(t, is a random variable F(x,0)=Px(osx], where x(() is the name of a process and x is the value taken f(x,1)= dF(x, n) f 8

16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 8 1 x( ) t is a random variable. ( , ) ( ) , where ( ) is the name of a process and is the value taken [ ] (,) ( ,) F xt P xt x xt x dF x t f xt dx = ≤ =