SET MEMBERSHIP ESTIMATION THEORY

Transcription

1 SET MEMBERSHIP ESTIMATION THEORY Michele TARAGNA Dipartimento di Elettronica e Telecomunicazioni Politecnico di Torino michele.taragna@polito.it Master Course in Mechatronic Engineering Master Course in Computer Engineering 0RKYQW / 0RKYOV Estimation, Filtering and System Identification Academic Year 07/08

2 Example: estimation of a resistance value N voltage-current measurements are performed on a real resistor, assuming that: its static characteristic is linear the device model is given by the Ohm s law v R = R i R the measurements are corrupted by an unknown noise e = [e,...,e N T The following system of linear equations is derived: v R, = R i R, +e v R, = R i R, +e v R,N. = R i R,N +e N 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

3 In matrix terms: v R, v R,. v R,N }{{} y = i R, i R,. i R,N }{{} Φ [R }{{} θ o + e e. e N }{{} e is in the standard form: y }{{} known data = F(θ o ) }{{} known function + e }{{} unknown noise F (θ o ) = Φ θ o = linear function of the unknown parameterθ o Goal: find an estimate ˆR ofrby means of an estimation algorithm (estimator)ψ applied to the data vectory: ˆR = ψ(y) = R 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

4 Least squares estimation errors θ o : true parameters that generated the data vectory Due to measurement noise,y = Φθ o +e Φθ o using the least squares algorithm as estimator: ˆθ = ( Φ T Φ ) Φ T y = ( Φ T Φ ) Φ T (Φθ o +e) = = ( Φ T Φ ) Φ T Φ }{{} I θ o + ( Φ T Φ ) Φ T e = θ o + ( Φ T Φ ) Φ T e ˆθ θ o = ( Φ T Φ ) Φ T e = estimation error e is not exactly known, but different assumptions may be made one : - random variable statistical estimation - componentwise bounded Set Membership estimation - energy bounded 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 3

5 Unknown But Bounded (UBB) errors e B e = uncertainty set Be = { ẽ R N : ẽ i ε,i =,...,N } { } = ẽ R N : ẽ = max ẽ i ε i=,...,n { { } Be= ẽ R N : ẽ T ẽ = N N ẽ i ε }= ẽ R N : ẽ = ẽ i ε i= i= e ε % e % e % e ε ε e e ε e 3 Assumption: B e is symmetric with respect to the origin ofr N 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 4 e 3

6 Problem: how to evaluate the uncertainty onˆθ induced by the uncertainty setb e? A = ( Φ T Φ ) Φ T = least squares operator : R N R n }{{}}{{} measurement space parameter space ˆθ θo = ( Φ T Φ ) Φ T e = Ae θ o = ˆθ Ae θ o EUS= ˆθ A[B e =Ay A[B e =A[y B e = Estimate Uncertainty Set y ε A = (Φ T Φ) - Φ T θ y e Φθ o y B e ε ε A = (Φ T Φ) - Φ T θ θ o EUS y y θ y Parameter space R 3 Measurement space R n N Note thatθ o EUS and that the distance betweenφθ o andy is not greater thanε 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 5 θ

7 TheEUS volume gives an idea of the estimation quality and, in particular, the Estimate Uncertainty IntervalsEUI j,j=,...,n, provide this measure: [ EUI j = min θ j, max θ j = [ˆθm j,ˆθ M j R θ EUS θ EUS }{{}}{{} ˆθ m j ˆθ M j the range of thej-th component of the estimate is such that: ˆθm j [θ o j ˆθ M j an upper bound on the estimation error of thej-th component is: ˆθ j [θ o j (ˆθ M j ˆθ m / j ) ˆθ is the symmetry center ofeus, becauseeus is the image of a symmetric set under a linear mapping EUI θ θ M θ θ m θ m θ θ EUS θ o θ M θ EUI 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 6

8 Evaluation ofeus The uncertainty set is a cube inr N centered in the origin: B e = { ẽ R N : ẽ i ε,i =,...,N } y = Φθ o +e the set of any possible measurement (called Measurement Uncertainty Set) is a cube inr N whose symmetry center is the data vectory: MUS = y B e = { ỹ R N : ỹ i y i ε,i =,...,N } R N the vertices ofmus are denoted byȳ k,k =,..., N Theorem: conv{θ,...,θ p } : EUS =A[MUS =conv { Aȳ k,k=,..., N} R n convex hull of the set{θ,...,θ p } is the smallest convex polyhedron (polytope) containingθ,...,θ p y y 6 y y 7 y3 y 3 y y 5 y B e y y 8 y 4 y y y 3 Measurement space R N A = (Φ T Φ) - Φ T θ EUS Ay 6 Ay θ Ay 3 Ay 7 Ay 4 Ay 8 Ay 5 Ay Parameter space R n 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 7 θ

9 Theorem: EUI j = whereˆθ m j = N k= Evaluation ofeui j [ˆθm j,ˆθ M j R a jk [ yk ε sign ( a jk ), ˆθ M j = ˆθ j ˆθ m j, A = [ a jk = (Φ T Φ) Φ T, ˆθ = [ˆθj = Ay Proof:ˆθ m j = min θ EUS θ j= min ỹ MUS (Aỹ) j = N = min a jk ỹ k = min ỹ: ỹ i y i εk= i=,...,n EUI N ỹ: ε ỹ i y i εk= i=,...,n θ θ M θ θ m a jk ỹ k = θ m min θ θ EUI EUS θ o θ M N ỹ:y i ε ỹ i y i +εk= i=,...,n and such a minimum is achieved byỹ k = y k εifa jk > 0, or byỹ k = y k +εifa jk < 0. SinceMUS = y Be is symmetric with respect to the data vectory, then EUS = A[MUS is symmetric with respect to the estimateˆθ = Ay and then: ˆθj = (ˆθ m j +ˆθ M / j ),j =,..., n ˆθ M j = ˆθ j ˆθ m j,j =,..., n θ a jk ỹ k 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 8

10 Description of ellipsoids LetΩ x be an ellipsoid inr N centered inx o : Ω x = { } x R N : (x x o ) T Σ x (x x o ) The form matrixσ x R N N is symmetric and positive definite it is invertible The directions of the main axes ofω x are given by the eigenvectorsu i ofσ x, which are orthogonal becauseσ x is positive definite The lengths of the semi-axes ofω x are given by λi (Σ x ), whereλ i (Σ x ) is thei-th eigenvalue ofσ x x u u u u x x o 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 9 x

11 Linear transformation of ellipsoids LetΩ x be an ellipsoid inr N centered inx o : Ω x = { x R N : (x x o ) T Σ x (x x o ) ε } and consider the linear transformation: z = Px R n, withp R n N,n < N Theorem: ifrank(p) = n, then { Ω z = P [Ω x = z R n : (z z o ) T Σ z (z z o ) ε } z o = Px o R n, Σ z = PΣ x P T R n n x Ω x x o P z P x o Ω z x 3 Space R N x x 3 x Space R n, n < N z 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 0

12 Evaluation ofeus The uncertainty set is a sphere inr N centered in the origin: B e = { ẽ R N : ẽ T ẽ ε } y = Φθ o +e the set of any possible measurement (called Measurement Uncertainty Set) is a sphere inr N whose symmetry center is the data vectory: { MUS = y Be = ỹ R N : (ỹ y) T (ỹ y) ε } R N Theorem: EUS =A [ } y Be = { θ R n :( θ ˆθ) T Φ T Φ( θ ˆθ) ε R n is an ellipsoid inr n withˆθ=ay as symmetry center and(φ T Φ) as form matrix y y 3 ε y Φθ o y B e y A = (Φ T Φ) - Φ T A = (Φ T Φ) - Φ T EUI θ θ M θ m θ m EUS θ M θ EUI Measurement space R N Parameter space R n y 3 y θ θ θ θ o 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

13 Theorem: EUS =A [ } y Be = { θ R n :( θ ˆθ) T Φ T Φ( θ ˆθ) ε R n is an ellipsoid inr n withˆθ=ay as symmetry center and(φ T Φ) as form matrix y y 3 ε y Φθ o y B e y A = (Φ T Φ) - Φ T A = (Φ T Φ) - Φ T EUI θ M θ m EUS θ EUI Measurement space R N Parameter space R n y 3 y Proof: by definition,eus is the linear mapping ofmus = y Be EUS = A [ y Be = { θ R n : ( θ Ay) T [AA T ( θ Ay) ε } ButAy = ˆθ,A = (Φ T Φ) Φ T and then: θ θ θ m θ θ θ o θ M through the matrixa: AA T = (Φ T Φ) Φ T [(Φ T Φ) Φ T T = (Φ T Φ) Φ T { Φ[(Φ T Φ) T} = = (Φ T Φ) Φ T Φ }{{} I [(Φ T Φ) T = (Φ T Φ) 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

14 Evaluation ofeui j Theorem: EUIj [ˆθ = j ε σ j, }{{} ˆθ j +ε σ j = }{{} ˆθ m j ˆθ M j [ˆθm j,ˆθ M j R σ j = [ (Φ T Φ) jj θ M θ EUS EUI θ θ θ o θ m θ m θ EUI θ M θ 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 3

15 Optimal (with minimal uncertainty) estimates Is theeus the smallest set containing the true parameterθ o? Are theeui j the smallest possible uncertainty intervals? Does the LS estimator provide the minimal uncertainty intervals? To answer all these questions, it is necessary to analyze the set of all the parameters that are consistent with both the data and the available information on noise Definition: a parameter θ is said to be feasible (or consistent) if(y Φ θ) B e } FPS = { θ R n : (y Φ θ) B e = Feasible Parameter Set = = set of all the parameters consistent with both the data and the information on noise and on the estimation problem FPS is independent of the estimation algorithm If data are generated by the true parameterθ o, thenθ o is feasible; in fact: y = Φθ o +e,e B e y Φθ o = e B e θ o FPS 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 4

16 Theorem: Relationship between FPS and EUS FPS EUS Proof: if θ FPS, then [ (y Φ θ) B e Φ θ y B e A Φ θ A[y B e = EUS [ ButA Φ θ = (Φ T Φ) Φ T Φ θ = θ and then θ EUS. The Parameter Uncertainty IntervalsPUI j,j =,...,n are defined as: [ [ PUI j = min θ j, max θ j = θj m,θj M R θ FPS θ FPS }{{}}{{} θj m θj M from the above theorem: PUI j EUI j,j =,...,n ˆθ m j θ m j [θ o j θ M j ˆθ M j 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 5

17 Evaluation offps andpui j If θ FPS, then(y Φ θ) B e = { ẽ R N : ẽ i ε,i =,...,N } (y Φ θ) i = FPS = { θ R n : ϕ T i : i-th row ofφ y i ϕ T θ i ε, i =,...,N y i ϕ T θ } i ε,i =,...,N i.e.,fps is a polytope (a convex polyhedron) generated by linear inequalities: y i ϕ T θ i ε ε y i ϕ T θ i ε y i ε ϕ T θ i y i +ε [ [ Moreover,PUIj = min θ j, max θ j = θj m,θj M R θ FPS }{{ θ FPS }}{{ } θj m θj M withθ m j andθ M j solutions of linear programming problems of the standard form: min x c T x with the constraint: Ax b 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 6

18 Evaluation offps andpui j Theorem: FPS = { θ R n : ( θ ˆθ) [ T Φ T Φ } ( θ ˆθ) ε α α = (y Φˆθ) T (y Φˆθ) = y Φˆθ ε = fitting error between measured outputs and estimated outputs a greater fitting error asmallerfps a lower uncertainty on parameters Moreover,PUI j [ˆθj = σ j ε α }{{,ˆθ j +σ j ε α }}{{ = } [ θ m j,θ M j R σ j = θ m j θ M j [(Φ T Φ) jj EUS EUI PUI M m FPS o m M PUI EUI 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 7

19 Optimal estimates Definition: given an estimateˆθ, the estimate errore(ˆθ) is given by: E(ˆθ) = sup θ ˆθ θ FPS Definition: an estimateˆθ opt is optimal if: Central estimate: ˆθC = [ˆθC j E(ˆθ opt ) E(ˆθ), ˆθ R n /, whereˆθ C j = (θj m +θj M ), j =,...,n the central estimate is optimal both ifb e = Be and ifb e = Be /, since: [θ o j ˆθ C j (θj M θj m ), j =,...,n ifb e = B e, the least squares estimateˆθ LS = (Φ T Φ) Φ T y is central ˆθ LS is optimal ifb e = B e, but in general it is not optimal ifb e = B e 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 8

20 Example: parametric estimation of a position transducer model 8 Position transducer 6 4 Voltage V z in V Position z in m The static characteristic of the position-voltage transducer is nearly linear in the range between.3 e 3.5 cm the characteristic can be linearly approximated by: V z = K t z +V o 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 9

21 In the linearity interval between.3 e3.5 cm: V z = K }{{} t unknown z + V o }{{} unknown The most relevant error occurs in the positionz measurement and it is not greater than0.5 mm to account for this error, the model equation can be rewritten as: z = K t V z V o K t +e where the unknown parameters are: θ =, θ = V o K t K t The N measurements taken in the linearity interval form a system of equations: z = V z, θ +θ +e z = V z, θ +θ +e z N. = V z,n θ +θ +e N V z,i : voltage provided by the transducer when the position value isz i 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 0

22 In matrix form: z z.. z N = V z, V z,.. V z,n θ θ + e e.. e N i.e., the estimation problem is in the standard form: y = Φ θ+e wherey R N,Φ R N,e R N and the unknown isθ R Using the Least Squares estimation algorithm: ˆθ = A y, witha = (Φ T Φ) Φ T ˆθ = [ ˆθ ˆθ = [ ˆK t = ˆθ = 549.6V/m, ˆVo = ˆθ ˆθ =.56V 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

23 Evaluation of the Estimate Uncertainty IntervalsEUI j e N Be = { ẽ N R N : ẽ i ε, i =,...,N }, ε = EUIj = [ˆθm j = min θ EUS θ j,ˆθ M j = max θ EUS θ j, j =, ˆθm j = min θ EUS θ j = N k= a jk [y k ε sign(a jk ) ˆθM j = max θ EUS θ j = N k= a jk [y k +ε sign(a jk ) = ˆθ j ˆθ m j [ˆθm,ˆθ M = [ , [ˆθm,ˆθ M = [.9 0,.39 0 [ ˆKm t, ˆK [ t M = /ˆθ M,/ˆθ m = [55.67, V/m m [ˆV o, ˆV [ o M = ˆθ M /ˆθ m, ˆθ m /ˆθ M = [ 3.703,.495 V 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification

24 Envelope of the static characteristics of models whose parameters θ are taken as the extremes of the Estimate Uncertainty IntervalsEUI j,j =, Experimental data (z, V z ) z min(θ M * Vz, θ m * Vz ) + θ m z = θ * V z + θ z max(θ M * Vz, θ m * Vz ) + θ M Position transducer 4 Voltage V z in V linearity interval Position z in m 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 3

25 Evaluation of the Parameter Uncertainty IntervalsPUI j } FPS = { θ R dim( θ) : y i [Φ θ i ε, i =,...,N [ PUIj = min θ FPS θ j, max θ FPS θ j EUIj, j =, The extremes ofpui j min θ FPS θ j= min M θ b ct θ max θ FPS θ j= min M θ b ( c)t θ [ [ θ m = min θ FPS θ,θ M = max,j =,, are solutions of the linear programming problems M= θ FPS θ [ Φ Φ θ m = min θ FPS θ,θ M = max θ FPS θ [ [ K m t,kt M = /θ M,/θ m [ [ V m o,vo M = θ M /θ m, θ m /θ M, b= [ y y +ε, c=j-th column ofi =[ , =[.596 0, = [54.0, V/m = [.735,.5 V 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 4

26 Envelope of the static characteristics of models whose parameters θ belong to the Feasible Parameter SetFPS 0 8 Experimental data (z, V z ) z = θ * V z + θ Position transducer 6 4 Voltage V z in V linearity interval Position z in m 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 5

27 Feasible Parameter SetFPS (continuous line) and set of estimates given by the extremes of Parameter Uncertainty IntervalsPUI j,j =, 0.09 Feasible Parameter Set FPS φ LS (y) 0.07 θ θ x 0 3 0RKYQW / 0RKYOV - Estimation, Filtering and System Identification 6

28 Essential references F. C. Schweppe, Uncertain Dynamics Systems. Englewood Cliffs, NJ: Prentice Hall, 973. M. Milanese, R. Tempo, A. Vicino (editors), Robustness in Identification and Control. New York: Plenum Press, 989. M. Milanese, A. Vicino, Optimal estimation theory for dynamic systems with set membership uncertainty: an overview, Automatica, vol. 7, no. 6, pp , 99. Special Issue on System Identification for Robust Control Design, IEEE Transactions on Automatic Control, vol. AC-37, no. 7, pp , 99. R. S. Smith, M. Dahleh (editors), The Modeling of Uncertainty in Control Systems, vol. 9 of Lecture Notes in Control and Information Sciences. London, UK: Springer-Verlag, 994. M. Milanese, J. Norton, H. Piet-Lahanier, É. Walter (editors), Bounding Approaches to System Identification. New York: Plenum Press, 996. J. R. Partington, Interpolation, Identification, and Sampling, vol. 7 of London Mathematical Society Monographs New Series. New York: Clarendon Press - Oxford, 997. A. Garulli, A. Tesi, A. Vicino (editors), Robustness in Identification and Control, vol. 45 of Lecture Notes in Control and Information Sciences. Godalming, UK: Springer-Verlag, 999. J. Chen, G. Gu, Control-Oriented System Identification: AnH Approach. New York: John Wiley & Sons, Inc., RKYQW / 0RKYOV - Estimation, Filtering and System Identification 7