H Estimation. Speaker : R.Lakshminarayanan Guide : Prof. K.Giridhar. H Estimation p.1/34

Transcription

1 H Estimation Speaker : R.Lakshminarayanan Guide : Prof. K.Giridhar H Estimation p.1/34

2 H Motivation The Kalman and Wiener Filters minimize the mean squared error between the true value and estimated values of any random process (Stationary in the case of wiener). Kalman extended the Wiener solution to non-stationary and time varying systems by incorporating a finite-dimensional state model into the estimation problem. x k+1 = A k x k + B k r k E(r k r l ) = δ kli y k = C k x k + D k r k E((x 0 x 0 )(x 0 x 0 ) ) = P 0 H Estimation p.2/34

3 H Motivation The Kalman filter operates and gives the best performance under the assumption of known covariance matrices of the process and observation noise, Q and R respectively, where Q = E[v k v k ], v k = B k r k and R = E[w k w k ], w k = D k r k The filter performance deteriorates considerability under modelling uncertainities (despite the fact that the process noise is introduced to accomodate for our ignorance of the underlying process) H Estimation p.3/34

4 Kalman Filter and Model Uncertainity Hence the Kalman Filter which is well tuned to a nominal plant behaves poorly for a perturbed one. Example : In a practical communications scenario, for a channel estimation problem, Additive noise variance could vary considerably. Modelling the doppler spectrum with a finite length AR process always has modelling uncertainities associated with it. A varying doppler introduces additional parametric uncertainities. H Estimation p.4/34

5 Beyond the Kalman Filter The goal is to design a filter that bounds the error norm of an entire class of perturbations. Various approaches to designing robust kalman filters: Set membership Kalman filtering. H filters. Design of Kalman filters based on Linear Matrix Inequalities.... H Estimation p.5/34

6 Transfer Function Norms H 2 norm : For any stable SISO linear systems with transfer function G(s), the H 2 -norm is defined as G 2 = ( 1 2π G(jw) 2 dw) 1/2 (1) From (1), G 2 can be interpreted as the average system gain over all frequencies. H Estimation p.6/34

7 Transfer Function Norms H norm :For stable SISO linear systems with transfer function G(s), the H 2 -norm is defined as G = sup w G(jw) (2) The H norm is the largest factor by which any sinusoid is magnified by the system. H Estimation p.7/34

8 Transfer Function Norms H norm in terms of the effect of G(s) on the space of input signals with bounded l 2 norms. Consider the system G(S) with input v(s) and output is given by z(s) = G(s)v(s) H Estimation p.8/34

9 Transfer Function Norms H norm in terms of the effect of G(s) on the space of input signals with bounded l 2 norms. Consider the system G(S) with input v(s) and output is given by z(s) = G(s)v(s) Now, Gv 2 = = ( 1 2π ( 1 2π G(jw)v(jw) 2 dw G(jw) 2 v(jw) 2 dw ) 1/2 ) 1/2 H Estimation p.9/34

10 Induced Norms Gv 2 sup w G(jw) ( 1 2π v(jw) 2 dw) 1/2 = G v 2 H Estimation p.10/34

11 Induced Norms Gv 2 sup w G(jw) ( 1 2π v(jw) 2 dw) 1/2 = G v 2 Hence G Gv 2 v 2,v 0 H Estimation p.11/34

12 Induced Norms Gv 2 sup w G(jw) ( 1 2π v(jw) 2 dw) 1/2 = G v 2 Hence G Gv 2 v 2,v 0 In fact, G = sup { } Gv 2,v 0 v 2 sup w {σ max (G(e jw ))} (3) Equation (3) is known as the induced second norm. sup w R σ max (G(jw)) is the maximum singular value of the transfer function matrix as a function of frequency. H Estimation p.12/34

13 A few Observations Therefore, the H -norm of G(s), namely G = sup w G(jw), is equivalent to the induced 2-norm of G(s), i.e., the maximum amplification of the energy of the input signal. Induced norms are used to specify the error bounds of an transfer operator. H Estimation p.13/34

14 Some Math... Vector Norms: The p-norm of a vector x = [x(1),x(2),...x(n))] R n as, for 1 p <, x p n x(j) p j=1 The infinity norm of x R n is defined as, (4) x max x(j) (5) j [1,n] H Estimation p.14/34

15 Norms for sequences For a sequence x = x 1,x 2,... with x i R n,i = 1, 2, 3,..., the p th and infinity norms of the sequence are defined respectively as, x p x(j) p p j=1 (6) x sup j x(j) (7) H Estimation p.15/34

16 Space of bounded sequences The space l n p for 1 p <, is defined as the p th and infinity norms of the sequence are defined respectively as, l n p {x = (x 1,x 2,...) x i R n,i = 1, 2, 3,..., x p < } (8) For p =, the space l is defined as l n {x = (x 1,x 2,...) x i R n,i = 1, 2, 3,..., x < } (9) H Estimation p.16/34

17 Induced Norms between normed spaces For an operator T, a mapping between two normed spaces X and Y, with norms. α and. β, the induced (α,β)) norm of T is given by T i(β,α) = sup x 0 T x β x α = sup T x β x α 1 = sup T x β x α =1 The maximum or supremum value of the norm is attained for a particular value of x. H Estimation p.17/34

18 Minimizing Induced Norms The induced norms represent the worst-case gain and measure of the maximum amplification that the mapping can exert on a bounded input signal. H Estimation p.18/34

19 Minimizing Induced Norms The induced norms represent the worst-case gain and measure of the maximum amplification that the mapping can exert on a bounded input signal. Minimizing induced norms amounts to minimizing the worst case effect of the transfer operator and guarantees the worst case result. H Estimation p.19/34

20 Transfer Operator T Every estimation method (G) has some estimation error associated with it. Consider the transfer operator T that maps input disturbances to estimation errors. T is function of the estimation method G. H Estimation p.20/34

21 Transfer Function, Transfer Operator H Estimation p.21/34

22 Robust Estimation Problem Objective: To design a filter G such that the induced norm of the mapping (T ) from the inputs r,x 0 and x 0 x 0 to the estimation error, e, is bounded for all possible perturbations (of parameters in the state and observation equations). Assumption: Perturbations and inputs (disturbance) are norm bounded. The time-varying state space model is x k+1 = A k x k + B k r k E(r k r l ) = δ kli e k = M k (x k x k ) E(r k (x 0 x 0 ) ) = 0 y k = C k x k + D k r k E((x 0 x 0 )(x 0 x 0 ) ) = P 0 v k = B k r k,q = E[v k v k ],w k = D k r k,q = E[w k w k ] H Estimation p.22/34

23 Robust Estimation Problem, Mathematically Knowledge of the Q and R assumed unknown. The cost to be minimized is J = T i2 sup (r,x 0 x 0 ) 1 M k (x k x k ) 2 r 2 + x 0 x 0 2 P 1 0 (10) J might not have a minima, ie., H problem has no solution. Consider the sub-optimal H problem of making J 1 = T i2 sup (r,x 0 x 0 ) 0 sup { r 2 + x 0 x 0 2 P 1 0 e 2 r 2 + x 0 x 0 2 P 1 0 1} γ e 2 (11) H Estimation p.23/34

24 Sub-Optimal Robust Estimation Problem, Mathematically (γ - a positive constant), incorporating a state space structure for F. Smaller γ is, better is the estimation strategy. Game theoretic Solution: Optimization problem solved in two steps. Maximize J 1 with respect to r,x 0. Minimize J 1 with respect to x 0. In short min x k max r,x 0 J 1, J 1 = e 2 (12) H Estimation p.24/34

25 Robust Estimation Problem, Mathematically Why maximize with respect to (r,x 0 ) before minimizing with respect to K(Filter Gain)? Answer: The supremum value of the induced norm is maximum only for a particular value of input. Choose the input sequence for which the cost attains supremum value (so maximize J 1 with respect to r,x 0 ). Then minimize it with respect to K. H Estimation p.25/34

26 Estimator Derivation Estimator structure: x k+1 = A k x k + K k (y k C k x k ) The estimation problem is : min x k max r,x 0 J 1, J 1 = e 2 given x k+1 = A k x k + B k r k r 2 + x 0 x 0 P (13) Solution Method: Augment the constraint cost function J 1 with the constraint with lagrange multipliers... H Estimation p.26/34

27 + (K k D k )(K k D k ) (14) H Estimator Equations The resulting filter equations are: x k+1 = x k + K k (y k C k x k ) K k = H k C k (C k H k C k + D kd k ) P k+1 = (I K k C k )H k (I K k C k ) H 1 k P 1 k γ 2 M k M k (15) H Estimation p.27/34

28 H Estimator Equation Discussion The set of equations describing H are just identical to that of the Kalman filter but for the term H k in the H filter in place of P k term in the Kalman filter. k - Measure of information about state prior to observation. P 1 In H estimator, H k = (P 1 k information available. γ 2 M k M k) 1 - less Smaller γ, lesser our faith in the model parameters. As γ, H k = P k. The H estimator Kalman estimator. γ γ, det(h k (γ )) = 0. More information cannot be extracted. P k+1 becomes indefinite. H Estimation p.28/34

29 Kalman Filter Equations x k+1 = x k + K k (y k C k x k ) K k = P k C k (C k P k C k + D kd k ) P k+1 = (I K k C k )P k (I K k C k ) + (K k D k )(K k D k ) (16) H Estimation p.29/34

30 The MMSE estimator minimizes E(e(n) e(n)) = R ee (0) Comparing H Estimation with = 1 2π = 1 2π = 1 2π +π π +π π +π π MMSE Estimation trace ( S ee (e jw ) ) dw trace ( G ve (e jw )G ve(e jw ) ) dw ( N i=1 σ 2 i ( Gve (e jw )) )) dw = T ve 2 2 (17) The equation (17) gives the 2-norm of the transfer operator. H Estimation p.30/34

31 Comparing H Estimation with MMSE Estimation Hence, MMSE estimator minimizes the 2-norm (average energy over all the frequencies) of the transfer function between the disturbance and the estimation error with respect to the estimator s transfer function. H Estimation p.31/34

32 Comparing H Estimation with MMSE Estimation The H estimator minimizes the induced 2-norm of the transfer function between the disturbance and the estimation error. It minimizes the H norm of the transfer function, namely T = sup σ max (H(jw)) (18) w R It minimizes the maximum singular value of the transfer function matrix as a function of frequency (or maximum gain from the input distubance to the estimation error) H Estimation p.32/34

33 Before I Wind-up... In a subsequent presentation, the following topics will be explored. H filter for systems with uncertain state models. Comparing the H and the Kalman filter for the OFDM channel estimation problem. Its relation to indefinite quadratic programming and risk-sensitive estimation. H Estimation p.33/34

34 Thank you H Estimation p.34/34