Probabilistic Optimal Estimation and Filtering Least Squares and Randomized Algorithms Fabrizio Dabbene 1 Mario Sznaier 2 Roberto Tempo 1 1 CNR - IEIIT Politecnico di Torino 2 Northeastern University Boston Workshop on Uncertain Dynamical Systems, Udine, Italy, 23-26 August 2011 Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 1 / 48
Motivation: Identification for Robust Control The classical approach to system identification is based on statistical assumptions about the measurement error, and provides estimates that have stochastic nature Worst-case identification, on the other hand, only assumes the knowledge of deterministic error bounds, and provides guaranteed estimates, thus being in principle better suited for robust control design However, a main limitation of such deterministic bounds lies in the fact that they often turn out being overly conservative, thus leading to estimates of limited use Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 2 / 48
Motivation: Identification for Robust Control A re-approahment We propose a re-approachement of the two paradigms: stochastic and worst-case, introducing probabilistically optimal estimates The main idea is to “exclude" sets of measure at most ǫ (accuracy) from the set of deterministic estimates We are decreasing the so-called worst-case radius of information at the expense of a probabilistic “risk." We compute a trade-off curve which shows how the radius of information decreases as a function of the accuracy Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 3 / 48
The IBC setting for systems ID and filtering Estimation problem: “Given an unknown element x, find an estimate of the function S ( x ) , based on a priori information K and on measurements of the function I ( x ) corrupted by additive noise q.” Ingredients (operators) Ingredients (sets) An information operator A problem element set X , I : X → Y with prior information Additive uncertainty/noise K ⊆ X y = I x + q A measurement space Y A solution operator A solution space Z S : X → Z Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 4 / 48
The IBC setting for systems ID and filtering Estimation problem: “Given an unknown element x, find an estimate of the function S ( x ) , based on a priori information K and on measurements of the function I ( x ) corrupted by additive noise q.” Ingredients (operators) Ingredients (sets) An information operator A problem element set X , I : X → Y with prior information Additive uncertainty/noise K ⊆ X y = I x + q A measurement space Y A solution operator A solution space Z S : X → Z Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 4 / 48
IBC Setting for System ID Estimation algorithm Estimation algorithm An algorithm A is a mapping (in general nonlinear) from Y into Z , i.e. A : Y → Z An algorithm provides an approximation A ( y ) of S x using the available information y ∈ Y of x ∈ K The outcome of such an algorithm is called an estimator and the notation ˆ z = A ( y ) is used Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 5 / 48
The IBC setting for systems ID and filtering Illustration of the considered framework Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 6 / 48
The setup of this talk The IBC setting for systems ID and filtering Problem element set X is R n The information operator I : X → Y is linear The uncertainty q ∈ Q ⊆ R m , where Q is a bounding set The solution set Z is R s and the solution operator S : X → Z is linear Assumption (Sufficient information) We assume that the information operator I is a one-to-one mapping, i.e. m ≥ n and rank I = n We assume for the sake of simplicity that the three sets X , Y , Z are equipped by the same ℓ p norm. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 7 / 48
Example System parameter identification Parameter identification problem which has the objective to identifying a linear system from noisy measurements. The problem elements are the input-output pairs ξ = ξ ( t , x ) of a dynamic system, parametrized by some unknown parameter vector x ∈ K ⊆ X and with given basis functions ϕ i ( t ) n � x i ϕ i ( t ) = Φ T ( t ) x ξ ( t , x ) = i = 1 Suppose then that m noisy measurements of ξ ( t , x ) are available for t 1 < t 2 < · · · < t m , Φ( t m )] T x + q . y = I x + q = [Φ( t 1 ) · · · (1) The solution operator is given by the identity, Sx = x and Z ≡ X . In this context, one usually assumes unknown but bounded errors | q i | ≤ R , i = 1 , . . . , m , that is Q = B ∞ ( R ) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 8 / 48
The consistency set I − 1 ( y ) A key role is played by following set, which represents the set of all problem elements x ∈ K ⊆ X compatible with (i.e. not invalidated by) the information I x , the uncertainty q and the bounding set Q Consistency set I − 1 ( y ) For given y ∈ Y , define I − 1 ( y ) . = { x ∈ K | there exists q ∈ Q : y = I x + q } (2) Under the sufficient information assumption, the set I − 1 ( y ) is bounded. For instance, in the previous example we have � � Φ( t m )] T x � ∞ ≤ R I − 1 ( y ) = x ∈ K : � y − [Φ( t 1 ) · · · In system identification, I − 1 ( y ) is sometimes referred to as parameter feasible set. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 9 / 48
The IBC setting for systems ID and filtering Our setup for this talk Illustration of the considered framework Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 10 / 48
Worst-Case Setting ... ... The Worst Case Setting Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 11 / 48
Worst-Case Setting Errors and optimal algorithms Given perturbed information y ∈ Y , the worst-case error is defined as r wc ( A , y ) . = x ∈I − 1 ( y ) �S x − A ( y ) � p . max This error is based on the available information y ∈ Y about x ∈ K , and it measures the approximation error between Sx and A ( y ) An algorithm A wc is called worst-case optimal if it minimizes the error o r wc ( A , y ) for any y ∈ Y o ( y ) . o , y ) . r wc = r wc ( A wc A r wc ( A , y ) = inf z wc = A wc A worst-case optimal estimator is given by ˆ o ( y ) o The minimal error r wc o ( y ) is called the (local) worst-case radius of information Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 12 / 48
Chevbychev center and central algorithms The Chebychev center z c ( H ) of a set H ⊂ Z and its “radius” r c ( H ) are defined as h ∈ H � h − z c ( H ) � . h ∈ H � h − z � . max = inf z ∈ Z max = r c ( H ) Optimal algorithms map data y into the Chebychev center of the set SI − 1 ( y ) , i.e. z wc z c ( SI − 1 ( y )) = ˆ o For this reason they are also called central algorithms For given set H , the ℓ p -Chebychev center x c ( H ) and radius r c ( H ) are the center and radius of the smallest ℓ p ball enclosing H . In general z c ( H ) may not be unique and not necessarily it belongs to H . if H is centrally symmetric then the origin is a Chebychev center of H Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 13 / 48
Chevbychev center and central algorithms The Chebychev center z c ( H ) of a set H ⊂ Z and its “radius” r c ( H ) are defined as h ∈ H � h − z c ( H ) � . h ∈ H � h − z � . max = inf z ∈ Z max = r c ( H ) Optimal algorithms map data y into the Chebychev center of the set SI − 1 ( y ) , i.e. z wc z c ( SI − 1 ( y )) = ˆ o For this reason they are also called central algorithms For given set H , the ℓ p -Chebychev center x c ( H ) and radius r c ( H ) are the center and radius of the smallest ℓ p ball enclosing H . In general z c ( H ) may not be unique and not necessarily it belongs to H . if H is centrally symmetric then the origin is a Chebychev center of H Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS 2011 13 / 48
Recommend
More recommend
