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Finite horizon anisotropy-based multicriteria filtering∗
Proceedings of the 8th IFAC Symposium on Robust Control Design Proceedings of the 8th IFAC Symposium on Robust Control Design Proceedings of the 8th IFAC Symposium on Robust Control Design July 8-11, 2015. Bratislava, Republic Proceedings of the 8th IFACSlovak Symposium on Robust Control Design July 8-11, 2015. Bratislava, Slovak Republic Available online at www.sciencedirect.com July 8-11, 2015. Bratislava, Slovak Republic July 8-11, 2015. Bratislava, Slovak Republic
ScienceDirect IFAC-PapersOnLine 48-14 (2015) 192–197
Finite horizon anisotropy-based Finite horizon anisotropy-based Finite horizon anisotropy-based Finite horizon anisotropy-based multicriteria filtering multicriteria filtering multicriteria filtering multicriteria filtering
Victor N. Timin Alexander P. Kurdyukov Victor Victor N. N. Timin Timin Alexander Alexander P. P. Kurdyukov Kurdyukov Victor N. Timin Alexander P. Kurdyukov V.A. Trapeznikov Trapeznikov Institute Institute of of Control Control Sciences, Sciences, RAS, RAS, 65 65 V.A. V.A. Trapeznikov Institute of Control Sciences, RAS, 65 Profsoyuznaya Str., 117997 Moscow GSP-4, Russia V.A. Trapeznikov Institute of Control Sciences, RAS, Profsoyuznaya Str., 117997 Moscow GSP-4, Russia 65 Profsoyuznaya Str., 117997 Moscow GSP-4, Russia ([email protected]), ([email protected]) Profsoyuznaya Str., 117997 Moscow GSP-4, Russia ([email protected]), ([email protected]), ([email protected]) ([email protected]) ([email protected]), ([email protected]) Abstract: Finite horizon anisotropy-based multicriteria filtering problem for the linear varying Abstract: Finite horizon anisotropy-based multicriteria filtering problem the varying Abstract: Finite horizon anisotropy-based multicriteria input filtering problem for for the linear linear varying discrete time system has been solved. Multidimensional disturbance consists of of Abstract: Finite horizon anisotropy-based multicriteria filtering problem for the varying discrete time system has been solved. Multidimensional input disturbance consistslinear of groups groups of discrete time system has been solved. Multidimensional input disturbance consists of groups of signals, each group is characterized by its inaccurately given information theory properties. For discrete time system has been solved. Multidimensional input disturbance consists of groups of signals, each group is characterized by its inaccurately given information theory properties. For signals, each group is characterized by its inaccurately given information theory properties. For that mathematical model, the bounded real lemma for an anisotropic norm has been proved. signals, each group is characterized by itsreal inaccurately informationnorm theory For that mathematical model, the bounded lemma forgiven an anisotropic hasproperties. been proved. that model, the bounded real for an anisotropic norm been proved. If themathematical subsystem norms to specific values, condition for multicriteria that mathematical model,are thelimited bounded real lemma lemma for a anisotropic norm has has proved. If norms are to specific values, aaansufficient sufficient condition for been multicriteria If the the subsystem subsystem norms are limited limited to An specific values,algorithm sufficient condition for multicriteria estimator existence has been obtained. estimation is based on recursion If the subsystem norms are limited to specific values, a sufficient condition for multicriteria estimator existence has been obtained. An estimation algorithm is based on recursion solving solving estimator existence has obtained. An algorithm is on recursion solving of linear inequalities the special estimator existencematrix has been been obtained.simultaneously An estimation estimationwith algorithm is based based of ona solving of linear difference difference matrix inequalities simultaneously with the inequality inequality of a recursion special type. type. of linear difference matrix inequalities simultaneously with the inequality of a special type. of linear difference matrix inequalities simultaneously with the inequality of a special type. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: robust robust filtering, filtering, anisotropic anisotropic norm, norm, Anisotropic-based Anisotropic-based Bounded Bounded Real Real Lemma, Lemma, Keywords: robust filtering, anisotropic norm, Anisotropic-based Bounded Real difference Riccati inequations Keywords: robust filtering, anisotropic norm, Anisotropic-based Bounded Real Lemma, Lemma, difference Riccati inequations difference difference Riccati Riccati inequations inequations 1. by 1. INTRODUCTION INTRODUCTION by the the a-anisotropic a-anisotropic norm norm of of the the system system defined defined as as its its 1. by the a-anisotropic norm of the system defined as maximum root mean square gain over such disturbances. 1. INTRODUCTION INTRODUCTION by the a-anisotropic norm of the system defined as its its maximum root mean square gain over such disturbances. maximum root mean square gain over such disturbances. In the limiting cases where a → 0 and a → ∞, the Let system have unknown estimated and directly observed maximum root mean square gain over such disturbances. In the limiting cases where a → 0 and a → ∞, the Let system have unknown estimated and directly observed In the limiting cases where a → 0 and a → ∞, the a-anisotropic norm gives the l2 induced Let have unknown estimated and directly observed outputs. The system is by external disturbance the limiting cases where aFrobenius → 0 andand a → ∞, the Let system system have unknown estimated directly observed In a-anisotropic norm gives the Frobenius outputs. The system is driven driven by an an and external disturbance a-anisotropic norm gives therespectively. Frobenius and and lll22 induced induced operator norm of the system outputs. The system is driven by an external disturbance that is time invariant Gaussian sequence. This sequence is a-anisotropic norm gives the Frobenius and 2 induced outputs. Theinvariant system is driven by an external disturbance operator norm of the system respectively. that is time Gaussian sequence. This sequence is norm of system respectively. that is is time time invariant Gaussian sequence. This sequence is operator produced from the noise sequence with zero operator norm of the the systemsystems, respectively. that invariant Gaussian is produced from the white white noise sequence. sequence This with sequence zero mean mean For discrete time invariant the estimator For discrete time invariant systems, the estimator estimator design design produced from the white noise sequence with zero mean and identity covariance matrix by the unknown linear time produced from the white noise sequence with zero mean For discrete time invariant systems, the design and identity covariance matrix by the unknown linear time problem minimizing the a-anisotropic norm of the error For discrete time invariant systems, the estimator design problem minimizing the a-anisotropic norm of the error and identity covariance matrix by the unknown linear time invariant shaping filter whose transfer function is in the and identity covariance the unknown minimizing the a-anisotropic norm of the error invariant shaping filter matrix whose by transfer functionlinear is intime the problem operator was solved by Vladimirov in (Vladimirov (2014)). problem minimizing the a-anisotropic norm of the error operator was solved by Vladimirov in (Vladimirov (2014)). invariant shaping filter whose transfer function is in the Hardy space H . For given estimator and shaping filter, 2 . For invariant shaping filter whose transfer function is in the operator was solved by Vladimirov in (Vladimirov (2014)). Hardy space H given estimator and shaping filter, 2 . For given estimator and shaping filter, operator was solved by Vladimirov in (Vladimirov (2014)). Hardy space H the effect of the disturbance on estimation be 2 Hardy space given estimator shapingcan filter, In recent recent years, years, an an approach approach to to the the construction construction of of time time 2 . For the effect of H the disturbance on the the and estimation can be In In recent years, an approach to the construction of time the effect of the disturbance on the estimation can be varying anisotropic filtering and control theory has measured by the ratio of the root mean square values In recent years, an approach to the construction of been time the effect of the disturbance on the estimation can be been measured by the ratio of the root mean square values varying anisotropic filtering and control theory has varying anisotropic filtering and control theory has been measured by the ratio of the root mean square values developing. Definitions of the anisotropy of the random of the signals. The aim of the estimator is to minimize varying anisotropic filtering and control theory has been measured by the mean issquare values developing. Definitions of the anisotropy of the random of the signals. Theratio aim of of the the root estimator to minimize developing. the Definitions of the the anisotropy of the the random of the signals. The of the estimator is to vector anisotropic norm of which this effect. Suppose that of a filter developing. Definitions of of the signals. The aim aim ofthe thechoice estimator to minimize minimize vector and and the anisotropic normanisotropy of control controlofsystem, system,random which this effect. Suppose that the choice of aa isshaping shaping filter this effect. Suppose that the choice of shaping filter vector and the anisotropic norm of control system, which correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the vector and the anisotropic norm of control system, which this effect. Suppose that the choice of a shaping filter belongs to an opponent whose aim is to maximize the correspond to the time varying anisotropic-based filtering correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the and control theories on finite horizon, were stated root mean square gain. If the set of the opponent is correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the and control theories on finite horizon, were stated in in root mean square gain. If the set of the opponent is control theories on horizon, were stated in root mean square gain. If the set of the opponent is the (Vladimirov al. (2006)). A bounded whole space H , then the worst damage to the estimation and control et theories on finite finite horizon, real were lemma stated for in root gain. the set of thetoopponent is the and (Vladimirov et al. (2006)). A bounded real lemma for wholemean spacesquare H22 , then theIfworst damage the estimation (Vladimirov et al. (2006)). A bounded real lemma for whole space H , then the worst damage to the estimation discrete time varying system was proved in (Maximov performance measured in terms of the root mean square 2 (Vladimirov al. (2006)). bounded for whole space H worst damage to the estimation discrete timeetvarying systemA was provedreal in lemma (Maximov 2 , then the performance measured in terms of the root mean square performancethe measured in could terms achieve of the the root root mean square square discrete time varying system was proved in (Maximov paper was for of gain opponent corresponds with discrete time This varying system wasbasis proved in solution (Maximov performance measured in terms of mean et al. al. (2011)). (2011)). This paper was the the basis for the the solution of gain which which the opponent could achieve corresponds with et et al. (2011)). This paper was the basis for the solution of gain which the opponent could achieve corresponds with special case of non-stationary filtering problem when the the H norm of the error operator relating the estimation ∞ et al. (2011)). This paper was the basis for the solution of gain which the opponent could achieve corresponds with special case of non-stationary filtering problem when the the H∞ norm of the error operator relating the estimation special case of non-stationary filtering problem when the the H norm of the error operator relating the estimation dimensions of the estimator output and input disturbance error with the disturbance. The result is numerically the ∞ caseofofthe non-stationary filtering problem when the the H∞ norm the error operator relating the estimation dimensions estimator output and input disturbance error with theofdisturbance. The result is numerically the special dimensions of of the the&estimator estimator output and and input input disturbance disturbance error with the disturbance. The is numerically the same as in the different situation where disturbances dimensions output error with the disturbance. The result result isthe numerically the coincide coincide (Yaesh (Yaesh & Stoica Stoica (2014)). (2014)). same as in the different situation where the disturbances same as in the different situation where the disturbances coincide (Yaesh & Stoica (2014)). are nonrandom square and (Yaesh & Stoica (2014)). same as in the different situation where thesequences disturbances are arbitrary arbitrary nonrandom square summable summable sequences and coincide The present paper spreads the ideas of (Yaesh & Stoica The present paper spreads the of (Yaesh are arbitrary nonrandom square summable sequences and estimation performance is measured by the l -gain of the 2 are arbitrary nonrandom square summable sequences and The present paper spreads the ideas ideas ofmulticriteria (Yaesh & & Stoica Stoica estimation performance is measured by the l2 -gain of the (2014)) to the general case and to the state The present paper spreads the ideas of (Yaesh & Stoica (2014)) to the general case and to the multicriteria state estimation performance is measured by the l -gain of the error operator (see for examples (Nagpal & Khargonekar 2 estimation performance is measured by the&l2Khargonekar -gain of the (2014)) error operator (see for examples (Nagpal to the general case and to the multicriteria state (Scherer et al. (1997)) of anisotropic-based filtering on (2014)) to the general case and to the multicriteria state (Scherer et al. (1997)) of anisotropic-based filtering on error operator (see for examples (Nagpal & Khargonekar (1991); Shaked (1990); Xie et al. (1991) )). On the other error operator (see for examples (Nagpal & On Khargonekar (1991); Shaked (1990); Xie et al. (1991) )). the other (Scherer et al. (1997)) of anisotropic-based filtering on finite horizon. (Scherer et al. (1997)) of anisotropic-based filtering on (1991); Shaked (1990); Xie et al. (1991) )). On the other finite horizon. hand, if the opponent is only allowed to produce Gaussian (1991); Shaked (1990); Xie et al. (1991) )). On the other hand, if the opponent is only allowed to produce Gaussian finite horizon. finite horizon. hand, if the opponent is only allowed to produce Gaussian white noise disturbances with scalar covariance matrix, hand, the opponent is only allowed produce Gaussian white ifnoise disturbances with scalartocovariance matrix, The The material material of of Sections Sections 2, 2, 3 3 and and 4 4 considers considers the the backbackwhite noise disturbances scalar covariance matrix, The material of Sections 2, 3 and 4 considers the then the above-mentioned ratio is the scaled H of ground of anisotropy-based theory for discrete time vary2 -norm white noise disturbances with with scalar covariance matrix, The material of Sections 2, 3 and 4 considers the backbackthen the above-mentioned ratio is the scaled H -norm of ground of anisotropy-based theory for discrete time vary2 -norm of then the above-mentioned ratio is the scaled H ground of anisotropy-based theory for discrete time the error operator (Diamond et al. (2001)). ing linear control systems. The material is based on pa2 thenerror the above-mentioned ratio is the scaled H2 -norm of ground of anisotropy-based theory for discrete timeonvaryvarythe operator (Diamond et al. (2001)). ing linear control systems. The material is based ing linear linear control systems. The Section material5 is isformulates based on on papathe error operator (Diamond et al. (2001)). per (Maximov et al. (2011)). the the error operator (Diamond et al. (2001)). ing control systems. The material based per (Maximov et al. (2011)). Section 5 formulates pathe Let us consider aa situation where the opponent is restricted Let us consider situation where the opponent is restricted per (Maximov et al. (2011)). Section 5 formulates the anisotropic norm bounded real lemma used below. In per (Maximov et al. (2011)). Section 5 formulates the Let us consider a situation where the opponent is restricted anisotropic norm bounded real lemma used below. In to produce disturbances whose anisotropy is bounded from Letproduce us consider a situation where the opponent is restricted to disturbances whose anisotropy is bounded from anisotropic norm bounded real lemma used below. In Section 6, the variant of anisotropic-based bounded real anisotropic norm bounded real lemma used below. In Section 6, the variant of anisotropic-based bounded real to produce disturbances whose anisotropy is bounded from above by a given nonnegative parameter. The anisotropy to produce disturbances whose anisotropy is bounded from above by a given nonnegative parameter. The anisotropy Section 6, the variant of anisotropic-based bounded real lemma in terms of matrix difference inequalities is preSection in 6, terms the variant of anisotropic-based bounded real lemma of matrix difference inequalities is preabove by a given nonnegative parameter. The anisotropy is a combined entropy theoretical measure of colouredabove by a given nonnegative parameter. Theofanisotropy lemma in in terms termsbasis of matrix matrix difference inequalities is prepre-7 is a combined entropy theoretical measure coloured- lemma sented; for results paper. of inequalities is sented; it it is is the the basis for the the difference results of of the the paper. Section Section 7 is aa combined entropy theoretical measure of ness and spatial nonroundness of a stationary Gaussian is combined entropy theoretical measure of colouredcolouredness and spatial nonroundness of a stationary Gaussian sented; it is the basis for the results of the paper. Section 7 formulates the multicriteria anisotropic suboptimal filterit isthe the multicriteria basis for the results of thesuboptimal paper. Section 7 ness and and spatial spatial nonroundness of a stationary stationary Gaussian sented; formulates anisotropic filtersequence. Then, the worst-case performance is described ness nonroundness of a Gaussian sequence. Then, the worst-case performance is described formulates the multicriteria anisotropic suboptimal filtering problem. In Section 8, the solution of above problem is formulates the multicriteria anisotropic suboptimal filtering problem. In Section 8, the solution of above problem is sequence. Then, the worst-case performance is described sequence. Then, the worst-case performance is described given. ing problem. problem. In Section Section 8, 8, the the solution solution of of above above problem problem is Section the The paper is supported by the Russian Foundation for Basic ing is given. SectionIn9 9 provides provides an an example example illustrating illustrating the result result The paper is supported by the Russian Foundation for Basic given. Section 9 provides an example illustrating the result of the solving the The paper is14-08-00069. supported by the Russian Foundation for Basic given. Section provides anfor example the result Research grantis of applying applying the9 algorithm algorithm for solvingillustrating the problem. problem. The paper supported by the Russian Foundation for Basic Research grant 14-08-00069. of applying the algorithm for solving the problem. Research grant 14-08-00069. of applying the algorithm for solving the problem. Research grant 14-08-00069. Copyright 2015 192 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015,IFAC IFAC (International Federation of Automatic Control) Copyright ©©2015 IFAC 192 Peer review underIFAC responsibility of International Federation of Automatic Copyright © 192 Copyright © 2015 2015 IFAC 192 Control. 10.1016/j.ifacol.2015.09.456
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2. CLASS OF SYSTEMS BEING CONSIDERED We consider a linear discrete time varying (LDTV) system F on a bounded time interval [0, N ]. Its n-dimensional state xk and r-dimensional output zk at time k are governed by the equations xk+1 = Ak xk + Bk wk ,
(1)
zk = Ck xk + Dk wk , (2) with initial condition x0 = 0, which are driven by the m-dimensional input wk . Here, Ak ∈ Rn×n , Bk ∈ Rn×m , Ck ∈ Rr×n , Dk ∈ Rr×m are matrices assumed to be known functions of time k. For any two moments of time s t, the values of the input and output signals W and Z on the interval [s, t] are assembled into the column-vectors ws zs .. .. W[s:t] := . , Z[s:t] := . . wt zt Since the state of the system is zero-initialized, then Z[0:t] = F[0:t] W[0:t] , where (3) F[s:t] := blocks j,k t (fjk ) is a block lower triangular matrix with (r×m)-blocks given by Cj Φj,k+1 Bk for j > k Dk for j = k . fjk = 0 otherwise Here, Φjk := Aj−1 × . . . ×Ak (4) is the state transition matrix from xk to xj for j k, with Φkk = In the identity matrix of order n. Since the matrix F[0:N ] completely specifies the system F on the time interval [0, N ], all the norms of F are those of F[0:N ] . In particular, the finite-horizon counterparts of the H2 and H∞ -norms are described by the Frobenius and operator norms of F[0:N ] as T F 2 := Tr(F[0:N F ∞ := σmax (F[0:N ] ), ] F[0:N ] ), where σmax (·) is the largest singular value of a matrix.
(5)
3. ANISOTROPY OF RANDOM VECTORS The relative entropy (Cover & Thomas (2006)) of a probability measure P with respect to another probability measure M on the same measurable space is defined by D(P M ) := E ln(dP/dM ). Here, P is assumed to be absolutely continuous with respect to M with density dP/dM , and E denotes the expectation in the sense of P . The relative entropy D(P M ), which is always nonnegative, only vanishes if P = M . D(P M ) will also be written as D(ξ η) or D(f g) if the probability measures P and M are distributions of random vectors ξ and η or are specified by their probability density functions (PDF’s) f and g with respect to a common measure. For any λ > 0, we denote by p,λ the -variate Gaussian PDF with zero mean and scalar covariance matrix λI : 2
p,λ (w) = (2πλ)−/2 e−|w|
/(2λ)
,
w ∈ R .
(6) 193
193
Let W be a square integrable absolutely continuous random vector with values in R and PDF f . Its relative entropy with respect to the Gaussian probability law (6) is computed as D(f p,λ ) = E ln(f (W )/p,λ (W )) = ln(2πλ) + E(|W |2 )/(2λ) − h(W ), (7) 2 where h(W ) := −E ln f (W ) = − R f (w) ln f (w)dw is the differential entropy of W . For what follows, the class of square integrable absolutely continuous R -valued random vectors is denoted by L2 . Definition 1. (Vladimirov et al. (1995, 2006)) The anisotropy A(W ) of a random vector W ∈ L2 is defined as the minimum relative entropy (7) of its PDF f with respect to the Gaussian PDF’s (6) with zero mean and scalar covariance matrices: A(W ) := inf D(f p,λ ) λ>0
=
ln(2πeE(|W |2 ) ) − h(W ). (8) 2
4. a-ANISOTROPIC NORM OF MATRICES
Let F ∈ Rs× be an arbitrary matrix. While the disturbance attenuation paradigm seeks to minimize the magnitude of the output Z := F W , the probability distribution of W can be regarded as the strategy of a hypothetical player aiming to maximize the root-mean-square gain of F with respect to W : R(F, W ) = E(|Z|2 )/E(|W |2 ). (9) 2 Here, the squared Euclidean norm | · | of a vector is interpreted as its “energy”, so that E(|W |2 ) and E(|Z|2 ) describe the average energy (or power) of the input and output of the operator F , respectively. The denominator E(|W |2 ) in (9) vanishes only in the trivial case, where W = 0 with probability one, which is excluded from consideration. The map F → R(F, W ) is a semi-norm in Rs× . It is a norm if and only if the matrix of second moments Σ := E(W W T ) of the random vector W is nonsingular. Indeed, since E(|W |2 ) = TrΣ and E(|Z|2 ) = Tr(F ΣF T ), the semi-norm properties follow from the representation of the root-mean-square gain (9) in terms of the Frobenius norm · 2 as R(F, W ) = F Ω 2 , where Ω := Σ/TrΣ. This also shows that R(F, W ) = 0 implies F = 0 if and only if Ω is positive definite which is equivalent to the positive definiteness of Σ. For any random vector W ∈ L2 , we quantify “nongenericity” of its probability distribution by the anisotropy A(W ). Accordingly, we assume that the disturbance player is constrained by the condition A(W ) a, where a is a given nonnegative parameter. Definition 2. For any a 0, the a-anisotropic norm of a matrix F ∈ Rs× is defined as an anisotropy-constrained upper envelope of the root-mean-square gains (9): |||F |||a := sup R(F, W ) : W ∈ L2 , A(W ) a . (10)
This definition closely follows the concept of an induced norm, with the only, though essential, difference being the constraint A(W ) a on the anisotropy (8). It is the
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latter point where the entropy theoretic considerations enter the construct of the a-anisotropic norm (10) thus making |||F |||a an anisotropy-constrained stochastic version of the induced operator norm F ∞ .
For any given matrix F ∈ Rs× , the a-anisotropic norm |||F |||a is a nondecreasing concave function of a 0 which satisfies √ F 2 = |||F |||0 |||F |||a lim |||F |||a = F ∞ . (11) a→+∞
The relations (11) show that the a-anisotropic norm occupies an intermediate unifying position between the scaled Frobenius norm and the induced operator norm. 5. ANISOTROPIC NORM BOUNDED REAL LEMMA Theorem 1. (Maximov et al. (2011)). Let F be a LDTV system with the state-space realization (1) - (2). Then its a-anisotropic norm on the time interval [0, N ] satisfies |||F |||a γ if and only if there exists q 0 such that for the matrices Rk ∈ Rn×n , with k = 0, . . . , N , governed by the difference Riccati equation T T Rk+1 = Ak Rk AT k + qBk Bk + Mk Sk Mk ,
+ qBk DkT )Sk−1 , Ck Rk CkT − qDk DkT ,
Mk = −(Ak Rk CkT
(12) (13)
Sk = Ir − (14) with the initial condition R0 = 0, the matrices S0 , . . . , SN are all positive definite and satisfy the inequality N ln det Sk m(N + 1) ln(1 − qγ 2 ) + 2a. (15) k=0
Prior to proving the theorem, note that the matrices S0 , . . . , SN , defined by (14), are all positive definite if and only if q < F −2 ∞ . For any such q, the left-hand side of (15) is nonpositive, since Sk Ir (and so, ln det Sk 0). Hence, any q satisfying the specifications of Theorem 1 must also satisfy the inequalities γ −2 1 − e−2a/(m(N +1)) q < γ −2 . (16) Here, the ratio α := a/(N +1) is the anisotropy production rate per time step. Therefore, if α significantly exceeds the dimension m of the input W , then (16) yields a relatively narrow localization of the candidate values for q about γ −2 . 6. ANISOTROPIC-BASED BOUNDED REAL LEMMA IN TERMS OF MATRIX INEQUALITIES.
k=0
with γ −2 (1 − e−2a/(m(N +1)) ) ≤ µ−2 < min (γ −2 , F −2 ∞ ).
Further we move from Riccati equation to Riccati inequality associated with this equation. Let us consider the inequality
Pk+1 Ak Pk ATk + Bk BkT + (Ak Pk CkT + Bk DkT )(µ2 Ir − −Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (20) with initial condition P0 = P0 = 0. Introduce a matrix (21) Sk = µ−2 (µ2 Ir − Ck Pk CkT − Dk DkT ), depending on the variable Pk in (20).
By analogy with (19), we consider the strict inequality for the determinant of matrix Sk N
k=0
ln det Sk ≥ m(N + 1) ln(1 −
γ2 ) + 2a. µ2
(22)
Taking in to account (21), the previous inequality has a form N
ln µ−2r (det(µ2 Ir − Ck Pk CkT − Dk DkT )) >
(23) γ2 > m(N + 1) ln (1 − 2 ) + 2a, µ with localization condition µ γ −2 (1 − e−2a/(m(N +1)) ) ≤ µ−2 < min (γ −2 , F −2 ∞ ). k=0
Let us introduce the matrix slack variable Ψk satisfying the inequality Ψk ≺ µ2 Ir − Ck Pk CkT − Dk DkT . (24)
Wherein the inequality (23) is transformed to N γ2 ln{µ−2r (det(Ψk )} > m(N + 1) ln (1 − 2 ) + 2a. (25) µ k=0
We perform the equivalent transformation of the Riccati equation (12 - 14) and the inequality (15). We introduce the change of variables Rk = q Pk and q = 1/µ2 . Then the matrices Sk in the equation (12-14) have the following form: Sk = µ−2 (µ2 Ir − Ck Pk CkT − Dk DkT ).
with initial condition P0 = 0, where the matrices Sk = SkT 0 satisfy the inequality N γ2 ln det Sk ≥ m(N + 1) ln(1 − 2 ) + 2a (19) µ
(17)
Using (17), the equation (12-14) is transformed into Pk+1 = Ak Pk ATk + Bk BkT + (Ak Pk CkT + Bk DkT )(µ2 Ir − −Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (18) 194
Consider the system (1), (2) given above. Denote |||F[0:N ] |||a an a-anisotropic norm of the system F[0:N ] on the interval [0, N ]. Problem 1. Find the conditions when |||F[0:N ] |||a < γ. The sufficient conditions for the limitation of an aanisotropic norm of the system F[0:N ] on interval k ∈ [0, N ] by given γ in terms of matrix inequalities are stated in the following theorem. Theorem 2. Let F[0:N ] with the state space realization (1), (2) be linear time varying system and real numbers a > 0 and γ > 0 are given. Then inequality |||F[0:N ] |||a < γ
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic Victor N. Timin et al. / IFAC-PapersOnLine 48-14 (2015) 192–197
holds true if there are matrices Pk = PkT 0 for k ∈ [1, N ] and Ψk = ΨTk 0 for k ∈ [0, N ] satisfying the inequalities below −Pk+1 0 Ak P k B k 2 0 −µ Ir Ck Pk Dk (26) Pk ATk Pk CkT −Pk 0 ≺ 0, T T Dk 0 −Im Bk Ψk − µ2 Ir Ck Pk Dk Pk CkT (27) −Pk 0 ≺ 0, T 0 −Ir Dk N
1
(det Ψk ) r > e
k=0
2a r
{µ2 (1 −
γ 2 m N +1 )r} , µ2
(28)
where the initial condition P0 = 0 and real number µ > 0 belongs to interval
Ψk − µ2 Ir Ck Dk CkT −Pk−1 0 ≺ 0. T 0 −Ir Dk
Consider the third inequality (28). We transform the righthand side of (28) to equivalent view
e
2a r
{µ2 (1 −
N
k=0
Execute a similarity transformation of a linear matrix inequality (26) by multiplying it on the right and on the left by nonsingular matrix blockdiag (I, I, Pk−1 , I) 0. We get the inequality
−Pk+1 0 Ak Bk 0 −µ2 Ir Ck Dk ≺ 0. −1 T ATk Ck −Pk 0 DkT 0 −Im BkT
(30)
An application of Schur’s lemma to the inequality (30) leads to the inequality of the form (20). For given µ, the existence condition of solution of inequality (20) implies the existence of the solution Pk for following Riccati difference equation Pk+1 = Ak Pk ATk + Bk BkT + Qk + (Ak Pk CkT + Bk DkT ) ×(µ2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (31) with initial condition P0 = P0 = 0.
By virtue of the fact that Qk = QTk 0, the condition of the monotony of the Riccati equation (Freiling & Ionescu (1999)) implies that the inequality Pk Pk 0 holds true. Pk is the solution of Riccati equation (18).
Execute a similarity transformation of a linear matrix inequality (27) by multiplying it on the right and on the left by nonsingular matrix blockdiag (I, Pk−1 , I) 0. We get the inequality 195
1 γ 2 m N +1 γ2 )r} = {e2a µ2r(N +1) (1 − 2 )m(N +1) } r . 2 µ µ (33)
Let us raise the inequality (28) to the power r
Proof.
Suppose for a given anisotropy level a > 0, anisotropy norm bound γ > 0 and scalar variable µ from the interval (29), the system of inequality (26) - (28) has the solution T Pk+1 = Pk+1 0 and Ψk = ΨTk 0 for all k = 0, . . . , N .
(32)
An application of Schur’s lemma to the inequality (32) leads to the inequality of the form (24) with variable Ψk .
γ −2 (1 − e−2a/(m(N +1)) ) < µ−2 < min (γ −2 , F0:N −2 ∞ ). (29) We shall show that the conditions of solution for solutions of inequalities (26) - (28) for each k = 0, . . . , N implies the existence of the Riccati difference equation (12 - 14) solutions when the inequality of a special form (15) for the matrix Sk (14) takes place.
195
det Ψk > e2a µ2r(N +1) (1 −
γ 2 m(N +1) ) . µ2
(34)
Take a logarithm of the both sides of (34): N
k=0
ln det Ψk > 2a + ln{µ2r(N +1) } + m(N + 1) ln (1 −
γ2 ). µ2 (35)
Then we carry ln{µ2r(N +1) } to the left-hand side and transform it. After transformation, we receive the inequality (25). The inequality (25) holds true for slack variable matrices Ψk , so this inequality will be done even for the inequality (23) with the matrices µ2 Ir − Ck Pk CkT − Dk DkT Ψk .
Taking into account Pk Pk 0 we receive
µ2 Ir −Ck Pk CkT −Dk DkT µ2 Ir −Ck Pk CkT −Dk DkT . (36)
If the inequality (23) holds true with matrices µ2 Ir − Ck Pk CkT − Dk DkT Ψk , then the inequality (19) will become true with matrices µ2 Ir − Ck Pk CkT − Dk DkT as well.
Thus the existence of the solution of the Riccati equation (18) when the inequality of special type (19) with matrices (17) follows from the existence of the solutions of inequalities (26)–(28). The Riccati equation (18) and inequality (19) are connected with equivalent transformations with the Riccati equation (12-14) and inequality (15). It means the limitation condition of the anisotropic norm in theorem 1 holds true.
7. FILTERING PROBLEM STATEMENT Consider a linear discrete time varying system F[0:N ] (37) you can see below
IFAC ROCOND 2015 196 July 8-11, 2015. Bratislava, Slovak Republic Victor N. Timin et al. / IFAC-PapersOnLine 48-14 (2015) 192–197
L x = A x + Bik wik , k+1 k k i=1 L yk = Cyk xk + Dyik wik , i=1 L z = C x + Dzik wik , zk k k
(37)
i=1
The filtering problem is to find an estimation Ξ = {ξk|k } of the estimated output Z = {zk } by the prehistory of the observable output measurement Y = {yj , j ≤ k}. An estimator E is found in a class of causal linear discrete time varying systems with following structure:
(Ak − Kk Cyk )ηk + Kk yk ,
η0 = 0,
simultaneously for all L channels. 8. FILTERING PROBLEM SOLUTION
where the state space vector xk ∈ Rn , the estimated output vector zk ∈ Rr , the input vector wk ∈ Rm , observation output vector yk ∈ Rp and matrices Ak ∈ Rn×n , Bik ∈ Rn×mi , Cyk ∈ Rp×n , Dyik ∈ Rp×mi , Czk ∈ Rr×n , Dzik ∈ Rr×mi , k ∈ [0, N ], initial condition xk=0 = x0 , r ≤ m and m = m1 + m2 + . . . + mL . Assume we know an anisotropy level ai of each input sequence Wi , A(Wi ) ≤ ai .
ηk+1 = Ak ηk + Kk (yk − Cyk ηk ) =
i = 1, ..., L on the set of estimators (39), find the estimator E of the output Z , providing satisfaction of the inequality |||Tz wi |||ai < γi , i = 1, ..., L, (40)
In a moment k, the i-th component εzi ,k|k value of the estimation error εz,k|k from i-th group of external disturbance wik is determined by the value of subsystem Tz wi ai anisotropic norm. A structure of matrices inequalities (26)(28) allows to use the theorem 2 directly for the searching of unknown gain matrices Kk and Mk in the estimator (38) for the multicriteria filtering problem solution. Theorem 3. Suppose the linear discrete time varying system F[0:N ] with the state space realization (37) on interval [0, N ] is given. Let the L real numbers γi > 0 and ai > 0 be. The estimator E of the output Z with state space realization (38) providing the fulfillment of the L inequalities (40) exists when there are the matrices Pik = PiTk 0, k ∈ [1, N ] with initial condition Pi0 = PiT0 = 0 and Φik = ΨTik 0, k ∈ [0, N ], satisfying of L inequalities −Pik+1 0 Ak Pik Bik 2 0 −µi Ir Czk Pik Dzik ≺ 0, Pik ATk Pik CzTk −Pik 0 DzTi 0 −Imi BiTk
(38)
ξk|k = Czk ηk + Mk (yk − Cyk ηk ) = (Czk − Mk Cyk )ηk + Mk yk .
k
Here ηk is the estimation of the system (37) state H = {ηk } at the moment k.
Ψik − µ2i Ir Czk Pik Dzik Pik CzT −Pik 0 ≺ 0, k 0 −Ir DzTi
The matrices Kk and Mk in estimator (38) of the output Z , k ∈ [0, N ], are unknown and have to be found.
From the equations of system (37) and the estimator (38), we can get the system Tεz w (39), identifying the estimation error of internal state Ex = X − H = {εxk = xk − ηk } and estimated output Ez = Z − Θ = {εzk = zk − ξk|k } on the interval [0, N ]. In the moment k, the estimation error εzi ,k|k from the external disturbance Wi from (38) and (39) is determined by the subsystem Tεz wi as follows:
εxi ,k+1 = (Ak − Kk Cyk )εxi ,k + (Bik − Kk Dyik )wik , εzi ,k|k = (Czk − Mk Cyk )εxi ,k + (Dzik − Mk Dyik )wik . (39)
Each subsystem Tεz wi with an external input wik and error estimation output εzi ,k|k , identifies an i-th channel. For all channels, each subsystem Tεz wi (39) depends on unknown equal gain matrices Kk and Mk in (38). Because of the linearity of the system and taking into account the relation εx,k =
L
(41)
εxi ,k ,
i=1
the common estimation error of the output εz,k|k in (39) in the moment k is the sum of each channel estimation L error εzi ,k|k in the moment k, so it means that εz,k = i=1 εzi ,k .
Problem 2. For given system F[0:N ] (37), input channel anisotropic levels ai > 0 and the threshold values γi > 0, 196
(42)
k
N
1
(det Ψik ) r > e
2ai r
k=0
{µ2i (1 −
γi2 mi N +1 ) r } , µ2i
(43)
where Ak = Ak − Kk Cyk , Czk = Czk −Mk Cyk , Bik = Bik − Kk Dyik , Dzik = Dzik −Mk Dyik and L real numbers µi > 0 < belong to the intervals γi−2 (1 − e−2ai /(mi (N +1)) ) < µ−2 i −2 γi for i = 1, ..., L. The proof of the previous theorem is based on the direct application of the theorem 2 to each channel. Remark 1. The inequality (43) used for the calculation of the slack variables Ψik+1 can be presented in recurrent form for i = 1, . . . , L k mi 1 1 (det Ψik+1 ) r > e2ai /r {µ2i (1−γi2 /µ2i ) r }k+2 / (det Ψik ) r . k=0
(44)
9. NUMERICAL EXAMPLE Let us consider a mathematical model of linear continues time invariant system, which is a damped pendulum. w x˙1 0 ω0 x1 00 , (45) = + + n 10 x˙2 −ω0 −2ξω0 x2 y = [0 1]
x1 x2
+ [0 1]
w n
,
(46)
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z = [0 1]
x1 x2
+ [0 0]
w n
(47)
,
where a damping parameter ξ = 0.2 and angular natural frequency ω0 = 10. An external disturbance w acts to the system input. The only state space vector component x2 is measured. This component is measured and observable as well. A discretization of the continuous system was carried out with sampling step ∆t = 0.01. For discrete time system, the design of an anisotropicbased multicriteria filter has been done. We have used the following anisotropy levels: A(w) ≤ 1 and A(n) ≤ 0.01. The bounds for anisotropic norms have been given as γ1 = γw = 0.25 and γ2 = γn = 0.05. A comparison of designed anisotropic-based filter has been carried out with the H2 filter. The H2 filter is the special case of the anisotropy-based filter if anisotropy level a=0. The H2 filter has been synthesized under the constraint of the threshold value γ = 0.25 for H2 norm of the map from the extended vector of external disturbances ν = (wT , nT )T to the estimation error ez . In the synthesis of H2 filter, the matrices Mk of estimator have been assumed to be zero. −4
4
x 10
k
K1
3 2 1 0 −1
0
0.5
1 Time, sec
1.5
2
0.5
1 Time, sec
1.5
2
−4
20
x 10
k
K2
15 10 5 0 −5
0
Fig. 1. Two components Kk1 and Kk2 of gain vector Kk The graphics of the coordinates Kk1 and Kk2 of twodimensional vector Kk depending on the time of t (t = k∆t, ∆t = 0.01, k ∈ [0, N ], N = 200) for the multicriteria anisotropic filter (solid line) and H2 filter (dashed line) are shown in Figure 1. The coordinates of the gain Kk in a case of the anisotropic filter for each discrete time moment k are larger than the coordinates of the gain Kk in a case of H2 filter. Because of the system (45-47) is time invariance, the coordinates of the gain Kk tend to steadystate values. The steady-state values for anisotropy-based filter are larger than H2 filter as well. This behavior of the coordinates of Kk characterizes ”intermediate” properties of the anisotropic estimator to be between the H2 and H∞ filters. CONCLUSION On finite horizon for the linear discrete time varying system with observable and estimated outputs, the problem of anisotropy-based robust filtering was solved. In terms of the matrix inequalities, a criterion for the boundedness of anisotropic norm of linear discrete time-varying system was formulated. The sufficient conditions for the existence of multi-criteria suboptimal anisotropic estimator with limited anisotropic norms of subsystems by a predetermined threshold values were received. The algorithm for 197
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the multicriteria suboptimal anisotropic estimator design was presented. REFERENCES Nagpal K.M., Khargonekar P.P. Filtering and smoothing in an H∞ -setting, IEEE Trans. Autom. Contr., V. 36, No 2, 1991, pp. 152–166. Diamond P., Vladimirov I., Kurdjukov A., and Semyonov A. Anisotropy-based perfomance analysis of linear dicrete time invariant control systems, Int. J. of Contr., 2001. V. 74, No 1, 2001, pp. 28–42. Vladimirov I. Anisotropy-based optimal filtering in linear discrete time invariant systems. Centre for Applied Dynamical Systems / [http://arxiv.org/abs/1412.3010] December 2014. T.M.Cover, and J.A.Thomas, Elements of Information Theory, Wiley, Hoboken, New Jersey, 2006. Boyd S., El Ghaoui L., Feron E., Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, 1994 I.G.Vladimirov, A.P.Kurdjukov, and A.V.Semyonov, Anisotropy of signals and the entropy of linear stationary systems. Doklady Mathematics, vol. 51, No. 3, 1995, pp. 388–390. Tchaikovsky M.M., Kurdyukov A.P., Timin V.N. Strict Anisotropic Norm Bounded Real Lemma in Terms of Inequalities Proceedings of the 18th IFAC World Congress, Milano (Italy), Aug. 28 - Sept. 2, 2011, pp. 2332–2337. Timin V.N., Tchaikovsky M.M., Kurdyukov A.P. A solution to anisotropic suboptimal filtering problem by convex optimization, Doklady Mathematics. vol. 85, No 3, 2012, pp.443–445. Timin V.N. Anisotropy-based suboptimal filtering for the linear discrete time-invariant systems, Automation and remote control, vol. 74, No 11, 2013, pp. 1773–1785. Vladimirov I.G., Diamond P., Kloeden P. Anisitropy-based robust performance analysis of finite horizon linear discrete time varying systems, Automation and remote control, vol. 67, No 8, 2006, pp.1265–1282. Maximov E,A., Kurdyukov A.P., Vladimirov I.G. Anisotropic Norm Bounded Real Lemma for Linear Time-Varying System. Preprint of the 18-th IFAC World Congress, Milano, 2011. Aug 28-Sept. 2, 2011, pp. 4701–4706. Yaesh I, Stoica A.-M. Linear Time-Varying Anisotropic Filtering and its Application to Nonlinear Systems State Estimation. Proceedings of European Control Conference, Strasburg, 2014. June24-27, 2014, pp. 975–980. Freiling G, Ionescu V. Time-varying discrete Riccati equation: some monotonicity results. Linear Algebra and Appl., vol. 286, 1999, pp. 135–148. Gershon E., Shaked U., Yaesh I.. H∞ Control and Estimation of State-Multiplicative Linear Systems, Springer, London, 2005. Scherer C.W., Gahinet P., and Chilali M. Multiobjective output-feedback control via LMI optimization, IEEE Trans. Autom. Contr., vol. 42, No 7, 1997, pp. 896–911. Shaked U. H∞ -minimum error estimation of linear stationary processes, IEEE Trans. Autom. Contr., vol.35, No 5, 1990, pp. 554–558. Xie L., de Souza C.E., Fu M. H∞ -estimation for discretetime linear uncertain systems, Int. J. Rob. Non. Cont. vol.1, 1991, pp. 111–123.
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Finite horizon anisotropy-based Finite horizon anisotropy-based Finite horizon anisotropy-based Finite horizon anisotropy-based multicriteria filtering multicriteria filtering multicriteria filtering multicriteria filtering
Victor N. Timin Alexander P. Kurdyukov Victor Victor N. N. Timin Timin Alexander Alexander P. P. Kurdyukov Kurdyukov Victor N. Timin Alexander P. Kurdyukov V.A. Trapeznikov Trapeznikov Institute Institute of of Control Control Sciences, Sciences, RAS, RAS, 65 65 V.A. V.A. Trapeznikov Institute of Control Sciences, RAS, 65 Profsoyuznaya Str., 117997 Moscow GSP-4, Russia V.A. Trapeznikov Institute of Control Sciences, RAS, Profsoyuznaya Str., 117997 Moscow GSP-4, Russia 65 Profsoyuznaya Str., 117997 Moscow GSP-4, Russia ([email protected]), ([email protected]) Profsoyuznaya Str., 117997 Moscow GSP-4, Russia ([email protected]), ([email protected]), ([email protected]) ([email protected]) ([email protected]), ([email protected]) Abstract: Finite horizon anisotropy-based multicriteria filtering problem for the linear varying Abstract: Finite horizon anisotropy-based multicriteria filtering problem the varying Abstract: Finite horizon anisotropy-based multicriteria input filtering problem for for the linear linear varying discrete time system has been solved. Multidimensional disturbance consists of of Abstract: Finite horizon anisotropy-based multicriteria filtering problem for the varying discrete time system has been solved. Multidimensional input disturbance consistslinear of groups groups of discrete time system has been solved. Multidimensional input disturbance consists of groups of signals, each group is characterized by its inaccurately given information theory properties. For discrete time system has been solved. Multidimensional input disturbance consists of groups of signals, each group is characterized by its inaccurately given information theory properties. For signals, each group is characterized by its inaccurately given information theory properties. For that mathematical model, the bounded real lemma for an anisotropic norm has been proved. signals, each group is characterized by itsreal inaccurately informationnorm theory For that mathematical model, the bounded lemma forgiven an anisotropic hasproperties. been proved. that model, the bounded real for an anisotropic norm been proved. If themathematical subsystem norms to specific values, condition for multicriteria that mathematical model,are thelimited bounded real lemma lemma for a anisotropic norm has has proved. If norms are to specific values, aaansufficient sufficient condition for been multicriteria If the the subsystem subsystem norms are limited limited to An specific values,algorithm sufficient condition for multicriteria estimator existence has been obtained. estimation is based on recursion If the subsystem norms are limited to specific values, a sufficient condition for multicriteria estimator existence has been obtained. An estimation algorithm is based on recursion solving solving estimator existence has obtained. An algorithm is on recursion solving of linear inequalities the special estimator existencematrix has been been obtained.simultaneously An estimation estimationwith algorithm is based based of ona solving of linear difference difference matrix inequalities simultaneously with the inequality inequality of a recursion special type. type. of linear difference matrix inequalities simultaneously with the inequality of a special type. of linear difference matrix inequalities simultaneously with the inequality of a special type. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: robust robust filtering, filtering, anisotropic anisotropic norm, norm, Anisotropic-based Anisotropic-based Bounded Bounded Real Real Lemma, Lemma, Keywords: robust filtering, anisotropic norm, Anisotropic-based Bounded Real difference Riccati inequations Keywords: robust filtering, anisotropic norm, Anisotropic-based Bounded Real Lemma, Lemma, difference Riccati inequations difference difference Riccati Riccati inequations inequations 1. by 1. INTRODUCTION INTRODUCTION by the the a-anisotropic a-anisotropic norm norm of of the the system system defined defined as as its its 1. by the a-anisotropic norm of the system defined as maximum root mean square gain over such disturbances. 1. INTRODUCTION INTRODUCTION by the a-anisotropic norm of the system defined as its its maximum root mean square gain over such disturbances. maximum root mean square gain over such disturbances. In the limiting cases where a → 0 and a → ∞, the Let system have unknown estimated and directly observed maximum root mean square gain over such disturbances. In the limiting cases where a → 0 and a → ∞, the Let system have unknown estimated and directly observed In the limiting cases where a → 0 and a → ∞, the a-anisotropic norm gives the l2 induced Let have unknown estimated and directly observed outputs. The system is by external disturbance the limiting cases where aFrobenius → 0 andand a → ∞, the Let system system have unknown estimated directly observed In a-anisotropic norm gives the Frobenius outputs. The system is driven driven by an an and external disturbance a-anisotropic norm gives therespectively. Frobenius and and lll22 induced induced operator norm of the system outputs. The system is driven by an external disturbance that is time invariant Gaussian sequence. This sequence is a-anisotropic norm gives the Frobenius and 2 induced outputs. Theinvariant system is driven by an external disturbance operator norm of the system respectively. that is time Gaussian sequence. This sequence is norm of system respectively. that is is time time invariant Gaussian sequence. This sequence is operator produced from the noise sequence with zero operator norm of the the systemsystems, respectively. that invariant Gaussian is produced from the white white noise sequence. sequence This with sequence zero mean mean For discrete time invariant the estimator For discrete time invariant systems, the estimator estimator design design produced from the white noise sequence with zero mean and identity covariance matrix by the unknown linear time produced from the white noise sequence with zero mean For discrete time invariant systems, the design and identity covariance matrix by the unknown linear time problem minimizing the a-anisotropic norm of the error For discrete time invariant systems, the estimator design problem minimizing the a-anisotropic norm of the error and identity covariance matrix by the unknown linear time invariant shaping filter whose transfer function is in the and identity covariance the unknown minimizing the a-anisotropic norm of the error invariant shaping filter matrix whose by transfer functionlinear is intime the problem operator was solved by Vladimirov in (Vladimirov (2014)). problem minimizing the a-anisotropic norm of the error operator was solved by Vladimirov in (Vladimirov (2014)). invariant shaping filter whose transfer function is in the Hardy space H . For given estimator and shaping filter, 2 . For invariant shaping filter whose transfer function is in the operator was solved by Vladimirov in (Vladimirov (2014)). Hardy space H given estimator and shaping filter, 2 . For given estimator and shaping filter, operator was solved by Vladimirov in (Vladimirov (2014)). Hardy space H the effect of the disturbance on estimation be 2 Hardy space given estimator shapingcan filter, In recent recent years, years, an an approach approach to to the the construction construction of of time time 2 . For the effect of H the disturbance on the the and estimation can be In In recent years, an approach to the construction of time the effect of the disturbance on the estimation can be varying anisotropic filtering and control theory has measured by the ratio of the root mean square values In recent years, an approach to the construction of been time the effect of the disturbance on the estimation can be been measured by the ratio of the root mean square values varying anisotropic filtering and control theory has varying anisotropic filtering and control theory has been measured by the ratio of the root mean square values developing. Definitions of the anisotropy of the random of the signals. The aim of the estimator is to minimize varying anisotropic filtering and control theory has been measured by the mean issquare values developing. Definitions of the anisotropy of the random of the signals. Theratio aim of of the the root estimator to minimize developing. the Definitions of the the anisotropy of the the random of the signals. The of the estimator is to vector anisotropic norm of which this effect. Suppose that of a filter developing. Definitions of of the signals. The aim aim ofthe thechoice estimator to minimize minimize vector and and the anisotropic normanisotropy of control controlofsystem, system,random which this effect. Suppose that the choice of aa isshaping shaping filter this effect. Suppose that the choice of shaping filter vector and the anisotropic norm of control system, which correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the vector and the anisotropic norm of control system, which this effect. Suppose that the choice of a shaping filter belongs to an opponent whose aim is to maximize the correspond to the time varying anisotropic-based filtering correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the and control theories on finite horizon, were stated root mean square gain. If the set of the opponent is correspond to the time varying anisotropic-based filtering belongs to an opponent whose aim is to maximize the and control theories on finite horizon, were stated in in root mean square gain. If the set of the opponent is control theories on horizon, were stated in root mean square gain. If the set of the opponent is the (Vladimirov al. (2006)). A bounded whole space H , then the worst damage to the estimation and control et theories on finite finite horizon, real were lemma stated for in root gain. the set of thetoopponent is the and (Vladimirov et al. (2006)). A bounded real lemma for wholemean spacesquare H22 , then theIfworst damage the estimation (Vladimirov et al. (2006)). A bounded real lemma for whole space H , then the worst damage to the estimation discrete time varying system was proved in (Maximov performance measured in terms of the root mean square 2 (Vladimirov al. (2006)). bounded for whole space H worst damage to the estimation discrete timeetvarying systemA was provedreal in lemma (Maximov 2 , then the performance measured in terms of the root mean square performancethe measured in could terms achieve of the the root root mean square square discrete time varying system was proved in (Maximov paper was for of gain opponent corresponds with discrete time This varying system wasbasis proved in solution (Maximov performance measured in terms of mean et al. al. (2011)). (2011)). This paper was the the basis for the the solution of gain which which the opponent could achieve corresponds with et et al. (2011)). This paper was the basis for the solution of gain which the opponent could achieve corresponds with special case of non-stationary filtering problem when the the H norm of the error operator relating the estimation ∞ et al. (2011)). This paper was the basis for the solution of gain which the opponent could achieve corresponds with special case of non-stationary filtering problem when the the H∞ norm of the error operator relating the estimation special case of non-stationary filtering problem when the the H norm of the error operator relating the estimation dimensions of the estimator output and input disturbance error with the disturbance. The result is numerically the ∞ caseofofthe non-stationary filtering problem when the the H∞ norm the error operator relating the estimation dimensions estimator output and input disturbance error with theofdisturbance. The result is numerically the special dimensions of of the the&estimator estimator output and and input input disturbance disturbance error with the disturbance. The is numerically the same as in the different situation where disturbances dimensions output error with the disturbance. The result result isthe numerically the coincide coincide (Yaesh (Yaesh & Stoica Stoica (2014)). (2014)). same as in the different situation where the disturbances same as in the different situation where the disturbances coincide (Yaesh & Stoica (2014)). are nonrandom square and (Yaesh & Stoica (2014)). same as in the different situation where thesequences disturbances are arbitrary arbitrary nonrandom square summable summable sequences and coincide The present paper spreads the ideas of (Yaesh & Stoica The present paper spreads the of (Yaesh are arbitrary nonrandom square summable sequences and estimation performance is measured by the l -gain of the 2 are arbitrary nonrandom square summable sequences and The present paper spreads the ideas ideas ofmulticriteria (Yaesh & & Stoica Stoica estimation performance is measured by the l2 -gain of the (2014)) to the general case and to the state The present paper spreads the ideas of (Yaesh & Stoica (2014)) to the general case and to the multicriteria state estimation performance is measured by the l -gain of the error operator (see for examples (Nagpal & Khargonekar 2 estimation performance is measured by the&l2Khargonekar -gain of the (2014)) error operator (see for examples (Nagpal to the general case and to the multicriteria state (Scherer et al. (1997)) of anisotropic-based filtering on (2014)) to the general case and to the multicriteria state (Scherer et al. (1997)) of anisotropic-based filtering on error operator (see for examples (Nagpal & Khargonekar (1991); Shaked (1990); Xie et al. (1991) )). On the other error operator (see for examples (Nagpal & On Khargonekar (1991); Shaked (1990); Xie et al. (1991) )). the other (Scherer et al. (1997)) of anisotropic-based filtering on finite horizon. (Scherer et al. (1997)) of anisotropic-based filtering on (1991); Shaked (1990); Xie et al. (1991) )). On the other finite horizon. hand, if the opponent is only allowed to produce Gaussian (1991); Shaked (1990); Xie et al. (1991) )). On the other hand, if the opponent is only allowed to produce Gaussian finite horizon. finite horizon. hand, if the opponent is only allowed to produce Gaussian white noise disturbances with scalar covariance matrix, hand, the opponent is only allowed produce Gaussian white ifnoise disturbances with scalartocovariance matrix, The The material material of of Sections Sections 2, 2, 3 3 and and 4 4 considers considers the the backbackwhite noise disturbances scalar covariance matrix, The material of Sections 2, 3 and 4 considers the then the above-mentioned ratio is the scaled H of ground of anisotropy-based theory for discrete time vary2 -norm white noise disturbances with with scalar covariance matrix, The material of Sections 2, 3 and 4 considers the backbackthen the above-mentioned ratio is the scaled H -norm of ground of anisotropy-based theory for discrete time vary2 -norm of then the above-mentioned ratio is the scaled H ground of anisotropy-based theory for discrete time the error operator (Diamond et al. (2001)). ing linear control systems. The material is based on pa2 thenerror the above-mentioned ratio is the scaled H2 -norm of ground of anisotropy-based theory for discrete timeonvaryvarythe operator (Diamond et al. (2001)). ing linear control systems. The material is based ing linear linear control systems. The Section material5 is isformulates based on on papathe error operator (Diamond et al. (2001)). per (Maximov et al. (2011)). the the error operator (Diamond et al. (2001)). ing control systems. The material based per (Maximov et al. (2011)). Section 5 formulates pathe Let us consider aa situation where the opponent is restricted Let us consider situation where the opponent is restricted per (Maximov et al. (2011)). Section 5 formulates the anisotropic norm bounded real lemma used below. In per (Maximov et al. (2011)). Section 5 formulates the Let us consider a situation where the opponent is restricted anisotropic norm bounded real lemma used below. In to produce disturbances whose anisotropy is bounded from Letproduce us consider a situation where the opponent is restricted to disturbances whose anisotropy is bounded from anisotropic norm bounded real lemma used below. In Section 6, the variant of anisotropic-based bounded real anisotropic norm bounded real lemma used below. In Section 6, the variant of anisotropic-based bounded real to produce disturbances whose anisotropy is bounded from above by a given nonnegative parameter. The anisotropy to produce disturbances whose anisotropy is bounded from above by a given nonnegative parameter. The anisotropy Section 6, the variant of anisotropic-based bounded real lemma in terms of matrix difference inequalities is preSection in 6, terms the variant of anisotropic-based bounded real lemma of matrix difference inequalities is preabove by a given nonnegative parameter. The anisotropy is a combined entropy theoretical measure of colouredabove by a given nonnegative parameter. Theofanisotropy lemma in in terms termsbasis of matrix matrix difference inequalities is prepre-7 is a combined entropy theoretical measure coloured- lemma sented; for results paper. of inequalities is sented; it it is is the the basis for the the difference results of of the the paper. Section Section 7 is aa combined entropy theoretical measure of ness and spatial nonroundness of a stationary Gaussian is combined entropy theoretical measure of colouredcolouredness and spatial nonroundness of a stationary Gaussian sented; it is the basis for the results of the paper. Section 7 formulates the multicriteria anisotropic suboptimal filterit isthe the multicriteria basis for the results of thesuboptimal paper. Section 7 ness and and spatial spatial nonroundness of a stationary stationary Gaussian sented; formulates anisotropic filtersequence. Then, the worst-case performance is described ness nonroundness of a Gaussian sequence. Then, the worst-case performance is described formulates the multicriteria anisotropic suboptimal filtering problem. In Section 8, the solution of above problem is formulates the multicriteria anisotropic suboptimal filtering problem. In Section 8, the solution of above problem is sequence. Then, the worst-case performance is described sequence. Then, the worst-case performance is described given. ing problem. problem. In Section Section 8, 8, the the solution solution of of above above problem problem is Section the The paper is supported by the Russian Foundation for Basic ing is given. SectionIn9 9 provides provides an an example example illustrating illustrating the result result The paper is supported by the Russian Foundation for Basic given. Section 9 provides an example illustrating the result of the solving the The paper is14-08-00069. supported by the Russian Foundation for Basic given. Section provides anfor example the result Research grantis of applying applying the9 algorithm algorithm for solvingillustrating the problem. problem. The paper supported by the Russian Foundation for Basic Research grant 14-08-00069. of applying the algorithm for solving the problem. Research grant 14-08-00069. of applying the algorithm for solving the problem. Research grant 14-08-00069. Copyright 2015 192 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015,IFAC IFAC (International Federation of Automatic Control) Copyright ©©2015 IFAC 192 Peer review underIFAC responsibility of International Federation of Automatic Copyright © 192 Copyright © 2015 2015 IFAC 192 Control. 10.1016/j.ifacol.2015.09.456
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2. CLASS OF SYSTEMS BEING CONSIDERED We consider a linear discrete time varying (LDTV) system F on a bounded time interval [0, N ]. Its n-dimensional state xk and r-dimensional output zk at time k are governed by the equations xk+1 = Ak xk + Bk wk ,
(1)
zk = Ck xk + Dk wk , (2) with initial condition x0 = 0, which are driven by the m-dimensional input wk . Here, Ak ∈ Rn×n , Bk ∈ Rn×m , Ck ∈ Rr×n , Dk ∈ Rr×m are matrices assumed to be known functions of time k. For any two moments of time s t, the values of the input and output signals W and Z on the interval [s, t] are assembled into the column-vectors ws zs .. .. W[s:t] := . , Z[s:t] := . . wt zt Since the state of the system is zero-initialized, then Z[0:t] = F[0:t] W[0:t] , where (3) F[s:t] := blocks j,k t (fjk ) is a block lower triangular matrix with (r×m)-blocks given by Cj Φj,k+1 Bk for j > k Dk for j = k . fjk = 0 otherwise Here, Φjk := Aj−1 × . . . ×Ak (4) is the state transition matrix from xk to xj for j k, with Φkk = In the identity matrix of order n. Since the matrix F[0:N ] completely specifies the system F on the time interval [0, N ], all the norms of F are those of F[0:N ] . In particular, the finite-horizon counterparts of the H2 and H∞ -norms are described by the Frobenius and operator norms of F[0:N ] as T F 2 := Tr(F[0:N F ∞ := σmax (F[0:N ] ), ] F[0:N ] ), where σmax (·) is the largest singular value of a matrix.
(5)
3. ANISOTROPY OF RANDOM VECTORS The relative entropy (Cover & Thomas (2006)) of a probability measure P with respect to another probability measure M on the same measurable space is defined by D(P M ) := E ln(dP/dM ). Here, P is assumed to be absolutely continuous with respect to M with density dP/dM , and E denotes the expectation in the sense of P . The relative entropy D(P M ), which is always nonnegative, only vanishes if P = M . D(P M ) will also be written as D(ξ η) or D(f g) if the probability measures P and M are distributions of random vectors ξ and η or are specified by their probability density functions (PDF’s) f and g with respect to a common measure. For any λ > 0, we denote by p,λ the -variate Gaussian PDF with zero mean and scalar covariance matrix λI : 2
p,λ (w) = (2πλ)−/2 e−|w|
/(2λ)
,
w ∈ R .
(6) 193
193
Let W be a square integrable absolutely continuous random vector with values in R and PDF f . Its relative entropy with respect to the Gaussian probability law (6) is computed as D(f p,λ ) = E ln(f (W )/p,λ (W )) = ln(2πλ) + E(|W |2 )/(2λ) − h(W ), (7) 2 where h(W ) := −E ln f (W ) = − R f (w) ln f (w)dw is the differential entropy of W . For what follows, the class of square integrable absolutely continuous R -valued random vectors is denoted by L2 . Definition 1. (Vladimirov et al. (1995, 2006)) The anisotropy A(W ) of a random vector W ∈ L2 is defined as the minimum relative entropy (7) of its PDF f with respect to the Gaussian PDF’s (6) with zero mean and scalar covariance matrices: A(W ) := inf D(f p,λ ) λ>0
=
ln(2πeE(|W |2 ) ) − h(W ). (8) 2
4. a-ANISOTROPIC NORM OF MATRICES
Let F ∈ Rs× be an arbitrary matrix. While the disturbance attenuation paradigm seeks to minimize the magnitude of the output Z := F W , the probability distribution of W can be regarded as the strategy of a hypothetical player aiming to maximize the root-mean-square gain of F with respect to W : R(F, W ) = E(|Z|2 )/E(|W |2 ). (9) 2 Here, the squared Euclidean norm | · | of a vector is interpreted as its “energy”, so that E(|W |2 ) and E(|Z|2 ) describe the average energy (or power) of the input and output of the operator F , respectively. The denominator E(|W |2 ) in (9) vanishes only in the trivial case, where W = 0 with probability one, which is excluded from consideration. The map F → R(F, W ) is a semi-norm in Rs× . It is a norm if and only if the matrix of second moments Σ := E(W W T ) of the random vector W is nonsingular. Indeed, since E(|W |2 ) = TrΣ and E(|Z|2 ) = Tr(F ΣF T ), the semi-norm properties follow from the representation of the root-mean-square gain (9) in terms of the Frobenius norm · 2 as R(F, W ) = F Ω 2 , where Ω := Σ/TrΣ. This also shows that R(F, W ) = 0 implies F = 0 if and only if Ω is positive definite which is equivalent to the positive definiteness of Σ. For any random vector W ∈ L2 , we quantify “nongenericity” of its probability distribution by the anisotropy A(W ). Accordingly, we assume that the disturbance player is constrained by the condition A(W ) a, where a is a given nonnegative parameter. Definition 2. For any a 0, the a-anisotropic norm of a matrix F ∈ Rs× is defined as an anisotropy-constrained upper envelope of the root-mean-square gains (9): |||F |||a := sup R(F, W ) : W ∈ L2 , A(W ) a . (10)
This definition closely follows the concept of an induced norm, with the only, though essential, difference being the constraint A(W ) a on the anisotropy (8). It is the
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latter point where the entropy theoretic considerations enter the construct of the a-anisotropic norm (10) thus making |||F |||a an anisotropy-constrained stochastic version of the induced operator norm F ∞ .
For any given matrix F ∈ Rs× , the a-anisotropic norm |||F |||a is a nondecreasing concave function of a 0 which satisfies √ F 2 = |||F |||0 |||F |||a lim |||F |||a = F ∞ . (11) a→+∞
The relations (11) show that the a-anisotropic norm occupies an intermediate unifying position between the scaled Frobenius norm and the induced operator norm. 5. ANISOTROPIC NORM BOUNDED REAL LEMMA Theorem 1. (Maximov et al. (2011)). Let F be a LDTV system with the state-space realization (1) - (2). Then its a-anisotropic norm on the time interval [0, N ] satisfies |||F |||a γ if and only if there exists q 0 such that for the matrices Rk ∈ Rn×n , with k = 0, . . . , N , governed by the difference Riccati equation T T Rk+1 = Ak Rk AT k + qBk Bk + Mk Sk Mk ,
+ qBk DkT )Sk−1 , Ck Rk CkT − qDk DkT ,
Mk = −(Ak Rk CkT
(12) (13)
Sk = Ir − (14) with the initial condition R0 = 0, the matrices S0 , . . . , SN are all positive definite and satisfy the inequality N ln det Sk m(N + 1) ln(1 − qγ 2 ) + 2a. (15) k=0
Prior to proving the theorem, note that the matrices S0 , . . . , SN , defined by (14), are all positive definite if and only if q < F −2 ∞ . For any such q, the left-hand side of (15) is nonpositive, since Sk Ir (and so, ln det Sk 0). Hence, any q satisfying the specifications of Theorem 1 must also satisfy the inequalities γ −2 1 − e−2a/(m(N +1)) q < γ −2 . (16) Here, the ratio α := a/(N +1) is the anisotropy production rate per time step. Therefore, if α significantly exceeds the dimension m of the input W , then (16) yields a relatively narrow localization of the candidate values for q about γ −2 . 6. ANISOTROPIC-BASED BOUNDED REAL LEMMA IN TERMS OF MATRIX INEQUALITIES.
k=0
with γ −2 (1 − e−2a/(m(N +1)) ) ≤ µ−2 < min (γ −2 , F −2 ∞ ).
Further we move from Riccati equation to Riccati inequality associated with this equation. Let us consider the inequality
Pk+1 Ak Pk ATk + Bk BkT + (Ak Pk CkT + Bk DkT )(µ2 Ir − −Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (20) with initial condition P0 = P0 = 0. Introduce a matrix (21) Sk = µ−2 (µ2 Ir − Ck Pk CkT − Dk DkT ), depending on the variable Pk in (20).
By analogy with (19), we consider the strict inequality for the determinant of matrix Sk N
k=0
ln det Sk ≥ m(N + 1) ln(1 −
γ2 ) + 2a. µ2
(22)
Taking in to account (21), the previous inequality has a form N
ln µ−2r (det(µ2 Ir − Ck Pk CkT − Dk DkT )) >
(23) γ2 > m(N + 1) ln (1 − 2 ) + 2a, µ with localization condition µ γ −2 (1 − e−2a/(m(N +1)) ) ≤ µ−2 < min (γ −2 , F −2 ∞ ). k=0
Let us introduce the matrix slack variable Ψk satisfying the inequality Ψk ≺ µ2 Ir − Ck Pk CkT − Dk DkT . (24)
Wherein the inequality (23) is transformed to N γ2 ln{µ−2r (det(Ψk )} > m(N + 1) ln (1 − 2 ) + 2a. (25) µ k=0
We perform the equivalent transformation of the Riccati equation (12 - 14) and the inequality (15). We introduce the change of variables Rk = q Pk and q = 1/µ2 . Then the matrices Sk in the equation (12-14) have the following form: Sk = µ−2 (µ2 Ir − Ck Pk CkT − Dk DkT ).
with initial condition P0 = 0, where the matrices Sk = SkT 0 satisfy the inequality N γ2 ln det Sk ≥ m(N + 1) ln(1 − 2 ) + 2a (19) µ
(17)
Using (17), the equation (12-14) is transformed into Pk+1 = Ak Pk ATk + Bk BkT + (Ak Pk CkT + Bk DkT )(µ2 Ir − −Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (18) 194
Consider the system (1), (2) given above. Denote |||F[0:N ] |||a an a-anisotropic norm of the system F[0:N ] on the interval [0, N ]. Problem 1. Find the conditions when |||F[0:N ] |||a < γ. The sufficient conditions for the limitation of an aanisotropic norm of the system F[0:N ] on interval k ∈ [0, N ] by given γ in terms of matrix inequalities are stated in the following theorem. Theorem 2. Let F[0:N ] with the state space realization (1), (2) be linear time varying system and real numbers a > 0 and γ > 0 are given. Then inequality |||F[0:N ] |||a < γ
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holds true if there are matrices Pk = PkT 0 for k ∈ [1, N ] and Ψk = ΨTk 0 for k ∈ [0, N ] satisfying the inequalities below −Pk+1 0 Ak P k B k 2 0 −µ Ir Ck Pk Dk (26) Pk ATk Pk CkT −Pk 0 ≺ 0, T T Dk 0 −Im Bk Ψk − µ2 Ir Ck Pk Dk Pk CkT (27) −Pk 0 ≺ 0, T 0 −Ir Dk N
1
(det Ψk ) r > e
k=0
2a r
{µ2 (1 −
γ 2 m N +1 )r} , µ2
(28)
where the initial condition P0 = 0 and real number µ > 0 belongs to interval
Ψk − µ2 Ir Ck Dk CkT −Pk−1 0 ≺ 0. T 0 −Ir Dk
Consider the third inequality (28). We transform the righthand side of (28) to equivalent view
e
2a r
{µ2 (1 −
N
k=0
Execute a similarity transformation of a linear matrix inequality (26) by multiplying it on the right and on the left by nonsingular matrix blockdiag (I, I, Pk−1 , I) 0. We get the inequality
−Pk+1 0 Ak Bk 0 −µ2 Ir Ck Dk ≺ 0. −1 T ATk Ck −Pk 0 DkT 0 −Im BkT
(30)
An application of Schur’s lemma to the inequality (30) leads to the inequality of the form (20). For given µ, the existence condition of solution of inequality (20) implies the existence of the solution Pk for following Riccati difference equation Pk+1 = Ak Pk ATk + Bk BkT + Qk + (Ak Pk CkT + Bk DkT ) ×(µ2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T (31) with initial condition P0 = P0 = 0.
By virtue of the fact that Qk = QTk 0, the condition of the monotony of the Riccati equation (Freiling & Ionescu (1999)) implies that the inequality Pk Pk 0 holds true. Pk is the solution of Riccati equation (18).
Execute a similarity transformation of a linear matrix inequality (27) by multiplying it on the right and on the left by nonsingular matrix blockdiag (I, Pk−1 , I) 0. We get the inequality 195
1 γ 2 m N +1 γ2 )r} = {e2a µ2r(N +1) (1 − 2 )m(N +1) } r . 2 µ µ (33)
Let us raise the inequality (28) to the power r
Proof.
Suppose for a given anisotropy level a > 0, anisotropy norm bound γ > 0 and scalar variable µ from the interval (29), the system of inequality (26) - (28) has the solution T Pk+1 = Pk+1 0 and Ψk = ΨTk 0 for all k = 0, . . . , N .
(32)
An application of Schur’s lemma to the inequality (32) leads to the inequality of the form (24) with variable Ψk .
γ −2 (1 − e−2a/(m(N +1)) ) < µ−2 < min (γ −2 , F0:N −2 ∞ ). (29) We shall show that the conditions of solution for solutions of inequalities (26) - (28) for each k = 0, . . . , N implies the existence of the Riccati difference equation (12 - 14) solutions when the inequality of a special form (15) for the matrix Sk (14) takes place.
195
det Ψk > e2a µ2r(N +1) (1 −
γ 2 m(N +1) ) . µ2
(34)
Take a logarithm of the both sides of (34): N
k=0
ln det Ψk > 2a + ln{µ2r(N +1) } + m(N + 1) ln (1 −
γ2 ). µ2 (35)
Then we carry ln{µ2r(N +1) } to the left-hand side and transform it. After transformation, we receive the inequality (25). The inequality (25) holds true for slack variable matrices Ψk , so this inequality will be done even for the inequality (23) with the matrices µ2 Ir − Ck Pk CkT − Dk DkT Ψk .
Taking into account Pk Pk 0 we receive
µ2 Ir −Ck Pk CkT −Dk DkT µ2 Ir −Ck Pk CkT −Dk DkT . (36)
If the inequality (23) holds true with matrices µ2 Ir − Ck Pk CkT − Dk DkT Ψk , then the inequality (19) will become true with matrices µ2 Ir − Ck Pk CkT − Dk DkT as well.
Thus the existence of the solution of the Riccati equation (18) when the inequality of special type (19) with matrices (17) follows from the existence of the solutions of inequalities (26)–(28). The Riccati equation (18) and inequality (19) are connected with equivalent transformations with the Riccati equation (12-14) and inequality (15). It means the limitation condition of the anisotropic norm in theorem 1 holds true.
7. FILTERING PROBLEM STATEMENT Consider a linear discrete time varying system F[0:N ] (37) you can see below
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L x = A x + Bik wik , k+1 k k i=1 L yk = Cyk xk + Dyik wik , i=1 L z = C x + Dzik wik , zk k k
(37)
i=1
The filtering problem is to find an estimation Ξ = {ξk|k } of the estimated output Z = {zk } by the prehistory of the observable output measurement Y = {yj , j ≤ k}. An estimator E is found in a class of causal linear discrete time varying systems with following structure:
(Ak − Kk Cyk )ηk + Kk yk ,
η0 = 0,
simultaneously for all L channels. 8. FILTERING PROBLEM SOLUTION
where the state space vector xk ∈ Rn , the estimated output vector zk ∈ Rr , the input vector wk ∈ Rm , observation output vector yk ∈ Rp and matrices Ak ∈ Rn×n , Bik ∈ Rn×mi , Cyk ∈ Rp×n , Dyik ∈ Rp×mi , Czk ∈ Rr×n , Dzik ∈ Rr×mi , k ∈ [0, N ], initial condition xk=0 = x0 , r ≤ m and m = m1 + m2 + . . . + mL . Assume we know an anisotropy level ai of each input sequence Wi , A(Wi ) ≤ ai .
ηk+1 = Ak ηk + Kk (yk − Cyk ηk ) =
i = 1, ..., L on the set of estimators (39), find the estimator E of the output Z , providing satisfaction of the inequality |||Tz wi |||ai < γi , i = 1, ..., L, (40)
In a moment k, the i-th component εzi ,k|k value of the estimation error εz,k|k from i-th group of external disturbance wik is determined by the value of subsystem Tz wi ai anisotropic norm. A structure of matrices inequalities (26)(28) allows to use the theorem 2 directly for the searching of unknown gain matrices Kk and Mk in the estimator (38) for the multicriteria filtering problem solution. Theorem 3. Suppose the linear discrete time varying system F[0:N ] with the state space realization (37) on interval [0, N ] is given. Let the L real numbers γi > 0 and ai > 0 be. The estimator E of the output Z with state space realization (38) providing the fulfillment of the L inequalities (40) exists when there are the matrices Pik = PiTk 0, k ∈ [1, N ] with initial condition Pi0 = PiT0 = 0 and Φik = ΨTik 0, k ∈ [0, N ], satisfying of L inequalities −Pik+1 0 Ak Pik Bik 2 0 −µi Ir Czk Pik Dzik ≺ 0, Pik ATk Pik CzTk −Pik 0 DzTi 0 −Imi BiTk
(38)
ξk|k = Czk ηk + Mk (yk − Cyk ηk ) = (Czk − Mk Cyk )ηk + Mk yk .
k
Here ηk is the estimation of the system (37) state H = {ηk } at the moment k.
Ψik − µ2i Ir Czk Pik Dzik Pik CzT −Pik 0 ≺ 0, k 0 −Ir DzTi
The matrices Kk and Mk in estimator (38) of the output Z , k ∈ [0, N ], are unknown and have to be found.
From the equations of system (37) and the estimator (38), we can get the system Tεz w (39), identifying the estimation error of internal state Ex = X − H = {εxk = xk − ηk } and estimated output Ez = Z − Θ = {εzk = zk − ξk|k } on the interval [0, N ]. In the moment k, the estimation error εzi ,k|k from the external disturbance Wi from (38) and (39) is determined by the subsystem Tεz wi as follows:
εxi ,k+1 = (Ak − Kk Cyk )εxi ,k + (Bik − Kk Dyik )wik , εzi ,k|k = (Czk − Mk Cyk )εxi ,k + (Dzik − Mk Dyik )wik . (39)
Each subsystem Tεz wi with an external input wik and error estimation output εzi ,k|k , identifies an i-th channel. For all channels, each subsystem Tεz wi (39) depends on unknown equal gain matrices Kk and Mk in (38). Because of the linearity of the system and taking into account the relation εx,k =
L
(41)
εxi ,k ,
i=1
the common estimation error of the output εz,k|k in (39) in the moment k is the sum of each channel estimation L error εzi ,k|k in the moment k, so it means that εz,k = i=1 εzi ,k .
Problem 2. For given system F[0:N ] (37), input channel anisotropic levels ai > 0 and the threshold values γi > 0, 196
(42)
k
N
1
(det Ψik ) r > e
2ai r
k=0
{µ2i (1 −
γi2 mi N +1 ) r } , µ2i
(43)
where Ak = Ak − Kk Cyk , Czk = Czk −Mk Cyk , Bik = Bik − Kk Dyik , Dzik = Dzik −Mk Dyik and L real numbers µi > 0 < belong to the intervals γi−2 (1 − e−2ai /(mi (N +1)) ) < µ−2 i −2 γi for i = 1, ..., L. The proof of the previous theorem is based on the direct application of the theorem 2 to each channel. Remark 1. The inequality (43) used for the calculation of the slack variables Ψik+1 can be presented in recurrent form for i = 1, . . . , L k mi 1 1 (det Ψik+1 ) r > e2ai /r {µ2i (1−γi2 /µ2i ) r }k+2 / (det Ψik ) r . k=0
(44)
9. NUMERICAL EXAMPLE Let us consider a mathematical model of linear continues time invariant system, which is a damped pendulum. w x˙1 0 ω0 x1 00 , (45) = + + n 10 x˙2 −ω0 −2ξω0 x2 y = [0 1]
x1 x2
+ [0 1]
w n
,
(46)
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic Victor N. Timin et al. / IFAC-PapersOnLine 48-14 (2015) 192–197
z = [0 1]
x1 x2
+ [0 0]
w n
(47)
,
where a damping parameter ξ = 0.2 and angular natural frequency ω0 = 10. An external disturbance w acts to the system input. The only state space vector component x2 is measured. This component is measured and observable as well. A discretization of the continuous system was carried out with sampling step ∆t = 0.01. For discrete time system, the design of an anisotropicbased multicriteria filter has been done. We have used the following anisotropy levels: A(w) ≤ 1 and A(n) ≤ 0.01. The bounds for anisotropic norms have been given as γ1 = γw = 0.25 and γ2 = γn = 0.05. A comparison of designed anisotropic-based filter has been carried out with the H2 filter. The H2 filter is the special case of the anisotropy-based filter if anisotropy level a=0. The H2 filter has been synthesized under the constraint of the threshold value γ = 0.25 for H2 norm of the map from the extended vector of external disturbances ν = (wT , nT )T to the estimation error ez . In the synthesis of H2 filter, the matrices Mk of estimator have been assumed to be zero. −4
4
x 10
k
K1
3 2 1 0 −1
0
0.5
1 Time, sec
1.5
2
0.5
1 Time, sec
1.5
2
−4
20
x 10
k
K2
15 10 5 0 −5
0
Fig. 1. Two components Kk1 and Kk2 of gain vector Kk The graphics of the coordinates Kk1 and Kk2 of twodimensional vector Kk depending on the time of t (t = k∆t, ∆t = 0.01, k ∈ [0, N ], N = 200) for the multicriteria anisotropic filter (solid line) and H2 filter (dashed line) are shown in Figure 1. The coordinates of the gain Kk in a case of the anisotropic filter for each discrete time moment k are larger than the coordinates of the gain Kk in a case of H2 filter. Because of the system (45-47) is time invariance, the coordinates of the gain Kk tend to steadystate values. The steady-state values for anisotropy-based filter are larger than H2 filter as well. This behavior of the coordinates of Kk characterizes ”intermediate” properties of the anisotropic estimator to be between the H2 and H∞ filters. CONCLUSION On finite horizon for the linear discrete time varying system with observable and estimated outputs, the problem of anisotropy-based robust filtering was solved. In terms of the matrix inequalities, a criterion for the boundedness of anisotropic norm of linear discrete time-varying system was formulated. The sufficient conditions for the existence of multi-criteria suboptimal anisotropic estimator with limited anisotropic norms of subsystems by a predetermined threshold values were received. The algorithm for 197
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