ISSN 0005-1179, Automation and Remote Control, 2016, Vol. 77, No. 1, pp. 1–20. © Pleiades Publishing, Ltd., 2016. Original Russian Text © V.N. Timin, A.P. Kurdyukov, 2016, published in Avtomatika i Telemekhanika, 2016, No. 1, pp. 5–29.

TOPICAL ISSUE

Suboptimal Anisotropic Filtering in a Finite Horizon V. N. Timin and A. P. Kurdyukov Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected], [email protected] Received March 30, 2015

Abstract—Consideration was given to the problem of robust stochastic filtering in a finite horizon for the linear discrete time-varying system. A random disturbance with inaccurately known probabilistic distribution is fed to the system input. Uncertainty of the input disturbance is defined in the information-theoretical terms by the anisotropy functional of a random vector. The sufficient condition for strict boundedness of the anisotropic norm of linear discrete timevarying system assigned by the threshold value (lemma of real boundedness) was proved in terms of the matrix inequalities. Sufficient conditions for boundedness of the anisotropic norm of two limiting cases of the anisotropy levels of the input disturbance (a = 0 and a → ∞) were established. A sufficient existence condition for the estimator guaranteeing boundedness of the anisotropic norm of the estimation error operator by the given threshold value was formulated and proved. Sufficient existence conditions for the estimators of two limiting cases of the anisotropy levels of input disturbance were obtained. The estimation algorithm relies on the recurrent solution of a system of matrix inequalities. DOI: 10.1134/S000511791601001X

1. INTRODUCTION The problem of estimating the state or a system output from the observed noisy output is one of the fundamental challenges to the control and digital signal processing. The classical result in the field of linear filtering was expounded in the R. Kalman paper [1]. The formulation of the problem of estimation in the state space (see, for example, [2]) in the Kalman filtering assumes that the model of process dynamics and statistical characteristics of the model noise and measurements are known precisely. At minimization of the quadratic performance criterion the optimal (Kalman) filter provides the minimal trace of the covariance matrix of the system state estimation error. Minimization of the quadratic optimality criterion formulated in the state space is known [3] to be operator-equivalent to the minimization of the H2 -norm of the estimation error operator. The Kalman filter (or the optimal H2 -filter [3]) designed for a given model is not robust [4] and can lose stability under small disturbances in the mathematical model of a plant. The H∞ -criterion for optimality is used in the case where the a priori information about the plant model and the statistical characteristics of noise in the model of process and measurement are not known precisely. The suboptimal filter designed from the criterion for boundedness of the H∞ criterion for optimality guarantees that in the worst case of the external disturbance the H∞ -norm of the operator mapping the input disturbance to the estimation error does not go beyond the given threshold. At the same time, the suboptimal H∞ -filter limits the trace of the covariance matrix of the estimation error. For this reason, the algorithms of H∞ -filtering (see, for example, [3, 5–8]) belong to the class of minimax algorithms minimizing the estimation error in the worst case of external disturbance. The strategies of Kalman linear quadratic Gaussian estimation and the theory of H∞ -estimation rely on different assumptions about the statistical characteristics of the input disturbances. In the problem of H2 -estimation, it is assumed that an external disturbance represents the Gaussian white noise. In the problem of H∞ -estimation, the signal of external disturbance is assumed to be unknown quadratic one summable with an arbitrary distribution. 1

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The optimal H2 -filters and H∞ -filters operate efficiently if the input signals belong to the classes assumed by the theory. The H2 -estimator provides unsatisfactory results in the case of strongly colored input disturbance and an H∞ -estimator intended for the worst case [9] under an input disturbance in the form of white or weakly colored noise. The so-called mixed H2 /H∞ -filters (see, for example, [10–15]) represent a line of designing filters that are less conservative than the H∞ -filters and more robust than the H2 -filters. The mixed H2 /H∞ approach minimizes the H2 -criterion under the given constraint for the H∞ -criterion. The methods of H2 -, H∞ -, and mixed H2 /H∞ -filtering come to solution of the Riccati equations or linear matrix inequalities (see, for example, [16, 17]). The trade-off between the H2 -optimal and H∞ -optimal filters is discussed in the problem of the so-called “generalized” H∞ -filtering [18] where the joint impact of the unknown initial conditions and the unmeasurable external disturbance on the estimation error is minimized. The minimax estimation and filtering under unknown covariances with the use of linear matrix inequalities are considered in [19]. For the linear time-varying systems, proposed were algorithms to design the time-varying filters based on the recurrent solution of the system of linear matrix difference inequalities [20]. The system of linear matrix inequalities should be solved at that at each estimation step. Another line of designing “intermediate” estimators for the linear discrete stationary systems relies on the anisotropy theory of stochastic optimization [21]. Design of the anisotropic estimator also comes to solving the Riccati equations [22] or the linear matrix inequalities [23, 24]. An approach to construction of a time-varying theory of anisotropic robust control and filtering has been developed in the last few years. The problem of anisotropic analysis of the linear time-varying discrete systems over a finite time interval was considered for the first time in [25] where defined were the notions of anisotropy and anisotropic system norm corresponding to the problems of the time-varying anisotropic theory. The problem of optimal anisotropic filtering in the finite horizon was solved in [26] by reduction to two difference Riccati equations in the direct and backward times. The following step in the construction of the time-varying anisotropic theory for the systems in the finite horizon was made in [27] where the necessary and sufficient conditions for boundedness of the anisotropic norm by the given threshold value were established. Relying on [27], the paper [28] solved the problem of anisotropic filtering in the suboptimal formulation in the finite horizon for a special case of equality of dimensionalities of the estimated output and external disturbance. For solution of the latter problem, [28] introduced an auxiliary variable representing the product of identity matrix and positive scalar value which leads to a significant increase in the conservatism. The present paper is structured so that the necessary information on the anisotropic theory of control and filtering is given brush treatment in Section 2. The criterion for boundedness of the anisotropic norm for the linear time-varying system in terms of the matrix inequalities is presented in Section 3. The problem of anisotropic filtering is formulated and solved in Section 4, and the existence conditions for the suboptimal anisotropic filter are formulated in Section 5. For the numerical example in Section 6, an anisotropic estimator is designed using the proposed algorithm. Comparison was carried out with the suboptimal H2 -filter and H∞ -filter. 2. NOTATION AND PRELIMINARY INFORMATION 2.1. Class of the Considered Systems Let us consider a discrete linear time-varying system F over the limited interval [0, N ] xk+1 = Ak xk + Bk wk ,

x0 = 0,

(1)

zk = Ck xk + Dk wk , AUTOMATION AND REMOTE CONTROL

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where k is the discrete time variable taking integer values over the interval [0, N ], xk ∈ Rn is the system state, zk ∈ Rr is the output, and wk ∈ Rm is the input disturbance. The real matrices of the system Ak , Ck , Bk , and Dk of the corresponding dimensions for m  r are known. For some time instants s and t (0  s  t  N ), the sequences of input and output signals w and z over the interval [s, t] are expanded into the column vectors Ws:t  [wsT , . . . , wtT ]T and Zs:t  [zsT , . . . , ztT ]T . If the initial state of the system is zero, then the system output is representable as Z0:t = F0:t W0:t , where Fs:t is the lower triangular matrix composed of r × m blocks fjk and defined as ⎧ ⎪ ⎨ Cj Φj,k+1 Bk

Fs:t = block (fjk ),

fjk =

sj j,kt

for j = k for j < k.

Dk 0

⎪ ⎩

for j > k (2)

For j  k, Φj,k = Aj−1 × . . . × Ak

(3)

is the transition matrix from the state xk to xj , provided that Φk,k = In , where In is the identity matrix of the order n. Since the matrix F0:N defines completely the system F over the time interval [0, N ] as a linear input-output operator from W0:N to Z0:N , all norms for the operator F are understood as those of the matrix F0:N . In particular, the counterparts of the H2 -norms and H∞ -norms in a finite horizon obey the Frobenius and spectral norms F0:N as ||F ||2 =



T F Tr(F0:N 0:N ),

||F ||∞ = σmax (F0:N ),

(4)

where σmax (·) is the maximal singular number of the matrix. 2.2. Anisotropy of the Random Vectors The relative entropy or informational Kullback–Leibler distance [29] of the probabilistic measure P relative to the probabilistic measure M over the same measured space is defined as 



D(P ||M ) = E ln

dP dM



with P assumed to be absolutely continuous relative to M with the density dP/dM (Radon– Nikodym derivative), where E is the expectation relative to P . The relative entropy is nonnegative and equal to zero if P = M . It is possible to represent D(P ||M ) as D(ξ||η) or D(f ||g), where P and M are the distributions of the random variables ξ and η, and f and g are the functions of the density of distribution probabilities. For any λ > 0, we denote by 

−l/2

pl,λ (w) = (2πλ)

wT w exp − 2λ



,

w ∈ Rl

(5)

the function of probability distribution density of the Gaussian vector with zero mean and scalar covariance matrix like λIl . AUTOMATION AND REMOTE CONTROL

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Let W be a quadratically integrable absolutely continuous random vector with values in Rl and the function of probability distribution density f . Then, its relative entropy with respect to the Gaussian law (5) is given by



D(f ||pl,λ ) = E ln

f (W ) pl,λ (W )



= −h(W ) +

E(|W |2 ) l ln(2πλ) + , 2 2λ

(6)

where h(W ) = −E ln f (W ) is the differential entropy W . The class of the quadratically integrable absolutely continuous random vectors in Rl is denoted by Ll2 . Definition 1 [21]. The anisotropy A(W ) of the random vector W ∈ Ll2 is defined as the minimal relative entropy (6) with respect to the Gaussian function of probability distribution density (5) with zero mean and scalar covariance matrix A(W ) = inf D(f ||pλ ) = λ>0

l ln(2πeE(|W |2 )/l) − h(W ). 2

(7)

2.3. Anisotropic Norm of Matrices Let F ∈ Rs×l be an arbitrary real matrix interpreted as a constant linear operator with the input interpreted as a quadratically integrable random vector W ∈ Rl which can be regarded as the input disturbance. The rms gain of the matrix F relative to the input W R(F, W ) =



E(|Z|2 )/E(|W |2 )

(8)

was introduced in [25]. The squared Euclidean norm | · |2 of the vector is interpreted here as its energy; therefore, E(|W |2 ) and E(|Z|2 ) describe, respectively, the averaged energy of the input and output of the operator F . Definition 2 [27]. For any a  0, the a-anisotropic norm of the matrix F ∈ Rs×l is given by 



|F |a = sup R(F, W ) : W ∈ Ll2 , A(w)  a .

(9)

Known is a fundamental property of the anisotropic norm that for any a  0 the anisotropic norm is a nondecreasing function in a and √ (10) ||F ||2 / l = |F |a=0  |F |a  lim |F |a = F ∞ . a→+∞

For the given system (1), with regard for the dimensionality l = m(N + 1) of the vector W0:N relations (10) for the family of a-anisotropic norms go over to 

||F ||2 / m(N + 1) = |F |a=0  |F |a  lim |F |a = F ∞ ,

(11)

a→+∞

where norms (4) are used as the limiting cases. For the discrete linear systems, calculation of the anisotropic norm in a finite horizon in the state space was reduced in [25] to the solution of three related equations such as the difference Riccati equation in the backward time, the algebraic equation comprising the matrix determinants, and the difference Lyapunov equation in the direct time. The procedure of anisotropic robust analysis is complicated by the presence of related difference equations with different ordering in time. The stated below theorem formulates a criterion for boundedness by the given value of the anisotropic norm (9) for the discrete linear time-varying system F with model in the state space (1) to which an operator like (2) corresponds. AUTOMATION AND REMOTE CONTROL

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Theorem 1 [27]. Let given be a discrete linear time-varying system F with model in the state space (1). Then, the a-anisotropic norm |F |a  γ exists over the interval [0, N ] if and only if there are —the matrices Rk = RkT  0 and Rk ∈ Rn×n , k = 0, . . . , N , satisfying the difference Riccati equation T T Rk+1 = Ak Rk AT k + qBk Bk + Mk Sk Mk ,

Mk 

+ qBk DkT )Sk−1 , Ck Rk CkT − qDk DkT

−(Ak Rk CkT

S k  Ir −

(12) (13) (14)

with the initial condition R0 = 0, —the matrices Sk = SkT  0 satisfying the inequality N 

ln det Sk  m(N + 1) ln(1 − qγ 2 ) + 2a,

(15)

k=0

—and a real number q satisfying the localization condition 



γ −2 1 − e−2a/(m(N +1))  q < min (γ −2 , F −2 ∞ ).

(16)

3. CRITERION FOR BOUNDEDNESS OF THE ANISOTROPIC NORM IN TERMS OF THE MATRIX INEQUALITIES We formulate a criterion for strict boundedness of the anisotropic norm in terms of the matrix inequalities related with an inequality associated with the difference Riccati equation and the inequality limiting the determinants of the positive definite matrices. Carry out an equivalent transformations of the Riccati equation (12)–(14) and inequality (15) and change the variables Rk = q Pk and q = 1/μ2 . Then, the matrix Sk of Eqs. (12)–(14) is given by Sk = μ−2 (μ2 Ir − Ck Pk CkT − Dk DkT ).

(17)

With regard for (17), Eqs. (12)–(14) go into T Pk+1 = Ak Pk AT k + Bk Bk

+ (Ak Pk CkT + Bk DkT )(μ2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T

(18)

with the initial condition P0 = 0, where the matrices Sk = SkT  0 satisfy the inequality 

N 

γ2 ln det Sk  m(N + 1) ln 1 − 2 μ k=0 for





+ 2a

(19)



γ −2 1 − e−2a/(m(N +1))  μ−2 < min (γ −2 , F −2 ∞ ). We go from the Riccati equation to the Riccati inequality. Consider the Riccati inequality T Pk+1  Ak Pk AT k + Bk Bk

+ (Ak Pk CkT + Bk DkT )(μ2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T with the initial condition P0 = P0 = 0. AUTOMATION AND REMOTE CONTROL

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(20)

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In what follows, we consider separately in the inequalities the initial step for k = 0 and the interval k ∈ [1, N ]. For k = 0 with the initial condition P0 = 0, the Riccati inequality (20) goes into an inequality on the positive definite matrix P1 : P1  B0 B0T + B0 D0T (μ2 Ir − D0 D0T )−1 B0T D0 ,

(21)

where the matrix μ2 Ir − D0 D0T is nondegenerate. Introduce the matrix Sk = μ−2 (μ2 Ir − Ck Pk CkT − Dk DkT )

(22)

depending on the variable Pk in inequality (20). By analogy with (19), we consider the strict inequality on the logarithm of the determinant of the matrix Sk 

N 

γ2 ln det Sk > m(N + 1) ln 1 − 2 μ k=0



+ 2a,

(23)

taking on with regard for (22) the form N 

ln μ

−2r



det(μ Ir − 2

k=0

Ck Pk CkT

−



Dk DkT )



γ2 > m(N + 1) ln 1 − 2 μ



+ 2a.

(24)

Introduce an auxiliary matrix variable Ψk satisfying the inequality Ψk ≺ μ2 Ir − Ck Pk CkT − Dk DkT .

(25)

For k = 0 with the initial condition P0 = 0, the inequality goes over to Ψ0 ≺ μ2 Ir − D0 D0T .

(26)

With regard for (25), inequality (24) is transformed into N  k=0



ln{μ

−2r

γ2 (det(Ψk )} > m(N + 1) ln 1 − 2 μ



+ 2a.

(27)

In the following theorem, in terms of the matrix inequalities over the time interval for k ∈ [0, N ] we formulate for the discrete linear time-varying system F a sufficient criterion for strict boundedness of the a-anisotropic norm by the given threshold value γ. Problem 1. Needed is to determine the conditions for which |F |a is less than γ over the interval k ∈ [0, N ]. Theorem 2 (sufficient conditions). Let given be the system F with a model in the state space (1) and the real numbers a > 0 and γ > 0. The condition |F |a < γ is satisfied for the a-anisotropic norm of the system F over the interval [0, N ] if for k = 1, . . . , N there exist the real (n × n) matrices Pk = PkT  0 with the initial condition P0 = 0 and, for k = 0, . . . , N , the real (r × r) AUTOMATION AND REMOTE CONTROL

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matrices Ψk = ΨT k  0 satisfying the inequalities ⎡

−Pk+1

⎢ ⎢ 0 ⎢ ⎢ T ⎣ Pk Ak

0

⎢ ⎢ ⎣

⎥ ⎥ ≺ 0, ⎥ ⎦

Dk ⎥

Pk CkT

−Pk

0

DkT

0

−Im

−Ir

0

1 r

(det Ψk ) > e

2a r

⎧ ⎨

⎥

0 ⎥ ⎦ ≺ 0,

−Pk

Pk CkT

(28)

⎤

Ψk − μ2 Ir Ck Pk Dk DkT

N 

⎤

Bk

−μ2 Ir Ck Pk

BkT

⎡

Ak Pk



μ2 1 −

(29)

γ2

 m ⎫N +1 r ⎬

μ2

⎭

,

(30)

γ max(F ∞ , γ) < μ <  . (1 − e−2a/(m(N +1)) )

(31)

k=0

⎩

where the real number μ > 0 is such that

Theorem 2 is proved in the Appendix. Remark 1. The matrix inequality (30) imposes a constraint on the geometric average of the the convex constraint since for any real symmetrical eigenvalues of the matrix Ψk . It defines  1 positive definite r × r matrix the function − det(·) r is convex. Moreover, the given convex set of constraints is representable as a linear matrix inequalities [30]. Remark 2. At the first step k = 0, the matrix inequalities (28) and (29) are reduced with regard for the initial condition P0 = 0 to the inequalities ⎡ ⎢ ⎢ ⎣

−P1

0

0

−μ2 I

B0T

D0T

B0

r

⎤

⎥ D0 ⎥ ⎦ ≺ 0,

Ψ0 − μ2 Ir D0 D0T

−Im

−Ir



≺ 0,

(32)

from which the desired matrices P1 and Ψ0 are determined if inequality (30) is satisfied. Remark 3. At each subsequent step k > 0, the matrix inequalities (28)–(30) are solved recurrently with respect to the matrices Pk+1 and Ψk . 3.1. Limiting Cases Consider the conditions of Theorem 2 for two important limiting cases where the level of anisotropy of the input signal is zero and tends to infinity. It follows from inequality (10) that on condition that the a-anisotropic norm is equal to zero it is equal to the scaled H2 -norm, and at tending to infinity it approaches the system H∞ -norm. Obviously, conditions (28)–(30) of Theorem 2 should come to the conditions for checking boundedness by the given number γ > 0, respectively, for the H2 -norm and H∞ -norm of system F . 3.1.1. Case of a = 0. Consider the case of zero-level anisotropy and demonstrate how the conditions of Theorem 2 are modified in this case. Lemma 1 (sufficient conditions). For the system F with model in the state space (1), the condition |F |a=0 < γ is satisfied for the given real numbers a = 0 and γ > 0 for the a-anisotropic norm AUTOMATION AND REMOTE CONTROL

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of system F over the interval [0, N ] if for k = 1, . . . , N there exist real (n×n) matrices Pk = PkT  0 with the initial condition P0 = 0 and for k = 0, . . . , N , real (r × r) matrices Φk = ΦT k  0 satisfying the inequalities ⎡

−Pk+1 Ak Pk

⎢ ⎢ Pk AT k ⎣

−Pk

0

0

−Im

BkT

⎡

−Φk

⎤

Bk

⎥

0 ⎥ ⎦ ≺ 0,

−Pk

DkT

(33)

⎤

Ck Pk Dk

⎢ ⎢ Pk C T k ⎣

⎥ ⎥ ≺ 0, ⎦

(34)

−Ir

0

trace Φk < mγ 2 .

(35)

Lemma 1 is proved in the Appendix. Remark 4. At the first step for k = 0 and with regard for the initial condition P0 = 0 the matrix inequalities (33)–(34) are transformed to the inequalities

−P1



B0

≺ 0,

B0T −Im

−Φ0 D0 −Ir

D0T



≺ 0,

(36)

from which the desired matrices P1 and Φ0 are determined under fulfilled condition (35). Remark 5. At each subsequent step k > 0, the matrix inequalities (33)–(35) are solved recurrently with respect to the matrices Pk+1 and Φk . 3.1.2. Case a → ∞. Consider the case where the level of anisotropy tends to infinity and demonstrate how in this case the conditions of Theorem 2 are modified. Lemma 2 (sufficient conditions). For the system F with model in the state space (1) and the given γ > 0 and a → ∞, the condition |F |a→∞ < γ is satisfied for the a-anisotropic norm of system F over the interval [0, N ] if for k = 1, . . . , N with the initial condition P0 = 0 there exist real (n × n) matrices Pk = PkT  0 satisfying the inequality ⎡

−Pk+1

0

⎢ ⎢ 0 ⎢ ⎢ T ⎣ Pk Ak

Ak Pk

Bk

−γ 2 Ir Ck Pk

⎥ ⎥ ≺ 0. ⎥ ⎦

Dk ⎥

Pk CkT

−Pk

0

DkT

0

−Im

BkT

⎤

(37)

Lemma 2 is proved in the Appendix. Remark 6. At the first step for k = 0 and with regard for the initial condition P0 = 0, the matrix inequality (37) is transformed into the inequality ⎡ ⎢ ⎢ ⎣

−P1

0

0

−γ 2 Ir

B0T

D0T

B0

⎤ ⎥

D0 ⎥ ⎦≺0

(38)

−Im

from which the desired matrix P1 is determined. Remark 7. At each subsequent step k > 0, the matrix inequality (37) is solved recurrently relative to the matrix Pk+1 . AUTOMATION AND REMOTE CONTROL

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4. FORMULATION OF THE FILTERING PROBLEM Let us consider a discrete linear time-varying system F given by xk+1 = Ak xk + Bk wk ,

xk=0 = 0,

yk = Cyk xk + Dyk wk ,

k ∈ [0, N ],

(39)

zk = Czk xk + Dzk wk with the state vector xk ∈ Rn , input vector of the external disturbance wk ∈ Rm , and measured and estimated output vectors yk ∈ Rp and zk ∈ Rr , respectively. Here, the matrices Ak ∈ Rn×n , Bk ∈ Rn×m , Cyk ∈ Rp×n , Dyk ∈ Rp×m , Czk ∈∈ Rr×n , Dzk ∈ Rr×m , provided that m  r. It is assumed that there exists a priori information about the statistical properties of the input sequence W = {wk } enabling one to constrain the level of anisotropy A(W )  a, where the number a > 0 defines the measure of statistical uncertainty of the input signal. The problem of filtering lies in determining the estimate Z = {zk|k } of the output Z = {zk } from the prehistory of measurements of the output Y = {yj , j  k}. Let E be an estimator of the output Z = {zk }. Then, zk|k = E(y0 , y1 , . . . , yk ). The error of estimation is defined as the difference Z = Z − Z = zk − zk|k . The estimator E is sought in the class of causal discrete linear time-varying systems of the following structure: k+1 = Ak x k + Kk (yk − Cyk x k ) = (Ak − Kk Cyk )x k + Kk yk , x k + Mk (yk − Cyk x k ) = (Czk − Mk Cyk )x k + Mk yk , zk|k = Czk x

0 = 0, x

(40)

k ∈ [0, N ],

 = {x k is the state estimate X k } of system (39) at the instant k. where x The sequences of the gain matrices Kk ∈ Rn×p and Mk ∈ Rr×n for k ∈ [0, N ] are unknown and liable for determination. Using the equations of system (39) and estimator (40), we determine in the state space the operator Tz˜w mapping over the interval [0, N ] the input sequence W = {wk } into the error of estimation the output sequence Z = {zk|k } and the error of estimation of the state of system  =X −X  = {x  k = xk − x k }. In the state space, it is given by X k+1 = (Ak − Kk Cyk )x k + (Bk − Kk Dyk )wk , x k + (Dzk − Mk Dyk )wk , zk|k = (Czk − Mk Cyk )x

x0 = 0,

(41)

k ∈ [0, N ].

Minimization or constraint by some optimality criterion of the value estimation error enables one to minimize or constrain the value of the error of estimation under the action on system (41) of an external disturbance W owing to the choice of a corresponding estimator E. In the present paper, it is the a-anisotropic norm of the operator Tz˜w that is used as the optimality criterion. The problem of suboptimal anisotropic filtering in the finite horizon is as follows. Problem 2. For the given system F with model in the state space (39), the anisotropy level a > 0 of the input disturbance W , and the threshold value γ > 0, needed is to determine the time-varying estimator E, if any, with model in the state space (40) such that the inequality |Tz˜w |a < γ

(42)

is satisfied. AUTOMATION AND REMOTE CONTROL

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5. SOLUTION OF THE FILTERING PROBLEM We formulate the sufficient existence condition for filter or estimator (40) under the given level a > 0 of anisotropy of input disturbance and the threshold value γ > 0 of the a-anisotropic norm of the operator of estimation error Tz˜w over the given interval of discrete time k ∈ [0, N ]. Theorem 3. Let for the system F with a model in the state space (39) given be real numbers a > 0 and γ > 0. The estimator E with realization in the state space (40) ensuring satisfaction of the inequality (42) for the interval [0, N ] exists if there are matrices P0 = 0, Pk = PkT  0 for k = 1, . . . , N and Ψk = ΨT k  0, k = 0, . . . , N , satisfying the matrix inequalities ⎡

−Pk+1

(Ak −Kk Cyk )Pk

0

Bk −Kk Dyk

⎤

⎢ ⎥ ⎢ 0 −μ2 Ir (Czk −Mk Cyk )Pk Dzk −Mk Dyk ⎥ ⎢ ⎥ ≺ 0, ⎢ ⎥ T T −Pk 0 ⎣Pk (Ak −Kk Cyk ) Pk (Czk −Mk Cyk ) ⎦

(Bk −Kk Dyk )T ⎡

(Dzk −Mk Dyk )T

Ψ k − μ 2 Ir

(Dzk − Mk Dyk )T N 

−Im

0

(Czk − Mk Cyk )Pk Dzk − Mk Dyk

⎢ ⎢ Pk (Cz − Mk Cy )T k k ⎣

1 r

(det Ψk ) > e

k=0

2a r

−Pk

0

0

−Ir

⎧ ⎨

μ2

⎩



γ2 1− 2 μ

(43)

⎤ ⎥ ⎥ ≺ 0, ⎦

(44)

 m ⎫N +1 r ⎬

(45)

⎭

and a real number μ > 0 such that γ . γ<μ< " −2a/m(N +1) 1−e

(46)

Proof of Theorem 3. The theorem is proved by the direct application to system (41) of the sufficient conditions for strict boundedness of the anisotropic norm (Theorem 2). Remark 8. The system of inequalities (43)–(46) defines the set of estimators guaranteeing boundedness of the a-anisotropic norm of the estimation error operator Tz˜w by the given threshold value γ. Remark 9. For each k, inequalities (43) and (44) are affine in the desired estimator matrices Kk and Mk . Remark 10. Inequality (45) for determination of the auxiliary variables Ψk is representable in the recurrent form as 1

(det Ψk+1 ) r > e2a/r {μ2 (1 − γ 2 /μ2 )m/r }k+2 /

k 

1

(det Ψk ) r .

(47)

k=0

Remark 11. At each step k, by solving the optimization problem min

Ψk ,Kk ,Mk

trace Pk+1

under constraints

(43), (44), (47)

(48)

it is possible to find on the set of solutions of a system of matrix inequalities (43)–(45) a solution with the minimal trace of the matrix Pk+1 [20]. Remark 12. The algorithm to calculate the desired filter sequences Kk and Mk boils down to the recurrent solution of the system of matrix inequalities (43), (44), (47) with the initial condition P0 = 0, where at solution of the optimization problem (48) the unknown matrices Pk+1 , Ψk , Kk , and Mk are calculated at each step k. AUTOMATION AND REMOTE CONTROL

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At the first step k = 0, inequalities (43) and (44) go over to ⎡ ⎢ ⎢ ⎣

−P1

0

B0 − K0 Dy0

0

−μ2 Ir

⎥

Dz0 − M0 Dy0 ⎥ ⎦ ≺ 0,

(B0 − K0 Dy0 )T (Dz0 − M0 Dy0 )T

⎤

−Im

Ψ 0 − μ 2 Ir

Dz0 − M0 Dy0

(Dz0 − M0 Dy0 )T

−Ir



≺ 0.

5.1. Limiting Cases For two limiting cases where the level of anisotropy of the input signal is zero and tends to infinity, we present the sufficient existence conditions for the filters from Theorem 3. 5.1.1. Suboptimal H2 -filter. For a = 0 and given γ, the a-anisotropic filter comes to the suboptimal H2 -filter, that is, to the filter bounding the scaled H2 -norm of system (41) by the given level γ. Upon approaching the minimal feasible value of γ → γ2opt , the suboptimal H2 -filter tends to the optimal H2 -filter. Lemma 3. Let the real numbers a = 0 and γ > 0 be given for the system F with realization in the state space (39). The estimator E realized in the state space (40) and guaranteeing satisfaction of inequality (42) for the interval [0, N ] exists, provided that there are matrices P0 = 0 and Pk = PkT  0, k = 1, . . . , N , and Φk = ΦT k  0, k = 0, . . . , N , satisfying the matrix inequalities ⎡

−Pk+1

⎢ ⎢ Pk (Ak − Kk Cy )T k ⎣

(Bk − Kk Dyk

⎡

(Ak − Kk Cyk )Pk Bk − Kk Dyk

)T

−Φk

⎢ ⎢ Pk (Cz − Mk Cy )T k k ⎣

−Pk

0

0

−Im

⎤ ⎥ ⎥ ≺ 0, ⎦

(Czk − Mk Cyk )Pk Dzk − Mk Dyk

(Dzk − Mk Dyk )T

trace

−Pk

0

0

−Ir

(49)

⎤ ⎥ ⎥ ≺ 0, ⎦

(50)

Φk < mγ 2 .

(51)

Proof. The proof of existence of the suboptimal H2 -filter relies on the direct application of Lemma 1 to system (41). Remark 13. On the set of solutions of the system of matrix inequalities (49)–(51), at each step k a solution with the minimal trace of the matrix Pk+1 can be found by solving the following optimization problem: min

Φk ,Kk ,Mk

trace Pk+1

under constraints

(49)–(51).

(52)

Remark 14. The algorithm to compute the desired sequences Kk and Mk of the filter comes to a recurrent solution of the system of matrix inequalities (49)–(51), where at each step k the unknown matrices Pk+1 , Φk , Kk , and Mk are calculated when solving the optimization problem (52). At the first step k = 0, inequalities (49)–(50) with the initial condition P0 = 0 go over to

−P1

B0 − K0 Dy0

(B0 − K0 Dy0 )T

−Im



≺ 0,

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(Dz0 − M0 Dy0 )T

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5.1.2. Suboptimal H∞ -filter. Under the anisotropy level a → ∞ and given γ, the a-anisotropic filter comes to the suboptimal H∞ -filter, that is, an equality bounding the H∞ -norm of system (41) by the given level γ. Lemma 4. Let for the system F realized in the state space (39) given be the real numbers γ > 0 and a → ∞. The estimator E realized in the state space (40) and guaranteeing satisfaction of inequality (42) for the interval [0, N ] exists, provided that there are matrices P0 = 0 and Pk = PkT  0, k = 1, . . . , N , satisfying the matrix inequality ⎡

−Pk+1

(Ak −Kk Cyk )Pk

0

Bk −Kk Dyk

⎤

⎢ ⎥ ⎢ 0 −γ 2 Ir (Czk −Mk Cyk )Pk Dzk −Mk Dyk ⎥ ⎢ ⎥ ≺ 0. ⎢ ⎥ T T −Pk 0 ⎣Pk (Ak −Kk Cyk ) Pk (Czk −Mk Cyk ) ⎦

(Bk −Kk Dyk )T

(Dzk −Mk Dyk )T

(53)

−Im

0

Proof. The proof of existence of the H∞ -filter relies on the direct application of Lemma 2 to system (41). Remark 15. At each step k = 0, . . . , N , by solving the optimization problem min

Kk ,Mk

trace Pk+1

under constraints

(53)

(54)

on the set of solutions of the linear matrix inequalities (53) one can find a solution with the minimal trace of the matrix Pk+1 . Remark 16. The algorithm to calculate the desired sequences Kk and Mk of the equality comes to the recurrent solution of the matrix linear inequality (53) with the initial condition P0 = 0, where at each step k = 0, . . . , N the unknown matrices Pk+1 , Kk , and Mk are calculated at solving the optimization problem (54). At the first step k = 0, the inequality (53) with the initial condition P0 = 0 goes over to ⎡ ⎢ ⎢ ⎣

−P1

0

0

−γ 2 Ir

(B0 − K0 Dy0

)T

B0 − K0 Dy0

(Dz0 − M0 Dy0

⎤ ⎥

Dz0 − M0 Dy0 ⎥ ⎦ ≺ 0.

(55)

−Im

)T

6. NUMERICAL EXAMPLE In the present section, an a-anisotropic filter is designed for a particular system using the proposed algorithm with its given parameters a, γ, and μ. The suboptimal H2 -, H∞ -, and a-anisotropic filters are compared. For simplicity and visualization of the results of the filter design algorithms, we consider a continuous stationary linear system like ⎛

⎞

⎡

⎤⎛

⎞

⎡

x˙1 −1 0 0 x1 1 ⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ 0⎦ ⎝ x2 ⎠ + 25 ⎣ 1 ⎝ x˙2 ⎠ = ⎣ 0 −2 x˙3 x3 0 0 −3 −1 ⎛

⎞

⎤

0   ⎥ w , 0⎦ n 0

  x1 ) * w ⎟ ⎜ , y = −1 2 1 ⎝ x2 ⎠ + 0 1 n x3 )

*

⎛

(56)

⎞

  x1 ) * w ⎟ ⎜ , z = 1 0 0 ⎝ x2 ⎠ + 0 0 n x3 )

*

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Components of the vector of the gain K.

which was slightly modified as compared with [31]. Only the external disturbance of model w arrives to the system input. Measurements are carried out in the measurement noise n. The continuous system is digitized with the step Δt = 0.01. For the observation interval N = 500, a suboptimal a-anisotropic filter was designed for the resulting discrete system. For determination of the filter coefficients, the problem of filtering was solved in the Matlab environment using the YALMIP language of modeling [32] and the SeDuMi toolbox [33]. Solution assumed that the anisotropy of the united external disturbance ν = (wT , nT )T of the system does not exceed A(ν)  0.001. The magnitude of γ bounding the a-anisotropic norm of system (41) for the discrete system (56) was taken equal to γa = 3.8. At that, the parameter μ was selected from the permissible localization interval μ = 16.25γa . Filter was designed for the case where the sequence of the filter coefficient Mk was assumed to be zero. For γ = γ2 = 1.6 and γ = γ∞ = 9.9, the suboptimal H2 -filter and H∞ -filter, respectively, were designed in a similar way. In the example at hand, the dimensionality of the measurement vector is p = 1; therefore, the matrix gain goes over to the three-dimensional vector. The results of designing the calculated sequence of the filter gain Kk in time tk = kΔt, k = 1, . . . , N , are plotted in the figure for Δ = 0.01 and N = 500. The graphs show the three-dimensional sequence of the gain Kk = (K1k , K2k , K3k )T for the suboptimal a-anisotropic filter (solid line), H2 -filter (dash-and-dot line), and H∞ -filter (dashed line). The components of the gain Kk tend to stationary values in virtue of stationarity considered system. For each time instant, the values of Kk repeat relations (10) for the norm, that is, the value of Kk for the H2 -filter is smaller than the values for the a-anisotropic filter which in turn are smaller than those of Kk for the H∞ -filter. Therefore, the behavior of the sequence of gain Kk of the suboptimal a-anisotropic filter characterizes the properties of the a-anisotropic filter that are in-between the H2 -filter and H∞ -filter. AUTOMATION AND REMOTE CONTROL

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7. CONCLUSIONS The problem of stochastic robust filtering in a finite horizon was solved for the linear discrete time-varying system with the observed and estimated outputs having inaccurately known probabilistic distribution of the input disturbance. The estimation error is characterized numerically by the anisotropic norm. The problem of determining the suboptimal a-anisotropic estimator comes to the problem of convex optimization. The criterion for boundedness of the anisotropic norm of the discrete linear time-varying system was formulated and proved in terms of the matrix inequalities. Consideration was given to the sufficient conditions for boundedness of the anisotropic norm in two limiting cases (a = 0 and a → ∞) of the anisotropy levels of the input disturbance. The sufficient existence conditions for the suboptimal anisotropic estimator constrained by the given threshold value of the anisotropic norm of the operator from the input disturbance to the estimation error were established. An algorithm to seek a suboptimal anisotropic estimator based on the recurrent solution of a system of matrix inequalities was presented. A numerical example illustrated the design of a suboptimal a-anisotropic filter. The suboptimal H2 -filter and H∞ -filter were compared with the suboptimal a-anisotropic filter. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 14-0800069. The authors should like to thank M.M. Tchaikovsky for careful reading of the manuscript and helpful remarks. APPENDIX Proof of Theorem 2. We prove that if a special-form inequality (19) is satisfied for the matrix Sk of the form (17), then existence of the solution of the difference Riccati equation (18) follows for each k = 0, . . . , N from the conditions for existence of the solutions of inequalities (28)–(30). Let the system of inequalities (28)–(30) be solvable for each k = 0, . . . , N relative to the real matrices Pk+1 and Ψk under the given level of anisotropy a > 0, threshold of boundedness of the norm γ > 0, and scalar variable μ from the interval (31). Consider the first inequality. Case of k = 0. Application of the Schur lemma to the first inequality (32) provides inequality (21). Case of k > 0. For all k > 0, execute such transformation for the linear matrix difference inequality (28). Multiply the inequality from left and right by a nondegenerate matrix like blockdiag(I, I, Pk−1 , I)  0 and obtain ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

−Pk+1

0

Ak

Bk

0

−μ2 Ir

Ck

Dk ⎥

AT k

CkT

−Pk−1

0

BkT

DkT

0

−Im

⎥ ⎥ ≺ 0. ⎥ ⎦

(A.1)

Triple application of the Schur lemma to inequality (A.1) provides inequality (20). For a given μ, it follows from the existence condition for solution of inequality (20) that there is a real matrix Qk = QT k  0 such that there exists solution Pk of the Riccati difference equation T Pk+1 = Ak Pk AT k + Bk Bk + Q k

+ (Ak Pk CkT + Bk DkT )(μ2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T

(A.2)

with the initial condition P0 = P0 = 0. AUTOMATION AND REMOTE CONTROL

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Since Qk = QT k  0 is satisfied for each k, it follows from the monotonicity condition for solution of the Riccati difference discrete equation [34] that Pk  Pk  0, where Pk is the solution of the Riccati equation (18). Consider the second inequality. Case of k = 0. Application of the Schur lemma to the second inequality (32) provides inequality (26). Case of k > 0. Apply such transformation to the linear matrix inequality (29) and multiply it from left and right by a nondegenerate matrix like blockdiag(I, Pk−1 , I)  0 to obtain ⎡ ⎢ ⎢ ⎣

Ψ k − μ 2 Ir

Ck

CkT

−Pk−1

DkT

0

Dk

⎤ ⎥

0 ⎥ ⎦ ≺ 0.

(A.3)

−Ir

Double application of the Schur lemma to the inequality (A.3) reduces it to inequality (25) relative to the auxiliary matrix variable Ψk . Consider the third inequality (30) bounding the determinant of the matrix Ψk . Transform the right side of the inequality 2a

er

⎧ ⎨ ⎩



μ2 1 −

γ2

 m ⎫N +1 r ⎬

μ2

⎭

⎧ ⎨

=

⎩



e2a μ2r(N +1) 1 −

γ2

m(N +1) ⎫ 1r ⎬ ⎭

μ2

.

(A.4)

Exponentiate the right and left sides of inequality (30) to the power r and obtain N 



det Ψk > e2a μ2r(N +1)

k=0

γ2 1− 2 μ

m(N +1)

.

(A.5)

By taking logarithm of both sides of inequality (A.5), we obtain N 



ln det Ψk > 2a + ln{μ

2r(N +1)

k=0



γ2 } + m(N + 1) ln 1 − 2 . μ

(A.6)

Transfer the addend ln{μ2r(N +1) } to the left side of the inequality and transform this side. Then, the inequality comes to (27) . Since inequality (27) is satisfied for the auxiliary matrix variable Ψk , it is all the more so is satisfied for inequality (24) with the matrix μ2 Ir − Ck Pk CkT − Dk DkT  Ψk . It follows from the condition Pk  Pk  0 that μ2 Ir − Ck Pk CkT − Dk DkT μ2 Ir − Ck Pk CkT − Dk DkT .

(A.7)

Since inequality (24) is satisfied with the matrix μ2 Ir − Ck Pk CkT −Dk DkT  Ψk , all the more so inequality (19) is satisfied with the matrix μ2 Ir − Ck Pk CkT − Dk DkT . Therefore, existence of the solution of the difference Riccati equation (18) follows from the existence of solutions of inequalities (28)–(30), provided that the special-form inequality (19) is satisfied for the matrix (17). Since the Riccati equation (18) and the special-form inequality (19) are related by equivalent transformations with the Riccati equation (12)–(14) and inequality (15), the condition for boundedness of the anisotropic norm formulated in Theorem 1 is satisfied in the strict sense, which is what we set out to prove. AUTOMATION AND REMOTE CONTROL

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Proof of Lemma 1. First, we prove that from the condition for boundedness of the a-anisotropic norm of system F over the interval [0, N ] defined by γ > 0 for a = 0 follows boundedness with the same number γ > 0 of the scaled system H2 -norm, that is, m(N1+1) F 2 < γ. Then, we demonstrate how in this case the matrix inequalities (28)–(30) are modified. Inequality (30) for a = 0 goes over to N 

⎧ ⎨

1 r

(det Ψk ) >

k=0



γ2 1− 2 μ

μ2

⎩

 m ⎫N +1 r ⎬

,

⎭

(A.8)

and after extraction of the root of (N + 1)st degree it is transformed into N 

(det Ψk )

1 r(N+1)



>μ

γ2 1− 2 μ

2

k=0

m r

.

(A.9)

We use the property of the determinant of block matrices, namely that N 

det Σ =

det Ψk ,

(A.10)

k=0

where ⎡ ⎢

Ψ0

0 ..

Σ=⎣

.

∗

⎤ ⎥ ⎦

(A.11)

ΨN

is the r × (N + 1) block matrix with r × r blocks Ψk = ΨT k  0 on the diagonal. Then, inequality (A.9) is representable as (det Σ)

1 r(N+1)



>μ

2

γ2 1− 2 μ

m r

.

(A.12)

Now we make use of the inequality between the geometric average and the arithmetic average for the eigenvalues of the matrix Σ, that is, 1

(det Σ) r(N+1) 

1 trace Σ. r(N + 1)

Then, (A.12) is replaced by the inequality 

γ2 1 trace Σ > μ2 1 − 2 r(N + 1) μ

m r

,

(A.13)

and with regard for Ψk ≺ μ2 Ir − Ck Pk CkT − Dk DkT all the more so it is satisfied for N 



trace (μ Ir − 2

k=0

Ck Pk CkT

−

Dk DkT )

> r(N + 1)μ

2

γ2 1− 2 μ

m r

.

By reducing the last inequality to N 



γ2 trace(Ck Pk CkT + Dk DkT ) > μ2 r(N + 1) 1 − 2 μ2 r(N + 1) − μ k=0 AUTOMATION AND REMOTE CONTROL

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we finally establish that ⎧ ⎨

N 



γ2 trace(Ck Pk CkT + Dk DkT ) < r(N + 1)μ2 1 − 1 − 2 ⎩ μ k=0

m ⎫ r ⎬ ⎭

,

r  m.

The left side of the inequality defines the square of the H2 –norm of system F over the interval [0, N ]. And since the inequality is satisfied for any r  m, and the maximum value of the right side is attained for r = m, we obtain for r = m that N  k=0

trace (Ck Pk CkT + Dk DkT ) < m(N + 1) γ 2 .

(A.14)

Let us see how inequality (28) is transformed. In virtue of the Schur lemma, the difference matrix linear inequality (28) is equivalent to the difference Riccati inequality T Ak Pk AT k − Pk+1 + Bk Bk

≺ (Ak Pk CkT + Bk DkT )(Ck Pk CkT + Dk DkT − μ2 Ir )−1 (Ak Pk CkT + Bk DkT )T ,

(A.15)

and the linear matrix inequality (29) is transformed into Ck Pk CkT + Dk DkT − μ2 Ir ≺ 0.

(A.16)

Existence of a solution of the difference Riccati inequality (A.15) with condition (A.16) ensures existence of the solution of the Lyapunov difference inequality with the matrix Pk = PkT  0 T Ak Pk AT k − Pk+1 + Bk Bk ≺ 0.

(A.17)

The matrices Pk = PkT  0 satisfying inequalities (A.17) and (A.14) enable satisfaction of the condition 1 F 2 < γ, m(N + 1)

"

(A.18)

which implies that the scaled H2 -norm of the system F is bounded over the interval [0, N ] by the given threshold value γ. We apply the Schur lemma to the Lyapunov difference inequality (A.17) and then carry out a similar transformation like blockdiag(I, Pk , I)  0. As the result, we get the matrix inequalities (33). Consequently, the matrix inequality (28) is modified into inequality (33). Inequality (29) becomes ineffective because it is the condition for existence of the Riccati inequality (A.15). Let us consider a modification of inequality (30) which is reduced to the inequality (A.14) on the H2 –norm of system F which is satisfied for sure if at each step trace (Ck Pk CkT + Dk DkT ) < m γ 2 .

(A.19)

Introduce an auxiliary matrix variable Φ = ΦT  0 satisfying the inequality Ck Pk CkT + Dk DkT ≺ Φ.

(A.20)

By applying twice the Schur lemma to inequality (A.20) and executing a homothetic dilatation like blockdiag(I, Pk , I)  0, we obtain the matrix linear inequality (34). Inequality (35) on the trace AUTOMATION AND REMOTE CONTROL

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of the auxiliary matrix variable Φ together with (34) ensures satisfaction of condition (A.18) for boundedness of the H2 -norm of system F . At the first step for k = 0, inequalities (A.17) and (A.20) go over to −P1 + B0 B0T ≺ 0, −Φ0 + D0 D0T ≺ 0. Application of the Schur lemma reduces them to the matrix linear inequalities (36), which is what we set out to prove. Proof of Lemma 2. Consider the case of a → ∞ which is possible if and only if F ∞  γ. −2a

Then, it follows from the localization inequality (31) of the variable μ that e m(N+1) → 0 and the right boundary of the interval tends to γ. The left boundary of interval (31) is equal to γ for any level of anisotropy a > 0. Therefore, the left and right boundaries of the interval of μ for a → ∞ tend to the same value of γ, that is, μ → γ and the interval “contracts” to a point. At that, with regard for the localization interval (31) 1−

lim

a→∞

e

1−

γ2 μ2

−2a m(N+1)

= lim



γ2 γ2

1−e

−2a m(N+1)

= 1.

−2a

a→∞



(A.21)

e m(N+1)

Inequality (30) placing lower limit to the product of the determinants of the matrices Ψk is transformed into N 

1 r

⎧ ⎨

(det Ψk ) >

k=0

⎩



e2a μ2r

γ2 1− 2 μ

m N +1 ⎫ 1r ⎬ ⎭

.

(A.22)

By exponentiating the right and left sides of inequality (A.22) to the power r, after conversion we obtain that μ

−2r(N +1)

N 



det Ψk >

e

2a m(N+1)



k=0

γ2 1− 2 μ

 m(N +1)

.

(A.23)

With regard for (A.21), inequality (A.23) goes over to μ−2r(N +1)

N 

det Ψk > 1.

(A.24)

k=0

Transform (A.24) to N 

det (μ−2 Ψk ) > 1.

(A.25)

k=0

Then, with regard for (25) and (22) inequality (A.25) takes form of the condition N 

ln det Sk > 0

(A.26)

k=0

which is satisfied for any symmetrical positive definite matrix Sk = μ−2 (μ2 Ir − Ck Pk CkT − Dk DkT ). Therefore, for a → ∞ inequalities (29) and (30) become ineffective and are eliminated from the solution of the problem. AUTOMATION AND REMOTE CONTROL

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It follows from the relations for norms (11) of the system F that for a → ∞ lima→+∞ |F |a = F ∞ . Consequently, for a → ∞ the variable μ → γ and inequality (28) go over to (37) which, therefore, is the criterion for boundedness of the H∞ -norm of the discrete linear time-varying system in terms of the difference matrix linear inequalities. By means of the nondegenerate matrix like blockdiag(I, I, Pk−1 , I)  0, we carry out homothetic dilatation for inequality (37), and then apply the Schur lemma to the resulting inequality with the result that obtained is the Riccati inequality T T T Pk+1  Ak Pk AT k + Bk Bk + (Ak Pk Ck + Bk Dk )

× (γ 2 Ir − Ck Pk CkT − Dk DkT )−1 (Ak Pk CkT + Bk DkT )T ,

k ∈ [0, N ],

(A.27)

with the initial condition P0 = 0. At the first step, for k = 0 inequality (A.27) goes over to P1  B0 B0T + B0 D0T (γ 2 Ir − D0 D0T )−1 D0 B0T .

(A.28)

Double application of the Schur lemma results in (38). Therefore, for a → ∞ inequalities (28)–(30) come to inequality (37) which is the criterion for boundedness of the H∞ -norm of system F on the interval [0, N ] defined by the threshold value of γ in terms of the matrix inequalities, which proves the lemma. REFERENCES 1. Kalman, R., A New Approach to Linear Filtering and Prediction Problems, Trans. ASME. J. Basic Eng., 1960, vol. 82D, pp. 34–45. 2. Anderson, B.D.O. and Moore, J.B., Optimal Filtering, Englewood Cliffs: Prentice Hall, 1979. 3. Hassibi, B., Sayed, A., and Kailath, T., Indefinite Quadratic Estimation and Control: A Unified Approach to H2 and H∞ Theories, Philadelphia: SIAM, 1999. 4. Doyle, J.C., Guaranteed Margins for LQG Regulators, IEEE Trans. Automat. Control, 1978, vol. AC-23, no. 4, pp. 756–757. 5. Simon, D., Optimal State Estimation: Kalman, Infinity, and Nonlinear Approaches, New Jersey: Wiley, 2006. 6. Nagpal, K.M. and Khargonekar, P.P., Filtering and Smoothing in an H∞ -setting, IEEE Trans. Automat. Control, 1991, vol. 36, no. 2, pp. 152–166. 7. Grimble, M.J. and Elsayed, A., Solution of the H∞ Optimal Linear Filtering Problem for Discrete-time Systems, IEEE Trans. Acoust. Speech Signal Proc., 1990, vol. 38, pp. 1092–1104. 8. Shaked, U. and Theodor, Y., H∞ -optimal Estimation: A Tutorial, in Proc. 31 IEEE CDC, 1992, pp. 2278–2286. 9. Theodor, Y., Shaked, U., and de Souza, C.E., A Game Theory Approach to Robust Discrete-time H∞ Estimation, IEEE Trans. Signal Proc., 1994, vol. 42, pp. 1486–1495. 10. Zhou, K., Glover, K., Bodenheimer, B., and Doyle, J., Mixed H2 and H∞ Performance Objectives I: Robust Performance Analysis, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1564–1574. 11. Doyle, J., Zhou, K., Gover, K., and Bodenheimer, B., Mixed H2 and H∞ Performance Objectives II: Optimal Control, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1575–1587. 12. Limebeer, D.J.N., Anderson, B.D.O., and Hendel, B., Mixed H2 /H∞ Filtering by the Theory of Nash Games, in Proc. Workshop on Robust Control, Tokyo, Japan, 1992, pp. 9–15. 13. Haddad, W.M., Bernstein, D.S., and Mustafa, D., Mixed-norm H2 /H∞ Regulation and Estimation: The Discrete-time Case, Syst. Control Lett., 1991, vol. 16, no. 4, pp. 235–247. AUTOMATION AND REMOTE CONTROL

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This paper was recommended for publication by O.A. Stepanov, a member of the Editorial Board AUTOMATION AND REMOTE CONTROL

Vol. 77

No. 1

2016