Optimal discrete-time H /γ 0 filtering and control under unknown covariances

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1 International Journal of Control ISSN: (Print) (Online) Journal homepage: Optimal discrete-time H filtering and control under unknown covariances Mark M. Kogan To cite this article: Mark M. Kogan (2015): Optimal discrete-time H filtering and control under unknown covariances, International Journal of Control, DOI: / To link to this article: Accepted author version posted online: 10 Sep Published online: 06 Oct Submit your article to this journal Article views: 24 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [Professor Mark M. Kogan] Date: 08 December 2015, At: 07:55

2 INTERNATIONAL JOURNAL OF CONTROL, Optimal discrete-time H filtering and control under unknown covariances Mark M. Kogan Department of Mathematics, Architecture and Civil Engineering State University, Nizhny Novgorod, Russia ABSTRACT New stochastic γ 0 and mixed H filtering and control problems for discrete-time systems under completely unknown covariances are introduced and solved. The performance measure γ 0 is the worst-case steady-state averaged variance of the error signal in response to the stationary Gaussian white zero-mean disturbance with unknown covariance and identity variance. The performance measure H is the worst-case power norm of the error signal in response to two input disturbances in different channels, one of which is the deterministic signal with a bounded energy and the other is the stationary Gaussian white zero-mean signal with a bounded variance provided the weighting sum of disturbance powers equals one. In this framework, it is possible to consider at the same time both deterministic and stochastic disturbances highlighting their mutual effects. Our main results provide the complete characterisations of the above performance measures in terms of linear matrix inequalities and therefore both the γ 0 and H optimal filters and controllers can be computed by convex programming. H optimal solution is shown to be actually a trade-off between optimal solutions to the H and γ 0 problems for the corresponding channels. 1. Introduction State estimation and control in stochastic formulations have been extensively investigated and have had many practical applications over the past decades. Among existing approaches, H 2 and H theories are two effective tools. If it is known that the disturbance is zero mean white noise with identity covariance or its power spectral densitymatrixisknownandrational,thentodesignthe standard Wiener Kalman filter or linear quadratic Gaussian (LQG) optimal controller, which minimises the variance of the estimation or control error, one should minimise the H 2 norm of the transfer matrix from the white noiseinputtotheerror.whenaprioristatisticalinformation on the external noise signals is unknown, then it is argued in Zhou, Glover, Bodenheimer, and Doyle (1994) that H norm of the above transfer matrix provides the worst-case variance of the error normalised by the variance of the noise input over all (wide-sense) stationary processeswithboundedvariance.asisalsowellknown, the H 2 and H filtering and control theories can also be motivated from a deterministic point of view. In particular, the H norm provides the smallest upper bound on the ratio between the L 2 norms of the error and the deterministic noise input. It is common to use the extra information provided by the stochastic model to introduce stochastic performance ARTICLE HISTORY Received 26 January 2015 Accepted 1 September 2015 KEYWORDS Discrete-time systems; mixed stochastic and deterministic disturbances; unknown covariances; performance measure γ 0 ; H filtering; H control measures within a robust control point of view. This originates multiobjective optimisation problems which have been actively studied in the literature (Bernstein & Haddad, 1989; Chen& Zhou, 2001; Doyle, Zhou, Glover, & Bodenheimer, 1994; Haddad, Bernstein, & Mustafa, 1991; Kaminer,Khargonekar,&Rotea,1993; Muradore & Picci, 2005; Yaesh&Shaked,1992). H 2 /H design enables development of filters and controllers that are robust with respect to uncertainty while minimising a bound on H 2 performance. The optimal linear filters and controllers have been widely used in many areas of applications. To implement them, complete specifications of statistical model parameters must be given. However, the selection of the covariance matrices of the measurement noise and disturbance process is often based on physical intuition and common sensesincetheyareeitherunknownorinexactlyknown in most practical cases. Several special cases of the minimax problem have been considered and solved for different model settings and different feasible sets of the statistics. Poor and Looze (1981) supposed that covariance matrices are contained in the convex and compact subsets of positive definite matrices having maximal elements and established that the minimax state estimator is given by the Kalman filter corresponding to these maximal elements. Calafiore and El Ghaoui (2001) derived a CONTACT Mark M. Kogan 2015 Taylor & Francis mkogan@nngasu.ru

3 2 M.M. KOGAN linear estimator that minimises a worst-case (with respect to the uncertainty in the noise model) measure of a posteriori parameter covariance matrix. In the paper (Bitar, Baeyens, Packard, & Poola 2010), it was shown that the estimation and filtering problems can be reformulated as semidefinite programmes when the measurement noise covariance matrix is unknown but confined to a convex polyhedral set with known vertices. In the recent paper (Kogan, 2014), the estimation problem under completely unknown covariances of random factors has been studied. The performance measure for a linear unbiased estimate has been defined as the worst-case variance of the estimation error normalised by the sum of variances of all random factors. It was shown that this performance measure is equal tothespectralnormofthe transfermatrix andtherefore the minimax estimate under unknown covariances of random factors can be computed in terms of linear matrix inequalities (LMIs). In the present paper, this approach is applied to the filtering and control problems under completely unknown covariances of random factors. We shall be interested in two classes of new filtering or control problems, which we shall refer to as the stochastic γ 0 problem and the mixed (deterministic and stochastic) H problem. The performance measure γ 0 is defined as the worst-case steady-state averaged variance of the error signal in response to the stationary Gaussian white zero-mean disturbance with identity variance. The worst-case is taken over all unknown nonzero covariance matrices of the stochastic disturbance. The performance measure H is defined as the worst-case power norm of the error signal in response to two input disturbances in different channels, one of which is the deterministic signal with a bounded energy and the other is the stationary Gaussian white zero-mean signal with a bounded variance provided the weighting sum of powers for input disturbances is equal to one. Our main results provide the complete characterisations of the above performance measures in terms of LMIs. Hence, both the γ 0 and H optimal filters and controllers minimising the appropriate criterion can be computed by convex programming. An illustrative example demonstrates that the H optimal filter is actually a trade-off between the appropriate H and γ 0 optimal filters. The structure of this paper is as follows. The performance measure γ 0 and its properties are presented in Section 2. Thedesignoftheγ 0 optimal filters and controllers in terms of LMIs is proposed in Section 3. The performance measure H and its characterisations are described in Section 4,whilethedesignoftheH optimal filters and controllers is presented in Section 5. An example is provided in Section 6 to illustrate our main results. A conclusion is finally drawn in Section Attenuation level γ 0 of stochastic signals Consider the discrete-time linear dynamic system described by the state equations x(t + 1) = Ax(t) + Bw(t), x(0) = x 0, z(t) = Cx(t) + Dw(t), where x(t) R n x is the state, z(t) R n z is the controlled output, A is the stable matrix, i.e., all its eigenvalues are strictly inside the unit circle of the complex plane, w(t) R n w is a stationary Gaussian white zero-mean random sequence of vectors with an unknown covariance matrix Ew(t)w T (t) K w,andx 0 is the random zero-mean initial state with an unknown covariance matrix Ex 0 x0 T = K 0.It is assumed that the initial state is uncorrelated with w(t) for all t. We define the attenuation level of stochastic signals for the system (1) as the worst-case steady-state averaged variance of the controlled output z(t)normalisedby theaveragedvarianceoftheinputw(t) over all nonzero covariance matrices K w, i.e., γ0 2 = sup J z, K w 0 J w J z = lim J w = lim (1) E z(t) 2, (2) E w(t) 2 = tr K w. To compute γ 0, we need to recall some well-known facts. Lemma 2.1: Given covariance matrix K w, the steady-state averaged variance of the controlled output in (1) is determined by the formulae J z = lim E z(t) 2 = tr (CQ 0 C T + DK w D T ) t = tr [K w (B T P 0 B + D T D)], where Q 0 = Q T 0 and P 0 = P T 0 0 is a solution to the equation AQA T Q + BK w B T = 0 (3) 0 is a solution to the equation A T PA P + C T C = 0. (4) Note also that if the system initial condition is x(0) = 0and,attimet = 0, we apply an arbitrary zero

4 INTERNATIONAL JOURNAL OF CONTROL 3 mean (wide-sense) stationary stochastic process with autocorrelation matrix R(τ) = Ew(τ)w(0) T and power spectral density matrix S(θ ) = i= z i R(i), z = e jθ, 0 θ < 2π, j = 1, then the controlled output approaches a zero mean stationary stochastic process whose the steady-state variance is J z = tr { 1 2π } H(e jθ )S(θ )H T (e jθ ) dθ, (5) 2π 0 where H(z) = C(zI A) 1 B + D is the transfer matrix of thesystem(1). Now we will show how to determine the attenuation level γ 0 through both the state space presentation of the system and its transfer matrix. First of all, a technical lemma will be stated. Lemma 2.2: Given matrix = T, the inequality tr ( K w ) 0 holds for all K w = Kw T 0 if and only if 0. Proof: Symmetric matrix can be presented as = nw i=1 λ ie i e T i,whereλ i and e i are eigenvalues and eigenvectors of,andvectorse i, i = 1,, n w form orthonormalised basis. Suppose that there exists λ j 0. Then, choosing K w = λ j e j e T j implies that tr (K w ) = λ 2 j 0, i.e. λ j = 0. Therefore, 0. Let 0 and the covariance matrix be presented as K w = μ i f i fi T,whereμ i 0andf i are eigenvalues and eigenvectors of K w.then,tr( K w ) = μ i fi T f i 0. Theorem 2.1: The performance measure γ 0 of the system (1) is equal to { 1 2π } γ0 2 = λ max H T (e jθ )H(e jθ ) dθ 2π 0 = λ max (B T P 0 B + D T D), (6) where P 0 = P0 T 0 is a solution to the Lyapunov Equation (4). The worst-case covariance matrix is equal to K = e max e T max,wheree max is the eigenvector associated with the maximum eigenvalue γ0 2 of the matrix BT P 0 B + D T D. Proof: We need to find the minimal value of γ 2 such that J z γ 2 tr K w K w 0. In view of Lemma 2.1, this condition may be rewritten as follows: tr [K w (B T P 0 B + D T D γ 2 I)] 0 K w 0. InviewofLemma2.2,weobtainthatγ0 2 = λ max(b T P 0 B + D T D).Moreover,forK = e max e T max we get tr [K (B T P 0 B + D T D γ0 2 I)] = 0. Since w(t) is a white noise process with an unknown covariance matrix K w and the power spectral density matrix S(θ) = K w, it follows from the definition and (5) that γ 2 0 is the minimal value of γ 2 such that {[ 1 2π J z γ 2 J w = tr H T (e jθ )H(e jθ ) dθ 2π 0 γ 2 I ] } K w 0 Kw 0. By applying Lemma 2.2, we arrive at the required result. Further, it follows from the monotonicity of the solution to the Lyapunov equation with respect to an additional positive semidefinite term on the right-hand side of the equation that the solution to the equation can be obtained with a given tolerance by solving the appropriatelmi.byapplyingtheschurlemma,wearriveatthe following propositions. Corollary 2.1: The performance measure γ 0 of system (1) can be computed in solving the problem min (7) γ 2 = γ0 2 subject to LMIs X XA 0 X C T 0, X XB 0 γ 2 I D T 0 (7) with respect to variables X = X T > 0, γ 2 > 0. Corollary 2.2: Given covariance matrix K w,thesteadystateaveragedvarianceofthecontrolledoutputcanbecomputed in solving the problem min (8) μ 2 subject to LMIs Y YA T 0 Y BK w 0, K w Y YC T 0 W DK w 0, trw μ 2 (8) K w with respect to variables Y = Y T > 0,Wandμ 2 > 0,orthe problem min (9) μ 2 subject to LMIs X XA 0 X C T 0, X XB 0 W D T 0, tr (K w W ) μ 2 (9) with respect to variables X = X T > 0,Wandμ 2 > 0.

5 4 M.M. KOGAN Remark 2.1: Since γ 2 0 = sup tr K w 1 J z and J z tr (CYC T + DK w D T ), where Y = Y T > 0isasolutiontoinequality AYA T Y + BK w B T 0, the worst-case covariance matrix K canalsobefoundasasolutiontotheproblem max (10) μ2 subject to LMIs AYA T Y + BK w B T 0, tr K w 1, tr (CYC T + DK w D T ) μ 2 (10) with respect to variables Y = Y T > 0, μ 2 > 0andK w = K T w 0. It is interesting to compare γ 0 with H 2 and H norms of the system transfer matrix. From (5) and standard properties of the H norm, it is easy to see that J z H 2 J w, H = sup H(e jθ ), θ [0,2π) where denotes the largest singular value. It is well known (see Zhou et al., 1994) that H 2 provides the smallest upper bound such that this inequality holds for arbitrary stationary processes with bounded variance, i.e., J z sup = H 2 S(θ ) 0 J. w Moreover, it is also well known that if w(t)isazeromean white process with S(θ) = I, i.e., K w = I and tr K w = n w, thenthesteady-statevarianceoftheoutputisequaltothe squared H 2 norm, where H 2 2 = tr { 1 2π H T (e jθ )H(e jθ ) dθ}. (11) 2π 0 Thus, we conclude that γ 0 H. Finally, comparing (6) and (11) results in 1 nw H 2 γ 0 H 2 and if n w = 1, then γ 0 = H 2. Thus, in contrast with the H 2 norm, which is equal to the steady-state variance of the controlled output in response to the white noise process with the identity covariance, the performance measure γ 0 corresponds to theworst-casesteady-statevarianceofthecontrolledoutput over all possible white noise processes with covariance matrices whose trace does not exceed one. 3. γ 0 -optimal stochastic filtering and control In this section, we consider filtering and control problems in which covariance matrices of stochastic disturbances and noises are completely unknown and synthesise the optimal filter and controller with respect to the performance measure γ γ 0 -optimal filtering Let a discrete-time linear dynamic system be described by the equations x(t + 1) = Ax(t) + B 1 w(t), x(0) = x 0, y(t) = C 2 x(t) + D 2 w(t), (12) z(t) = C 1 x(t), where x(t) R n x is the state, y(t) R n y is the measured output, z(t) R n z is the controlled output, w(t) R n v is a stationary Gaussian white zero-mean random sequence of vectors containing both disturbance process and measurement noise with unknown covariance Ew(t)w T (t) K w, x 0 is the random zero-mean initial state. It is assumed that the initial state is uncorrelated with w(t)forallt.note that when the disturbance process and measurement noisedonotcorrelatewitheachother,oneshouldsetw(t) = col (w 1 (t), w 2 (t)). In this case, matrices B 1, D 2 and K w will be of the form (B 11 0), (0 D 22 )anddiag(k w1, K w2 ), respectively. To estimate the state and the controlled output, consider the filter (one-step predictor) described by the equations x f (t + 1) = Ax f (t) + L[y(t) C 2 x f (t)], z f (t) = C 1 x f (t), x f (0) = 0, (13) where x f (t) R n x isthefilterstateandl is the gain matrix. This leads to the following difference equation for estimation errors e(t) = x(t) x f (t) and e z (t) = z(t) z f (t): where e(t + 1) = A c e(t) + B w w(t), e(0) = x 0 e z (t) = C 1 e(t), (14) A c = A LC 2, B w = B 1 LD 2. (15) Let γ 0 (L) denote the attenuation level of stochastic signals w(t) for system (14) with respect to the estimation error e z (t). We define γ 0 -optimal filter under which γ 0 (L)

6 INTERNATIONAL JOURNAL OF CONTROL 5 is minimal, i.e., min L γ 0 (L) = γ 0 (L 0 ). This filtering problem can be regarded as a game between an observer and the stochastic disturbance with the cost function to achieve the goal 0 (L; K w ) = J ez γ 2 J w min max 0 (L; K w ) 0 L K w for a least value of γ>0. By substituting the matrices A c and B c into LMIs (7) and introducing the new variable Z = XL, itiseasyto check that by Corollary 2.1 the γ 0 -optimal filter is characterised as follows. Theorem 3.1: The gain matrix of the γ 0 -optimal filter for system (12) is computed as L 0 = X 1 Z, where X = X T > 0 and Z are solutions to the problem min (16) γ 2 = γ0 2(L 0) subject to LMIs X XA ZC 2 0 X C1 T 0, X XB1 ZD 2 γ 2 0. (16) I Remark 3.1: In view of Remark 2.1, it is easy to see that theworst-casecovariancematrixforfilter(13)witha givengainmatrixl canbefoundasasolutiontotheproblem max (17) μ2 subject to LMIs (A LC 2 )Y (A LC 2 ) T Y + (B 1 LD 2 )K w (B 1 LD 2 ) T 0, tr K w 1, tr (C 1 YC1 T ) μ2 (17) with respect to variables Y = Y T > 0, μ 2 > 0andK w = K T w 0. Note that due to Corollary 2.2, the gain matrix L K of thesteady-statekalmanfilter,whichminimises J ez = lim E e z (t) 2 provided that disturbance w(t) has a given covariance matrix K w,canbecomputedasl K = X 1 Z,whereX = X T > 0andZ aresolutionstotheproblemmin (18) μ 2 subject to LMIs X XA ZC 2 0 X C1 T 0, X XB1 ZD 2 0, tr (K W w W ) μ 2. (18) 3.2 γ 0 -optimal control Let a discrete-time linear dynamic system be described by the equations x(t + 1) = Ax(t) + B 1 w(t) + B u u(t), z(t) = C 1 x(t) + D u u(t), x(0) = x 0, (19) where x(t) R n x is the state, u(t) R n u is the control input, z(t) R n z is the controlled output, w(t) R n v is a stationary Gaussian white zero-mean random sequence of vectors with unknown covariance Ew(t)w T (t) K w and x 0 is the random zero-mean initial state. It is assumed that the initial state is uncorrelated with w(t)forallt. The γ 0 -optimal control problem is to determine a linear state-feedback u(t) = 0 x(t) such that the closed-loop system is stable and its attenuation level of stochastic signals w(t) is minimal, i.e., min γ 0( ) = γ 0 ( 0 ). By substituting the matrix of the closed-loop system into (7), multiplying the first and second inequalities by diag (X 1, X 1, I) anddiag(x 1, I, I), respectively, from the left and from the right, introducing the new variables Y = X 1 and Z = Y, itcanbeshownthatthelmisin Corollary 2.1 result in the following characterisation of the γ 0 -optimal controller. Theorem 3.2: The gain matrix of the γ 0 -optimal controller for system (19) is computed as 0 = ZY 1,whereY = Y T > 0 and Z are solutions to the problem min (20) γ 2 = γ 2 0 ( 0) subject to LMIs Y AY + B u Z 0 Y YC1 T + ZT D T u 0, Y B1 γ 2 0. (20) I

7 6 M.M. KOGAN Remark 3.2: Note that due to Corollary 2.2, the gain matrix K of the LQG controller, which minimises J z = lim E z(t) 2 under disturbances with a given covariance matrix K w, can be computed as K = ZY 1,whereY = Y T > 0andZ aresolutionstotheproblemmin (21) μ 2 subject to LMIs Y YA T + Z T B T u 0 Y B 1 K w 0, Y YC T 1 + Z T D T u 0, trw μ 2. W (21) 4. Attenuation level γ,0 of deterministic and stochastic signals In this section, we consider at the same time both stochastic and deterministic signals highlighting their mutual effects. Consider an internally stable system described by the equations x(t + 1) = Ax(t) + B w w(t) + B v v(t), x(0) = x 0 z(t) = Cx(t) + D w w(t) + D v v(t), (22) whereallthenotationsarethesameasin(1)plusv(t) is the deterministic signal. The transfer matrix of this systemfromcol(w, v)toz is denoted as follows: Let now H(z) = (H w (z) H v (z)), H w (z) = C(zI A) 1 B w + D w, H v (z) = C(zI A) 1 B v + D v. s 2 P = lim s(t) 2 be the power norm and P ={s : s 2 P < } the set of signals with bounded power. Note that all signals in l 2 with s 2 2 = s(t) 2 < have zero power, so actually it is a seminorm. Assuming that v P, then the stochastic component of the controlled output z will be a Gaussian stationary random signal, and therefore ergodic. It follows that the sample paths of z will have finite power norm (with probability one). We define the attenuation level of deterministic and stochastic signals for system (22) as the worst-case power norm of the controlled output over all admissible deterministic and stochastic signals with bounded powers, i.e., γ 2,0 = sup v 2 P +r2 w 2 P 0 z 2 P v 2 P + r2 w 2, (23) P where r 2 0 is a weighting coefficient. We refer to this performance measure as the H norm, because if we ignore w(t) then the norm induced on H from v to z is the H norm of the transfer matrix H v (z) (see Doyle et al., 1994), i.e., z P sup = H v 2 v P 0 v = γ 2 (H v ), P whereas if we ignore v(t) andsetr = 1, then J z = z 2 P, J w = w 2 P andthenorminducedonh from w to z is the γ 0 (H w )normofthetransfermatrixh w (z). Hence, when both v and w act on the system, the induced norm γ,0 = γ,0 (H v, H w )willbeamixtureofthesenorms. Note that when v(t) 0andK w = I, z is the output of an asymptotically stable system driven by the white noise with identity intensity and, therefore, the equalities z 2 P = J z = H w 2 2 hold. It immediately follows from the definition that γ,0 (H v, H w ) max{γ (H v ), r 1 γ 0 (H w )}. The following theorem characterises the performance measure γ,0 (H v, H w )intermsofthesolutiontothericcati equation. Theorem 4.1: The H norm of system (22) is less than aprescribedvalueofγ if and only if where X γ = X T γ λ max (B T w X γ B w + D T w D w)<γ 2 r 2, (24) 0 is the solution to the Riccati equation A T XA X + C T C + (B T v XA + DT v C)T (γ 2 I B T v XB v D T v D v ) 1 (B T v XA + DT v C) = 0 (25) such that γ 2 I B T v X γ B v D T v D v > 0 and matrix A v = A + B v (γ 2 I B T v X γ B v D T v D v ) 1 (B T v X γ A + D T v C) (26) is stable. Proof: Let X γ bethesolutionto(25)andv(x) = x T X γ x. Calculating the increment V along any trajectory of the system, taking the P seminorm and completing the

8 INTERNATIONAL JOURNAL OF CONTROL 7 square, we have z 2 P γ 2 v 2 P = lim + D T w D w)w(t) lim w T (t)(b T w X γ B w [v(t) v (t)] T (γ 2 I B T v X γ B v D T v D v )[v(t) v (t)], where v (t) = (γ 2 I B T v X γ B v D T v D v ) 1 (B T v X γ A + D T v C)x(t). Then by Lemma 2.2, condition (24) implies that tr [(B T w X γ B w + D T w D w)k w ] γ 2 r 2 tr K w, and hence γ,0 2 <γ2. Let now γ,0 <γ.sinceγ γ,0,then H v <γ and, hence, it follows from KYP lemma (Green & Limebeer, 1995;Yakubovich,1962) that there exists a stabilising solution X γ = Xγ T > 0 of the Riccati Equation (25). Then, we have z 2 P γ 2 v 2 P = tr [(BT w X γ B w + D T w D w)k w ]. (27) We will show that matrix X γ satisfies inequality (24). Suppose this is not the case, i.e., there exists a 0 such that a T (B T w X γ B w + D T w D w)a γ 2 r 2 a 2. Choose the covariance matrix K w = aa T to obtain z 2 P γ 2 v 2 P = at (B T w X γ B w + D T w D w)a γ 2 r 2 tr K w. This implies γ,0 2 γ 2 which contradicts the assumption. Thus, B T w X γ B w + D T w D w <γ 2 r 2 I and, consequently, λ max (B T w X γ B w + D T w D w)<γ 2 r 2. Corollary 4.1: The H norm of system (22) is less than aprescribedvalueofγ if and only if there exists a matrix X = X T > 0 such that LMIs A T XA X A T XB v C T B T v XB v γ 2 I D T v < 0, B T w XB w + D T w D w < r 2 γ 2 I (28) are feasible. Remark 4.1: It is important to highlight that if X is the solution to the first inequality in (28) at the minimal value of γ 2 equal to γ 2 and BT w X B w + D T w D w <γ 2 r2 I, i.e., r 2 >γ 2(H v )λ max (B T w X B w + D T w D w),thenγ,0 2 = γ 2. This means that the performance measure γ,0is the trade-off between γ (H v )andγ 0 (H w )ifandonlyifthe following inequality holds r 2 r 2 = γ 2 (H v )λ max (B T w X B w + D T w D w). (29) Now, let us consider a question concerning the worst-case disturbances. We will show that, in the case when γ,0 2 (H v, H w )>γ 2 (H v ), the supremum in (23) is attained. Theorem 4.2: Let the inequality (29) hold. Then γ,0 2 (H v, H w ) is attainable under a worst-case disturbance v (t) = (γ 2 I B T v X B v D T v D v ) 1 (B T v X A + D T v C)x(t), (30) where X is the stabilising solution of the Riccati Equation (25) with γ = γ, 0 and x(t) is the solution of the system x(t + 1) = A v x(t) + B w w (t), x(0) = x 0 with matrix A v given in (26) for γ = γ, 0,andw (t) is a worst-case white noise process with covariance matrix K = e max e T max,wheree max is the eigenvector associated with the maximum eigenvalue γ,0 2 r2 of the matrix B T w X B w + D T w D w. Proof: According to Remark 4.1 in the case under consideration,wehaveγ <γ,0 and, hence, there exists X, the stabilising solution of the Riccati Equation (25) for γ = γ,0.thenforthelinearfeedbackdisturbance (30), the equality z 2 P γ 2 v 2 P = tr [(BT w X B w + D T w D w)k w ] holds and hence, z 2 P v 2 P + r2 tr K w = γ 2,0 + tr [K w(b T w X B w + D T w D w γ 2,0 r2 I)] v 2 P + r2 tr K w, where tr [K w (B T w X B w + D T w D w γ 2,0 r2 I)] 0dueto z 2 P v 2 P + r2 tr K w γ 2,0. (31) Now, we will show that there exists K w of the form K w = aa T such that tr [K w (B T w X B w + D T w D w γ,0 2 r2 I)] = a T (B T w X B w + D T w D w γ,0 2 r2 I)a = 0. To that end, suppose that a T (B T w X B w + D T w D w γ,0 2 r2 I)a < 0foralla 0. Then according to Theorem 4.1, the H norm of the system (22) will be less than γ,0, which contradicts to the assumption. This means that for K w = e max e T max, where (B T w X B w + D T w D w)e max = γ,0 2 r2 e max, the inequality (31) becomes the equality. Note that the signal v (t) is not unique. Every signal v (t) + g(t)withg(t) l 2 yields thesamesupremumin(31).

9 8 M.M. KOGAN Corollary 4.2: Suppose that (29) holds. Then H norm of the system (22) can be computed in solving the problem min (32) γ 2 subject to LMIs X XA XB v 0 X 0 C T γ 2 I D T v X XB w 0 γ 2 r 2 I D T w with respect to variables X = X T > 0, γ 2 > 0. 0, 0 (32) Note that from (32), it follows γ 2,0 (H v, H w )> λ max (D T v D v ). Besides, γ 2 (H v ) can be computed to a high accuracy as a minimum value of γ 2 for which the first inequality in (32) is feasible, while γ 2 0 (H w) can be obtained in solving the problem min (32) γ 2,whereone should exclude the third block row and column in the first inequality (32) and set r = 1inthesecondinequality. 5. H -optimal stochastic filtering and control In this section, we consider filtering and control problems in which both deterministic and stochastic disturbances act on the system and synthesise the optimal filter and controller with respect to the performance measure H. 5.1 H -optimal filtering Let a discrete-time linear dynamic system be described by the equations x(t + 1) = Ax(t) + B 1 w(t) + B 2 v(t), x(0) = x 0, y(t) = C 2 x(t) + D 21 w(t) + D 22 v(t), (33) z(t) = C 1 x(t), where x(t) R n x is the state, y(t) R n y is the measured output, z(t) R n z is the controlled output, w(t) R n v is a stationary Gaussian white zero-mean random sequence of vectors containing both disturbance process and measurement noise with unknown covariance Ew(t)w T (t) K w, v(t) P is the deterministic disturbance and x 0 is the random zero-mean initial state. It is assumed that the initial state is uncorrelated with w(t)for all t.we consider afilterdescribedbytheequations(13)andobtainthefollowing equations for estimation errors: e(t + 1) = A c e(t) + B w w(t) + B v v(t), e(0) = x 0 e z (t) = C 1 e(t), (34) where A c = A LC 2, B w = B 1 LD 21, B v = B 2 LD 22. (35) Let γ,0 (L) denote the H norm of the system (34) with respect to the estimation error e z (t). We define H -optimal filter under which γ,0 (L) is minimal, i.e., min L γ,0 (L) = γ,0 (L,0 ). This filtering problem can be regarded as a game between an observer and the joint disturbances with the cost function,0 (L; v, w) = e z 2 P γ 2 ( v 2 P + r2 w P ) to achieve the goal min L max,0(l; v, w) 0 v P,w P foraleastvalueofγ>0. By substituting the matrices A c, B w and B v into LMIs (32) and introducing the new variable Z = XL, itiseasy tocheckthatbycorollary4.2theh -optimal filter is characterised as follows. Theorem 5.1: ThegainmatrixoftheH -optimal filter for system (33) is computed as L, 0 = X 1 Z, where X = X T > 0 and Z are solutions to the problem min (36) γ 2 subject to LMIs X XA ZC 2 XB 2 ZD 22 0 X 0 C1 T γ 2 I 0 < 0, X XB1 ZD 21 γ 2 r 2 < 0. (36) I 5.2 H -optimal control Consider a system be described by the equations x(t + 1) = Ax(t) + B 1 w(t) + B 2 v(t) + B u u(t), z(t) = C 1 x(t) + D u u(t), x(0) = x 0, (37) where u(t) R n u is the control input, z(t) R n z is the controlled output, w(t) R n v is a stationary Gaussianwhitezero-meanrandomsequenceofvectorswith unknown covariance Ew(t)w T (t) K w, v(t) P is the deterministic disturbance and x 0 istherandomzeromeaninitialstate.itisassumedthattheinitialstateis uncorrelated with w(t)forallt.

10 INTERNATIONAL JOURNAL OF CONTROL 9 The H -optimal control problem is to determine a linear state-feedback u(t) =,0 x(t) suchthatthe closed-loop system is stable and its attenuation level of stochastic and deterministic signals is minimal, i.e., min γ,0( ) = γ,0 (,0 ). J z (t) By substituting the matrix of the closed-loop system into (32), multiplying the first and second inequalities by diag (X 1, X 1, I, I)anddiag(X 1, I, I), respectively, from theleftandfromtheright,introducingthenewvariables Y = X 1 and Z = Y, itcanbeshownthatthelmisin Corollary 4.2 result in the following characterisation of the optimal controller. Theorem 5.2: The gain matrix of the H -optimal controller for system (37) is computed as, 0 = ZY 1,where Y = Y T > 0 and Z are solutions to the problem min (38) γ 2 subject to LMIs Y AY + B u Z B 2 0 Y 0 YC T γ 2 I 0 < 0, Y B1 γ 2 r 2 < 0. (38) I 6. Illustrative example To illustrate quantitative ratio between different filters, we choose to consider a system described by the equations ( x(t + 1) = v(t) 2 ) x(t) w(t) y(t) = (1 0)x(t) + (0 0 1)w(t) + v(t). (39) According to Theorem 5.1, we computed the gain matrix of H optimal filter for r 2 = 0.6 as well as, by ignoring one of disturbances, the gain matrices of γ 0 and H optimal filters L T,0 L T 0 L T = ( ), = ( ), = ( ), for which γ 2,0 (L,0) = , γ 2 0 (L 0) = , γ 2 (L ) = Note that H norms of the system (34) with γ 0 and H optimal filters are equal to γ 2,0 (L 0) = and γ 2,0 (L ) = Figure 1. Evolutions of J z (t)forh (solid line), γ 0 (dotted line) and H (dashed line) filters. For comparison, we present on Figure 1 simulations results of the averaged variance J z (t)forthefilteringerrors with these H (solid line), γ 0 (dotted line) and H (dashed line) optimal filters when t 1 J z (t) = (1/t) e z (i) 2, v(t) = e 0.001t cos(πt/10), i= K w = Note that it is not necessarily to have the better result for the H optimal filter under an arbitrary covariance of the stochastic disturbance; however, the squared attenuation level of deterministic and stochastic disturbances for this filter is minimal and is equal to Shown on Figure 2 are the maximal singular value plots for system (34) (without the stochastic disturbance) σ[h(e jθ )] Figure 2. Maximal singular value plots for the system (34) with H (solid line), γ 0 (dotted line) and H (dashed line) filters. t θ

11 10 M.M. KOGAN with these filters. The figure illustrates that H filter (dashed line) provides almost constant amplitude of the error regardless of the input angle frequency. At the intervals 0 <θ<0.5 and 5.7 <θ<6.28, H and γ 0 filters (solid and dotted lines) have lower quality than the H filter, but H filter is better than γ 0 filter. At the interval 0.5 <θ<5.7, picture is just opposite: the best is γ 0 filter, slightly inferior to him H filter, and the worst is H filter. It should be added that γ 0 norms of the systems with these filters are as follows: γ0 2 (L,0) = , γ0 2 (L 0) = , γ0 2 (L ) = This shows that the γ 0 -norm of the system with the H -optimal filter is slightly greater than the minimal value of the γ 0 -norm for the γ 0 -optimal filter. Thus, by choosing properly the value of the weighting coefficient r 2,onecansynthesisetheH -optimal filter that provides a trade-off between the H and γ 0 optimal filters. 7. Conclusion In this paper, we have solved the discrete-time optimal filtering and control problems under both stochastic disturbances with unknown covariances and mixed deterministic and stochastic disturbances with unknown covariances. The synthesis of the optimal filters and controllers is based on introducing new performance measures γ 0 and H.Theperformancemeasureγ 0 is characterised by the worst-case averaged steady-state variance of the error normalised by the variance of the white noise disturbance with an unknown covariance matrix. The performance measure H is characterised by the the worst-case power norm of the error in response to two input disturbances in different channels, oneofwhichisthedeterministicsignalwithabounded energy and the other is the stationary Gaussian white zero-mean signal. It has been shown that the optimal filters and controllers can be designed by solving LMIs. In this framework, it is possible to consider stochastic and deterministic disturbances simultaneously and to analyze their mutual effects. Acknowledgements This work was supported in part by the Russian Foundation for Basic Research under Grants and The paper is prepared within the terms of the research project 3021 financed by the Ministry of Education and Science of the Russian Federation within the basic part of the Research Assignment. Disclosure statement No potential conflict of interest was reported by the author. Funding Russian Foundation for Basic Research [grantnumber ], [grant number ]. References Bernstein, D.S., & Haddad, W.M. (1989). LQG control with an H performance bound: A Riccati equation approach. IEEE Transactions on Automatic Control, 34, Bitar, E., Baeyens, E., Packard, A., & Poola, K. (2010). Linear minimax estimation for random vectors with parametric uncertainty. In Proceedings of American Control Conference (pp ). Baltimore, MD: IEEE. Calafiore, G., & El Ghaoui, L. (2001). Minimum variance estimation with uncertain statistical model. In Proceedings of the 40th IEEE Conference on Decision and Control (pp ). Orlando, FL: IEEE. Chen, X., & Zhou, K. (2001). Multiobjective H 2 /H control design. SIAM Journal on Control and Optimization, 40(2), Doyle, J., Zhou, K., Glover, K., & Bodenheimer, B. (1994). Mixed H 2 and H performance objectives II: Optimal control. IEEE Transactions on Automatic Control, 39, Green, M., & Limebeer, D.J.N. (1995). Linear robust control. Englewood Cliffs, NJ: Prentice Hall. Haddad, W.M., Bernstein, D.S., & Mustafa, D. (1991). Mixednorm H 2 /H regulation and estimation: The discrete-time case. Systems and Control Letters, 16, Kaminer, I, Khargonekar, P.P., & Rotea, M.A. (1993). Mixed H 2 /H control for discrete-time systems via convex optimization. Automatica, 29(1), Kogan, M.M. (2014). LMI based minimax estimation and filtering under unknown covariances. International Journal of Control, 87(6), Muradore, R., & Picci, G. (2005). Mixed H 2 /H control: The discrete-time case. Systems and Control Letters, 54,1 13. Poor, H.V., & Looze, D.P. (1981). Minimax state estimation for linear stochastic systems with noise uncertainty. IEEE Transactions on Automatic Control, 26, Yaesh, I., & Shaked, U. (1992). Game theory approach to state estimation of linear discrete-time processes and its relation to H -optimal estimation. International Journal of Control, 55(6), Yakubovich, V.A. (1962). Solution of certain matrix inequalities in the stability theory of nonlinear control systems. Soviet Mathematics Doklady, 3, Zhou, K., Glover, K., Bodenheimer, B., & Doyle, J. (1994). Mixed H 2 and H performanceobjectivesi:robustperformance analysis. IEEE Transactions on Automatic Control, 39,