Linear Estimation in Krein Spaces

1s IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 1, JANUARY 1996 Linear Estimation in ein Spaces- eory Babak Hassibi, Ali H. Sayed, Member, IEEE, and Thomas Kailath, Fellow, IEEE Abstract- The authors develop a self-contained theory for that while quadratic forms in Hilbert space always have linear estimation in Krein spaces. The derivation is based on minima (or maxima), in Krein spaces one can assert only that simple concepts such as projections and matrix factorizations they will always have stationary points. Further conditions will and leads to an interesting connection between Krein space projection and the recursive computation of the stationary points have to be met for these to be minima or maxima. We explore of certain second-order (or quadratic) forms. The authors use this by first considering the problem of finding a vector k to the innovations process to obtain a general recursive linear stationarize the quadratic form (z - k*y,z - k*y),where (., .) estimation algorithm. When specialized to a state-space structure, is an indefinite inner product, * denotes conjugate transpose, the algorithm yields a Krein space generalization of the celebrated y is a collection of vectors in a Krein space (which we can Kalman filter with applications in several areas such as Hw- filtering and control, game problems, risk sensitive control, and regard as generalized random variables), and z is a vector adaptive filtering. outside the linear space spanned by the y. If the Gramian matrix R, = (y,y) is nonsingular, then there is a unique stationary point kGy, given by the projection of z onto the I. INTRODUCTION linear space spanned by the y; the stationary point will be N some recent explorations, we have found that H" esti- a minimum if, and only if, R, is strictly positive definite as mation and control problems and several related problems well. In a Hilbert space, the nonsingularity of R, and its strict (risk-sensitive estimation and control, finite memory adaptive positive definiteness are equivalent properties, but this is not filtering, stochastic interpretation of the KYP lemma, and true with y in a Krein space. others) can be studied in a simple and unified way by relating Now in the Hilbert space theory it is well known (moti- them to Kalman filtering problems, not in the usual (stochastic) vated by a Bayesian approach to the problem) that a certain Hilbert space, but in a special kind of indefinite metric space deterministic quadratic form J(z,y), where now z and y known as a Krein space (see, e.g., [9], [lo]). Although the are elements of the usual Euclidean vector space, is also two types of spaces share many characteristics, they differ in minimized by kGy with exactly the same k as before. In the special ways that turn out to mark the differences between the Krein-space case, kgy also yields a stationary point of the linear-quadratic-Gaussian (LQG) or H2 theories and the more corresponding deterministic quadratic form, but now this point recent theories. The connections with the conventional H" will be a minimum if, and only if, a different condition, not Kalman filter theory will allow several of the newer numerical 4 > 0, but R, - R,,R;lR,, > 0, is satisfied. In Hilbert algorithms, developed over the last three decades, to be applied space, unlike Krein space, the two conditions for a minimum to the H" theories [22]. hold simultaneously (see Corollary 3 in Section IV). This In this paper the authors develop a self-contained theory for simple distinction turns out to be crucial in understanding the linear estimation in Krein spaces. The ensuing theory is richer difference between H2 and H" estimation, as we shall show than that of the conventional Hilbert space case which is why in detail in of this series of papers. it yields a unified approach to the above mentioned problems. Part I1 In this first part, we continue with the general theory by Applications will follow in later papers. The remainder of the paper is organized as follows. We exploring the consequences of assuming that {z,y} are based introduce Krein spaces in Section I1 and define projections on some underlying state-space model. The major ones are in Krein spaces in Section 111. Contrary to the Hilbert space a reduction in computational effort, O(Nn3)versus O(N3), case where projections always exist and are unique, the Krein- where N is the number of observations and n is the number space projection exists and is unique if, and only if, a certain of states and the possibility of recursive solutions. In fact, Gramian matrix is nonsingular. In Section IV, we first remark it will be seen that the innovations-based derivation of the Hilbert space-Kalman filter extends to Krein spaces, except that now the Riccati variable P,, and the innovations Gramian Manuscript received March 4, 1994; revised June 16, 1995. Recommended by Associate Editor at Large, B. Pasik-Duncan. This work was supported Re+ are not necessarily positive (semi)definite. The Krein in part by the Advanced Research Projects Agency of the Department of space-Kalman filter continues to have the interpretation of Defense monitored by the Air Force Office of Scientific Research under performing the triangular factorization of the Gramian matrix Contract F49620-93-1-0085 and in part by a grant from NSF under award MIP-9409319. of the observations, R,; this reduces the test for R, > 0 to B. Hassibi and T. Kailath are with the Information Systems Laboratory, recursively checking that the Re,%> 0. Stanford University, Stanford, CA 94305 USA. Similar results are expected for the corresponding indefinite A. H. Sayed is with the Department of Electrical and Computer Engineering, University of Califomia, Santa Barbara, CA 93106 USA. quadratic form. While global expressions for the station- Publisher Item Identifier S 0018-9286(96)00386-8. ary point of such quadratic forms and of the minimization 0018-9286/96$05.00 0 1996 IEEE HASSIBI et al.: LINEAR ESTIMATION IN KREIN SPACES-PART I 19 condition were readily obtained, as previously mentioned, Definition 1 (Krein Spaces): An abstract vector space recursive versions are not easy to obtain. Dynamic pro- {K,(., .)} that satisfies the following requirements is called gramming arguments are the ones usually invoked, and they a Krein Space: turn out to be algebraically more complex than the simple i) K is a linear space over C, the complex numbers. innovations (Gram-Schmidt orthogonalization) ideas available ii) There exists a bilinear form (., .) E C on IC such that in the stochastic (Krein space) case. a) (Y,4= by)*. Briefly, given a possibly indefinite quadratic form, our b) (ax + by,z) = a(x,z) + b(y,z) approach is to associate with it (by inspection) a Krein-space for any x,y,z E a, b E C, and where * denotes model whose stationary point will have the same gain IC; as for K, complex conjugation. the deterministic problem. The Kalman filter (KF) recursions iii) The vector space admits a direct orthogonal sum can now be invoked and give a recursive algorithm for the K: decomposition stationary point of the deterministic quadratic form; moreover, the condition for a minimum can also be expressed in terms of IC=K+$K- quantities easily related to the basic Riccati equations of the Kalman filter. These results are developed in Sections V and such that {K,, (.,.)} and {IC-, -(.,.)} are Hilbert VI, with Theorems 5 and 6 being the major results. spaces, and While it is possible to pursue many of the results of this paper in greater depth, the development here is sufficient to (X,Y)= 0 solve several problems of interest in estimation theory. In the companion paper [l], we shall apply these results to H" for any x E IC+ and y E IC-. and risk-sensitive estimation and to finite memory adaptive Remarks: filtering. In a future paper we shall study various dualities and 1) Recall that Hilbert spaces satisfy not only i), ii)-a), and apply them to obtain dual (or so-called complementary) state- ii)-b) above, but also the requirement that space models and to solve the H2, H", and risk-sensitive control problems. We may mention that using these results (x,z)> 0 when z # 0. we have also been able to develop the (possibly) numeri- 2) The fundamental decomposition of defines two pro- cally more attractive square root arrays and Chandrasekhar K jection operators P+ and P- such that recursions for H" problems [22], to study robust adaptive filtering [23], to obtain a stochastic interpretation of the P+K=K+ and P-K=K-. Kalman-Yacubovich-Popov lemma, and to study convergence issues and obtain steady-state results. The point is that the Therefore, for every x E IC we can write many years of experience and intuition gained from the LQG or H2 theory can be used as a guide to the corresponding x=P+x+P-x=x++z~,x* €IC*. H" results. Note that for every x E IC+, we have (z,z)2 0, but the converse is not true: (2, 2 does not necessarily A. Notation x) 0 imply that x E IC+. A remark on the notation used in the paper. Elements in 3) A vector x E K will be said to be positive if (z,x) > 0, a Krein space are denoted by bold face letters, and elements neutral if (x,x)= 0, or negative if (z,x)< 0. Corre- in the Euclidean space of complex numbers are denoted by spondingly, a subspace M c IC can be positive, neutral, normal letters.

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