The Kalman–Bucy filter  is the optimal state estimator for linear continuous-time systems with white Gaussian noises. It is well known that the optimal state estimation error covariance attained by the Kalman–Bucy filter in steady state satisfies the algebraic Riccati equation [2, 3, 4], which, however, does not provide an analytical expression of the error covariance. Over the years, various upper bounds and lower bounds have been obtained for the error covariance (see, e.g.,  and the references therein). Meanwhile, nonrecursive algebraic solutions have been obtained as well (see, e.g., [6, 3, 7] and the references therein).
In this paper, we develop an alternative approach towards this problem from the viewpoint of the Bode integral . Bode integral was the first and arguably the best known result to analyze the fundamental limitations of feedback control systems using analytic function theory [9, 10, 11, 12, 13, 14, 15]. In particular, the Bode integral is implied by the Poisson–Jensen formula . This line of research has been of continuing interest to the control community [17, 18], and was related to information theory in recent years as well [19, 20, 21, 22, 23, 24, 25, 26, 27]; such results together with  inspired us to reexamine the Kalman–Bucy filter (in steady-state) using analytic function theory, e.g., the Poisson–Jensen formula, which will be seen to play an essential role in our analysis.
In particular, we first develop from the Poisson–Jensen formula a counterpart of Jensen’s formula, based on which we obtain an analytical expression of the trace of the division of the optimal output estimation error covariance over the noise covariance by the Kalman–Bucy filter; this is given in terms of the plant dynamics, e.g., the unstable poles, and the noise statistics in a frequency-domain integral characterization. The analysis mainly concerns examining the algebraic Riccati equation associated with the Kalman–Bucy filter using analytic function theory.
We compare our result with the existing related results in [29, 30, 31, 32, 33], and our integral characterization is seen to be the most generic and reduces to all of them. Specifically, the case when the plant is stable and the plant output is a scalar process was considered in . The authors of  then analyzed the case when the plant output is scalar but the plant is not necessarily stable. The following results in 
considered the case when the process noise is relatively small in variance (and in the limit, zero) compared with the observation noise. Most recently, the case when the plant output is a vector process while the observation noise is with an identity covariance matrix was investigated in. In addition, discussions on the dual problem in control can be found in [30, 32].
The remainder of the paper is organized as follows. Section II introduces the technical preliminaries. Section III introduces a counterpart of Jensen’s formula. In Section IV, we examine the Kalman–Bucy filter using algebraic function theory and presents an integral characterization of the optimal output estimation error covariance. Relevant discussions and interpretations are also presented. Concluding remarks are given in Section IV.
In this section, we introduce some relevant notions from stochastic processes, and briefly review the basic properties of the Kalman–Bucy filter.
Ii-a Notations and Basic Concepts
In this paper, we consider real-valued continuous zero-mean random variables and vectors, as well as continuous-time stochastic processes. We denote random variables and vectors using boldface letters. The logarithmis defined with base , and all functions are assumed to be measurable.
A zero-mean stochastic process is said to be (asymptotically) stationary  if for any , the (asymptotic) correlation matrix
exists. The (asymptotic) power spectrum of is then defined as
It can be verified that is positive semidefinite. In the scalar case, we denote by . For the inversion,
Moreover, the (asymptotic) covariance matrix of is given by
In the scalar case, reduces to (asymptotic) variance . Moreover, if is (asymptotically) white, then is a constant for all , and
where is the Dirac delta function.
Ii-B The Kalman–Bucy Filter
where is the state to be estimated, is the system output, is the process noise, and is the measurement noise. The system matrices are and , and is assumed to be detectable. Suppose that and are white Gaussian with covariance matrices
respectively, and that the initial state is Gaussian with covariance satisfying . Furthermore, , , and are assumed to be mutually uncorrelated.
The Kalman–Bucy filter is given by
where , , , and . Herein, denotes the Kalman gain, which is given by
where denotes the state estimation error covariance matrix as
Herein, is obtained using the Riccati equation
It is known that the Kalman–Bucy filtering system converges, i.e., the estimator is asymptotically stable, the state estimation error and the output estimation error are asymptotically stationary, and is asymptotically white, under the assumption that the system is detectable. Moreover, in steady state, the optimal state estimation error covariance matrix
attained by the Kalman–Bucy filter satisfies the algebraic Riccati equation
whereas the steady-state Kalman gain is given by
In addition, the optimal steady-state output estimation error covariance matrix is found to be
where denotes the true value of the system output (in comparison, denotes the measured value of the system output). Moreover, when , (II-B) reduces to
It is also worth mentioning that
Iii A Counterpart of Jensen’s Formula
In this section, we first examine the Poisson–Jensen formula (see, e.g., Appendix C of ) for a special class of analytic functions, and obtain a formula that will be essential for the rest of the paper.
be a rational transfer function for which the numerator polynomial and denominator polynomial are both of order . Suppose that all the poles of are stable and that has no zeros on the imaginary axis. In addition, suppose that
where denote the zeros of .
See Appendix A.
Note that in the discrete-time case (for the unit disk), Jensen’s formula is a consequence of the more general Poisson–Jensen formula . Likewise in the continuous-time case (for the half plane), (7) is also a consequence of the more general Poisson–Jensen formula (39). As such, (7) as well as the subsequent (2) may be viewed as the continuous-time (half-plane) counterparts of the discrete-time (unit-disk) Jensen’s formula.
More generally, when not necessarily all the poles of are stable, we can obtain the following result based on Proposition 1.
be a rational transfer function for which the numerator polynomial and denominator polynomial are both of order . Suppose that has no zeros on the imaginary axis and that
where denote the zeros of and denote its poles.
See Appendix B.
Iv Optimal Error Covariance by the Kalman–Bucy Filter
We now study the optimal output estimation error covariance matrix by the continuous-time Kalman–Bucy filter, which is determined by the plant dynamics and the noise statistics in an integral characterization.
The optimal output estimation error covariance matrix by the Kalman–Bucy filter satisfies
and denote the eigenvalues of
denote the eigenvalues of.
See Appendix C.
On the right-hand side of (1), the term
denotes the power spectrum of ; in general, however, needs not to be stable, and hence the function
is a Popov function (see, e.g., , for a detailed discussion).
One may refer to Appendix A of  for further details on this.
Iv-a Interpretation from the Viewpoint of Bode Integral
can also be obtained as a consequence of the (continuous-time) Bode integral ; it is interesting to discover that such a relation exists between these two top equations , namely algebraic Riccati equation and Bode integral, of the control field. Specifically, the estimator may be viewed as a feedback system, and thus the Bode integral can be obtained for its sensitivity from to (see Fig. 1). Indeed, if we denote
then (IV-A) can be rewritten as
where denote the poles of . Herein, we have used the fact that
which holds since the system is detectable (all the unstable modes of the system are observable) and thus the unstable poles of correspond to the eigenvalues of with real parts larger than zero. It is also worth mentioning that when it is further assumed that , (19) reduces to
to hold is that all the entries of have at least two more poles than zeros, which is not satisfied in the case of Kalman–Bucy filtering as discussed in this paper.
Iv-B A More Explicit Expression of (1)
Equality (1) can be rewritten as
where denote the zeros of and denote its poles.
By invoking (2), it holds that
it follows from  (Appendix A) that
Concerning the terms in (2), it holds in general that
since the unstable poles of must belong to the set of eigenvalues with real parts larger than zero, while not all such eigenvalues of may be present in the set of unstable poles of . Note that this is different from the case of , as discussed in the proof of Theorem 1; in such a case, since the system is detectable (all the unstable modes of the system are observable), the set of unstable poles of is exactly the same as the set of eigenvalues of with real parts larger than zero.
Iv-C Some Straightforward Corollaries
Using the fact that
we could obtain lower and upper bounds on based on (1) as
One might also compare our bounds with those in, e.g., , which, however, goes beyond the scope of this paper.
We now consider some special cases of Theorem 1.
If , and suppose that and are white Gaussian with covariance matrix and variance , respectively, then
If , and suppose that and are white Gaussian with variances and , respectively, then
Iv-D Relation to Existing Results
We now present a list of comparisons with existing results in the related works. It will be seen that our result in (1) is the most generic and reduces to all the listed ones, although they have adopted different approaches.
Indeed, equation (32) is referred to as the Yovits–Jackson formula in . In a broad sense, the Yovits–Jackson formula can be viewed as the continuous-time counterpart of the Kolmogorov–Szegö formula [4, 35, 41, 42, 43] in the discrete-time case. For the multiple-input multiple-output (MIMO) case, when is stationary ( is stable), (1) reduces to
As we pointed out in , our formula therein generalizes the Kolmogorov–Szegö formula and the Wiener–Masani formula to the non-stationary case, and correspondingly in this paper, we have generalized the Yovits–Jackson formula to the MIMO case as well as the non-stationary case.
In this paper, we have shown that the trace of the division of the optimal output estimation error covariance over the noise covariance attained by the Kalman–Bucy filter can be explicitly expressed in terms of the plant dynamics and noise statistics in a frequency-domain integral characterization. We have also discussed the relation of our integral characterization to the Bode integral. Possible future research directions include analysis of other classes of algebraic Riccati equations.
V-a Proof of Proposition 1
It follows from the Poisson–Jensen formula for the half plane  that for every point , ,
where denote the nonminimum-phase zeros of and denote the complex conjugate of . Hence, taking , we have
It is known  that
We finally consider the remaining term in (V-A):
Denote , and . Since , we have . In addition, as , we have
It thus follows that the limit (42) always exists; it is always finite, although it can be when . This completes the proof.
V-B Proof of Proposition 2
where is chosen as a polynomial of satisfying the following conditions:
is with the same order as those of and ;
all the zeros of are nonminimum phase;
there is no pole-zero cancellation between and or between and ; and
the following relation holds:
Therefore, the zeros of are given by the zeros of , while the poles of are given by the zeros of . On the other hand, the poles of both and are stable. Then, using Proposition 1, we arrive at
This completes the proof.
V-C Proof of Theorem 1
For all , it holds that
Hence, (2) can be rewritten as