780 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 11, NOVEMBER 2005

Adapt the Steady-State Kalman Gain Using theNormalized Autocorrelation of Innovations

Bo Han and Xinggang Lin, Senior Member, IEEE

Abstract—A discrete linear time-varying stochastic systemwith scalar measurement data is considered for which neithermeasurement noise variance nor power of process noise is known.A novel adaptive Kalman filter that gradually approaches theoptimum steady-state gain is proposed in this letter. Differentfrom previous adaptation schemes, our algorithm adjusts theKalman gain depending on the normalized autocorrelation of theprediction error sequence of the suboptimal filter. We also presenthow to choose appropriate latency time and sample window lengthin the adaptation process. In our experiments, the filter showsadvantages over several other methods under the same conditions.

Index Terms—Adaptive Kalman filtering, steady-state Kalmanfilter.

I. INTRODUCTION

KALMAN filtering requires complete a priori knowledgeof both the system equations and of the statistics of the

noise sources affecting the system. However, in practical situ-ations, the statistics are seldom exactly known time invariants.It is well known that using a wrong model in the design of aKalman filter can lead to large estimation errors or even to a di-vergence. In this letter, we assume that the system equations andform of the process noise covariance are exactly given, while thepower of each noise is determined by an unknown time-varyingpositive scalar multiplier, respectively.

Since the 1960s, various adaptive Kalman filters have beenproposed. However, as is stated in [1], in the past several years,only a few publications in the area of adaptive Kalman filteringcan be found in the literature. Some methods for estimating thenoise statistics are briefly reviewed in [2]. The Bayesian esti-mation method is impractically time consuming; in maximumlikelihood estimation, the iteration diverges in many cases, andthey are mainly designed for stationary processes. In [3], themeasurement noise covariance is not properly estimated, andthe process noise covariance is coarsely modified to provide anegative feedback. The optimality criterion of the problem ismodified in [4]; at the same time, the filtering problem is trans-formed into a smoothing one. In some schemes, such as in [1]and [5], only uncertainty of the process noise or of the mea-surement noise is considered. Several methods, including inno-

Manuscript received January 26, 2005; revised July 9, 2005. This workwas supported by the National Science Foundation of China under Project60472028. The associate editor coordinating the review of this manuscript andapproving it for publication was Dr. Patrick A. Naylor.

B. Han is with Institute of Image and Graphics, Department of Elec-tronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:hanb02@mails.tsinghua.edu.cn).

X. Lin is with Department of Electronic Engineering, Tsinghua University,Beijing 100084, China (e-mail: xglin@tsinghua.edu.cn).

Digital Object Identifier 10.1109/LSP.2005.856870

vation correlation [6], modified covariance matching [7], andone-bit adjustment [8], will be compared to our algorithm inexperiments.

The rest of this letter is organized as follows. Section IIconcretely states the problem. How to adapt the steady-stateKalman gain toward its optimum is detailed in Section III.Section IV discusses the appropriate latency time and samplewindow length in the process of adaptation. Some experimentalresults and comparisons are exhibited in Section V. Section VIconcludes the letter.

II. STATEMENT OF THE PROBLEM

We consider a discrete linear stochastic system described by(1), shown at the bottom of the next page.

Here, is the state vector; is the measurement value.is the state transition matrix; is the measurement matrix;they are exactly given constants. is the process noise;is the measurement noise; they are mutually independent andare modeled as Gaussian white noises. is a known positivesemidefinite matrix; and are un-known time-varying scalars. We assume that the pairis detectable; the pair is stabilizable, where

(the superscript means matrix transpose.) TheKalman filtering process can be depicted by

(2)

Here, is the prediction error (or residual); it is referred toas innovation when the filter has no model error. is called theKalman gain. is the estimate of based on the observationset . Here, the tilde means one-step pre-diction. The error covariance matrix is defined as follows. Here,the operator means expectation

(3)

We assume that the sequence is ergodic so that time-domainstatistics can be used to adapt the filter. We also assume nosudden change in the second-order statistics of innovationsequence; thus, the sequence can be considered short-termstationary.

The problem is to adapt , based on the output of a subop-timal Kalman filter, to minimize .

III. ADAPT THE STEADY-STATE KALMAN GAIN

If and are known constants (denoted by and ), theproblem can be solved by a standard Kalman filter. It is wellknown that the Kalman filter will enter a steady state as time

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HAN AND LIN: ADAPT THE STEADY-STATE KALMAN GAIN 781

goes by; will converge to a constant , which is the uniquesolution of the following algebraic Riccati equation:

(4)

Simultaneously, will converge to a constant , which iscalled the steady-state Kalman gain

(5)

Since Kalman filtering is insensitive to the error in initial stateestimation, we can adopt a specific , so that .Directly using in the filtering process will result in a time-invariant observer that is called a steady-state Kalman filter.

From the definition, it is obvious that is determined byand ; we can obtain an analytical expression of by solving(4). It is proved that does not change when and aremultiplied by the same scalar [9]; so, it is a function of the“noise-to-signal ratio” . Performance of the filter lieson ; thus, it is completely determined by .

In our defined conditions, true divergence—error covariancebecoming unbounded—can never occur with any incorrectnoise-to-signal ratio [10]. Here, the prime means anestimate that may differ form the optimal value. Also, in thiscase, the matrix is exponentially stable, i.e.,all of its eigenvalues are inside the unit circle. We can callthe “fading” matrix because

(6)

If and keep invariant for a period of time, and we keepusing a certain steady-state gain , then the actual error co-variance will converge to , which is the unique solutionof the following algebraic Lyapunov equation [10]:

(7)

reaches its minima iff , i.e., is ideal. Whendiffer greatly from for a long period, the estimation error

may become intolerably large; this phenomenon is called “ap-parent” divergence. Therefore, we seek to resize toward itsoptimal value, depending on the output of the suboptimal filter,so as to prevent apparent divergence and to minimize estimationerror.

Our adaptation of is based on the innovation sequence ,which is a zero-mean Gaussian sequence. When the error co-variance has converged, we have

(8)

Now, we define the normalized autocorrelation, on which ouradaptation is based, as follows:

(9)

It is obvious that . It is well known that if, the innovation sequence is white when the initial error can

be ignored. Moreover, we also assume that in the model

(10)

Putting (8) and (5) into (9), we obtain (11). It can be foundfrom (4) and (7) that is a function of and . We will provethat for any given is strictly monotone with respect to .The subscript will be omitted throughout the proof

(11)

First, once is given, the right-side term of in (11) is fullydetermined.

Since just changing the values of and have no influenceon , we suppose that

then and (12)

Now, can be disassembled as follows [10]:

where (13)

The matrix , which is the unique solution of the followingequation, is also determined by :

(14)

Thus, the left-side term of in (10) becomes

(15)

For that and , so and. Now we can express the partial derivative as follows:

(16)

So, the sign of (16) given is decided by

(17)

Thus, (17) cannot be zero or (10) will not be satisfied. Be-cause (17) is continuous with respect to , its sign does notchange with respect to . This completes the proof.

wherewhere

(1)

782 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 11, NOVEMBER 2005

This feature is our foundation of adapting . Given a nonop-timal , the actual ratio can be computed as follows:

(18)

Because is estimated statistically, there may be error in itsvalue. To avoid vibration, or even divergence of , we choosea mild speed so that each adaptation step will not be too drastic.So, the adaptation will be accomplished by

sgn

(19)

IV. CHOOSE APPROPRIATE LATENCY TIME

AND SAMPLE WINDOW LENGTH

As stated in the last section, we can use statistics of the in-novation sequence to adapt only when has converged, i.e.,the initial error can be ignored, and the process can be consid-ered stationary in a short period. Now, we will discuss how longit takes the filter to “forget” the initial error

(20)

As shown in (20), the effect of the initial error will be weak-ened at a step of the fading matrix. Since is exponentiallystable, converges to a zero matrix. An appropriate legacytime corresponding to is determined as follows, where isa small positive number such as 0.01:

(21)

The estimate of may differ from the corresponding expec-tation. Statistically, the longer the sample window, the smallerthe error. At the same time, we hope to use a short windowfor fast adaptation. We adopt a simple method for deciding thesample window length .

As is known, when is the product of twoindependent identically distributed (i.i.d.) zero-mean Gaussianvariables, is a distributed variable. We defineconfidence region of the normalized autocorrelation asfollows ( can be set, such as 0.95):

(22)

Our philosophy is that, for a given , we have to ensure thatpossibility of large adaptation error is small enough. To sim-plify the problem, we only consider the situation that .

Fig. 1. Performance comparison of four algorithms in different conditions.

, which is a function of , is determined as follows. Here,we assume that the length must be longer than to ensurestatistical validity of the time average quantities. See (23) at thebottom of the page.

V. EXPERIMENTAL RESULTS AND COMPARISONS

Now, we will take the following linear kinematics model as anumerical example:

and

(24)

For the ease of signal generation and comparison, we per-form experiments on stationary sequences. They cover a 120-dBrange of noise-to-signal dynamics. For each noise condition, wegenerate 1000 independent sequences, of which each has 10 000time-domain samples. The performance criterion is set as thestatistical value in (25). Here, is the corresponding theoret-ical result obtained by the optimal Kalman filter. The smaller thevalues (with 1 being the minimum), the better the performance

(25)

Fig. 1 compares the performance of four adaptive Kalman fil-tering schemes in different conditions. Their performances aregood and very close when is small, while our algorithm showspredominance as becomes large. Each of the parameters inschemes in [6]–[8] is set as well as that in the reference papers.The algorithm in [6] often yields negative estimates of or ,

(23)

HAN AND LIN: ADAPT THE STEADY-STATE KALMAN GAIN 783

so it can hardly be applied to a model with unknown noise co-variance forms; such wrong adaptations are canceled in our ex-periments. Keep in mind that for the schemes in [7] and [8], theactual measurement noise variance needs to be exactly known.

Our scheme is extremely computationally efficient comparedwith that in [6]. Steady-state gain is used to save the burden ofiterative computation in most steps, and all parameters throughthe adaptation are searched in a lookup table. Given a certainmodel, it is convenient to construct such a table using softwaresuch as Maple.

VI. CONCLUSION

The experimental results prove our algorithm to be efficientand applicable to a wide range of noise conditions. Furthermore,when the normalized autocorrelation is small enough, we canalso use (8) to estimate the noise covariances as a byproduct.Finally, our algorithm is computationally economical. Althoughthe scheme is limited to scalar measurement situations, it seemsimportant since numerous applications belong to this category.

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