228 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

Nu-Gap Model Reduction in the Frequency Domain

Aivar Sootla

AbstractThe nu-gap metric was originally developed to evaluate ro-bustness of a plant-controller feedback loop and it has many attractiveproperties. For example, performance of the closed loop changes continu-ously and stability is robust with respect to small perturbations of the plant(or the controller) in the nu-gap metric. In light of these properties, one canstate that the nu-gap metric provides a good measure of distance betweensystems in a closed loop setting. This is very useful in model approximation,which is the focus of this technical note. The presented nu-gap approxima-tion method is based on semidefinite programming and frequency responsematching, which allows to extend the method to account for frequency-de-pendent weights in the objective function. The frequency-weighted exten-sion is the major advantage of the presented method in comparison withother nu-gap model reduction methods. This extension is applied to ap-proximation of controllers obtained by loop shaping and illustrated on anumerical example.

Index TermsController reduction, model reduction, nu-gap metric,semidefinite programming.

I. INTRODUCTION

Model reduction of linear systems is well studied in the literatureandmany efficient methods have been derived. The methods deliveringthe best approximation quality in the norm are balanced trunca-tion [1] and optimal Hankel model reduction [2]. These methods alsoguarantee stability, however, computationally very expensive. Krylovmethods, cf. [3], on the other hand, are considerably cheaper, but gen-erally provide only local approximations. It is important to remark thatthe Krylov method [4] provides locally optimal approximations in the

norm, if the resulting approximation is stable.Most of the standard model reduction methods measure the approxi-

mation error in the or norms, that is, they measure the distancein the open loop setting. In a closed loop setting these norms usually donot reflect the distance adequately. An early attempt to create a morereliable metric in the closed loop setting was the introduction of the gapmetric in [5], followed by many papers including [6], [7] and [8]. In thelast reference the -gap metric is introduced. This metric induces theweakest topology in which stability of a closed loop is maintained forsufficiently small perturbations and performance varies continuously.This is a valuable property in controller reduction, since the controllerswhich are sufficiently close in the -gap metric will stabilize the nom-inal plant and the performance of the loops will be similar as well.Literature in -gap model reduction is not as rich as in the

case. An approximation method have been already proposed in the first-gap paper [8], where low-order models are obtained one step at atime. It means, if the order of the full order model is , then firstly op-timal in the -gap approximation of order is obtained. Here, isthe multiplicity of the smallest Hankel singular value of a normalizedright graph symbol of the model. After that, optimal orderapproximation to order model is calculated and so on down to

Manuscript received September 26, 2011; revised April 12, 2012; acceptedMay 10, 2013. Date of publication November 18, 2013; date of current versionDecember 19, 2013. This work was supported by the Swedish Research Councilwith the Linnaeus Grant for Lund Center for Complex Engineering Systems.Recommended by Associate Editor Z. Wang.The author was with the Department of Automatic Control, Lund University,

Lund 221 00, Sweden and is now with the Department of Bioengineering, Impe-rial College London, London SW7 2AZ, UK (e-mail: a.sootla@imperial.ac.uk).Digital Object Identifier 10.1109/TAC.2013.2290817

the required reduction order. In [9] a similar method is proposed, withmore freedom to choose optimal approximations.In this note, a single-input-single-output model reduction algorithm

is discussed, as well as, its extension to the frequency-weightedproblem. The presented method has its roots in [10], [11], which arealso related to [12], [13]. As opposed to the previous works in -gapmodel reduction [8], [9], which use state-space representations tocompute low-order approximation, the presented algorithm employsthe frequency domain data. The use of frequency domain data hassome advantages, e.g., a straightforward frequency-weighted exten-sion. The frequency-weighted version is applied to approximation ofcontrollers, which are obtained using the loop shaping procedure(cf. [14], [15]). The frequency-weighted -gap approximation is ableto further reduce the order of the controller in comparison with itsnon-weighted counterpart without sacrificing performance levels. Thisis done by successfully exploiting the structure of the controller designwhich is based on adding frequency weights.The rest of the note is organized as follows: in Section II the -gap

metric is briefly introduced. Section III is devoted to the algorithmderivation. In Section IV extension to the frequency-weighted problemand its application to controller reduction is discussed, which is illus-trated on a numerical example in Section V.

Notation

Let denote the space of discrete-time by rationaltransfer matrices and denote . The normis computed as , where is the max-imum singular value of a matrix and is the complex identity. Ifis stable, then this expression computes its norm . Thefunction is called the winding number of and it is equalto the number of times the curve encircles the origin. Let

be the normalized right graph symbol of a plant, where and is the normalized right coprime fac-

torization of , and finally

if stabilizesotherwise

II. PRELIMINARIES

Without loss of generality consider only discrete-time models. Thecontinuous-time models are discretized using the bilinear transforma-tion while pre-warping around a specific frequency . Due to bijec-tivity of the bilinear transformation, one can view the discrete-timevariable as a basis function in the continuous-time domain

.Definition 1: Let , be rational transfer matrices, and ,

be their right normalized graph symbols. Let

(1)

be called the winding number condition. Let , :be the functions such that

if (1) is satisfiedotherwise

0018-9286 2013 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014 229

Note that there is no assumption on the open-loop stability of and, which proves useful in many applications. The function is

called -gap and it is a metric. Moreover, the stability margindefines a ball around in the -gap metric, which contains only plantsstabilized by the controller ; performance is changing continuouslydepending on the plant, if the plant is perturbed in the -gap metric.These properties can be formalized in the following statement, whichis a combination of results from [15]:Proposition 1:1) The function is a metric.2) Robustness (Remark 3.11 to Theorem 3.10). Given and

i) is stable for all satisfying if, andonly if,

ii) is stable for all satisfying if, and onlyif,

3) Performance (Theorem 3.19). For any , , such thatand are stable,

4) Approximation. This result is based on Theorem 8.6. Let be aplant of order and be a set of the transfer functions whichhave a minimal realization of order smaller than , then

where is a -th largest Hankel singular value of the right nor-malized graph symbol of .

Proposition 1 can be restated to describe controller properties instead.For instance, it can be shown that the changes in performance are con-tinuous given small perturbations in the controller.

III. LOW-ORDER APPROXIMATION IN THE -GAP METRIC

Let and the elements of be scalar, not necessarily stable,transfer functions and consider the problem

Let , be the normalized right coprime factorization of . Let ,be finite impulse response (FIR) filters with a fixed order , i.e.,

and . Let also and be a rightcoprime factorization of . In the new variables , , and , thefunction is computed as

The winding number condition (1) is equivalent to

(2)

and the equivalence is shown in a straightforward manner using theproperties of the winding number function. Finally, -gap model re-duction in the new variables is formulated as follows:

(3)

(4)

Note by construction. In this formulationis a full order model, and parametrizes its th order approxi-mation. The usual approach to minimization is rewriting thenorm constraint (4) with an infinite number of constraints for everyfrequency

(5)

The obtained program is not generally convex even for the scalar-valued functions. The main result of this technical note is rewriting(5) in an equivalent formulation, from which a quasi-convex programcan be obtained. Introduce new variables ,

and yet another formulation as

(6)

Theorem 1: Assume and is the normalized coprime factorizationof the plant , then: The minima and of the programs (5) and (6) are equal , are the solutions to (5), if and only if, , , , arethe solutions to (6) for some and .

Before proceeding with the proof of the theorem, it is important toremark how (6) is going to be used. A quasi-convex formulation isobtained in two steps: removing the coprimeness constraint on , andfixing variables , in the program.Removing the coprimeness constraint on and in (6) is a relax-

ation; however in the authors experience, numerical examples providecoprime and . This can occur due to the following reasoning. Forand to be not-coprime, they should have precisely the same zeros,which is a measure zero subset of all admissible and . Thereforenumerical solutions typically do not reach not-coprime and . Ifand are nearly not-coprime, a possible remedy would be loweringthe order .The winding number condition in (6) is enforced on the variablesand . Therefore after fixing those, the norm constraint becomes

semidefinite and the program becomes quasi-convex. This is a key ob-servation in the presented approach and it is discussed in detail in thenext subsection, while now the main result is to be shown using thefollowing lemma:Lemma 1: Assume that a rational function has no poles on

the unit circle. If is strictly positive for all , then

Proof: If has no poles on the unit circle andis positive for all , then the function

does not encircle the origin and is equal tozero. Using the properties of the winding number function it followsthat: .And, finally, .

230 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

Proof of Theorem 1: First, let us show that for alland . Let and . Sincefor any complex number , we have

The term appears on both sides of the norm constraint and can beeliminated without affecting the solution. Note that has no zeros onthe unit circle according to constraints above, whereas has no poleson the unit circle by construction. Now we can apply Lemma 1 andreplace the conditions and on the unit circle with

and on the unit circle. This gives us (5) and, finally,.

To prove the converse let and be an optimal solution to (5).The functions and also satisfy the constraints of (6) withand . Therefore .The second part of the theorem is shown using the same arguments.

If , , , , are the solutions of (6), then by construction ,, are the solutions to (5). On the other hand, if , , and are

the solutions to (5), then they also satisfy the constraints of (6) with, and .

A. Tractable Algorithm and Implementation

Let the program (6) without the coprimeness constraints on , and, be called the relaxed program (6), which is the basis for the pre-

sented below model reduction method. After the functions andsatisfying the winding number condition are found, the relaxed pro-gram (6) with fixed and becomes quasi-convex and can be solvedusing standard tools. Moreover, the proof of Theorem 1 shows thatand can be interpreted as a guess on variables and . This is ex-ploited by iterating over and , while setting and onevery step. This approach to the non-convex program (i.e., the relaxedversion of (6)) is a standard heuristic, which gives suboptimal solutionsto (6). Generally, there is no guarantee that this heuristic provides so-lutions close to the optimal one. However, in the studied case andare closely related to and , which provides a good reason to performsuch iterations.An obvious choice for the initial and is computing , which

is as close as possible to in the -gap. However, the relaxed problem(6) is non-convex and there is no guarantee that a better in terms ofquality initial guess will provide a better final result. Therefore, it maynot be beneficial to compute the initial guess with a method as [9],which provides very close to , but is numerically expensive.Hence a simpler choice of the initial point is proposed and it is moti-vated by the fourth bullet of Proposition 1. Let be anyoptimal Hankel approximation of the right normalized graph symbol

of , where , and are FIR filters or order . Now simplyset initial equal to and initial to .Frequency dependent constraints in (6) may be imposed for all the

frequencies at once using the Kalman-Yakubovitch-Popov lemma (cf.[16]). To provide a computationally cheaper program the constraintsmay be enforced on a frequency grid , where is big enoughto avoid over-fit. Since the -gap metric emphasizes the behavioraround the gain cross-over frequencies, it is reasonable to make thegrid denser around those frequencies. The -gap model reductionmethod is summarized in Algorithm 1. The algorithm is implemented

using the interior-point solver SEDUMI [17] and YALMIP [18]. Sincethe program (7) is quasi-convex, it is solved using bisection. To savecomputational effort bisection is performed in variableinstead of . This also gives upper and lower bounds for the bisectionprocedure (1 and 0, respectively). The stopping criterion is based onand convergence in is attained. Since the previous solutions andare used on the next step, is always bigger or equal to . Onthe other hand is bounded from below, which entails convergencein .

Algorithm 1 -gap model reduction

Inputs: Full order plant , approximation order , tolerance ,domain of

Output: Reduced order plant

Compute the normalized right coprime factorization of

Compute -th order Hankel approximation of, where , and are FIR filters.

Set , and

repeat

Set and solve

(7)

where and

Set , and ,

until

B. Computational Complexity

Let be the order of the full order model and be the order ofapproximation and let . There are two main contributors tocomputational complexity: the computation of the initial guess and theoptimization problem. The computation of the initial guess involvesnormalized coprime factorization and Hankel approximation, whichare performed using Riccati and Lyapunov equations. Therefore com-plexity is floating point operations (flops).The overall cost of an optimization program when solved with SE-

DUMI does not exceed flops, where is thenumber of decision variables and is the number of rows in the in-equalities [19]. In the case, when the constraints are enforced on a fre-quency grid, and , where is the number orfrequency points in the grid. Computing the frequency samples costsat most . Therefore, in this case, the cost of the optimization pro-gram does not exceed . If the constraints are enforced using theKYP lemma for all the frequencies, then and .This gives the overall cost of .In comparison with other methods, the presented one has a compet-

itive cost. The methods from [9] and [15] require solving Riccatior Lyapunov equations, respectively. Therefore the total cost of the al-gorithms is .

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014 231

IV. FREQUENCY-WEIGHTED EXTENSIONAND CONTROLLER REDUCTION

In this section the -gap model reduction is applied to the controllerreduction problem. Specifically, the controllers obtained using theloop shaping (cf. [14], [20]) are considered. The first step in theloop shaping is the introduction of the shaped plant ,where the weights and are chosen according to desiredspecifications. After that the central controller , which minimizes

, is obtained. The actual controller is computedas . In the studied case and are scalar, therefore aone-sided weight can be considered without loss of generality and

.In order to obtain a reduced order controller one can approximateby a controller of order

(8)

In this case the actual reduced order controller is . Abetter way to address the low-order controller design is to introducedirectly as a variable. Note that only the shaped loop has a conve-

nient form , which allows the use of the -gap metricfor controller approximation. Therefore the only possibility of approx-imating the controller, while having as a variable, is as follows:

(9)

The program (8) can be viewed as (9) with an additional constraint. In (9) the structure of is relaxed and this grants

more flexibility in the decision variable choice. Hence the frequency-weighted approach can provide controllers of lower order in compar-ison with the central controller approximation, while keeping similarperformance levels. As a final remark, note that andthe problem (9) is essentially the frequency-weighted -gap reductionproblem

A. Frequency-Weighted -Gap Reduction

As discussed above, is the problem to solve.In order to account for the frequency-weights in Algorithm 1, it is re-quired to redesign the initial point computation and the program (7),which is solved inside the loop. In preparation of this technical notedifferent approaches to both problems were considered and the listedheuristics provided the most acceptable results.Let , be the normalized right coprime factors of , and, be the normalized right coprime factors of . Sim-

ilarly to the non-weighted algorithm, the initial guess is found by ap-proximating a right graph symbol of . However, in this case theweight in the approximated graph symbol must be preserved, hencethe transfer matrix is approximated. This transfer ma-trix is a right graph symbol of , but its norm is smaller or equal toone, instead of being equal to one for the non-weighted computationof the initial guess. Finally, the computation of the initial guess is per-formed as follows:

(10)

where , are stable functions of order , , , are FIR filtersand the initial guess is and .

TABLE IAPPROXIMATION ERRORS OF VARIOUS METHODS IN THE -GAPMETRIC AND EFFECT ON THE CLOSED LOOP PERFORMANCE

To finalize the algorithm, let , be the normalized right co-prime factors of and change the program (7) accordingly

(11)

where and .

V. NUMERICAL EXAMPLE. ROBUST STABILIZATION OF CART

The main goal of this example (which is taken form [15]) is to illus-trate that the proposed method is competitive in comparison with theknown techniques in the non-weighted case, but has an additional ad-vantage in the form of the frequency-weighted extension.For the loop-shaping procedure the same weight as in [15] cannot be

used, since in (10) and (11) the weight has to be invertible. Hence, twozeros at are added, which results in a slightly worse disturbanceattenuation. The plant and the shaping weight are chosen as

In comparison with [15] the controller design procedure is also sim-plified. The controller is obtained using Theorem 16.1 from [20] with

, providing . The reader may notice thatoverall this design is less robust to changes in than the one providedin [15], however, this design will still colorfully illustrate the results of-gap controller reduction. The continuous-timemodels are discretizedwhile pre-warping around the frequency , which is roughlythe cut-off frequency of the closed loop.

A. Approximation of the Central Controller

The results of -gap approximation of the central controller are pre-sented in Table I. The parameter for [9] is chosen as a zero matrix.The notation [9]+Alg 1 signifies that the initial guess in Algorithm 1 isobtained using [9]. Finally, the relative closed loop error is defined asfollows:

The tolerance for Algorithm 1 is set to , all the constraints areenforced on an equidistant frequency grid with 0.02 between the sam-ples. The algorithm converges after at most 6 iterations, which occursfor order 4.

232 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

The results of [9] and Algorithm 1 with different initial guesses arequite similar for all the orders. Moreover, the results are quite similarin both the -gap approximation quality and the relative closed looperror.Another important remark is that Algorithm 1 with the initial guess

[9] does not provide consistently better solutions than Algorithm 1witha simple initial guess. This happens because the problem is non-convexwith possible multiple local minima. Hence while starting with a betterinitial guess in terms of approximation quality, there is no guaranteethat the final result will still be better in terms of approximation quality.Surprisingly, for order 3 Algorithm 1 provides the same controller (co-efficients of the transfer functions match) with both initial guesses.An interesting detail appears when the order of approximation is set

to 2. The initial guess provides the low-order controller on the dis-tance 0.45 in the -gap metric from the original one. This second orderapproximation does not stabilize the shaped plant . However, afterapplying Algorithm 1 a stabilizing controller is obtained already afterthe first iteration. This controller lies on the distance 0.29 in the -gapmetric.While the performance of the closed loop is not satisfactory (therelative closed loop error is larger than 4), the loop is stabilized. Themethod [9] also provides a stabilizing controller with the -gap error0.29 and the relative closed loop error 2.47.According to Table I, the relative closed loop error of [9]+Alg 1

can be worse than the relative closed loop error of [9]. This meansthat the performance of the initial guess (model obtained with [9]) isbetter than the performance of the model obtained with [9]+Alg 1.This seems counterintuitive, however, the minimization objective isthe -gap metric not the relative closed loop error. It is certainly pos-sible that there are controllers and such that

, but

To conclude this discussion, the performance is changing continuouslywith respect to small changes of the controller in the -gap metric (ac-cording to Proposition 1), however, there is no guarantee that the per-formance will change monotonically.

B. Weighted Approximation of the Controller

Since the actual controller for the plant is , the order of the ac-tual controller is plus the order of , which is equal to 6. To obtain aneven lower order design the distance is minimizedover . Now, is the controller for the plant , not for the shapedplant . The grid for this example is also equidistant with 0.005between two consecutive samples, and . All the obtainedcontrollers of orders less than 6 destabilize the systems, and for orderslarger than 7 the match is almost perfect even after the initial guesscomputation. For order 6 approximation 33 iterations are required, fororder 7 approximation 19. The number of iterations grows signifi-cantly, mainly since the problem is now more complicated due to theweight.For order 6 the presented algorithm provides relative closed loop

error , and for order 7 . This is a significantimprovement in comparison with the initial guess valuesand for orders 6 and 7, respectively. The improvement inthe -gap distance is even larger against for order6 and against for order 7. This improvementis not surprising, since in the weighted case there are no guaranteeson the initial guess quality. Note that the relative closed loop error ofcontroller of order 6 is comparable to of order 6, which resultsin order 12 controller . Hence, the frequency-weighted -gapmodel reduction is able to provide a much lower order approximation

than its non-weighted counterpart without sacrificing performance orrobustness.

VI. CONCLUSION

This technical note is devoted to the -gap model reduction algo-rithm. The main asset of the proposed approach is the use of the fre-quency domain data, which grants more flexibility. For example, thefrequency-weighted version of the proposed approach is obtained, andit is not clear if it is possible with [9]. The frequency-weighted ex-tension is validated on a numerical example, where reduced order con-trollers of significantly smaller orders were obtainedwithout sacrificingperformance levels. Other extensions are also easier to perform in thefrequency domain setting, since then the only obstacle is to formulatethe extension as a convex constraint.

ACKNOWLEDGMENT

The author would like to thank Dr. Sandberg for bringing attention tothis problem, Dr. Ranzter for valuable comments, and the anonymousreferees, whose comments contributed to the improvement of this tech-nical note.

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[3] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, ser.Advances in Design and Control. Singapore: SIAM, 2005.

[4] S. Gugercin, A. C. Antoulas, and C. Beattie, model reduction forlarge-scale linear dynamical systems, SIAM J.Matrix Anal. Appl., vol.30, no. 2, pp. 609638, 2008.

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[9] G. Buskes and M. Cantoni, A step-wise procedure for reduced orderapproximation in the -gap metric, in Proc. 47th IEEE Conf. DecisionControl, Dec. 2008, pp. 48554860.

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[17] J. F. Sturm, Using SeDuMi 1.02, A MATLAB toolbox for optimiza-tion over symmetric cones, Optim. Methods Softw., vol. 1112, pp.625653, 1999.

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Modified Bryson-Frazier Smoother Cross-Covariance

William R. Martin, Senior Member, IEEE

AbstractThe Expectation Maximization algorithm can be used to es-timate Kalman filter parameter matrices. This requires smoother resultsand typically the Rauch-Tung-Striebel smoother is used for this purpose.If either of the state transition or plant noise matrices require estimation,a sequence of single time-step smoother cross-covariance matrices, basedon the Rauch-Tung-Striebel smoother formulation, are also required. Themodified Bryson-Frazier smoother could be used as the smoother for theExpectation Maximization algorithm, but to estimate the state transitionor plant noise matrix, this smoother would first need a method to gen-erate the sequence of single time-step smoother cross-covariance matricesbased on the modified Bryson-Frazier formulation. This paper developsan expression for thesemodified Bryson-Frazier cross-covariancematrices,thus allowing the modified Bryson-Frazier smoother to be substituted forthe Rauch-Tung-Striebel smoother in the Expectation Maximization algo-rithm. The modified Bryson-Frazier smoother formulation requires inver-sion only of filter innovation matrices, which are generally much smallerthan state error covariance matrices, which allows both greater computa-tional speed and better numerical accuracy due to smaller matrix size. Thisformulation can be used even when covariance and transition matrices aresingular.

Index TermsEstimation, expectation maximization, Kalman filtering,linear dynamic system, smoothing.

I. INTRODUCTIONKalman filtering applications occasionally arise where the filter

designer has imperfect knowledge of some, many or all of the filterparameters for various filter matrices due to a variety of reasons;the physics is not tractable, there are too many filter states, the costinvolved in determining parameters is too high, or perhaps one hasonly an educated guess for values of Markov time constants, initialstate uncertainties, plant or measurement noise, or measurementmatrix coefficients.Under these conditions, it is possible to use the Expectation Max-

imization (EM) algorithm [1] to refine these Kalman filter parameterestimates. The application of the EM algorithm in this context requiresthe use of smoothing techniques. The Appendix in [1] shows the equa-tions for the filter, smoother, and cross-covariance required for theEM algorithm. We note that although [1] references Jazwinski [2] asthe source for the filter and smoother equations, we also recognize

Manuscript received March 31, 2011; revised November 09, 2011 andNovember 10, 2011; accepted June 27, 2012. Date of publication June 19,2013; date of current version December 19, 2013. Recommended by AssociateEditor Z. Wang.The author is with the Applied Physics Laboratory, Johns Hopkins University,

Laurel, MD 20723 USA (e-mail: william.martin@jhuapl.edu).Digital Object Identifier 10.1109/TAC.2013.2270074

Appendix equations (A8A10) [1, p. 263] as the discrete-time fixed-interval Rauch-Tung-Striebel (RTS) [3], [4, pp. 163164] smootherequations.In order to provide an alternate smoothing technique for the EM al-

gorithm, this paper develops the single time-step modified Bryson-Fra-zier (mBF) cross-covariance matrix equation for the mBF smoother.The paper is organized as follows. Section II investigates param-

eter estimation using the EM algorithm with the RTS smoother,with Section II-A showing the Kalman filter equations, Section II-Bshowing the RTS smoother equations, and Section II-C showing theRTS cross-covariance equation and EM algorithm as given in [1].Section III discusses the mBF smoother equations and Section IVshows the mBF equations with cross-covariance. Section V derives themBF cross-covariance, with Section V-A developing truth, predictionerror, and innovation equations and showing two innovation expecta-tion properties, Section V-B demonstrating the relationship betweenmBF adjoint vectors and matrices and defining filter cross-covariance,Section V-C reviewing the orthogonality principle, and Section V-Dderiving the mBF cross-covariance using results from the previoussubsections. Conclusions are given in Section VI.

II. PARAMETER ESTIMATION USING THE EM ALGORITHM

A. Kalman Filter EquationsThe time interval over which the filtering and smoothing will take

place is denoted by . measurements will occur from timesthrough . Fixed-interval filtering and smoothing will be used exclu-sively; define the Kroneker delta by for and .The system model is given by

(1)(2)

where and are the true system state and measurementvectors at time . Vectors , and are independentzero-mean Gaussian random variables with plant noise covari-ance matrix , measurement noise covariancematrix , a priori state error covariance ma-trix and state with

, , , is the statetransition matrix from time to , and is the measurementmatrix relating the state to the measurement at time . The Kalmanfilter time extrapolation equations are

(3)

(4)

where is the extrapolated state estimate at time given mea-surements from time through time , is the extrapolatedstate error covariance of and following [1], we takeand to simplify notation. The Kalman filter update equa-tions at measurement time are given by

(5)

(6)(7)(8)(9)

where is the innovation vector, is the innovation covariance ma-trix, is the Kalman (optimal) gain matrix, is the updated state

0018-9286 2013 IEEE