Copyright [2006] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this letter, we consider a multiobjective filtering problem for uncertain linear continuous time-invariant systems subject to error variance constraints. A linear filter is used to estimate a linear combination of the system states. The problem addressed is the design of a filter such that, for all admissible parameter uncertainties, the following three objectives are simultaneously achieved: 1) the filtering process is P-stable, i.e., the poles of the filtering matrix are located inside a parabolic region; 2) the steady-state variance of the estimation error of each state is not more than the individual prespecified value; and 3) the transfer function from exogenous noise inputs to error state outputs meets the prespecified H∞ norm upper-bound constraint. An effective algebraic matrix inequality approach is developed to derive both the existence conditions and the explicit expression of the desired filters. An illustrative example is used to demonstrate the usefulness of the proposed design approach

Fang, J

Wang, Z

IEEE Signal Processing Letters

English

Brunel University Research Archive

IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 3, MARCH 2006 137Robust H Filter Design With VarianceConstraints and Parabolic Pole AssignmentZidong Wang, Senior Member, IEEE, and Jian’an FangAbstract—In this letter, we consider a multiobjective filteringproblem for uncertain linear continuous time-invariant systemssubject to error variance constraints. A linear filter is used toestimate a linear combination of the system states. The problemaddressed is the design of a filter such that, for all admissibleparameter uncertainties, the following three objectives are si-multaneously achieved: 1) the filtering process is -stable, i.e.,the poles of the filtering matrix are located inside a parabolicregion; 2) the steady-state variance of the estimation error ofeach state is not more than the individual prespecified value; and3) the transfer function from exogenous noise inputs to error stateoutputs meets the prespecified norm upper-bound constraint.An effective algebraic matrix inequality approach is developed toderive both the existence conditions and the explicit expression ofthe desired filters. An illustrative example is used to demonstratethe usefulness of the proposed design approach.Index Terms—Algebraic matrix inequality, error variance con-straints, filtering, Kalman filtering, pole assignment.I. INTRODUCTIONTHE FILTERING problem has been playing an importantrole in signal processing and control engineering. Amongvarious filtering schemes, the celebrated Kalman filtering ap-proach minimizes the norm of the estimation error, underthe assumptions that an exact model is available and the noiseprocesses have exactly known statistical properties. To improvethe robustness of Kalman filters, in the past decade, the robustperformance of the designed filters has become an impor-tant issue (see, e.g., [2], [3], [10], and [11]). On the other hand,it is quite common in filtering problems, such as the tracking ofa maneuvering target and recognition of flight paths from mul-tiple sources, to have performance objectives that are directlyexpressed as upper bounds on the variances of the estimationerror [6]. These prescribed variance restrictions may not be min-imal but should meet engineering requirements, and therefore,the obtained filters are often nonunique [7]–[9], [12].It is well known that, by constraining the poles of the filteringmatrix to lie inside a prescribed region in the open left-halfManuscript received June 6, 2005; revised September 13, 2005. This workwas supported in part by the Engineering and Physical Sciences ResearchCouncil (EPSRC) of the U.K. under Grant GR/S27658/01, in part by theNuffield Foundation of the U.K. under Grant NAL/00630/G, and in part bythe Alexander von Humboldt Foundation of Germany. The associate editorcoordinating the review of this manuscript and approving it for publication wasProf. Amy E. Bell.Z. Wang is with the Department of Information Systems and Computing,Brunel University, Uxbridge, UB8 3PH, U.K., and is also with the School ofInformation Sciences and Technology, Donghua University, Shanghai 200051,China (e-mail: Zidong.Wang@brunel.ac.uk).J. Fang is with the School of Information Sciences and Technology, DonghuaUniversity, Shanghai 200051, China.Digital Object Identifier 10.1109/LSP.2005.862618plane, the filter designed would have expected transient perfor-mance. Besides, regional pole assignment can also provide indi-rect tolerance against plant uncertainties. In the past few years,the filter design problem with regional pole placement has re-ceived initial research attention (see, e.g., [11] and referencestherein). As indicated in [4], for linear time-invariant contin-uous systems, a parabolic region is directly related to the max-imum percent overshoot and the rise time. Therefore, assigningthe poles of the filtering matrix inside a prespecified parabolicregion would guarantee satisfactory transient behavior of the fil-tering dynamics. It should be pointed out that a parabolic regioncannot be simply represented by the intersections of linear ma-trix inequality (LMI) regions. Hence, the methods developed in[11] are not applicable for parabolic pole assignment.Motivated by the above discussion, in this letter, it is our in-tention to deal with the robust filtering problem for un-certain linear continuous time-invariant systems with both errorvariance and parabolic pole constraints, so that the resulting fil-tering process will be provided with expected transient property,steady-state error variance constraint, and disturbance rejectionbehavior, in the presence of parameter uncertainties.Notation: and denote the -dimensional Eu-clidean space and the set of all real matrices, respec-tively. The superscript “ ” denotes the transpose. The notation(respectively, ), where and are sym-metric matrices, means that is positive semi-definite(respectively, positive definite). is the identity matrix withcompatible dimension. stands for the mathematical expec-tation operator with respect to the given probability measure .and represent the real and imaginary parts of thecomplex number , respectively.II. PROBLEM FORMULATION AND ASSUMPTIONSConsider the following class of linear uncertain continuous-time systems(1)(2)where , , and , , , and are knownconstant matrices. is a zero-mean Gaussian white noiseprocess with covariance . The initial state has themean and covariance and is uncorrelated with .and are real-valued perturbation matrices satisfying(3)1070-9908/$20.00 © 2006 IEEEAuthorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 06:47 from IEEE Xplore.  Restrictions apply.138 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 3, MARCH 2006where is a real uncertain matrix satisfying ,and , , and are known constant matrices of appropriatedimensions.Assumption 1: The system matrix is Hurwitz stable, andthe matrix or is of full-row rank.The linear full-order filter is given by(4)where denotes the state estimation, and and are filterparameters to be determined.The estimation error covariance in the steady state is denotedby , where, if the limit exists. By definingand considering (1), (2), and (4), we obtain the following aug-mented system:(5)When the system (5) is robustly asymptotically stable, thesteady-state covariance defined by(6)exists and satisfies the following Lyapunov matrix equation:(7)Fact 1: Let and be two positive scalars. If acomplex number satisfies(8)then is located within a parabolic region in the left-hand side of the complex plane where the vertex is at , asshown in Fig. 1.Assume that the error state outputs are represented by ,where is a known constant matrix of appropriate dimension.Given a desired parabolic pole region , we are now readyto state the variance-constrained filtering problem consid-ered for uncertain continuous systems. Our objective is to seekthe filter parameters and such that for all admissible pa-rameter uncertainties, the following three requirements are si-multaneously satisfied:1) The system (5) is -stable, i.e., all poles of the filteringmatrix remain within the parabolic pole region.2) The steady-state error covariance meets(9)Fig. 1. Parabolic region P(a; b).where means the th diagonal element of , i.e., thesteady-state variance of the th state.denotes the prespecified steady-state error estimationvariance constraint on the th state and can be determinedby the practical performance requirements.3) The norm of the transfer functionfrom disturbancesto error state outputs (or ) sat-isfies the constraint , where is theknown error state output matrix, , whereand denotesthe largest singular value of ; and is a given positiveconstant.III. MAIN RESULTS AND PROOFSLemma 1: [7] Let , , , , and be real matricesof appropriate dimensions with and satisfying. Then, for any scalars andwhere , we have: 1); and 2).The main results of this letter are given as follows, whichshow that the -stability, performance, and the steady-stateconstraints on the filtering process are closely related to the pos-itive definite solutions to a pair of Riccati-like matrix equations.For notational simplicity, we first make the followingdefinitions:(10)(11)(12)(13)(14)(15)Authorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 06:47 from IEEE Xplore.  Restrictions apply.WANG AND FANG: ROBUST FILTER DESIGN WITH VARIANCE CONSTRAINTS 139Theorem 1: Assume that the norm upper-boundand the parabolic region are given. Let andbe sufficiently small constants and be anarbitrary orthogonal matrix. If there exist scalars ,, and a matrix such that and theRiccati-like matrix equations(16)(17)have positive definite solutions and , respec-tively, where , , , , , and are defined in (10)–(13),then with the parameters determined by(18)the filter (4) will be such that, for all admissible perturbationsand : 1) the filtering matrix satisfies the par-abolic pole constraint ; 2); and 3) the steady-state error covariance exists and meets.Proof: We first show that , where is defined in(15). It follows from Lemma 1 that(19)(20)Set Block-diag . After tedious calculation, wehave from (19) and (20) that(21)where(22)(23)(24)Equation (16) means that , and the ex-pression of in (18) implies . From Assumption 1,we know that is invertible. Now, substituting the expressioninto (24), we can verify that(25)where and are defined in (13) and (14), respectively.Note that (17) is actually the same as. Using the expression of in (18) and fact ,we obtain from (25) thatNow, we have the conclusion that . Let us firstprove that .1) Denote . Let ,and it follows from (15) that(26)Let be an eigenvalue of and be theassociated eigenvector. Then, we have(27)where denotes the complex conjugate transpose. Pre-multiplying and post-multiplying (26) by and , re-spectively, yield(28)or , which implies fromFact 1 that the eigenvalues of are situated insidethe parabolic region .2) Rearrange (26) as(29)where .Then, the proof of can be completed by astandard manipulation of (29); see [7] for more details.3) Equation (26) can be further transformed into(30)where. Notice that is asymptot-ically stable. Subtract (7) from (30) to give, orequivalentlywhich means that and . The proof ofTheorem 1 is then completed.In view of Theorem 1, if the positive definite solutionsand to (16) and (17) exist, and meets, , we will have the following conclusions:1) the augmented system (5) is -stable; 2) ;and 3) , . Hence, with thefilter (4) whose parameters and are determined by (18), theAuthorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 06:47 from IEEE Xplore.  Restrictions apply.140 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 3, MARCH 2006variance-constrained robust filtering gain design task willbe accomplished.Finally, a solution to the addressed filter design problem isgiven as a corollary from Theorem 1.Corollary 1: Let the desired parabolic pole region ,the disturbance attenuation constraint , and thesteady-state error variance constraintsbe given. If there exist positive definite matrices andsuch that the conditions of Theorem 1 are all satisfied,then a desired filter gain for the addressed filter designproblem can be obtained by (18).Remark 1: In practical applications, it is very desirable todirectly solve Riccati matrix equations (16) and (17), subjectto the constraint , and then ob-tain the expected filter parameters readily from (18). We cansee that the key step in designing the expected filters is to con-sider the solvability of the Riccati-like equations (16) and (17).When we deal with (16) and (17), the local numerical searchingalgorithms suggested in [1] are very effective for a relativelylow-order model. A related discussion of the solving algorithmsfor Riccati-like equations can also be found in [5].IV. NUMERICAL EXAMPLEIn this section, a simple design example is presented to illus-trate the usefulness and exibility of the theory developed in thisletter.Consider a linear continuous-time uncertain stochasticsystem (1) and (2), with parameters as follows:It is desired to design robust filter (4) such that: 1) the poles ofthe filtering matrix are all constrained to lie inside theparabolic region ; 2) the transfer function fromdisturbances to error state outputs satisfies the con-straint ; and 3) the steady-state covarianceexists and satisfies , .By setting , , and, we solve the Riccati-like matrix equations (16) and(17) and obtainClearly, . Then, for orthogonal “ma-trices” and , the corresponding expected filterparameters in these two cases can be obtained from (18), respec-tively, as the following:It is easy to verify that all specified performance requirementsare achieved.V. CONCLUSIONIn this letter, we have considered a parabolic pole and vari-ance-constrained robust filtering problem for linear contin-uous-time systems. It has been shown that this filtering problemcan be converted into an auxiliary problem that is related to thesolutions of two Riccati-like matrix equations. The existenceconditions and the analytical expression of desired estimatorshave been characterized, and a numerical example has been ex-ploited to show the effectiveness of the proposed design method.Note that only sufficient conditions have been obtained in ourmain results, and one of our future research topics would behow to reduce the conservatism of the design.REFERENCES[1] E. Beran and K. Grigoriadis, “A combined alternating projections andsemidefinite programming algorithm for low-order control design,” inProc. 13th IFAC World Congress, vol. C, San Francisco, CA, 1996, pp.85–90.[2] H. Gao and C. Wang, “Comments and further results on ‘A descriptorsystem approach to H control of linear time-delay systems’,” IEEETrans. Autom. Control, vol. 48, no. 3, pp. 520–525, Mar. 2003.[3] H. Gao, J. Lam, L. Xie, and C. Wang, “New approach to mixedH =H filtering for polytopic discrete-time systems,” IEEE Trans.Signal Process., vol. 53, no. 8, pp. 3183–3192, Aug. 2005.[4] N. Kawasaki and E. Shimemura, “Determining quadratic weighting ma-trices to locate poles in a specified region,” Automatica, vol. 19, pp.557–560, 1983.[5] A. Saberi, P. Sannuti, and B. M. Chen,H Optimal Control, ser. Systemsand Control Engineering. London, U.K.: Prentice-Hall Int., 1995.[6] R. E. Skelton and T. Iwasaki, “Liapunov and covariance controllers,” Int.J. Control, vol. 57, pp. 519–536, 1993.[7] Z. Wang and B. Huang, “Robust H =H filtering for linear systemswith error variance constraints,” IEEE Trans. Signal Process., vol. 48,no. 8, pp. 2463–2467, Aug. 2000.[8] Z. Wang, B. Huang, and P. Huo, “Sampled-data filtering with error co-variance assignment,” IEEE Trans. Signal Process., vol. 49, no. 3, pp.666–670, Mar. 2001.[9] Z. Wang, D. W. C. Ho, and X. Liu, “Variance-constrained filtering foruncertain stochastic systems with missing measurements,” IEEE Trans.Autom. Control, vol. 48, no. 7, pp. 1254–1258, Jul. 2003.[10] L. Xie, L. Lu, D. Zhang, and H. Zhang, “Improved robust H and Hfiltering for uncertain discrete-time systems,” Automatica, vol. 40, no. 5,pp. 873–880, 2004.[11] F. Yang and Y. S. Hung, “Robust mixedH =H filtering with regionalpole assignment for uncertain discrete-time systems,” IEEE Trans. Cir-cuits Syst. I, Reg. Papers, vol. 49, no. 8, pp. 1236–1241, Aug. 2002.[12] E. Yaz and R. E. Skelton, “Continuous and discrete state estimation witherror covariance assignment,” in Proc. IEEE Conf. Decision Control,Brighton, U.K., 1991, pp. 3091–3092.Authorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 06:47 from IEEE Xplore.  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Robust H∞ filter design with variance constraints and parabolic pole assignment

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