Optimal Unbiased Filtering via Linear Matrix Inequalities

Transcription

1 Proceedings of the American Control Conference Albuquerque, New Mexico June a03-3a $10.00 o I 997 AACC Optimal Unbiased Filtering via Linear Matrix Inequalities James T. Watson, Jr. and Karolos M. Grigoriadis' Department of Mechanical Engineering University of Houston Houston, TX Abstract Solutions to the optimal H, and L2 - L, unbiased reduced-order filtering problems are obtained in terms of Linear Matrix Inequalities (LMIs). The order of the optimal filter is equal to the number of measurements. Both continuous-time and discrete-time results are presented. An explicit parametrization of all optimal unbiased filters is provided in terms of a free contractive matrix. 1. Introduction The filtering problem is to estimate the states (or a linear combination of the states) of a system using past measurements. The celebrated Kalman filter provides a recursive algorithm to minimize the error variance of the state estimation when the power spectral density of the process and the measurement noise is known [6],[19],[17]. During the last three decades, Kalman filtering techniques have found widespread applications in aerospace guidance, navigation and control problems [9]. Recently, H, filtering has received considerable attention since, in contrast to the traditional Kalman filtering, it does not require knowledge of the statistical properties of the noise. The objective here is to minimize the energy of the estimation error for the worst-case bounded energy disturbance [3], [14], [12]. An overview of H, filtering can be found in [16]. Standard Kalman and Ha filtering approaches result in filters of order equal to the order of the system. Reduced-order filters, i.e. filters of order lower than the order of the system, are often desirable to reduce the complexity and computational burden of the real-time filtering process 1111, [l],[8]. Unbiased reduced-order filtering problems were considered in [13] and [ll] following an algebraic approach. In this paper, a Linear Matrix Inequality formulation is presented for the H, and L2 - L, unbiased reduced-order filtering. Solutions are obtained by solving Linear Matrix Inequality optimization problems and explicit parametrizations of all optimal unbiased filters are provided in terms of a contractive matrix. An algebraic approach proposed in [lo], [4], [18] is utilized. The notation to be used in this paper is as fol- 'Corresponding author. karolosquh. edu lows: The H, norm llglla of a rational transfer function G(s) is defined as llgll, w a(.) denotes the maximum singular value of a matrix. For a discrete-time transfer function G(r), the H, norm is ((G((, The L2 norm of a vec- tor valued function f (t) is ( l f l l ~ = ~ {STfT(t)f(t)dt}'12 and the L, norm is I l f l l ~, =sup {ft(t)f(t)}ll2. For a discrete-time vector-valued signal f(k), the 12 norm is llflllz = {CFfT(k)f(k)}'/2 and the 1, norm is Ilflll, =sup {ft(k)f(k)}"2. denotes the transpose k of a real matrix and (e)' denotes the Moore-Penrose generalized inverse of a matrix. The matrix norm (1. 11 is the maximum singular value of a matrix, that is IIAIJ = {X,,,(AAT)}1/2. Given a real n x m matrix B with rank T, an orthogonal complement BL is defined as the (possibly non-unique) (n- T ) x n matrix that satisfies BIB = 0 and BLBLT > 0. Hence, BL can be computed from the singular value decomposition of B as follows: BL = TU; T is an arbitrary nonsingular matrix and U2 is defined by the singular value decomposition The standard notation >, 2 (<, 2) is used to denote the positive (negative) definite and semidefinite ordering of matrices. The proofs of the results have been omitted. 2. Continuous-Time Unbiased Reduced-Order Filtering Consider an nth-order linear, time-invariant, continuoustime system with a state-space representation X = Ax+Bw y = Cx+Dw z = Lx x(t) E R" is the state vector, y(t) E Rp is the measured output and z(t) E R' is the signal to be estimated T < n. Here, w(t) E R' is a disturbance vector containing both process and measurement noise. The matrices t 2825 Authorized licensed use limited to: University of Houston. Downloaded on June 30, 2009 at 10:33 from IEEE Xplore. Restrictions apply.

2 A, B, C, D and L are real and of appropriate dimensions. The case of known initial condition x(0) = 50 is considered and without loss of generality it is assumed that xo = 0. The filtering problem is to estimate the signal z from the measurement y. We consider rth-order linear time-invariant filters of the following form 9= HE + Jy (4) 9(t) E RT is the estimate of z(t). The filter matrices H and J are real matrices of appropriate dimensions. The estimation error is equal to e = z - 2 = Lx - 9 and the estimation error dynamics is given by (5) t! = He + (LB - JD)w + (LA - HL - JC)x. (6) Stability of the error dynamics requires that H is Hurwitz. Unbiasedness of the filter requires that the estimation error dynamics be independent of the system state x, that is the following condition is satisfied [ 131, [ 111 LA-HL-JC=O. ( 7) Hence, the unbiased filter has an observer-type structure. By defining the following augmented matrices Using the results of Lemma 1, the unbiased filter error dynamics can be written in terms of the free matrix parameter Z as i: = Aee + Bew (13) Ae = de1 + 2Ae2, Be = Be1 + 2Be2 (14) and the known matrix parameters Ael, Ae2, Be1 and Be2 are defined by de1 = LAC'Z, de2 = (I -,CL+)Z, (15) Bel = LB + LAL'D, Be2 = (I - LLc+)D. (16) 3. Optimal H, Filtering The H, optimal filtering problem is to find a filter F to minimize the worst-case estimation error energy IlellL2 for all bounded energy disturbances w, that is llell Lz min sup -. 7 WEL2-{0} IlWllLZ (17) Hence, the optimal H, filter minimizes the energy gain of the system from the disturbance w to the estimation error e. This problem is equivalent to the following H, norm minimization problem mp IITwe(lo0 L J L J the unbiased filter error dynamics equation (6) can be written as 6 = FTe + (LB + FD)w (9) and the unbiasedness condition (7) as FL = LA (10) the augmented matrix F contains the unknown filter parameters H and J. Necessary and sufficient conditions for the existence of an unbiased filter and a parametrization of all solutions are obtained by the following result which follows from the general solution of a linear matrix equation [15]. Lemma 1 There exists an unbiased reduced-order filter (4) for the system (1)-(3) ij and only if LA(I - P L) = 0. (11) If this condition is satisfied, all unbiased reduced-order filter matrix parameters F are parametrized by 3 = LAL+ + 2(1- LL+) (12) 2 is a free matrix parameter of appropriate dimensions. T,, is the transfer function from the disturbance w to the estimation error e. The y-suboptimal H, filtering problem is to find (if it exists) a filter F such that IlTwellm < Y (18) y is a given positive scalar. The solution of the 7- suboptimal H, unbiased filtering problem is provided by the following result. Theorem 2 There exasts an rth-order unbiased filter to solve the y-suboptimal H, filtering problem if and only if the unbiasedness condition (11) is satisfied and there exists a matrix P > 0 such that If these conditions are satisfied, all filter parameters 2 in (13)-(14) that provide unbiased 7-suboptimal H, filters are parametrized as follows Z = -R-lrT@ATQ + Q1/2LQ1/2 (20) r=[ :]]A=[ Ae2 Be2], (21) 2826

3 R and L are free parameters subject to Q, = (rr-lrt -e)-' > 0, R > 0, 1 1 ~ 1 1 < 1 (23) and 52 and Q are defined by Sl = R-l - R-lI'T(Q,ATQA@)I', Q = (A@A)-'. (24) Condition (19) is a Linear Matrix Inequality on the variable P > 0, that is, the unbiased y-suboptimal H, filtering problem is an LMI feasibility problem. The op timal unbiased If, filter is obtained by solving the LMI optimization problem minimize y (25) P>O subject to the constraint (19). The optimal filter parameter 2 in (14) can also be computed as the solution of the following LMI with respect to 2 given the optimal values of P and y from problem (25), Ae and Be are defined by (14). This matrix inequality results from the Bounded Real Lemma [2], [18]. 4. Optimal L2 - L, Filtering The optimal LZ - L, filtering problem is to find a filter T to minimize the worst-case estimation error peak value llelllm over all bounded energy disturbances &, that is llell L, min sup -. 3 wel2-{o} IIWIILZ (27) The y-suboptimal L2 - L, filtering problem is to find (if it exists) a filter T such that sup - IlellL- <Y (28) WE&-(0) II~IILZ y is a given positive scalar. Hence, the y-suboptimal LZ - L, filtering condition (28) guarantees that the peak value of the filtering error will be bounded by yllwll~~ for any disturbance w with bounded energy. The solution of the y-suboptimal LZ - L, unbiased filtering problem is provided by the following result. Theorem 3 There exists an rth-order unbiased filter to solve the y-suboptimal L2-L, filtering problem if and only if the unbiasedness condition (11) is satisfied and there exists a matrix P > I such that If these conditions are satisfied, all filter parameters Z in (13) that provide unbiased y-suboptimal L2 - L, filters are parametrized as 2 = -R-lrT@.RTQ + Q1/2LQ1/2 (30) and Q,, R and L are free parameters subject to Q, = (rr-lrt- e)-' > 0, R > 0, ll~ll< 1 (33) and Cl and 9 are defined by 52 = R-l - R-lI'T(@ATQA@)I', Q = (h@a)-l. (34) Condition (29) is an LMI on the variables P and y. The optimal Lz - L, unbiased filtering problem is solved by solving the LMI optimization problem minimize y (35) P> I subject to the constraint (29). As in the H, case, an optimal filter parameter 2 can also be computed as the solution of an LMI problem, that is, solve PA,+AFP PBe <o [ 1 BTP -721 with respect to 2, A,, Be are defined by (14) and P and y are obtained from (35). 5. Discrete-Time Unbiased Reduced-order Filtering Next, the discrete-time unbiased reduced-order filtering problem is examined: Given a nth-order linear, timeinvariant, discrete-time system ~ ( k + 1) = Az(k) + Bw(k) (37) y(k) = CX(k) +Dw(k) (38) z(k) = Ls(k) (39) x(k) E R", y(k) E RP, z(t) E RT and w(t) E R', we seek to obtain an rth-order discretetime filter F, r < n, of the form f(k + 1) = Hf(k) + Jy(k) (40) to minimize the estimation error e(k) = z(k) - f(k) = Lz(k) - 3(k). (41) As in the continuous-time case, an unbiased discrete-time filter exists if and only if the unbiasedness condition (11) is satisfied and the unbiased filter error dynamics is given by e(k -t 1) = Aee(k) + BeW(k) (42) A, and Be are defined as in (14)-(15) Authorized licensed use limited to: University of Houston. Downloaded on June 30, 2009 at 10:33 from IEEE Xplore. Restrictions apply.

4 The optimal discrete-time H, filtering problem corresponds to solving min sup - llel112 W Z2-{0} IlWlllZ (43) subject to (45). The optimal filter parameter 2 in (14) can be also computed as the solution of the following LMI with respect to 2 which is equivalent to the H, optimization mf IITweIlm Twe is the transfer function from w to e. The y- suboptimal H, filtering problem is to find (if it exists) a filter F such that IlTwellOo < (44) y is a given positive scalar. As in the continuous time case, the discretetime H, filtering minimizes the energy of the estimation error for all bounded energy disturbance signals. Theorem 4 There exists a discrete-time rth-order unbiased filter to solve the discrete-time y-suboptimal H, filtering problem if and only if the unbiasedness condition (11) is satisfied and there exists a matrix P > I such that P - AF1 PAe1-I -Az1 PBe1 >O. (45) If these conditions are satisfied, all filter parameters 2 in (14) that provide unbiased y-suboptimal H, filters are parametrized as IT given the optimal values of P and y from problem (52), A, and Be are defined by (14). This matrix inequality results from the discrete Bounded Real Lemma [lo],[4], P81. The optimal discretetime 12 - I, filtering problem is to find a filter.f to minimize the worst-case estimation error peak value Ilelll, over all bounded energy disturbances w, that is min sup - Ilell1, w l2-{0} 114ll2 (54) The y-suboptimal Z2-1, filtering problem is to find (if it exists) a filter F such that sup - Ilell1, <Y (55) IlwIlz2 W Zz-{O} y is a given positive scalar. Theorem 5 There exists a discrete-time rth-order unbiased filter to solve the y-suboptimal12-i, filtering problem if and only if the unbiasedness condition (11) is satisfied and there exists a matrix P > I such that P - AT1 PAe1 -A; PB,~ IT > O. (56) If these conditions are satisfied, all filter parameters 2 in (42) that provide discrete-time, unbiased y-suboptimal12-1, filters are parametrized as 2 = 2 1 -k (57) U is any matrix such that llull < 1, and P-l 0 P-1 0.=[ 0 ++=[ 0 I1. (51) The optimal discretetime unbiased tained by solving the LMI optimization minimize y P>I J H, filter is ob- P-1 0 R=[ 0 +I]* 2828

5 The optimal filter parameter 2 in (14) can also be computed as the solution of the following LMI with respect to 2 P-1 Ae t?, (63) given the optimal values of P and y from problem (52), A, and Be are defined by (14). 6. Numerical Example Consider the following state-space model of a double inte grator: L 1 L J 1 0 = [o 1]x+[: OiIl R,]- z = [o 11 The disturbance vector w contains both process and me& surement noises. We seek to design optimal first-order H, and Lz - L, unbiased filters to estimate the statevariable 22 = z from the noisy measurements y. The optimal H, unbiased filter is obtained using the results of Theorem 2, as follows: B= [ ] y, and the optimal H,-norm error is yopt = The optimal Lz - L, unbiased filter is obtained using the results of Theorem 3, as follows E= [ ] y, and the optimal LZ - L, error is Yopt = Conclusion An explicit characterization of the solutions to the unbiased optimal H, and LZ - L, filtering problem was provided. Necessary and sufficient conditions were obtained in terms of LMIs and a parametrization of all reduced-order filters was derived in terms of a contractive matrix. Both continuous-time and discretetime results were presented. References Bettayeb, M. and D. Kavranoglu, Reduced order H, filtering, Proc Amer. Control Conf., Baltimore, Maryland, Boyd, S, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM Studies in Appl. Mathematics, Philadelphia, Doyle, J., K. Glover, P. Khargonekar and B. Francis, State-space solutions to standard H2 and H, control problems, IEEE Trans. Automatic Control, Vol. 34, pp , [4] Gahinet, P. and P. Apkarian, A linear matrix inequality approach to H, control, Int. J. Robust Nonlinear Control, Vol. 4, pp , [5] Gahinet, P., A. Nemirovski, A. Laub, and M. Chilali, LMI Control Toolbox For Use with MATLAB, The Mathworks Partner Series, Massachusetts, [6] Gelb, A., Kasper, J. FF., Jr., Nash, R. A., Jr., Price, C. F., and Sutherland, A. A., Applied Optimal Estimation, MIT Press, Cambridge, Massachusetts, [7] El Ghaoui, L. and P. Gahinet, Rank minimization under LMI constraints: A framework for output feedback problems. In Proc. European Control Conf., Groningen, The Netherlands, [8] Grigoriadis, K., and Watson, J., Jr., Reduced-order H, Filtering via Linear Matrix Inequalities, In Proc. 13th IFA C World Congress, San Francisco, California, Also, IEEE Trans. of Aerospace and Electronic Systems, to appear. [9] Hutchinson, C. E., The Kalman Filter Applied to Aerospace and Electronic Systems, IEEE Trans. Aerosp. Electronic Systems, Vol. AES-20, No. 4, [lo] Iwasaki, T. and R. Skelton, All controllers for the general H, control problem: LMI existence conditions and state space formulas, Automatica, Vol. 30, pp , [Ill Kim, H., C. Sims and K. Nagpal, Reduced order filtering in an H, setting, In Proc Amer. Control Conf., Chicago, Illinois, [12] Limebeer, D. and U. Shaked, New results in H, filtering,! In Proc Math. Theory Networks Sys., Kobe, Japan, [13] Nagpal, K. Helmick, R. and Sims, C., Reduced-Order Estimation: Part 1. Filtering, Int. J. Control, Vol. 45, NO. 6, pp , [14] Nagpal, K. and P. Khargonekar, Filtering and Smoothing in an H, Setting, IEEE Trans. Automatic Control, Vol. 36, pp , [15] bo, C., and Mitra S., Generalized Inverse of Matrices ant its Applications, John Wiley & Sons, New York, [16] Shaked, U. and Y. Theodor, H,-optimal estimation: A tutorial, In Proc Conf. Dec. Control, Tucson, Arizona, [17] Siouris, G., An Engineering Approach to Optimal Control and Estination Theory, John Wiley & Sons, New York, [18] Skelton, R. T. Iwasaki and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. Taylor & Francis, London, [19] Sorenson, H. (Ed.), Kalman Filtering: Theory and Applications, IEEE Press, New York,