Discrete-Time H Gaussian Filter

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1 Proceedings of the 17th World Congress The International Federation of Automatic Control Discrete-Time H Gaussian Filter Ali Tahmasebi and Xiang Chen Department of Electrical and Computer Engineering, University of Windsor, Windsor, Ontario, Canada, 9B 3P4 Corresponding Author, Department of Electrical and Computer Engineering, University of Windsor, Windsor, Ontario, Canada, 9B 3P4 ( xchen@uwindsor.ca) Abstract: A discrete-time signal estimator for systems subject to both white noise and bounded-power disturbance signals is developed. Sufficient and necessary conditions for the robust optimal filter are proved and the resulting filter gain is characterized by a set of two cross-coupled Riccati equations. 1. ITRODUCTIO It is generally known that signal estimation for the dynamic systems is one of the most important problems in engineering applications [Anderson, 1979, Petersen and Savkin, 1999]. The popular Kalman filter[anderson, 1989], also known as H 2 filter, is an optimal design that is based on the stochastic noise model with known power spectral densities. However, this technique may be very sensitive to changes in system parameters or other disturbances with unknown spectral densities. For such cases, a better choice is to use an H filter, which is developed specifically to address model uncertainty [Fu et al., 1992, agpal et al., 1991], and different techniques have been well developed and applied for different systems (see for example [Gao and Wang, 2003, i and Fu, 1997, Xu and Chen, 2002] and the references therein). Although H filter usually provides much better robustness than H 2 filter, it, in general, does not guarantee desired optimal performance if it is applied to a system affected by stochastic noise. Clearly, a mixed H 2 /H filter design scheme that can combine the strengths of these two estimation methods in a systematic way is highly desirable. Several methods have been proposed to carry out the robust optimal filter design and a few examples are given here for different approaches to this problem. One method is to convert the problem to an auxiliary minimization one [Bernstein et al., 1989]. In this approach, an H 2 estimation error is minimized which is subject to a prespecified H constraint. This constraint is introduced in the optimization process by a Riccati equation whose solution leads to an upper bound on the H 2 error variance. In [Rotstein et al., 1996], [Halder et al., 1997] and [Khargonekar et al., 1996], the mixed H 2 /H filters are obtained using convex programming characterization. For systems with norm-bounded parameter uncertainties, the problem is solved in [Wang et al., 2000] and [Wang et al., 1999] by using Riccati-like equations, where the transfer function from the noise inputs to error state outputs meets an H -norm upper bound constraint. For discrete-time polytopic systems, [Palhares et al., 2001] obtains the mixed H 2 /H filters by solving a set of linear matrix inequalities (MIs), while [Gao et al., 2005] uses the parameter-dependent stability idea and finds a filter that depends on the parameters, which are assumed This work is supported in part by SERC Grant. to reside in a polytope and be measurable online. A time domain game theoretic approach is proposed in [Theodor et al., 1996] which improves the H 2 performance of the central H filter while satisfying the required H performance. For systems with polytope-bounded uncertainty, a pole-placement design strategy is proposed in [Goncalves et al., 2006] which utilizes an optimization algorithm in the space of filter parameters. In [Chen and Zhou, 2002], utilizing the game approach, a new formulation called H Gaussian filter is proposed, and it is shown that the robust optimal filter can be obtained by solving a set of cross-coupled Riccati equations. The result is a Kalmantype filter for uncertain plants and is characterized by the choice of the disturbance attenuation level γ. One advantage of this approach is that optimal state estimation is achieved at the presence of the worst case model uncertainty. Therefore, it clearly reflects the trade-off between the inherently conflicting H 2 and H performances. Motivated by the approach in [Chen and Zhou, 2002], in this paper, the ash game methodology is adopted to derive a mixed H 2 /H filter in discrete time. The design is based on a constrained optimization problem and is characterized by two cross-coupled Riccati equations. As it can be seen, obtaining the discrete-time counterpart of the continuous procedure is not so straightforward. An optimal filter gain is characterized by an equation consisting of the plant parameters and the solutions to the Riccati equations. The remaining part of this paper is organized as follows: Section II presents some definitions and preliminary results required for obtaining the main solution; in Section III, the problem formulation and the design of the discrete-time mixed H 2 /H filter is provided; the conclusion can be found in Section IV. otations: Throughout this work, if a is a vector and A is a matrix of arbitrary dimensions, a T and A T represent their transposes, respectively. We use the notation D := {z : z = 1} to describe the points on the unit circle in the complex plane. Furthermore, when working with a discrete time signal u (k) (scalar- or vector-valued), for simplicity of presentation, we denote δu := u (k+1), and then the time variable k will be omitted. represents the Euclidian norm of a vector /08/$ IFAC / KR

2 2. PREIMIARIES The following theorem presents the solution to this problem. Only the proof of the sufficiency part is given here. The neces- part of the proof can also be carried out, however, it consists Definition 1. (Bounded power signal). Consider the given discrete-sittime real vector stochastic signal u (k) : of more derivations that could not be included due to page limit. u (k) = [u 1(k) u 2(k) u m(k) ] T R m k Z, where u i(k), i = 1,..., m are real stationary discrete random processes, we define the mean and autocorrelation matrices, respectively, as: E{u} := [E{u 1(k) } E{u 2(k) } E{u m(k) }] T, R uu(n) = lim E{u (k+n) u T (k) }. where n is an integer. The Fourier transform of R uu(n), or the power spectral density of u (k), is: Theorem 4. For the constrained optimization problem stated above, suppose (C, A) is detectable. If there is a solution P 0 for: P = AP A T (BD T + AP C T )(R + CP C T ) 1.(DB T + CP A T ) + BB T, i.e., A (AP C T +BD T )(R +CP C T ) 1 C is stable, where R + CP C T > 0, then J() achieves the minimum value at: = (AP C T + BD T )(R + CP C T ) 1. (4) S uu = 1 2π k= R uu(k) e jωk. A stationary stochastic vector signal is said to have bounded power if: both R uu and S uu exist; lim E{ u (k) 2 2 } <. Definition 2. (P-norm). et P be the set of all signals with bounded power, we define the seminorm: u 2 P = lim E{ u (k) 2 }. (1) Conversely, let (C, A) be detectable. If there are 1 and P 1 0, where A + 1 C is stable and P 1 solves: P 1 = (A + 1 C)P 1 (A + 1 C) T + (B + 1 D)(B + 1 D) T, such that J() is minimized at 1, then there is a P 0 solving (4), where R + CP C T > 0. Moreover, the optimal can be found as: = (AP C T + BD T )(R + CP C T ) 1, if A + C is Hurwitz. proof(sufficiency) For any for which A+C is stable, there is a P 0 solving: P = (A + C)P(A + C) T + (B + D)(B + D) T. Definition 3. (Mutually uncorrelated signals). Two stochastic vector signals u 1 and u 2 are said to be mutually uncorrelated if: E{u 1(k1)u T 2(k 2) } = 0, k 1, k 2 Z. Constrained Optimization On the other hand, since P is a stabilizing solution, so A+ C is stable, for = (AP C T +BD T )(R+CP C T ) 1. Using (4), (3) can be rewritten as: P =APA T + APC T T + CPA T + CPC T T (R + CP C T ) T CP A T AP C T T + (R + CP C T ) T (R + CP C T ) T AP A T + P + R T. The constrained optimization problem presented in this section plays an important role in the main derivations of this work. Given A R n n, B R n r, C R p n, D R p r and R = DD T > 0, let be any consistent matrix such that A + C is stable in discrete time sense and define the index function: J() = trace(qpq T ), (2) where Q is any constant weighting matrix, and P = P T 0 satisfies: P = (A + C)P(A + C) T + (B + D)(B + D) T. (3) The constrained optimization problem is stated as follows: find (, P ) where A+ C is stable, such that J() is minimized at, i.e.: min J() = min P trace(qpqt ) = trace(qp Q T ), where (, P) and (, P ) are all subject to constraint (3). If we define P = P P, then the above expression can be simplified into: P =(A + C) P(A + C) T + ( )(R + CP C T )( ) T. From this yapunov equation, it is obvious that P 0 and also P = 0 if and only if =. Hence: J() J( ) = trace(q PQ T ) 0, or in other words, J() achieves the minimum value at, which concludes the proof for the sufficiency condition. 3. DISCRETE-TIME H FITER DESIG Consider the filter design problem in Fig. 1. For the plant G, described by: 13534

3 Theorem 5. et the plant G be described by the equation set (5), where w and w 0 are assumed to be uncorrelated, and the cost functions J 1 and J 2 are defined as (9) and (10). If there are stabilizing solutions P 1 0 and P 2 0 to: Fig. 1. H Gaussian filter δx = Ax + B 0 w 0 + B 1 w, x (0) = 0, z 0 = C 0 x, (5) z = C 1 x, y = C 2 x + D 20 w 0, where w is a bounded power signal and w 0 is a white noise signal. The following standard assumptions are made: (A1) (C 2, A) is detectable ; (A2) R 0 := D 20 D20 T > 0 ; A λi B0 (A3) has full row rank, λ D. C 2 D 20 The goal is to find a filter: F : y [ ẑ ẑ 0 ], (6) where ẑ and ẑ 0 are estimates of z and z 0, respectively. The filter is to be designed as: δˆx = Aˆx + (C 2ˆx y), ẑ 0 = C 0ˆx, (7) ẑ = C 1ˆx, where is the filter gain to be calculated. Define the following error variables: e := z ẑ, e 0 := z 0 ẑ 0, e x := x ˆx, (8) and the cost functions as: J 1 (F, w, w 0 ) = γ 2 w 2 P e 2 P, (9) J 2 (F, w, w 0 ) = e 0 2 P. (10) The discrete-time H Gaussian filter design problem is stated as follows: Find a filter F in the form (7) and a worst disturbance signal w such that they achieves: J 1 (F, w, w 0 ) J 1 (F, w, w 0 ), J 2 (F, w, w 0 ) J 2 (F, w, w 0 ). for any bounded power signal w w and any other admissible F. Combining the equations of the plant and the filter and implementing e x := x ˆx, derive: δe x = (A + C 2 )e x + (B 0 + D 20 )w 0 + B 1 w, e 0 = C 0 e x, e = C 1 e x. The design is presented in the following theorem. (11) P 1 =ĀT P 1 Ā + γ 2 Ā T P 1 B 1 (I γ 2 B T 1 P 1 B 1 ) 1 B T 1 P 1 Ā + C T 1 C 1, P 2 =A F P 2 A T F (B 0D T 20 + A F P 2 C T 2 )(R 0 + C 2 P 2 C T 2 ) 1.(D 20 B T 0 + C 2 P 2 A T F) + B 0 B T 0, where (I γ 2 B T 1 P 1 B 1 ) > 0, R 0 + C 2 P 2 C T 2 > 0 and: Ā = A + C 2, 1 = I γ 2 B 1 B T 1 P 1, (12) (13) A F = (I + γ 2 B 1 B T 1 P )A + γ 2 B 1 B T 1 P C 2. Then by choosing that satisfies: = (A F P 2 C T 2 + B 0 D T 20)(R 0 + C 2 P 2 C T 2 ) 1, (14) the filter F : δˆx = (A + C 2 )ˆx y, ẑ 0 = C 0ˆx, ẑ = C 1ˆx, and the worst disturbance signal: achieve: w = γ 2 B1 T P Āe x, J 1 (F, w, w 0 ) J 1 (F, w, w 0 ), J 2 (F, w, w 0 ) J 2 (F, w, w 0 ). Conversely, if there exists a filter F, with a worst disturbance signal w, such that for the system without white noise, we have: 0 < J 1 (F, w, 0) J 1(F, w, 0), w w, and a worst disturbance signal w at the presence of white noise, such that: J 1 (F, w, w 0 ) J 1 (F, w, w 0 ), J 2 (F, w, w 0 ) J 2 (F, w, w 0 ), then, there exist stabilizing solutions P 1 0 and P 2 0 to: P 1 =ĀT P 1 Ā + γ 2 Ā T P 1 B 1 (I γ 2 B T 1 P 1B 1 ) 1 B T 1 P 1Ā + C T 1 C 1, P 2 =A F P 2 A T F (B 0D T 20 + A F P 2 C T 2 )(R 0 + C 2 P 2 C T 2 ) 1.(D 20 B T 0 + C 2P 2 A T F ) + B 0B T 0, where (I γ 2 B T 1 P 1B 1 ) > 0 and R 0 + C 2 P 2 C T 2 > 0. Moreover, the optimal value of the filter gain satisfies: = (A F P 2 C T 2 + B 0D T 20 )(R 0 + C 2 P 2 C T 2 ) 1. proof (Sufficiency) If we choose the filter gain, that satisfies: = (A F P 2 C T 2 + B 0 D T 20)(R 0 + C 2 P 2 C T 2 ) 1, completing the squares, using (12), we have: 13535

4 J 1 (F, w, w 0 ) =γ 2 w 2 P e 2 P =γ 2 w w 2 P lim trace ( (B 0 + D 20 ) T.P 1 AE{xw0 T } ) =γ 2 w w 2 P, where w = γ 2 B1 T P Āe x, which is bounded since A + C 2 + B 1 w is stable. ext it is shown that J 1 achieves the minimum value at the given. et 1 be any filter gain such that both A + 1 C and A + 1 C 2 + B 1 w are stable. Substituting the above w in the plant-filter equations (11), we get: δe x = ( A + 1 C 2 + γ 2 B 1 B T 1 P (A + 1C 2 ) ) e x + (B D 20 )w 0 =A e x + B w 0, e 0 =C 0 e x. The first difference equation above can be solved as: and k 1 e x(k) = A k j 1 B w 0(j), j=0 J 2 (F, w, w 0 ) = e 0 2 P= lim where = lim E 1 E{e T x C T 0 C 0 e x } { k k 1 w0(i) T BT i=0 j=0.(a T ) k i 1 C T 0 C 0 A k j 1 B w 0(j) } = lim = lim k 1 k 1 i=0 j=0. δ (i j) B T (AT )k j 1 C T 0 ] k 1 i=0.(a T ) k i 1 C0 T ] =trace(c 0 Y C T 0 ) Y = A i B B(A T T ) i, i=0 trace[c 0 A k i 1 B trace[c 0 A k i 1 B B T which is the solution of the yapunov equation A Y A T Y + B B T = 0. Then, by Theorem 4, the solution to this constrained optimization problem, is to satisfy: = (A F P 2 C T 2 + B 0 D T 20)(R 0 + C 2 P 2 C T 2 ) 1, where P 2 is the solution to (13). (ecessity) First, for the system without white noise (w 0 = 0), suppose there exists a filter F and a signal w such that they achieve: 0 < J 1 (F, w, 0) J 1(F, w, 0), w w. In other words, for the linear operator R e w, defined as: δe x = Āe x + B 1 w, e = C 1 e x, it holds that R e w < γ. Then, by the bounded real lemma [de Souza and Xie, 1992], there exists a P 1 0, solving (12) and the worst disturbance signal is w = γ 2 B1 T P Āe x. ext, including the white noise signal into the system, it can be seen that: J 1 (F, w, w 0) =γ 2 w w 2 P lim 1 =γ 2 w w 2 P, trace [ (B 0 + D 20 ) T.P 1 AE{xw T 0 } ] which means that the worst disturbance signal at the presence of the white noise is w = γ 2 B1 T P Āe x. ow, substituting the w into the equation set (11), we get: δe x = ( Ā + γ 2 B 1 B1 T P Ā ) e x + (B D 20 )w 0 = A e x + B w 0 e 0 = C 0 e x Similar to the proof of sufficiency, we can write: where J 2 (F, w, w 0 ) = e 0 2 P= trace(c 0 Y C T 0 ) Y = A i B B T (AT )i i=0 and by Theorem 4, is to satisfy: = (A F P 2 C T 2 + B 0 D T 20)(R 0 + C 2 P 2 C T 2 ) 1, and P 2 is the solution to (13). 4. IUSTRATIVE EXAMPE Consider the following dynamic system: δx = x + w z = [ ] x, y = [ ] x + [ ]w w,

5 First, considering only the H performance, assume the filter gain = [ 2 3 ] T. Fixing γ = 1.5, there exists a solution P 1 0 to: P 1 =(A + C 2 ) T P 1 (A + C 2 ) + C T 1 C 1 + γ 2 (A + C 2 ) T.P 1 B 1 (I γ 2 B T 1 P 1B 1 ) 1 B T 1 P 1(A + C 2 ). This filter achieves T ew = , where T ew represents the transfer function from w to e, and therefore satisfying the H requirement. In this case, the worst disturbance signal is characterized by w = 0.444B1 TP (A + C 2)e x = K w e x. However, when the noise signal w 0 is added, the optimal performance of the system in the worst case is then calculated by: J 2 = trace(c 1 C1 T P 2) = , where P 2 is the solution to the discrete-time yapunov equation: P 2 =(A + C 2 + B 1 K w )P 2 (A + C 2 + B 1 K w ) T + (B 0 + D 20 )(B 0 + D 20 ) T. It is clear that the performance of this filter in presence of noise is not desirable. On the other hand, a Kalman filter can be calculated that satisfies the H 2 optimal performance requirement. This filter can be found by solving the Riccati equation: Fig. 2. Error behavior at the presence of white noise signal w 0 for the closed -loop system with filter (dashed) and with the multi-objective filter (solid). P 2 =AP 2 A T (B 0 D T 20 + AP 2C T 2 )(R 0 + C 2 P 2 C T 2 ) 1.(D 20 B T 0 + C 2P 2 A T ) + B 0 B T 0, leading to a filter gain: = (AP 2 C2 T + B 0 D20)(R T 0 + C 2 P 2 C2 T ) = The optimal performance of the system with this filter then becomes: J 2 = trace(c 1 C T 1 P 2 ) = , which in fact is much lower than obtained when only the H performance was considered. ow, designing a multi-objective filter using Theorem 5 and fixing γ = 2.5, results to solutions to Riccati equations (12) and (13) as: P 1 = [ and a filter gain: that satisfies (14). ] > 0, P 2 = = > 0, , (15) For the closed-loop system consisting this filter, the cost functions are: J 1 = , J 2 = ote that although the index J 2 is worse than the Kalman filter, but is still much improved compared to the system with only an H filter. On the other hand, as can be seen in Figure 2, the error performance of the closed-loop system is much better in the presence of the white noise signal w 0 when the multiobjective filter is used, compared to the filter that only satisfies the H performance. Fig. 3. Singular value diagram for T ew for the system with multi-objective filter. Figure 3 shows the singular value diagram for the transfer function T ew of the system with filter gain (15) which, as expected, meets the disturbance attenuation of γ = COCUSIOS In this paper, we have developed a direct design method for a robust optimal signal estimator in discrete-time domain. This method provides a filter that can be capable of achieving robust performance against system model uncertainties, as well as optimal performance against white noise. The mixed H 2 /H filter can be obtained by solving a set of cross-coupled Riccati equations. REFERECES B.D.O. Anderson, and J.B. Moore. Optimal Filtering. Englewood Cliffs, Prentice-Hall,

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