An IQC Approach to Robust Estimation against Perturbations of Smoothly Time-Varying Parameters

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1 Proceedings of the 47th EEE Conference on Decision and Control Cancun, Mexico, Dec 9-11, 8 An QC Approach to Robust Estimation against Perturbations of Smoothly Time-Varying Parameters Joost Veenman and Hakan Köroğlu and Carsten W Scherer Delft Center for Systems and Control, Delft University of Technology Mekelweg 68 CD Delft The Netherlands Phone : {87171,85899} Fax : joostveenman@gmailcom, {hkoroglu,cwscherer}@tudelftnl Abstract n this paper the design of robust H -estimators for the class of linear systems that depend rationally on ratebounded uncertain time-varying parameters is addressed The uncertainties are characterized by dynamic QCs and new LM conditions are formulated that allow for a nontrivial extension of the estimation problem with a weighting filter at the output n order to show the power of the dynamic QC approach, a numerical example is included in which the result is compared with alternative approaches based on parameter-dependent Lyapunov functions The effectiveness of the additional weighting filter is illustrated in a second numerical example NTRODUCTON During the last decade, the robust H -estimator synthesis problem for linear systems that are subject to white-noise disturbances and norm-bounded parameter uncertainties has been studied intensively The problem can be formulated as the design of a linear time-invariant estimator that guarantees a norm-bound on the asymptotic variance of the estimation error Some early works, [1], [], proposed solutions based on Riccati equations and later in [3], [4] methods based on LM techniques were introduced for systems with parametric uncertainties that can be captured by a polytopic region t is well known that these methods might be rather conservative since the uncertain parameters are allowed to vary arbitrarily fast For this reason, there have been various attempts to reduce conservatism by using parameter-dependent Lyapunov functions PDLF n [5] and [6], methods were suggested for systems that depend affinely on uncertain time-invariant parameters Only recently, in [7], [8], [9], these methods were generalized to the case in which the parameters are allowed to be time-varying Despite the fact that the results were considerably improved, they all still might yield conservative results A second drawback is that the system matrices are restricted to depend affinely on the uncertain parameters An alternative framework for stability and performance analysis is the dynamic integral quadratic constraint QC approach, which was introduced in [1] QCs are very useful in describing different types of uncertainties One could think, eg, of uncertain time-delays, multiple static nonlinearities or uncertain LT dynamics Our special attention is dedicated to QC tests for smoothly time-varying parametric uncertainties with bounded rates-of-variation, as was shown in [11], [1], and as was recently extended in [13] and [14] Until very recently, the dynamic QC-framework was mostly suitable for analysis purposes only The essential difficulty in synthesis based on dynamic QCs was the characterization of stability of the closed-loop system Recently this problem was completely resolved in [15] A necessary and sufficient positivity condition was derived that can be imposed on the synthesis LMs in order to enforce stability of the corresponding closed-loop system t was also shown in [15] that this result yields a complete solution for the L -gain and H -estimator synthesis problems if the uncertainties are described by dynamic QCs The first goal of this paper is to merge the new results of [15] with the special QC test for uncertain smoothly time-varying parameters from [13] and to show that we can systematically reduce conservatism with respect to the PDLF based approaches for a broader class of systems The second goal is to show that the estimator synthesis problem from [15] can be further generalized to the use of a weighting filter at the output With this filter, we can improve performance in a desired frequency band of interest at the cost of a possible degradation of performance outside the desired frequency band After introducing some preliminaries in Section, we formally state the problem in Section Then in Section V, we give a short recap of QC theory and describe the special QC test for smoothly time-varying parameter uncertainties as it was presented in [13] This is then merged with an QC test for H -performance n Section V, we state and prove the main result of this paper, namely, a solution in terms of LM conditions for the robust estimation problem with a weighting filter at the output for systems that depend rationally on uncertain smoothly time-varying parameters n Section V, two numerical examples are given n the first example, we show that it is indeed possible to further reduce conservatism with respect to the PDLF approaches that are known from the literature The second example illustrates the effect of using a weighting filter that emphasizes a certain frequency band in the estimation error We conclude the paper with some final remarks /8/$5 8 EEE 533

2 NOTATON AND PRELMNARES The symbol Le n [, is used to denote the extended space of R n -valued functions that are square integrable on finite intervals L n [, Le[, n represents the space of R n -valued functions, g : [, R n, of finite energy g : g, g : gt dt The Fourier transform of gt is denoted by ĝiω By an operator we mean a map G : Le a [, L e b [, that takes one L e[, space into another one RL m n RH m n will be used to denote the space of all real-rational and proper and stable matrix functions that have no poles on the imaginary axis closed right-half complex-plane We will view time-varying uncertain parameters as static linear time-varying LTV operators L s acting by multiplication on signals, where L s : { : [, R} The variation of is defined by another static LTV operator V d dt d dt acting on differentiable signals and being identical to multiplication with V t : d t/dt assuming t is differentiable For a precise definition see [13] The preceding concept of a timevarying operator and its rate-of-variation motivates to introduce the concept of the so called region-of-variation ROV R : { t, V t,t [, } where we assume R to be compact and star-shaped with star-center zero, ie τr R, τ [, 1] We describe the set of static LTV operators that vary in this region as L R : { L s : t, V t R, t [, } PROBLEM FORMULATON Consider the system interconnection in Figure 1 where G RH represents a nominal and stable LT system that admits a minimal realization of the form G : G qp G zp G yp G qw G zw G yw A B p B w C q D qp D qw C z D zp D zw C y D yp D yw, with A R n n, B p R n np, B w R n nw, C q R nq n, C z R nz n and C y R ny n The plant is subject to perturbations of static LTV operators Ls np nq, with the set representing the uncertainty For notational simplicity, we consider a single but repeated LTV uncertainty r : r δ where { r δ : δ L R }, R R r interacts with G through a linear fractional transformation for which we require the inverse D qp r to exist for all r Since D qp and are known this can be easily verified on beforehand The main goal in robust estimation is the synthesis of a stable LT system E : C E s A E B E + D E that dynamically processes the measurement y R ny to provide an estimate z e R nz of the signal z R nz with a guaranteed performance level in the H -sense For general operators, there are several definitions of the H - norm, which are not necessarily equivalent see eg [16] and references therein We will employ a signal based interpretation of the H -norm For this purpose we define the real vector w 1 T R n w w p G Fig 1 q z y E + e W z e Robust estimation problem with 1 being located at the th position Let e f be the response of the system to an impulsive input in the direction of w We say that the squared H -norm of the map w e f is less then γ > if we have n w 1 e f < γ, r 1 n this paper we extend the robust estimation problem from [15] with a given stable weighting filter W : C W s A W B W + D W, where A W R nw nw, B W R nw nz, C W R nz nw With this filter we can emphasize a certain frequency band of interest Then we can improve performance within the frequency band of interest at the cost of a possible degradation of performance outside the frequency band of interest Let us now formally state the robust estimation problem: Problem 1: Given the plant G, the weighting filter W and the compact region R, design a stable LT filter E such that for all r the squared H -norm from w e f is less then γ > V ROBUST STABLTY AND H -PERFORMANCE FOR LTV-SYSTEMS Recall that an uncertainty is said to satisfy the QC defined by the multiplier Π RL nq+np nq+np if the following condition holds true: q q, Π, q L nq q q n applications one constructs a whole family of multipliers that are parameterized as Π Ψ PΨ with a suitable set of symmetric matrices P and with a typically tall Ψ RH n ψ n p such that the QC holds for all Π Π and all t is then well known from [1] that stability of the feedback interconnection G qp and amounts to verifying whether the inverse D qp exists for all and whether there exists an ǫ > and a P P T P for which Gqp iω Ψiω PΨiω Gqp iω e f ǫ, ω R Since we are concerned with LTV perturbations we employ the QC-test from [13] Let us therefore consider the following variant of the swapping lemma: Lemma 1 Swapping Lemma: Consider the perturbations r : r δ, l : l δ, k : k δ and the transfer 534

3 matrix H RH l r, and let H B and H C be defined based on a minimal realization of H as [ AH B H H C H D H ],H B [ AH B H ] [ ] AH,H C, C H where A H R nh nh, B H R nh r and C H R nh l Then we have that H HC r l H V k H B V k HB }{{}}{{}}{{}}{{} H le re le H re t is now possible to arrive at the following test from [13], which basically is a reformulation of the test from [1] Theorem 1: Let G qp RH nq np be an arbitrary stable transfer matrix and let us be given a compact region R R Then for all r the feedback interconnection of G qp and r is stable if there exist a matrix P P T and an ǫ > such that with G qp : G qp r k and ΠΨ Hre P11 P PΨ 1 Hre H le P1 T P, H le the following FD and LM respectively hold true: Gqp iω Gqp iω Πiω ǫ, ω R T P, δ, υ R, Θ δ, υ Θ δ, υ 3 where Θ δ, υ : diag l δ, k υ Remark 1: n order to render condition 3 tractable, several relaxation schemes are suggested in [13] n our numerical examples we used the zeroth order Pólya relaxation Let us hence be given the set of vertices {δ i, υ i } m i1 R and let the polytopic region be defined as { m } P {δ i,υ i } m : α i1 i δ i, υ i m : α i 1, α i [, 1], i1 i1 such that the ROV R is now defined through the given set of vertices with Θ i : diag l δ i, k υ i Then 3 is satisfied if there exists a P P T for which Θ i T P Θ j, i 1,, m, j i,,m Remark : The size of P heavily depends on the McMillan degree of H For a basis function of the form Hs 1 s + α s + α µ T r, we can obtain a minimal realization as α 1 1 [ ] AH B H C H D H r 1 α 1 µ For this realization we then have that lµ+1r and kµr n order to be able to establish a test for H -performance as well, we introduce the following notation: H re Gqp H re G qw à B 1 B B H le W G zp WG zw W C 1 D11 D1 C D1 D Ẽ G yp G yw C F1 F à 11 à 1 B11 B1 B13 à B1 B C11 D11 D1 C 1 C D1 D Ẽ C3 F1 F A W B W C z B W D zp B W D zw B W A H B H C q B H D qp B H D qw A H B H A B p B w C H D H C q D H D qp D H D qw C H D H C W D W C z D W D zp D W D zw D W C y D yp D yw where G qp G qp, Gzp G zp, Gyp G yp t is now easily verified that the weighted system interconnected with the estimator admits the following realization: A B 1 B C 1 D 11 D 1 : 4 C D 1 D Ã+ BD E C BCE B1 + BD E F1 B + BD E F B E C AE C 1 B E F1 D11 B E F D1 C +ẼD E C Ẽ C E D1 +ẼD E F1 D +ẼD E F Let us also introduce the symmetric matrix variable X and the selection matrix Φ which are respectively partitioned as X 11 X 15 X, Φ T, X15 T X 55 where Φ is compatible with to the block structure of X and where the sizes of X 11 X 55 are equal to the sizes of A W, A H, A H, A, A E respectively We now can state the following feasibility test for robust stability and H -performance: Theorem : There exists a stable estimator E such that for all r the interconnection of Figure 1 is stable and the squared H -norm from w e is rendered less then γ >, if D 1 and D and if there exist matrices P P T, X X T and Z Z T for which 3 as well as the following conditions hold true: T X A X B 1 C 1 D 11 C D 1 P }{{} M P X A X B 1 C 1 D 11 C D 1, 5 535

4 TraceZ < γ, 6 B T X B Z, 7 Φ T XΦ 8 Proof: Since R is star-shaped with center zero, 3 implies that the left-upper block of P is positive semidefinite With a simplified version of the recent result in [15] we infer that 8 and 5 imply stability of A and, with the KYP-lemma, the validity of, which in turn proves robust stability of the feedback interconnection of G qp and r Now in order to prove robust H -performance, let us describe the closed-loop trajectories of system 4 for an impulsive input in the direction of w as ẋ q f p f e f : A C 1 B 1 D 11 C D 1 x p e, x B w, 9 where x t, p et and e f t are the th evolutions of respectively the closed-loop state x, the signal p e which is the extension of the input-signal p and the estimation error signal e Then after having right- and left-multiplied 7 and 5 with w and x t T, p e tt T and their transposes respectively, we can integrate from zero to infinity to infer q f t p f t T P q f t p f t dt+ e f tt e f tdt 1 x T Xx w T Zw Now observe that the filtered signals ˆp f and ˆq f are respectively defined through the relations ˆp f H leˆp e and ˆq f H reˆq where ˆq G qpˆp e We can then invoke Parseval s theorem, in order to transform the first term of 1 into ˆq f ˆp f P ˆq f ˆp f dω ˆq ˆq Π Since this is nonnegative for all r, we conclude ˆp e ˆp e dω e f tt e f tdt wt Zw, w 11 Summing over then leads to 1 thanks to 6 V MAN RESULTS n this section we present a solution in terms of LMs for the robust H -estimator synthesis problem, extended with a weighting filter at the output Let us hence introduce the matrix variables T11 T T 1 T1 T T, T 1 X 11 X 14 X, R X14 T X 44 T11, T 3 S, T, where RR T is partitioned according to the block structure of X X T and where the sizes of T 11 T T 11, T 1, T T T are equal to the sizes of à 11, à 1, à respectively Then we are ready to state the main result of this paper Theorem 3: There exists a stable estimator E such that for all r the interconnection of Figure 1 is stable and the squared H -norm from w to e is rendered less than γ >, if there exist matrices K, L, M, N, P P T, R R T, X X T, T 1, T, T 3, Z Z T, for which 3, 6 as well as the following conditions hold true: D H D qw, D W D zw ND yw, 1 R T 3 R T1 T T T B 1 + BN F R T 1 R X X B +L F, B TT 1+ F TNT BT BT X+ F TLT Z 13 O T M P O, 14 where O T1 TÃT + B M T1 TÃ+ BN C T T B BN F 1 K X B 1 +L F 1 C 1 C1 D1 C T +Ẽ M C +ẼN C D +ẼN F 1 With solutions of these LMs, one can then construct T11 + T 1 T T 1 T T 1 T Y T T 1 T T, 15 and non-singular U and V such that XY + UV T With K KT 1 XÃY and M MT 1, the estimator realization matrices A E, B E, C E, D E can then be chosen as AE B E U X B K L V T C E D E M N CY 16 R Proof: Let R : and observe that by the elimination lemma 8 is equivalent to the existence of a R R T for which X R Moreover, since RB, 7 can be equivalently rewritten as R X X B B T X Z 17 Let us now perform the congruence transformation and the bijective variable transformation that are well known to linearize the nominal output-feedback control problem see eg [17] For this we introduce the new variables K, L, M and N which are defined through the following equality: K L X ÃY + M N + U X B where X U X U T, X Y V V T AE B E V T C E D E CY, Y, Y V T Then 17 and 5 can be equivalently rewritten as Y RY Y Y R B + BN F RY R X X B +L F, 18 B T + F TNT BT BT X+ F TLT Z 536

5 and Õ T M P Õ, 19 respectively, where Õ ÃY + BM Ã+ BN C B1 + BN F 1 K XÃ+L C X B 1 +L F 1 C 1 Y C1 D1 C Y +ẼM C +ẼN C D +ẼN F 1 Since 18 and 19 are not yet affine in all the variables, we perform the additional congruence transformation with T :diagt 1, with compatible dimensions of that was suggested in [18] and as employed in [19] in a different context Then the following relations hold true: Y T 1 T, T1 TY T 1 T 3, RY T 1 R, KT 1 K, MT 1 M, T1 T B B, C1 Y T 1 C 1, T1 T ÃT Ã11 T 11 Ã 1 Ã11T 1 + T 1 Ã T Ã Observe that is affine in the variables T 11, T 1, T and that the second congruence transformation yields 13 and 14 Moreover, applying the two suggested congruence transformations to 8 yields T T Y T X R YT T3 R T T 1 R T 1 R X R, which is implied by 13 Finally observe that D 1 and D are zero if and only if 1 is satisfied Remark 3: By applying the Schur complement with respect to the lower right identity block of MP, we can transform 14 into a genuine LM Remark 4: Though we have shown that it is possible to extend the estimation problem of [15] by applying a weighting-filter at the output for the class of LTV uncertainties with bounded rates of variation, the result can be easily extended to any type of uncertainty that can be suitably described by QCs of the form Π Ψ PΨ, with appropriate adaptation of the stability condition from [15] Remark 5: t is straightforward to arrive at a solution in terms of LMs for the robust L -estimator synthesis problem nstead of 1, we need to guarantee e f, e f γ ǫ w, w for some ǫ > Conditions 1, 6 and 13 can then be disregarded and we only need to extend 14 with the appropriate performance matrices and impose 1 to enforce closed-loop stability Remark 6: The matrix variable S as well as the matrix variables K and M can be eliminated especially for analysis purposes According to our experience, though, elimination of S might deteriorate a numerically reliable construction of the estimator V NUMERCAL EXAMPLES We will first show that we can reduce conservatism with respect to the approaches based on PDLFs We especially focus on the recent results of [8], [9] where the proposed H -filter design methods are based on PDLFs that depend quadratically on the uncertain parameters t should be mentioned that the stochastic interpretation of the H -norm, which is used in [7], [8], [9], is equivalent to the signal based interpretation we are using such that we are allowed to compare our results We hence took the following numerical example from [8], [9]: q z y p w 1 Figure displays the squared H -norm from w e W for the plant 1 with δt 14 and for increasing values of µ length of basis µ and υ rate-bound for the methods from [8], [9] and those in this paper The lines QC, µ and PDLF correspond to the standard quadratic stability tests where δt is allowed to vary arbitrarily fast The lines PDLF 1 and PDLF represent the results of [8] and respectively use a Lyapunov function that depends affinely and quadratically on δ The line PDLF 3 corresponds to the results of [9] and is not a complete solution in terms of LMs, due to a bilinear term Hence, for this example, the QC approach clearly outperforms all the PDLF approaches for sufficiently large µ e QC, µ QC, µ 1 QC, µ QC, µ 3 PDLF PDLF 1 PDLF PDLF Rate of Variation υt for δt < 14 Fig Numerical Example 1 n our second example we illustrate the possibility to further improve the estimation results by employing a weighting filter at the output for the mass-spring-damper system that is displayed in Figure 3 Mass m 1 is disturbed by an external w 1 c 1 k 1 m 1 x 1 c k x m t Fig 3 Mass-spring-damper system with uncertain time-varying m force w 1 and mass m is uncertain and time-varying The objective is to estimate position x based on the measurement x 1 which is corrupted by measurement noise w Observe, with k 3 k 1 +k, c 3 c 1 +c, p q, δt < 1 and 537

6 m ˆm +W δt, that G can be written as 1 z 1 k3 k m 1 m 1 c3 c 1 m 1 m 1 m 1 k q ˆm k c ˆm ˆm c ˆm W ˆm p y k ˆm k c w ˆm ˆm c ˆm W ˆm For our illustration, we used the numerical values m 1 1, ˆm, c 1 1, c 1, k 1 1, k 1, W 15 and a parameter trajectory δtsin 1 4 t By plotting υ δ vs δ we observe that the ROV is an ellipse that can be covered by a polytope according to the left upper-plot of Figure 4 We used the weighting filter W 5s+5 5s+1, µ3, and a pulse signal as the input disturbance w 1 with an amplitude of 1 that was repeated every 1 sec with a random direction and a pulse width of 1% of the period Finally we used a band-limited white-noise signal with variance of 1 for the disturbance input w n the lower plot of Figure 4, the estimation error is shown for respectively the nominal design ie δ, the robust design and the robust design with a weighting filter at the output Clearly the estimator that was designed with the weighting filter at the output leads to an error with smaller low frequency content υ δ et 1 1 δt, υt Time sec Nominal, Robust, Robust+Weight Time sec Fig 4 Estimation error e for sinusoidal perturbations δt V CONCLUDNG REMARKS We have given an LM solution for an extension of the robust H -estimator synthesis problem with a weighting filter at the output, and for systems that depend rationally on smoothly time-varying parameter uncertainties which are characterized with dynamic QCs Generalizations to any type of uncertainties that can be suitably described by dynamic QCs are straightforward, once the QC multipliers are factorized in the form assumed in this paper Though the results were formulated for H -estimation, the technique readily extends to L -gain performance We illustrated through two numerical examples how to systematically reduce conservatism with respect to approaches based on parameter dependent Lyapunov functions and how to benefit from an additional weighting filter at the performance output ACKNOWLEDGEMENT The second and third authors are supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs REFERENCES [1] U Shaked and C E de Souza, Robust minimum variance filtering, EEE Transactions on Signal Processing, vol 43, pp , 1995 [] Y Theodor and U Shaked, Robust discrete-time minimum-variance filtering, EEE Transactions on Signal Processing, vol 44, pp , 1996 [3] J C Geromel, Optimal linear filtering under parameter uncertainty, EEE Transactions on Signal processing, vol 47, pp , 1999 [4] C E de Souza and A Trofino, An LM approach to the design of robust H filters, in Advances in Linear Matrix nequality Methods in Control, L E Ghaoui and S Niculescu, Eds SAM,, ch 9, pp [5] J C Geromel, M C de Oliveira, and J Bernussou, Robust filtering of discrete-time linear systems with parameter-dependent Lyapunov functions, Proceedings of the 38 th Conference on Decision and Control, pp , 1999 [6] K A Barbosa, C E de Souza, and A Trofino, Robust H filtering for uncertain linear systems: LM based methods with parametric Lyapunov function, System & Control Letters, vol 54, pp 51 6, 5 [7] K A Barbosa and A Trofino, Robust H filtering for linear system with uncertain time-varying parameters, Proceedings of the 41 th EEE Conference on Decision an Control, pp , [8] C E de Souza, K A Barosa, and A Trofino, Robust filtering for linear systems with convex-bounded uncertain time-varying parameters, EEE Transactions on Automatic Control, vol 5, pp , 7 [9] X Jun and X Lihua, An improved approach to robust H and H filter design for uncertain linear system with time-varying parameters, Proceedings of the 6 th Chinese Control Conference, pp , 7 [1] A Megretski and A Rantzer, System analysis via integral quadratic constraints, EEE Transactions on Automatic Control, vol 4, pp , 1997 [11] U Jönsson and A Rantzer, Systems with uncertain parameters - 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