Robust finite-frequency H2 analysis

Technical report from Automatic Control at Linköpings universitet Robust finite-frequency H2 analysis Alfio Masi, Ragnar Wallin, Andrea Garulli, Anders Hansson Division of Automatic Control E-mail: masi@dii.unisi.it, ragnarw@isy.liu.se, garulli@dii.unisi.it, hansson@isy.liu.se 2th October 2 Report no.: LiTH-ISY-R-2973 Accepted for publication in 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA Address: Department of Electrical Engineering Linköpings universitet SE-58 83 Linköping, Sweden WWW: http://www.control.isy.liu.se AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.

Abstract Finite-frequency H 2 analysis is relevant to a number of problems in which a priori information is available on the frequency domain of interest. This paper addresses the problem of analyzing robust finite-frequency H 2 performance of systems with structured uncertainties. An upper bound on this measure is provided by exploiting convex optimization tools for robustness analysis and the notion of finite-frequency Gramians. An application to a comfort analysis problem for an aircraft aeroelastic model is presented. Keywords: Optimization, Control theory

Avdelning, Institution Division, Department Datum Date Division of Automatic Control Department of Electrical Engineering 2--2 Språk Language Rapporttyp Report category ISBN Svenska/Swedish Licentiatavhandling ISRN Engelska/English Examensarbete C-uppsats D-uppsats Övrig rapport Serietitel och serienummer Title of series, numbering ISSN 4-392 URL för elektronisk version http://www.control.isy.liu.se LiTH-ISY-R-2973 Titel Title Robust finite-frequency H2 analysis Författare Author Alfio Masi, Ragnar Wallin, Andrea Garulli, Anders Hansson Sammanfattning Abstract Finite-frequency H 2 analysis is relevant to a number of problems in which a priori information is available on the frequency domain of interest. This paper addresses the problem of analyzing robust finite-frequency H 2 performance of systems with structured uncertainties. An upper bound on this measure is provided by exploiting convex optimization tools for robustness analysis and the notion of finite-frequency Gramians. An application to a comfort analysis problem for an aircraft aeroelastic model is presented. 2 Nyckelord Keywords Optimization, Control theory

Robust finite-frequency H 2 analysis Alfio Masi, Ragnar Wallin, Andrea Garulli, Anders Hansson Abstract Finite-frequency H 2 analysis is relevant to a number of problems in which a priori information is available on the frequency domain of interest. This paper addresses the problem of analyzing robust finite-frequency H 2 performance of systems with structured uncertainties. An upper bound on this measure is provided by exploiting convex optimization tools for robustness analysis and the notion of finite-frequency Gramians. An application to a comfort analysis problem for an aircraft aeroelastic model is presented. I. INTRODUCTION Research on robust H 2 analysis has been carried out for roughly forty years. A survey of the field is given in. Algorithms based on Riccati equations or linear matrix inequalities are presented in for instance 2, 3, 4, 5, 6, 7, 8 and 9. However, in certain applications it may not be so informative to compute an H 2 measure over the entire frequency range. If the disturbances and all other input signals have a limited frequency range, it is in many cases more relevant to consider a measure computed over this limited range only. In new measures for stability and performance analysis of aircrafts were suggested. One of the measures was for turbulence response and is H 2 -like and formulated over a limited frequency range. Measures like this are used in the aerospace industry today. Other areas where robust finite-frequency H 2 analysis can be applied are structural dynamics, acoustics and colored noise disturbance rejection. How to compute the H 2 norm over a finite frequency interval, for a system without uncertainty, is described in. A short review of the procedure is given in Section II-A. The key step is to compute the finite-frequency observability Gramian. This is accomplished by first computing the regular observability Gramian and then scale it by a system dependent matrix. Hence, when we consider robust H 2 analysis over a finite frequency interval we do not only have to compute an estimate of the worst case observability Gramian but we also have to find a well defined system corresponding to this Gramian. Most of the methods referenced above, for computing the robust H 2 norm, do not deliver such a system. The contribution of this paper is to provide an upper bound on the robust finite-frequency H 2 performance of systems with structured uncertainties, by combining the notion of finite-frequency Gramians with convex optimization tools commonly used in robust control. A. Masi and A. Garulli are with Dipartimento di Ingegneria dell Informazione, Università di Siena, via Roma 56, 53 Siena, Italy. E-mail: {garulli,masi}@dii.unisi.it R. Wallin and A. Hansson are with the Department of Electrical Engineering, Linköping University, SE-58 83 Linköping, Sweden. E-mail: {ragnarw,hansson}@isy.liu.se The rest of the paper is organized as follows. In Section II the problem formulation and some background information is given. In Section III it is shown how to compute an upper bound on the finite-frequency H 2 norm. Dynamic scaling can be introduced to tighten the upper bound. The procedure for doing so is presented in Section IV. In Section V, the proposed technique is validated by means of a numerical example and an application concerning comfort analysis for an aeroelastic aircraft model. Finally, conclusions are drawn and presented in Section VI. II. PROBLEM FORMULATION AND PRELIMINARIES Consider the state-space system: { ẋ(t) = Ax(t)+Bw(t) z(t) = Cx(t), () and denote its transfer function G(s). The H 2 norm of () is defined as G(jω) 2 2 = 2π Tr{G(jν) G(jν)}dν = { } Tr B T e ATt C T Ce At B dt ( ) } = Tr {B T e ATt C T Ce At dt B } {{ } W o = Tr { B T W o B }, (2) where W o is the observability Gramian of the system. To compute the observability Gramian we can solve the Lyapunov equation A T W o +W o A+C T C =. (3) Using Parseval s identity we can also express the observability Gramian as: W o = 2π H (ν)c T CH(ν)dν, (4) where H(ν) = (jνi A). In this paper, we are interested in the finite-frequency H 2 norm, defined as G(jω) 2 2, ω = 2π ω ω A. Finite-frequency observability Gramian Tr{G(jν) G(jν)}dν. (5) To study the robust finite-frequency H 2 problem, it is necessary to introduce the notion of finite-frequency Gramians. The finite-frequency observability Gramian in the

frequency range ( ω, ω) is defined as W o ( ω) = 2π ω ω H (ν)c T CH(ν)dν. (6) The next lemma provides a way to compute W o ( ω) in (6), in terms of the regular observability Gramian W o. Lemma : The finite-frequency observability Gramian can be computed as W o ( ω) = L( ω) W o +W o L( ω), (7) where W o is the regular observability Gramian computed as in (3) and L( ω) = 2π ω ω H(ν)dν = j 2π ln (A+j ωi)(a j ωi). (8) Proof: See. By following the ideas in, it is also possible to obtain a finite-frequency Gramian in a generic frequency range (ω,ω). B. Robust H 2 norm estimation Consider the system in Linear Fractional Representation (LFR): ẋ(t) = Ax(t)+B q q(t)+b w w(t) p(t) = C p x(t)+d pq q(t) z(t) = C z x(t)+d zq q(t) q(t) = (δ)p(t), (9) where (δ) = diag(δ I s,...,δ nδ I snδ ). () δ i R si denotes thei-th component of the uncertainty vector δ,n δ is the number of uncertain parameters,x R n,z R p, q R d, p R d, with d = n δ i= s i, w R m and A, B q, B w, C p, C z, D pq, D zq, are real matrices of appropriate dimensions. Define C = Cp C z Dpq, D = D zq In view of analysis of the LFR system (9), the matrix A is assumed to be Hurwitz and the pair (C,A) is assumed to be observable. Without loss of generality, let us define := { (δ) : δ i, for i =,...,n δ }. Let us now introduce the transfer function matrix ( ) A B M (jω) M M(jω) = 2 (jω) q B w = C M 2 (jω) M 22 (jω) p D pq. C z D zq () For a simpler notation we drop the dependence on ω. Then, the transfer function matrix from w to z is given by the upper LFT 2. S(M; ) = M 22 +M 2 (I M ) M 2. (2) The aim of this paper is to compute the robust finitefrequency H 2 norm of system (9), i.e. sup S(M; ) 2 2, ω= 2π sup Tr ω ω S(M; ) S(M; )dν (3) By exploiting the results in 7-8, it is possible to compute an upper bound on the regular robust H 2 norm, i.e. sup S(M; ) 2 2= 2π sup Tr S(M; ) S(M; )dν. Let us by X denote the set of all Hermitian block-diagonal operators which commute with. These operators are often called scaling matrices. First we only consider constant scaling matrices, but in Section IV we extend the analysis to include frequency-dependent scaling matrices. Lemma 2: The system S(M, ) has robust H 2 norm less than γ 2 if there exist a matrix X X, Hermitian matrices P, P + R n n and a Hermitian matrix W o R n n, satisfying P >, X >, AP +P A T +B qxb T q CP +DXB T q AP+ +P+AT +B qxbq T CP + +DXBq T Wo I >, I P + P P C T +B qxd T DXD T X I P +C T +B qxd T DXD T X I <, <, Tr { } BwW T ob w < γ 2. (4) Proof: See 7-8. Problem (4) is a SemiDefinite Program (SDP) and can be solved by applying standard convex optimization tools 5. The condition provided by Lemma 2 is in general conservative, since it guaranties robust H 2 performance for complex nonlinear time-varying uncertainties 8, while we only consider real linear time-invariant ones. If the conditions of Lemma 2 are satisfied, then is also possible to find a state-space representation of a system given by a spectral factorization related to the solution of (4). The procedure is described in the following Lemma. Lemma 3: Let P +, P, X, W o, be the solution of (4). Define ˆM = ( ) ˆM ˆM2 such that ( ) ˆM ˆM2 X 2 ˆM = = ˆM 2 ˆM22 I with M given by (). Then, X 2 M I (5) ˆM = X 2 M X 2 <, (6) and there exists a spectral factor N RH, N RH, such that I ˆM ˆM = NN, and N ˆM2 2 2 < γ 2.

A state-space realization of N ˆM2 is given by: N A+(P C T +B q XD T )R C B w ˆM2 = R 2C where R = X I, (7) DXD T. (8) Moreover, W o is the observability Gramian of N ˆM2. Proof: See 7-8. III. AN UPPER BOUND ON THE ROBUST FINITE-FREQUENCY H 2 NORM The results from Lemma 2 and Lemma 3 allow us to derive an upper bound on the frequency gains for all frequencies ω and for all uncertainties. This upper bound can be used to obtain an estimate of the robust finite-frequency H 2 norm. In the next theorem we present how to compute this estimate, based on the results above. Theorem 3.: LetP +,P,X,W o, be the solution of (4). Then an upper bound on the robust finite-frequency H 2 norm is given by: sup S(M; ) 2 2, ω Tr { Bw(W T o L( ω)+l( ω) } W o )B w, (9) where L( ω) = j 2π ln (A+j ωi)(a j ωi), (2) with A = A+(P C T +B q XD T )R C, (2) and R as in (8). Proof: Let P +, P, X, W o, be the solution of (4) and set Y = (N ˆM2 ) (N ˆM2 ), (22) where N and ˆM 2 are defined according to Lemma 3. By means of simple algebraic manipulations we get Y = (N ˆM2 ) (N ˆM2 ) = ˆM ( 2 I ˆM ) ˆM ˆM2 = ˆM 2 I + ˆM (I ˆM ˆM ) ˆM ˆM 2. From (23) and (6), one has that: { I ˆM ˆM > (23) ˆM 2 ˆM 2 Y + ˆM 2 ˆM (I ˆM ˆM ) ˆM ˆM2 =. (24) Let us define T = ( I I ˆM ˆM ) ˆM ˆM2 I By exploiting (24), one gets ˆM T ˆM I ˆM ˆM2 ˆM 2 ˆM ˆM 2 ˆM2 Y ˆM = ˆM I. T. (25) (26) It then follows that ˆM ˆM ˆM = ˆM ˆM ˆM2 ˆM 2 ˆM ˆM 2 ˆM2 I Y. (27) Now, let q = X 2q, p = X 2p. By pre- and post-multiplying (27) by q(jω) w(jω) and its adjoint, one obtains z(jω) 2 + p(jω) 2 q(jω) 2 +w(jω) Y(jω)w(jω), ω. (28) Since and X 2 commute, one has q = X 2q = X 2 X 2 p = p. By recalling that is a contractive operator, we have q(jω) 2 p(jω) 2, and hence (28) leads to z(jω) 2 = w(jω) S(M, ) S(M, )w(jω) w(jω) Y(jω)w(jω), ω,. (29) Since (29) holds for any w(jω), we also have S(M, )(jω) S(M, )(jω) Y(jω), ω,. (3) Since (3) holds for all frequencies, due to Lemma and (7)-(8) in Lemma 3 one has that ω sup S(M, ) 2 2, ω Tr{Y(jω)}dω 2π ω = N ˆM2 2 2, ω = Tr { } Bw(W T o L( ω)+l( ω) W o )B w where L( ω) is given by (2)-(2). IV. EXTENSION TO DYNAMIC SCALING (3) In the proof of Theorem 3., Y(jω) plays the role of an upper bound on S(M, )(jω) S(M, )(jω) across all frequencies and all uncertainties described by. As suggested in the previous section, the use of frequencydependent scaling matrices allows one to reduce the conservatism of this upper bound. Let us introduce a dynamic scaling of the form X(s) = Ψ(s)XΨ(s) = { C ψ (si A ψ ) I } X{ }, (32) where A ψ R n ψ n ψ and C ψ R d n ψ are fixed matrices of appropriate dimension, X R (n ψ+d) (n ψ +d) is the free basis of parameters, and { } denotes what can be deduced by symmetry. Notice that X must be chosen so that X(s) X. The couple (C ψ,a ψ ) has to be observable and A ψ stable. As shown in 3, by setting M = (M,M 2 ), and by replacing X by X(jω) the frequency dependent inequality (27) can be rewritten as M(jω)X(jω)M(jω) M 2(jω) X(jω) I M 2(jω) Y(jω) <, (33)

where the upper left block of (33) can be expressed as X(ω) M I (jω)x(jω)m (jω) { } { } Ψ X = { } M Ψ X{ } I I M 2 Ψ { } X M Ψ Ψ = X { }. M 2 Ψ I I (34) By introducing the transfer function matrix C(sI Ã) Bq + D = ( M Ψ Ψ M 2 Ψ ) (35) and setting Υ = I T, (34) can be reformulated as { C(sI Ã) I } Bq X X { } D Υ I (36) Notice that, from () and (32), the matrices of the realization (35) can be expressed as à = A BqC ψ A ψ A ψ C = C DC ψ Bq = Bq I I Cψ I D = D (37) where à Rñ ñ, Bq Rñ d, C R (p+d) ñ, D R (p+d) d, ñ = 2n ψ +n, and d = 2n ψ +2d. By introducing the affine function of X Bq Π(X) = D Υ = Π Π 2, X X I Bq D Υ Π T 2 Π 22 (38) with Π Rñ ñ, Π 2 Rñ (d+p), and Π 22 R (d+p) (d+p), the following lemma can be stated. Lemma 4: The system S(M, ) has robust H 2 norm less than γ 2 if there exist a matrix X R (n ψ+d) (n ψ +d) such that X(s) X, Hermitian matrices P, P + Rñ ñ, Q R n ψ n ψ, and a Hermitian matrix W o Rñ ñ, satisfying P >, Q >, Aψ Q+QA T ψ QC T ψ X <, C ψ Q ÃP +P à T P CT Π(X) <, CP ÃP+ +P + à T P + CT Π(X) <, CP + Wo I >, I P + P { Tr B T w } Bw Wo < γ 2. T (39) Proof: See 7. By following the same approach adopted in Section III for constant scalings, it is possible to introduce a suitable system which bounds the H 2 performance of the considered LFR (9)-() from above. Given the solution of the SDP (39), the bounding system has a state-space realization Ñ M2 = à (Π 2 P CT )Π C Bw 22, Π /2 22 C (4) and its observability Gramian is given by W o. This allows one to formulate the extension of Theorem 3. to the case of dynamic scalings. Theorem 4.: LetP +,P,X, Wo, be the solution of (39). Then an upper bound on the robust finite-frequency H 2 norm is given by: sup S(M; ) 2 2, ω { T Bw Tr ( W Bw o L( ω)+ L( ω) Wo ) where and }, (4) L( ω) = j 2π ln (Ã+j ωi)(ã j ωi), (42) à = à (Π 2 P CT )Π 22 C. (43) V. NUMERICAL EXAMPLES In this section we discuss the application of the presented techniques to two case studies: an accademic example, for which the actual robust finite-frequency H 2 norm is known, and an example derived from the clearance of flight control laws of civil aircraft. All SDPs have been solved by SDPT3 4 under YALMIP 5. Example 5.: Consider system (9), with 2.5.5 5.25.5.5 A =.5 5 B q = 5 B w = C = D = 5 (44) (δ) = δi 2 and δ. This system is known to have robust H 2 norm equal to γ 2 =.53, attained for δ =.25. Figure reports the gain plots of the system for different values of the uncertain parameter δ, while Figure 2 shows the finite-frequency H 2 norm obtained by numerical integration for different values of ω and δ. It can be observed that the contribution to the H 2 norm from the high-frequency peak at rad/s is not negligible. Let us assume that we want to compute the robust finite-frequency H 2 norm of the system

.8 ω = 5 2.5 sup S(M, ) 2 2, ω Theorem 4., n ψ = Theorem 4., n ψ = Theorem 4., n ψ = 2 G(jω) 2.6 G(jω) 2 2, ω.4.2.5 ω = 5 2 2 ω rad/s FIG. Example 5.: Gain plots for different values of δ. 2 2 ω rad/s FIG. 3 Example 5.: finite-frequency robust H 2 norm. In this analysis, filters have been considered of the form S(M, ) 2 2, ω.5.5.5 δ.5 2 ω rad/s 4 2 FIG. 2 Example 5.: Frequency limited H 2 norm for different values of ω and δ. with ω = 5 rad/s, in order to skip the contribution of the high-frequency peak. It can be checked that the worst-case finite-frequency H 2 norm is still attained for δ =.25 and takes the value sup S(M, ) 2 2,5 =.899. In the analysis the dynamic scaling Ψ(s) in (32) has been chosen as Ψ(s) = (s p) n ψ (s p) (s+p) n ψ I nψ 2 2 I (s+p) n ψ 2... (s+p) I 2 I 2 (45) with p = 5. Figure 3 reports the true value of the robust finite-frequency H 2 norm (thick line) and the upper bounds returned by Theorem 4. for different degrees n ψ of the dynamic scaling (45). An alternative approach for approximating the finitefrequency H 2 norm is to low-pass filter the system output. F(s) = p n φ (s+ p) n φ, (46) where p is devised in such a way that the gain at ω ω = 5 rad/s is decreased by less than %. In Table I the results obtained by the application of Lemma 4 on the filtered system are reported, while Table II present the n φ 2 3 4 5 n ψ = 2.26.966.9652.9649.9648.9648 n ψ =.8836.8254.87.849.893.873 n ψ = 2.8482.82.7997.7994.7993.7994 n ψ = 3.842.7939.7929.793.793.798 TABLE I Example 5.: Robust H 2 norm for the system with lowpass filter. results obtained by the application of Theorem 4. on the non filtered system. n ψ 2 3 G 2 2,5.263.2233.953.93 TABLE II Example 5.: finite-frequency robust H 2 norm returned by Theorem 4.. It can be noticed that the computation of the regular robust H 2 norm for the filtered system leads to more conservative results with respect to those provided by robust finitefrequency H 2 norm techniques described in Theorem 4., even for high orders of the low-pass filter. Example 5.2: In this example, we consider a model of a civil aircraft including both rigid and flexible body dynamics. A comfort criterion concerning the effect of wind turbulences on different points of the aircraft is formulated. In 6 it has been shown that the problem can be cast as an H 2 performance analysis problem for an extended system, including an approximated Von Karman filter modeling the wind spectrum and output filters accounting for the type of turbulence field. The uncertain parameter δ corresponds to

the level of fullness of the fuel tanks and it is normalized in the range,. The resulting overall uncertain system is modeled as an LFR (9)-() with n = 2 states and block of size d = 4. The considered model has been derived in the frequency range between and 5 rad/s and has no physical meaning outside this range. Hence, the comfort criterion boils down to robust finite-frequency H 2 computation for the considered LFR with ω = 5 rad/s. Figure 4 reports the gain plot of S(M, ) for different values of the parameter δ. Figure 5 shows the values of the robust finite-frequency H 2 G(jω) 2 3 2.5 2.5.5 ω = 5 2 2 3 ω rad/s FIG. 4 Example 5.2: Gain plots for different values of δ. norm obtained by applying the technique of Theorem 3. with constant scalings in portions of the uncertainty interval, (partitioning the uncertainty interval allows one to reduce the conservatism). The asterisks in Figure 5 represent the values of the H 2 norm obtained by numerical integration, for fixed values of the uncertain parameter δ. S(M, ) 2 2, ω.8.6.4.2.8.6.4.2 Theorem 3., ω = 5 S(M, ) 2 2,5.5.5 δ VI. CONCLUSIONS The paper has proposed a technique for the computation of an upper bound to the robust finite-frequency H 2 performance of systems affected by structured uncertainties. The main result is formulated in terms of a convex optimization problem and the use of dynamic scaling matrices is exploited in order to reduce the conservatism of the bound. The present work has been motivated by the evaluation of a comfort criterion for an aeroelastic model of an aircraft, but the same technique can be exploited in a wide variety of applications in which a priori information is available on the frequency spectrum of the input signals. Future research will concern the investigation of suitable criteria for choosing the basis functions for the dynamic scaling matrices and the study of efficient strategies for partitioning the parametric uncertainty domain, in order to achieve a good trade-off between conservatism and computational burden. REFERENCES F. Paganini and E. Feron, Advances in linear matrix inequalitiy methods in control, ch. Linear matrix inequality methods for robust H 2 analysis: A survey with comparisons. SIAM, 2. 2 A. A. Stoorvogel, The robust H 2 problem: A worst case design, IEEE Transactions on Automatic Control, vol. 38, pp. 358 37, 993. 3 M. A. Peters and A. A. Stoorvogel, Mixed H 2 /H control in a stochastic framework, Linear Algebra and its Applications, vol. 25 26, pp. 97 996, 994. 4 T. Iwasaki, Robust performance analysis for systems with normbounded time-varying structured uncertainty, in Proceedings of the 994 American Control Conference, 994. 5 S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, 994. 6 E. Feron, Analysis of robusth 2 performance using multiplier theory, SIAM Journal on Control & Optimization, vol. 35, no., pp. 6 77, 997. 7 F. Paganini, State space conditions for robust H 2 analysis, in Proceedings of the 997 American Control Conference, vol. 2, pp. 23 234, 997. 8 F. Paganini, Convex methods for robust H 2 analysis of continuoustime systems, IEEE Transactions on Automatic Control, vol. 44, no. 2, pp. 239 252, 999. 9 M. Sznaier, T. Amishima, P. A. Parrilo, and J. Tierno, A convex approach to robust H 2 performance analysis, Automatica, vol. 38, pp. 957 966, 22. M. R. Anderson, A. Emami-Naeni, and J. H. Vincent, Measures of merit for multivariable flight control, Tech. Rep. WL-TR-94-35, Systems Control Technology Inc, Palo Alto, California, USA, 99. W. Gawronski, Advanced structural dynamics and active control of structures. Springer Verlag, 24. 2 G. Dullerud and F. Paganini, A course in robust control theory: a convex approach. Springer Verlag, 2. 3 A. Giusto, H and H 2 robust design techniques for static prefilters, in IEEE Conference on Decision and Control, 35 th, Kobe, Japan, pp. 237 242, 996. 4 K. C. Toh, M. J. Todd, and R. H. Tütüncü, SDPT3 - a Matlab software package for semidefinite programming, Optimization Methods and Software, no., pp. 545 58, 999. 5 J. Löfberg, YALMIP : A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, (Taipei, Taiwan), 24. http://control.ee.ethz.ch/ joloef/yalmip.php. 6 C. Papageorgiou and R. Falkeborn, COFCLUO deliverable D2.4.2 - Review and analysis of models and criteria, tech. rep., University of Linkoping (Sweden), April 28. http://er-projects.gf.liu.se/ COFCLUO. FIG. 5 Example 5.2: robust finite-frequency H 2 analysis.