H 2 optimal model reduction - Wilson s conditions for the cross-gramian

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1 H 2 optimal model reduction - Wilson s conditions for the cross-gramian Ha Binh Minh a, Carles Batlle b a School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet, Hanoi, Vietnam b Department of Applied Mathematics IV and the Institute of Industrial and Control Engineering, Universitat Politècnica de Catalunya, Spain Abstract In optimality conditions for the H 2 approximation of linear MIMO systems were written in terms of the controllability and observability Gramians. In this note we obtain those conditions using the cross-gramian. This approach has some advantages from the computational point of view. Keywords: cross-gramian, model order reduction, H2-norm, Wilson s optimal conditions, tangential interpolation. Introduction Model order reduction (MOR is a technique widely used for simulation and control design of large scale dynamical systems. The goal of MOR is to obtain a system with a greatly reduced number of states but the same inputs and outputs of the original one, with an err typically measured by means of a suitable norm of the transfer function, as small as possible for a given degree of the reduction; see 234 for a review of available techniques and examples of application. In, the gradients of the H 2 -norm of the error system were computed in terms of the controllability and observability Gramian matrices of the original systems, and the resulting stationary points, which can be used to select the reduced order model system, were interpreted in terms of tangential interpolation conditions. In this paper we compute those stationary points in terms of the cross-gramian matrix, and perform a similar analysis. Consider a linear time-invariant system represented by ẋ(t Ax(t Bu(t, y(t Cx(t, (. where (A, B, C R n n R n m R p n, x(t R n, u(t R m and y(t R p. We assume that system (. is a square system, i.e. the number of inputs is equal to the number of outputs, m p. The transfer function of system (., G(s : C(sI A B, s C, is then a square matrix. This work was supported by Vietnamese National Foundation for Science and Technology Development (NAFOSTED under grant , and by Spanish CICYT project DPI addresses: minh.habinh@hust.edu.vn; ha.b.minh@gmail.com (Ha Binh Minh, carles.batlle@upc.edu (Carles Batlle H 2 -norm: The H 2 -norm of G(s is defined as ( G(s 2 H 2 : trace G( jω G( jωdω. 2π It is known that the H 2 -norm of G(s can be computed via the Gramians of the system, see e.g. 4, G(s 2 H 2 trace(b T QB trace(cpc T trace(crb, (.2 where P, Q are the controllability and observability Gramians, which satisfy the Lyapunov matrix equations AP PA T BB T 0, A T Q QA C T C 0, and R is the cross-gramian 5, which obeys the Sylvester matrix equation AR RA BC 0. It can be shown 6789 that, for SISO systems and also for symmetric MIMO systems, i.e. those such that G T (s G(s, the different Gramians are related by R 2 PQ. H 2 -norm of error system using the cross Gramian: Let Ĝ(s : Ĉ(sI Â B be the transfer function of the reduced system that we want to construct, with (Â, B, Ĉ R r r R r m R p r, with p m and r n fixed (ideally, r n. The error system, which has transfer function E(s : G(s Ĝ(s, has the realization {A e, B e, C e } : { A 0 0 Â, B B, C Ĉ }. R X Let R e : be the cross Gramian associated with Y R E(s, which satisfies A e R e R e A e B e C e 0, if written in Preprint submitted to Elsevier December 9, 203

2 block form, A 0 R X R X A 0 0  Y R Y R 0  C Ĉ B B 0. (.3 It follows from the above equation that AR RA BC 0, (.4  R R BĈ 0, (.5 AX X BĈ 0, (.6 ÂY YA BC 0. (.7 The H 2 -norm of the error system E(s can be computed via the cross Gramian R e by means of Using E(s 2 H 2 trace(c e R e B e ( trace R X C Ĉ Y R B B trace(crb CX B ĈYB Ĉ R B. trace(cx B trace(x BC trace(x ( ÂY Y A trace(xây XY A trace(yâx YAX BC (by (.7 trace(y (AX X trace(ybĉ } {{ } trace( ĈY B, we can rewrite the H 2 -norm of E(s as BĈ (by (.6 E(s 2 H 2 trace(crb 2CX B Ĉ R B (.8 trace(crb 2ĈYB Ĉ R B. (.9 Our goal is to write down the equations that (Â, B, Ĉ must satisfy in order to be an stationary point of E(s 2 H 2. We will proceed along the lines originally proposed by Wilson.0 2. Wilson s optimal conditions for the cross Gramian Assume that G(s is a square system, asymptotically stable and in minimal realization. Theorem 2.. The gradients ÂJ, B J, Ĉ J of are given by J(Â, B, Ĉ : E(s 2 H 2 ÂJ 2( R 2 YX, B J 2(Ĉ R CX, ĈJ 2( R B YB, (2. where R, X, Y satisfy equations (.5-(.7. Proof. By perturbing  to  Â, the corresponding X, Y, R will change to X X, Y Y, R R, which satisfy the following equation A 0 R X X 0   Y Y R R R X X A 0 Y Y R R 0   C Ĉ B B 0. (2.2 By comparing equations (.3 and (2.2, one gets that Â, R, X, Y must satisfy the equations A X X  X  0, (2.3  Y Y A ÂY 0, (2.4  R R R   R 0. (2.5 Using now (.8 for the H 2 -norm of E(s, the first-order perturbation J with respect to  can be computed as J J( Â, B, Ĉ J(Â, B, Ĉ trace(crb 2C(X X B Ĉ( R R B trace(crb 2CX B Ĉ R B 2trace(C X B trace(ĉ R B. (2.6 The first term in (2.6 can be rewritten as trace(c X B trace( X BC while the second one becomes trace( X ( ÂY YA BC (by (.7 trace( X ÂY YA X trace(y X  YA X trace(y ( X  A X X  (by (2.3 trace(yx Â, (2.7 trace(ĉ R B trace( R BĈ trace( R (  R R BĈ (by (.5 trace( R R R R trace( R R R R trace( R ( R  R R   R (by (2.5 trace( R( R   R 2trace( R 2 Â. (2.8 J 2trace(( R 2 YX  ( R 2 YX, Â, 2

3 and thus ÂJ 2( R 2 YX. Next, we compute B J by perturbing B to B B. The corresponding Y, R will be perturbed to Y Y, R R, satisfying A 0 R X 0  Y Y R R R X A 0 Y Y B B B R R 0  C Ĉ 0. (2.9 By comparing the two equations (.3 and (2.9, one sees that B, R, Y must solve the equations  Y Y A B C 0, (2.0  R R BĈ 0. (2. Using again (.8, the first-order perturbation J with respect to B can be computed as J J(Â, B B, Ĉ J(Â, B, Ĉ trace(crb 2C(X X B Ĉ( R R ( B B trace(crb 2CX B Ĉ R B trace(2cx B Ĉ R B Ĉ R B trace(ĉ R } {{ B } neglected due to second-order term trace(2cx B Ĉ R B Ĉ R B. (2.2 The third term in (2.2 is trace(ĉ R B trace( R BĈ which yields trace( R (  R R BĈ (by (.5 trace( R R R R trace( R R  R R trace( ( R  R BĈ trace( BĈ R (by (2. R trace(ĉ R B. (2.3 J 2trace((Ĉ R CX B 2(Ĉ R CX, B, B J 2(Ĉ R CX. Finally, a computation similar to that of B J shows that the first-order perturbation J with respect to Ĉ is given by ĈJ 2( R B YB. 3 The next result takes advantage of Theorem 2. to present an implicit solution for the reduced system (Â, B, Ĉ. Theorem 2.2. At every stationary point of J where R is invertible, one has the identities where  W T AV, B W T B, Ĉ CV, W T V I n, W : Y T ( R T, V : X R. Proof. At a stationary point of J one has using Theorem 2., ÂJ 0, B J 0, Ĉ J 0, R 2 YX 0, Ĉ R CX 0, R B YB 0. If we define W : Y T ( R T and V : X R, then W T V R Y ( X R I } {{ } n, W T B R Y W T B W T B, Ĉ C ( X R CV. } {{ } V V It remains to prove that  W T AV. To this end, we first multiply by W T both sides of equation (.6 to get that W T AX W T X W T BĈ 0. Now, noting that X V R, W T V I n, it follows that W T AV R W T V R W T B Ĉ 0, I n B (W T AV R R BĈ 0. By comparing with equation (.5, thus implies that, indeed,  W T AV. The result in theorem 2.2 defines Â, B, Ĉ as functions of R, X and Y, i.e. (Â, B, Ĉ G( R, X, Y : ( R YAX R, R YB, CX R. (2.4 However, the R, X, Y, in turn, depend on Â, B, Ĉ through the 3 Sylvester-like matrix equations (.5, (.6 and (.7, i.e. ( R, X, Y F(Â, B, Ĉ. (2.5 Hence, stationary points of the H 2 -norm of the error system are fixed points of the map H G F and, in principle, they could be computed iteratively. This is the same situation than the controllability and observability Gramian approach in but, in that case, there are 6 Lyapunov matrix equations involved in each iteration, so the computational advantages of our method could be important for very large scale systems.

4 3. Tangential interpolation In this section we prove that the stationary points of the H 2 -norm of the error system computed in terms of the cross- Gramian define a tangential interpolation of the full system error function at the mirror poles of the reduced system, as was done in for the controllability and observability Gramian approach. Assume that Ĝ(s has realization {Â, B, Ĉ} with  diagonalizable, and that all poles of Ĝ(s are distinct, i.e. there exists a non-singular S such that S ÂS diag( λ,..., λ r. The transfer function Ĝ(s can be written as Ĝ(s Ĉ(sI  B (ĈS (si S ÂS (S B s λ (ĈS... (S B s λ r r ĉ i b i, i s λ i where ĈS : ĉ... ĉ r, and S B : b T... b T r T. Note that the rows of S and the columns of S are, respectively, left and right eigenvectors of Â. if S : t T... tr T T and S : s... s r, where ti, s i, i,..., r are column vectors in R r, we have the following identities:  λ i ti T, tt B i b T i, i,..., r t T i Âs i λ i s i, Ĉs i ĉ i, i,..., r. Theorem 3.. Let Ĝ(s r ĉ i b i i have distinct first-order s λ i poles. Then, 2 ( B Js i G( λ i Ĝ( λ i ĉ i, (3. 2 tt i ( Ĉ J b T i G( λ i Ĝ( λ i, (3.2 2 tt i (  Js i b T i 2 tt i (  Js j d G(s Ĝ(s ĉ i, (3.3 ds s λi 2( λ i λ j t T i ( ĈJĉ j b T i ( B Js j. (3.4 Proof. Let y T i : ti T Y. Multiplying from the left both sides of (.7 by ti T one gets that ti T  Y ti T YA tt B i C 0, λ i ti T b T i y T i (A λ i I b T i C. (3.5 Similarly, let x i : Xs i and multiply from the right both sides of (.6 by s i to obtain AXs i X Âs i B Ĉs i 0, λ i s i ĉ i 4 (A λ i Ix i Bĉ i, (3.6 Similar computations can be done for equation (.5. If p T i : t T i R and q i : Rs i, one gets that ti T  R t T R i t T B i Ĉ 0, λ i ti T b T i  R R Âs i B Ĉs i 0, λ i s i ĉ i p T i ( λ i I b T i Ĉ, (3.7 ( λ i Iq i Bĉ i. (3.8 It then follows from (3.5, (3.6, (3.7, (3.8 that y T i b T i C(A λ i I, (3.9 x i (A λ i I Bĉ i, (3.0 p T i b T i Ĉ( λ i I, (3. q i ( λ i I Bĉ i. (3.2 2 ( B Js i (Ĉ R CXs i Ĉ Rs i C Xs i q i x i Ĉ( λ i I } {{ B ĉ } i C(A λ i I B } {{ } Ĝ( λ i G( λ i G( λ i Ĝ( λ i ĉ i, 2 tt i ( Ĉ J tt i ( R B YB ti T R B ti T Y B p T i y T i b T i Ĉ( λ i I } {{ B b } T i C(A λ i I B } {{ } Ĝ( λ i b T i G( λ i Ĝ( λ i, G( λ i which prove (3. and (3.2. In order to prove (3.3 and (3.4, we first compute 2 tt i (  Js j as 2 tt i (  Js j t T i ( R R YXs j ti T R Rs j p T q j i ti T Y y T i Xs j x j b T i Ĉ( λ i I ( λ j I Bĉ j } {{ }} {{ } ( p T i ( q j b T i C(A λ i I (A λ j I Bĉ j } {{ }} {{ } y T x j i Ĉ( b T i λ i I ( λ j I B C(A λ i I (A λ j I B ĉ j. ĉ i

5 Now using d ds G(s C(sI A 2 d B and dsĝ(s Ĉ(sI Â 2 B, we obtain, for the case i j, that 2 tt i ( Â Js Ĉ(Â i b T i λ i I 2 B C(A λ i I 2 B ĉ i b T d i G(s Ĝ(s ĉ i. ds s λi If i j, the identity (M λ i I (M λ j I (M λ i I (M λ j I allows us to write 2 tt i ( Â Js Ĉ j b T i (Â λ i I (Â λ j I B } {{ } applying the above identity C (A λ i I (A λ j I B ĉ j } {{ } applying the above identity Ĉ ( (Â λ i I (Â λ j I B b T i C ( (A λ i I (A λ j I B ĉ j b T i (Ĝ( λ i Ĝ( λ j ( G( λ i G( λ j ĉ j (3.3 ( b T i G( λ i Ĝ( λ i ĉ j } {{ } 2 tt i ( Ĉ J (by (3.2 which concludes the proof. b T i ( G( λ j Ĝ( λ j ĉ j } {{ } 2 ( B Js i (by (3. 2( λ i λ j ti T ( Ĉ Jĉ j b T i ( B Js j, Equations (3., (3.2 and (3.3 define the tangential interpolation conditions for a MIMO system, while (3.4 expresses a redundancy condition of the parameters of (Â, B, Ĉ (see for a discussion. Equation (3.4, in the form (3.3, can also be interpreted as the equality of the secants of G and Ĝ through two mirror poles of Ĝ. 4. Conclusions We have obtained conditions for the stationary points of the H 2 -norm of the error system associated to a general linear MIMO system. Our conditions are formulated in terms of cross-gramians, instead of controllability and observability Gramians as in, and we have interpreted them in terms of tangential interpolation conditions. This cross-gramian-based solution may be useful for largescale systems, as the number of matrix equations that must be 5 solved at each step of the iterative procedure that must be carried on to find the stationary point is half that of the algorithm based on the controllability and observability Gramians. Furthermore, very powerful numerical algorithms are available to solve the Sylvester-type matrix equations that appear at each iteration. 234 A possible extension of our approach for the case of nonlinear systems could consider either the analytical, PDE-based definition of the cross-gramian developed in 5, or the empirical one presented in 6, which relies on numerical data obtained from simulations or experiments. Unstable linear systems would also be another interesting generalization, using the ideas developed in 78. References P. van Dooren, K. A. Gallivan, and P. A. Absil. H2-optimal model reduction of MIMO systems. Appl. Math. Lett., 2(2: , A. C. Antoulas. Approximation of Large-scale Dynamical Systems. Advances in design and control. Society for Industrial and Applied Mathematics, W.H.A. Schilders, J. Rommes, and H.A. van der Vorst. Model Order Reduction: Theory, Research Aspects and Applications: Theory, Research Aspects and Applications. European Consortium for Mathematics in Industry. Springer, S. Gugercin, A. C. Antoulas, and C. A. Beattie. H2 Model Reduction for Large-Scale Linear Dynamical Systems. SIAM J. Matrix Analysis Applications, 30(2: , K.V. Fernando and H. Nicholson. On the structure of balanced and other principal representations of SISO systems. Automatic Control, IEEE Transactions on, 28(2:228 23, K.V. Fernando and H. Nicholson. Minimality of SISO linear systems. Proceedings of the IEEE, 70(0:24 242, A.J. Laub, L. M. Silverman, and M. Verma. A note on cross-grammians for symmetric realizations. Proceedings of the IEEE, 7(7: , K.V. Fernando and H. Nicholson. On a fundamental property of the cross- Gramian matrix. Circuits and Systems, IEEE Transactions on, 3(5: , K.V. Fernando and H. Nicholson. On the cross-gramian for symmetric MIMO systems. Circuits and Systems, IEEE Transactions on, 32(5: , D.A. Wilson. Optimum solution of model-reduction problem. Electrical Engineers, Proceedings of the Institution of, 7(6:6 65, 970. K. A. Gallivan, A. Vandendorpe, and P. van Dooren. Model Reduction of MIMO Systems via Tangential Interpolation. SIAM J. Matrix Analysis Applications, 26(2: , G. Golub, S. Nash, and C. Van Loan. A Hessenberg-Schur method for the problem AX XB C. Automatic Control, IEEE Transactions on, 24(6:909 93, D. C. Sorensen and A. C. Antoulas. The Sylvester equation and approximate balanced reduction. Linear Algebra and its Applications, (0:67 700, U. Baur and P. Benner. Cross-Gramian based model reduction for datasparse systems. ETNA. Electronic Transactions on Numerical Analysis electronic only, 3: , T.C. Ionescu and J. M A Scherpen. Nonlinear cross Gramians and gradient systems. In Decision and Control, th IEEE Conference on, pages , C. Himpe and M. Ohlberger. Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems, October K. Zhou, G. Salomon, and E. Wu. Balanced realization and model reduction for unstable systems. International Journal of Robust and Nonlinear Control, 9(3:83 98, H.R. Shaker. Generalized cross-gramian for linear systems. In Industrial Electronics and Applications (ICIEA, 202 7th IEEE Conference on, pages , 202.