Data-driven model order reduction of linear switched systems

Transcription

1 Data-driven model order reduction of linear switched systems IV Gosea, M Petreczky, AC Antoulas arxiv: v1 mathna 15 Dec 2017 Abstract The Loewner framework for model reduction is extended to the class of linear switched systems One advantage of this framework is that it introduces a trade-off between accuracy and complexity Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system Hence, another advantage of the framework is that it does not require the initial system matrices More exactly, the data used in this framework consists in frequency domain samples of inputoutput mappings of the original system The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems A key role is played by the coupling matrices, which ensure the transition from one active mode to another 1 Introduction Model order reduction MOR seeks to transform large, complicated models of time dependent processes into smaller, simpler models that are nonetheless capable of representing accurately the behavior of the original process under a variety of operating conditions The goal is an efficient, methodical strategy that yields a dynamical system evolving in a substantially lower dimension space hence requiring far less computational resources for realization, yet retaining response characteristics close to the original system Such reduced order models could be used as efficient surrogates for the original model, replacing it as a component in larger simulations Hybrid systems are a class of nonlinear systems which result from the interaction of continuous time dynamical sub-systems with discrete events More precisely, a hybrid system is a collection of continuous time dynamical systems The internal variable of each dynamical system is governed by a set of differential equations Each of the separate continuous time systems are labeled as a discrete mode The transitions between the discrete states may result in a jump in the continuous internal variable Linear switched systems in short LSS constitute a subclass of hybrid systems; the main property is that these systems switch among a finite number of linear subsystems Also, the discrete events interacting with the sub-systems are governed by a piecewise continuous function called the switching signal Data-Driven System Reduction and Identification Group, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106, Magdeburg, Germany gosea@mpi-magdeburgmpgde Centre de Recherche en Informatique, Signal et Automatique de Lille CRIStAL, UMR CNRS 9189, CNRS, Ecole Centrale de Lille, France mihalypetreczky@ec-lillefr Department of Electrical and Computer Engineering, Rice University, 6100 Main St, MS-366, Houston, TX 77005, USA aca@riceedu 1

2 Hybrid and switched systems are powerful models for distributed embedded systems design where discrete controls are routinely applied to continuous processes However, the complexity of verifying and assessing general properties of these systems is very high so that the use of these models is limited in applications where the size of the state space is large To cope with complexity, abstraction and reduction are useful techniques In this paper we analyze only the reduction part In the past years, hybrid and switched systems have received increasing attention in the scientific community For a detailed characterization of this relatively new class of dynamical systems, we refer the readers to the books 24, 37, 38 and 16 Such systems are used in modeling, analysis and design of supervisory control systems, mechanical systems with impact, circuits with relays or ideal diodes The study of the properties of hybrid systems in general and switched systems in particular is still the subject of intense research, including the problems of stability see 13 and 37, realization including observability/controllability see 30 and 31, analysis of switched DAE s see 26 and 39 and numerical solutions see 17 Recently, considerable research has been dedicated to the problem of MOR for linear switched systems The most prolific method that has been applied is balanced truncation or some sort of gramian based derivation of it Techniques that are based on balancing have been considered in the following: 15, 11,8, 36, 27, 33 and 29 Also, another class of methods involve matching of generalized Markov parameters known also as time domain Krylov methods such as the ones in 7 and 6; H type of reduction methods were developed in 41, 9 and 42 Finally, we mention some publications that are focused on the reduction of discrete LSS, such as 40 and 10 A linear switched system involves switching between a number of linear systems the modes of the LSS Hence, to apply balanced truncation techniques to a switched linear system, one may seek for a basis of the state space such that the corresponding modes are all in balanced form It may happen that some state components of the LSS are difficult to reach and observe in some of the modes yet easy to reach and observe in others In that case, deciding how to truncate the state variables and obtain a reduced order model is not trivial A solution to this problem is proposed in 27 where it turns out that the average gramian can be used to obtain a reduced order model This method will be used as a comparison tool for our new MOR method In the sequel we exclusively consider interpolatory MOR methods and in particular the Loewner framework applied to LSS Roughly speaking, in the linear case, interpolatory methods seek reduced models whose transfer function matches that of the original system at selected frequencies For the nonlinear case, these methods require appropriate definitions of transfer functions In this paper, we focus on generalizing the Loewner Framework for reducing linear switched systems The presentation is tailored to emphasize the main procedure for a simplified case of LSS ie only two modes and LTI s in SISO format activating in both modes The paper is organized as follows In the next section, we review the formal definition of continuous-time linear switched systems Furthermore, we introduce the generalized transfer functions for LSS as input-output mappings in frequency domain Section 3 includes a brief introduction of the Loewner framework for linear systems In Section 4, we introduce the Loewner framework for LSS with two modes In section 5, we generalize most of the results in the previous section for the case of LSS with D 2 modes Finally, in Section 6, we discuss the applicability of the new introduced method for reducing LSS In this sense, by means of three numerical examples one of which large scale, we compare the results obtained by applying the Loewner method against the method in 27 In Section 7, we present a summary of the findings and the conclusions 2

3 2 Linear switched systems Definition 21 A continuous time linear switched system LSS is a control system of the form: E σt ẋt = A σt xt + B σt ut, xt = x 0, Σ : 1 yt = C σt xt, where Q = 1, 2,, D}, D > 1, is a set of discrete modes, σt is the switching signal, u is the input, x is the state, and y is the output The system matrices E q, A q R nq nq, B q R nq m, C q R p nq, where q Q, correspond to the linear system active in mode q Q, and x 0 is the initial state We consider the E q matrices to be invertible Furthermore, the transition from one mode to another is made via the so called switching or coupling matrices K q1,q 2 R nq 2 nq 1 where q1, q 2 Q Remark 21 The case for which the coupling is made between identical modes is excluded, Hence, when q 1 = q 2 = q, consider that the coupling matrices are identity matrices, ie K q,q = I nq The notation Σ = n 1, n 2,, n D, E q, A q, B q, C q q Q}, K qi,q i+1 q i, q i+1 Q}, x 0 is used as a short-hand representation for LSS s described by the equations in 1 The vector n = n1 n 2 n D is the dimension order of Σ The linear system which is active in the q th mode of Σ is denoted with Σ q and it is described by where 1 q D E q x q t = A q x q t + B q ut, xt k = x k, Σ k : yt = C q x q t The restriction of the switching signal σt to a finite interval of time 0, T can be interpreted as finite sequence of elements of Q R + of the form: νσ = q 1, t 1 q 2, t 2 q k, t k, where q 1,, q k Q and 0 < t 1 < t 2 < < t k R +, t t k = T, such that for all t 0, T we have: q 1 if t 0, t 1, q 2 if t t 1, t 1 + t 2, σt = q i if t t t i 1, t t i 1 + t i, q k if t t t k 1, t t k 1 + t k In short, by denoting T i := t t i 1 + t i, T 0 := 0, T k := T, write: q 1 if t 0, T 1, σt = q i if t T i 1, T i, i > 2 Denote by P CR +, R n, P c R +, R n, the set of all piecewise-continuous, and piecewise-constant functions, respectively Definition 22 A tuple x, u, σ, y, where x : R + D i=1 Rn i, u P CR +, R m, σ P c R +, Q, y P CR +, R p is called a solution, if the following conditions simultaneously hold: 3 2

4 1 The restriction of xt to T i 1, T i is differentiable, and satisfies E qi ẋt = A qi xt + But 2 Furthermore, when switching from mode q i to mode q i+1 at time T i, the following holds E qi+1 3 Moreover, for all t R, yt = C σt xt holds lim t Ti x qi+1 t = K qi,q i+1 x qi T i The switching matrices K qi,q i+1 allow having different dimensions for the subsystems active in different modes For instance, the pencil A qi, E qi R nq i nq i, while the pencil A qi+1, E qi+1 R nq i+1 nq i+1 where the values nqi and n qi+1 need not be the same If the K qi,q i+1 matrices are not explicitly given, it is considered that they are identity matrices The input-output behavior of an LSS system can be formalized in time domain as a map fu, σt This particular map can be written in generalized kernel representation as suggested in 32 using the unique family of analytic functions: g q1,,q k : R k + R p and h q1,,q k : R k + R p m with q 1,, q k Q, k 1 such that for all pairs u, σ and for T = t 1 + t t k we can write: fu, σt = g q1,q 2,,q k t 1, t 2,, t k + k ti i=1 0 h qi,q i+1,,q k t i τ, t i+1,, t k uτ + T i 1 dτ, where the functions g, h are defined for k 1, as follows, g q1,q 2,,q k t 1, t 2,, t k = C qk eãq k t k Kqk 1,q k eãq k 1 t k 1 Kqk 2,q k 1 K q1,q 2 eãq 1 t 1 x 0, 3 h q1,q 2,,q k t 1, t 2,, t k = C qk eãq k t k Kqk 1,q k eãq k 1 t k 1 Kqk 2,q k 1 K q1,q 2 eãq 1 t 1 B1 4 Note that, for the functions defined in 3 and 4 we consider the E qi matrices to be incorporated into the A qi and B qi matrices ie à qi = E 1 q i A qi, Bqi = E 1 q i B qi Moreover, the transformed coupling matrices are written accordingly K qi,q i+1 = E 1 q i+1 K qi,q i+1 In the rest of the paper, the LSS we treat are assumed to have zero initial conditions, ie, x 0 = 0 Hence, only the h functions in 4 are relevant for characterizing the input-output mapping f The behavior of the input-output mappings in frequency domain is in turn characterized by a series of multivariate rational functions obtained by taking the multivariable Laplace transform of the regular kernels in 4, as for H q1 s 1 = C q1 Φ q1 s 1 B q1, H q1,q 2 s 1, s 2 = C q1 Φ q1 s 1 K q2,q 1 Φ q2 s 2 B q2, H q1,q 2,q 3 s 1, s 2, s 3 = C q1 Φ q1 s 1 K q2,q 1 Φ q2 s 2 K q3,q 2 Φ q3 s 3 B q3, For k 1, write the level k generalized transfer function associated to the switching sequence q 1, q 2,, q k, and evaluated at the points s 1, s 2, s k as, H q1,q 2,,q k s 1, s 2,, s k = C q1 Φ q1 s 1 K q2,q 1 Φ q2 s 2 K qk,q k 1 Φ qk s k B qk, 5 where Φ q s = se q A q 1, q j 1, 2,, D}, 1 j k and k 3 These functions are the generalized transfer functions of the linear switched system Σ Their definition is similar to the ones corresponding to bilinear systems see 3 By using their samples, we are able to directly come up with reduced switched models that interpolate the original model - generalization of the Loewner framework to LSS 4

5 We construct LSS reduced models by means of matching samples of input-output mappings corresponding to the original LSS system and evaluated at finite sampling points as opposed to other approaches - see 6 and 7, where the behavior at infinity is studied instead, ie by matching Markov parameters For the explicit derivation of these types of transfer functions which is based on the so-called Volterra series representation we refer the readers to 34 3 Interpolatory MOR methods and the Loewner framework Consider a full-order linear system defined by E R n n, A R n n, B R n m, C R p n, and its transfer function Hs = CsE A 1 B Given left interpolation points: µ j } q j=1 C, with left tangential directions: l j } q j=1 Cp, and right interpolation points: λ i } k i=1 C, with right tangential directions: r i } k i=1 C m, find a reduced-order system Ê, Â, ˆB, Ĉ, such that the resulting transfer function, Ĥs is a tangential interpolant to Hs: l T j Ĥµ j = l T j Hµ j, j = 1,, q, and Ĥλ ir i = Hλ i r i, i = 1,, k 6 Interpolation points and tangent directions are selected to realize appropriate MOR goals If instead of state space data, we are given input/output data, the resulting problem is hence modified Given a set of input-output response measurements specified by left driving frequencies: µ j } q j=1 C, using left input directions: l j } q j=1 Cp, producing left responses: v j } q j=1 Cm, and right driving frequencies: λ i } k i=1 C, using right input directions: r i } k i=1 C m, producing right responses: w i } k i=1 C p, find low order system matrices Ê, Â, ˆB, Ĉ, such that the resulting transfer function, Ĥs, is an approximate tangential interpolant to the data: l T j Ĥµ j = v T j, j = 1,, q, and Ĥλ i r i = w i, i = 1,, k 7 31 Overview of the Loewner framework for linear systems The approach we discuss in this section is data driven After collecting input/output eg frequency response measurements for some appropriate range of frequencies, we construct models which fit or approximately fit the data and have reduced dimension The key is that, larger amounts of data than necessary are collected and the essential underlying system structure is extracted appropriately Thus an advantage of this approach is that it can provide the user with a trade-off between accuracy of fit and complexity of the model The Loewner framework was developed in a series of papers; for details we refer the reader to 1, as well as 25, 23, 22, 4, 19, 20 For a recent overview see 5 32 The Loewner pencil We will formulate the results for the more general tangential interpolation problem We are given the right data: λ i ; r i, w i, i = 1,, k, and the left data: µ j ; l T j, v T j, j = 1,, q; it is assumed for simplicity that all points are distinct The dimensions are as in 6, 7 The data can be organized as follows: the right data: Λ = diag λ 1,, λ k C k k, R = r 1,, r k C m k, W = w 1,, w k C p k, 5

6 and the left data: M = diag µ 1,, µ q C q q, L T = l 1,, l q C q p, V T = v 1,, v q C q m Then, the associated Loewner pencil, consists of the Loewner and shifted Loewner matrices The Loewner matrix L C q k, is defined as: v1 T r 1 l T 1 w 1 v1 µ 1 λ 1 T r k l T 1 w k µ 1 λ k L = vq T r 1 l T q w 1 µ q λ 1 v T q r k l T q w k µ q λ k L satisfies the Sylvester equation LΛ ML = VR LW Suppose that the underlying transfer function is Hs = CsE A 1 B, and define the generalized observability/controllability matrices: Cµ 1 E A 1 O =, R = λ 1 E A 1 B λ k E A 1 B 8 Cµ q E A 1 It readily follows that the Loewner matrix can be factored as L = OER The shifted Loewner matrix L s C q k, is defined as: µ 1 v1 T r 1 l T 1 w 1λ 1 µ µ 1 λ 1 1 v1 T r k l T 1 w kλ k µ 1 λ k L s = µ qvq T r 1 l T q w 1λ 1 µ q λ 1 µ qv T q r k l T q w kλ k µ q λ k L s satisfies the Sylvester equation L s Λ ML s = MVR LWΛ, and can be factored in terms of the generalized controllability/observabilty matrices as L s = OAR Finally notice that the following relations hold: V = CR, W = OB 33 Construction of reduced order models We will distinguish two cases namely, the right amount of data and the more realistic redundant amount of data cases The following lemma covers the first case Lemma 31 Assume that k = q, and let L s, L, be a regular pencil, such that none of the interpolation points λ i, µ j are its eigenvalues Then E = L, A = L s, B = V, C = W, is a minimal realization of an interpolant of the data, ie, the rational function Hs = WL s sl 1 V, interpolates the data the conditions in 7 are hence matched If the pencil L s, L is singular we are dealing with the case of redundant data In this case if the following assumption is satisfied: L rank xl L s = rank = rank L L s = r k, 9 L s for all x λ i } µ j }, we consider the following SVD factorizations: L L L s = Y 1 S 1 X T 1, = Y 2 S 2 X T 2, 10 where Y 1, X 2 C k k By selecting the first r columns of the matrices Y 1 and X 2, we come up with projection matrices Y, X C k r The following result is used in practical applications 6 L s

7 Lemma 32 A realization E, A, B, C of an approximate interpolant is given by the system matrices E = Y T LX, A = Y T L s X, B = Y T V, C = WX Hence, the rational function Hs = WXY T L s X sy T LX 1 Y T V approximately matches the data the conditions in 7 are approximately fulfilled, ie Hλ i r i = w i + ɛ r i and l T j Hµ j = v T j + ɛ l j T, where the residual errors are collected in the vectors ɛ r i and ɛ l j Thus, if we have more data than necessary, we can consider L s, L, V, W, as a singular model of the data An appropriate projection then yields a reduced system of order k see 25, 2 A direct consequence is that the Loewner framework offers a trade-off between accuracy and complexity of the reduced order system, by means of the singular values of L Remark 31 For an error bound that links the quality of approximation to the singular values of the Loewner pencil which is valid only at the interpolation points µ j and λ i, we refer the readers to 5 4 The Loewner framework for LSS - the case D=2 The characterization of linear switched systems by means of rational functions suggests that reduction of such systems can be performed by means of interpolatory methods In the following we will show how to generalize the Loewner framework to LSS by interpolating appropriately defined transfer functions on a chosen grid of frequencies interpolation points As for the linear case, the given set of sampling interpolation points is first partitioned into the two following categories: left interpolation points: µ j } l j=1 C and and right interpolation points: λ i } k i=1 C In this paper we consider only the case of SISO linear switched systemshence the left and right tangential directions can be considered to be scalar ie taking the value 1 Since the transfer functions which are going to be matched are not single variable functions anymore they depend on multiple variables as described in, the structure of the interpolation points used in the new framework is going to change Instead of having singleton values as in Section 3, we will use instead n-tuples that include multiple singleton values For simplicity of the exposition, we first consider the simplified case D = 2 the LSS system switches between two modes only This situation is encountered in most of the numerical examples in the literature we came across Nevertheless, all the results presented in this section can be generalized for higher number of modes in a more or less straightforward way as presented in Section 5 Depending on the switching signal σt, we either have, Σ 1 : E 1 ẋ 1 t = A 1 x 1 t + B 1 ut, yt = C 1 x 1 t or Σ 2 : E 2 ẋ 2 t = A 2 x 2 t + B 2 ut, yt = C 2 x 2 t where dimσ 1 = n 1 ie x 1 R n 1 and E 1, A 1 R n 1 n 1, B 1, C T 1 R n 1 and also dimσ 2 = n 2 ie x 2 R n 2 and E 2, A 2 R n 2 n 2, B 2, C T 2 R n 2 Notice that we allow both the two subsystems to be written in descriptor format having possibly singular E matrix Denote, for simplicity, with K 1 the coupling matrix when switching from mode 1 to mode 2 instead of K 1,2 and, with K 2, the coupling matrix when switching from mode 2 to mode 1 instead of K 2,1 with K 1 R n 2 n 1 and K 2 R n 1 n 2 The generalized transfer functions are defined as where Φ q s = se q A q 1, q 1, 2}, Level 1 7

8 H 1 s 1 = C 1 Φ 1 s 1 B 1 H 2 s 2 = C 2 Φ 2 s 2 B 2 Level 2 H 1,2 s 1, s 2 = C 1 Φ 1 s 1 K 2 Φ 2 s 2 B 2 H 2,1 s 2, s 1 = C 2 Φ 2 s 2 K 1 Φ 1 s 1 B 1 Level 3 H 1,2,1 s 1, s 2, s 3 = C 1 Φ 1 s 1 K 2 Φ 2 s 2 K 1 Φ 1 s 3 B 1 H 2,1,2 s 1, s 2, s 3 = C 2 Φ 2 s 1 K 1 Φ 1 s 2 K 2 Φ 2 s 3 B 2, Definition 41 Consider the two LSS, ˆΣ = n 1, n 2, Êi, Âi, ˆB i, Ĉi} 2 i=1, ˆK i,j } 2 i,j=1, 0 and Σ = n 1, n 2, Ēi, Āi, B i, C i } 2 i=1, K i,j } 2 i,j=1, 0 These systems are said to be equivalent if there exist non-singular matrices Z L j and Z R j so that Ē j = Z L j ÊjZ R j, Ā j = Z L j ÂjZ R j, Bj = Z L j ˆB j, Cj = ĈjZ R j, j 1, 2} and also K 1 = Z L 2 ˆK 1 Z R 1, K2 = Z L 1 ˆK 2 Z R 2 In this configuration, one can easily show that the transfer functions defined above are the same for each LSS and for all sampling points s k 41 The generalized controllability and observability matrices Consider a LSS system Σ as described in 1 with dimσ k = n k for k = 1, 2 and let K 1 R n 2 n 1 and K 2 R n 1 n 2 be the coupling matrices Before stating the general definitions, we first clarify how the newly introduced matrices are constructed through a simple self-explanatory example Example 41 Consider 10 left interpolation points µ 1 1,, µ 1 4, µ 2 1,, µ 2 6 } which are written in nested multi-tuple format corresponding to each mode of the LSS: Mode 1 : µ 1 1 = Mode 2 : µ 1 2 = 1 µ, 1 1 µ 2, µ µ, 2 1 µ 1, µ 1 4 µ 2 1 =, µ 2 2 = 2 µ 1, 2 µ 2, µ 2 3, 2 µ 1, µ 2 4, µ µ 2, 2 µ 1, µ 2 4, 2 µ 2, µ 2 3, µ 2 6 We explicitly write the generalized observability matrices O 1 and O 2 as follows: O 1 = C 1 Φ 1 µ 1 1 C 2 Φ 2 µ 1 2 K 1 Φ 1 µ 1 3 C 1 Φ 1 µ 2 1 C 2 Φ 2 µ 2 2 K 1 Φ 1 µ 2 3 C 1 Φ 1 µ 2 1 K 2 Φ 2 µ 2 4 K 1 Φ 1 µ 2 5, O 2 = C 2 Φ 2 µ 1 2 C 1 Φ 1 µ 1 1 K 2 Φ 2 µ 1 4 C 2 Φ 2 µ 2 2 C 1 Φ 1 µ 2 1 K 2 Φ 2 µ 2 4 C 2 Φ 2 µ 2 2 K 1 Φ 1 µ 2 3 K 2 Φ 2 µ 2 6 8

9 Definition 42 Given a non-empty set Q, denote with Q i the set of all i th tuples with elements from Q Introduce the concatenation of two tuples composed of elements symbols α 1,, α i, and β 1,, β j from Q as the mapping : Q i Q j Q i+j with the following property: α1, α 2,, α i β1, β 2,, β j = α1, α 2, α i, β 1, β 2, β j, Remark 41 In the following we denote the l th element of the ordered set µ i j with µ i j l where j Q and i 1 For instance, µ := µ 2 1, µ 2 4, µ 2 5 For simplicity, use the notation H 1,2,1 µ 2 1, µ 2 4, µ instead of H 1,2,1 µ 1, µ 2 4, µ 2 5 Definition 43 We define the nested right multi-tuples: } } λ 1 = λ 1 1, λ 2 1,, λ k 1, λ 2 = λ 1 2, λ 2 2,, λ k 2 composed of sets of right i th tuples: i λ 1, i λ 3, λ i 2, i λ λ i 5, λ i 4, λ i 1, 1 =, λ i 2 = i λ 2m i 3,, λ i 4, λ i 1, i λ 2m i 1, λ i 2m i 2,, λ i 3, λ i 2 i λ, 2 i λ 4, λ i 1, i λ 6, λ i 3, λ i 2, i λ 2m i 2,, λ i 3, λ i 2, i λ 2m i, λ i 2m i 3,, λ i 4, λ i That is, we construct the right tuples based on the following recurrence relations where λ i 1 1 = i i λ 1 and λ 2 1 = λ i 2 λ i 1 g = λ i i 2g 1 λ 2 g 1, λ i 2 g = λ i i 2g λ 1 g 1, 13 for λ i 2g 1 C, 1 < g m i, i = 1,, k, m i 1 so that the equality m m k = k holds Also, introduce the nested left multi-tuples } µ 1 = µ 1 1, µ 2 1,, µ l 1, µ 2 = } µ 1 2, µ 2 2,, µ l 2 14 composed of sets of left j th tuples j µ 1, j µ 2, µ j 3, j µ µ j 1, µ j 4, µ j 5, 1 = j µ 1, µ j 4,, µ j 2p j 3, j µ 2, µ j 3,, µ j 2p j 2, µ j 2p j 1 j µ 2, j µ 1, µ j 4, j µ µ j 2, µ j 3, µ j 6, 2 = j µ 2, µ j 3,, µ j 2p j 2, j µ 1, µ j 4,, µ j 2p j 3, µ j 2p j 15 9

10 That is, we construct the left tuples based on the following recurrence relations where µ j 1 1 = j j µ 1 and µ 2 1 = µ j 2 µ j 1 h = µ j 2 h 1 µ j j 2h 1, µ 2 h = µ j 1 h 1 µ j 2h, 16 for µ j 2h 1 C, 1 < h p j, j = 1,, l, p j 1 so that p p l = l Given that the following conditions are satisfied for all i = 1,, k and g = 1,, m i, λ i 2g 1 / eiga 1, E 1, λ i 2g / eiga 2, E 2 17 associate the following matrices to the set of right tuples in 12, as R i 1 = Φ 1 λ i 1 B 1, Φ 1 λ i 3 K 2 Φ 2 λ i 2 B 2,, Φ 1 λ i 2m i 1 K 2 K 1 Φ 1 λ i 3 K 2 Φ 2 λ i 2 B 2, R i 2 = Φ 2 λ i 2 B 2, Φ 2 λ i 4 K 1 Φ 1 λ i 1 B 1,, Φ 2 λ i 2m i K 1 K 2 Φ 2 λ i 2 K 1 Φ 1 λ i 1 B 1, where i = 1,, k and R i q C nq m i is attached to Λ i q for q 1, 2} The matrices: R 1 = R 1 1, R 2 1,, R k 1 C n1 k, R 2 = R 1 2, R 2 2,, R k 2 C n 2 k 18 are defined as the generalized controllability matrix of the LSS system Σ, associated with the right multi-tuple λ Similarly, assuming that the following conditions are satisfied for all j = 1,, l and h = 1,, p j, µ j 2h 1 / eiga 1, E 1, µ j 2h / eiga 2, E 2 19 associate the following matrices to the set of right tuples in 15, as C 1 Φ 1 µ j 1 C 2 Φ 2 µ j 2 K 1 Φ 1 µ j 3 O j 1 = O j 2 = Cp j n 1, j = 1,, l, C 2 Φ 2 µ j 2 K 1 Φ 1 µ j 3 K 2 K 1 Φ 1 µ j 2p j 1 C 2 Φ 2 µ j 1 C 1 Φ 1 µ j 1 K 2 Φ 2 µ j 4 Cp j n 2, j = 1,, l, C 1 Φ 1 µ j 1 K 2 Φ 2 µ j 2 K 1 K 2 Φ 2 µ j 2p j and the generalized observability matrices: O 1 = O 1 1 O l 1 Cl n 1, O 2 = O 1 2 O l 2 Cl n

11 Definition 44 For ν 1, 2}, let Q ν,+ and Q +,ν be the ordered sets containing all tuples that can be constructed with symbols from the alphabet Q = 1, 2} and that start and respectively end with the symbol ν Also, no two consecutive characters are allowed to be the same Hence, explicitly write the new introduced sets as follows: Q 1,+ = 1, 1, 2, 1, 2, 1, }, Q 2,+ = 2, 2, 1, 2, 1, 2, }, 21 Q +,1 = 1, 2, 1, 1, 2, 1, }, Q +,2 = 2, 1, 2, 2, 1, 2, } 22 Remark 42 In the following we denote the l th element of the ordered set Q ν,+ with Q ν,+ l For example, one writes Q 1,+ 4 := 1, 2, 1, 2 Moreover, we have Q +,2 3 Q 1,+ 2 = 2, 1, 2, 1, 2 The compact notation H Q +,1µ 1 2 is used instead of H 2,1 µ 2, µ 3, where µ 1 2 := µ 2, µ 3 Definition 45 Let the i th unit vector be denoted with e i = 0, 1,, 0 T R k In some contexts we may use the alternative notation e i,k to stress the fact the vector has dimension k Also let 0 k,l R k l be an all zero matrix Hence, use the notation 0 k = 0, 0,, 0 T R k for the zero valued vector of size k In the sequel, denote with Ĥ the generalized transfer functions corresponding to a LSS ˆΣ Definition 46 We say that a LSS ˆΣ = k, k, Êi, Âi, ˆB i, Ĉi} 2 i=1, ˆK i,j } 2 i,j=1, 0 matches the data associated with the right tuples λ 1 a,, λ k a } as well as left tuples µ 1 b,, µ l b }, a, b = 1, 2 and corresponding to the original LSS Σ = n 1, n 2, E i, A i, B i, C i } 2 i=1, K i,j } 2 i,j=1, 0, if the following 2k 2 + 2k relations H Q +,1 hµ j 1 h = ĤQ +,1 hµ j 1 h, H Q +,2 hµ j 2 h = ĤQ +,2 hµ j 2 h, H Q 1,+ gλ i 1 g = ĤQ 1,+ gλ i 1 g, H Q 2,+ gλ i 2 g = ĤQ 2,+ gλ i 2 g, H Q +,1 h Q 2,+ gµ j 1 h λ i 2 g = ĤQ +,1 h Q 2,+ gµ j 1 h λ i 23 2 g H Q +,2 h Q 1,+ gµ j 2 h λ i 1 g = ĤQ +,2 h Q 1,+ gµ j 2 h λ i 1 g hold for j = 1,, k, h = 1,, p j and i = 1,, k, g = 1,, m i, where p 1 + p 2 + p k = m 1 + m 2 + m k = k The following lemma extends the rational interpolation idea for linear systems approximation to the linear switched system case Lemma 41 Interpolation of LSS Let Σ = n 1, n 2, E i, A i, B i, C i } 2 i=1, K i,j } 2 i,j=1, 0 be a LSS of order n 1, n 2 An order k reduced LSS ˆΣ = k, k, Êi, Âi, ˆB i, Ĉi} 2 i=1, ˆK i,j } 2 i,j=1, 0 is constructed using the projection matrices chosen as in 18 and 20 for l = k, ie X 1 = R 1, X 2 = R 2 and Y T 1 = O 1, Y T 2 = O 2 Additionally assume rankr i = ranko i = k, i 1, 2} The reduced matrices corresponding to the I st subsystem ˆΣ 1 are computed as, Ê 1 = Y T 1 E 1 X 1, Â 1 = Y T 1 A 1 X 1, ˆB1 = Y T 1 B 1, Ĉ 1 = C 1 X 1, ˆK1 = Y T 2 K 1 X 1, 24 while the reduced matrices corresponding to the II nd subsystem ˆΣ 2 can also be computed as, Ê 2 = Y T 2 E 2 X 2, Â 2 = Y T 2 A 2 X 2, ˆB2 = Y T 2 B 2, Ĉ 2 = C 2 X 2, ˆK2 = Y T 1 K 2 X 2 25 It follows that the reduced-order system ˆΣ matches the data of the system Σ as it was previously introduced in Definition 46 11

12 Proof of Lemma 41 For simplicity, assume that we have one set of right multi-tuples, and one set of left multi-tuples with the same number of interpolation points k for each mode Moreover let k be an even positive number For the first mode, write down the interpolation nodes as follows: λ1 = λ 1, λ3, λ 2,, λ2k 1,, λ 3, λ 2 }, µ 1 = µ 1, µ2, µ 3,, µ2, µ 3,, µ 2k 1 } 26 For the second mode, write down the interpolation nodes as follows: λ2 = λ 2, λ4, λ 1,, λ2k,, λ 2, λ 1 }, µ 2 = µ 2, µ1, µ 4,, µ1, µ 4,, µ 2k } 27 This corresponds to the case for which l = k, l = k = 1 and m 1 = p 1 = k It follows that the interpolation conditions stated in Definition 46, can be rewritten by taking into account the aforementioned simplification as, H Q 2k conditions: +,1 jµ 1 j = ĤQ +,1 jµ 1 j, j 1,, k} 28 H Q +,2 jµ 2 j = ĤQ +,2 jµ 2 j H Q 2k conditions: 1,+ iλ 1 i = ĤQ 1,+ iλ 1 i, i 1,, k} 29 H Q 2,+ iλ 2 i = ĤQ 2,+ iλ 2 i k 2 conditions: H Q +,1 j Q 2,+ iµ 1 j λ 2 i = ĤQ +,1 j Q 2,+ iµ 1 j λ 2 i, 30 k 2 conditions: H Q +,2 j Q 1,+ iµ 2 j λ 1 i = ĤQ +,2 j Q 1,+ iµ 2 j λ 1 i 31 With the assumptions in 26 and 27, it follows that the associated generalized controllability and observability matrices defined previously in 18 and 20, are rewritten as: R 1 = Φ 1 λ 1 B 1, Φ 1 λ 3 K 2 Φ 2 λ 2 B 2,, Φ 1 λ 2k 1 K 2 K 2 Φ 2 λ 2 B 2 C n k, R 2 = Φ 2 λ 2 B 2, Φ 2 λ 4 K 1 Φ 1 λ 1 B 1,, Φ 2 λ 2k K 1 K 1 Φ 1 λ 1 B 1 C n k, C 1 Φ 1 µ 1 C 2 Φ 2 µ 2 C 2 Φ 2 µ 2 K 1 Φ 1 µ 3 C 1 Φ 1 µ 1 K 2 Φ 2 µ 4 O 1 = C 2 Φ 2 µ 2 K 1 Φ 1 µ 3 K 1 Φ 1 µ 2k 1, O 2 = C 1 Φ 1 µ 1 K 2 Φ 2 µ 4 K 2 Φ 2 µ 2k with both O 1, O 2 C k n Additionally, introduce the notation ˆΦ i s = sê Â 1 From 24 and 25, using that X i = R i for i = 1, 2, it readily follows that: a ˆΦ1 λ 1 ˆB 1 = e 1 and b ˆΦ1 λ 2i 1 ˆK 2 e i 1 = e i, i = 2,, k, c ˆΦ2 λ 2 ˆB 2 = e 1 and d ˆΦ2 λ 2i ˆK 1 e i 1 = e i, i = 2,, k These equalities imply the right-hand conditions in 29 Similarly, from 24 and 25, using that Yj T = O j for j = 1, 2, it follows that: e C 1 ˆΦ1 µ 1 = e T 1 and f e T j 1K 2 ˆΦ2 µ 2j = e T j, j = 2,, k, g C 2 ˆΦ2 µ 2 = e T 1 and h e T j 1K 1 ˆΦ1 µ 2j 1 = e T j, j = 2,, k, which imply left-hand conditions in 28 Finally, with X = R, Y T = O, and combining a-h, all interpolation conditions in 30 and 31 are hence satisfied Remark 43 For instance, in Example 42, the conditions stated in 48 are satisfied 12

13 411 Sylvester equations for O and R The generalized controllabilty and observability matrices satisfy Sylvester equations To state the corresponding result we need to define the following quantities First introduce the matrices R = e T 1,m 1 e T 1,m R 1 k, L T = e T k 1,p 1 e T 1,p R 1 l, 32 l and the block-shift matrices S R S L = blkdiag J m1,, J mk, = blkdiag J T p 1,, J T p l where J u = Ru u Finally we arrange the left interpolation points in the diagonal matrices as, M 1 = blkdiag M 1 1, M 2 1,, M l 1, M 2 = blkdiag M 1 2, M 2 2,, M l 2, 34 where M j 1 = diag µ j 1, µ j 3,, µ j 2p j 1 and M j 2 = diag µ j 2, µ j 4,, µ j 2p j ; we used the MATLAB notation blkdiag which outputs a block diagonal matrix with each input entry as a block Also arrange the right interpolation points in the diagonal matrices: Λ 1 = blkdiag Λ 1 1, Λ 2 1,, Λ l 1, Λ 2 = blkdiag Λ 1 2, Λ 2 2,, Λ l 2, 35 where Λ i 1 = diag λ i 1, λ i 3,, λ i 2m i 1 and Λ i 2 = diag λ i 2, λ i 4,, λ i 2m i In the definitions above, ie we analyzed the general case, ie, the assumptions made in were no longer valid We are now ready to state the following result Lemma 42 The generalized controllability matrices R 1, R 2 defined by 18, satisfy the following Sylvester equations: A 1 R 1 + K 2 R 2 S R + B 1 R = E 1 R 1 Λ 1, 36 A 2 R 2 + K 1 R 1 S R + B 2 R = E 2 R 2 Λ 2 Proof of Lemma 42 Assume again, for simplicity of the proof, that the assumptions made in are valid Hence, we have one set of right multi-tuples for each of the two modes with same number of interpolation points k with k even Multiplying the first equation in 36 on the right with the first unit vector e 1 we obtain: A 1 R B 1 = λ 1 E 1 R 1 1 R 1 1 = λ 1 E 1 A 1 1 B 1 = Φ 1 λ 1 B 1 37 where R j i is the j th column of R i with j k and i 1, 2} Thus the first column of the matrix which is the solution of the first equation in 36 is indeed equal to the first column of the generalized controllability matrix R 1 Multiplying the second equation in 36 on the right with the first unit vector e 1 we obtain: A 2 R B 2 = λ 2 E 2 R 1 2 R 1 2 = λ 2 E 2 A 2 1 B 2 = Φ 2 λ 2 B 2 38 Thus the first column of the matrix which is the solution of the second equation in 36 is indeed equal to the first column of the generalized controllability matrix R 2 Multiplying first equation in 36 on the right with the j th unit vector e j, we obtain: A 1 R j 1 + K 2 R j 1 2 = λ 2j 1 E 1 R j 1 R j 1 = λ 2j 1 E 1 A 1 1 K 2 R j

14 Multiplying second equation in 36 on the right with the j th unit vector e j, we obtain: A 2 R j 2 + K 1 R j 1 1 = λ 2j E 2 R j 2 R j 2 = λ 2j E 2 A 2 1 K 1 R j From 39 and 40 we write the following linear recursive system of equations: R j 1 = Φ 1 λ 2j 1 K 2 R j 1 2 R j 2 = Φ 2 λ 2j K 1 R j 1 1 with initial conditions 37 and 38 Hence, by solving the coupled system of equations, we indeed conclude that R 1 and R 2 matrices satisfying 36 are the generalized controllability matrices defined in 18 for this particular choice of the right interpolation points This proof can be nevertheless adapted from the simplified case in to the more general case treated in Definition 43 Remark 44 The generalized Sylvester equations in 36 can be compactly written as only one generalized Sylvester equation in the following way D D A D R D + K R D SR + B D R D = E D R D Λ D 42 where X D X1 0 0 K2 0 = for X R, A, B, E, R, Λ} and K =, S DR 0 X 2 K 1 0 = SR S R 0 Proposition 41 The pair of generalized Sylvester equations in 36 has unique solutions if the right interpolation points are chosen so that the Sylvester operator T L R = I A D Λ D E D + SR K, is invertible, ie have no zero eigenvalues where denotes the Kronecker product Remark 45 The motivation behind this subsection is closely related to building parametrized reduced order models The idea is that, one can also use only one sided interpolation conditions either left as in 28 or right as in 29 to reduce the original LSS The other degrees of freedom are given by free parameters Further development of these Sylvester equation was studied in 3 in Section 44 for the case of bilinear systems Lemma 43 The generalized observability matrices O 1 and O 2 defined by 20 satisfy the following generalized Sylvester equations: O 1 A 1 + S L O 2 K 1 + LC 1 = M 1 O 1 E 1 O 2 A 2 + S L O 1 K 2 + LC 2 = M 2 O 2 E 2 43 Proof of Lemma 43 Similar to the proof of Lemma 42 Remark 46 The generalized Sylvester equations in 43 can be compactly written as only one generalized Sylvester equation in the following way where X D X1 0 = 0 X 2 D D D D D 41 O D A D + SL O D K + L D C D = M D O D E D 44 0 for X O, C, L, M, } and S DL = SL S L 0 Proposition 42 The pair of generalized Sylvester equations in 36 has unique solutions if the right interpolation points are chosen so that the Sylvester operator T T T L R = A D I E D M D + K S is invertible, ie have no zero eigenvalues where denotes the Kronecker product 14 D L D

15 42 The generalized Loewner pencil Definition 47 Given a linear switched system Σ as defined in 1, let R 1, R 2 } and O 1, O 2 } be the controllability and observability matrices defined in 18, 20 respectively, and associated with the multi-tuples in 11, 14 respectively The Loewner matrices L 1 and L 2 are defined as L 1 = O 1 E 1 R 1, L 2 = O 2 E 2 R 2 45 Additionally, the shifted Loewner matrices L s1 and L s2 are defined as L s1 = O 1 A 1 R 1, L s2 = O 2 A 2 R 2 46 Also define the quantities W 1 = C 1 R 1 W 2 = C 2 R 2, V 1 = O 1 B 1 V 2 = O 2 B 2 and Ξ 1 = O 2 K 1 R 1 Ξ 2 = O 1 K 2 R 2 47 Remark 47 In general, the Loewner matrices defined above need not have only real entries For instance, it may happen that the samples points are purely imaginary values on the jω axis In this case, we refer the readers to Section 431 in 3 We propose a similar method to enforce all system matrices have only real entries In short, the sampling points have to be chosen as complex conjugate pairs; after the data is arranged into matrix format, use projection matrices as in equation 426 in 3 to multiply the matrices in 45, 46 and 47 to the left and to the right In this way, the LSS does not change as pointed out in Definition 41 Remark 48 Note that L k and L sk where k 1, 2}, as defined above, are indeed Loewner matrices, that is, they can be expressed as divided differences of appropriate transfer function values of the underlying LSS see the following example Example 42 Given the LSS described by C j, E j, A j, B j D = 2 and j 1, 2}, consider the ordered tuples of left interpolation points: µ1, µ 2, µ 3 }, µ2, µ 1, µ 4 } and right interpolation points λ 1, λ 3, λ 2 }, λ2, λ 4, λ 1 } The associated generalized observability and controllability matrices are computed as follows C O 1 = 1 Φ 1 µ 1 C, O C 2 Φ 2 µ 2 K 1 Φ 1 µ 3 2 = 2 Φ 2 µ 2 C 1 Φ 3 µ 1 K 2 Φ 2 µ 4 R 1 = Φ 1 λ 1 B 1 Φ 1 λ 3 K 2 Φ 2 λ 2 B 2, R2 = Φ 2 λ 2 B 2 Φ 2 λ 4 K 1 Φ 1 λ 1 B 1 The projected Loewner matrices can be written in terms of the samples in the following way: L 1 = L 2 = H 1 µ 1 H 1 λ 1 µ 1 λ 1 H 1,2 µ 1,λ 2 H 1,2 λ 3,λ 2 µ 1 λ 3 H 2,1 µ 2,µ 3 H 2,1 µ 2,λ 1 µ 3 λ 1 H 2,1,2 µ 2,µ 3,λ 2 H 2,1,2 µ 2,λ 3,λ 2 µ 3 λ 3 H 2 µ 2 H 2 λ 2 µ 2 λ 2 H 2,1 µ 2,λ 1 H 2,1 λ 4,λ 1 µ 2 λ 4 H 1,2 µ 1,µ 4 H 1,2 µ 1,λ 2 µ 4 λ 2 H 1,2,1 µ 1,µ 4,λ 4 H 1,2,1 µ 1,λ 4,λ 1 µ 4 λ 4 = O 1 E 1 R 1, = O 2 E 2 R 2 15

16 The projected shifted Loewner matrices can also be written in terms of the samples as: L s1 = L s2 = µ 1 H 1 µ 1 λ 1 H 1 λ 1 µ 1 λ 1 µ 1 H 1,2 µ 1,λ 2 λ 3 H 1,2 λ 3,λ 2 µ 1 λ 3 µ 3 H 2,1 µ 2,µ 3 λ 1 H 2,1 µ 2,λ 1 µ 3 λ 1 µ 3 H 2,1,2 µ 2,µ 3,λ 2 λ 3 H 2,1,2 µ 2,λ 3,λ 2 µ 3 λ 3 µ 2 H 2 µ 2 λ 2 H 2 λ 2 µ 2 λ 2 µ 2 H 2,1 µ 2,λ 1 λ 4 H 2,1 λ 4,λ 1 µ 2 λ 4 µ 4 H 1,2 µ 1,µ 4 λ 2 H 1,2 µ 1,λ 2 µ 4 λ 2 µ 4 H 1,2,1 µ 1,µ 4,λ 4 λ 4 H 1,2,1 µ 1,λ 4,λ 1 µ 4 λ 4 = O 1 A 1 R 1, = O 2 A 2 R 2, The same property applies for the V i and W j vectors and Ξ j matrices: H V 1 = 1 µ 1 H = O H 2,1 µ 2, µ 3 1 B 1, V 2 = 2 µ 2 = O H 1,2 µ 1, µ 4 2 B 2, W 1 = H 1 λ 1 H 1,2 λ 3, λ 2 = C 1 R 1, W 2 = H 2 λ 2 H 2,1 λ 4, λ 1 = C 2 R 2, H Ξ 1 = 2,1 µ 2, λ 1 H 2,1,2 µ 2, λ 3, λ 2 = O H 1,2,1 µ 1, µ 4, λ 1 H 1,2,1,2 µ 1, µ 4, λ 3, λ 2 2 K 1 R 1, H Ξ 2 = 1,2 µ 1, λ 2 H 1,2,1 µ 1, λ 4, λ 1 = O H 2,1,2 µ 2, µ 3, λ 2 H 2,1,2,1 µ 2, µ 3, λ 4, λ 1 1 K 2 R 2 It readily follows that, given the original system Σ, a reduced LSS of order two can be obtained without computation matrix factorizations or solves as: Ê k = OER, Â = OAR, ˆN = ONR, ˆB = OB, Ĉ = CR This reduced system matches sixteen moments of the original system, namely: four of H 1 /H 2 : H 1 µ 1, H 2 µ 2, H 1 λ 1, H 2 λ 2, three of H 1,2 : H 1,2 µ 1, µ 4, H 1,2 µ 1, λ 2, H 1,2 λ 3, λ 2, three of H 2,1 : H 2,1 µ 2, µ 3, H 2,1 µ 2, λ 1, H 2,1 λ 4, λ 1, one of H 1,2,1,2 : H 1,2,1,2 µ 1, µ 4, λ 3, λ 2 one of H 2,1,2,1 : H 2,1,2,1 µ 2, µ 3, λ 4, λ 1 48 ie in total 22k + k 2 = 16 moments are matched using this procedure 421 Properties of the Loewner pencil We will now show that the quantities defined earlier satisfy various equations which generalize the ones in the linear or bilinear case The equations that are be presented in this section are used to automatically find the Loewner and shifted Loewner matrices by means of solving Sylvester equations instead of building the divided difference matrices from the computed samples at the sampling points 16

17 Proposition 43 The Loewner matrix L 1 and the shifted Loewner matrix L s1 corresponding to mode 1 satisfy the following relations where L, R, Λ k, M k, S L, S R are given in 32,33 and34: L s1 = L 1 Λ 1 + V 1 R + Ξ 2 S R 49 L s1 = M 1 L 1 + LW 1 + S L Ξ 1 50 The Loewner matrix L 2 and the shifted Loewner matrix L s2 corresponding to mode 2 satisfy the following relations: L s2 = L 2 Λ 2 + V 2 R + Ξ 1 S R 51 L s2 = M 2 L 2 + LW 2 + S L Ξ 2 52 Proof of Proposition 43 By multiplying the first equation in 36 with O 1 to the left we obtain: O 1 A 1 R 1 + O 1 K 2 R 2 S R + O 1 B 1 R = O 1 E 1 R 1 Λ 1 L s1 + Ξ 2 S R + V 1 R = L 1 Λ 1 and hence relation 49 is proven Similarly we prove 51 By multiplying the first equation in 43 with R 1 to the right we obtain: O 1 A 1 R 1 + S L O 2 K 1 R 1 + LC 1 R 1 = M 1 O 1 E 1 R 1 L s1 + S L Ξ 1 + LW 1 = M 1 L 1 and hence relation 50 is proven Similarly we prove 52 Proposition 44 The Loewner matrices L 1 and L 2 satisfy the following Sylvester equations: M 1 L 1 L 1 Λ 1 = V 1 R LW 1 + Ξ 2 S R S L Ξ 1, 53 M 2 L 2 L 2 Λ 2 = V 2 R LW 2 + Ξ 1 S R S L Ξ 2 54 Proof of Proposition 44 By subtracting equation 49 from 50 we directly obtain 53 and also, by subtracting equation 51 from 52 we directly obtain 54 Proposition 45 The shifted Loewner matrices L s1 and L s2 satisfy the following Sylvester equations: M 1 L s1 L s1 Λ 1 = M 1 V 1 R LW 1 Λ 1 + M 1 Ξ 2 S R S L Ξ 1 Λ 1, 55 M 2 L s2 L s2 Λ 2 = M 2 V 2 R LW 2 Λ 2 + M 2 Ξ 1 S R S L Ξ 2 Λ 2 56 Proof of Proposition 43 By subtracting equation 49 after being multiplied with M 1 to the left from equation 50 after being multiplied with Λ 1 to the right, we directly obtain 55 Similar procedure is applied to prove 56 Remark 49 The right hand side of the equations contains constant 0/1 matrices ie R, L, S R, S L as well as matrices ie V j, W j, Ξ j, j 1, 2} which are directly constructed by putting together the given samples values as pointed out in Example 42 17

18 43 Construction of reduced order models As we already noted, the interpolation data for the LSS case is significantly different than the one used for the linear case without switching, as higher order transfer function values are matched as shown in the previous sections However, the rest of the procedure remains more or less unchanged Lemma 44 Assume that k = l and that none of the interpolation points λ i, µ j are eigenvalues of the pencils L s1, L 1 and L s2, L 2 Moreover, consider the Loewner matrices L 1 and L 2 to be invertible Then, a realization of a reduced order LSS ˆΣ that matches the data of the original LSS Σ as introduced in Definition 46 is given by the following matrices, Ê1 = L 1, Â 1 = L s1, ˆB1 = V 1, Ĉ 1 = W 1, Ê 2 = L 2, Â 2 = L s2, ˆB2 = V 2, Ĉ 2 = W 2 and ˆK1 = Ξ 1, ˆK2 = Ξ 2 If k = n, then the proposed realization is equivalent to the original one as in Definition 41 Proof of Lemma 44 This result directly follows from Lemma 41 by taking into consideration the notations introduced in In the case of redundant data, at least one of the pencils L sj, L j is singular for j 1, 2}, and hence construct pairs of projectors X j, Y j corresponding to mode j similar to 10 The MOR procedure for approximate data matching is presented as follows Procedure 1 Consider the rank revealing singular value factorization of the Loewner matrices, L j = S j O T Y j Ỹ j O S Xj Xj = Yj S j X T j + Ỹj S j XT j, 57 j where Y j, X j R k r j and S j R r j r j The projected system matrices corresponding to subsystem ˆΣ j are computed as, Ê j = Y T j L j X j, Â j = Y T j L sj X j, ˆBj = Y T j V j, Ĉ j = W j X j, for j 1, 2} Moreover, the projected coupling matrices are computed in the following way ˆK 1 = Y T 2 Ξ 1 X 1, ˆK2 = Y T 1 Ξ 2 X 2 By choosing r j as the numerical rank of the Loewner matrix L j ie the largest neglected singular value corresponding to index r j +1 is less than machine precision ɛ, ensure that the Êj matrices are not singular Hence, construct a reduced order LSS denoted with ˆΣ, that approximately matches the data of the original LSS Σ If the truncated singular values are all 0 the ones on the main diagonal of the matrices S j, then the matching is exact We provide a qualitative rather than quantitative result for the projected Loewner case The quality of approximation is directly linked to the singular values of the Loewner pencils which represent an indicator of the desired accuracy For linear systems with no switching, an error bound is provided in 5 as a quantitative measure The dimensions of the subsystems ˆΣ 1 and ˆΣ 2, corresponding to the reduced order LSS, need not be the same ie r 1 r 2 In this case the coupling matrices are not square anymore The projectors are computed via singular value factorization of the Loewner matrices The use of the Drazin or Moore-Penrose pseudo inverses also holds as shown in 2 18

19 5 The Loewner framework for linear switched systems - the general case In this section we are mainly concerned with generalizing some of the results presented in Section 4 Most of the findings can be smoothly extended to the cases with more complex switching patterns more modes The key for this is enforcing a cyclic structure of the interpolation framework, so that, everything can be written in matrix equation format Definition 51 Let Γ and Θ be finite sets of tuples so that Γ, Θ prefix closure property, ie q 1, q 2,, q i, λ 1,, λ i Γ q 2,, q i, λ 2,, λ i Γ i 2 and Θ has the suffix closure property, ie Q k C k so that Γ has the q 1, q 2,, q j, µ 1,, µ j Θ q 1,, q j 1, µ 1,, µ j 1 Θ j 2 Now consider the specific subset Γ q for any q Q of the set Γ in the following way: Γ q = q 1, q 2,, q i, λ 1,, λ i Γ q 1 = q, i δ Γ }, δ Γ = max w /2 w Γ Denote the cardinality of Γ q with k q = cardγ q and explicitly enumerate the elements of this set as follows: Γ q = w q 1, w q 2,, w q kq } Consider the following function mapping r : Γ q C nq 1 that maps a word form Γ q into a column vector of size n q : rq, q 2,, q i, λ 1,, λ i = Φ q λ 1 K q2,qφ q2 λ 2 K qi,q i 1 Φ qi λ i B qi Now we are ready to construct the reachability matrix R q corresponding to the mode q of the system Σ as follows: R q = rw q 1 rw q 2 rw q kq C nq kq 58 Similarly, define the specific subset Θ q for any q Q of the set Θ in the following way: Θ q = q 1, q 2,, q j, µ 1,, µ j Γ q j = q, j δ Θ }, k=1 δ Θ = max w /2 w Θ Consider the cardinality of Θ q to be the same as the one of Γ q, ie k q = cardθ q Although this additional constraint is not necessarily needed, we would like to enforce the construction of reduced systems with square matrices A k and E k Next we explicitly enumerate the elements of this set as follows: Θ q = v q 1, v q 2,, v q kq } Consider the following mapping o : Θ q C 1 nq that maps a word form Θ q into a row vector of size n q : oq 1, q 2,, q j 1, q, µ 1,, µ j = C q1 Φ q1 µ 1 K q2,q 1 Φ q2 µ 2 K q,qj 1 Φ q µ j Now we are ready to construct the observability matrix O q C kq nq corresponding to the mode q of the system Σ as follows O q = ov q 1 T ov q 2 T ov q kq T T C k q n q 59 Consider the following example to show how the general procedure is extended from the linear case no switching to the case when switching occurs 19

20 Example 51 Take D = 3 3 active modes and hence Q = 1, 2, 3} The following interpolation points are given: s 1, s 2,, s 18 } C The first step is to partition this set into two disjoint subsets each having 9 points: left interpolation points : µ 1, µ 2,, µ 9 } right interpolation points : λ 1, λ 2,, λ 9 } The set Γ is composed of three subsets Γ = Γ 1 Γ2 Γ3 which are defined in a cyclic way by imposing the previously defined suffix closure property, as follows Γ 1 = 1, λ 1, 1, 3, λ 4, λ 3, 1, 3, 2, λ 7, λ 6, λ 2 } Γ 2 = 2, λ 2, 2, 1, λ 5, λ 1, 2, 1, 3, λ 8, λ 4, λ 3 } Γ 3 = 3, λ 3, 3, 2, λ 6, λ 2, 3, 2, 1, λ 9, λ 5, λ 1 } To the sets Γ j, we attach the reachability matrices R j defined as follows: R 1 = Φ 1 λ 1 Φ 1 λ 4 K 3,1 Φ 3 λ 3 B 3 Φ 1 λ 7 K 3,1 Φ 3 λ 6 K 2,3 Φ 2 λ 2 B 2, R 2 = Φ 2 λ 2 Φ 2 λ 5 K 1,2 Φ 1 λ 1 B 1 Φ 2 λ 8 K 1,2 Φ 1 λ 4 K 3,1 Φ 3 λ 3 B 3, R 3 = Φ 3 λ 3 Φ 3 λ 6 K 2,3 Φ 2 λ 2 B 2 Φ 3 λ 9 K 2,3 Φ 2 λ 5 K 1,2 Φ 1 λ 1 B 1 In the same manner, the set Θ is composed of three subsets Θ = Θ 1 Θ2 Θ3 which are again defined in a cyclic way by imposing the previously defined prefix closure property, as follows Θ 1 = 1, µ 1, 3, 1, µ 3, µ 4, 1, 2, 1, µ 1, µ 5, µ 7 } Θ 2 = 2, µ 2, 1, 2, µ 1, µ 5, 2, 3, 2, µ 2, µ 6, µ 8 } Θ 3 = 3, µ 3, 2, 3, µ 2, µ 6, 3, 1, 3, µ 3, µ 4, µ 9 } To the sets Θ i, we attach the observability matrices O i defined as follows: O 1 = C 1 Φ 1 µ 1 C 3 Φ 3 µ 3 K 1,3 Φ 1 µ 4 C 1 Φ 1 µ 1 K 2,1 Φ 2 µ 5 K 1,2 Φ 1 µ 7, O 2 = C 2 Φ 2 µ 2 C 1 Φ 1 µ 1 K 2,1 Φ 2 µ 5 C 2 Φ 2 µ 2 K 3,2 Φ 3 µ 6 K 2,3 Φ 2 µ 8 20

21 O 3 = C 3 Φ 3 µ 3 C 2 Φ 2 µ 2 K 3,2 Φ 3 µ 6 C 3 Φ 3 µ 3 K 1,3 Φ 1 µ 4 K 3,1 Φ 3 µ 9 51 Sylvester equations for R q and O q In this section we would like to generalize the results presented in Lemma 42 and Lemma 43, and hence extend the framework to a general number of operational modes denoted with D Definition 52 Introduce the special concatenation of tuples composed of mixed elements symbols that are from two different sets for example Q and C as the mapping with the following property: α1 β 1 α2 β 2 = α1 α 2 β1 β 2, where α k Q i k and βk C j k for k = 1, 2 Definition 53 For g, i = 1,, D, let S g i = S g i 1 S g i k g R k i k g be constant matrices that contain only 0/1 entries constructed so that S g i 1 = 0 ki and for u = 2,, k g, we write: S g e u 1,ki, if i u = λ C, st w g u = g, λ w u 1 i, 60 0 ki, else Also, introduce the matrices R i and Λ i that are defined similarly as in 32 and 35, ie, R i = e T 1,m 1 e T 1,m R 1 k i, Λ k i = blkdiag Λ 1 i, Λ 2 i,, Λ k i R k i k i, 61 where the diagonal matrices Λ a i, a = 1,, k contain the right interpolation points associated to mode i For general cyclic structure incorporated of the set Γ, it follows that the reachability matrices R i R n i k i, 1 i D satisfy the following system of generalized Sylvester equations: A 1 R 1 + A 2 R 2 + A D R D + D i=1 D i=1 D i=1 K i,1 R i S 1 i + B 1 R 1 = E 1 R 1 Λ 1 K i,2 R i S 2 i + B 2 R 2 = E 2 R 2 Λ 2 K i,d R i S D i + B D R D = E D R D Λ D 62 Note that S i i = 0 ki,k i, and if k 1 = k 2 = = k D = k, the above defined matrices S g i following equality g Q : D i=1 S g i = blkdiag J m1,, J mk satisfy the 63 21

22 where J l is the Jordan block of size l defined in 33 To directly find R g, g = 1, 2, 3 for the case presented in Example 51, we have to solve the following system of coupled generalized Sylvester equations A 1 R 1 + K 3,1 R 3 S B 1 R = E 1 R 1 Λ 1 A 2 R 2 + K 1,2 R 1 S B 2 R = E 2 R 2 Λ 2 A 3 R 3 + K 2,3 R 2 S B 3 R = E 3 R 3 Λ 3 where: Λ 1 = λ λ λ 7, Λ 2 = λ λ λ 8 R = 1 0 0, S 1 3 = S 2 1 = S 3 2 =, Λ 3 = λ λ λ 9 This corresponds to the case k 1 = k 2 = k 3 = 3, k = 1 and m 1 = 3 T Definition 54 For h, j = 1,, D, let T h h T h T T j = j 1 T j kh R l h l j be constant matrices that contain only 0/1 entries constructed so that T h T j 1 = 0lj and for v = 2,, k g, we write:,, h T e v 1,kj, if µ C, st w v T j v = h = w v 1 j h, µ, 0 lj, else 64 Also, introduce the following matrices L j T = e T 1,p 1 e T 1,p R 1 l j, M l j = blkdiag M 1 j, M 2 j,, M l j R l j l j, 65 where the diagonal matrices M v j for v = 1,, l j contain the left interpolation points associated to mode j For general cyclic structure incorporated by definition in the set Θ, one can conclude that the observability matrices O j R l j n j, 1 j D satisfy the following system of generalized Sylvester equations: O 1 A 1 + O 2 A 2 + O D A D + D j=1 D j=1 D j=1 T 1 j O j K 1,j + L 1 C 1 = M 1 O 1 E 1 T 2 j O j K 2,j + L 2 C 2 = M 2 O 2 E 2 T D j O j K D,j + L D C D = M D O D E D Note that T j j = 0 lj,l j, and if l 1 = l 2 = = l D = l, the square matrices T h j following equality, h Q: D j=1 T h j = blkdiag R l l satisfy the J p1,, J pl T 67

23 Again to find the matrices O h, h = 1, 2, 3 in Example 51, it is required to solve the following system of coupled generalized Sylvester equations O 1 A 1 + T 1 3 O 3 K 1,3 + T 1 2 O 2 K 1,2 + LC 1 = M 1 O 1 E 1 O 2 A 2 + T 2 1 O 1 K 2,1 + T 2 3 O 3 K 2,3 + LC 2 = M 2 O 2 E 2 O 3 A 3 + T 3 2 O 2 K 3,2 + T 3 1 O 1 K 3,1 + LC 3 = M 3 O 3 E 3 where: M 1 = µ µ µ 7 T 1 3 = T 2 1 = T 3 2 =, M 2 = µ µ µ 8, M 3 =, T 1 2 = T 2 3 = T 3 1 = µ µ µ ,, L = e 1 This corresponds to the case l 1 = l 2 = l 3 = 3, l = 1 and p 1 = 3 Note that the relation in 67 hold, ie, T T 1 3 = T T 2 3 = T T 3 2 = J T 3 52 The Loewner matrices For the case of linear switched systems with D active modes, the generalization of the Loewner framework includes one important feature Instead of only one pair of Loewner matrices as in the linear case without switching which is covered in Section 3, we define a pair of Loewner matrices for each individual active mode; hence in total D pairs of Loewner matrices Definition 55 Given a linear switched system Σ, let R i i Q} and O j j Q} be the controllability and observability matrices associated with the multi-tuples Γ i and Θ j The Loewner matrices L i i Q} are defined as L 1 = O 1 E 1 R 1, L 2 = O 2 E 2 R 2,, L D = O D E D R D 68 Additionally, the shifted Loewner matrices L si i Q} are defined as L s1 = O 1 A 1 R 1, L s2 = O 2 A 2 R 2,, L sd = O D A D R D 69 Also introduce the matrices i, j Q W i = C i R i, V j = O j B j, and Ξ i,j = O j K i,j R i Remark 51 The number of Loewner matrices, shifted Loewner matrices, W i row vectors and V j column vectors is the same as the number of active modes ie D On the other hand, the number of matrices Ξ i,j increases quadratically with D ie in total D 2 matrices Remark 52 Note that the matrices L i and L si as defined in 68 and 69 for i 1, 2,, D} are indeed Loewner matrices, that is, they can be expressed as divided differences of generalized transfer function values of the underlying LSS Proposition 51 The Loewner matrices L h satisfy the following Sylvester equations: M h L h L h Λ h = V h R LW h + 23 D j=1 Ξ j,h S h j T h j Ξ h,j, h Q 70

24 Proposition 52 The shifted Loewner matrices L sh satisfy the following Sylvester equations: M h L sh L sh Λ h = M h V h R LW h Λ h + D j=1 M h Ξ j,h S h j T h j Ξ h,j Λ h, h Q 71 Remark 53 The proof of the results stated in is performed in a similar manner as for the results obtained for the special case D = 2 in Section 4 ie for Construction of reduced order models The general procedure for the case with D switching modes is more or less similar to the one covered in Section 43 where D = 2 Lemma 51 Let L j be invertible matrices for j = 1,, D, such that none of the interpolation points λ i, µ k are eigenvalues of any of the Loewner pencils L sj, L j Then, it follows that the matrices Êj = L j, Â 1 = L sj, ˆBj = V j, Ĉ j = W j, ˆKi,j = Ξ i,j }, i, j 1,, D} form a realization of a reduced order LSS ˆΣ that matches the data of the original LSS Σ If k j = n j for j = 1,, D, the proposed realization is equivalent to the original one The concept of a LSS matching the data of another LSS in the case D > 2 is formulated in a similar manner as to the case D = 2 ie, which is covered in Definition 46 Also, the definition of equivalent LSS for the case D > 2 is formulated similarly as to Definition 41 In the case of redundant data, at least one of the pencils L sj, L j is singular for j 1,, D} The main procedure is presented as follows Procedure 2 Consider the rank revealing singular value factorization of the Loewner matrices: L j = S j O T Y j Ỹ j O S Xj Xj = Yj Σ j X T j + Ỹj S j XT j 72 j where X j, Y j R k j r j, S j R r j r j and j = 1,, D} Here, choose r j as the numerical rank of the Loewner matrix L j ie the largest neglected singular value corresponding to index r j + 1 is less than machine precision ɛ The projected system matrices computed as Ê j = Y T j L j X j, Â j = Y T j L sj X j, ˆBj = Y T j V j, Ĉ j = W j X j, for j 1,, D} and the projected coupling matrices computed in the following way ˆK i,j = Y T j Ξ i,j X i, i, j 1,, D}, form a realization of a reduced order LSS denoted with ˆΣ that approximately matches the data of the original LSS Σ Each reduced subsystem ˆΣ j has dimension r j, j 1,, D} Remark 54 If the truncated singular values are all 0 the ones on the main diagonal of the matrices S j, then the interpolation is exact 24

25 6 Numerical experiments In this section we illustrate the new method by means of three numerical examples We use a certain generalization of the balanced truncation BT method for LSS as presented in 27 to compare the performance of our new introduced method The main ingredient of the BT method is to compute the the controllability and observability gramians P i and Q i where i 1, 2,, D} as the solutions of the following Lyapunov equations: 61 Balanced Truncation A i P i E T i + E i P i A T i + B i B T i = 0 A T i Q i E i + E T i P i A i + C T i C i = 0 In 27 it has been shown that, if certain conditions are satisfied, the technique of simultaneous balanced truncation can be applied to switched linear systems Hence, in some special cases, the existence of a global transformation matrix V bal is guaranteed; it follows that: V bal PVbal T = V T 1 bal QVbal = U i 75 where U i are diagonal matrices Although conceptually attractive as a MOR method, in general the conditions are rather restrictive in practice This motivates the search for a more general MOR approach for the case where simultaneous balancing cannot be achieved The problem of finding a balancing transformation for a single linear system can be formulated as finding a nonsingular matrix such that the following cost function is minimized see 1: fv = tracevpv T + V T QV 1 76 For the class of LSS with distinct operational modes, we hence have to minimize not one but a number of D cost functions: f i V = tracevp i V T + V T Q i V 1, i 1, 2,, D} 77 If the conditions of Corollary IV3 from 27 hold, simultaneous balancing is possible, and there exists a transformation V which simultaneously minimizes f i for all i = 1, 2,, D Instead of having D functions as in 77, one can introduce a single overall cost function ie the average of the cost functions of the individual modes Define the function f av as in 27: f av V = 1 D tracevp i V T + V T Q i V 1 = tracevp av V T + V T Q av V 1 78 D where i=1 P av = 1 D D P i, i=1 Q av = 1 D D Q i, 79 In the case of LSS, the BT method computes a basis where the sum of the sum of the eigenvalues of P i and Q i over all modes is minimal Hence, minimizing the proposed overall cost function provides a natural extension of classical BT to the case of LSS It follows that the transformation Ṽ that minimizes the cost function in 78 is precisely the one which balances the pair P av, Q av of average gramians By applying Ṽ to the individual modes and truncating, a reduced order model is obtained After applying the transformation Ṽ, the new state space representations of the individual modes need not be balanced Nevertheless, as stated in 27, it is expected to be relatively close to being balanced 25 i=1

26 62 First example As first example we consider the simple model of an evaporator vessel from 28 There is a constant inflow of liquid fin into a tank and an outflow that depends on the pressure in the tank and the Bernoulli resistance R b To keep the level of fluid in the evaporator vessel at or below a pre-specified maximum, an overflow mechanism is activated when the level of fluid L in the evaporator exceeds the threshold value L th This causes a flow through a narrow pipe with resistance R p and inertia I that builds up flow momentum p The system is modeled in two distinct operation modes: mode 1, where there is no overflow the fluid level is below the overflow level, and mode 2, where the overflow mechanism is active The ordinary differential equations describing the system in the two operation modes are given by I 0 ṗ Rp 0 ṗ 0 = + mode 1 0 C I 0 0 C L ṗ L 0 1/R b Rp 1 = 1 1/R b L ṗ L f in 0 + f in mode 2 Supposing the system is initially in mode 1, the inflow causes the tank to start filling, which causes an outflow through resistance Rb In this mode the outflow through the narrow pipe is zero If L exceeds the level L th, a switch from mode 1 to mode 2 occurs at the point in time when L = L th Figure 1: Schematic of the evaporator vessel In the following, use the parameters R b = 1, R p = 05, I = 05, C = 15, f in = 025, L th = 008 and compute the following system matrices: Mode 1 : A 1 = 0 1, B 1 =, C = Mode 2 : A 2 = 1 1, B 2 =, C 1 2 = First consider the following tuples of left and right interpolation nodes for each mode: λ 1 = 15, 2, 1} λ 2 = 1, 15, 15}, µ 1 = 2, 0, 05} µ 2 = 0, 2, 05} Hence, following the procedure described in Section 4, we recover the following system matrices: Mode 1 : Ê 1 = 1 1, Â 1 = 1 1, ˆB1 = 5, Ĉ 1 1 =

27 Mode 2 : Ê 2 = , Â 2 = , ˆB2 = , Ĉ 2 = Note that the recovered realization is equivalent to the original one no reduction has been enforced - the task was to recover the initial system The coupling matrices are also computed: ˆK = 13 17, ˆK2 = Second example For the next experiment, consider the CD player system from the SLICOT benchmark examples for MOR see 12 This linear system of order 120 has two inputs and two outputs We consider that, at any given instance of time, only one input and one output are active the others are not functional due to mechanical failure For instance, consider mode j to be activated whenever the j th input and the j th output are simultaneously failing where j 1, 2} In this way, we construct a LSS system with two operational modes Both subsystems are stable SISO linear systems of order 120 This initial linear switched system Σ will be reduced by means of the Loewner framework to obtain ˆΣ L and balanced truncation method proposed in 27 to obtain ˆΣ B The frequency response of each original subsystem is depicted in Fig 2 Frequency response of the original LSS 10 5 I st subsystem II nd subsystem Frequencyω Figure 2: Frequency response of the original subsystems For the Loewner method, we choose 160 logarithmically distributed interpolation points inside 10 1, 10 5 j Fig 3 shows the decay of the singular values of the Loewner matrices corresponding to both subsystems We notice that the 70 th singular values attain machine precision For ˆΣ L we decide to truncate at order k = 28 for both reduced systems The same truncation order is chosen for ˆΣ B Next we compare the quality of approximation of the frequency response corresponding to the original system with the responses of the two reduced systems In Fig 4 the relative error in frequency domain is depicted for both MOR methods Loewner and BT Notice that the Loewner method produces better results especially in the range of the selected sampling points Also, compare the time domain response of the original linear switched system against the ones corresponding to the two reduced models We use a sinusoidal signal as the control input The switching times are randomly chosen within 0,10s The blue rectangular signal in Fig 5 represents the switching signal Notice that the output of the LSS is well approximated for both MOR methods, as it can be seen in the lower part of Fig 5 27

28 10 0 Singular value decay of the Loewner matrix I st subsystem II nd subsystem Figure 3: Decay of the singular values of the Loewner matrices Error in frequency domain - mode 1 Loewner BT Frequencyω Error in frequency domain - mode 2 Loewner BT Frequencyω Figure 4: Frequency domain approximation error Finally, by inspecting the time domain error between the original response and the responses coming from the two reduced models depicted in Fig 6, we notice that the Loewner method generally produces better results The error curve corresponding to our proposed method is two orders of magnitude below the error curve corresponding to the BT method for most of the points on the time axis 64 Third example For the last experiment, consider a large scale LSS system constructed as in 21 from the original machine stand example given in 14 In this example, the system variability is induced by a moving tool slide on the guide rails of the stand see Fig 7 The aim is to determine the thermally driven displacement of the machine stand structure Following the model setting in 14, consider the heat equation with Robin boundary conditions Using a finite element FE discretization and denoting the external influences as the system input z, we obtain the dynamical heat model E th ẋt = A th txt + B th tzt 80 describing the deformation independent evolution of the temperature field x with the system matrices E th, A th t and B th t The variability of the model is described by time dependent matrices A th t and B th t This leads directly to the linear time varying system described by 80 Since model reduction for LTV systems is a highly storage consuming procedure, the authors 28

29 Switching signal σt Original LSS 2 x Time domain simulation 104 Reduced LSS - Loewner Reduced LSS - BT Times sec Figure 5: Time domain simulation Time domain simulation error Loewner BT Times sec Figure 6: Time domain approximation error in 21 exploit properties of the spatially semi-discretized model to set up a LSS consisting of LTI subsystems only As described in 14, the guide rails of the machine stand are modeled as 15 equally distributed horizontal segments see Fig 7 Any of these segments is said to be completely covered by the tool slide if its midpoint lies within the height of the slide On the other hand, each segment whose midpoint is not covered is treated as not in contact and therefore the slide always covers exactly 5 segments at each time This in fact allows the stand to reach 11 distinct, discrete positions given by the model restrictions In this way, one can define the subsystems of the LSS as follows: E th ẋ = A l th Σ l : x + Bl th zl y = Cx, 81 where l 1,, 11} Note that the change of the input operator Btht is hidden in the input z itself, since it is sufficient to activate the correct boundary parts by choosing the corresponding columns in Bth via the input z l Therefore, the input operator B th t := B th becomes constant and the input variability is represented by the input z l : zi l z i, segment i is in contact, :=, i = 1,, , otherwise, Here, z i R is the thermal input as described in 21 The only varying part influencing the model reduction process left in the dynamical system is the system ma trix A th t := A l th 29

30 Figure 7: Schematic of the tool slide on the guide rails of the stand After the finite element discretization was performed, we are given a SLS with 11 modes Each subsystem has dimension n = The E and C matrices are the same for all modes of the SLS The B matrices have 6 columns corresponding to different inputs and the C matrix has 9 rows corresponding to different outputs For all the experiments performed, we take into consideration only two active modes the first and the fifth This corresponds to the particular case of D = 2 covered in Section 4 Also, although our new introduced method can be easily generalized to the MIMO case, we consider only for simplicity reasons the first input and the first output for each of the two modes the measurements used in the Loewner framework are hence scalar values Both subsystems are stable linear systems of order in sparse format This initial large scale LSS will be again reduced as in the second example by means of the Loewner method and of the balanced truncation method proposed in 27 The frequency response of each original subsystem is depicted in Fig 8 Frequency response of the LSS modes 1 and 5 I st subsystem II nd subsystem Frequencyω Figure 8: Frequency response of the original subsystems For the Loewner method, we choose 200 logarithmically spaced interpolation points inside 10 5, 10 3 j The decay of the singular values of the Loewner matrices corresponding to both subsystems can be viewed in Fig 9 We notice that already the 70 th singular values attain machine precision For the Loewner reduced order LSS ie Σ 1, we decide to truncate at order k 1 = 26 for both subsystems which are stable LTI s The same truncation order is chosen for the reduced order model computed via BT Next we compare the quality of approximation of the frequency response 30