OPTIMAL H 2 MODEL REDUCTION IN STATE-SPACE: A CASE STUDY

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OPTIMAL H 2 MODEL REDUCTION IN STATE-SPACE: A CASE STUDY RLM Peeters, B Hanzon, D Jibetean Department of Mathematics, Universiteit Maastricht, PO Box 66, 6200 MD Maastricht, The Netherlands, E-mail: ralfpeeters@mathunimaasnl CWI, Kruislaan 43, PO Box 94079, 090 GB Amsterdam, The Netherlands, E-mail: hanzon@cwinl, djibetean@cwinl Keywords: Model reduction, H 2 -approximation, global optimization, rational optimization Abstract A state-space approach to H 2 model reduction of stable systems is pursued, using techniques from global rational optimization, and aimed at finding global optima First, a concentrated version of the H 2 criterion is derived, by optimizing analytically with respect to a subset of parameters Next, a bound is developed, which is shown to be sharp for reduction to models of order The choice of parameterization is such as to keep all these criteria rational To illustrate the approach, a nontrivial example of a multivariable stable system of order 4 was designed, together with an approximation of order 2 Detailed analysis shows the approximation to be the global optimum among all stable system of order 2, constituting the first such system described in the literature The fact that it also yields a global optimum to the bound, which happens to be sharp at the optimum, leads to a couple of questions for further research Introduction One of the important problems in the field of modelling and systems identification is the model reduction problem It is not uncommon in practical applications to arrive at models of high complexity, either directly as a result of the identification procedure when using a high order (black-box model on available input-output data, or indirectly as a result of the interconnection of a number of subsystems, each of which is modelled individually Models of lower complexity, however, usually lend themselves much better for model analysis and for the application of popular techniques for control system design Thus, there is a clear practical interest in the problem of finding good lower order approximate models to a given model of high order One of the important aspects to deal with this problem lies in the choice of a criterion to measure the difference between two models Various choices for such criteria are described in the literature In this paper we focus attention on one of the most well-known criteria for stable systems, the H 2 criterion, which allows for a straightforward interpretation in system theoretic terms However, one of the unattractive characteristics of the H 2 criterion is that it may well possess a number of local, nonglobal optima This makes it a hazardous task to apply numerical techniques only to determine an H 2 -optimal reduced-order model to a given system, such as the usual iterative local search techniques for multivariate function minimization (like steepest descent, Newton s method, the Gauss-Newton method, etc Depending on the choice of parameterization, the H 2 model reduction problem can be formulated as a rational optimization problem This opens up the possibility to apply techniques from the field of global rational optimization to determine a global optimum to this problem Such techniques often consist of a mix of methods from (constructive algebra and numerical analysis, and they may put high demands on the available computer hardware and software The current state-of-the-art of these methods is such that only examples of a rather limited complexity can be handled satisfactorily in all detail However, despite such limitations, these methods are still quite valuable eg in test situations where exact results are required or when theoretic results and conjectures are being analyzed In this paper we pursue a state-space approach along such lines First, the H 2 model reduction problem is formulated and a concentrated version of it is derived, by optimizing analytically with respect to a number of parameters which happen to enter the problem in a linear least squares manner Next, a bound on this concentrated H 2 criterion is derived, which is shown to be sharp in the case of approximation by a model of order To illustrate the approach we have designed an example of a stable system of order 4 with two inputs and two outputs, which we intend to approximate by a stable system of order 2 For this particular example, which leads to a rational optimization problem, the available software was unable to handle the concentrated H 2 criterion due to its complexity, but it did return a global optimum for the much simpler bound Subsequent analysis shows that this global optimum for the bound also happens to constitute a global optimum for the concentrated H 2 criterion This leads to a number of new research questions by which we conclude the paper 2 The discrete time H 2 model reduction problem We consider the space of linear time-invariant systems in discrete time which are stable and have strictly proper rational transfer functions We deal with the real multivariable situation, where the systems have m inputs and p outputs

It is supposed that a system Σ of McMillan degree n is given, stemming from this class The H 2 model reduction problem consists of finding an approximation ˆΣ which is of a prescribed order k < n and of which the H 2 -distance between Σ and ˆΣ is as small as possible Here, the H 2 criterion is given by (cf, eg, [3]: { } V = Σ ˆΣ 2 H 2 = trace (H i Ĥi(H i Ĥi T ( where H i denotes the i-th Markov matrix of the transfer function H(z = H iz i of Σ (and likewise for Ĥi For fixed k n, it is known that among all systems of order k no approximation of Σ of order < k exists which constitutes a local minimum of the H 2 criterion See also [, 2] Therefore, one may require the order of ˆΣ to be equal to k or to be k, depending on what is convenient; relaxation of this constraint will not lead to different solutions 3 A state space approach to H 2 model reduction We will pursue a state space approach It is assumed that a minimal realization of order n (ie, state space dimension n is available for Σ, which is denoted by (A, B, C It then holds that H(z = C(zI n A B and H i = CA i B Likewise, ˆΣ will be represented by a (minimal state space matrix triple (Â, ˆB, Ĉ of order k Stability of Σ and ˆΣ corresponds to asymptotic stability of the dynamical matrices A and Â, which means that all eigenvalues are in the open complex unit disk Substitution of the expressions for the Markov matrices in terms of the matrices of the state space realizations (A, B, C and (Â, ˆB, Ĉ makes it possible to rewrite the H 2 criterion as follows: { ( V = trace C Ĉ ( ( } P P 2 C T P2 T (2 P 3 ĈT where P, P 2 and P 3 denote the unique solutions of the following discrete time Lyapunov and Sylvester matrix equations: P AP A T = BB T, (3 P 2 AP 2  T = B ˆB T, (4 P 3 ÂP 3ÂT = ˆB ˆB T (5 (See for instance [4, 3] An equivalent dual formulation of the H 2 criterion is offered by the expression { ( V = trace B T ˆB ( ( } T Q Q 2 B Q T 2 Q 3 ˆB (6 where Q, Q 2 and Q 3 denote the unique solutions of: Q A T Q A = C T C, (7 Q 2 A T Q 2  = C T Ĉ, (8 Q 3 ÂT Q 3  = ĈT Ĉ (9 It is noted that the entries of the matrix solutions of the Lyapunov and Sylvester equations involved are rational in terms of the entries of Â, ˆB and Ĉ Consequently, the H 2 criterion V is rational in these entries too This shows how the H 2 model reduction problem can be formulated as a rational optimization problem 4 On the parameterization of (Â, ˆB, Ĉ The H 2 criterion is obviously insensitive to a change of basis of the state space This creates parameter redundancy in the formulation of V above, if all the entries of Â, ˆB and Ĉ are left free To remove this excess freedom, (pseudo-canonical parameterizations of Â, ˆB and Ĉ can be employed, as described in the literature, for which the entries of these matrices depend in a (usually rather simple rational way on the parameters The use of such parameterizations preserves rationality of V For most of our purposes attention can be restricted, without loss of generality, to a single generic parameter chart for the matrix triple (Â, ˆB, Ĉ A convenient parameter chart is for instance offered by the following structured matrices (see [, 3]: 0 0  = 0 0, ˆB =, 0 0 Ĉ = (0 Here the starred entries denote parameters to be chosen freely under the sole constraint that stability of  holds Note that observability is already built in Controllability (and hence minimality of (Â, ˆB may be required to hold too, but as we have indicated at the end of Section 2, this is not really an issue This parameter chart involves k(m + p parameters, which precisely agrees with the manifold dimension of the space of (strictly proper systems of order k An alternative, dual parameter chart which has controllability built in instead of observability, is easily constructed in a likewise fashion Other parameterizations may have their virtues too For instance, one may employ Schur parameters to build the stability property of  directly into the parameterization in a way which allows one to get rid of any restrictions on the parameter domain (Cf, eg, [2] and [9] Such an approach can be combined with the choice of parameterization described above, and can be achieved in a way which leaves V rational In that case the H 2 model reduction problem is reformulated as an unconstrained rational optimization problem for which one may hope to find a finite number of stationary points 5 The concentrated H 2 criterion From Eqn (6 it is not difficult to see that the criterion V is of the linear least squares form with respect to the matrix ˆB if  and Ĉ are kept fixed This makes it possible to optimize for ˆB

analytically in terms of Â, Ĉ and (A, B, C As it happens, a unique minimizing solution for ˆB for this restricted linear least squares problem exists, provided that (Â, Ĉ is observable It is given by ˆB = Q 3 QT 2 B ( Note that in view of the choice of parameter chart advocated above, this allows one to eliminate km parameters Substitution into (6 returns what we will call the concentrated H 2 criterion, denoted V c and given by: V c = trace { B T Q B } trace { B T Q 2 Q 3 QT 2 B } (2 This criterion is to be minimized with respect to  and Ĉ, which enter the expression through Q 2 and Q 3 Since the first term of V c is entirely determined by the given system Σ (it denotes the squared H 2 norm of Σ, viz Σ 2 H 2 we may focus on the second term instead To this end we define the criterion W c by: W c = trace { B T Q 2 Q 3 QT 2 B } (3 It is noted that: (i W c is to be maximized with respect to  and Ĉ; (ii W c 0, and since also V c 0, a natural uniform upper bound on W c is provided by Σ 2 H 2 ; (iii W c = ˆΣ 2 H 2, so the aim is to maximize the norm of the approximation (Â, ˆB, Ĉ, but notably under the constraint that ˆB satisfies Eqn (, which involves Â, Ĉ and (A, B, C; (iv W c can still be constructed to be an unconstrained rational function of the remaining parameters Of course, from (2 a similar (dual approach can be followed to optimize V with respect to only Ĉ, for fixed  and ˆB In this case the unique minimizing value for Ĉ is given by Ĉ = CP 2 P 3 (4 Substitution into (2 leads to an alternative concentrated H 2 criterion Ṽc, which is given by Ṽ c = trace { CP C T } trace { CP 2 P 3 P T 2 C T } (5 This criterion is to be minimized with respect to  and ˆB In this case it is natural to introduce the associated criterion W c according to W c = trace { CP 2 P 3 P T 2 C T } (6 which is to be maximized with respect to  and ˆB The criterion Ṽc involves only kp parameters A choice between the two alternatives V c and Ṽc may be based on the number of inputs m and the number of outputs p 6 A doubly concentrated H 2 criterion for k = Above it was noted that the H 2 criterion V is a linear least squares criterion with respect to the sets of parameters in ˆB or in Ĉ individually, but not simultaneously Nevertheless it would be nice if V could be optimized analytically for a fixed choice of  with respect to all of the parameters in ˆB and Ĉ, in which case the number of free parameters in the resulting doubly concentrated H 2 criterion would drop to just n This actually turns out to be possible in case approximation is carried out with systems of order (ie, k = In this context the following theorem is useful (for a proof, cf [0] Theorem 6 Let Q be a fixed n n Hermitian matrix Consider the (real-valued expression r(p = trace{qp (P P P } (7 where P ranges over the set of n k matrices of full column rank k n Then: (i The (globally maximal value of r(p is equal to the sum of the k largest eigenvalues of Q (ii This maximal value is attained for any matrix P of which the column space is spanned by k independent eigenvectors of Q corresponding to these k largest eigenvalues To see how this theorem can be applied, we reconsider the expression W c = trace { B T Q 2 Q 3 QT 2 B } In case k =, it is noted that Q 3 and  are actually scalar and Q 2 is of size n They are given by Q 2 = (I n ÂAT C T Ĉ, (8 Q 3 = ĈT Ĉ/( Â2 (9 Therefore W c = { ( Â2 trace B T (I n ÂAT C T } { Ĉ(ĈT Ĉ Ĉ T C(I n ÂA B = trace ( Â2 C(I n ÂA BB T (I n ÂAT C T Ĉ(ĈT Ĉ Ĉ } T This last expression is in a suitable form to apply the above theorem to The role of Q is played by the matrix ( Â2 C(I n ÂA BB T (I n ÂAT C T and that of P is played by Ĉ Thus, the doubly concentrated H 2 criterion is given by W d = λ max { ( Â2  2 H( H( T }, (20 which is to be maximized with respect to the scalar real parameter  over the open interval (, 7 A bound on the H 2 criterion To apply the previous approach also to the situation with k > we first note that Q 2 and Q 3 can be written as products of observability matrices of infinite size: where O = C CA CA 2 Q 2 = O T Ô, (2 Q 3 = ÔT Ô, (22 Ĉ ĈÂ, Ô = ĈÂ2 (23

Then W c can be rewritten as { } W c = trace [O BB T T O ]Ô (ÔT T Ô Ô T, (24 which resembles the form of Eqn (7 There are two problems with this attempt to apply Theorem 6 ( The matrices playing the role of Q and P are of infinite size (2 The matrix Ô, which plays the role of P, cannot be chosen freely: it involves the fixed matrix  and it has the structure of an observability matrix (A closely related problem is that the matrix playing the role of Q currently does not involve  at all It is possible to overcome the first problem, essentially by noting that high matrix powers of A and  can be expressed as linear combinations of low matrix powers according to the theorem of Cayley-Hamilton One way to work this out is offered by the approach based on Faddeev sequences to solve Lyapunov and Sylvester equations, see [7] We have the following definitions and notation Definition 7 Let X be a matrix of size n n, and y a vector of size n Then: (i The characteristic polynomial det(zi n X of X is denoted by χ X (z = z n + χ z n + + χ n z + χ n (25 (ii The controller companion form matrix associated with X is denoted by X c and defined by χ χ n χ n 0 0 X c = (26 0 0 (iii The Faddeev sequence of X is denoted by {X 0, X,, X n } and defined recursively by X 0 := I n, (27 X k := XX k trace{xx k } I n k (28 Equivalently, it holds (for k = 0,,, n that X k = X k + χ X k + + χ k X + χ k I n (29 (iv The Faddeev reachability matrix of (X, y is denoted by F X (y and defined by F X (y = ( X 0 y X y X n y (30 It then follows (see again [7] that the matrices Q 2 and Q 3 are given by Q 2 = Q 3 = F A T (c i ˆ (FÂT (ĉ i T, (3 FÂT (ĉ i (FÂT (ĉ i T, (32 where (for i =,, p the column vectors c i and ĉ i denote the transposed rows of C and Ĉ, respectively, whence: c T ĉ C = T, Ĉ =, (33 c T p and where the matrices ˆ (of size n k and (of size k k are the unique solutions of the associated (highly structured Sylvester and Lyapunov equations in controller companion form: ĉ T p ˆ A c ˆ  T c = e e T, (34 Âc ÂT c = e e T (35 It is noted that the expression for Q 3 now involves a finite number of Faddeev reachability matrices of finite size, but also the positive definite symmetric matrix To deal with this, we introduce the following matrices: Then: GÂT (ĉ i := FÂT (ĉ i /2, (of size k k, (36 G A T (c i := F A T (c i ˆ /2, (of size n k (37 Q 2 = Q 3 = G A T (c i (GÂT (ĉ i T, (38 GÂT (ĉ i (GÂT (ĉ i T (39 The matrices GÂT (ĉ i and G A T (c i depend linearly on the entries of ĉ i and c i, respectively Therefore, these matrices can also be rewritten as: GÂT (ĉ i = ( ˆM ĉ i ˆMk ĉ i, (40 G A T (c i = ( M c i M k c i, (4 with ˆM,, ˆM k of size k k depending only on Â, and M,, M k of size n k depending only on  and A Consequently, we find: Q 2 = Q 3 = j= M j c i ĉ T i j= ˆM j ĉ i ĉ T i Finally, we therefore may write: where ˆM T j = ˆM T j = M j C T Ĉ ˆM j T, (42 j= j= ˆM j Ĉ T Ĉ ˆM T j (43 Q 2 = M(C ˆM(ĈT, (44 Q 3 = ˆM(Ĉ ˆM(ĈT, (45 M(C = ( M C T M k C T, (of size n kp, (46 ˆM(Ĉ = ( ˆM Ĉ T ˆMk Ĉ T, (of size k kp (47

Substitution of these expressions into the expression for W c yields: W c = trace { [M(C T BB T M(C] ˆM(ĈT ( ˆM(Ĉ } ˆM(ĈT ˆM( Ĉ (48 This shows how the infinite size problem has been solved The structure problem, however, still exists for the matrix ˆM(Ĉ, which apart from Ĉ also still depends on  In view of Theorem 6 we conclude that the sum of the k largest eigenvalues of the pk pk matrix Z = M(C T BB T M(C constitutes an upper bound for the maximally achievable value of W c It is noted that this matrix Z depends on the given system (A, B, C and on  As we have seen, this bound is actually sharp for k = 8 Case study: globally optimal H 2 model reduction of a multivariable system from order 4 to order 2 The set-up above has been applied to study a nontrivial multivariable example with 2 inputs and 2 outputs, where a given system Σ of order 4 is approximated by another system ˆΣ of order 2 Use has been made of tools from constructive algebra, computer algebra software and numerical software (Maple, Mathematica, Matlab The methods employed for global rational optimization are described in [5, 8, 9] The system Σ involved in this example is given by: A = 0 0 0 0 0 0 0 0 0 0 8 2 C = 4, B = ( 0 0 0 0 0 2 3 4 383 2080 279 040 839 8320 37 460 49 33280 The second order approximation ˆΣ is given by: ( ( 0  =, ˆB = 2 3 4, Ĉ = 4 9 0 6 4 99 280, (49 ( 0 (50 The design of this example will be discussed in more detail elsewhere It is such that the system ˆΣ is guaranteed to constitute a stationary point (on the manifold of systems of order 2 of the H 2 criterion V associated with Σ, a property which is easily verified However, no further aspects of the concentrated H 2 criterion W c nor of its bound W d have been used in the design of this example The Hessian of V at ˆΣ can be computed to be positive definite, which implies that ˆΣ yields a local minimum of V A similar statement holds for the concentrated H 2 criterion V c To show that ˆΣ constitutes a global minimum of V we intend to employ techniques from global rational optimization Using the generic chart for (Â, ˆB, Ĉ indicated in Section 4, Eqn (0, the criterion V is rational and involves 8 free parameters The numerator polynomial has total degree 8 and consists of 655 terms; the denominator polynomial has total degree 6 and consists of 27 terms After the first concentration step, which eliminates the matrix ˆB, the concentrated criterion W c is still rational and involves 4 free parameters The numerator polynomial has total degree 4 and consists of 705 terms; the denominator polynomial has total degree 2 and consists of 277 terms (but can be factored to become more concise This problem is still too complex to be handled by our software, using tools from constructive algebra The upper bound W d The second concentration step, which eliminates the matrix Ĉ, produces an upper bound W d on the concentrated criterion W c It turns out that this bound is again rational and involves only 2 free parameters The numerator polynomial has total degree 8 and consists of 42 terms and the denominator polynomial has total degree 6 and consists of 28 terms (and can be factored to become more concise It has the following explicit form: W d = (( a 2 (790785444 + 3296829824a 692769920a 2 2897270720a 3 + 8884528a 4 +438089920a 5 8864576a 6 + 82098966a 2 +6894389696a a 2 5524025360a 2 a 2 235600232a 3 a 2 6509328a 4 a 2 5653808a 5 a 2 + 60932736a 6 a 2 +4986490756a 2 2 + 573263544a a 2 2 324024308a 2 a 2 2 58392552a 3 a 2 2 27035000a 4 a 2 2 + 30466368a 5 a 2 2 +560025568a 3 2 + 2067608a a 3 2 + 24475432a 2 a 3 2 85974588a 3 a 3 2 5724440a 4 a 3 2 + 00342727a 4 2 +3579240a a 4 2 + 2800888a 2 a 4 2 30466368a 3 a 4 2 +9843237a 5 2 + 575746a a 5 2 + 424888a 2 a 5 2 +5940378a 6 2 + 766592a a 6 2 + 952074a 7 2/ (08600(6 + 4a + a 2 2 (4 2a 2 + 4a 2 + a 2 2 2, (5 where the characteristic polynomial of  is denoted by det(zi 2  = z2 + a z + a 2 Investigation of the upper bound W d shows a number of interesting properties for this example (i The expression for W d is rational, despite the fact that square roots appear in intermediate calculations and that the two largest eigenvalues of a 4 4 matrix have to be summed (ii The bound W d is not sharp everywhere on the domain of stable matrices Â, but it happens to be sharp at the intended approximation ˆΣ (iii The bound W d can be handled by our software We are able to compute all of its stationary points and to deter-

mine its global maximum It turns out that there is a single global maximum for W d within the stability domain; it is located at the intended approximation ˆΣ Together with (ii this proves that ˆΣ is the unique globally optimal H 2 approximation of order 2 of Σ (iv Using the techniques of [8, 9], the rational bound W d is transformed into a polynomial expression when investigating global maximality at the approximation ˆΣ This polynomial should then be nonnegative everywhere on the stability domain As it happens, the polynomial is actually nonnegative on the whole of R 2, which means that there is no need to employ Schur parameters (The use of Schur parameters would double the total degrees of the numerator and denominator polynomials involved, and largely increase the number of terms A popular technique to show nonnegativity of a polynomial is by writing it as a sum of squares of polynomials, for which (numerical semi-definite programming software is available in Matlab Although not every nonnegative polynomial admits such a decomposition, many polynomials do and counterexamples are not easily constructed However, the available software breaks down on the polynomial of the present example, indicating no such decomposition to exist! Therefore, other (symbolic, exact methods from constructive algebra were used to establish nonnegativity of this polynomial As it turns out, the polynomial has 7 global minima, of which only one is in the feasible area 9 Conclusions We have employed a state space approach to derive a bound on the achievable quality of a k-th order H 2 approximation to a given stable system of order n For k = this bound is sharp and reduces the model reduction problem to a rational optimization problem in a single variable 2 We have presented a nontrivial multivariable example of a stable system with two inputs and two outputs of order 4 and an associated unique globally optimal H 2 - approximation of order 2 To the best of our knowledge, this is the first such example reported in the literature (The cases for which global optimality results of this kind are available are usually limited either to the scalar case, or to the case k =, or to the case k = n, see for instance [6] 3 We have pursued an ad hoc technique, involving exact methods from constructive algebra, to establish global optimality of the approximation of the example The success of this approach gives rise to a couple of research questions: (i Why is the bound sharp at the approximation ˆΣ? (ii Why does the bound have a global maximum at the approximation ˆΣ? References [] L Baratchart, On the topological structure of inner functions and its use in identification, in: Analysis of Controlled Dynamical Systems, Lyon, France, 990, Progress in Systems and Control Theory, Vol 8, Birkhauser, pp 5 59, 990 [2] L Baratchart and M Olivi, Inner-unstable factorization of stable rational transfer functions, in: Progress in System and Control Theory: Modeling, Estimation and Control of Systems with Uncertainty, Vol 0, Birkhäuser, pp 22 39, 99 [3] P Fulcheri and M Olivi, Matrix Rational H 2 - Approximation: A Gradient Algorithm Based On Schur Analysis, SIAM Journal on Control and Optimisation, Vol 36, No 6, pp 203 227, 998 [4] B Hanzon, Identifiability, Recursive Identification and Spaces of Linear Dynamical Systems, CWI Tracts 63 and 64, Centrum voor Wiskunde en Informatica (CWI, Amsterdam, 989 [5] B Hanzon and D Jibetean, Global minimization of a multivariate polynomial using matrix methods, to appear in: Journal of Global Optimization, 2003 (Also available as: CWI report PNA-R009, CWI, Amsterdam, 200 [6] B Hanzon and JM Maciejowski, Constructive algebra methods for the L 2 problem for stable linear systems, Automatica, Vol 32, No 2, pp 645 657, 996 [7] B Hanzon and RLM Peeters, A Faddeev Sequence Method for Solving Lyapunov and Sylvester Equations, Linear Algebra and its Applications, Vol 24-243, pp 40 430, 996 [8] D Jibetean, Global optimization of rational multivariate functions, CWI report PNA-R020, CWI, Amsterdam, 200 [9] D Jibetean, Algebraic Optimization with Applications to System Theory, PhD Thesis, Vrije Universiteit, Amsterdam, 2003 [0] P Lancaster and M Tismenetsky, The Theory of Matrices, 2nd ed New York: Academic Press, 985 [] AJM van Overbeek and L Ljung, On-line Structure Selection for Multi-Variable State-Space Models, Automatica, Vol 8, pp 529 543, 982 [2] RLM Peeters, B Hanzon and M Olivi, Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm, Proceedings of the ECC 99, Karlsruhe, Germany, 3/08/999 03/09/999, paper F0662, 999 [3] RLM Peeters, System Identification Based on Riemannian Geometry: Theory and Algorithms, Tinbergen Institute Research Series, Vol 64 Thesis Publishers, Amsterdam, 994

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