Hankel Optimal Model Order Reduction 1

Transcription

1 Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both the theory and an algorithmi side of Hankel optimal model order redution. 9.1 Problem Formulation and Main Results This setion formulates the main theoretial statements of Hankel optimal model redution, inluding the famous Adamyan-Arov-Krein (AAK) theorem The Optimization Setup Let G = G(s) be a matrix-valued funtion bounded on the jθ-axis. The task of Hankel ˆ optimal model redution of G alls for finding a stable LTI system G of order less than a given positive integer m, suh that the Hankel norm H of the differene = G Ĝ is minimal. Remember that Hankel norm of an LTI system with transfer matrix = (s), input w, and output v, is defined as the L2 gain of the assoiated Hankel operator H, i.e. as the maximum of the future output energy integral ( ) 1/2 v(t) 2 dt 0 subjet to the onstraints 0 w(t) = 0 for t 0, w(t) 2 dt 1. 1 Version of Marh 29, 2004

2 2 While the standard numerial algorithms of Hankel optimal model redution will be defined here for the ase when G is a stable ausal finite order LTI system defined by a ontrollable and observable state spae model ẋ = Ax + Bw, y = Cx + Dw, (9.1) some useful insight an be obtained by studying general statements when G = G(s) is not neessarily a rational transfer matrix The i-th largest singular value of a Hankel operator Let G = G(s) be a matrix-valued funtion bounded on the jθ-axis. For an integer k > 0 let us say that ψ = ψ k is the k-th singular number of H G if ψ k is the minimum of the L2 gain of H G over the set of all linear transformations of rank less than k (this definition works for arbitrary linear transformations, not only for Hankel operators). When G is defined by a minimal state spae model (9.1) with m states and a Hurwitz matrix A, ψ i = 0 for i > m, and the first m largest singular numbers s i are square roots of the orresponding eigenvalues of W o W, where W o, W are the ontrollability and observability Grammians of (9.1). For some non-rational transfer matries, an analytial or numerial alulation of ψ i may be possible. For example, the i-th largest singular number of H G, where G(s) = exp( s), equals 1 for all positive i The AAK Theorem The famous Adamyan-Arov-Krein theorem provides both a theoretial insight and (taking a onstrutive proof into aount) an expliit algorithm for finding Hankel optimal redued models. Theorem 9.1 Let G = G(s) be a matrix-valued funtion bounded on the jθ-axis. Let ψ 1 ψ 2... ψ m 0 be the m largest singular values of H G. Then ψ m is the minimum of G Ĝ over the set of all stable systems G ˆ of order less than m. In other words, approximating Hankel operators by general linear transformations of rank less than m annot be done better (in terms of the minimal L2 gain of the error) than approximating it by Hankel operators of rank less than m. The proof of the theorem, to be given in the next setion for the ase of a rational transfer matrix G = G(s), is onstrutive, and provides a simple algorithm for alulating the Hankel optimal redued model.

3 H-Infinity quality Hankel optimal redued models It is well established by numerial experiments that Hankel optimal redued models usually offer very high H-Infinity quality of model redution. A somewhat onservative desription of this effet is given by the following extension of the AAK theorem. Theorem 9.2 Let G = G(s) be a stable rational funtion. Let ψ 1 ψ 2... ψ m > 0 be the m largest singular values of H G. Let ψ m > ψ m+1 > ψ m+2 > > ψ r = 0 be the ordered sequene of the remaining singular values of H G, eah value taken one, without repetition. Then there exists a Hankel optimal redued model G ˆ of order less than m suh that r G G ˆ ψ m + ψ k. k=m+1 Just as in the ase of the basi AAK theorem, the proof of Theorem 9.2 is onstrutive, and hene provides an expliit onstrution of the redued model with the proven properties. In pratie, the atual H-Infinity norm of model redution error is muh smaller. It is important to remember that the Hankel optimal redued model is never unique (at least, the D terms do not have any effet on the Hankel norm, and hene an be modified arbitrarily). The proven H-Infinity model redution error bound is guaranteed only for a speially seleted Hankel optimal redued model.

4 4 9.2 Proof of the AAK theorem The fat that ψ m is a lower bound for the Hankel norm model redution error follows from the definition of ψ m and from the fat that rank of the Hankel operator assoiated with a stable system equals its order. This setion ontains a rather expliit onstrution of a redued model of order less than m for whih the Hankel norm of the approximation error equals ψ m. To avoid inessential mathematial ompliations, we onsider only the ase when G is defined by a minimal (ontrollable and observable) finite dimensional state spae model ẋ = Ax + Bw, y = Cx + Dw (9.2) with a Hurwitz matrix A. The starting point of the proof is the fat that, aording to the expliit formulae for singular value deomposition of Hankel operators based of the ontrollability and observability Grammians W W 1, W o of system (9.2), the matrix W o ψm 2 has not more than m 1 positive eigenvalues. The proofs is similar to the one given in the derivation of H- Infinity suboptimal ontrol, and also uses the generalized Parrot s theorem in ombination with a modified version of the KYP lemma Upper bounds for Hankel norms The following simple observation is frequently helpful when workin with Hankel operators. Theorem 9.3 If rational transfer matries G +, G of same dimensions are suh that all poles of G + have negative real part, and all poles of G have positive real part, then G + H G + + G. Sine the Hankel operator assoiates with a transfer matrix G = G + + G is defined by the stable part G + of G, and the Hankel norm never exeeds L-Infinity norm, the statement is true. In fat, Theorem 9.3 is the simple side of the so-alled Nehari theorem, whih laims that G + H is the minimum of G + + G over the set of all anti-stable G. The non-trivial part of the Nehari theorem will be a by-produt of the proof of the AAK theorem. Example 9.1 Adding G = 1/2 to G + = 1/(s + 1) yields G = (1 s)/(2 + 2s) (Infinity-norm G = 0.5 equals the Hankel norm of 1/(s + 1)).

5 KYP lemma for L-Infinity norm approximation A simple but important observation given by the KYP lemma is that a ertifiate of an L-Infinity bound H δ for a given transfer matrix H(s) = d + (si a) 1 b is delivered by a symmetri matrix p = p suh that δ 2 w 2 x + dw 2 2 x p(a x + bw) 0 x, w. (9.3) Note that a does not have to be a Hurwitz matrix here. When system (9.2) is approximated by system G r (not neessarily stable) with state spae model ẋ r = A r x r + B r w, y r = C r x r + D r w, (9.4) the approximation error dynamis system with input w and output δ = y y r has state spae model x = a x + bw, δ = x + dw, where [ ] [ ] x Ax + Bw x =, a x + bw =, x + dw = Cx + Dw C r x r D r w. (9.5) A r x r + B r w x r Let = (s) denote the transfer matrix from w to δ. Aording to the KYP lemma, the inequality δ an be established by finding a symmetri matrix [ ] p = p p = 11 p 12 (9.6) p 21 p 22 suh that (9.3) holds. In view of Theorem 9.3, for Hankel model order redution it is important to keep trak of the order of the stable part of G r. Tis is made possible by the following observation. Theorem 9.4 If (9.3) holds for a, b,, d, p defined by (9.5),(9.6), and p 22 has less than m positive eigenvalues then the order of the stable part of G r is less than m. Proof Let V = V + +V be the diret sum deomposition of the state spae {x r } into the stable observable subspae V + of A r with respet to C r, and the ompementary A r invariant subspae V. In a system of oordinates assoiated with this deomposition, matries C r, A r, p 22 have the blok form [ ] [ ] [ a + 0 p ++ p C + r = +, A r =, p 0 a 22 =, p + p

6 6 where the pair ( +, a + ) is observable. Substituting w = 0 into (9.3) yields whih, in partiular, implies pa + a p, p ++ a + + a + p Hene p ++ > 0 and the number of positive eigenvalues of p 22 is at least as large as the dimension of a +. Finally, note that the dimension of a + is not smaller than the order of the stable part of G r Hankel optimal redued models via Parrot s Theorem The observations made in the previous two subsetions suggest the following approah to onstuting a redued model G ˆ of system G given by (9.2): simply find a matrix p = p in (9.6) suh that p 22 has less than n + m positive eigenvalues, and (9.3) holds for a, b,, d defined by (9.5) with some A r, B r, C r, D r. Then G, ˆ defined as the stable part of D r + C r (si A r ) 1 B r, will satisfy G Ĝ H δ. Note that, one p is fixed, the existene of A r, B r, C r, D r satisfying the requirements an be heked via the generalized Parrot s theorem (same as used in the derivation of H-Infinity suboptimal ontrollers). Theorem 9.5 Let ψ : R n R m R k R be a quadrati form whih is onave with respet to its seond argument, i.e. Then an m-by-k matrix L suh that ψ(0, g, 0) 0 g R m. (9.7) ψ(f, Lh, h) 0 f R n, h R k (9.8) exists if and only if the following two onditions are satisfied: (a) for every h R k there exists g R m suh that inf ψ(f, g, h) > ; (9.9) f R n (b) the inequality holds for all f R n, h R k. sup ψ(f, g, h) 0 (9.10) g R m

7 7 In appliation to Hankel optimal model redution, set [ ] [ ] [ ] θ x r A r B f = x, g =, h =, L = r, w C r D r y r [ ] [ ] x Ax + Bw ψ(f, g, h) = 2 p + δ 2 w 2 Cx + Dw y r 2. (9.11) θ x r Note that ψ is onave with respet to g. It turns out that one onvenient seletion for p is [ ] W δ 2 W 1 o W o p = δ 2 W 1 δ 2 W 1. (9.12) W o W o While formally there is no need to explain this hoie (one just has to verify that onditions (a),(b) of Theorem 9.5 are satisfied), there is a lear line of reasoning here. To ome to this partiular hoie of p, note first that (9.9) implies ψ(f, 0, 0) 0, whih means for ψ defined by (9.11). Hene p 11 A + A p 11 C C p 11 W o. Similarly, under the simplifyin assumption that p is invertible, (9.10) means Aq 11 + q 11 A δ 2 BB, where q 11 is the upper left orner of p 1. Hene Sine we have 1 q 11 δ 2 W. 1 q 11 = p 11 p 12 p 22 p 21, 1 1 p 12 p 22 p 21 = p 11 δ 2 q 11. Sine our desire is to minimize the number of positive eigenvalues of p 22, it is natural to use the minimal possible values of p 11 and q 11, whih suggests using p 11 = W o, q 11 = δ 2 W, p 12 = p 21 = p 22 = W o δ 2 W 1.

8 8 With the proposed seletion of p, the quadrati form ψ an be represented as ψ(f, g, h) = 2x [W o Bw + A (W o δ 2 W 1 )x r + (W o δ 2 W 1 )θ C y r ] + ψ(g, h). Hene ondition (a) an be satisfied if the equation W o Bw + A (W o δ 2 W 1 )x r + (W o δ 2 W 1 )θ C y r = 0 has a solution (θ, y r ) for every pair (w, x r ). In order to prove this, it is suffiient to show that every vetor ψ suh that also satisfies Note that by the definition, ψ C = 0, ψ (W o δ 2 W 1 ) = 0 (9.13) ψ W o B = 0, ψ A (W o δ 2 W 1 ) = 0. (9.14) (W o δ 2 W 1 )A + A (W o δ 2 W 1 ) + C C = δ 2 W 1 BB W 1. Multiplying this by ψ on the left and ψ on the right and using (9.13) yields ψ B W 1 = 0 and hene (9.14) follows. Similarly, the quadrati form ψ an be represented as W 1 2 ψ(f, g, h) = (x+x r ) (W o δ 2 W 1 )(Ax+Bw θ)+2δ 2 x (Ax+Bw)+δ 2 w Cx y r 2, whih is unbounded with respet to θ if (x + x r ) (W o δ 2 W 1 ) = 0 and is made non negative by seleting y r = Cx r otherwise.