6 Optimal Model Order Reduction in the Hankel Norm

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1 6 Optimal Model Order Reduction in the Hankel Norm In this lecture, we discuss optimal model reduction in the Hankel norm. There is no known solution to the optimal model reduction problem in the H -norm. Since there is a connection between the Hankel norm and the H -norm, the presented method is still interesting to us, and gives a lot of insight. The analysis also establishes a link to SVD. 6.1 The Hankel Norm and the Hankel Operator TheHankelnormofasystemG = (A,B,C,D) H is definedby G 2 0 y(t) 2 dt 0 H := sup 0 u L 2 (,0] u(t)2 dt, where y(t) = Ce A(ts) Bu(s)ds. The Hankel norm tells how much energy can be transferred from past inputs into future outputs throughthesystemg. InExercise3.4b),weshowedthat G H = λ max (PQ) = σ 1. Onedefinesthe HankeloperatorΓ G of thesystemg by, Γ G : L 2 (,0] L 2 [0, ) : (Γ G u)(t) = 0 Ce A(ts) Bu(s)ds, t > 0. Theinducednormof Γ G isequaltothehankelnormofg, Γ G = G H. A nice propertyof Γ G is that it has finite rank, with the rank being equal to the minimal number of states required to realize the input-output map G, Rank Γ G = n. Furthermore,onecan makean SVDofΓ G, withthedyadicexpansion (Γ G u)(t) = σ i u i (t)(v i,u) L2 (,0], (6.1) where σ i are the Hankel singular values of G, and the singular vectors are v i L 2 (,0], u i L 2 [0, ). Maybeonewouldthinkit is morenaturaltoconsidertheoperatorgdirectly, G : L 2 (, ) L 2 (, ) : (Gu)(t) = t Ce A(ts) Bu(s)ds+Du(t). This operator is generally not of finite rank, however, and has then no finite dyadic expansion such as (6.1). This make further analysis difficult. SinceonecanmakeanSVDofΓ G,onecanessentiallyapplytheSchmidt-Mirskytheoremtoprove that Γ G Γ Gr = GG r H σ r+1, (6.2) forallg r oforderr. FromthedefinitionoftheHankelnormitshouldalsobeclearthat G H G for all G H. Hence, the bound (6.2) also holds for the H -norm, as has already been stated in earlier lectures. Relation between thehankelnormandthel -norm To proceed, we need to introduce more system spaces, namely L, H, and H (r). A transfer function G(s) belongstol if, and onlyif, G := sup G(jω) ω 33

2 is finite. Hence G(s) may have poles in both C + and C, but not on the imaginary axis. A transfer function G(s) belongs to H if, and only if, G(s) H. Hence, unstable systems with no stable poles belong to H. A transfer function belongs to H (r) if it belongs to L and it has at most r polesin thecomplex lefthalf plane, C. We can now state the following fundamental theorems. Theorem12 (Nehari). Supposethat G H, andf H. Then GF L, and min GF = G H (= σ 1 ). F H Hence,oneinterpretationof G H isthatitistheminimumdistancebetweengandacompletely unstablesystem,measuredin thel -norm. The following important extension to Nehari s theorem is available. Theorem13 (Adamjan-Arov-Krein). Suppose that G H andq H (r). Then GQ L,and where σ r+1 isthe (r +1)-th largest Hankelsingular value ofg. min GQ = σ r+1, (6.3) Q H (r) DenoteanoptimalQ H (r)byq,i.e., GQ = σ r+1. Astable/anti-stabledecomposition ofq can beperformedsuchthatq = G r +F,whereG r H,degG r r,andf H. Itfollows that GG r = GQ +F GQ + F = σ r+1 + F. (6.4) If we can show that the L -norm of the unstable termf is small, then the stable part of Q can be a goodcandidate forh modelreduction. A boundon F is derivedin Section6.3. Notethatthestable partg r ofq solvestheoptimalhankelnormapproximation problem,since GG r H = min GG r F = min GQ = σ r+1, F H Q H (r) and we know that a lower bound of the problem is σ r+1, see (6.2). Note also that the direct term D r ofg r isnotdeterminedbyhankelnormapproximationsinceitdoesnotappearinthehankelnorm. 6.2 State-SpaceFormulas forconstruction of Q In this section, we give state-space formulas for the computation of an optimal Hankel approximation Q whoseexistenceis guaranteedby Theorem13. Suppose the system G = (A,B,C,D) H is of order n, and that it is square (m = p). The formulas for non-square systems are slightly more complicated, and are given in Linear Robust Control. Assumetherealization ofgis suchthat thegramians taketheform [ P1 0 P = 0 σ r+1 I l ], Q = [ Q1 0 0 σ r+1 I l wherelisthemultiplicityofthesingularvalueσ r+1. Onecanchooseapermutedbalancedrealization of G, for example. Conformably to P, Q, partition the state-space realization of G, [ ] [ ] A11 A A = 12 B1, B =, C = [ ] C A 21 A 22 B 1 C 2. 2 One can prove (Exercise 6.3) that under the given assumptions, there is a unitary matrix U R p m, U T U = I, suchthat B 2 = C2 T U. (6.5) 34 ],

3 Also define E 1 := Q 1 P 1 σ 2 r+1 I. Arealization of theoptimal approximation Q isnow givenby  = E1 1 r+1a T 11 +Q 1 A 11 P 1 σ r+1 C1 T UB1 T ) (6.6) ˆB = E1 1 1B 1 +σ r+1 C1 T U) (6.7) Ĉ = C 1 P 1 +σ r+1 UB1 T (6.8) ˆD = D σ r+1 U. (6.9) The transfer function Q (s) = (Â, ˆB,Ĉ, ˆD) is of order n l and belongs to H (r). In fact, it has exactlyr stablepolesandnrlunstablepoles. WeverifybelowthatE(s) := G(s)Q (s)satisfies E(s) E(s) = σ 2 r+1 I, wheree(s) = E(s) T, andthus E = σ r+1. Notethat E(jω) = σ r+1 for all ω. The given Q solves the optimization problem in Theorem 13. This is generally not the unique solution, however. All the solutions(also in the non-square G case) are parameterized in Linear Robust Control using a linear fractional transformation. Toverify thate(s) E(s) = σr+1 2 I,weusethefollowing lemma. Lemma1. LetG = (A,B,C,D). If there isasymmetric matrix P suchthat PA T +AP +BB T = 0 PC T +BD T = 0 D T D = γ 2 I, then G(s) G(s) = γ 2 I andg L. Conversely, if G = (A,B,C,D) isminimalandg(s) G(s) = I,then there is a symmetric matrix P that satisfies the above conditions. If werealize E(s) = G(s)Q (s) with [ ] [ A 0 BˆB] A e =, B e =, C e = [ C 0  Ĉ], D e = D ˆD, the assumptions of Lemma 1 are satisfied using P 1 0 I P e = 0 σ r+1 I l 0 I 0 E1 1 1 as P,and γ = σ r H ErrorBounds In this section, we make the standing assumption that the Hankel singular values of G are distinct. This simplifies the notation. The results can be strengthened if some singular values are identical, just as in balanced truncation. Using the error bound for balanced truncation, the H -norm of G can be bounded using the Hankel singular values. If Theorem 6 is applied and all states are truncated, we obtain GG( ) 2 35 σ i.

4 Optimal Hankel norm approximation can be used to improve this general bound. If we construct the optimal Hankel approximation G n1 as Q using the formulae from (6.6) to (6.9), we obtain G G n1 = σ n. The Gramians of the system G n1 are given by P 1 E 1 and E1 1 Q 1, and hence the Hankelsingular values ofg n1 are λ i (P 1 E 1 E1 1 Q 1) = λ i (P 1 Q 1 ) = σ i (G), i = 1,...,n1 (the remaining Hankel singular values). We can repeat this procedure on G n1, and removehankelsingularvaluesonebyone. Intheendonlyadirectterm D = G 0 remains. Applying the triangle inequality, we have G D = (GG n1 )+(G n1 G n2 )+...+(G 1 D) σ i. (6.10) Hence,therealwaysexistsaconstant DthatwillshifttheNyquistdiagramofGsothatitiscontained in a circle of radius equal to the sum of the distinct Hankel singular values. If we use G( ) as the directterm,thenweneedtomake theradiusafactor twolarger. Let us now return to the bound (6.4). We want an a priori bound on F since that yields a boundon GG r. Wehave that F H is of ordernr1 (remember themultiplicity l = 1). Furthermore,F H and F = F. Thefollowing lemmacan bederived. Lemma 2. Assume that σ i (G) are distinct, and let the optimal approximation Q H (r) of G be Q = G r +F, suchthat G r,f H. Then σ i (F ) σ i+r+1 (G), i = 1,...,nr 1. Using (6.10) and Lemma 2 on F, there is a D such that F D n i=r+2 σ i(g). It follows thatthis D yields GG r D = GG r F +F D GQ + F D = σ r+1 + F D Wesumtheresultsup in thefollowingtheorem. i=r+1 Theorem 14. Suppose G H is of order n, and has distinct Hankel singular values σ i. Then for all approximations G r H of order r,it holds that σ r+1 GG r H GG r. Furthermore, there exists G r H of order r, given by the stable part of the transfer function Q = (Â, ˆB,Ĉ, ˆD) H (r), and D suchthat σ i. GG r H = Γ G Γ Gr = σ r+1 GG r D σ i. i=r+1 Inparticular, when r = n1, asolution to theoptimal H -norm approximation problem is obtained. 6.4 Suggested Reading Chapter 10 in Linear Robust Control gives a thorough and readable presentation of Hankel norm approximation. All the stated results above are proved there. 36

5 6.5 Exercises EXERCISE 6.1 (Antoulas [2004]) LetthemodelGbegiven by G(s) = s+1 s 6 +3s 5 +5s 4 +7s 3 +5s 2 +3s+1. Compute optimal Hankel norm approximations G r of G, and the corresponding Q for r = 3 and r = 5. Also plot the Bode diagrams of the approximation errors G Q and G G r, and compute thel /H -normof theerrors. EXERCISE 6.2 Derive a state-space algorithm that performs a stable/anti-stable decomposition of state-space modelsg = (A,B,C,D) L. EXERCISE 6.3 Provethat thereis always aunitary U thatsatisfies(6.5). (Hint: ShowfirstthatB 2 B T 2 = CT 2 C 2.) EXERCISE 6.4 Prove that all transfer functions G H of degree n (with distinct Hankel singular values) can be expanded as G(s) = D +σ 1 E 1 (s)+σ 2 E 2 (s)+...+σ n E n (s), wheree i H areall-pass,i.e.,e i (s) E i (s) = I,andG r (s) = D+σ 1 E 1 (s)+...+σ r E r (s)isoforder r. EXERCISE 6.5 (Extra) Prove that the following lower bound on the weighted approximation criterion holds, min W o(gg r )W i σ r+1 ([M o GM i ] + ), G r H,degG r r whereσ r+1 ([M o GM i ] + ) is computedas follows: Computethe(unstable) spectralfactors M o and M i, M o (s) M o (s) = W o (s) W o (s), M i (s)m i (s) = W i (s)w i (s), such that M o (s),m o (s) 1,M i (s),m i (s) 1 have their poles in the the open right half half plane C +. Here[P] + denotesthestable partofp. 37