H 2 optimal model approximation by structured time-delay reduced order models

Transcription

1 H 2 optimal model approximation by structured time-delay reduced order models I. Pontes Duff, Charles Poussot-Vassal, C. Seren 27th September, GT MOSAR and GT SAR meeting /49

2 Table of Contents Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 2/49

3 Outlines Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 3/49

4 Introduction Preliminaries H2 I/O delay H2 -state delay Conclusions and perspectives Large-scale models Some motivating examples in the simulation & control domains... Digitalized and computer-based modeling and studies are crucial steps for any system / concept or physical phenomena understanding 4/49 I. Pontes Duff et al. [Onera] H2 optimal model approximation by structured time-delay reduced order models

5 Introduction Preliminaries H2 I/O delay H2 -state delay Conclusions and perspectives Large-scale models Some motivating examples in the simulation & control domains... Digitalized and computer-based modeling and studies are crucial steps for any system / concept or physical phenomena understanding Problem: involved numerical dynamical models are too complex Due to finite machine precision, computation burden and memory management, I difficulties with system simulation, analysis, optimization, controller design I results are not accurate, time consumption I inappropriate actual numerical tools 4/49 I. Pontes Duff et al. [Onera] H2 optimal model approximation by structured time-delay reduced order models

6 Large-scale models... e.g. in aeronautics rigid behaviour Physical system fluid mechanics, structure, etc. Differential Algebraic Equations (DAEs) or Rational Functions Partial Differential Equations (PDEs) Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) H(s) = H 1 (s)+h d e τs +... p + ρ g = ρ a ρ t +.(ρ v ) = 0 Highly accurate and/or flexible aircraft Spacecraft, launcher, satellites, Fluid dynamics physics (Navier and Stokes) 5/49

7 Large-scale models... e.g. in aeronautics rigid behaviour Physical system fluid mechanics, structure, etc. Differential Algebraic Equations (DAEs) or Rational Functions Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) H(s) = H 1 (s)+h d e τs +... Discretisation finite elements, finite volume, etc. Partial Differential Equations (PDEs) p + ρ g = ρ a ρ t +.(ρ v ) = 0 Highly accurate and/or flexible aircraft Spacecraft, launcher, satellites, Fluid dynamics physics (Navier and Stokes) 5/49

8 Large-scale models... e.g. in aeronautics rigid behaviour Physical system fluid mechanics, structure, etc. Differential Algebraic Equations (DAEs) or Rational Functions Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) H(s) = H 1 (s)+h d e τs +... Discretisation finite elements, finite volume, etc. Partial Differential Equations (PDEs) p + ρ g = ρ a ρ t +.(ρ v ) = 0 simulation, control, analysis, optimisation, etc. Large-scale Highly accurate and/or flexible aircraft Spacecraft, launcher, satellites, Fluid dynamics physics (Navier and Stokes) 5/49

9 Large-scale models... e.g. in aeronautics rigid behaviour Physical system fluid mechanics, structure, etc. Differential Algebraic Equations (DAEs) or Rational Functions Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) H(s) = H 1 (s)+h d e τs +... Discretisation finite elements, finite volume, etc. Partial Differential Equations (PDEs) p + ρ g = ρ a ρ t +.(ρ v ) = 0 simulation, control, analysis, optimisation, etc. Large-scale objective: alleviate numerical burden Highly accurate and/or flexible aircraft Spacecraft, launcher, satellites, Fluid dynamics physics (Navier and Stokes) 5/49

10 Introduction Topics addressed in this presentation : H 2 -model approximation with I/O delay structure. H 2 optimality conditions for reduced state-delay systems. 6/49

11 Introduction Topics addressed in this presentation : H 2 -model approximation with I/O delay structure. H 2 optimality conditions for reduced state-delay systems. Goal: find a nth order rational model approximation in the form Ĥ(s) = Ĉ(sÊ Â) 1 ˆB 6/49

12 Introduction Topics addressed in this presentation : H 2 -model approximation with I/O delay structure. H 2 optimality conditions for reduced state-delay systems. Delay Goal: find a nth order rational I/O delay structured model approximation in the form Ĥ d (s) = ˆ o(s)ĉ(sê Â) 1 ˆB ˆ i (s) where ˆ i (s) = diag(e sˆτ 1,..., e sˆτnu ) and ˆ o(s) = diag(e sˆγ 1,..., e sˆγny ) 6/49

13 Benchmarks Example 1: Ladder network 1 { Eẋ(t) = Ax(t) + Bu(t) G Ladder := y(t) = Cx(t) (1) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

14 Benchmarks Example 1: Ladder network Hankel Singular Values (State Contributions) Stable modes 1 State Energy State Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

15 Benchmarks Example 1: Ladder network 1 Impulse response 0.6 Ladder Network Amplitude (Volts) Time (ms) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

16 Benchmarks Example 1: Ladder network 1 Impulse response Balanced Truncation Order = Amplitude (Volts) Time (ms) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

17 Benchmarks Example 1: Ladder network 1 Impulse response Balanced Truncation Order = Amplitude (Volts) Time (ms) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

18 Benchmarks Example 1: Ladder network 1 Impulse response Balanced Truncation Order = Amplitude (Volts) Time (ms) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

19 Benchmarks Example 1: Ladder network 1 Impulse response Balanced Truncation Order = Amplitude (Volts) Time (ms) Finite dimensional model of order N = 100. It has an intrinsic input-delay behavior. 1 Gugercin, S., Polyuga R., G., Beattie C., van der Schaft, A., "Structure-preserving tangential interpolation for model reduction of port-hamiltonian systems", Automatica 48(9): /49

20 PDE St-Venant equations... toward linearisation Benchmarks St-Venant 2 S t + Q x Q t + (Q2 /S) + gs H x x = 0 = gs(i J), (1) x [0; L] is the spatial variable, H(x, t) the water depth, S(x, t) the wetted area, Q(x, t) the discharge... Step 1: Apply linearisation at (H 0, Q 0 ), which are both x dependent. Step 2: Apply Laplace transformation around equilibrium. Step 3: Find solutions of h(s, x), q(s, x) and identify coefficient with boundary conditions 2 Dalmas, V., Robert, G., Poussot-Vassal, C., Pontes Duff, I. and Seren, C., "Parameter dependent irrational and infinite dimensional modelling and approximation of an open-channel dynamics", in Proceedings of the 15 th European Control Conference, (ECC 16), Aalborg, Denmark, July, /49

21 where H(s, x, Q 0 ) = [ G e(s, x, Q 0 ) ] [ ] q G s(s, x, Q 0 ) e(s) q s(s) Benchmarks St-Venant (2) G e(s, x, Q 0 ) = λ 1(s)e λ 2(s)L+λ 1 (s)x λ 2 (s)e λ 1(s)L+λ 2 (s)x B 0 s(e λ1(s)l e λ2(s)l ) G s(s, x, Q 0 ) = λ 1(s)e λ1(s)x λ 2 (s)e λ 2(s)x B 0 s(e λ1(s)l e λ2(s)l ) (3) Irrational transfer function. infinite model order. 9/49

22 Gain [db] Phase [rad] H(s, x, Q 0 ) = [ G e(s, x, Q 0 ) ] [ ] q G s(s, x, Q 0 ) e(s) q s(s) Benchmarks St-Venant (2) Frequency [Hz] Frequency [Hz] Frequency [Hz] Frequency [Hz] Experience and simulations shows I/O-delay behavior. Try to search I/O delay approximation. 9/49

23 Outlines Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 10/49

24 Delay-free H 2 model approximation problem formulation Model approximation Mathematical optimization Objectives: find a reduced order system Ĥ for which: the approximation error is small; and the stability is preserved based on a procedure computationally stable and efficient. The quality of the approximation can be evaluated using some mathematical norms. For any given system G of order N N, let find Ĥ defined by: s.t.: Ĥ := {Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) J = G Ĥ 2 is minimum optimisation problem to solve 11/49

25 Recall: G, Ĥ H2 = 1 2π Preliminaries on H 2 model approximation + G(iω)Ĥ(iω)dω Mathematical formulation Find Ĥ of order n << N which minimizes: Delay-free H 2 model approximation Ĥ := arg min Ĥ H 2 dim(ĥ) = n G Ĥ H2 (3) 12/49

26 Recall: G, Ĥ H2 = 1 2π Preliminaries on H 2 model approximation + G(iω)Ĥ(iω)dω Mathematical formulation Find Ĥ of order n << N which minimizes: Rational Interpolation: Given shift points σ 1,..., σ r C find Ĥ = (Ê, Â, ˆB, Ĉ) s.t. H(σ j ) = Ĥ(σ j ) d ds H(s) s=σ j = Delay-free H 2 model approximation Ĥ := arg min Ĥ H 2 dim(ĥ) = n G Ĥ H2 (3) d ds Ĥ(s) s=σ j 12/49

27 Preliminaries on H 2 model approximation H 2 optimality conditions 3 Let n ˆφ k Ĥ(s) = s ˆλ k k=1 delay-free H 2 -optimality conditions (SISO) If Ĥ is a local optimum of H 2 approximation problem, then Ĥ( ˆλ k ) = G( ˆλ k ) Ĥ ( ˆλ k ) = G ( ˆλ k ) (4) for k = 1,..., n. 3 S. Gugercin, A.C. Antoulas and C. Beattie, " H 2 model reduction for large-scale linear dynamical systems", SIAM Journal on matrix analysis and applications, vol. 30, no. 2, pp , /49

28 Preliminaries on H 2 model approximation H 2 optimality conditions 4 Given a system G H 2, initial shift points {σ 1,..., σ r} it=it+1 Rational interpolation (Êit, Âit, ˆB it, Ĉit) New shift points {σ 1,..., σ r} = { ˆλ 1,..., ˆλ r} no convergence? Λ (Âit, Êit) = (ˆλ 1,..., ˆλ r) reduced order system Ĥ := (Êit, Âit, ˆB it, Ĉit) Point-fixed iterative techniques: IRKA, TF-IRKA,... Rational interpolation: Krylov subspaces, Loewner framework,... 4 S. Gugercin, A.C. Antoulas and C. Beattie, " H 2 model reduction for large-scale linear dynamical systems", SIAM Journal on matrix analysis and applications, vol. 30, no. 2, pp , /49

29 Outlines Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 15/49

30 H 2 optimal model reduction with I/O delay structure Let Ĥ = (Ê, Â, ˆB, Ĉ, 0) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) (5) whose transfer function is Ĥ(s) = Ĉ(sÊ Â) 1 ˆB (6) H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ Ĥ := arg min G Ĥ H2. Ĥ H 2,dim(Ĥ) r Problem formulation and goals 16/49

31 H 2 optimal model reduction with I/O delay structure Let Ĥ = (Ê, Â, ˆB, Ĉ, 0) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) (5) whose transfer function is Ĥ(s) = Ĉ(sÊ Â) 1 ˆB (6) H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ Ĥ := arg min G Ĥ H2. Ĥ H 2,dim(Ĥ) r Problem formulation and goals Let Ĥ d = (Ê, Â, ˆB, Ĉ, ˆ i (s), ˆ o(s)) be defined as: { Êẋ(t) = Âx(t) + ˆB ˆ i (u(t)) y(t) = ˆ o(ĉx(t)) (7) whose transfer function is Ĥ d (s) = ˆ o(s)ĉ(sê Â) 1 ˆB ˆ i (s) (8) where ˆ i (s) = diag(e sˆτ 1,..., e sˆτnu ) and ˆ o(s) = diag(e sˆγ 1,..., e sˆγny ) I/O Delay H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ d = (Ê, Â, ˆB, Ĉ, ˆ i (s), ˆ o(s)) Ĥ d := arg min G Ĥ d H2. Ĥ d H 2,dim(Ĥ d ) r 16/49

32 H 2 optimal model reduction with I/O delay structure Let Ĥ = (Ê, Â, ˆB, Ĉ, 0) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) (9) Problem formulation and goals Let Ĥ d = (Ê, Â, ˆB, Ĉ, τ) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t τ) y(t) = Ĉx(t) (11) whose transfer function is Ĥ(s) = Ĉ(sÊ Â) 1 ˆB (10) whose transfer function is Ĥ d (s) = Ĉ(sÊ Â) 1 ˆBe τs (12) H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ Ĥ := arg min G Ĥ H2. Ĥ H 2,dim(Ĥ) r model ap- (SISO) Input-Delay H 2 proximation Given a system G H 2, the goal is to find a system Ĥ d = (Ê, Â, ˆB, Ĉ, τ) Ĥ d := arg min G Ĥ d H2. Ĥ d H 2,dim(Ĥ d ) r 17/49

33 H 2 optimal model reduction with I/O delay structure Input delay H 2 inner product Given a stable N-th order system G H 2, find a reduced n-th order (such that n N) stable input-delay delays model Ĥ d = (Ê, Â, ˆB, Ĉ, ˆτ) which minimizes J 2 = G Ĥ d 2 H 2. (13) Suppose that Ĥ d = Ĥe sˆτ, and both models have the pole residue decomposition Then, G(s) = N j=1 ψ j s µ j and Ĥ(s) = n ˆφ k s ˆλ k k=1 J 2 = G Ĥ d 2 H 2 = G 2 H 2 2 G, Ĥe sτ H2 + Ĥe sτ 2 H 2. (14) Compute H 2 inner product in the presence of a input delay. 18/49

34 H 2 optimal model reduction with I/O delay structure Input delay H 2 inner product Given a stable N-th order system G H 2, find a reduced n-th order (such that n N) stable input-delay delays model Ĥ d = (Ê, Â, ˆB, Ĉ, ˆτ) which minimizes J 2 = G Ĥ d 2 H 2. (13) Suppose that Ĥ d = Ĥe sˆτ, and both models have the pole residue decomposition Then, G(s) = N j=1 ψ j s µ j and Ĥ(s) = n ˆφ k s ˆλ k k=1 J 2 = G Ĥ d 2 H 2 = G 2 H 2 2 G, Ĥe sτ H2 + Ĥe sτ 2 H 2. (14) Compute H 2 inner product in the presence of a input delay. 18/49

35 Recall : G, Ĥ H2 = 1 2π First, Ĥe sτ H2? H 2 optimal model reduction with I/O delay structure + G(iω)Ĥ(iω)dω. H 2 -norm input-delay invariance Let Ĥ H 2 and τ > 0. Then Ĥe sτ H 2 and Input delay H 2 inner product Ĥ H2 = Ĥe sτ H2 Proof. 2π He sτ 2 H 2 = H(jω)e jω H(jω)e jω dω = H(jω)H(jω)dω = 2π H 2 H 2 19/49

36 Recall : G, Ĥ H2 = 1 2π First, Ĥe sτ H2? H 2 optimal model reduction with I/O delay structure + G(iω)Ĥ(iω)dω. H 2 -norm input-delay invariance Let Ĥ H 2 and τ > 0. Then Ĥe sτ H 2 and Input delay H 2 inner product Ĥ H2 = Ĥe sτ H2 Proof. 2π He sτ 2 H 2 = H(jω)e jω H(jω)e jω dω = H(jω)H(jω)dω = 2π H 2 H 2 19/49

37 Recall : G, Ĥ H2 = 1 2π First, Ĥe sτ H2? H 2 optimal model reduction with I/O delay structure + G(iω)Ĥ(iω)dω. H 2 -norm input-delay invariance Let Ĥ H 2 and τ > 0. Then Ĥe sτ H 2 and Input delay H 2 inner product Ĥ H2 = Ĥe sτ H2 Proof. 2π He sτ 2 H 2 = H(jω)e jω H(jω)e jω dω = H(jω)H(jω)dω = 2π H 2 H 2 19/49

38 H 2 optimal model reduction with I/O delay structure Recall : If both systems are real, G, Ĥ H2 = 1 2π Delay-less H 2 -inner product expression Input delay H 2 inner product + G(iω)Ĥ(iω)dω. Let G H 2 to be a strictly proper real model, φ C and ˆλ C. Then G, φ s ˆλ H 2 = G( ˆλ)φ. 20/49

39 H 2 optimal model reduction with I/O delay structure Recall : If both systems are real, G, Ĥ H2 = 1 2π Delay-less H 2 -inner product expression Input delay H 2 inner product + G(iω)Ĥ(iω)dω. Let G H 2 to be a strictly proper real model, φ C and ˆλ C. Then G, φ s ˆλ H 2 = G( ˆλ)φ. Proof Cauchy s residues theorem 20/49

40 H 2 optimal model reduction with I/O delay structure Input delay H 2 -inner product expression Let G H 2 to be a strictly proper real model expressed by Let τ > 0 and ˆλ C. Then G(s) = N j=1 G, e sτ s ˆλ H 2 = ψ j s µ j. N ψ j e µ j τ Input delay H 2 inner product. (15) ˆλ µ j j=1 The expression depends on the pole residue decomposition of G. The delay "break the structure" of G. 21/49

41 H 2 optimal model reduction with I/O delay structure Input delay H 2 -inner product expression Let G H 2 to be a strictly proper real model expressed by Let τ > 0 and ˆλ C. Then G(s) = N j=1 G, e sτ s ˆλ H 2 = ψ j s µ j. N ψ j e µ j τ Input delay H 2 inner product. (15) ˆλ µ j j=1 The expression depends on the pole residue decomposition of G. The delay "break the structure" of G. 21/49

42 H 2 optimal model reduction with I/O delay structure Input delay H 2 inner product Proof Cauchy s theorem NOT POSSIBLE!! Because of exponential growth e sτ 22/49

43 H 2 optimal model reduction with I/O delay structure Input delay H 2 inner product Proof Cauchy s theorem Use other contour encircling the poles of G 22/49

44 H 2 optimal model reduction with I/O delay structure Finally we are able to characterize the inner product. Input delay H 2 inner product computation Input delay H 2 inner product Let G, Ĥ be two SISO systems in H 2 whose respective transfer functions G(s) = N j=1 ψ j s µ j and Ĥ(s) = n k=1 ˆφ k s ˆλ k, and let τ > 0. Hence, if Ĥ d = Ĥe sτ, the inner product G, Ĥ d H2 is given by: G, Ĥ d H2 = N Ĥ( µ j )ψ j e τµ j. (16) j=1 23/49

45 H 2 optimal model reduction with I/O delay structure Finally we are able to characterize the inner product. Input delay H 2 inner product computation Input delay H 2 inner product Let G, Ĥ be two SISO systems in H 2 whose respective transfer functions G(s) = N j=1 ψ j s µ j and Ĥ(s) = n k=1 ˆφ k s ˆλ k, and let τ > 0. Hence, if Ĥ d = Ĥe sτ, the inner product G, Ĥ d H2 is given by: G, Ĥe sτ H2 = N j=1 Ĥ( µ j)ψ j e τµ j = G, Ĥ H2 (16) where N ψ j e G(s) τµ j = s µ j j=1 23/49

46 H 2 optimal model reduction with I/O delay structure Let G(s) = 1 s+1 = ψ 1 and H(s) = s µ s+2 = φ s λ. Delay-free case: Input delay H 2 inner product φ G, H H2 = ψh( µ) = ψ µ λ = 1 3 = φ ψ λ µ = φg( λ) = H, G H 2 H 2 inner product can be computed using pole-residues decomposition of G or H. 24/49

47 H 2 optimal model reduction with I/O delay structure Let G(s) = 1 s+1 = ψ 1 and H(s) = s µ s+2 = φ s λ. Delay-free case: Input delay H 2 inner product φ G, H H2 = ψh( µ) = ψ µ λ = 1 3 = φ ψ λ µ = φg( λ) = H, G H 2 H 2 inner product can be computed using pole-residues decomposition of G or H. Input-delay case: Let τ = 1. By noticing that, G, Ĥe s H2 = Ge s, Ĥ H2, one apply the symmetric version as follows : 1 3 e 1 = G, Ĥe s H2 = Ge s τ ˆλ, Ĥ H2 ˆφG( λ)e = }{{} ˆφ ψ ˆλ µ e λτ = 1 3 e2. incorrect symmetric version Symmetric version of the H 2 inner product does not provide the same result any more. 24/49

48 H 2 optimal model reduction with I/O delay structure Let G(s) = 1 s+1 = ψ 1 and H(s) = s µ s+2 = φ s λ. Delay-free case: Input delay H 2 inner product φ G, H H2 = ψh( µ) = ψ µ λ = 1 3 = φ ψ λ µ = φg( λ) = H, G H 2 H 2 inner product can be computed using pole-residues decomposition of G or H. Input-delay case: Let us compute H 2 inner product between He s and G using the extended formula: G, He sτ H2 = ψh( µ)e τµ φ = ψ µ λ eτµ = 1 3 e 1. Which modifies the optimality conditions. 24/49

49 Recall: G(s) = N k=1 H 2 optimal model reduction with I/O delay structure ψ k s µ k and Ĥ(s) = n k=1 delay-free H 2 -optimality conditions Input delay H 2 Optimality conditions ˆφ k s ˆλ k If Ĥ is a local optimum of H 2 problem, then (interpolation condition on G). for k = 1,..., n. Ĥ( ˆλ k ) = G( ˆλ k ) Ĥ ( ˆλ k ) = G ( ˆλ k ) (17) 25/49

50 G(s) = N k=1 H 2 optimal model reduction with I/O delay structure ψ k s µ k and Ĥ(s) = n k=1 Input-delay H 2 -optimality conditions ˆφ k s ˆλ k 5 Input delay H 2 Optimality conditions If Ĥ d = Ĥe sˆτ is a local optimum of input delay H 2 problem, then (interpolation condition on G) Ĥ( ˆλ k ) = G( ˆλ k ) Ĥ ( ˆλ k ) = G ( ˆλ k ) (18) for k = 1,..., n where G(s) is given by G(s) = N k=1 ψ k e µ kτ s µ k and (delay condition) N ( n ) φ j µ k ψ k µ k + ˆλ e µ kτ = 0. (19) j k=1 j=1 5 I. Pontes Duff Pereira, C. Poussot-Vassal and C. Seren, "Optimal H 2 model approximation based on multiple input/output delays systems", arxiv preprint arxiv: /49

51 G(s) = N k=1 H 2 optimal model reduction with I/O delay structure ψ k s µ k and Ĥ(s) = n k=1 Input-delay H 2 -optimality conditions ˆφ k s ˆλ k 5 Input delay H 2 Optimality conditions If Ĥ d = Ĥe sˆτ is a local optimum of input delay H 2 problem, then (interpolation condition on G) Ĥ( ˆλ k ) = G( ˆλ k ) Ĥ ( ˆλ k ) = G ( ˆλ k ) (18) for k = 1,..., n where G(s) is given by G(s) = N k=1 ψ k e µ kτ s µ k and (delay condition) N ( n ) φ j µ k ψ k µ k + ˆλ e µ kτ = 0. (19) j k=1 j=1 5 I. Pontes Duff Pereira, C. Poussot-Vassal and C. Seren, "Optimal H 2 model approximation based on multiple input/output delays systems", arxiv preprint arxiv: /49

52 H 2 optimal model reduction with I/O delay structure IO-dIRKA Algorithm IO-dIRKA Given G, compute pole residue decomposition. Then, initial shift points {σ 1,..., σ r} and τ > 0 it=it+1 IRKA for fixed τ (Êit, Âit, ˆB it, Ĉit) satisfying condition on G it convergence? reduced order system Ĥ := (Êit, Âit, ˆB it, Ĉit, τ it ) no For τ it fixed construct G it For fixed Ĥ it Optimize τ 27/49

53 Amplitude Mismatch error H 2 optimal model reduction with I/O delay structure IO-dIRKA Algorithm ψ Take G delay (s) = s 2 + 2ξω 0 s + ω0 2 e τs, where τ = 2, ω 0 = 1 and ξ = 1/4. Loewner framework for uniformly spaced interpolation points iω k, k = 1,..., 100 G = Ĉ(sÊ Â) 1 ˆB of order N = 34, a delay-free model interpolating G delay. IRKA for n = 2,..., 8; I/O IRKA for n = Impulse response G, N = 34 ^Hd, n = 2, = = 2 (IO-dIRKA) ^H, n = 2 (IRKA) 10 0 Impulse response error ^Hd, n = 2, = = 2 with mean error " = e-06 (IO-dIRKA) ^H, n = 2 with mean error " = e-02 (IRKA) ^H, n = 3 with mean error " = e-03 (IRKA) ^H, n = 3 (IRKA) ^H, n = 4 with mean error " = e-04 (IRKA) 0.4 ^H, n = 4 (IRKA) ^H, n = 5 (IRKA) ^H, n = 6 (IRKA) 10-2 ^H, n = 5 with mean error " = e-04 (IRKA) ^H, n = 6 with mean error " = e-05 (IRKA) ^H, n = 7 with mean error " = e-05 (IRKA) ^H, n = 7 (IRKA) ^H, n = 8 with mean error " = e-05 (IRKA) 0.2 ^H, n = 8 (IRKA) Time (s) Time (s) 28/49

54 H 2 optimal model reduction with I/O delay structure Ladder network benchmark Impulse response 0.6 Ladder Network Amplitude (Volts) Time (ms) Original model of order =100. H 2 optimal delay-free approximations n = 6, 12 and /49

55 H 2 optimal model reduction with I/O delay structure Ladder network benchmark Impulse response Order = Amplitude (Volts) Time (ms) Original model of order =100. H 2 optimal delay-free approximations n = 6, 12 and /49

56 H 2 optimal model reduction with I/O delay structure Ladder network benchmark Impulse response Order = Amplitude (Volts) Time (ms) Original model of order =100. H 2 optimal delay-free approximations n = 6, 12 and /49

57 H 2 optimal model reduction with I/O delay structure Ladder network benchmark Impulse response Order = Amplitude (Volts) Time (ms) Original model of order =100. H 2 optimal delay-free approximations n = 6, 12 and /49

58 H 2 optimal model reduction with I/O delay structure Ladder network benchmark Impulse response Ladder system G, N = 100 Ĥ2 0 (IRKA, n = 20) Ĥd (IO-dIRKA, n = 6) 0.2 Amplitude (Volts) Time (ms) Original model of order =100. H 2 optimal delay-free approximations n = 20. H 2 optimal input-delay approximations n = 6, τ opt = 19.27s. 30/49

59 H 2 optimal model reduction with I/O delay structure EDF Rhin flow benchmark Irrational model filtered. Loewner exact interpolation n = 103 (filtered and stable). 31/49

60 H 2 optimal model reduction with I/O delay structure EDF Rhin flow benchmark Impulse response of the models. This example illustrates the benefit of delay structured reuced order models for specific transport phenomena. 32/49

61 Outlines Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 33/49

62 H 2 optimality conditions for reduced state-delay systems Let Ĥ = (Ê, Â, ˆB, Ĉ, 0) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) (20) whose transfer function is Ĥ(s) = Ĉ(sÊ Â) 1 ˆB (21) H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ Ĥ := arg min G Ĥ H2. Ĥ H 2,dim(Ĥ) r Problem formulation and goals 34/49

63 H 2 optimality conditions for reduced state-delay systems Let Ĥ = (Ê, Â, ˆB, Ĉ, 0) be defined as: { Êẋ(t) = Âx(t) + ˆBu(t) y(t) = Ĉx(t) (20) Problem formulation and goals Let Ĥ d = (Ê, Â, ˆB, Ĉ, τ) be defined as: { Êẋ(t) = Âx(t τ) + ˆBu(t) y(t) = Ĉx(t) (22) whose transfer function is Ĥ(s) = Ĉ(sÊ Â) 1 ˆB (21) H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ Ĥ := arg min G Ĥ H2. Ĥ H 2,dim(Ĥ) r whose transfer function is Ĥ d (s) = Ĉ(sÊ Âe τs ) 1 ˆB (23) Single-state delay H 2 model approximation Given a system G H 2, the goal is to find a system Ĥ d = (Ê, Â, ˆB, Ĉ, τ) Ĥ d := arg min G Ĥ d H2. Ĥ d H 2,dim(Ĥ d ) r 34/49

64 H 2 optimality conditions for reduced state-delay systems One dimension single-state delay problem Problem formulation and goals Given G H 2 and a fixed τ > 0, find a reduced order single state delay model order 1, { ẋ(t) = ˆαx(t τ) + ˆφu(t) Ĥ d := y(t) = x(t) whose transfer function is given by such that: Ĥ d (s) = ˆφ s ˆαe sτ H 2. (24) ˆφ G Ĥ d H2 = min G ( ˆφ, ˆα) R 2 s ˆαe sτ H 2 (25) Reduced order system is defined only by two real parameters. It has infinitely many poles. What are the optimality conditions? Are they interpolation conditions? 35/49

65 H 2 optimality conditions for reduced state-delay systems Problem formulation and goals Recall: if G & Ĥ have semi-simple poles, they read s.t.: n ˆφ i Ĥ(s) = s ˆλ (26) i i=1 H 2 -optimality conditions If both G and Ĥ are H 2 and Ĥ is a local minimum of the H 2 approximation problem, then the following interpolation equalities hold: { G( ˆλ i ) = Ĥ( ˆλ i ) G ( ˆλ i ) = Ĥ, i = 1... r (27) ( ˆλ i ) In Equation (27), ˆλ i corresponds to the i th pole of Ĥ. What are the poles of Ĥ d? From now on, the model Ĥ d will be decomposed as Ĥ d (s) = ˆφP τ (s) = ˆφ s ˆαe sτ 36/49

66 H 2 optimality conditions for reduced state-delay systems Spectral decomposition of Ĥ d (s) 6 The Lambert function W k The Lambert function W k (s) is a multivalued (except at 0) complex function associating for the k th complex branch, a complex number W k (s) such that : s = W k (s)e W k(s), k Z (28) i.e., given a s C, for each complex branch. Equation (28) has one solution in the k-th complex branch, namely W k (s). 6 Corless, Robert M., et al., "On the LambertW function", Advances in Computational mathematics 5.1 (1996): /49

67 H 2 optimality conditions for reduced state-delay systems Spectral decomposition of Ĥ d (s) 6 The Lambert function W k The Lambert function W k (s) is a multivalued (except at 0) complex function associating for the k th complex branch, a complex number W k (s) such that : s = W k (s)e W k(s), k Z (28) i.e., given a s C, for each complex branch. Equation (28) has one solution in the k-th complex branch, namely W k (s). Example Poles of P(s) = 1 s+e s : λ k + e λ k = 0 λ k e λ k = 1 λ k = W k ( 1) for k Z Poles of P(s). 6 Corless, Robert M., et al., "On the LambertW function", Advances in Computational mathematics 5.1 (1996): /49

68 H 2 optimality conditions for reduced state-delay systems Spectral decomposition of Ĥ d (s) Spectral decomposition of H d The model Ĥ d has infinite poles which can be computed using the Lambert function as follows : λ k = 1 τ W k(τ ˆα), for k Z. (29) Moreover, if Ĥ d = ˆφP τ, the infinite partial fraction decomposition of P τ = is given by 1 P τ (s) = φ k where φ k = s λ k k= 1 s ˆαe sτ τλ k. (30) H convergence 7, and H 2 -weak convergence. 7 Partington, JR and Glover, K and Zwart, HJ and Curtain, Ruth F, "L approximation and nuclearity of delay systems", Systems & control letters 10,1 (1988): /49

69 H 2 optimality conditions for reduced state-delay systems Spectral decomposition of P 2 τ. Let P τ = 1 s ˆαe sτ H 2. Then Spectral decomposition of Ĥ d (s) P 2 τ (s) = 1 ψ k (s λ k ) 2 + ρ 1 k s λ k (31) k= where ψ k = 1 (1 + τλ k ) 2 and ρ k = 2τ 2 λ k (1 + τλ k ) 3. 39/49

70 H 2 optimality conditions for reduced state-delay systems Spectral H 2-inner product : Spectral H 2 -inner product : Let F H 2 and P τ = 1 s ˆαe sτ H 2. Then : F, P τ H2 = k= φ k F( λ k ), (32) where φ k = 1 1+τλ k. In addition, F, P τ 2 H2 = k= ρ k F( λ k ) ψ k F ( λ k ) (33) where ψ k = 1 (1+τλ k ) 2 and ρ k = 2τ 2 λ k (1+τλ k ) 3. 40/49

71 H 2 optimality conditions for reduced state-delay systems Let E( ˆφ, ˆα) be the H 2 error, i.e., Single-state delay H 2-optimality conditions: E( ˆφ, ˆα) = G Ĥ d 2 H 2 = G Ĥ d, G Ĥ d H2 Partial derivatives: The partial derivative of the H 2 analytically by : error E with respect to the parameters are given { E ˆφ = 2 G Ĥ d, P τ H2 E = 2 ˆφ ˆα ˆατ G Ĥ d, P τ + P 2 τ H2 Proof Sketch: E Θ = 2 G Ĥ d, Ĥ d Θ H 2. (34) 41/49

72 H 2 optimality conditions for reduced state-delay systems Single-state delay H 2-optimality conditions: Single-state delay H 2 -optimality conditions version 1 ˆφ Let Ĥ d = s ˆαe sτ H 2 and G H 2. Let us suppose also that G H 2. If Ĥ d is the best H 2 approximation of G, then : G, P τ H2 = Ĥ d, P τ H2 (35) G, P τ + P 2 τ H2 = Ĥ d, P τ + P 2 τ H2 (36) 42/49

73 H 2 optimality conditions for reduced state-delay systems Single-state delay H 2-optimality conditions: 8 Single-state delay H 2 -optimality conditions version 2 ˆφ Let Ĥ d = s ˆαe sτ H 2 and G H 2. Let us suppose also that G H 2. If Ĥ d is the best H 2 approximation of G, then : k= G( λ k )φ k = k= Ĥ d ( λ k )φ k, (37) k= G ( λ k )(φ k ψ k ) + k= G( λ k)ρ k = k= Ĥ d ( λ k)(φ k ψ k ) + k= Ĥd( λ k )ρ k (38) where λ k, for k Z, are the poles of Ĥ d, φ k = 2τ 2 λ k (1+τλ k ) τλ k, ψ k = 1 (1+τλ k ) 2 and ρ k = Generalized interpolation conditions. if τ = 0, G( ˆα) = Ĥ( ˆα) and G ( ˆα) = Ĥ ( ˆα). 8 Pontes Duff, I., Gugercin, S., Beattie, C., Poussot-Vassal, C. and Seren, C., "H 2 -optimality conditions for reduced time-delay systems of dimension one", in Proceedings of the 13th IFAC Workshop on Time Delay Systems, /49

74 Let H 2 optimality conditions for reduced state-delay systems G(s) = 10 s s Find ˆφ R and ˆα ( π/2, 0) which minimizes : Application where H d (s) = ˆφ s ˆαe s. E( ˆφ, ˆα) = G H d 2 H 2 = E( ˆφ, ˆα) 2 H 2 44/49

75 Hence, where B = H 2 optimality conditions for reduced state-delay systems { E( ˆφ, ẋ(t) = Ax(t) + Aτ x(t τ) + Bu(t) ˆα) := y(t) = Cx(t) A = [ ] 10/9 [ ] 10/9 ˆφ and C = [ 1 ] , A τ = Use delay Lyapunov equations 9 to compute the Norm [ 0 ] ˆα MATLAB function fminunc ˆα ˆφ Verify generalized interpolation condition by truncation. Application 9 Jarlebring, E., Vanbiervliet, J., and Michiels, W., "Characterizing and computing the H2 norm of timedelay systems by solving the delay lyapunov equation.", Automatic Control, IEEE Transactions on, 56(4), /49

76 H 2 optimality conditions for reduced state-delay systems Application N S 1,G,N S 1,Ĥd,N S 2,G,N S 2,Ĥd,N S 1,G,N = N 1 G( λ k )φ k N 1 Ĥ d ( λ k )φ k = S 1,Ĥd,N k= N k= N S 2,G,N = S 2,Ĥd,N = N 1 k= N G ( λ k )(φ k ψ k ) + N 1 k= N N 1 k= N G( λ k )ρ k Ĥ N 1 d ( λ k )(φ k ψ k ) + Ĥ d ( λ k )ρ k. k= N Error H φ α /49

77 Outlines Introduction Preliminaries on H 2 model approximation H 2 optimal model reduction with I/O delay structure H 2 optimality conditions for reduced state-delay systems Conclusions and perspectives 47/49

78 Conclusions and perspectives Scientific contributions H 2 optimal model reduction with I/O structure : I/O H 2 inner product expression, interpolation conditions on G, I/O IRKA algorithm, application to industrial benchmark. Single-state delay H 2 -optimality conditions :H 2 spectral inner product expression, generalized interpolation conditions. Future work I/O delay 1)Isometry structure (In progress - conjoint work with Christoph Zimmer -TU Berlin) state-delay 1) Generalize interpolation conditions to general framework. (In progress (conjoint work with S. Gugercin and C. Beattie - VT)) 2) Find a way to deal with them. 48/49

79 H 2 optimal model approximation by structured time-delay reduced order models I. Pontes Duff, Charles Poussot-Vassal, C. Seren 27th September, GT MOSAR and GT SAR meeting /49