Page 375 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 375
IONAL APPROXIMATION FOR DATA-DRIVEN MODELING AND COMPLEXITY
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
Structure-Preserving Interpolation for Bilinear Systems
Steffen W. R. Werner, werner@mpi-magdeburg.mpg.de
Max Planck Institute for Dynamics of Complex Technical Systems, Germany
Coauthors: Peter Benner, Serkan Gugercin
The modeling of natural processes as population growth, mechanical structures and fluid dy-
namics, or stochastic modeling often results in bilinear time-invariant dynamical systems
k (1)
Ex˙ (t) = Ax(t) + Njx(t)uj(t) + Bu(t),
j=1
y(t) = Cx(t),
with E, A, Nj ∈ Rn×n, for j = 1, . . . , m, B ∈ Rn×m and C ∈ Rp×n. The aim of model
reduction for (1) is the reduction of related computational resources, like time and memory for
the simulation of (1), by the reduction of internal states n, while approximating the input-to-
output behavior of the system. Often related to the underlying applications, bilinear systems (1)
can have special structures that one wants to preserve in the reduced-order model as, e.g., in
case of bilinear mechanical systems
mm (2)
M q¨(t) + Dq˙(t) + Kq(t) = Np,jq(t)uj(t) + Nv,jq˙(t)uj(t) + Buu(t),
j=1 j=1
y(t) = Cpq(t) + Cvq˙(t),
with M, D, K, Np,j, Nv,j ∈ Rn×n, for j = 1, . . . , m, Bu ∈ Rn×m and Cp, Cv ∈ Rp×n.
In case of linear systems, structured-preserving interpolation of the underlying transfer func-
tion in the frequency domain can be used to efficiently construct reduced-order models with the
same structure as the original system [1].
We present an extension of the structure-preserving interpolation framework to the bilin-
ear system case, which we describe in the frequency domain by general multivariate transfer
functions
k−1
Gk(s1, . . . , sk) = C(sk)K(sk)−1 (Imj−1 ⊗ N (sk−j))(Imj ⊗ K(sk−j)−1) (3)
j=1
× (Imk−1 ⊗ B(s1)),
for k ≥ 1 and where N (s) = N1(s) . . . Nm(s) with the matrix functions C : C → Cp×n,
K : C → Cn×n, B : C → Cn×m, Nj : C → Cn×n for j = 1, . . . , m. We develop a projection-
based, structure-preserving interpolation framework for bilinear systems associated with (3)
that allows the efficient construction of reduced-order structured bilinear systems.
References
[1] C. A. Beattie and S. Gugercin. Interpolatory projection methods for structure-
preserving model reduction. Syst. Control Lett., 58(3):225–232, 2009.
doi:10.1016/j.sysconle.2008.10.016.
373
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
Structure-Preserving Interpolation for Bilinear Systems
Steffen W. R. Werner, werner@mpi-magdeburg.mpg.de
Max Planck Institute for Dynamics of Complex Technical Systems, Germany
Coauthors: Peter Benner, Serkan Gugercin
The modeling of natural processes as population growth, mechanical structures and fluid dy-
namics, or stochastic modeling often results in bilinear time-invariant dynamical systems
k (1)
Ex˙ (t) = Ax(t) + Njx(t)uj(t) + Bu(t),
j=1
y(t) = Cx(t),
with E, A, Nj ∈ Rn×n, for j = 1, . . . , m, B ∈ Rn×m and C ∈ Rp×n. The aim of model
reduction for (1) is the reduction of related computational resources, like time and memory for
the simulation of (1), by the reduction of internal states n, while approximating the input-to-
output behavior of the system. Often related to the underlying applications, bilinear systems (1)
can have special structures that one wants to preserve in the reduced-order model as, e.g., in
case of bilinear mechanical systems
mm (2)
M q¨(t) + Dq˙(t) + Kq(t) = Np,jq(t)uj(t) + Nv,jq˙(t)uj(t) + Buu(t),
j=1 j=1
y(t) = Cpq(t) + Cvq˙(t),
with M, D, K, Np,j, Nv,j ∈ Rn×n, for j = 1, . . . , m, Bu ∈ Rn×m and Cp, Cv ∈ Rp×n.
In case of linear systems, structured-preserving interpolation of the underlying transfer func-
tion in the frequency domain can be used to efficiently construct reduced-order models with the
same structure as the original system [1].
We present an extension of the structure-preserving interpolation framework to the bilin-
ear system case, which we describe in the frequency domain by general multivariate transfer
functions
k−1
Gk(s1, . . . , sk) = C(sk)K(sk)−1 (Imj−1 ⊗ N (sk−j))(Imj ⊗ K(sk−j)−1) (3)
j=1
× (Imk−1 ⊗ B(s1)),
for k ≥ 1 and where N (s) = N1(s) . . . Nm(s) with the matrix functions C : C → Cp×n,
K : C → Cn×n, B : C → Cn×m, Nj : C → Cn×n for j = 1, . . . , m. We develop a projection-
based, structure-preserving interpolation framework for bilinear systems associated with (3)
that allows the efficient construction of reduced-order structured bilinear systems.
References
[1] C. A. Beattie and S. Gugercin. Interpolatory projection methods for structure-
preserving model reduction. Syst. Control Lett., 58(3):225–232, 2009.
doi:10.1016/j.sysconle.2008.10.016.
373
