Summer School on Numerical Linear Algebra for Dynamical and
High-Dimensional Problems
Trogir
October 10-15, 2011
Solution of Computational Problems for
Descriptor Systems
Matthias Voigt
Computational Methods in Systems and Control Theory
Max Planck Institute for Dynamics of Complex Technical Systems
Magdeburg, Germany
MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
NETWORK THEORY
Max Planck Institute Magdeburg
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Descriptor Systems
Given: LTI descriptor system
Σ:
(
E ẋ(t) = Ax(t) + Bu(t),
y (t) = Cx(t) + Du(t),
E , A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m ,
descriptor vector x(t) ∈ Rn , input vector u(t) ∈ Rm , output vector
y (t) ∈ Rp .
Assumptions: λE − A is regular, i.e. det(λE − A) 6≡ 0.
Transfer function
G (s) = C (sE − A)−1 B + D
Max Planck Institute Magdeburg
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Additional Structures
Bounded realness
G has no poles with nonnegative real parts,
I − G (iω)G H (iω) < 0 for all values ω ∈ R.
Positive realness
G has no poles with positive real parts,
G (iω) + G H (iω) < 0 for any iω that is not a pole of G with ω ∈ R,
if iω or ∞ is a pole of G , then it is simple and the relevant residue
matrix is positive semidefinite Hermitian.
General frequency domain inequalities (FDIs)
H Q
(iωE − A)−1 B
I
ST
Max Planck Institute Magdeburg
S
R
(iωE − A)−1 B
< 0, Q = Q T , R = R T .
I
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Connection to Linear Matrix Inequalities
Bounded real lemma
Under certain conditions: G is bounded real if and only if the LMI
T
A X + XTA + CTC XTB + CTD
4 0, E T X = X T E < 0 is feasible.
BT X + DT C
DT D − I
Positive real lemma
Under certain conditions: G is positive real if and only if the LMI
T
A X + XTA XTB − CT
4 0, E T X = X T E < 0 is feasible.
BT X − C
−D T − D
Kalman-Yakubovič-Popov lemma
Under certain conditions: FDI holds for all iω if and only if the LMI
T
A X + XTA + Q XTB + S
< 0, E T X = X T E is feasible.
BT X + ST
R
Max Planck Institute Magdeburg
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Problems under Consideration
Weakening equivalence conditions
The conditions which are needed to state equivalence are quite strong at
the moment =⇒ weaking this conditions to make it more practical.
Passivity enforcement
Bounded realness/positive realness/FDI are natural properties of
real-world systems. Often these are lost due to modeling
errors/approximation errors =⇒ restore this by introducing small errors
to system’s matrices (can be done by perturbation of purely imaginary
eigenvalues of skew-Hamiltonian/Hamiltonian matrix pencils (next
slide)).
Max Planck Institute Magdeburg
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Skew-Hamiltonian/Hamiltonian Matrix Pencils
Properties
F G
R
block structure: λS − H = λ
−
H FT
T
skew-Hermitian G , H, and Hermitian S, T ,
S
−R T
with
Hamiltonian eigensymmetry (symmetry
with respect to imaginary axis (and
the real axis in the real case)),
pencils λS̃− H̃ :=J P T J T (λS − H) P
0 I
with J =
are again sH/H,
−I 0
structured Schur form:
S1 S2
H
T T
J Q J (λS − H) Q = λ
− 1
0 S1T
0
H2
,
−H1T
with orthogonal Q, upper triangular S1 , and H1 ; however existence
cannot be guaranteed, can be solved by structured embedding.
Max Planck Institute Magdeburg
Matthias Voigt, Solution of Computational Problems for Descriptor Systems
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Problems under Consideration
Structure-preserving balancing
Goal: find a simple J -congruence transform such that:
in the transformed pencil a maximum number of eigenvalues has
been isolated (i.e., part of the pencil is in structured Schur form),
in the remainder, the rows and columns of the pencil are as close in
norm as possible (to improve numerical accuracy).
Perturbation theory for purely imaginary eigenvalues
For many applications the interesting eigenvalues of
skew-Hamiltonian/Hamiltonian pencils are the purely imaginary ones.
Question: How to move these eigenvalues off the imaginary axis by
structured perturbations?
might be possible by arbitrarily small perturbations if some of the
purely imaginary eigenvalues are not simple,
depends on the skew-Hamiltonian/Hamiltonian Kronecker canonical
form.
Max Planck Institute Magdeburg
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