ml_ct_dss_lqgbt

LQG balanced truncation for descriptor systems.

Contents

Syntax

[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_lqgbt(A, B, C, D, E)
[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_lqgbt(A, B, C, D, E, opts)
[rom, info] = ml_ct_dss_lqgbt(sys)
[rom, info] = ml_ct_dss_lqgbt(sys, opts)

Description

This function computes the generalized linear-quadratic Gaussian
balanced truncation for a descriptor system of the form
    E*x'(t) = A*x(t) + B*u(t),                                      (1)
       y(t) = C*x(t) + D*u(t).                                      (2)
Therefore, first an additive decomposition of the system is performed
using the matrix disk function, such that
         [ Ei  0 ]       [ Ai  0 ]                        [ Ci ]
    E2 = [       ], A2 = [       ], B2 = [ Bi, Bp ], C2 = [    ],
         [ 0  Ep ]       [ 0  Ap ]                        [ Cp ]
with (Ei, Ai, Bi, Ci, D) belonging to the polynomial part and
(Ep, Ap, Bp, Cp, 0) belonging to the strictly proper part.
Now, the two generalized continuous-time Riccati equations
    Ap*Pp*Ep' + Ep*Pp*Ap' + Bp*Bp'
        - (Ep*Pp*Cp' + B*M') * inv(Rb) * (Ep*Pp*Cp' + B*M')' = 0,
    Ap'*Qp*Ep + Ep'*Qp*Ap + Cp'*Cp
        - (Bp'*Qp*Ep + M'*C)' * inv(Rc) * (Bp'*Qp*Ep + M'*C) = 0
are solved for the reduction of the strictly proper part, with
    Rb = I + M*M',
    Rc = I + M'*M,
where M = D - Ci * inv(Ai) * Bi. Also, the two generalized
discrete-time Lyapunov equations
    Ai*Pi*Ai' - Ei*Pi*Ei' - Bi*Bi' = 0,
    Ai'*Qi*Ai - Ei'*Qi*Ei - Ci'*Ci = 0
are solved for the reduction of the polynomial part. As result, a
reduced-order system of the form
    Er*x'(t) = Ar*x(t) + Br*u(t),                                   (3)
        y(t) = Cr*x(t) + Dr*u(t)                                    (4)
is computed, such that for the original transfer function G with the
(right) coprime factorization G = N*inv(M) and the reduced-order
transfer function Gr with an r-th order strictly proper part and the
(right) coprime factorization Gr = Nr*inv(Mr) it holds
    ||[N; M] - [Nr; Mr]||_{\infty} <= 2 * (Hsvp(r+1)/sqrt(1
        + Hsvp(r+1)^2) + ... + Hsvp(n)/sqrt(1 + Hsvp(n)^2)),
with Hsvp the vector containing the characteristic LQG singular values
of the system.

Input

A    - matrix from (1) with dimensions n x n
B    - matrix from (1) with dimensions n x m
C    - matrix from (2) with dimensions p x n
D    - matrix from (2) with dimensions p x m
sys  - structure or state-space object, containing the descriptor
       system's matrices:

Entry
Meaning
A
matrix from (1) with dimensions n x n
B
matrix from (1) with dimensions n x m
C
matrix from (2) with dimensions p x n
D
matrix from (2) with dimensions p x m
E
matrix from (1) with dimensions n x n

opts - structure, containing the following optional entries:

Parameter
Meaning
Beta
nonnegative scalar, used as shift of the in Bass' algorithm for better conditioning if StabMethod == 'lyap' is chosen only used if RiccatiSolver = 'newton'
default: 0.1
caredlopts
structure, containing the optional parameters for the Riccati equation sign function solver, only used if RiccatiSolver = 'sign', see ml_caredl_sgn
default: struct()
careopts
structure, containing the optional parameters for the computation of the continuous-time algebraic Riccati equations, only used if RiccatiSolver = 'newton', see ml_care_nwt_fac
default: struct()
DecompEig
positive scalar, overestimation of the absolute value of the largest finite eigenvalue of s*E - A, if set, replaces the computation with DecompTol
default: []
DecompTol
nonnegative scalar, tolerance multiplied with the largest singular value of E to determine the smallest non-quasi-zero singular value of E
default: log(n)*eps
gdlyapopts
structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyap_smith_fac
default: struct()
ImproperTrunc
nonnegative scalar, tolerance multiplied with the largest proper Hankel singular value of the system to truncate the improper part
default: log(n)*eps
Index
nonnegative integer, index of the descriptor system used to set an upper bound on the size of the reduced improper part, Inf if unknown
default: Inf
infdecopts
structure, containing the optional parameters for the decomposition of the finite and infinite parts of the system using the disk function and subspace extraction method, see ml_disk and ml_getqz
default: struct()
Method
character array, determining algorithm for the computation of the reduced-order model
  • 'sr' - square-root method
  • 'bfsr' - balancing-free square-root method
default: 'sr'
Order
positive integer, order of the resulting reduced-order model chosen by the user if 'order' is set for OrderComputation
default: min(10,length(Hsv)) + Nu + Ni
OrderComputation
character array, determining the method for the computation of the size of the reduced-order model
  • 'order' - take explicit order
  • 'tolerance' - using absolute error bound
default: 'tolerance'
RiccatiSolver
character array, determining the solver for the dual Riccati equations
  • 'newton' - Newton iteration
  • 'sign' - dual sign function method
default: 'sign'
stabdecopts
structure, containing the optional parameters for the decomposition of the stable and unstable parts of the system using the disk function and subspace extraction method, , only used if RiccatiSolver = 'newton'see ml_disk and ml_getqz
default: struct()
StabMethod
character array, determining the method of stabilization of the system, only used if RiccatiSolver = 'newton'
  • 'abe' - partial stabilization with the algebraic Bernoulli equation
  • 'lyap' - partial stabilization with Bass' algorithm
default: 'abe'
stabmethodopts
structure, containing the optional parameters for the sign function based Lyapunov or Bernoulli equation solver used for the stabilization, only used if RiccatiSolver = 'newton', see ml_abe_sgn or ml_lyap_sgn
default: struct()
Tolerance
nonnegative scalar, tolerance used for the computation of the size of the reduced-order model by an absolute error bound if 'tolerance' is set for OrderComputation
default: 1.0e-02

Output

Ar   - matrix of (3) with dimensions r x r
Br   - matrix of (3) with dimensions r x m
Cr   - matrix of (4) with dimensions p x r
Dr   - matrix of (4) with dimensions p x m
Er   - matrix of (3) with dimensions r x r
rom  - structure or state-space object, containing the reduced-order
       descriptor system:

Entry
Meaning
A
matrix from (3) with dimensions r x r
B
matrix from (3) with dimensions r x m
C
matrix from (4) with dimensions p x r
D
matrix from (4) with dimensions p x m
E
matrix from (3) with dimensions r x r

info - structure, containing the following information:

Entry
Meaning
AbsErrBound
computed error bound for the absolute error of the (right) coprime factorization in H-infinity norm
Hsvi
a vector, containing the computed Hankel singular values of the improper part of the system
Hsvp
a vector, containing the computed characteristic LQG singular values of the proper part of the system
infoCARE_C
structure, containing information about the Newton solver for the regulator Riccati equation, see ml_care_nwt_fac
infoCARE_O
structure, containing information about the Newton solver for the filter Riccati equation, see ml_care_nwt_fac
infoCAREDL
structure, containing information about the sign function solver for the dual Riccati equations, see ml_caredl_sgn
infoGADTF
structure, containing information about the additive decomposition of the system into its infinite, finite stable and finite anti-stable parts, see ml_ct_dss_adtf
infoGDLYAP_C
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper controllability Gramian, see ml_gdlyap_smith_fac
infoGDLYAP_O
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper observability Gramian, see ml_gdlyap_smith_fac
infoPARTSTAB_C
structure, containing information about the partial stabilization used for the controllability Riccati equation, see ml_ct_dss_partstab
infoPARTSTAB_O
structure, containing information about the partial stabilization used for the observability Riccati equation, see ml_ct_dss_partstab
Nf
Dimension of the finite reduced part in the reduced-order model
Ni
Dimension of the improper part in the reduced- order model

Reference

P. Benner, T. Stykel, Model order reduction for differential-algebraic equations: A survey, in: A. Ilchmann, T. Reis (Eds.), Surveys in Differential-Algebraic Equations IV, Differential-Algebraic Equations Forum, Springer International Publishing, Cham, 2017, pp. 107--160. https://doi.org/10.1007/978-3-319-46618-7_3

See Also

ml_ct_ss_lqgbt | ml_morlabopts