ml_ct_ss_lqgbt

LQG balanced truncation for standard systems.

Contents

Syntax

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

Description

This function computes the linear-quadratic Gaussian balanced
truncation for a standard system of the form
    x'(t) = A*x(t) + B*u(t),                                        (1)
     y(t) = C*x(t) + D*u(t).                                        (2)
Therefor, the two algebraic Riccati equations
    A*P  + P*A' + B*B' - (P*C' + B*D') * inv(Rb) * (P*C' + B*D')' = 0,
    A'*Q + Q*A  + C'*C - (B'*Q + D'*C)' * inv(Rc) * (B'*Q + D'*C) = 0,
are solved for the Gramians P and Q, with
    Rb = I + D*D',
    Rc = I + D'*D.
As result, a reduced-order system of the form
    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 r-th order transfer
function Gr with the (right) coprime factorization Gr = Nr*inv(Mr) it
holds
    ||[N; M] - [Nr; Mr]||_{\infty} <= 2 * (Hsv(r+1)/sqrt(1
        + Hsv(r+1)^2) + ... + Hsv(n)/sqrt(1 + Hsv(n)^2)),
with Hsv 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 standard
       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

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()
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
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'
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()
stabsignmopts
structure, containing the optional parameters for the matrix sign function used for the decomposition into stable and anti-stable system parts, only used if RiccatiSolver = 'newton', see ml_signm
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
UnstabDim
integer, dimension of the deflating anti-stable subspace in the control and filter Riccati equations, only used if RiccatiSolver = 'newton', negative if unknown
default: -1

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
rom  - structure or state-space object, with the following entries:

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

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
Hsv
a vector, containing the computed characteristic LQG singular values
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
infoPARTSTAB_C
structure, containing information about the partial stabilization used for the controllability Riccati equation, see ml_ct_ss_partstab
infoPARTSTAB_O
structure, containing information about the partial stabilization used for the observability Riccati equation, see ml_ct_ss_partstab
N
Dimension of 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_dss_lqgbt | ml_morlabopts