ml_dt_dss_lqgbt

LQG balanced truncation for discrete-time desc. systems.

Contents

Syntax

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

Description

This function computes the generalized linear-quadratic Gaussian balanced truncation for a discrete-time descriptor system of the form

   E*x(t+1) = 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 discrete-time Riccati equations

   Ap*Pp*Ap' - Ep*Pp*Ep' + Bp*Bp' - (Ap*Pp*Cp' + Bp*M')
       * inv(Cp*Pp*Cp' + Rb) * (Ap*Pp*Cp' + Bp*M')' = 0,
   Ap'*Qp*Ap - Ep*Qp*Ep' + Cp'*Cp - (Bp'*Qp*Ap + M'*Cp)'
       * inv(Bp*Qp*Bp' + Rc) * (Bp'*Qp*Ap + M'*Cp) = 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+1) = 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

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

Parameter
Meaning
Beta
nonnegative scalar, used as shift of the in Bass' algorithm for better conditioning
default: 0.1
daredlopts
structure, containing the optional parameters for the Riccati equation structure-preserving doubling solver, only used if RiccatiSolver = 'sda', see ml_daredl_sda_fac
default: struct()
dareopts
structure, containing the optional parameters for the computation of the continuous-time algebraic Riccati equations, only used if RiccatiSolver = 'newton', see ml_dare_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
gdlyapdlopts
structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyapdl_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(Hsvp)) + 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
  • 'sda' - dual structure-preserving doubling
default: 'sda'
stabdecopts
structure, containing the optional parameters for the decomposition of the stable and unstable parts of the system using the sign function and subspace extraction method, only used if RiccatiSolver = 'newton', see ml_signm and ml_getqz
default: struct()
stabmethodopts
structure, containing the optional parameters for the Smith iteration based Lyapunov equation solver used for the stabilization, only used if RiccatiSolver = 'newton', see ml_dlyap_smith
default: struct()
StoreProjection
{0, 1}, used to disable/enable storing of the computed projection matrices W and T
default: 0
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

Note: Parameters marked with {!} may also be a cell array containing multiple arguments. In this case an cell array of the same size is returned with one entry computed for each input argument and the marked fields of the info struct are cells as well. When multiple arguments are given as cells, they are expected to have the same length.

Output

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

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
infoDARE_C
structure, containing information about the Newton solver for the regulator Riccati equation, see ml_dare_nwt_fac
infoDARE_O
structure, containing information about the Newton solver for the filter Riccati equation, see ml_dare_nwt_fac
infoDAREDL
structure, containing information about the structure-preserving doubling solver for the dual Riccati equations, see ml_daredl_sda_fac
infoADTF
structure, containing information about the additive decomposition of the system into its infinite, finite stable and finite anti-stable parts, see ml_dt_dss_adtf
infoGDLYAPDL
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper Gramians, see ml_gdlyapdl_smith_fac
infoPARTSTAB_C
structure, containing information about the partial stabilization used for the controllability Riccati equation, see ml_dt_dss_partstab
infoPARTSTAB_O
structure, containing information about the partial stabilization used for the observability Riccati equation, see ml_dt_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
T
{!}
projection matrix used as right state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1
W
{!}
projection matrix used as left state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1

Reference

M. R. Opmeer, R. F. Curtain, Linear quadratic Gaussian balancing for discrete-time infinite-dimensional linear systems, SIAM J. Control Optim., 43(4), 1196--1221. https://doi.org/10.1137/S0363012903431189

See Also

ml_dt_ss_lqgbt | ml_morlabopts