ml_ct_dss_hinfbt
H-infinity balanced truncation for descriptor systems.
Contents
Syntax
[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_hinfbt(A, B, C, D, E) [Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_hinfbt(A, B, C, D, E, opts)
[rom, info] = ml_ct_dss_hinfbt(sys) [rom, info] = ml_ct_dss_hinfbt(sys, opts)
Description
This function computes the H-infinity 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' - beta * (Ep*Pp*Cp' + B*M') * inv(Rb) * (Ep*Pp*Cp' + B*M')' = 0, Ap'*Qp*Ep + Ep'*Qp*Ap + Cp'*Cp - beta * (Bp'*Qp*Ep + M'*C)' * inv(Rc) * (Bp'*Qp*Ep + M'*C) = 0
are solved for the reduction of the strictly proper part, with beta = (1 - GAM^(-2)) and
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 sqrt(beta)*G = N*inv(M) and the reduced-order transfer function Gr with an r-th order strictly proper part and the (right) coprime factorization sqrt(beta)*Gr = Nr*inv(Mr) it holds
1/sqrt(beta) *||[N; M] - [Nr; Mr]||_{\infty} <= 2 * (Hsvp(r+1)/sqrt(1 + (beta*Hsvp(r+1))^2) + ... + Hsvp(n)/sqrt(1 + (beta*Hsvp(n))^2)),
with Hsvp a vector containing the characteristic H-infinity singular values of the system.
Note: 1) The parameter GAM refers to the optimal cost gamma computed during the H-infinity controller construction of the LQG normalized system with feed-through term
[ s*E - A | B 0 B ] [ --------------- ] [ C | 0 0 0 ]. [ 0 | 0 0 I ] [ C | 0 I D ]
Still, there is no known implementation of the Gamma-iteration for descriptor systems in MATLAB. In case of estimating/computing an optimal cost parameter GAM_OPT, the GAM > GAM_OPT can be chosen arbitrarily. By default, the maximum of 1 and the given GAM is disturbed by 0.01% before used for computations. 2) In the case of GAM == Inf, the H-infinity balanced truncation is identical to the LQG balanced truncation.
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_fac 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 |
Gamma | positive scalar, used for the scaling of the Riccati equations, see GAM above default: Inf |
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, if 0 no improper balanced truncation is performed 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
|
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
|
RiccatiSolver | character array, determining the solver for the dual Riccati equations
|
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() |
StabMethod | character array, determining the method of stabilization of the system, only used if RiccatiSolver = 'newton'
|
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_cabe_sgn or ml_lyap_sgn 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 |
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 |
Gamma | positive scalar, the disturbed scaling parameter of the computations |
Hsvi | a vector, containing the computed Hankel singular values of the improper part of the system |
Hsvp | a vector, containing the computed characteristic H-infinity 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_fac |
infoADTF | 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 |
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_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 |
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 |
See Also