functionbalanceABC

Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation

Extends from Modelica.Icons.Function (Icon for functions).

Information

Syntax

(scale,As,Bs,Cs) = Matrices.balanceABC(A,B,C);
(scale,As,Bs)    = Matrices.balanceABC(A,B);
(scale,As,,Cs)   = Matrices.balanceABC(A,C=C);

Description

This function returns a vector scale, such that with T=diagonal(scale) system matrix S_scale 

          |inv(T)*A*T, inv(T)*B|
S_scale = |                    |
          |       C*T,     0   |

has a better condition as system matrix S 

    |A, B|
S = |    |
    |C, 0|

that is, conditionNumber(S_scale) ≤ conditionNumber(S). The elements of vector scale are multiples of 2 which means that this function does not introduce round-off errors.

Balancing a linear dynamic system in state space form 

der(x) = A*x + B*u
    y  = C*x + D*u

means to find a state transformation x_new = T*x = diagonal(scale)*x so that the transformed system is better suited for numerical algorithms.

Example

import Modelica.Math.Matrices;

A = [1, -10,  1000; 0.01,  0,  10; 0.005,  -0.01,  10];
B = [100, 10; 1,0; -0.003, 1];
C = [-0.5, 1, 100];

(scale, As, Bs, Cs) := Matrices.balanceABC(A,B,C);
T    = diagonal(scale);
Diff = [Matrices.inv(T)*A*T, Matrices.inv(T)*B;
        C*T, zeros(1,2)] - [As, Bs; Cs, zeros(1,2)];
err  = Matrices.norm(Diff);

-> Results in:
scale = {16, 1, 0.0625}
norm(A)  = 1000.15, norm(B)  = 100.504, norm(C)  = 100.006
norm(As) = 10.8738, norm(Bs) = 16.0136, norm(Cs) = 10.2011
err = 0

The algorithm is taken from

H. D. Joos, G. Grübel:
RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which is based on the balance function from EISPACK.

Inputs

TypeNameDefaultDescription
Real[:,size(A, 1)]ASystem matrix A
Real[size(A, 1),:]Bfill(0.0, size(A, 1), 0)System matrix B (need not be present)
Real[:,size(A, 1)]Cfill(0.0, 0, size(A, 1))System matrix C (need not be present)

Outputs

TypeNameDefaultDescription
Real[size(A, 1)]scalediagonal(scale)=T is such that [inv(T)*A*T, inv(T)*B; C*T, 0] has smaller condition as [A,B;C,0]
Real[size(A, 1),size(A, 1)]AsBalanced matrix A (= inv(T)*A*T )
Real[size(A, 1),size(B, 2)]BsBalanced matrix B (= inv(T)*B )
Real[size(C, 1),size(A, 1)]CsBalanced matrix C (= C*T )

Revisions

  • Sept. 14, 2014 by Martin Otter: Implemented.