This information is part of the Modelica Standard Library maintained by the Modelica Association.

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.

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

A	Type: Real[:,size(A, 1)] Description: System matrix A
B	Default Value: fill(0.0, size(A, 1), 0) Type: Real[size(A, 1),:] Description: System matrix B (need not be present)
C	Default Value: fill(0.0, 0, size(A, 1)) Type: Real[:,size(A, 1)] Description: System matrix C (need not be present)

scale	Type: Real[size(A, 1)] Description: diagonal(scale)=T is such that [inv(T)*A*T, inv(T)*B; C*T, 0] has smaller condition as [A,B;C,0]
As	Type: Real[size(A, 1),size(A, 1)] Description: Balanced matrix A (= inv(T)*A*T )
Bs	Type: Real[size(A, 1),size(B, 2)] Description: Balanced matrix B (= inv(T)*B )
Cs	Type: Real[size(C, 1),size(A, 1)] Description: Balanced matrix C (= C*T )