.Modelica.Math.Matrices.balanceABC

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.

Interface

function balanceABC
  extends Modelica.Icons.Function;
  input Real A[:, size(A, 1)] "System matrix A";
  input Real B[size(A, 1), :] = fill(0.0, size(A, 1), 0) "System matrix B (need not be present)";
  input Real C[:, size(A, 1)] = fill(0.0, 0, size(A, 1)) "System matrix C (need not be present)";
  output Real scale[size(A, 1)] "diagonal(scale)=T is such that [inv(T)*A*T, inv(T)*B; C*T, 0] has smaller condition as [A,B;C,0]";
  output Real As[size(A, 1), size(A, 1)] "Balanced matrix A (= inv(T)*A*T )";
  output Real Bs[size(A, 1), size(B, 2)] "Balanced matrix B (= inv(T)*B )";
  output Real Cs[size(C, 1), size(A, 1)] "Balanced matrix C (= C*T )";
end balanceABC;

Revisions

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