riccond (Robust Control Toolbox)

Robust Control Toolbox

riccond

Condition numbers of continuous algebraic Riccati equation.

Syntax

```
[tot] = riccond(a,b,qrn,p1,p2)
```

Description
Riccond provides the condition numbers of continuous Riccati equation. The input variable qrn contains the weighting matrix

for the Riccati equation

where P = P2/P1 is the positive definite solution of ARE, and [P2; P1] spans the stable eigenspace of the Hamiltonian

Several measurements are provided:

Frobenius norms norAc, norQc, norRc of matrices A_c, Q_c, and Rc, respectively.
condition number conR of R.
condition number conP1 of P1.
Arnold and Laub's Riccati condition number (conArn) [1].
Byers condition number (conBey) [2].
residual of Riccati equation (res).

The output variable tot puts the above measurements in a column vector

 tot= [norA,norQ,norRc,conR,conP1,conBey,res]'

For an ill-conditioned problem, one or more of the above measurements could become large. Together, these measurements give a general sense of the Riccati problem conditioning issues.

Algorithm
Arnold and Laub's Riccati condition number is computed as follows [1]:

where A_cl= A_c - R_cP and

Byers' Riccati condition number is computed as [2]

References
[1] W. F. Arnold, III and A. Laub, "Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations," Proceedings of the IEEE, Vol. 72, No. 12, Dec. 1984.

[2] R. Byers, "Hamiltonian and Symplectic Algorithms for the Algebraic Riccati Equation," Ph.D. dissertation, Dept. of Comp. Sci., Cornell University, Ithaca, NY, 1983.

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