Mu Analysis and Synthesis Toolbox  h2norm, hinfnorm

Calculate the H2, H norms of a SYSTEM matrix

Syntax

````out` = h2norm(sys)
out = hinfnorm(sys,`tol`)
```

Description
`h2norm` calculates the 2-norm of a stable, strictly proper SYSTEM matrix. The output is a scalar, whose value is the 2-norm of the system.

The output from `hinfnorm` is a 1 x 3 vector, `out`, which is made up (in order) of a lower bound for ||`sys`||·, an upper bound for ||`sys`||·, and a frequency, o, at which the lower bound is achieved. The ||·||· norm calculation is an iterative process and requires a test to stop. The variable `tol` specifies the tolerance used to calculate the ||`sys`||·. The iteration stops when

(the current upper bound) (1 + `tol`) x (the current lower bound).

The default value of `tol` is 0.001.

Algorithm
The H2` `norm of a SYSTEM follows from the solution to the Lyapunov equation.

AX + XA' + BB' = 0,

with ||`sys`||2 = trace (CXC').

Calculation of the H `norm` requires checking for j axis eigenvalues of a Hamiltonian matrix, H , which depends on a parameter . If H has no j axis eigenvalues, then the ||·||· `norm` of the SYSTEM matrix is less than . If the matrix H does have j axis eigenvalues, then these occur at the frequencies where the transfer matrix has a singular value (not necessarily the maximum) equal to . By iterating, the value of the ||·||· `norm` can be obtained.

Reference
Boyd, S., K. Balakrishnan and P. Kabamba, "A bisection method for computing the H norm of a transfer matrix and related problems," Math Control Signals and Systems, 2(3), pp. 207-219, 1989.

Boyd, S., and K. Balakrishnan, "A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its H norm," Systems and Control Letters, vol. 15-1, 1990.

Bruinsma, O., and M. Steinbuch, "A fast algorithm to compute theH norm of a transfer function matrix," Systems and Control Letters, vol. 14, pp. 287-293, 1990.

`hinfsyn`, `h2syn`, `ric_eig`, `ric_schr` getiv, sortiv, tackon h2syn 