Algorithms for computing the optimal H∞ norm

A gradient method for computing the optimal norm for a general H_∞ control problem is presented. This method is much faster than a bisection method and the additional cost of computing the gradient is small. Convergence is predicated on the smoothness of the spectral radius of the product of certain Riccati solutions. Hybrid bisection-gradient methods can be used in the nonsmooth case. The problem of the ill-conditioning of the Riccati equations as the solution approaches the optimal value is also addressed. It is shown that the required gradient can be formed using invariant subspaces of the associated Hamiltonians rather than explicitly forming the Riccati solutions