H-Infinity Norm Calculation via a State Space Formulation

There is much interest in the design of feedback controllers for linear systems that minimize the H-infty norm of a specific closed-loop transfer function.  The H-infty optimization problem initiated by Zames (1981), \cite{zames1981feedback}, has received a lot of interest since its formulation.  In H-infty control theory one uses the H-infty norm of a stable transfer function as a performance measure.  One typically uses approaches in either the frequency domain or a state space formulation to tackle this problem.  Frequency domain approaches use operator theory, J-spectral factorization or polynomial methods while in the state space approach one uses ideas similar to LQ theory and differential games.  One of the key computational issues in the design of H-infty optimal controllers is the determination of the optimal H-infty norm.  That is, determining the infimum of r for which the H-infty norm of the associated transfer function matrix is less than r.  Doyle  et al (1989), presented a state space characterization  for the sub-optimal H-infty control problem.  This characterization requires that the unique stabilizing solutions to  two Algebraic Riccati Equations are positive semi definite as well as satisfying a spectral radius coupling condition.  In this work, we describe an algorithm by Lin et al(1999),  used to calculate the H-infty norm for the state feedback and output feedback control problems.  This algorithm only relies on standard assumptions and divides the problem into three sub-problems. The first two sub-problems rely on algorithms for the state feedback problem formulated in the frequency domain as well as a characterization of the optimal value in terms of the singularity of the upper-half of  a matrix created by the stacked basis vectors of the invariant sub-space of the associated Hamiltonian matrix.  This characterization is verified through a bisection or secant method.  The third sub-problem relies on the geometric nature of the spectral radius of the product of the two solutions to the Algebraic Riccati Equations associated with the first two sub-problems.  Doyle makes an intuitive argument that the spectral radius condition will fail before the conditions involving the Algebraic Riccati Equations fail.  We present numerical results where we demonstrate that the Algebraic Riccati Equation conditions fail before the spectral radius condition fails.