  We consider the linear matrix inequality (LMI) problem of $H_\infty$ output
feedback control problem for a generalized plant whose control input, measured
output, disturbance input, and controlled output are scalar. We provide an
explicit form of the optimal value. This form is the unification of some
results in the literature of $H_\infty$ performance limitation analysis. To
obtain the form of the optimal value, we focus on the non-uniqueness of
perpendicular matrices, which appear in the LMI problem. We use the null
vectors of invariant zeros associated with the dynamical system for the
expression of the perpendicular matrices. This expression enables us to reduce
and simplify the LMI problem. Our approach uses some well-known fundamental
tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and
matrix completion. We use these techniques for the simplification of the LMI
problem. Also, we investigate the structure of dual feasible solutions and
reduce the size of the dual. This reduction is called a facial reduction in the
literature of convex optimization.
