This paper considers the design of a stabilizing static output feedbac
k gain which keeps the linear quadratic (LQ) performance index less th
an a specified number (we call this an 'LQ suboptimal controller'). Ex
istence of such a controller is shown to be equivalent to the existenc
e of a positive-definite matrix P such that P satisfies two linear mat
rix inequalities (LMIs) while P-1 satisfies another LMI. All LQ subopt
imal controllers are explicitly parameterized by the freedom in the ch
oice of the positive-definite matrix P satisfying the LMIs, and an arb
itrary positive scalar and an arbitrary matrix of fixed dimension with
a norm bound. A modified version of the min/max algorithm is given to
find a positive-definite solution P to the LMIs.