A linear matrix inequality approach to<i>H</i>_∞control

Abstract The continuous‐ and discrete‐time H ∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI‐based parametrization of all H ∞ ‐suboptimal controllers, including reduced‐order controllers. The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H ∞ controllers and bear important connections with the controller order and the closed‐loop Lyapunov functions. Thanks to such connections, the LMI‐based characterization of H ∞ controllers opens new perspectives for the refinement of H ∞ design. Applications to cancellation‐free design and controller order reduction are discussed and illustrated by examples.