Linear Matrix Inequalities

The origin of Linear Matrix Inequalities (LMIs) goes back as far as 1890, although they were not called this way at that time, when Lyapunov showed that the stability of a linear system \( {\bf {\dot x}} = {\bf {Ax}} \) is equivalent to the existence of a positive definite matrix P, which satisfies the matrix inequality\( \bf {{A^T}P} + \bf {PA} < \bf {0}, \) expression which will be clarified below. The term “Linear Matrix Inequality” was coined by Willems in the 1970’s to refer to this specific LMI, in connection with quadratic optimal control. Due to the lack of good computers as well as of efficient algorithms to solve them, the LMIs did not receive a great deal of consideration from control and system researchers until the late 1980’s, when Nesterov and Nemirovsky developed interior-point methods that allowed solving elegantly LMI problems. New algorithms appeared then, triggering a renewed interest in this subject.