LMIs in Control/Matrix and LMI Properties and Tools/Matrix Inequalities and LMIs

Matrix Inequality Definition-1 A Matrix Inequality, , in the variable is an expression of the form , where and , Linear Matrix Inequality Definition-2 A Linear Matrix Inequality, , in the variable is an expression of the form , where and , Bilinear Matrix Inequality Definition-3 A Bilinear Matrix Inequality (BMI), , in the variable is an expression of the form where , and , , Example Consider the matrices and , where . It is desired to find a symmetric matrix satisfying the inequality where . The elements of are the design variables in this problem, and although equation is indeed an LMI in the matrix , it does not look like the LMI in definition 3. For simplicity, let us consider the case of so that each matrix is of dimension , and Writing the matrix in terms of a basis , yields Note that the matrices are linearly independent and symmetric, thus forming a basis for the symmetric matrix . The matrix inequality in equation can be written as Defining and yields which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation rather than the scalar form of definition 2. External Links LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet. LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd. LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.