What Are Linear Matrix Inequalities?

Linear matrix inequalities (LMIs) are constraints of the form F(x) = F0 + x1F1 + ... + xmFm ≥ 0, where the Fi are symmetric matrices, x is a vector of real decision variables, and the inequality means that F(x) must be positive semidefinite. The constraint is simultaneously linear in the decision variables x and expresses a requirement on the eigenvalues of a matrix, making it far more expressive than a standard linear inequality while retaining convexity. Many problems in control theory, structural optimization, and statistics that appear intractable when written in other forms become efficiently solvable once cast as LMIs. The study of LMIs as a systematic tool for control engineering was advanced significantly by the 1994 monograph of Boyd, El Ghaoui, Feron, and Balakrishnan, Linear Matrix Inequalities in System and Control Theory, which unified decades of scattered results into a coherent computational framework.

The conceptual roots of LMIs reach back to the 1940s, with the Lyapunov stability criterion for linear systems providing the earliest example. A linear time-invariant system is stable if and only if there exists a positive definite matrix P satisfying the Lyapunov inequality A^T P + PA < 0, a condition that is itself an LMI in P. The key enabling development that made LMIs computationally accessible was the maturation of interior-point methods for semidefinite programming (SDP) in the early 1990s, which produced polynomial-time algorithms capable of solving LMI feasibility and optimization problems to arbitrary precision.

Mathematical Formulation

An LMI constraint defines a convex feasible set: the set of all decision vectors x satisfying F(x) ≥ 0 is a convex subset of R^m. Multiple LMI constraints can be combined into a single block-diagonal LMI by stacking the matrices. Feasibility problems ask whether any x satisfies the constraints; optimization problems minimize a linear objective over the feasible set, yielding a semidefinite program (SDP). The MIT tutorial on linear and bilinear matrix inequalities provides a systematic treatment of how Lyapunov inequalities, H-infinity norm bounds, and other control-theoretic conditions all translate into the standard LMI form. Schur complement lemmas are a central manipulation tool: they convert a condition involving a matrix inverse into an equivalent LMI, a step that appears repeatedly in H-infinity and LQR formulations.

Convex Optimization Connection

Because the feasible set of an LMI is convex and closed, semidefinite programming inherits the strong duality and optimality guarantees of convex optimization. Interior-point methods such as the primal-dual algorithm of Nesterov and Nemirovskii solve SDPs in time polynomial in the problem size and the required precision. Software packages including SeDuMi, SDPT3, and MOSEK implement these solvers, while modeling layers like YALMIP and CVX allow users to specify LMI constraints symbolically and invoke solvers automatically. This computational accessibility transformed LMIs from a theoretical curiosity into a routine engineering tool. Unlike Riccati equation approaches to optimal control, which produce a single solution, LMI formulations allow multiple performance specifications to be combined into a single optimization, including bounds on H2 and H-infinity norms simultaneously.

Applications in Control Design

LMI methods address a wide range of control problems for linear systems and uncertain systems. Robust stability analysis under structured or unstructured uncertainty reduces to checking LMI feasibility over a polytope of possible plant matrices. Controller synthesis for mixed H2/H-infinity performance, gain-scheduled control for parameter-varying systems, and state-feedback or output-feedback stabilization all admit LMI formulations. The Springer article on LMI techniques in optimal control surveys how these formulations extend to descriptor systems and time-delay systems.

Applications

Linear matrix inequalities have applications across a range of engineering and scientific fields, including:

• Robust stability and performance analysis of uncertain linear systems
• H-infinity and mixed H2/H-infinity controller synthesis
• Gain-scheduled control for linear parameter-varying plants
• Structural optimization with stiffness and frequency constraints
• Filter design with passivity and bounded-real constraints