Linear Matrix Inequality (LMI) in Control Engineering

Linear Matrix Inequality (LMI) is a powerful convex optimization framework widely used in modern control system analysis and design. An LMI condition expresses a constraint requiring that a matrix, which depends affinely on decision variables, is positive semidefinite. Because LMI problems can be solved efficiently using numerical solvers, they have become a standard tool for stability analysis, controller synthesis, observer design, and performance optimization in control engineering.

LMI-based methods offer several important advantages. First, multiple design objectives — such as stability guarantees, disturbance attenuation measured by the H-infinity or l2-induced norm, pole placement in desired regions, and input/output constraints — can be handled simultaneously within a single optimization problem. Second, LMI formulations naturally accommodate system uncertainties, including polytopic and norm-bounded uncertainties, enabling robust controller design. Third, various control structures — state feedback, dynamic output feedback, observer-based feedback, and gain-scheduled control — can all be addressed through appropriate LMI formulations.

Key mathematical tools that support LMI-based design include the Schur complement for converting nonlinear matrix inequalities into LMI form, the Elimination Lemma (Projection Lemma) for reducing the number of decision variables, the Generalized KYP Lemma for specifying frequency-domain performance in targeted frequency ranges, and the S-variable approach for reducing conservatism in robust control problems.

In our research, LMI optimization plays a central role across multiple topics, including Model Error Compensator gain design for systems with polytopic uncertainties, multi-rate observer and controller synthesis using cyclic reformulation, and robust invariant set estimation for switched systems.

Blog Articles on LMI 

For detailed explanations with mathematical formulations, MATLAB code examples, and references to key literature, see the following blog articles:

• Design of controller parameters using linear matrix inequalities in MATLAB — Introduction to LMI fundamentals: stability analysis, H-infinity performance, and state feedback controller design with MATLAB Robust Control Toolbox. Covers continuous-time, discrete-time, and l2-induced norm minimization problems.

• Advanced LMI Techniques in Control System Design — Advanced topics including Schur complement, Elimination Lemma, S-variable approach, Generalized KYP Lemma, polytopic robust control, and dynamic output feedback design. Includes comprehensive literature references.

MATLAB Code Examples

The following MATLAB scripts demonstrate basic LMI formulations for control system analysis and design. The Robust Control Toolbox is required. Source code is also available on GitHub.

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