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This article considers the problem of designing lead, lag, and lead-lag compensators for multi-input–multi-output (MIMO) linear systems under the $H_{\infty }$ performance measure. This is the first time that the lead, lag and lead-lag compensators are generalized to the MIMO cases by preserving their classical compensator structures. Theoretical results on the stability analysis and synthesis of MIMO systems under the $H_{\infty }$ control performance with the proposed lead, lag, and lead-lag compensators are obtained. Then relevant algorithms for designing lead, lag, and lead-lag compensators are provided to determine the compensator parameters. Differing from traditional design methods, which mostly rely on some trial-and-error procedures, the proposed methods are algorithmic and the compensators can be synthesized systematically. Illustrative examples are used to demonstrate the effectiveness and advantages of the proposed methods. [ABSTRACT FROM AUTHOR]