Design of the low-order controllers by the $H_\infty$ criterion: A parametric approach

Consideration was given to the problem of describing all stabilizing controllers of a given structure (for example, the the PID-controllers) satisfying the $H_\infty$ criterion. Controllers of a certain family were defined by the parameters $\mathbf{k}$, and in the parameter space a domain corresponding to the desired criteria was specified. Two approaches were proposed where (i) the desired domain is represented as an intersection of the admissible sets or (ii) its boundary is determined analytically. The two-parameter case is of special importance because it allows one to make use of the graphical mathematics.