Robust Control Toolbox™

Further Mathematical Background

Efficient interior-point algorithms are now available to solve the three generic LMI problems (8-2)-(8-4) defined in Three Generic LMI Problems. These algorithms have a polynomial-time complexity. That is, the number N() of flops needed to compute an -accurate solution is bounded by

M N3 log(V/)

where M is the total row size of the LMI system, N is the total number of scalar decision variables, and V is a data-dependent scaling factor. Robust Control Toolbox™ software implements the Projective Algorithm of Nesterov and Nemirovski [20], [19]. In addition to its polynomial-time complexity, this algorithm does not require an initial feasible point for the linear objective minimization problem (8-3) or the generalized eigenvalue minimization problem (8-4).

Some LMI problems are formulated in terms of inequalities rather than strict inequalities. For instance, a variant of (8-3) is

Minimize c^Tx subject to A(x) < 0.

While this distinction is immaterial in general, it matters when A(x) can be made negative semi-definite but not negative definite. A simple example is

(3-5)

Such problems cannot be handled directly by interior-point methods which require strict feasibility of the LMI constraints. A well-posed reformulation of (8-5) would be

Minimize c^Tx subject to x 0.

Keeping this subtlety in mind, we always use strict inequalities in this manual.

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LMIs and LMI Problems

References

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