# LMIs in Control/Matrix and LMI Properties and Tools/Iterative Convex Overbounding

Iterative convex overbounding is a technique based on Young’s relation that is useful when solving an optimization problem with a BMI constraint.

**The System**Edit

Consider the matrices , where S and R are design variables in the BMI given by

**The LMI:**Iterative Convex OverboundingEdit

Suppose that S_{0} and R_{0} are known to satisfy (1). The BMI of (1) is implied by the LMI

where Φ(R,S) = B(RDS_{0}+R_{0}DS-R_{0}DS_{0})C, W>0 is an arbitrary matrix, D=UV, and the matrices U and
V^{T} have full column rank. The LMI of (2) is equivalent to the BMI of (1)
when R = R_{0} and S = S_{0}, and is therefore non-conservative for values of R and S and are close to
the previously known solutions R_{0} and S_{0}.

Alternatively, the BMI of (1) is implied by the LMI

where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and V^{T} have full column rank.
Again, the LMI of (4) is equivalent to the BMI of (2) when R = R_{0} and S = S_{0}, and is
therefore non-conservative for values of R and S and are close to the previously known solutions
R_{0} and S_{0}.

**Conclusion:**Edit

A benefit of convex overbounding compared to a linearization approach, is that in addition to
ensuring conservatism or error is reduced in the neighborhood of R = R_{0} and S = S_{0}, the LMIs
of (2) and (3) imply (1). Iterative convex overbounding is particularly useful when used to solve an optimization problem with BMI constraints.

## External LinksEdit

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- Young’s Relation