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

Consider the matrices $Q=Q^{T}\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},R\in \mathbb {R} ^{m\times p},D\in \mathbb {R} ^{p\times q},S\in \mathbb {R} ^{q\times r}andC\in \mathbb {R} ^{r\times n}$, where S and R are design variables in the BMI given by

${\begin{aligned}\qquad Q+BRDSC+C^{T}S^{T}D^{T}R^{T}B^{T}<0\qquad (1)\\\end{aligned}}$

The LMI:Iterative Convex Overbounding

Suppose that S₀ and R₀ are known to satisfy (1). The BMI of (1) is implied by the LMI

${\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&B(R-R0)U&C^{\text{T}}(S-S0)^{\text{T}}V^{\text{T}}\\*&W^{\text{-1}}&0\\*&*&-W\\\end{bmatrix}}<0\qquad (2)$

where Φ(R,S) = B(RDS₀+R₀DS-R₀DS₀)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 = R₀ and S = S₀, and is therefore non-conservative for values of R and S and are close to the previously known solutions R₀ and S₀.

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

${\begin{bmatrix}Q+\phi (R,S)+\phi ^{\text{T}}(R,S)&Z^{\text{T}}U^{\text{T}}(R-R0)^{\text{T}}B^{\text{T}}+V(S-S0)C\\*&Z\\\end{bmatrix}}<0\qquad (3)$

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 = R₀ and S = S₀, and is therefore non-conservative for values of R and S and are close to the previously known solutions R₀ and S₀.

Conclusion:

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 = R₀ and S = S₀, 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 Links

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