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.
Consider the matrices , where S and R are design variables in the BMI given by
The LMI:Iterative Convex OverboundingEdit
Suppose that S0 and R0 are known to satisfy (1). The BMI of (1) is implied by the LMI
where Φ(R,S) = B(RDS0+R0DS-R0DS0)C, W>0 is an arbitrary matrix, D=UV, and the matrices U and VT 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
where Z > 0 is an arbitrary matrix, D = UV, and the matrices U and VT 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.
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.
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