LMIs in Control/A Newton-like method for solving rank constrained linear matrix inequalities

Introduction
The rank constrained LMI problem is a natural as well as important generalization of this problem. It is a non-convex feasibility problem defined by LMI constraints together with an additional matrix rank constraint.

The system
Find x Rm such that F (x) := F0 + m i=1 xiFi 0, (1) G(x) := G0 + mi=1 xiGi 0, (2) rank G(x) ¡= r. Let Sn + = X Sn — X0 and, for each integer s, let Sn +(s) = X Sn — X0, rank(X) = s. De�fine Mr = SnF + × r s=0 SnG + (s) = (X, Y ) SnF × SnG —X0, Y 0, rank(Y )r and L = (X, Y ) SnF × SnG — (X, Y ) = (F (x), G(x)) for some x Rm. Problem 1 can be stated in the following equivalent form.

B(AX + XAT + 2X)BT I 0, CT(Y A + ATY + 2Y )CTT I 0, [ X I I Y ] I 0, rank ([ X I I Y ] I ) n + nc

Theorem 1.1
Sn +(s) is a connected smooth manifold of dimen- sion 1 2 s(2n s + 1). The tangent space of Sn +(s) at an element X is TXSn +(s) = X + XT — Rn×n

Theorem 1.2
Sn +(s) is a connected smooth manifold of dimen- sion 1 2 s(2n s + 1). The tangent space of Sn +(s) at an element X is TXSn +(s) = X + XT — Rn×n

Conclusion
Like all other algorithms that attempt to solve the rank constrained LMI problem, convergence from an arbitrary initial condition is not guaranteed. Though the convergence properties of the algorithm are not yet completely understood, as demonstrated by the experiments, the algorithm can be quite effective.

Reference
K.M. Grigoriadis et al. Low-order control design for LMI problems using alternating projection meth�ods Automatica (1996)