LMIs in Control/Minimum-Rank Solutions of Linear Matrix Equations/Rank minimization problem: general form

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The information on this page is referenced from: "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization" by Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo, published in SIAM Review 2010, Volume 52, Issue 3, pages 471-501.

Systems are very often represented by matrices. The rank of a matrix offers some very useful information about the system it represents. For instance: a state space representation of a system is controllable if its state matrix is full rank.

In some circumstances, finding a simpler model of a system is useful. For example; finding a lower-degree statistical model for a random process, discovering a low-order realization of a linear state space system model, and a finding lower order controllers for certain systems. In some circumstances, this reduction of order can be achieved by minimizing the rank of the appropriate matrix.

The System
This problem is solvable only in certain circumstances. In general however, if the set of all models that can be used to describe the system and that satisfy the desired constraints is a convex set, $$\mathcal{C}$$, then the system $$\mathcal{X}$$ may be suitable for the LMI's application.

One specific case that is solvable is the case where the nuclear norm (the sum of the singular values of the matrix) is minimized over a set of feasible models or designs that are affine in the matrix variable $$\mathcal{X}$$. All cases discussed in the pages of this chapter fall into this category.

The Data
The inputs for this LMI are the linear map $$\mathcal{A} : \mathcal{R}^{m*n} to \mathcal{R}^{p} $$ required to be affine in the matrix variable $$\mathcal{X} \in \mathcal{R}^{m * n}$$ and the vector $$\mathcal{b} \in \mathcal{R}^{p}$$.

The LMI: General Form of the Rank Minimization Problem


\begin{align} \text{minimize} \; \text{rank} X &\\ \text{subject to} \; \mathcal{A}(X) = b &\\ \end{align}$$

Conclusion:
The result of this LMI is the guaranteed minimum rank representation of the system described by $$\mathcal{X}$$.

Implementation

 * - A downloadable matlab code implementation of this LMI using YALMIP.

Related LMIs
Links to other closely-related LMIs