LMIs in Control/Matrix and LMI Properties and Tools/Convexity of LMIs

Definition

A set, $𝒮$, in a real inner product space is convex if for all $x,y\in 𝒮$ and $\alpha \in ℝ$, where $0\le \alpha \le 1$, it holds that $\alpha x+(1-\alpha )y\in 𝒮$.

Lemma 1.1

The set of solutions to an LMI is convex.

That is, the set $𝒮=\{x\in {ℝ}^m\mid F(x)\le 0\}$ is a convex set, where $F:{ℝ}^m⟶{𝕊}^n$ is an LMI.

Lemma 1.2

An LMI, $F:{ℝ}^m⟶{𝕊}^n$, in the variable $x\in {ℝ}^m$ is an expression of the form

$F(x)=F_0+\sum _{i=1}^m{x}_i{F}_i\le 0$

where $x^T=[x_1\cdots x_m]$ and $F_i\in {𝕊}^n$, $i=0,\dots ,m$.

Proof

Consider $x,y\in {ℝ}^m$ and $\alpha \in [0,1]$, and suppose that $x$ and $y$ satisfy Lemma 1.2.

The LMI $F:{ℝ}^m⟶{𝕊}^n$ is convex, since

$F(\alpha x+(1-\alpha )y)=F_0+\sum _{i=1}^m(\alpha x_i+(1-\alpha )y_i)F_i$

$\begin{array}{cc}F(\alpha x+(1-\alpha )y)& =F_0+\sum _{i=1}^m(\alpha x_i+(1-\alpha )y_i)F_i\\ & =F_0-\alpha F_0+\alpha F_0+\alpha \sum _{i=1}^m{x}_i{F}_1+(1-\alpha )\sum _{i=1}^m{y}_i{F}_i\\ & =\alpha F_0+\alpha \sum _{i=1}^m{x}_i{F}_i+(1-\alpha )F_0+(1-\alpha )\sum _{i=1}^m{y}_i{F}_i\\ & =\alpha F(x)+(1-\alpha )F(y)\\ \end{array}$

Convexity of LMI

From Lemma 1.1, it is known that an optimization problem with a convex objective function and LMI constraints is convex.

The following is a non-exhaustive list of scalar convex objective functions involving matrix variables that can be minimized in conjunction with LMI constraints to yield a semi-definite programming (SDP) problem.

• $𝒥(x)=\frac{1}{2}{x}^{T}PX+q^{T}x+r$, where $x,q\in {ℝ}^n$, $P\in {𝕊}^n$, $P>0$, and $r\in ℝ$.

1. Special case when $q=0$ and $r=0:𝒥(x)=\frac{1}{2}{x}^{T}Px$, where$x\in {ℝ}^n$, $P\in {𝕊}^n$, and $P>0$.
2. Special case when $P=$2$\cdot 1$, $q=0$, and $r=0:𝒥(x)=x^{T}x=\Vert x{\Vert }_2^2$, where $x\in {ℝ}^n$.

• $𝒥(X)=tr(X^{T}PX+Q^{T}X+X^{T}R+S)$, where $X$, $Q$, $R\in {ℝ}^{n\times m}$, $P\in {𝕊}^n$, $S\in {ℝ}^{n\times n}$, and $P\ge 0$.

1. Special case when $Q=R=0$ and $S=0:𝒥(X)=tr(X^{T}PX)$, where $X\in {ℝ}^{n\times m}$, $P\in {𝕊}^n$, and $P\ge 0$.
2. Special case when $P=1$, $Q=R=0$ and $S=0:𝒥(X)=tr(X^{T}X)=\Vert X{\Vert }_F^2$, where $X\in {ℝ}^{n\times m}$.
3. Special case when $P=0$, $R=0$ and $S=0:𝒥(X)=tr(Q^{T}X)$, where $X$, $Q\in {ℝ}^{n\times m}$.
4. Special case when $P=1$, $Q=R=0$, $S=0$, and $X\in {𝕊}^n:𝒥(X)=tr(X^2)$, where $X\in {𝕊}^n$.

• $𝒥(X)=\log (\det (X^{-1}))=\log (\det (X))$, where $X\in {𝕊}^n$ and $X>0$.

Relative Definition of a Matrix

The definiteness of a matrix can be found relative to another matrix.

For example,

Consider the matrices $A\in {𝕊}^n$ and $B\in {𝕊}^n$. The matrix inequality $A<B$ is equivalent to $A-B<0$ or $B-A<0$.

Knowing the relative definiteness of matrices can be useful.

For example,

If in the previous example we have $A<B$ and also know that $A>0$, when we know that $B>0$.

This follows from $0<A<B$.

Strict and Non-strict Matrix Inequalities

A strict matrix inequality can be converted to a non-strict matrix inequality.

For example,

$A>0$ is implied by $A\ge ϵ1$, where $ϵ\in {ℝ}_{>0}$. Similarly, $B<0$ is implied by $B\le -ϵ1$, where $ϵ\in {ℝ}_{>0}$

Converting a strict matrix inequality into a non-strict matrix inequality is useful when working with LMI solvers that cannot handle strict constraints.

Concatenation of LMIs

A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI.

For example,

satisfying the LMIs $A<0$ and $B<0$ is equivalent to satisfying the concatenated LMI

$\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]<0$

More generally, satisfying the LMIs $A_i<0$, $i=1,\dots ,n$ is equivalent to satisfying the concatenated LMI $diag\{A_1,\dots ,A_n\}<0$.

External Links

• 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.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.