## LMI Applications

Finding a solution *x* to the LMI system

A(x) < 0 | (1) |

is called the *feasibility problem*. Minimizing a convex objective
under LMI constraints is also a convex problem. In particular, the *linear objective
minimization problem*:

Minimize *c ^{T}x* subject to

A(x) < 0 | (2) |

plays an important role in LMI-based design. Finally, the *generalized eigenvalue
minimization problem*

Minimize *λ* subject to

$$\begin{array}{l}A\left(x\right)<\lambda B\left(x\right)\\ B\left(x\right)>0\\ C\left(x\right)>0\end{array}$$ | (3) |

is quasi-convex and can be solved by similar techniques. It owes its name to the fact that
is related to the largest generalized eigenvalue of the pencil
(*A*(*x*),*B*(*x*)).

Many control problems and design specifications have LMI formulations [9]. This is especially true for Lyapunov-based analysis and design, but also for
optimal LQG control, *H*^{∞} control,
covariance control, etc. Further applications of LMIs arise in estimation, identification,
optimal design, structural design [6], [7], matrix scaling problems, and so on. The main strength of LMI formulations is the
ability to combine various design constraints or objectives in a numerically tractable manner.

A nonexhaustive list of problems addressed by LMI techniques includes the following:

Robust stability of systems with LTI uncertainty (µ-analysis) ([24], [21], [27])

Robust stability in the face of sector-bounded nonlinearities (Popov criterion) ([22], [28], [13], [16])

Lyapunov stability of parameter-dependent systems ([12])

Input/state/output properties of LTI systems (invariant ellipsoids, decay rate, etc.) ([9])

Multi-model/multi-objective state feedback design ([4], [17], [3], [9], [10])

Robust pole placement

Optimal LQG control ([9])

Control of stochastic systems ([9])

Weighted interpolation problems ([9])

To hint at the principles underlying LMI design, let's review the LMI formulations of a few typical design objectives.

### Stability

The stability of the dynamic system

$$\dot{x}=Ax$$

is equivalent to the feasibility of the following problem:

Find *P* =
*P** ^{T}* such that

*A*

^{T}*P*+

*P*

*A*< 0,

*P*>

*I.*

This can be generalized to linear differential inclusions (LDI)

$$\dot{x}=A(t)x$$

where *A*(*t*) varies in the convex envelope of a set
of LTI models:

$$A\left(t\right)\in C\text{o}\left\{{A}_{\text{1}},\dots ,{A}_{n}\right\}=\left\{{\displaystyle \sum _{i=1}^{n}{a}_{i}{A}_{i}:{a}_{i}\ge 0,{\displaystyle \sum _{i=1}^{N}{a}_{i}=1}}\right\}.$$

A sufficient condition for the asymptotic stability of this LDI is the feasibility of

Find *P* =
*P** ^{T}* such that $${A}_{i}^{T}P+P{A}_{i}<0,\text{\hspace{1em}}P>I$$.

### RMS Gain

The random-mean-squares (RMS) gain of a stable LTI system

$$\{\begin{array}{c}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

is the largest input/output gain over all bounded inputs
*u*(*t*). This gain is the global minimum of the
following linear objective minimization problem [1], [25], [26].

Minimize γ over *X* = *X*^{T} and γ such that

$$\left(\begin{array}{ccc}{A}^{T}X+XA& XB& {C}^{T}\\ {B}^{T}X& -\gamma I& {D}^{T}\\ C& D& -\gamma I\end{array}\right)<0$$

and

$$X>0.$$

### LQG Performance

For a stable LTI system

$$G\text{\hspace{0.17em}}\{\begin{array}{c}\dot{x}=Ax+Bw\\ y=Cx\end{array}$$

where *w* is a white noise disturbance with unit covariance, the LQG or
*H*_{2} performance
∥*G*∥_{2} is defined by

$$\begin{array}{c}{\Vert G\Vert}_{2}^{2}:\text{\hspace{0.17em}}=\underset{T\to \infty}{\mathrm{lim}}E\left\{\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int}}{y}^{T}\left(t\right)y\left(t\right)dt}\right\}\\ =\frac{1}{2\pi}{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}{G}^{H}\left(j\omega \right)G\left(j\omega \right)d\omega .}\end{array}$$

It can be shown that

$${\Vert G\Vert}_{2}^{2}=\mathrm{inf}\left\{\text{Trace}\left({\text{CPC}}^{T}\right):AP+P{A}^{T}+B{B}^{T}<0\right\}.$$

Hence $${\Vert G\Vert}_{2}^{2}$$ is the global minimum of the LMI problem. Minimize Trace
(*Q*) over the symmetric matrices *P*,*Q*
such that

$$AP+P{A}^{T}+B{B}^{T}<0$$

and

$$\left(\begin{array}{cc}Q& CP\\ P{C}^{T}& P\end{array}\right)>0.$$

Again this is a linear objective minimization problem since the objective Trace
(*Q*) is linear in the decision variables (free entries of
*P*,*Q*).