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Finding a solution *x* to the LMI system

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

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

Minimize *λ* subject to

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.

The stability of the dynamic system

is equivalent to the feasibility of the following problem:

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

This can be generalized to linear differential inclusions (LDI)

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

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

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

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

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

and

For a stable LTI system

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

It can be shown that

Hence
is
the global minimum of the LMI problem. Minimize Trace (*Q*)
over the symmetric matrices *P*,*Q* such
that

and

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

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