LMIs and LMI Problems

A linear matrix inequality (LMI) is any constraint of the form

(3-1)

where

• x = (x1, . . . , xN) is a vector of unknown scalars (the decision or optimization variables)
• A0, . . . , AN are given symmetric matrices
• < 0 stands for "negative definite," i.e., the largest eigenvalue of A(x) is negative

Note that the constraints A(x) > 0 and A(x) < B(x) are special cases of (8-1) since they can be rewritten as -A(x) < 0 and A(x) - B(x) < 0, respectively.

The LMI (8-1) is a convex constraint on x since A(y) < 0 and A(z) < 0 imply that . As a result,

• Its solution set, called the feasible set, is a convex subset of RN
• Finding a solution x to (8-1), if any, is a convex optimization problem.

Convexity has an important consequence: even though (8-1) has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of LMI constraints can be regarded as a single LMI since

• 

where diag (A1(x), . . . , AK(x)) denotes the block-diagonal matrix with
A1(x), . . . , AK(x) on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity.

In most control applications, LMIs do not naturally arise in the canonical form (8-1), but rather in the form

L(X1, . . . , Xn) < R(X1, . . . , Xn)

where L(.) and R(.) are affine functions of some structured matrix variables X1, . . . , Xn. A simple example is the Lyapunov inequality

(3-2)

where the unknown X is a symmetric matrix. Defining x1, . . . , xN as the independent scalar entries of X, this LMI could be rewritten in the form (8-1). Yet it is more convenient and efficient to describe it in its natural form (8-2), which is the approach taken in the LMI Lab.

Three Generic LMI Problems

Finding a solution x to the LMI system

(3-3)

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

(3-4)

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

(3-5)

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])
• Quadratic stability of differential inclusions ([15], [8])
• 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])
• Robust H control ([11], [14])
• Multi-objective H synthesis ([18], [23], [10], [18])
• Design of robust gain-scheduled controllers ([5], [2])
• 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

• 

is equivalent to the feasibility of

Find P = P^T such that A^T P + P A < 0, P > I.

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

RMS Gain

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 over X = X^T and such that

• 

LQG Performance

For a stable LTI system

• 

where w is a white noise disturbance with unit covariance, the LQG or H2 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

• 

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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Linear Matrix Inequalities

Further Mathematical Background

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