Robust pole placement in LMI regions

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

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Robust Pole Placement in LMI Regions
Mahmoud Chilali, Pascal Gahinet, and Pierre Apkarian, Associate Member, IEEE

Abstract— This paper discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality
(LMI) regions, a class of convex regions of the complex plane that
embraces most practically useful stability regions. The focus is on
linear systems with static uncertainty on the state matrix. For this
class of uncertain systems, the notion of quadratic stability and
the related robustness analysis tests are generalized to arbitrary
LMI regions. The resulting tests for robust pole clustering are all
numerically tractable because they involve solving linear matrix
inequalities LMI’s and cover both unstructured and parameter
uncertainty.
These analysis results are then applied to the synthesis of
dynamic output-feedback controllers that robustly assign the
closed-loop poles in a prescribed LMI region. With some conservatism, this problem is again tractable via LMI optimization.
In addition, robust pole placement can be combined with other
performance, to capture
control objectives, such as 2 or
realistic sets of design specifications. Physically motivated examples demonstrate the effectiveness of this robust pole clustering
technique.

H

H1

I. INTRODUCTION

S

TABILITY is a minimum requirement for control systems.
In most practical situations, however, a good controller
should also deliver sufficiently fast and well-damped time responses. A customary way to guarantee satisfactory transients
is to place the closed-loop poles in a suitable region of the
complex plane. We refer to this technique as regional pole
placement, by contrast with pointwise pole placement, where
the poles are assigned to specific locations in the complex
plane. For example, fast decay, good damping, and reasonable
controller dynamics can be imposed by confining the poles in
the intersection of a shifted half-plane, a sector, and a disk
[18], [1], [4], [5]. Regional pole assignment has also been
considered in conjunction with other design objectives, such
or
performance [20], [8], [28], [9], [32].
as
Because real systems always involve some amount of uncertainty, it is natural to worry about the robustness of pole
clustering, i.e., whether the poles remain in the prescribed
region when the nominal model is perturbed. Such robustness
issues have been thoroughly studied in the context of pointwise
pole placement [23], [22], [25]. In comparison, few results
are available on robust regional pole clustering. These results
include a Lyapunov approach to compute explicit robustness
bounds for pole clustering in a disk [10] and extensions of the
Manuscript received August 20, 1997; revised August 20, 1998. Recommended by Associate Editor, M. Dahleh.
M. Chilali was with INRIA Rocquencourt, 78153 Le Chesnay Cedex,
France.
P. Gahinet is with The MathWorks, Inc., Natick, MA 01760 USA (e-mail:

).
P. Apkarian is with CERT-ONERA, 31055 Toulouse Cedex, France (e-mail:
).
Publisher Item Identifier S 0018-9286(99)09613-0.

notion of quadratic stability to robust pole placement in a disk
or a sector [3], [16], [15].
This paper extends these results to more general clustering
regions and to structured uncertainty. The regions considered
here are the linear matrix inequality (LMI) regions introduced
in [9]. This class of regions covers a large variety of useful clustering regions, including half-planes, disks, sectors,
vertical/horizontal strips, and any intersection thereof. The
following analysis and synthesis problems are addressed:
• robustness of pole clustering within a given LMI region
in the face of unstructured or parameter uncertainty in
the state matrix;
• synthesis of output-feedback controllers that robustly assign the closed-loop poles in some arbitrary LMI region
(assuming static and unstructured uncertainty on the plant
matrices).
With some conservatism, these problems are reduced to solving LMI’s. Because LMI’s can be solved numerically using
efficient optimization algorithms, such as those described in
[29], [30], [6], and [35], or implemented in [14] and [2],
our approach yields practical analysis and synthesis tools for
robust regional pole placement. See [7] for an overview of the
applications of LMI techniques in control theory.
This paper is organized as follows. Section II recalls the
definition of LMI regions and key results on pole clustering in LMI regions. Section III contains the main result, a
generalization of the Bounded Real Lemma to arbitrary LMI
regions. This result gives a sufficient condition in terms of
LMI’s for robust pole clustering within a given LMI region.

Section IV shows how some standard robustness analysis
tests for parameter uncertainty can be generalized to LMI
regions and illustrates the performance of the resulting robust
pole clustering tests on a realistic example. Section V applies
the results in Section III to the synthesis of output-feedback
controllers that robustly assign the closed-loop poles in a
given LMI region. This section also shows how to combine
robust pole clustering with other synthesis objectives using
the multi-objective design framework developed in [26], [33],
[32]. Finally, Section VI demonstrates the effectiveness of this
approach on a physically motivated design example.
II. BACKGROUND
This section recalls the basics of LMI regions and some
useful properties of Kronecker products.
A. Notation
and denote the sets of real and complex numbers, restands for the open left half-plane.
spectively. The notation

0018–9286/99$10.00  1999 IEEE


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

For a complex matrix
and
transpose of

denotes the Hermitian

is defined as

For Hermitian matrices,
means
is positive
means
is positive semidefinite.
definite and
In symmetric block matrices, we use as an ellipsis for terms
induced by symmetry, e.g.,

Finally, we use the shorthand

..
.

..

.

..

.

..

.

• conic sector with apex at the origin and inner angle

:

Key facts about LMI regions include [9] the following.
• Intersections of LMI regions are LMI regions.
• Any convex region symmetric with respect to the real axis
can be approximated by an LMI region to any desired
accuracy.
is -stable, i.e., has all its eigenvalues
• A real matrix
in the LMI region , if and only if a symmetric matrix
exists such that

..
.

(2)
This result can be seen as a generalization of the Lyapunov theorem because for the usual stability region
, (2) reduces to

B. Kronecker Products
The Kronecker product is an important tool for the subsequent analysis. Recall the Kronecker product of two matrices
and
is a block matrix
with generic block entry
, that is,

The following properties of the Kronecker product are easily
established [17]:

Pole clustering in LMI regions can be formulated as an

LMI optimization problem, a convex semidefinite program
that is easily tractable with recently available interior-point
techniques. Moreover, it is possible to combine such pole
clustering specifications with other design objectives while
preserving tractability [9], [32].
III. ROBUSTNESS OF POLE CLUSTERING IN LMI REGIONS

The eigenvalues of
are the pairwise products
of the eigenvalues of
and . As a result,
the Kronecker product of two positive-definite matrices is a
positive-definite matrix. Finally, the singular values of
consist of all pairwise products
of singular values
and .
of

The notions of robust and quadratic stability are useful tools
for analyzing the stability of uncertain state-space models [7],
[24]. These notions are now generalized to pole clustering in
arbitrary LMI regions, and a counterpart of the Bounded Real
Lemma is derived for LMI regions. Although our analysis
is restricted to static (real or complex) uncertainty, its implications for more general classes of uncertainty (dynamic or
time-varying) are briefly discussed at the end of the section.
A. Robust and Quadratic

-Stability

Consider the uncertain linear system

C. LMI Regions
An LMI region is any subset
can be defined as

of the complex plane that

(3)

(1)

depends fractionally on the normwhere the state matrix
bounded uncertainty matrix

and
are real matrices such that
where
matrix-valued function

. The

is called the characteristic function of . Below are a few
examples of LMI regions:
:
;
• half-plane
with radius :
• disk centered at

(4)
. The value

corresponds to the nominal
with
and the parameter
defines the level of
state matrix
uncertainty. Although the uncertain model (3) is physically
meaningful only for real uncertainty , we also consider
the complex case because of its connection with dynamic
uncertainty (see Section III-D).
Let
(5)


CHILALI et al.: POLE PLACEMENT IN LMI REGIONS

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be
is
of
,

any LMI region, and suppose the nominal state matrix
-stable, i.e., has all its eigenvalues in . The question
interest here is as follows: Given some uncertainty level
do the poles of
remain in
for all
satisfying
?

Definition 3.1 (Robust -Stability): The uncertain system
lie
(3)–(4) is robustly -stable if the eigenvalues of
for all admissible uncertainties .
in
Similar definitions can be found in [7] and [24] for the
usual stability region and in [4] for uncertain polynomials.
Nonconservative assessment of robust -stability is difficult,
for the
except in special cases, e.g., complex unstructured
open left half-plane. Although conservative, the following
notion of quadratic -stability proves more practical for
analysis and synthesis purposes.
Definition 3.2 (Quadratic -Stability): Given any LMI region defined by (5), the uncertain system (3), (4) is said to
be quadratically -stable if a real symmetric matrix
exists such that

We are now ready to state the main result, a sufficient
LMI-based condition for quadratic -stability.
Theorem 3.3: The system (3) with uncertainty

is quadratically -stable if matrices
exist such that

and

(10)

(11)
with the notation

(6)
such that
.
for all complex matrices
is -stable if and only if
Recall from Section II-C that
exists such that
. Hence, quadratic
-stability implies robust -stability, but the converse is
generally false because quadratic -stability requires a single
satisfies
for all admissible ’s. The
assumption “ real” incurs no loss of generality and is motivated by the tractability of the synthesis problem discussed
in Section V.
is the open left half-plane, it is well known that
When
is equivalent to quadratic starobust stability for complex
bility for real or complex [24], which in turn is completely
characterized by the Bounded Real Lemma:
The uncertain system (3), (4) is quadratically stable
if and only if
exists such that

Proof: See Appendix A.
The inequalities (10), (11) are LMI’s with unknown matrices
and . Hence, testing this sufficient condition numerically
can be tackled efficiently with LMI solvers. The matrix
plays the role of Lyapunov matrix, and
can be viewed as

a scaling matrix that accounts for the block-diagonal structure
in the relation
(see proof in Appendix
of
A). The variable also accounts for the nonuniqueness of the
. Specifically, replacing
by
factorization
is equivalent to replacing
by
.
The size of is typically small because for most useful LMI
in (8) has rank less than three.
regions, the matrix
It is insightful to explicitate the LMI (10) for well-known
regions, such as the left half-plane and the disk.
and
• The open left half-plane corresponds to
(i.e.,
). Taking
, (10) reduces to

(7)
Using a bilinear shift [8], this result remains true for vertical
half-planes and disks centered on the real axis. Next, the
Bounded Real Lemma condition for quadratic stability can
actually be generalized to arbitrary LMI regions.
B. Main Result
with characteristic function

Given an LMI region

which coincides with the Bounded Real Lemma inequaland redefining
ity (7) up to dividing by the scalar
as
.
with center
and radius corre• The disk
sponds to

(8)
factorize the matrix

as
(9)

have full column rank (such a factorization is
where
has rank , both
easily obtained from the SVD of ). If
and
are
matrices.

Because
take

has rank one,
is again scalar and we can
without loss of generality. The LMI constraint

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

(10) then reads as

—————

————- ——- ——-

——-

—————

————- ——- ——-

——-

By a Schur complement with respect to the block (1, 1),
this is equivalent to

which is simply the discrete-time version of the Bounded
. This
Real Lemma applied to the system
has all its eigenvalues
result stems from the fact that
if and only if
is stable in the

in
discrete-time sense, i.e., has all its eigenvalues in the unit
disk.
C. Intersections of LMI Regions
In practical applications, LMI regions are often specified as
the intersection of elementary regions, such as conic sectors,
,
disks, or vertical half-planes. Given LMI regions
the intersection

has characteristic function

If quadratic -stability is of interest, Theorem 3.3 should
. When
be applied to the overall characteristic function
robust -stability is the primary concern, however, it is more
efficient and less conservative to test quadratic stability for
independently. Indeed, this process
each elementary region
,
guarantees robust stability with respect to each region
which in turn establishes robust -stability.
elementary
More specifically, if is the intersection of
LMI regions with characteristic functions

a sufficient condition for robust
bounded uncertainty

-stability against norm-

is the existence, for each region
such that

, of a pair of matrices

(12)

Fig. 1. Robustness analysis interconnection.

The LMI feasibility problems (12)
should be
because no coupling
solved independently for each region
exists between the constraints for each region. By contrast,
amounts to jointly
applying Theorem 3.3 directly to
and
solving all LMI’s (12) with
as variables. This method is clearly more
costly and more conservative because of the requirement that
satisfy (12) for all regions .
a single
D. Comments on Quadratic

-Stability

Theorem 3.3 gives a sufficient condition for quadratic
-stability in the face of complex and unstructured uncermust be real
tainty. As mentioned earlier, the uncertainty

for the uncertain model
(13)
is the open left halfto be physically meaningful. When
plane and robust stability is of interest, the quadratic stability
test is known to be conservative for real uncertainty . It is
therefore legitimate to question the value of Theorem 3.3 as a
tool for assessing robust -stability.
While acknowledging conservatism for this particular uncertainty model, we now briefly review other benefits of quadratic
-stability that strengthen its practical appeal. Rewrite (13) as

(14)
. Then,
is simply the
and let
closed-loop matrix for the feedback loop of Fig. 1, and robust
-stability is therefore equivalent to requiring the closed-loop
satisfying
.
poles remain in for all
is the open left half-plane, equivalence exists
When
between [24]:
• quadratic stability;
;
• robust stability against complex with
where
;
•
• robust stability against stable dynamic uncertainty
satisfying

;
• feasibility of the Bounded Real Lemma LMI (7).
Similar connections between quadratic -stability, robust
-stability against dynamic uncertainty, and Theorem 3.3 can
be established for general LMI regions. Specifically, for
analytic in
(i.e., -stable), define the
norm with


CHILALI et al.: POLE PLACEMENT IN LMI REGIONS

respect to

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as

Proof: Multiplying (15) left and right by
, respectively, we get for all

Straightforward adaptations of the small gain and generalized
Nyquist theorems [25] lead to the following results.
Theorem 3.4: The following properties are equivalent.
is robustly -stable for static complex uncertainty
•
satisfying
.
.
•

• The closed-loop system
is robustly -stable against dynamic uncertainties
that are -stable and satisfy
.
has no poles on the boundary of
and
• If
, the nominal poles of
remain in . More precisely, the number of poles of
in
is always equal to the number of nominal
in .
poles (all in ) plus the number of poles of
These results indicate quadratic -stability (and the related
test in Theorem 3.3) also provides some robustness against
dynamic uncertainty, which is desirable in practice. Also, for
general regions, -stability is difficult to handle numerically
, the boundary of .
as it requires an exhaustive sweep of
In this respect, quadratic -stability provides tractable, though
possibly conservative, means for checking robust -stability.
E. Time-Varying Uncertainties
The proof of Theorem 3.3 remains valid when the uncertainty is time varying; i.e.,

Although the notion of “pole” disappears for linear timevarying systems, the generalized Bounded Real Lemma of
Theorem 3.3 still provides the following guarantees:
at all time ;
• -stability of the matrix
is con• exponential decay of the transients whenever
with

tained in some stable half-plane
.
The second property is a consequence of the following lemma.
Lemma 3.5: Consider an LMI region with characteristic
and suppose the dynamical
function
system

Recalling
ing by

and

, and divid, this process leads to

which, by definition of , ensures
.
This lemma shows, for quadratic -stable time-varying
systems, stability and decay rate are essentially determined by
. It says
time-invariant considerations, i.e., whether
little more, however, about transient behaviors. Can we also
expect well-damped responses by choosing an adequate conic
sector? Does a disk prevent fast dynamics? Such extensions
to the time-varying case remain open for future research.
IV. PARAMETER UNCERTAINTY
This section discusses refinements of the previous robustness analysis results when the uncertainty is structured. The
main motivation is the assessment of robust -stability in
the face of parameter uncertainties. As is usual when dealing with structured uncertainty, the resulting tests are only
sufficient conditions for robust pole clustering in a given

LMI region . Our analysis technique relies on the use of
similar to the parametera parameter-dependent matrix
dependent Lyapunov functions used in [19], [13], [11] for
regular robust stability analysis. Such approaches have proven
to be significantly less conservative than quadratic stability for
time-invariant parameter uncertainty.
The analysis below deals with the same basic uncertain
is real and structured; i.e.,
model (3), but now assumes
(16)
where the ’s denote the (normalized) uncertain parameters.
the hypercube, in which
We denote by
ranges according to (16), and by
the set of
vertices of this hypercube; that is,

To stress the dependence on the parameter vector
uncertain state matrix is written as

, the
(17)

is quadratically

-stable, i.e.,

exists such that

The relevant dimensionality parameters are defined by

(15)

for all time

. Then, the quadratic function
satisfies, for all

(18)

For such parameter uncertainty, robust pole clustering in
the LMI region
(19)


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

is equivalent to the existence of symmetric matrices
parametrized by such that

(20)
To enforce tractability of (20), we restrict the search of
to matrices with affine dependence on
functions

where

Proof: See Appendix B.
This theorem provides a test for robust -stability that

involves solving a finite set of LMI’s and is therefore tractable.
Applications to some aeronautics systems suggest it can be
sharp. In its most general form, this test can be computationally
demanding for high-order systems with multiple uncertainties.
With additional conservatism, the computational cost can be
reduced as follows.
• Use symmetric and skew-symmetric scalings in place of
and .
the general and unconstrained scalings
and
and
Specifically, impose
consistent with the uncertainty
make the structure of
structure
(23)

Two robust -stability tests are derived next using such
’s. The first test applies to general linear-fractional
affine
on whereas the second test is restricted
dependence of
. These results are strongly
to affine dependence
related to the general integral quadratic constraint framework
developed by Megretski and Rantzer in [27, pp. 825–826]. For
simplicity, our results are stated for a single Lyapunov matrix
regardless of the complexity of the LMI region. For LMI
regions that are intersections of elementary LMI regions ,
sharper tests can be obtained by using independent Lyapunov

for each
as indicated in Section III-C.
matrices
A. General Parameter Dependence
Theorem 4.1: Given the parameter uncertainty specified
in (19), a full-rank factorization
by (16), the LMI region
of , and the notation (18), the uncertain matrix
in (17) is robustly -stable if the following matrices
and
(with
exist: symmetric matrices
), scaling matrices
, and
(all
such that
in

That is, require
matrices of the form (23).
• Set some of the matrices

and

for all

to zero.

B. Affine Parameter Dependence
in (17), the uncertain state matrix depends

When
affinely on the parameters :
(24)
For such uncertain systems, the following robust -stability
test is easily derived using the multiconvexity technique develis multiconvex
oped in [13]. Recall a function
when it is convex with respect to each of its variables separately. For differentiable functions, this property is equivalent
to requiring the Hessian of has nonnegative diagonal entries.
Theorem 4.2: Given the parameter uncertainty specified
by (16), the uncertain system with state matrix (24) is robustly
-stable if symmetric matrices
and scalars
exist such that

(25)
(26)
(27)
————- ————-

——-

————- ————-

——-

and, for all vertices

—-

— —-

hold at all vertices

of

and for

, with

(21)

Proof: Condition (26) ensures, for any , the quadratic
function of

(22)

is multiconvex in the ’s. Using the same argument as in
[13], (25) holds over the entire hypercube if it holds at the
vertices.

of the uncertainty hypercube


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TABLE I
VARIABLE DESCRIPTION

C. Robust Analysis Application
The analysis techniques developed in this section are applied to a realistic missile example (see [34] for additional
details and insights). The purpose is to determine admissible
uncertainty levels for which stability and adequate damping
are preserved.
The dynamics of the controlled missile roll axis are described by

Fig. 2. Closed-loop poles of A( ) + B ( )KC for some parameter values
( 1 ;  2 )
1; 0:5; 0; 0:5; 1 2 .

2 f0 0

g

where
and the meaning of the different variables is given in Table I.
The output-feedback gain matrix
is given and has been
designed using eigenspace techniques. The parameters
and
represent uncertainties whose effects on the missile
dynamics are reflected in the matrices
and
. Numerical
values for these matrices can be found in Appendix C.
The objective is to estimate, in the parameter space
,
the largest square
where closed-loop stability is

maintained, and the largest square where closed-loop damping
for the missile roll axis. The
is adequate, that is,
uncertain parameters
and
enter affinely in the statespace matrices, so the techniques of Theorems 4.1 and 4.2 are
both applicable. The closed-loop pole locations for parameter
values in the set
are plotted in Fig. 2. Clearly, both stability and damping
constraints
are violated for some
pairs in
this uncertainty set.
The shaded area in Fig. 3 shows the region in the parameter
where closed-loop stability is maintained. This
space
area has been computed using an exhaustive search over a fine
grid in the parameter space. Based on the results of Theorem
4.1, we estimate the largest stability square using either fixed
matrices. As expected, a fixed
or parameter-dependent
leads to a conservative answer (dashed square in Fig. 3). In
contrast, using a parameter-dependent
provides a sharp
estimate of the largest allowable uncertainty (solid square).
Similarly, Fig. 4 shows the uncertainty region where adeis maintained, and the dashed and solid
quate damping
lines delimit the estimated safe regions using Theorem 4.2

Fig. 3. Stability region estimates with fixed (dotted) and parameter-dependent (solid) X matrices.

Fig. 4. Adequate damping region estimates with fixed (dotted) and parameter-dependent (solid) X matrices.

with fixed or parameter-dependent
matrices. The damping
is captured by the conic LMI region
constraint

Again, the estimate based on parameter-dependent
provides a sharp answer.

matrices

V. OUTPUT-FEEDBACK SYNTHESIS
This last section shows how to use our main analysis
result (Theorem 3.3) for synthesis purposes. Specifically, we


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

consider the problem of computing an output-feedback controller that robustly assigns the closed-loop poles in a prescribed LMI region . For tractability reasons, the discussion
is restricted to unstructured uncertainty.
The problem statement is as follows. Consider the uncertain
state-space model

Theorem 5.1: A full-order output-feedback controller
and a matrix
exist such that (30) holds if and only if

symmetric matrices
and
and matrices
two
and
exists such that

(31)

and (32), shown at the bottom of the page, where
(28)
where

and the static uncertainty
. Given the LMI region

satisfies

we are interested in designing a full-order dynamic controller
(29)
that robustly assigns the closed-loop poles in .
because this
Without loss of generality, assume
amounts to a mere change of variable in the controller matrices
and considerably simplifies the formulas. The closed-loop
matrix is

If these LMI’s are feasible, an th-order controller that rois
bustly assigns the closed-loop poles in

where the matrices
are derived as follows.
and
such that
• Compute any square matrices
where

• Solve the following linear equations for
:

, and

(33)
From Theorem 3.3 with
, a sufficient condition for
is the existence
quadratic (hence, robust) -stability of
such that
of
(30)

is a full-rank factorization of . This
where
matrix inequality is not jointly convex in and the controller
matrices. It can be reduced, however, to a convex LMI
problem by using the linearizing change of controller variables
introduced in [26], [33], [9]. This change leads to the following
synthesis result.

Proof: The proof involves the changes of variable introduced in [26], [33], and [9] and is omitted for brevity.

Inequalities (31) and (32) are LMI’s in the variables
and
that can be solved numerically
using LMI optimization software [14]. Theorem 5.1 therefore
provides a tractable (but somewhat conservative) approach to
robust pole assignment in LMI regions.
is the intersection of several eleRemark 5.2: When
as discussed in Section III-C, the
mentary LMI regions
synthesis LMI’s (31), (32) must be written for each region
variables, and the resulting set of LMI’s
using the same
must be solved jointly. Indeed, the synthesis problem is no

(32)


CHILALI et al.: POLE PLACEMENT IN LMI REGIONS

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longer convex when a different
is used for each
(this
prevents using the linearizing change of variable). The extra
conservatism introduced by this additional restriction is modest
in most applications.

The linearized longitudinal dynamics of the missile are
described by

A. Mixed Design Specifications
From a practical viewpoint, enforcing quadratic -stability
is rarely sufficient because most design problems are essentially multi-objective. For instance, realistic designs are likely
or
(loop shaping) objectives in addition to
to include
robust pole assignment for transient tuning. Fortunately, LMIbased synthesis can accomodate a rich variety of closed-loop
specifications within a single LMI optimization problem, as
shown in [32]. This problem is achieved with some conservatism, but has proven effective in a number of applications.
The basic requirement is that a single closed-loop Lyapunov
should account for all design
function
specifications.
As an immediate extension of the results in [26], [33], and
[32], it is easy to mix quadratic -stability with other objecor
performance, passivity constraints,
tives, such as
bounds on the impulse response, etc. As an example, we
can combine a quadratic -stability objective as captured by
-norm bound on some input/output
Theorem 5.1 with an
channels of the closed-loop system. For instance, if the nominal plant is described by

the (nominal) closed-loop transfer function
can be further constrained to

from

to

by combining the LMI conditions (31) and (32) for robust pole
assignment with the additional LMI constraint, shown in (34)
at the bottom of the page.

where
and
denote the angle of attack, pitch
rate, vertical accelerometer measurement, and fin deflection,
and
are auxiliary signals
respectively. The variables
used to model variations of the aerodynamic coefficients for
ranging between 0 and 20 . The parameter uncertainty
has been normalized; that is,
.
that meets
We need to design a dynamic compensator
the following specifications:
• settling time of 0.2 s with minimal overshoot and zero
in resteady-state error for the vertical acceleration
sponse to a step command;
• adequate high-frequency rolloff for noise attenuation and
to withstand neglected dynamics and flexible modes;
• maximum deflection of two (in normalized units) imposed
;
on the control signal
• time-domain specifications must be met over the uncer.
tainty range
To attack this problem, we use the feedback structure

denotes the reference accelerasketched in Fig. 5. Here,
denote the weighted tracking error and
tion signal, and
control input, respectively. An integrator is introduced in the
acceleration channel to enforce zero steady-state error. The
full compensator is therefore given by

To incorporate bounds on the size of unmodeled dynamics and
penalize tracking error, we use the weighting filters (see [31]):

VI. DESIGN EXAMPLE
This section illustrates the benefits of robust pole placement
in LMI regions through a missile autopilot design example.
The problem setup comes from [31] and [21], where additional
motivations and details can be found. It has been slightly simplified to focus on aspects relevant to the technique proposed
in this paper.

First, we perform a standard
-optimal design in which
gain between the inputs
we minimize the closed-loop
and the outputs
. This process is meant
to enforce high-frequency rolloff as well as stability and
performance for all admissible uncertainties . The step
controller are depicted in
responses of the resulting (pure)

(34)

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Fig. 5. Synthesis structure.

Fig. 6. LMI region.

Fig. 7 for
(nominal) and
(perturbed). Although
this first design could be deemed acceptable, it suffers from
up to 30% overshoot in the perturbed transient responses.
To improve transient behavior, we add a robust pole clustering constraint to achieve better damping across the uncertainty
range. Specifically, we require robust pole clustering in the
LMI region represented in Fig. 6. This region is defined as
the intersection of the following:
• disk with center zero and radius 1500 (to prevent fast
dynamics);
and angle
• shifted conic sector with apex at
. Its characteristic function is

and the corresponding

and

Fig. 7. Pure

H1 design—nominal and perturbed (1 = 61) step responses.
TABLE II
CONTROLLER ZERO-POLE-GAIN DESCRIPTION

matrices are

This particular region is chosen to achieve differential damping
at low and high frequency (the damping constraint takes effect
). Because the
constraint already enforces
for
closed-loop stability, it is inconsequential that this LMI region
intersects the right half-plane.
The resulting synthesis problem is multi-objective because
norm subject to
it involves minimizing the closed-loop
robust pole clustering in the selected region. In the LMI
framework, this problem is attacked by minimizing the closedloop gain subject to the LMI constraints of Theorem 5.1

for robust pole clustering and the LMI constraint (34) for the
” (see Section V-A for details).
constraint “closed-loop gain
This LMI optimization problem was solved with [14] and
produced the compensator

with zero-pole-gain description in Table II. The corresponding
step responses are shown in Fig. 8. The transients are smoother
control for both nominal
than those obtained with pure
and perturbed plants. More importantly, thanks to the disk

constraint, this result is achieved with significantly slower


CHILALI et al.: POLE PLACEMENT IN LMI REGIONS

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APPENDIX A
Proof of Theorem 3.3: Assume
without loss of
is always invertible because
generality. First, observe
(10) implies

which, from
and the properties of eigenvalues
and hence
of Kronecker products, secures
for all admissible .
By definition, quadratic -stability holds if and only if
exists such that, for all
satisfying

Fig. 8. Final design—nominal and perturbed

(1 = 61) step responses.

Equivalently, the inequality

(35)

should hold for any nonzero vector and admissible
, this result amounts to requiring
For fixed

.

whenever
(36)
Fig. 9. Rolloff in final design.

controller dynamics. Indeed, the fastest mode in the pure
controller is
, whereas it is only
in the
multi-objective controller. These improvements are secured
performance because
without tangible degradation of the
.
both designs have nearly optimal performance
Finally, Fig. 9 shows the final controller has adequate rolloff
properties.

is the unique solution
where

Observing
of the equation

(37)
An equivalent and simpler characterization of

is

Now,
any

ensures, for

together with
matrix

VII. CONCLUSION
Tractable analysis and synthesis techniques have been derived for robust pole placement in LMI regions. For analysis
with unstructured uncertainties, the Bounded Real Lemma
characterization of quadratic stability has been generalized
to pole clustering in arbitrary LMI regions. For parameter
uncertainty, two robust -stability tests have been derived
that rely on scaling and multiconvexity techniques. Although
both tests provide only sufficient conditions, they have proven
sharp in a number of applications. Finally, we have proposed
a tractable LMI-based approach to the synthesis of outputfeedback controllers that robustly assign the closed-loop poles
in a prescribed LMI region. Combination of robust pole
assignment with other closed-loop design specifications has
also been discussed.

Consequently, a sufficient condition for (36) to hold is that
whenever
(38)
or, equivalently, from the expression (37) for
whenever

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

Using a standard -procedure argument [36], [12], this condition is in turn equivalent (up to rescaling ) to the single
LMI constraint

inequalities (40) are collectively equivalent to

(39)
Finally, a Schur complement with respect to the block (3,3)
of (10) shows the equivalence between (39) and (10).
Remark A.1: Theorem 3.3 is nonconservative when
has rank one
. Indeed,
is then characterized
and coincides with the set of
by the relation
for some
.
vectors such that
Because the gap between these two sets is the only source
of conservatism in the proof, the LMI constraints (10), (11)
then become necessary and sufficient for quadratic -stability
against complex unstructured uncertainty. No scaling is needed
incurs no conservatism.
in this case; i.e., setting

whenever

(42)
Now, it is easily shown that

APPENDIX B
Proof of Theorem 4.1: First, observe for
valued function

, the matrix-

is concave in . As a result, (22) holds for all
if it
holds at the vertices of .
To establish well posedness, i.e, invertibility of
for all
, suppose
is singular for some , and
such that
.
consider
• Premultiplying and postmultiplying (21) by the full-rank
and its transpose shows the
matrix
matrix

————–

—-

— —

Using this simpler characterization, we can show any
satisfies

–

—-

–

—-

———-

———–

is negative definite
and
• Premultiplying and postmultiplying (22) by
its transpose shows the same matrix is positive definite,
a contradiction.
The remainder of the proof parallels that of Theorem 3.3.
-stability for the uncertain system (17) is guaranteed if
satisfies, for all
and

Using the expressions of
direct calculations show, for fixed

with the notation

—-

–

——— —–

— —-

———-

——–

—–

(40)

(simply premultiply and postmultiply (22) by
transpose). Consequently

(41)

(43) holds

and
,
, the -dependent

and a sufficient condition for (40) is (42) with

(43)

and its

replaced


CHILALI et al.: POLE PLACEMENT IN LMI REGIONS

by
, the latter being equivalent to (21) by a standard procedure argument [36], [12].
To complete the proof, we need to show (41) holds for all
. Suppose
for some
and nonzero .
Then, the left-hand side in (40) evaluates to zero when setting
, a contradiction. Hence,
cannot
be singular over , which together with
guarantees (41).
APPENDIX C

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H1

H1

Mahmoud Chilali, photograph and biography not available at the time of
publication.


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 12, DECEMBER 1999

Pascal Gahinet received the degree from the Ecole Polytechnique in 1984 and
from the E.N.S.T.A., Paris, France, in 1986, and the M.S. degree in electrical
and computer engineering and the Ph.D. degree in system theory from the
University of California, Santa Barbara, in 1987 and 1989, respectively.
He was with the French National Research Institute in Computer and
Control Sciences (I.N.R.I.A.) from 1990 to 1995, taught control at E.N.S.T.A.
from 1991 to 1994, and served as a Consultant for THOMSON-CSF and
A´erospatiale during this period. He has been with The MathWorks, Inc., since
April 1996 and is currently Senior Project Leader for the control and advanced
control product developments. His research interests include robust control

theory, linear matrix inequalities, numerical linear algebra, and numerical
software for control. He is coauthor of the LMI Control Toolbox and the
Control System Toolbox for use with MATLAB.

Pierre Apkarian (A’94) received the degree in engineering from the Ecole Sup´erieure d’Informatique,
Electronique, Automatique, Paris, in 1985, the M.S.
degree and “Diplˆome d’Etudes Appronfondies” in
mathematics from the University of Paris VII, in
1985 and 1986, and the Ph.D. degree in control
engineering from the “Ecole Nationale Sup´erieure
de I’A´eronautique et de Espace” (ENSAE) in 1988.
He has been a Research Scientist at ONERACERT and Associate Professor at ENSAE since
1988. His research interests include robust and gainscheduling control theory, linear matrix inequality techniques, mathematical
programming, and applications in aeronautics.
Dr. Apkarian is a member of SIAM.