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ABSTRACT: 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 (LMIs) 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 control
objectives, such as H<sub>2</sub> or H<sub>∞</sub> performance, to
capture realistic sets of design specifications. Physically motivated
examples demonstrate the effectiveness of this robust pole clustering
technique
IEEE Transactions on Automatic Control 01/2000; · 2.11 Impact Factor
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ABSTRACT: The authors present a comprehensive application of linear
fractional transformation and polytopic control techniques to the
control of an arm-driven inverted pendulum, The particular interest of
this application lies in the fact that all ingredients of the design
problem have to be taken into account; from the specifications up to the
constraints inherent to real-world implementations. In this context, it
has been shown that currently available synthesis methodologies, such as
μ and LPV techniques, may fail to provide acceptable answers, A major
obstacle is undoubtedly the implementation constraint that puts hard
limitations on the controller dynamics, These limitations are generally
difficult to handle within the usual formulation of LPV control
techniques. It has been shown that a suitable extension of these
techniques including LMI region constraints on the closed-loop dynamics
can overcome this difficulty. When implementable, it has been observed
that LPV controllers outperform fixed μ controllers both in
robustness and performance. These observations were confirmed by
simulations but more importantly by a number of records on the physical
experiment
IEEE control systems 03/1999; · 2.49 Impact Factor
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ABSTRACT: In this paper, we discuss some ideas for improving the efficiency
and accuracy of numerical methods for solving algebraic Riccati
equations (AREs) based on invariant or deflating subspace methods. The
focus is on AREs for which symmetric solutions exist, and our methods
apply to both standard linear-quadratic-Gaussian (or H<sub>2</sub>) AREs
and to so-called H<sub>∞</sub>-type AREs arising from either
continuous-time or discrete-time models. The first technique is a new
symmetric representation of a symmetric Riccati solution computed from
an orthonormal basis of a certain invariant or deflating subspace. The
symmetric representation does not require sign definiteness of the
Riccati solution. The second technique relates to improving algorithm
efficiency. Using a pencil-based approach, the solution of a Riccati
equation can always be reformulated so that the deflating subspace whose
basis is being sought corresponds to eigenvalues outside the unit
circle. Thus, the natural tendency of the QZ algorithm to deflate these
eigenvalues last, and hence, to appear in the upper left blocks of the
appropriate pencils, then reduces the amount of reordering that must be
done to a Schur form
IEEE Transactions on Automatic Control 10/1997; · 2.11 Impact Factor
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ABSTRACT: This paper presents an overview of a linear matrix inequality
(LMI) approach to the multiobjective synthesis of linear output-feedback
controllers. The design objectives can be a mix of H<sub>∞</sub>
performance, H<sub>2</sub> performance, passivity, asymptotic
disturbance rejection, time-domain constraints, and constraints on the
closed-loop pole location. In addition, these objectives can be
specified on different channels of the closed-loop system. When all
objectives are formulated in terms of a common Lyapunov function,
controller design amounts to solving a system of linear matrix
inequalities. The validity of this approach is illustrated by a
realistic design example
IEEE Transactions on Automatic Control 08/1997; · 2.11 Impact Factor
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ABSTRACT: This paper investigates the use of linear parameter-varying (LPV)
systems in the analysis and controller design for a nuclear pressurised
water reactor. The synthesis technique incorporates a priori bounds on
the rate of variation of the parameter, which leads to less conservative
designs. The synthesis procedure is formulated as a linear matrix
inequality (LMI)
Decision and Control, 1996., Proceedings of the 35th IEEE; 01/1997
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ABSTRACT: In this paper, the problem of robust stability of systems subject
to parametric uncertainties is considered. Sufficient conditions for the
existence of parameter-dependent Lyapunov functions are given in terms
of a criterion which is reminiscent of, but less conservative than,
Popov's stability criterion. An equivalent frequency-domain criterion is
demonstrated. The relative sharpness of the proposed test and existing
stability criteria is then discussed. The use of parameter-dependent
Lyapunov functions for robust controller synthesis is then considered.
It is shown that the search for robustly stabilizing controllers may be
limited to controllers with the same order as the original plant. A
possible synthesis procedure and a numerical example are then discussed
IEEE Transactions on Automatic Control 08/1996; · 2.11 Impact Factor
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ABSTRACT: This paper addresses the design of state- or output-feedback
H<sub>∞</sub> controllers that satisfy additional constraints on
the closed-loop pole location. Sufficient conditions for feasibility are
derived for a general class of convex regions of the complex plane.
These conditions are expressed in terms of linear matrix inequalities
(LMIs), and the authors' formulation is therefore numerically tractable
via LMI optimization. In the state-feedback case, mixed H<sub>2</sub>/H
<sub>∞</sub> synthesis with regional pole placement is also
discussed. Finally, the validity and applicability of this approach are
illustrated by a benchmark example
IEEE Transactions on Automatic Control 04/1996; · 2.11 Impact Factor
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ABSTRACT: This paper presents new tests to analyze the robust stability
and/or performance of linear systems with uncertain real parameters.
These tests are extensions of the notions of quadratic stability and
performance where the fixed quadratic Lyapunov function is replaced by a
Lyapunov function with affine dependence on the uncertain parameters.
Admittedly with some conservatism, the construction of such
parameter-dependent Lyapunov functions can be reduced to a linear matrix
inequality (LMI) problem and hence is numerically tractable. These
LMI-based tests are applicable to constant or time-varying uncertain
parameters and are less conservative than quadratic stability in the
case of slow parametric variations. They also avoid the frequency sweep
needed in real-μ analysis, and numerical experiments indicate that
they often compare favorably with μ analysis for time-invariant
parameter uncertainty
IEEE Transactions on Automatic Control 04/1996; · 2.11 Impact Factor
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ABSTRACT: In this paper, the problem of robust stability of systems subject
to parametric uncertainties is considered. Sufficient conditions for the
existence of parameter-dependent Lyapunov functions are given in terms
of a criterion which is reminiscent of but less conservative than
Popov's stability criterion. It is shown how the so-called S-procedure
plays a crucial role in the derivation of this criterion. A comparison
with existing stability criteria is done. An equivalent frequency-domain
criterion is given. Extensions to cover slowly time-varying systems and
robust performance are given
American Control Conference, 1995. Proceedings of the; 07/1995
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ABSTRACT: An important class of linear time-varying systems consists of
plants where the state-space matrices are fixed functions of some
time-varying physical parameters θ. Small gain techniques can be
applied to such systems to derive robust time-invariant controllers.
Yet, this approach is often overly conservative when the parameters
θ undergo large variations during system operation. In general,
higher performance can be achieved by control laws that incorporate
available measurements of θ and therefore “adjust” to
the current plant dynamics. This paper discusses extensions of
H<sub>∞</sub> synthesis techniques to allow for controller
dependence on time-varying but measured parameters. When this dependence
is linear fractional, the existence of such gain-scheduled H<sub>∞
</sub> controllers is fully characterized in terms of linear matrix
inequalities. The underlying synthesis problem is therefore a convex
program for which efficient optimization techniques are available. The
formalism and derivation techniques developed here apply to both the
continuous- and discrete-time problems. Existence conditions for robust
time-invariant controllers are recovered as a special case, and
extensions to gain-scheduling in the face of parametric uncertainty are
discussed. In particular, simple heuristics are proposed to compute such
controllers
IEEE Transactions on Automatic Control 06/1995; · 2.11 Impact Factor
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ABSTRACT: A new test of robust stability/performance is proposed for linear
systems with uncertain real parameters. This test is an extension of the
notion of quadratic stability where the fixed quadratic Lyapunov
function is replaced by a Lyapunov function with affine dependence on
the uncertain parameters. Admittedly with some conservatism, the
construction of such parameter-dependent Lyapunov functions can be
reduced to an linear matrix inequality (LMI) problem, hence is
numerically tractable. This LMI-based test can be used for both fixed or
time-varying uncertain parameters and is always less conservative than
the quadratic stability test whenever the parameters cannot vary
arbitrarily fast. Its also completely bypasses the frequency sweep
required in real μ-analysis
Decision and Control, 1994., Proceedings of the 33rd IEEE Conference on; 01/1995
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ABSTRACT: This paper describes a new MATLAB-based toolbox for control design
via linear matrix inequality (LMI) techniques. After a brief review of
LMIs and of some of their applications to control, the toolbox contents
and capabilities are presented
Decision and Control, 1994., Proceedings of the 33rd IEEE Conference on; 01/1995
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ABSTRACT: This paper is concerned with the application of advanced linear
parameter-varying (LPV) techniques to the global control of a missile.
The LPV technique considered in this paper is an extension of the
standard H<sub>∞</sub> synthesis technique to the case where the
plant depends affinely on a time-varying vector θ(t). Working in
the class of LPV plants, the proposed methodology produces an LPV
controller. That is, a controller which is automatically
“gain-scheduled” along the trajectories of the plant. LPV
controllers solutions to the problem are characterized via a set of
Riccati linear matrix inequalities (LMI) which can be solved using
convex programming. The missile under consideration is a very demanding
plant. The power and advantages of the proposed methodology as an
efficient tool to handle the global performances and robustness of the
missile on its whole operating range are demonstrated
Decision and Control, 1994., Proceedings of the 33rd IEEE Conference on; 01/1995
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ABSTRACT: In many control problems, the design constraints have natural formulations in terms of linear matrix inequalities (LMI). When no analytical solution is available, such problems can be attacked by solving the LMIs via convex optimization techniques. This paper describes the polynomial-time projective algorithm for the numerical solution of LMIs. Simple geometrical arguments are used to clarify the strategy and convergence mechanism of the projective method. A complexity analysis is provided, and applications to two generic LMI problems are discussed.
American Control Conference, 1994; 08/1994
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ABSTRACT: This paper is concerned with H <sub>∞</sub>-like control of a class of linear parameter-varying (LPV) plants. Here the state-space entries of the plant are assumed to depend affinely on a time-varying vector θ of real parameters which is measured in real-time. These parameter measurements are incorporated in the control law to optimize the performance and robustness of the closed-loop system. The resulting controller is therefore time-varying and automatically "gain-scheduled" along the parameter trajectories. Complete solvability conditions are obtained for continuous- and discrete-time systems in terms of linear matrix inequalities (LMI) and a physically motivated example demonstrates the advantages and performance of the proposed methodology.
American Control Conference, 1994; 08/1994
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ABSTRACT: This paper presents the software package LMI-LAB for the
manipulation and resolution of linear matrix inequalities (LMI). Fairly
general systems of LMI's can be handled as well as two important
optimization problems under LMI constraints. The polynomial-time
projectile method of Nesterov and Nemirovsky is used to solve the
underlying convex optimization programs. Several benchmark examples
demonstrate that the complexity and running time of these algorithms are
by no means prohibitive. This confirms that LMI formulations constitute
a computationally viable and reasonable approach to control system
design
Decision and Control, 1993., Proceedings of the 32nd IEEE Conference on; 01/1994
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[show abstract]
[hide abstract]
ABSTRACT: An important class of linear time-varying systems consists of
plants where the state-space coefficients are fixed functions of certain
time-varying physical parameters θ. Small gain techniques can be
applied to such systems to derive robust time-invariant controllers.
Yet, this approach is often unduly conservative when the parameters
θ undergo large variations during system operation. In
particular, higher performance can be achieved by control laws which
incorporate available measurements of θ and therefore
“adjust” to the current plant dynamics. This paper extends H
<sub>∞</sub>-like synthesis techniques to allow for controller
dependence on the time-varying plant parameters θ. The dependence
on θ is restricted to be linear fractional. The resulting
parameter-dependent output feedback problem is reformulated as a robust
performance problem with structured uncertainty and solved by elementary
state-space manipulations. Feasibility is characterized in terms of
linear matrix inequalities which can be solved by convex optimization
techniques. Finally a characterization of time-invariant robust
controllers is obtained as a special case
Decision and Control, 1993., Proceedings of the 32nd IEEE Conference on; 01/1994
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ABSTRACT: The continuous- and discrete-time H<sub>∞</sub> control
problems are solved via elementary manipulations on linear matrix
inequalities (LMI). Two interesting new features emerge through this
approach: solvability conditions valid for both regular and singular
problems, and an LMI-based parametrization of all
H<sub>∞</sub>-suboptimal controllers, including reduced-order
controllers. The solvability conditions involve Riccati inequalities
rather than the usual indefinite Riccati equations. Alternatively, these
conditions can be expressed as a system of three LMI's. Efficient convex
optimization techniques are available to solve this system. Moreover,
its solutions parametrize the set of H<sub>∞</sub> controllers and
bear important connections with the controller order and the closed-loop
Lyapunov functions. Thanks to such connections, the LMI-based
characterization of H<sub>∞</sub> controllers opens new
perspectives for the refinement of H<sub>∞</sub> design.
Applications to cancellation-free design and controller order reduction
are discussed and illustrated by examples
Decision and Control, 1993., Proceedings of the 32nd IEEE Conference on; 01/1994
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ABSTRACT: This paper is concerned with scaled H<sub>∞</sub> synthesis. Scaling is introduced to take into account structured problems in Linear Fractional form where the structured aspect comes from uncertainties and/or performance objectives. A convex characterisation to compute constant (frequency independent) scaling of transfer functions is presented. It is combined with H<sub>∞</sub> synthesis to provide a μ synthesis methodology. The proposed μ synthesis procedure is based on D - K iterations and is demonstrated for the multimodel synthesis problem of a flexible beam.
American Control Conference, 1993; 07/1993
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ABSTRACT: A few implementation methods are proposed to improve the
reliability of existing state-space algorithms for
L <sub>∞</sub>-norm computation. The themes successively
addressed include matrix pencil implementations, scaling/balancing and
robustness to ill-conditioning of the Hamiltonian spectrum
Decision and Control, 1992., Proceedings of the 31st IEEE Conference on; 02/1992
