1,336 research outputs found

### Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems

The Jensen's inequality plays a crucial role in the analysis of time-delay
and sampled-data systems. Its conservatism is studied through the use of the
Gr\"{u}ss Inequality. It has been reported in the literature that fragmentation
(or partitioning) schemes allow to empirically improve the results. We prove
here that the Jensen's gap can be made arbitrarily small provided that the
order of uniform fragmentation is chosen sufficiently large. Non-uniform
fragmentation schemes are also shown to speed up the convergence in certain
cases. Finally, a family of bounds is characterized and a comparison with other
bounds of the literature is provided. It is shown that the other bounds are
equivalent to Jensen's and that they exhibit interesting well-posedness and
linearity properties which can be exploited to obtain better numerical results

### Convex lifted conditions for robust stability analysis and stabilization of linear discrete-time switched systems

Stability analysis of discrete-time switched systems under minimum dwell-time
is studied using a new type of LMI conditions. These conditions are convex in
the matrices of the system and shown to be equivalent to the nonconvex
conditions proposed by Geromel and Colaneri. The convexification of the
conditions is performed by a lifting process which introduces a moderate number
of additional decision variables. The convexity of the conditions can be
exploited to extend the results to uncertain systems, control design and
$\ell_2$-gain computation without introducing additional conservatism. Several
examples are presented to show the effectiveness of the approach.Comment: 9 pages, 3 figure

### Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

Linear Parameter-Varying (LPV) systems with jumps and piecewise
differentiable parameters is a class of hybrid LPV systems for which no
tailored stability analysis and stabilization conditions have been obtained so
far. We fill this gap here by proposing an approach relying on the
reformulation of the considered LPV system as an impulsive system that will
incorporate, through a suitable state augmentation, information on both the
dynamics of the state of the system and the considered class of parameter
trajectories. Conditions for the stability of the hybrid system, and hence that
of the associated LPV system, under both constant and minimum dwell-time are
established. Those results are based on the use of a clock- and
parameter-dependent Lyapunov function that is enforced to be decreasing along
the flow and the jumps of the system. An interesting adaptation of this result
consists of a minimum dwell-time stability condition for LPV switched impulsive
systems with time-varying dimensions. The minimum dwell-time stability
condition is notably shown to naturally generalize and unify the well-known
quadratic and robust stability criteria all together. Those conditions are then
adapted to address the stabilization problem via timer-dependent and a timer-
and/or parameter-independent (i.e. robust) state-feedback controllers. Finally,
the last part addresses the stability of LPV systems with jumps under a range
dwell-time condition which is then used to provide stabilization condition for
LPV systems using a sampled-data state-feedback gain-scheduled controller. The
obtained stability and stabilization conditions are formulated as
infinite-dimensional semidefinite programming problems which are solved using a
relaxation approach based on sum of squares programming. Examples are given for
illustration of the results.Comment: 24 pages, 6 figures, 1 tabl

### Stability and $L_1/\ell_1$-to-$L_1/\ell_1$ performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

Solutions to the interval observation problem for delayed impulsive and
switched systems with $L_1$-performance are provided. The approach is based on
first obtaining stability and $L_1/\ell_1$-to-$L_1/\ell_1$ performance analysis
conditions for uncertain linear positive impulsive systems in linear fractional
form with norm-bounded uncertainties using a scaled small-gain argument
involving time-varying $D$-scalings. Both range and minimum dwell-time
conditions are formulated -- the case of constant and maximum dwell-times can
be directly obtained as corollaries. The conditions are stated as
timer/clock-dependent conditions taking the form of infinite-dimensional linear
programs that can be relaxed into finite-dimensional ones using polynomial
optimization techniques. It is notably shown that under certain conditions, the
scalings can be eliminated from the stability conditions to yield equivalent
stability conditions on the so-called "worst-case system", which is obtained by
replacing the uncertainties by the identity matrix. These conditions are then
applied to the special case of linear positive systems with delays, where the
delays are considered as uncertainties. As before, under certain conditions,
the scalings can be eliminated from the conditions to obtain conditions on the
worst-case system, coinciding here with the zero-delay system -- a result that
is consistent with all the existing ones in the literature on linear positive
systems with delays. Finally, the case of switched systems with delays is
considered. The approach also encompasses standard continuous-time and
discrete-time systems, possibly with delays and the results are flexible enough
to be extended to cope with multiple delays, time-varying delays,
distributed/neutral delays and any other types of uncertain systems that can be
represented as a feedback interconnection of a known system with an
uncertainty.Comment: 35 pages; 13 figures. arXiv admin note: text overlap with
arXiv:1801.0378

### Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints

Impulsive systems are a very flexible class of systems that can be used to
represent switched and sampled-data systems. We propose to extend here the
previously obtained results on deterministic impulsive systems to the
stochastic setting. The concepts of mean-square stability and dwell-times are
utilized in order to formulate relevant stability conditions for such systems.
These conditions are formulated as convex clock-dependent linear matrix
inequality conditions that are applicable to robust analysis and control
design, and are verifiable using discretization or sum of squares techniques.
Stability conditions under various dwell-time conditions are obtained and
non-conservatively turned into state-feedback stabilization conditions. The
results are finally applied to the analysis and control of stochastic
sampled-data systems. Several comparative examples demonstrate the accuracy and
the tractability of the approach.Comment: 12 pages, 2 figures, 6 table

### Co-design of aperiodic sampled-data min-jumping rules for linear impulsive, switched impulsive and sampled-data systems

Co-design conditions for the design of a jumping-rule and a sampled-data
control law for impulsive and impulsive switched systems subject to aperiodic
sampled-data measurements are provided. Semi-infinite discrete-time
Lyapunov-Metzler conditions are first obtained. As these conditions are
difficult to check and generalize to more complex systems, an equivalent
formulation is provided in terms of clock-dependent (infinite-dimensional)
matrix inequalities. These conditions are then, in turn, approximated by a
finite-dimensional optimization problem using a sum of squares based
relaxation. It is proven that the sum of squares relaxation is non conservative
provided that the degree of the polynomials is sufficiently large. It is
emphasized that acceptable results are obtained for low polynomial degrees in
the considered examples.Comment: 27 pages; 5 figure

### Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints: L1- and Linfinity-gains characterization

Copositive linear Lyapunov functions are used along with dissipativity theory
for stability analysis and control of uncertain linear positive systems. Unlike
usual results on linear systems, linear supply-rates are employed here for
robustness and performance analysis using L1- and Linfinity-gains. Robust
stability analysis is performed using Integral Linear Constraints (ILCs) for
which several classes of uncertainties are discussed. The approach is then
extended to robust stabilization and performance optimization. The obtained
results are expressed in terms of robust linear programming problems that are
equivalently turned into finite dimensional ones using Handelman's Theorem.
Several examples are provided for illustration.Comment: Accepted in the International Journal of Robust and Nonlinear Contro

### Sign properties of Metzler matrices with applications

Several results about sign properties of Metzler matrices are obtained. It is
first established that checking the sign-stability of a Metzler sign-matrix can
be either characterized in terms of the Hurwitz stability of the unit
sign-matrix in the corresponding qualitative class, or in terms the negativity
of the diagonal elements of the Metzler sign-matrix and the acyclicity of the
associated directed graph. Similar results are obtained for the case of Metzler
block-matrices and Metzler mixed-matrices, the latter being a class of Metzler
matrices containing both sign- and real-type entries. The problem of assessing
the sign-stability of the convex hull of a finite and summable family of
Metzler matrices is also solved, and a necessary and sufficient condition for
the existence of common Lyapunov functions for all the matrices in the convex
hull is obtained. The concept of sign-stability is then generalized to the
concept of Ker$_+(B)$-sign-stability, a problem that arises in the analysis of
certain jump Markov processes. A sufficient condition for the
Ker$_+(B)$-sign-stability of Metzler sign-matrices is obtained and formulated
using inverses of sign-matrices and the concept of $L^+$-matrices. Several
applications of the results are discussed in the last section.Comment: 29 page

### A class of $L_1$-to-$L_1$ and $L_\infty$-to-$L_\infty$ interval observers for (delayed) Markov jump linear systems

We exploit recent results on the stability and performance analysis of
positive Markov jump linear systems (MJLS) for the design of interval observers
for MJLS with and without delays. While the conditions for the $L_1$
performance are necessary and sufficient, those for the $L_\infty$ performance
are only sufficient. All the conditions are stated as linear programs that can
be solved very efficiently. Two examples are given for illustration.Comment: 11 pages; 2 figure

### Stability and performance analysis of linear positive systems with delays using input-output methods

It is known that input-output approaches based on scaled small-gain theorems
with constant $D$-scalings and integral linear constraints are non-conservative
for the analysis of some classes of linear positive systems interconnected with
uncertain linear operators. This dramatically contrasts with the case of
general linear systems with delays where input-output approaches provide, in
general, sufficient conditions only. Using these results we provide simple
alternative proofs for many of the existing results on the stability of linear
positive systems with discrete/distributed/neutral time-invariant/-varying
delays and linear difference equations. In particular, we give a simple proof
for the characterization of diagonal Riccati stability for systems with
discrete-delays and generalize this equation to other types of delay systems.
The fact that all those results can be reproved in a very simple way
demonstrates the importance and the efficiency of the input-output framework
for the analysis of linear positive systems. The approach is also used to
derive performance results evaluated in terms of the $L_1$-, $L_2$- and
$L_\infty$-gains. It is also flexible enough to be used for design purposes.Comment: 34 page

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