## Section: New Results

### New techniques of observation and control

#### Interval observers

Participants : Frédéric Mazenc, Silviu-Iulian Niculescu, Olivier Bernard.

The *interval observer* method is a recent state estimation technique.
It is used in particular in biological contexts, where taking into account
the presence of uncertainties is essential. We have completed the theory of
the linear interval observers in several works.

1. The contribution of the work [26] is twofold.
A first part of our work is devoted to the problem of exhibiting
necessary and sufficient conditions which guarantee that, for a time-invariant
linear system of dimension 2, a time-invariant linear
and exponentially stable interval observer can be constructed.
In the second part of the work, when these conditions
are violated, we have shown that one can still construct exponentially stable linear interval
observers but these interval observers have the
remarkable feature of being *time-varying*.
Thus, we managed to give a complete picture of the difficulties and
solutions which arise for systems of dimension two.
To illustrate the power of our approach, we have applied it to a chaotic
system which is known to be highly sensitive to uncertainties in the
initial conditions.

2. The work [27] (see also [54] ) presents a solution to the problem of constructing exponentially stable interval observers for any time-invariant exponentially stable system. This result, which is constructible, relies on two crucial steps. The first step consists in transforming, through a time-invariant change of coordinates, the system under consideration into a system of the Jordan form. The second consists in determining a time-varying change of coordinates which transforms the system in Jordan form into a cooperative time-invariant system (recall that a linear system is cooperative if it is associated to a matrix whose off-diagonal entries are nonnegative).

3. In [97] , we investigated
the problem of constructing interval observers for exponentially stable
linear systems with point-wise delays. First, we proved that classical
interval observers for systems without delays are *not robust* with
respect to the presence of delays that appear in a specific structure
location, no matter how small the delay is.
Next, we have shown that, in general, for linear systems
classical interval observers endowed with a point-wise
delay are not satisfactory because they are exponentially unstable.
Finally, we have designed interval observers of a new type.
Our construction relies on framers that incorporate *distributed
delay* terms. These framers are interval observers when the delay
is smaller than an upper bound that we have estimated.

#### Finding positive solutions of systems with delay

Participants : Frédéric Mazenc, Silviu-Iulian Niculescu.

In [34] (see also [59] ), a new technique of design of feedbacks for systems with pointwise delays is developed. It relies on the introduction of an operator which has remote connections with the ones used in reduction model approaches. Using this operator, one succeeds, in many cases, to rewrite the closed-loop system we consider into the interconnection of an ordinary differential equation with an integral equation. An important advantage of this representation, is that it allows to derive simple conditions, in terms of initial conditions, ensuring that the resulting solutions of the closed-loop systems are positive. It is worth mentioning that our wish to determine positive solutions for systems with delay had several strong motivations: when this objective is reached, one can easily solve more general problems: in particular, we have shown that our new result can be used to generate solutions which respect to more general constraints than the constraint of sign and solutions which can be compared between each other, which is useful when is only available an approximate knowledge of the initial condition and an estimation of the state variables at each instant is desirable.

#### Lyapunov-based results

Participants : Frédéric Mazenc, Claudio De Persis, Michael Malisoff, Marcio De Queiroz, Olivier Bernard.

We did some works in which strict Lyapunov functions play a central role.

1. *Systems with quantized feedback.*

Quantized control systems are systems in which the control law is a piece-wise constant function of time taking values in a finite set. The design of quantized control systems is based on a partition of the state space. The aim of [18] was to design control laws for general families of quantized time-delay control systems. Our approach relies on the construction of Lyapunov-Krasowskii functionals and provided with quantized feedbacks which are parametrized with respect to the quantization density. Our approach leads to a set of conditions to design quantized control systems which are robust with respect to delays.

2. *Strict Lyapunov functions under LaSalle conditions*.

In [33] , we provided new techniques for building explicit global strict Lyapunov functions for broad classes of periodic time-varying nonlinear systems satisfying LaSalle conditions. Our new constructions are simpler than the designs available in the literature. We illustrated our work using the Lotka-Volterra model, which plays a fundamental role in bioengineering. We used our strict Lyapunov constructions to prove robustness of the Lotka-Volterra tracking dynamics to uncertainty in the death rates.

3. *Adaptive control*.

In [30] and [56] , we studied adaptive tracking problems for nonlinear systems with unknown control gains. We constructed controllers that yield uniform global asymptotic stability for the error dynamics, hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrated our work using a brushless DC motor turning a mechanical load. We quantified the effects of time-varying uncertainties on the motor electric parameters.

#### Predictive control

Participant : Sorin Olaru.

1. In [39] , new results have been obtained toward predictive control design for nonlinear systems upon optimization-in-the-loop techniques for an air ventilation problem. From a methodological point of view, the use of local embeddings of the nonlinearity led to polytopic differential inclusions [19] . This were further used for the nonlinear predictive control synthesis. A generic procedure which deals with the state-space partitioning for a multi-model description and subsequently obtain the control law by means of LMIs have been presented in [19] . An interesting aspect of this result is the explicit formulation of the control law in terms of patchy feedback gains.

2. On NCS related topics, several results can be mentioned with respect to the construction of polytopic embeddings for linear systems affected by variable time delays. In [21] , the Cayley-Hamilton approach was investigated while in [48] a comparison is made with respect to the alternative methods (Taylor series approximations, truncations, Jordan normal forms). For the same class of systems, several results have been reported for the adaptation of predictive control techniques for their constraints handling mechanisms (see [52] and [53] ). For the stability point of view, an interesting aspect is the characterization of terminal invariant sets [53] , a research subject which receives currently a renewed attention.

3. The fragility of proportional-derivative controllers has been studied in relation with robotics/tele-operation application in the two recent publications [50] , [51] .

#### Comparison systems

Participants : André Fioravanti, José Geromel [Unicamp] , Rubens Korogui [Unicamp] .

A new idea to the study of -stability and filter [92] and control [93] designs can be obtained from the Rekasius transformation of the delay term in frequential term. After the relations of stability and norms between the comparison and real systems are obtained, classical techniques involving Riccati equation from the theory of LTI systems are used to derive infinite-dimensional filters and controllers for time-delay systems. All implementation issues are discussed in order to provide a new and easy to implement technique.

#### Discrete Markov jump systems

Participants : André Fioravanti, José Géromel [Unicamp] , Alim Gonçalves [Unesp] .

In the last years, the interest on
networked control has enormously increased. Actually, communication
networks inherently introduce packet dropouts, quantisation errors,
time-delays and limited bandwidths. Up to now, the most successful way
to model packet dropouts are stochastic markovian systems, since they
are the basic mathematical tool for the models of many different
networks. In [22] and [23] , the H_{2} and
-filtering,
and in [67] , the state feedback design problems of those
systems are addressed.

#### Stochastic optimization in energy production systems

Participants : Henri Borsenberger, Philippe Dessante, Jorge Luis Reyespesantez, Guillaume Sandou.

Continuing the collaboration with the Energy Department of Supélec, the use of robust optimization methods has allowed the computation of energy production control laws taking into account various uncertainties on the plant. Main uncertainties, which have been taken into account, are the consumer load prediction, the maximum unit production level and the production costs. Results exhibit a more robust technical behavior together with a decrease of global operation costs [13] , [41] , [66] .

#### Use of metaheuristic optimization methods for automatic control

Participants : Gilles Duc, Bianca Minodora Heiman, Saïd Ighobriouen, Gabriela Raduinea, Guillaume Sandou, Mohamed Yagoubi.

The development of generic methodologies for automatic control based on metaheuristic methods has led to several promising results. Among them are the automatic computation of weigthing filters and the design of static output feedbacks using Particle Swarm Optimization, and the use of ant colony optimization for the identification of nonlinear systems. Some promising results have also been obtained using multiobjective Particle Swarm Optimization [49] , [73]