“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

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Stochastic Processes And Filtering Theory

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Introduction To Infinite Dimensional Linear Systems Theory

https://hal.archives-ouvertes.fr/hal-01363448/document

Recent developments on the stability of systems with aperiodic sampling: An overview

Hybrid data-driven fuzzy active disturbance rejection control for tower crane systems

https://hal.archives-ouvertes.fr/hal-01426572/document

Using exponential time-varying gains for sampled-data stabilization and estimation

Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a state-coordinate transformation. This study proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous- and discrete-time frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.

/pdf/overview-of-linear-time-invariant-interval-observer-design-2kxk82r016.pdf

Overview of linear time-invariant interval observer design: towards a non-smooth optimisation-based approach

The problem of state observation, based on spatially-sampled output measurements, is addressed for a class of infinite dimensional systems, modelled by a semi-linear heat equation augmented with a structured uncertain part involving a set of unknown parameters. An adaptive observer is designed that provides online estimates of the system (spatially distributed) state and unknown parameters based on sampled data (in space). Sufficient conditions for the observer to be exponentially convergent are established. These include an ad-hoc persistent excitation condition as well as a condition on how the observer gain must be selected in relation with the space sampling interval.

Adaptive Observer for a Class of Parabolic PDEs

The problem of observer design is addressed for output-injection nonlinear systems. A major difficulty with this class of systems is that the state equation involves an output-dependent term that is explicitly dependent on unknown parameters. As the output is only accessible to measurement at sampling times, the output-dependent term turns out to be (almost all time) subject to a double uncertainty, making previous adaptive observers inappropriate. Presently, a new hybrid adaptive observer is designed and shown to be exponentially convergent under ad-hoc conditions.

/pdf/sampled-data-adaptive-observer-for-a-class-of-state-affine-2qoepo69mb.pdf

Sampled-Data Adaptive Observer for a Class of State-Affine Output-Injection Nonlinear Systems

This paper proposes a key feature for vision based automatic landing of a UAV: the addition to a given control law of the constraint that a given ground point be maintained in the camera field of view (FoV). This feature is an application case of an output constraint method recently developed and available for any nonlinear system. This method is based on an observation made for relative degree one systems, and extended iteratively to any nonlinear system. The landing aircraft application case considered in this article consists of the design of a control law for an aircraft to maintain a given ground point in the camera field of view for visual servoing purpose. Simulation results confirm the effectiveness of the approach.

Laurent Burlion

Papers

Using exponential time-varying gains for sampled-data stabilization and estimation

Overview of linear time-invariant interval observer design: towards a non-smooth optimisation-based approach

Adaptive Observer for a Class of Parabolic PDEs

Sampled-Data Adaptive Observer for a Class of State-Affine Output-Injection Nonlinear Systems

Keeping a ground point in the camera field of view of a landing UAV