Nicolas Hudon

Flatness-Based Extremum-Seeking Control Over Periodic Orbits

In this note, an extremum-seeking controller is developed to steer a periodic system to orbits that maximize a functional of interest for a class of differentially flat nonlinear systems. The problem is posed as a real-time optimal trajectory generation problem in which the optimal orbit is computed using an extremum-seeking approach. Using the flatness property of the dynamics, the original dynamic optimization problem is transformed to a parameterized optimization problem which can be solved in real-time to approximate the optimal orbit. The control algorithm provides tracking of the optimal orbit. A drug delivery control problem is considered to demonstrate the application of the technique.

Dissipativity-based decentralized control of interconnected nonlinear chemical processes

Abstract This paper presents an approach, based on dissipative systems theory, to the analysis and control design of interconnected nonlinear processes. The objective is to design distributed feedback controllers to achieve plant-wide stability. Extensions of classical results on the stability of large-scale interconnected systems lead to input–output dissipativity constraints for each subsystems, encoded as supply rates from input to output interconnecting ports. For each subsystem, a parameterized nonlinear feedback controller is designed using nonlinear dissipative inequalities to ensure that the aforementioned dissipativity constraints are met in closed-loop. One focus of this paper is the design of domination-based nonlinear feedback controllers to meet the above interconnection constraints. This paper also presents new results on the construction of storage functions for control affine systems, as a generalization of physics-based approaches to dissipative systems theory. Control of interconnected chemical reactors with a recycle stream is presented throughout the paper to demonstrate the proposed construction.

Representation and Control of Brayton—Moser Systems using a Geometric Decomposition

Abstract This paper considers the problem of representing a sufficiently smooth control affine system as a structured Brayton–Moser system and to use the obtained structure to stabilize a desired equilibrium of the system. The present note proposes a geometric decomposition technique to express a given vector field as a Brayton–Moser system with desired structure. The proposed method is based on a decomposition of a differential one-form that encodes the divergence of a given vector field into its exact and anti-exact components, by using a homotopy operator, and into its co-exact and anti-coexact components, by introducing a dual homotopy operator. This enables one to compute, via integration, the potential and the structure generating the drift vector field. By identification of the obtained structure and a desired structure, it is therefore possible to study the feedback realization problem of the Brayton–Moser structure and feedback stabilization of control affine systems. Application of the proposed constructive approach to the control of the three-dimensional rigid-body problem is presented to illustrate the propose approach.

A thermodynamic approach towards Lyapunov based control of reaction rate

Abstract This paper proposes a Lyapunov-based approach for the control of reaction rate involved in chemical reactors through the use of irreversible thermodynamics. More precisely, the reaction rate is structurally derived as a nonlinear function of the reaction force in order to ensure the inherent non-negative definiteness property of the irreversible entropy production due to the reaction. On this basis, it allows to cover a large class of reaction rates described by the mass-action-law. As a consequence, the control of the reaction rate consists in controlling the reaction force through the support of an affinity-related storage function to operate the entire system at a desired operating point. Besides, the convergence condition is given.

Stabilization of Nonlinear Systems via Potential-based Realization

This paper considers the problem of representing a sufficiently smooth nonlinear system as a structured potential-driven system and to exploit the obtained structure for the design of nonlinear state feedback stabilizing controllers. The problem has been studied in recent years for systems modeled as structured potential-driven systems, for example gradient systems, generalized Hamiltonian systems and systems given in Brayton-Moser form. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique to re-express a given vector field into a desired potential-driven form. The decomposition method is based on the Hodge decomposition theorem, where a one-form associated to the given vector field is decomposed into its exact, co-exact, and harmonic parts.

Stabilization of nonlinear systems via potential-based realization

This technical note considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation for stability analysis and state feedback controller design. These problems have been studied in recent years for particular classes of potential-driven systems. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique, based on the Hodge decomposition theorem, to re-express a given vector field into a potential-driven form. Using the proposed decomposition technique, stability conditions are developed based on the convexity of a computed potential. Finally, stabilization is studied in the context of the proposed decomposition by reshaping the Hessian matrix of the obtained potential using damping feedback.

Geometric decomposition and potential-based representation of nonlinear systems

This paper considers the problem of representing a sufficiently smooth nonlinear dynamical as a structured potential-driven system. The proposed approach is based on a decomposition of a differential one-form that encodes the divergence of the given vector fields into its exact and anti-exact components, and into its co-exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems, and generalized Hamiltonian systems are given to illustrate the proposed approach.

Feedback Stabilization of Metriplectic Systems

Abstract This paper considers the problem of stabilizing control affine systems where the drift dynamics is generated by a metriplectic structure. These systems can be viewed as an extension of generalized (or dissipative) Hamiltonian systems, where two potentials, interpreted as generalized energy and entropy, are generating the dynamics. The proposed approach consists in generating, by homotopy centered at an equilibrium of the system, a mixed potential for the metriplectic system, and in using the obtained potential to construct a damping state feedback controller. Stability of the closed-loop system is then considered.

Stability and feedback stabilization for a class of mixed potential systems

This paper studies the problems of stability analysis and feedback stabilization design for a class of control affine systems where the drift dynamics is generated by a metriplectic structure. Those systems are composed of a conserved part and a dissipative part and appear, for example, in non-equilibrium thermodynamics. They can be viewed as an extension of generalized (or dissipative) Hamiltonian systems, where two potentials, interpreted as generalized energy and entropy, are generating the dynamics. The proposed approach consists in constructing, by homotopy centered at an equilibrium of the system, a potential for the metriplectic system that can be used as a Lyapunov function candidate for the system, and in using the obtained potential to construct damping state feedback controllers. Stability of the closed-loop system is then considered.

Passivity and Passive Feedback Stabilization for a Class of Mixed Potential Systems

Abstract This paper studies the input-output properties of a class of control affine systems where the drift dynamics is generated by a metriplectic structure. Those systems, related to generalized (or dissipative) Hamiltonian systems, are generated by a conserved component and a dissipative component and appear, for example, in non-equilibrium thermodynamics. In non-equilibrium thermodynamics, the two potentials generating the dynamics are interpreted as generalized energy and generalized entropy, respectively. In this note, passivity and passive feedback stabilization of this class of systems are studied, with the output function taken as the gradient of the conserved component of the dynamics, and the proposed storage function is computed using the dissipative (metric) component of the dynamics.