Hector Ramirez

Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback

It is shown that a strictly-input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain. This follows since the controller imposes exponential dissipation of the total energy. The result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled PDEs and ODEs. The result is specialized to port-Hamiltonian control systems and a simplified DNA-manipulation process is used to illustrate the result.

On the passivity based control of irreversible processes

Irreversible port-Hamiltonian systems (IPHS) have recently been proposed for the modelling of irreversible thermodynamic systems. On the other hand, a classical result on the use of the second law of thermodynamics for the stabilization of irreversible processes is the celebrated thermodynamic availability function. These frameworks are combined to propose a class of Passivity Based Controller (PBC) for irreversible processes. An alternative formulation of the availability function in terms of internal energy is proposed. Using IPHS a matching-condition, which is interpreted in terms of energy-shaping, is derived and a specific solution that permits to assign a desired closed-loop structure and entropy rate is proposed. The approach can be compared with Interconnection and Damping Assignment-PBC, this method however leads in general to thermodynamically non-coherent closed-loop systems. In this paper a system theoretic approach is employed to derive a constructive method for the control design. The closed-loop system is in IPHS form, hence it can be identified with a thermodynamic system and the control parameters related with thermodynamic variables, such as the reaction rates in the case of chemical reactions. A generic non-linear non-isothermal continuous stirred tank reactor is used to illustrate the approach.

On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems

This paper is concerned with the energy shaping of 1-D linear boundary controlled port-Hamiltonian systems. The energy-Casimir method is first proposed to deal with power preserving systems. It is shown how to use finite dimensional dynamic boundary controllers and closed-loop structural invariants to partially shape the closed-loop energy function and how such controller finally reduces to a state feedback. When dissipative port-Hamiltonian systems are considered, the Casimir functions do not exist anymore (dissipation obstacle) and the immersion (via a dynamic controller)/reduction (through invariants) method cannot be applied. The main contribution of this paper is to show how to use the same ideas and state functions to shape the closed-loop energy function of dissipative systems through direct state feedback i.e. without relying on a dynamic controller and a reduction step. In both cases, the existence of solution and the asymptotic stability (by additional damping injection) of the closed-loop system are proven. The general theory and achievable closed-loop performances are illustrated with the help of a concluding example, the boundary stabilization of a longitudinal beam vibrations.

Feedback equivalence of input–output contact systems

Abstract Control contact systems represent controlled (or open) irreversible processes which allow us to represent simultaneously the energy conservation and the irreversible creation of entropy. Such systems systematically arise in models established in Chemical Engineering. The differential-geometric of these systems is a contact form in the same manner as the symplectic 2-form is associated to Hamiltonian models of mechanics. In this paper we study the feedback preserving the geometric structure of controlled contact systems and render the closed-loop system again as a contact system. It is shown that only a constant control preserves the canonical contact form, hence a state feedback necessarily changes the closed-loop contact form. For strict contact systems, arising from the modelling of thermodynamic systems, a class of state feedback that shapes the closed-loop contact form and contact Hamiltonian function is proposed. The state feedback is given by the composition of an arbitrary function and the control contact Hamiltonian function. The similarity with structure preserving feedback of input–output Hamiltonian systems leads to the definition of input–output contact systems and to the characterization of the feedback equivalence of input–output contact systems. An irreversible thermodynamic process, namely the heat exchanger, is used to illustrate the results.

Modelling and control of multi-energy systems: An irreversible port-Hamiltonian approach

Abstract In recent work a class of quasi port Hamiltonian system expressing the first and second principle of thermodynamics as a structural property has been defined: Irreversible port-Hamiltonian system. These systems are very much like port-Hamiltonian systems but differ in that their structure matrices are modulated by a non-linear function that precisely expresses the irreversibility of the system. In a first instance irreversible port-Hamiltonian systems are extended to encompass coupled mechanical and thermodynamical systems, leading to the definition of reversible–irreversible port Hamiltonian systems. In a second instance, the formalism is used to suggest a class of passivity based controllers for thermodynamic systems based on interconnection and Casimir functions. However, the extension of the Casimir method to irreversible port-Hamiltonian systems is not so straightforward due to the “interconnection obstacle”. The heat exchanger, a gas-piston system and the non-isothermal CSTR are used to illustrate the formalism.

Energy shaping of boundary controlled linear port Hamiltonian systems

In this paper, we consider the asymptotic stabilization of a class of one dimensional boundary controlled port Hamiltonian systems by an immersion/reduction approach and the use of Casimir invariants. We first extend existing results on asymptotic stability of linear infinite dimensional systems controlled at their boundary to the case of stable Port Hamiltonian controllers including some physical constraints as clamping. Then the relation between structural invariants, namely Casimir functions, and the controller structure is computed. The Casimirs are employed in the selection of the controllers Hamiltonian to shape the total energy function of the closed loop system and introduce a minimum in the desired equilibrium configuration. The approach is illustrated on the model of a micro manipulation process with partial-actuation on one side of the spatial domain.

Passivity Based Control of Irreversible Port Hamiltonian Systems

The frameworks of thermodynamic availability function and irreversible port Hamiltonian systems are used to derive passivity based control strategies for irreversible thermodynamic systems. An energy based availability function is defined using as generating function the internal energy. This is a variation with respect to previous works where the total entropy usually corresponds to the generating function. The specific structure of irreversible port-Hamiltonian systems then permits to elegantly derive stability conditions for open and closed thermodynamic systems. The results are illustrated on two classical thermodynamic examples: The heat exchanger and the continuous stirred tank reactor.

Irreversible port Hamiltonian systems

Abstract A class of quasi port Hamiltonian system expressing the first and second principle of thermodynamic as a structural property is defined, namely Irreversible PHS. The IPHS is defined by: a generating function that for physical systems corresponds to the total energy; a constant skew-symmetric structure matrix that represents the network structure of the system; a non-linear function that depends on the states and co-states and on the Poisson bracket of the generating function and some entropy function. For physical systems this Poisson bracket defines the thermodynamic driving force. The IPHS is completed with input and output ports. IPHS encompasses a large set of thermodynamic systems, including heat exchangers and chemical reactors. The non-isothermal CSTR is used to illustrate the formalism.

Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold

This technical note addresses the structure preserving stabilization by output feedback of conservative input-output contact systems, a class of input-output Hamiltonian systems defined on contact manifolds. In the first instance, achievable contact forms in closed-loop and the associated Legendre submanifolds are analysed. In the second instance the stability properties of a hyperbolic equilibrium point of a strict contact vector field are analysed and it is shown that the stable and unstable manifolds are Legendre submanifolds. In the third instance the consequences for the design of stable structure preserving output feedback are derived: in closed-loop one may achieve stability only relatively to some invariant Legendre submanifold of the closed-loop contact form and furthermore this Legendre submanifold may be used as a control design parameter. The results are illustrated along the technical note on the example of heat transfer between two compartments and a controlled thermostat.

Exponential stability of a class of PDE's with dynamic boundary control

We show that a finite dimensional strictly passive linear controller exponentially stabilizes a large class of partial differential equations which are actuated through its boundaries on a one dimensional spatial domain. This is achieved by extending existing results on exponential stability of boundary control system with static boundary control to the case with dynamic boundary control. The approach is illustrated on a physical example.