1,721,098 research outputs found
A Discrete-Time Formulation of Nonlinear Distributed-Parameter Port-Hamiltonian Systems
No full textThis paper introduces a new framework of nonlinear, discrete-time, boundary control systems (BCSs) in the port-Hamiltonian form. The contribution is twofold. We start with a discrete-time approximation of a nonlinear port-Hamiltonian BCS, i.e. a dynamical system modelled by a nonlinear partial differential equation with boundary actuation and sensing. The most important feature is that the discretisation is performed in time only so that the “distributed nature” of the state is preserved. By approximating the gradient of the Hamiltonian density with its discrete gradient, the obtained sampled dynamics inherit the passivity of the original one. Besides, we prove that, under mild conditions, it is well-posed, i.e. the “next” state always exists. The second contribution deals with control design. More precisely, we have determined sufficient conditions for the plant dynamics and a static output feedback gain to make the closed-loop system asymptotically stable
Trajectory Tracking for Discrete-Time Port-Hamiltonian Systems
Full text linkThis letter presents a regulator for nonlinear, discrete-time port-Hamiltonian systems that lets the state track a reference signal. Similarly to continuous-time approaches, the synthesis is based on the mapping via state-feedback of the open-loop error system to a target one in port-Hamiltonian form, and with an asymptotically stable origin that corresponds to the perfect tracking condition. The procedure is formally described by a matching equation that, in continuous-time, turns out to be a nonlinear partial differential equation (PDE). This is not the case for sampled-data systems, so an algebraic approach is proposed. The solution is employed to construct a dynamical regulator that performs an “approximated” mapping. The stability analysis relies on Lyapunov arguments
Control by interconnection beyond the dissipation obstacle of finite and infinite dimensional port-Hamiltonian systems
Full text linkBy exploiting the properties of the geometric structure of a port-Hamiltonian system, a general methodology for the definition of a new control port that allows to solve the so-called dissipation obstacle within the control by interconnection framework is discussed. This approach can be applied to a large class of port-Hamiltonian systems, both in the lumped, and in the distributed parameter cases. It is also shown how the limitations of the control by interconnection and energy-shaping via Casimir generation can be removed by interconnecting the controller to a different control port, i.e. how it is possible to compute a new passive output that is instrumental for removing the intrinsic constraints imposed by the dissipative structure of the system
Control Design for a Class of Discrete-Time Port-Hamiltonian Systems
No full textThe paper aims at extending the continuous-time energy-shaping plus damping injection control design technique to deal with a class of nonlinear, discrete-time port-Hamiltonian systems. For such systems, the gradient of the Hamiltonian function in the continuous-time dynamics is replaced by a discrete gradient, thus leading to a state equation in implicit form. Its well-posedness is studied both in the autonomous and non-autonomous cases to determine when the dynamical equation admits a solution for the next state. Based on this analysis, the extension of the energy-shaping plus damping injection control methodology is discussed. At first, it is supposed that the control action depends on the discrete gradient of an energy function. Then, this hypothesis is removed, and an algebraic solution to the matching equation is proposed to enlarge the class of stabilising controllers
Brayton-moser formulation of infinite dimensional port-hamiltonian systems with application to boundary control
No full textIn this paper, for a class of distributed port-Hamiltonian systems defined on a one-dimensional spatial domain, an equivalent Brayton-Moser formulation is provided. The dynamic is expressed as a gradient equation with respect to a new storage function, the 'mixed-potential,' with the dimensions of power. The system is then passive with respect to a supply rate that is related to the reactive power, and that depends on the boundary port variables and on their time derivatives. This equivalent representation is the starting point for the development of boundary control laws able to shape the mixed-potential function. Differently from energy-balancing control schemes, this technique allows to deal with pervasive dissipation in the system in an effective way. The general theory is illustrated with the help of an example, the boundary stabilisation of a transmission line with internal dissipation
Control by Interconnection of Mixed Port Hamiltonian Systems
No full textIn this note, the regulation problem for mixed finite and infinite dimensional port Hamiltonian systems (m-pH systems) is discussed. A m-pH system results from the power conserving interconnection of finite and infinite dimensional systems in port Hamiltonian form. In particular, the system given by the interconnection of two finite dimensional systems, one of which is the controller, by means of an infinite dimensional connection is studied. The proposed control methodology is a generalization to the infinite dimensional case of a well-established passivity-based control technique for finite-dimensional port Hamiltonian systems, the control by interconnection and energy shaping, according to which the open-loop energy function is shaped so that a minimum in the desired configuration is introduced. This procedure is possible once the state variable of the controller is related to the state variable of the plant by constraining the state of the closed-loop system on a structural invariant (defined by a set of Casimir functions). In this way, the energy function of the controller, which is freely assignable, becomes a function of the configuration of the plant and, then, it can be easily shaped in order to solve the regulation problem
Exponential Stabilisation of Port-Hamiltonian Boundary Control Systems via Energy-Shaping
Full text linkThis paper is concerned with the exponential stabilisation of a class of linear boundary control systems (BCS) in port-Hamiltonian form through energy-shaping. Starting from a first feedback loop that is in charge of modifying the Hamiltonian function of the plant, a second control loop that guarantees exponential convergence to the equilibrium is designed. In this way, a major limitation of standard energy-shaping plus damping injection control laws applied to linear port-Hamiltonian BCS, namely the fact that only asymptotic convergence is assured, has been removed
Stabilization of Unstable Distributed Port-Hamiltonian Systems in Scattering Form
Full text linkIn this letter, we consider the exponential stabilization of a distributed parameter port-Hamiltonian system interconnected with an unstable finite-dimensional linear system at its free end and control input at the opposite one. The infinite-dimensional system can also have in-domain anti-damping. The control design passes through the definition of a finite-dimensional linear system that “embeds” the response of the distributed parameter model, and that can be stabilized by acting on the available control input. The conditions that link the exponential stability of the latter system with the exponential stability of the original one are obtained thanks to a Lyapunov analysis. Simulations are presented to show the pros and cons of the proposed synthesis methodology
Dissipativity-based boundary control of linear distributed port-Hamiltonian systems
No full textThe main contribution of this paper is a general synthesis methodology of exponentially stabilising control laws for a class of boundary control systems in port-Hamiltonian form that are dissipative with respect to a quadratic supply rate, being the total energy the storage function. More precisely, general conditions that a linear regulator has to satisfy to have, at first, a well-posed and, secondly, an exponentially stable closed-loop system are presented. The methodology is illustrated with reference to two specific stabilisation scenarios, namely when the (distributed parameter) plant is in impedance or in scattering form. Moreover, it is also shown how these techniques can be employed in the analysis of more general systems that are described by coupled partial and ordinary differential equations. In particular, the repetitive control scheme is studied, and conditions on the (finite dimensional) linear plant to have asymptotic tracking of generic periodic reference signals are determined
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