Brayton-Moser Equations Research Articles

Passivity and Power Based Control of a Robot with Parallel Architecture

AbstractThis paper presents the design and implementation of two Passivity-based controllers (PBC) to the problem of trajectory tracking of a robot manipulator with a parallel architecture, one called IDA-PBC (Interconnection and Damping Assignment) and the other called Power-Shaping. For the first controller is used the model system described as a Port-controlled Hamiltonian (PCH), which provides the system be described in a appropriate form for applying balancing of energy. For the second controller, it is used a model described by Brayton-Moser equations, which describes the system for implementing the balancing of power. Through these two methods, at the end, it is obtained the same controller known as PD with gravity compensation, which is implemented in the manipulator in study.

Power balancing for a new class of non-linear systems and stabilization of RLC circuits

In this paper the method of power shaping, as recently introduced for the stabilization of non-linear RLC circuits, is generalized to a larger class of systems showing similarities (and important differences) with the class of port-controlled Hamiltonian systems. Other than for port-controlled Hamiltonian systems, the stabilization of these new systems is not stymied by a ‘dissipation obstacle’ and, in fact, every power-shaping controller is power balancing as well. It is shown that the power-shaping controller can be realized as a port-controlled Hamiltonian system connected by means of a gyrator to the plant. The theoretical results are applied to the class of non-linear RLC circuits described by Brayton–Moser's equations, and a physical implementation of the controllers in terms of standard electrical circuit elements is given.

Geometric modeling of nonlinear RLC circuits

In this paper, the dynamics of nonlinear RLC circuits including independent and controlled voltage or current sources is described using the Brayton-Moser equations. The underlying geometric structure is highlighted and it is shown that the Brayton-Moser equations can be written as a dynamical system with respect to a noncanonical Dirac structure. The state variables are inductor currents and capacitor voltages. The formalism can be extended to include circuits with elements in excess, as well as general noncomplete circuits. Relations with the Hamiltonian formulation of nonlinear electrical circuits are clearly pointed out.

Tuning of Passivity-Preserving Controllers for Switched-Mode Power Converters

Nonlinear passivity-based control (PBC) algorithms for power converters have proved to be an interesting alternative to other, mostly linear, control techniques. The control objective is usually achieved through an energy reshaping process and by injecting damping to modify the dissipation structure of the system. However, a key question that arises during the implementation of the controller is how to tune the various control parameters. From a circuit theoretic perspective, a PBC forces the closed-loop dynamics to behave as if there are artificial resistors-the control parameters-connected in series or in parallel to the real circuit elements. In this paper, a solution to the tuning problem is proposed that uses the classical Brayton-Moser equations. The method is based on the study of a certain "mixed-potential function" which results in quantitative restrictions on the control parameters. These restrictions seem to be practically relevant in terms stability, overshoot and nonoscillatory responses. The theory is exemplified using the elementary single-switch buck and boost converters.

ON NONLINEAR RLC NETWORKS: PORT-CONTROLLED HAMILTONIAN SYSTEMS DUALIZE THE BRAYTON-MOSER EQUATIONS

In this paper it is shown that the recently proposed port-controlled Hamiltonian systems with dissipation precisely dualize the classical Brayton-Moser equations. As a consequence, useful and important properties of the one framework can be translated to the other. For both frameworks a novel method is proposed to deal with networks containing capacitor-only loops or inductor-only cutsets using the Lagrange multiplier. This leads to the notion of implicit Brayton-Moser equations. Furthermore, the form and existence of the mixed-potential function is rederived from an external port point of view.

A thermodynamical approach to noise in non-linear networks

The physical assumptions on which linear fluctuation theory is based do not apply to internal noise in non-linear systems. As a consequence, there is still no complete theory for internal (thermal) noise in non-linear electrical networks. In this paper the authors apply Stratonovich's 'non-linear non-equilibrium thermodynamics' to describe thermal noise in reciprocal non-linear RLC-networks. As an example the (deterministic) Brayton-Moser equations are treated. A stochastic description in terms of master equations for the one-time probability density related to the Markov processes of the fluctuating electrical quantities is obtained. This decription is a generalization of Nyquist's theorem to non-linear resistors. When the non-linear resistances are represented by Taylor polynomials, different master equations are obtained for different degrees of non-linearity. A transformation of variables is used to apply the theory to non-linear dynamical network elements. To consider networks including independent sources, the theory is enlarged to open thermodynamical systems. The results are used to derive approximate noise-equivalent circuits for non-linear resistors. In this sense, this paper proves the approximate validity of 'conventional' noise source models and explicitly fixes the stochastic properties of these noise sources (as far as possible).

A Hamiltonian formulation for complete nonlinear RLC-networks

In this brief a new Lagrangian and Hamiltonian formulation for complete nonlinear RLC-networks is given. The formalism makes use of the explicit construction of Brayton and Moser's "mixed potential function" (1964) and therefore explicitly reveals the close connection between the Brayton-Moser equations and Hamiltonian formalisms for dissipative systems. On the basis of our Hamiltonian formalism the examination of noise by means of nonequilibrium statistical mechanics is possible. Furthermore stability propositions can be made that will be essentially different from those given by Brayton and Moser. Because of their complexity these and other possible applications of the formalism are not the subject of this brief.

The generalized Lagrange formulation for nonlinear RLC networks

Based on the concept of generalized Euler-Lagrange equations, this paper develops a Lagrange formulation of RLC networks of considerably broad scope. It is shown that the generalized Lagrange equations along with a set of compatibility constraint equations represents a set of governing differential equations of order equal to the order of complexity of the network. In this method the generalized coordinates include capacitor charges and inductor fluxes and the generalized velocities are comprised of an independent set of capacitor voltages and inductor currents. The generalized Hamilton equations are also developed and the connection with the Brayton-Moser equations is established.

Derivation of the Brayton–Moser equations from a topological mixed potential function

A topological mixed potential function, P * T, is defined and shown to be an exact differential. From P * T, the Brayton-Moser equations are obtained from topological relationships of the network and the constitutive relationships of the elements. A relationship between P * T and Brayton-Moser's mixed potential function, Q, is developed.

Formulation of the State-space Equations for Non-linear Networks

ABSTRACT The formulation of state-space equations is discussed for a general class of nonlinear networks, and a block diagram derived summarizing the general non-linear network state-space relationships. Certain special forms of the nonlinear state-space equations are then derived for restricted classes of network; these are the Brayton-Moser equations, the Lagrangian and co-Lagrangian equations and the Hamiltonian equations.