Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

From External to Internal System Decompositions

The recently obtained approach to the construction of state maps, which is directly based on the linear differential operator describing the system, is shown to lead to an immediate and insightful relation between external and internal decompositions and symmetries of a linear system. This is applied to the decomposition of a system into its controllable and uncontrollable part (in the state space representation commonly referred to as the Kalman decomposition), and to the correspondence between external and internal symmetries

A graphic condition for the stability of dynamical distribution networks with flow constraints

We consider a basic model of a dynamical distribution network, modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge, subject to unknown but constant inflows and outflows. In [1] we showed how a distributed proportionalintegral controller structure, associating with every edge of the graph a controller state, regulates the state variables of the vertices, irrespective of the unknown constant inflows and outflows, in the sense that the storage variables converge to the same value (load balancing or consensus). In many practical cases, the flows on the edges are constrained. The main result of [1] is a sufficient and necessary condition, which only depend on the structure of the network, for load balancing for arbitrary constraint intervals of which the intersection has nonempty interior. In this paper, we will consider the question about how to decide the steady states of the same model as in [1] with given network structure and constraint intervals. We will derive a graphic condition, which is sufficient and necessary, for load balancing. This will be proved by a Lyapunov function and the analysis the kernel of incidence matrix of the network. Furthermore, we will show that by modified PI controller, the storage variable on the nodes can be driven to an arbitrary point of admissible set.Comment: submitted to MTNS 201

Constrained proportional integral control of dynamical distribution networks with state constraints

This paper studies a basic model of a dynamical distribution network, where the network topology is given by a directed graph with storage variables corresponding to the vertices and flow inputs corresponding to the edges. We aim at regulating the system to consensus, while the storage variables remain greater or equal than a given lower bound. The problem is solved by using a distributed PI controller structure with constraints which vary in time. It is shown how the constraints can be obtained by solving an optimization problem.Comment: CDC201

On interconnections of infinite-dimensional port-Hamiltonian systems

Network modeling of complex physical systems leads to a class of nonlinear systems called port-Hamiltonian systems, which are defined with respect to a Dirac structure (a geometric structure which formalizes the power-conserving interconnection structure of the system). A power conserving interconnection of Dirac structures is again a Dirac structure. In this paper we study interconnection properties of mixed finite and infinite dimensional port-Hamiltonian systems and show that this interconnection again defines a port-Hamiltonian system. We also investigate which closed-loop port-Hamiltonian systems can be achieved by power conserving interconnections of finite and infinite dimensional port-Hamiltonian systems. Finally we study these results with particular reference to the transmission line

Traffic control in coherence-multiplexed networks

Coherence multiplexing (CM) is a relatively unknown form of optical CDMA, which is particulary suitable in medium bit rate, short-range optical networks like LANs. The main purpose of the technique is to allow multiple users to transmit through a common optical fiber simultaneously. When this number is too large, however, the BER will become unacceptably high. Therefore a protocol is needed to control the traffic. In this paper several protocols are presented. An adapted version of synchronous TDMA, two new protocols and a central control unit will be proposed and discussed. Finally, the protocols will be compared with respect to performance and practical implementation aspects

Nonsquare Spectral Factorization for Nonlinear Control Systems

This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions can be easily constructed. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are give

Matching in the method of controlled Lagrangians and IDA-passivity based control

This paper reviews the method of controlled Lagrangians and the interconnection and damping assignment passivity based control (IDA-PBC)method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange system, respectively Hamiltonian system, by searching for a stabilizing structure preserving feedback law. The conditions under which two Euler-Lagrange or Hamiltonian systems are equivalent under feedback are called the matching conditions (consisting of a set of nonlinear PDEs). Both methods are applied to the general class of underactuated mechanical systems and it is shown that the IDA-PBC method contains the controlled Lagrangians method as a special case by choosing an appropriate closed-loop interconnection structure. Moreover, explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms. The $\lambda$-method as introduced in recent papers for the controlled Lagrangians method transforms the matching conditions into a set of linear PDEs. In this paper the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.\u