172 research outputs found
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
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.
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
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
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 -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
Plug-and-play Solvability of the Power Flow Equations for Interconnected DC Microgrids with Constant Power Loads
In this paper we study the DC power flow equations of a purely resistive DC
power grid which consists of interconnected DC microgrids with constant-power
loads. We present a condition on the power grid which guarantees the existence
of a solution to the power flow equations. In addition, we present a condition
for any microgrid in island mode which guarantees that the power grid remains
feasible upon interconnection. These conditions provide a method to determine
if a power grid remains feasible after the interconnection with a specific
microgrid with constant-power loads. Although the presented condition are more
conservative than existing conditions in the literature, its novelty lies in
its plug-and-play property. That is, the condition gives a restriction on the
to-be-connected microgrid, but does not impose more restrictions on the rest of
the power grid.Comment: 8 pages, 2 figures, submitted to IEEE Conference on Decision and
Control 201
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