arXiv subjects
Publications and source records attributed to Arjan van der Schaft.
The standard dissipation inequality for passivity is extended from storage functions to general Lagrange subspaces. This is shown to have some interesting consequences. A classical factorization result for passive systems is extended to this generalized case, making use of the newly defined concept of the Hamiltonian lift of a DAE system.
In this note we address the problem of indirect adaptive (regulation or tracking) control of nonlinear, input affine dissipative systems. It is assumed that the supply rate, the storage and the internal dissipation functions may be expressed as nonlinearly parameterized regression equations where the mappings (depending on the unknown parameters) satisfy a monotonicity condition -- this encompasses a large class of physical systems, including passive systems. We propose to estimate the system parameters using the "power-balance" equation, which is the differential version of the classical dissipation inequality, with a new estimator that ensures global, exponential, parameter convergence under the very weak assumption of interval excitation of the power-balance equation regressor. To design the indirect adaptive controller we make the standard assumption of existence of an asymptotically stabilizing controller that depends -- possibly nonlinearly -- on the unknown plant parameters, and apply a certainty-equivalent control law. The benefits of the proposed approach, with respect to other existing solutions, are illustrated with examples.
As shown in earlier work, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended to pairs of linear differential operators defining a formally skew-adjoint relation between flows and efforts. Furthermore it is shown how the underlying repeated integration by parts operation can be streamlined by the use of two-variable polynomial calculus. Dirac structures defined by formally skew adjoint operators and differential operator effort constraints are treated within the same framework. Finally it is sketched how the approach can be also used for Lagrangian subspaces on bounded domains.
The power conserving interconnection of port-thermodynamic systems via their power ports results in another port-thermodynamic system, while the same holds for the rate of entropy increasing interconnection via their entropy flow ports. Control by interconnection of port-thermodynamic systems seeks to control a plant port-thermodynamic system by the interconnection with a controller port-thermodynamic system. The stability of the interconnected port-thermodynamic system is investigated by Lyapunov functions based on generating functions for the submanifold characterizing the state properties as well as additional conserved quantities. Crucial tool is the use of canonical point transformations on the symplectized thermodynamic phase space.
In the contact-geometric formulation of classical thermodynamics distinction is made between the energy and entropy representation, which can be resolved by taking homogeneous coordinates for the intensive variables. This results in a geometric formulation on the cotangent bundle of the manifold of extensive variables, where all geometric objects are homogeneous in the cotangent variables. The resulting geometry is studied in-depth. Additional homogeneity with respect to the extensive variables, corresponding to the classical Gibbs-Duhem relation, is treated within the same geometric framework.
In this two-part paper we develop a unifying framework for the analysis of the feasibility of the power flow equations for DC power grids with constant-power loads. In Part I of this paper we present a detailed introduction to the problem of power flow feasibility of such power grids, and the associated problem of selecting a desirable operating point which satisfies the power flow equations. We introduce and identify all long-term voltage semi-stable operating points, and show that there exists a one-to-one correspondence between such operating points and the constant power demands for which the power flow equations are feasible. Such operating points can be found by solving an initial value problem, and a parametrization of these operating points is also obtained. In addition, we give a full characterization of the set of all feasible power demands, and give a novel proof for the convexity of this set. Moreover, we present a necessary and sufficient LMI condition for the feasibility of a vector of power demands under small perturbation, which extends a necessary condition in the literature.
In this two-part paper we develop a unifying framework for the analysis of the feasibility of the power flow equations for DC power grids with constant-power loads. Part II of this paper explores further implications of the results in Part I. In particular, we refine several results in Part I to obtain a necessary and sufficient condition for the feasibility of nonnegative power demands, which is cheaper to compute than the necessary and sufficient LMI condition in Part I. Moreover, we prove two novel sufficient conditions, which generalize known sufficient conditions for power flow feasibility in the literature. In addition, we prove that the unique long-term voltage semi-stable operating point associated to a feasible vector of power demands is a strict high-voltage solution. A parametrization of such operating points, which is dual to the parametrization in Part I, is also obtained, along with a parametrization of the boundary of the set of feasible power demands.
The purpose of this paper is two-fold. First, to make clear (and de-mystify) the basic concepts of classical thermodynamics, and thus to enable the integration of thermodynamics within systems modeling and control. Second, to demonstrate that systems and control theory provides a natural context for the formulation and understanding of classical thermodynamics. This is not so surprising since classical thermodynamics, firmly rooted in engineering with questions such as the maximal efficiency of steam engines, deals from the very start with systems in interaction with their surrounding (by heat flow, mechanical work, flow of matter, etc.). In particular, it will be shown that dissipativity theory is key in the formulation and interpretation of the First and Second Law of thermodynamics. Also a geometric view on the state properties and the dynamics of thermodynamic systems will be emphasized, thereby unifying and simplifying different representations of thermodynamic systems. On the other hand, I will also argue that thermodynamics motivates paradigm shifts within systems and control; in particular, the use of non-minimal state space formulations, and a geometric view on them. Furthermore, while systems and control theory has been primarily based on linear systems with quadratic cost criteria, in line with basic system models in electrical and mechanical engineering (RLC-circuits, mass-spring-damper systems, etc.), thermodynamics necessitates to go beyond this linear-quadratic paradigm.
In this paper, we provide a graphic formulation of non-isothermal reaction systems and show that a non-isothermal detailed balanced network system converges (locally) asymptotically to the unique equilibrium within the invariant manifold determined by the initial condition. To model thermal effects, the proposed modeling approach extends the classical chemical reaction network by adding two parameters to each direct (reaction) edge, depicting, respectively, the instantaneous internal energy change after the firing of the reaction and the variation of the reaction rate with respect to the temperature. For systems possessing thermodynamic equilibria, our modeling approach provides a compact formulation of the dynamics where reaction topology and thermodynamic information are presented simultaneously. Finally, using this formulation and the Legendre transformation, we show that non-isothermal detailed balanced network systems admit some fundamental properties: dissipativeness, the detailed balancing of each equilibrium, the existence and uniqueness of the equilibrium, and the asymptotic stability of each equilibrium. In general, the analysis and results of this work provide insights into the research of non-isothermal chemical reaction systems.
It is rigorously proved that ideal memcapacitors and meminductors are not passive or lossless devices, nor are they satisfying the weaker notion of cyclo-passivity, which arises when dropping the requirement of non-negativity of the storage function. Equivalently, this implies that there exist excitation profiles that allow to extract more energy from the device than it is supplied with; so that their energy conversion efficiency exceeds 100%. This means that ideal memcapacitors and meminductors constitute so-called overunity systems. An illustrative mechanical analogue is provided that explicitly confirms this property. Hence, the question arises if ideal memcapacitors and meminductors will just remain some mathematical toys or artefacts.
A consequence of the Second Law of thermodynamics is that no thermodynamic system with a single heat source at constant temperature can convert heat into mechanical work in a recurrent manner. First we note that this is equivalent to cyclo-passivity at the mechanical port of the thermodynamic system, while the temperature at the thermal port of the system is kept constant. This leads to the general system-theoretic question which systems with two power ports have similar behavior: when is a system cyclo-passive at one of its ports, while the output variable at the other port (such as the temperature in the thermodynamic case) is kept constant? This property is called `one-port cyclo-passivity', and entails, whenever it holds, a fundamental limitation to energy transfer from one port (where the output is kept constant) to the other port. Sufficient conditions for one-port cyclo-passivity are derived for systems formulated in general port-Hamiltonian form. This is illustrated by a variety of examples from different {(multi-)}physical domains; from coupled inductors and capacitor microphones to synchronous machines.
Starting from a symmetrization and extension of the basic definitions and results of dissipativity theory we obtain new results on cyclo-dissipativity; in particular their external characterization and description of the set of storage functions.
In this work we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots (FJRs) in the port-Hamiltonian (pH) framework. The proposed design method, called virtual contraction based control (v-CBC), combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity and the existence of (incrementally) passive maps.
After recalling standard nonlinear port-Hamiltonian systems and their algebraic constraint equations, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in Barbero-Linan et al., and extending the linear theory as developed in Van der Schaft et al., Beattie et al.. The resulting new type of algebraic constraints equations are called Lagrange constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.
This paper introduces assume/guarantee contracts on continuous-time control systems, hereby extending contract theories for discrete systems to certain new model classes and specifications. Contracts are regarded as formal characterizations of control specifications, providing an alternative to specifications in terms of dissipativity properties or set-invariance. The framework has the potential to capture a richer class of specifications more suitable for complex engineering systems. The proposed contracts are supported by results that enable the verification of contract implementation and the comparison of contracts. These results are illustrated by an example of a vehicle following system.
Since the 1970s contact geometry has been recognized as an appropriate framework for the geometric formulation of the state properties of thermodynamic systems, without, however, addressing the formulation of non-equilibrium thermodynamic processes. In Balian & Valentin (2001) it was shown how the symplectization of contact manifolds provides a new vantage point; enabling, among others, to switch between the energy and entropy representations of a thermodynamic system. In the present paper this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, as already largely present in the literature, appears to be elegant and effective. This culminates in the definition of port-thermodynamic systems, and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.
Output impedances are inherent elements of power sources in the electrical grids. In this paper, we give an answer to the following question: What is the effect of output impedances on the inductivity of the power network? To address this question, we propose a measure to evaluate the inductivity of a power grid, and we compute this measure for various types of output impedances. Following this computation, it turns out that network inductivity highly depends on the algebraic connectivity of the network. By exploiting the derived expressions of the proposed measure, one can tune the output impedances in order to enforce a desired level of inductivity on the power system. Furthermore, the results show that the more "connected" the network is, the more the output impedances diffuse into the network. Finally, using Kron reduction, we provide examples that demonstrate the utility and validity of the method.
We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.