Feedback controllers for port-Hamiltonian systems reveal an intrinsic inverse optimality property since each passivating state feedback controller is optimal with respect to some specific performance index. Due to the nonlinear port-Hamiltonian system structure, however, explicit (forward) methods for optimal control of port-Hamiltonian systems require the generally intractable analytical solution of the Hamilton-Jacobi-Bellman equation. Adaptive dynamic programming methods provide a means to circumvent this issue. However, the few existing approaches for port-Hamiltonian systems hinge on very specific sub-classes of either performance indices or system dynamics or require the intransparent guessing of stabilizing initial weights. In this paper, we contribute towards closing this largely unexplored research area by proposing a time-continuous adaptive feedback controller for the optimal control of general time-continuous input-state-output port-Hamiltonian systems with respect to general Lagrangian performance indices. Its control law implements an online learning procedure which uses the Hamiltonian of the system as an initial value function candidate. The time-continuous learning of the value function is achieved by means of a certain Lagrange multiplier that allows to evaluate the optimality of the current solution. In particular, constructive conditions for stabilizing initial weights are stated and asymptotic stability of the closed-loop equilibrium is proven. Our work is concluded by simulations for exemplary linear and nonlinear optimization problems which demonstrate asymptotic convergence of the controllers resulting from the proposed online adaptation procedure.
We study $H_\infty$ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model.
We consider nonlinear electrical circuits for which we derive a port-Hamiltonian formulation. After recalling a framework for nonlinear port-Hamiltonian systems, we model each circuit component as an individual port-Hamiltonian system. The overall circuit model is then derived by considering a port-Hamiltonian interconnection of the components. We further compare this modelling approach with standard formulations of nonlinear electrical circuits
We present an adaptive sampling strategy for the optimization-based structure preserving model order reduction (MOR) algorithm developed in [Schwerdtner, P. and Voigt, M. (2020). Structure preserving model order reduction by parameter optimization, Preprint arXiv:2011.07567]. This strategy reduces the computational demand and the required a priori knowledge about the given full order model, while at the same time retaining a high accuracy compared to other structure preserving but also unstructured MOR algorithms. A numerical study with a port-Hamiltonian benchmark system demonstrates the effe…
The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together with their generating functions. This gives rise to new classes of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cy…
We study port-Hamiltonian systems with two external ports, and the strategies and limitations for conversion of energy from one port into the other. It turns out that, apart from the cyclo-passivity of port-Hamiltonian systems, this is related to the internal connection structure of port-Hamiltonian systems. A source of motivation for energy conversion is provided by thermodynamics, in particular the Carnot theory of conversion of thermal into mechanical energy. This is extended to general port-Hamiltonian systems which are satisfying structural conditions on their topology; thus generalizing …
Discrete-Time Models for Implicit Port-Hamiltonian Systems
Cinvestav del IPN（墨西哥国立理工学院高级研究中心）；McGill Centre for Intelligent Machines（麦吉尔大学智能机器中心）；St.Petersburg State University（圣彼得堡国立大学）；UPMC Paris 6, UMR7222, Institut de Systèmes et de Robotique（巴黎第六大学，UMR7222，系统与机器人研究所）
We consider the problem of minimizing the supplied energy of infinite-dimensional linear port-Hamiltonian systems and prove that optimal trajectories exhibit the turnpike phenomenon towards certain subspaces induced by the dissipation of the dynamics.
The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for a port-Hamiltonian systems with invertible feedthrough as another port-Hamiltonian system on the same state space is given. It is shown that the zero dynamics for any port-Hamiltonian system with commensurate wave speeds are well-defined, and are also a port-Hamiltonian system. Examples include wave equations wit…
Stochastic port-Hamiltonian systems on infinite-dimensional spaces governed by Itô stochastic differential equations (SDEs) are introduced and some properties of this new class of systems are studied. They are an extension of stochastic port-Hamiltonian systems defined on a finite-dimensional state space. The concept of well-posedness in the sense of Weiss and Salamon is generalized to the stochastic context. Under this extended definition, stochastic port-Hamiltonian systems are shown to be well-posed. The theory is illustrated on an example of a vibrating string subject to a Hilbert space-va…
Nonlinear port-Hamiltonian system identification from input-state-output data (ISO-pHNN)
Université Marie et Louis Pasteur, SUPMICROTECH, CNRS, institut FEMTO-ST（玛丽和路易·巴斯德大学、SUPMICROTECH、法国国家科学研究中心、FEMTO-ST研究所）；Institute for Mathematical Modeling, Analysis and Computational Mathematics, University of Wuppertal（伍珀塔尔大学数学建模、分析与计算数学研究所）
This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian embedding of the partial differential-algebraic systems. We show that a spatially discretized network model describing the advection of the internal energy density with respect to an underlying incompressible stationary Euler-type hydrodynamics can be considered as a parameter-dependent finite-dimensional port-Hamiltonian system. Moreover, we present an infinite-di…
Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.
It is a universal phenomenon that the state and input of the continuous stirred tank reactor (CSTR) systems are both disturbed. This paper proposes a (state, input)-disturbed port-Hamiltonian framework that can be used to model and further designs a stochastic passivity based controller to asymptotically stabilize in probability the (state, input)-disturbed CSTR (sidCSTR) systems. The opposite entropy function and the availability function are selected as the Hamiltonian for the model and control purposes, respectively. Furthermore, the proposed (state, input)-disturbed port-Hamiltonian model …
We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this end, after reducing the problem to an ODE with feed-through term, we derive an input-state turnpike towards a subspace for optimal control of generalized port-Hamiltonian ordinary differential equations. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. By means of the port-Hamiltonian structure, we…
The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is de…
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The dynamic iteration is monotone in the sense that the error is decreasing, it does not require any stability condition and is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
We consider the problem of minimizing the entropy, energy, or exergy production for state transitions of irreversible port-Hamiltonian systems subject to control constraints. Via a dissipativity-based analysis we show that optimal solutions exhibit the manifold turnpike phenomenon with respect to the manifold of thermodynamic equilibria. We illustrate our analytical findings via numerical results for a heat exchanger.
We study port-Hamiltonian systems on a familiy of intervals and characterise all boundary conditions leading to $m$-accretive realisations of the port-Hamiltonian operator and thus to generators of contractive semigroups. The proofs are based on a structural observation that the port-Hamiltonian operator can be transformed to the derivative on a familiy of reference intervals by suitable congruence relations allowing for studying the simpler case of a transport equation. Moreover, we provide well-posedness results for associated control problems without assuming any additional regularity of th…
The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity, KYP inequalities and a~finite available storage is investigated.
Motivated by recent work in this area we expand on a generalization of port-Hamiltonian systems that is obtained by replacing the Hamiltonian function representing energy storage by a general Lagrangian subspace. This leads to a new class of algebraic constraints in physical systems modeling, and to an interesting class of DAE systems. It is shown how constant Dirac structures and Lagrangian subspaces allow for similar representations, and how this leads to descriptions of the DAE systems entailing generalized Lagrange multipliers.
Infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domain with full boundary control and without internal damping are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result shows that well-posed port-Hamiltonian systems, with state space $L^2((0,1);\mathbb C^n)$ and input space $\mathbb C^n$, are exactly controllable.
The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.
This paper studies the problem of frequency regulation in power grids, while maximizing the social welfare. Two price-based controllers are proposed; the first one an internal-model-based controller and the second one based on a continuous gradient method for optimization. Both controllers can be implemented in a fully distributed fashion, with freedom in choosing a controller communication network. As a result, two real-time dynamic pricing models described by port- Hamiltonian systems are obtained. By coupling with the port- Hamiltonian description of the physical network we obtain a closed-…
We consider networks of infinite-dimensional port-Hamiltonian systems $\mathfrak{S}_i$ on one-dimensional spatial domains. These subsystems of port-Hamiltonian type are interconnected via boundary control and observation and are allowed to be of distinct port-Hamiltonian orders $N_i \in \mathbb{N}$. Wellposedness and stability results for port-Hamiltonian systems of fixed order $N \in \mathbb{N}$ are thereby generalised to networks of such. The abstract theory is applied to some particular model examples.
Electric circuits are usually described by charge- and flux-oriented modified nodal analysis. In this paper, we derive models as port-Hamiltonian systems on several levels: overall systems, multiply coupled systems and systems within dynamic iteration procedures. To this end, we introduce new classes of port-Hamiltonian differential-algebraic equations. Thereby, we additionally allow for nonlinear dissipation on a subspace of the state space. Both, each subsystem and the overall system, possess a port-Hamiltonian structure. A structural analysis is performed for the new setups. Dynamic iterati…
We give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port-Hamiltonian systems (pHS). The first-order optimality system for the port-Hamiltonian system-constrained optimal control problem is formally derived. Then we propose a gradient-based algorithm to find optimal controls. The port-Hamiltonian system formulation naturally conserves flow and supports a wide array of further mod…
Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (s…
We will give general sufficient conditions under which a controller achieves robust regulation for a boundary control and observation system. Utilizing these conditions we construct a minimal order robust controller for an arbitrary order impedance passive linear port-Hamiltonian system. The theoretical results are illustrated with a numerical example where we implement a controller for a one-dimensional Euler-Bernoulli beam with boundary controls and boundary observations.
This paper offers a geometric framework for modeling port-Hamiltonian systems on discrete manifolds. The simplicial Dirac structure, capturing the topological laws of the system, is defined in terms of primal and dual cochains related by the coboundary operators. This finite-dimensional Dirac structure, as discrete analogue of the canonical Stokes-Dirac structure, allows for the formulation of finite-dimensional port-Hamiltonian systems that emulate the behaviour of the open distributed-parameter systems with Hamiltonian dynamics.
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 alge…
When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exac…
We establish an exponential stabilization result for linear port-Hamiltonian systems of first order with quite general, not necessarily continuous, energy densities. In fact, we have only to require the energy density of the system to be of bounded variation. In particular, and in contrast to the previously known stabilization results, our result applies to vibrating strings or beams with jumps in their mass density and modulus of elasticity.
Boundary feedback stabilisation of linear port-Hamiltonian systems on an interval is considered. Generation and stability results already known for linear feedback are extended to nonlinear dissipative feedback, both to static feedback control and dynamic control via an (exponentially stabilising) nonlinear controller. A design method for nonlinear controllers of linear port-Hamiltonian systems is introduced. As a special case the Euler-Bernoulli beam is considered.
We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissipation mapping and the in/output mapping. In contrast to the well-known representation on the basis of the underlying Stokes-Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed …
This contribution deals with energy-based in-domain control of systems governed by partial differential equations with spatial domain up to dimension two. We exploit a port-Hamiltonian system description based on an underlying jet-bundle formalism, where we restrict ourselves to systems with 2nd-order Hamiltonian. A certain power-conserving interconnection enables the application of a dynamic control law based on structural invariants. Furthermore, we use various examples such as beams and plates with in-domain actuation to demonstrate the capability of our approach.
We consider an operator-theoretic approach to linear infinite-dimensional port-Hamiltonian systems. In particular, we use the theory of system nodes by Staffans to formulate a~suitable concept for port-Hamiltonian systems, which allows a unifying approach to systems with boundary as well as distributed control and observation. The concept presented in this article is further neither limited to parabolic nor hyperbolic systems, and it also covers partial differential equations on multi-dimensional spatial domains. Our presented theory is substantiated by means of several physical examples.
Port-Hamiltonian systems theory provides a systematic methodology for the modeling, simulation and control of multi-physics systems. The incorporation of algebraic constraints has led to a multitude of definitions of port-Hamiltonian differential-algebraic equations (DAE) systems. This paper presents extensions of results in Gernandt, Haller & Reis (2021) and Mehrmann & Van der Schaft (2022) in the context of maximally monotone structures and shows that any such space can be written as composition of a Dirac and a resistive structure. Furthermore, appropriate coordinate representations are pre…
In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of appropriate differential operators. For this system class, energy-based control by means of Casimir functionals as well as energy balancing is analysed and demonstrated using a nonlinear Euler-Bernoulli beam.
Based on Pontryagin Maximum Principle (PMP), this paper establishes a generalized PMP aiming at control system with with extra input/output terms. The paper details the adaptive target and gives a proof of the generalized theorem. Transformation and application of this specific method is showed and thereafter an example explained its feasibility.
We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by (formally) setting the second-order term zero. Further…
