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 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-…
In this paper we provide a unifying energy-based approach to the modeling, analysis and control of power systems and markets, which is based on the port-Hamiltonian framework. Using a primal-dual gradient method applied to the social welfare problem, a distributed dynamic pricing algorithm in port-Hamiltonian form is obtained. By interconnection with the physical model a closed-loop port-Hamiltonian system is obtained, whose properties are exploited to prove asymptotic stability to the optimal points. This result is extended such that also general nodal power constraints are included into the …
In this paper, we discuss the state feedback output regulation problem (SFRP) for infinite-dimensional linear control systems with infinite-dimensional exosystems. Under the polynomial stabilizability assumption, sufficient and necessary conditions are given for the solvability of the SFRP. The solvability of this problem is characterized in terms of the solvability of a pair of linear regulator equations. An application of the solvability of the SFRP for polynomial stable SISO system is given.
This paper studies robust output tracking and disturbance rejection for boundary controlled infinite-dimensional port--Hamiltonian systems including second order models such as the Euler--Bernoulli beam. The control design is achieved using the internal model principle and the stability analysis using a Lyapunov approach. Contrary to existing works on the same topic no assumption is made on the external well-posedness of the considered class of PDEs. The results are applied to robust tracking of a piezo actuated tube used in atomic force imaging.
This paper presents a performance-regulation method for a class of stochastic timed event-driven systems aimed at output tracking of a given reference setpoint. The systems are either Discrete Event Dynamic Systems (DEDS) such as queueing networks or Petri nets, or Hybrid Systems (HS) with time-driven dynamics and event-driven dynamics, like fluid queues and hybrid Petri nets. The regulator, designed for simplicity and speed of computation, is comprised of a single integrator having a variable gain to ensure effective tracking under time-varying plants. The gain's computation is based on the I…
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.
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…
We present and solve a Linear Quadratic Regulator (LQR) for the boundary control of the beam equation. We use the simple technique of completing the square to get an explicit solution. By decoupling the spatial frequencies we are able to reduce an infinite dimensional LQR to an infinte family of two two dimensional LQRs each of which can be solved explicitly.
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…
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.
An adaptive funnel control method is considered for the regulation of the output for a class of nonlinear infinite-dimensional systems on real Hilbert spaces. After a decomposition of the state space and some change of variables related to the Byrnes-Isidori form, it is shown that the funnel controller presented in (Berger et al., 2020) achieves the control objective under some assumptions on the nonlinear system dynamics, like well-posedness and Bounded-Input-State Bounded-Output (BISBO) stability. The theory is applied to the regulation of the temperature in a chemical plug-flow tubular reac…
This paper considers the robust cooperative output regulation for a network of parabolic PDE systems. The solution of this problem is obtained by extending the cooperative internal model principle from finite to infinite dimensions. For a time-invariant digraph describing the communication topology, a two-step backstepping approach is presented to systematically design cooperative state feedback regulators. They allow to solve both the leader-follower and the leaderless output synchronization problem in the presence of disturbances and model uncertainty for a finite-dimensional leader. Solvabi…
This paper is concerned with the regulation control of a one-dimensional reaction-diffusion equation in the presence of a state-delay in the reaction term. The objective is to achieve the PI regulation of the right Dirichlet trace with a command selected as the left Dirichlet trace. The control design strategy consists of the design of a PI controller on a finite dimensional truncated model obtained by spectral reduction. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the proposed control design procedure achieves both the exponential …
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.
This paper is concerned with the Proportional Integral (PI) regulation control of the left Neu-mann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage on the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unsta…
We consider an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way. This path-dependency is the novelty of the model, and leads to an infinite dimensional stochastic optimal control problem. We solve the problem completely, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. To the best of ou…
This thesis deals with the formulation and analysis of two systems of conservation laws defined on two complementary intervals and coupled by some moving interface as a single infinite-dimensional port-Hamiltonian system. This approach may be regarded as an extension of the well-established mathematical framework of boundary port-Hamiltonian systems, i.e., infinite-dimensional Hamiltonian systems interacting with its physical environment. In order to keep track of the power flow at the position of the interface, we introduce interface port variables. We provide a well-posed port-Hamiltonian mo…
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.
In this paper, we consider infinite-dimensional port-Hamiltonian systems with in-domain actuation by means of an approach based on Stokes-Dirac structures as well as in a framework that exploits an underlying jet-bundle structure. In both frameworks, a dynamic controller based on the energy-Casimir method is derived in order to stabilise certain equilibrias. Moreover, we propose distributed-parameter observers deduced by exploiting damping injection for the observer error. Finally, we compare the approaches by means of an in-domain actuated vibrating string and show the equivalence of the cont…
This paper is concerned with a stochastic linear-quadratic (LQ) optimal control problem on infinite time horizon, with regime switching, random coefficients, and cone control constraint. To tackle the problem, two new extended stochastic Riccati equations (ESREs) on infinite time horizon are introduced. The existence of the nonnegative solutions, in both standard and singular cases, is proved through a sequence of ESREs on finite time horizon. Based on this result and some approximation techniques, we obtain the optimal state feedback control and optimal value for the stochastic LQ problem exp…
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.
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.
We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi and Prosdocimi ("Optimal portfolio choice with path dependent labor income: the infinite horizon case", SIAM Journal on Control and Optimization, 58(4), 1906-1938.) and Dybvig and Liu ("Lifetime consumption and investment: retirement and constrained borrowing", Journal of Economic Theory, 145, pp. 885-907). In particular, in Biffis, Gozzi and Prosdocimi the influence of past wages on the future ones is modelled linearly in the evolution equation of labo…
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…
The general context of this work is the feedback control of an infinite-dimensional system so that the closed-loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction-diffusion equation might be either open-loop stable or unstable. The proposed control strategy goes as follows. First, …
We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discrete-time feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and…
A new technique for performance regulation in event-driven systems, recently proposed by the authors, consists of an adaptive-gain integral control. The gain is adjusted in the control loop by a real-time estimation of the derivative of the plant-function with respect to the control input. This estimation is carried out by Infinitesimal Perturbation Analysis (IPA). The main motivation comes from applications to throughput regulation in computer processors, where to-date, testing and assessment of the proposed control technique has been assessed by simulation. The purpose of this paper is to re…
This paper studies the robustness of policy iteration in the context of continuous-time infinite-horizon linear quadratic regulation (LQR) problem. It is shown that Kleinman's policy iteration algorithm is inherently robust to small disturbances and enjoys local input-to-state stability in the sense of Sontag. More precisely, whenever the disturbance-induced input term in each iteration is bounded and small, the solutions of the policy iteration algorithm are also bounded and enter a small neighborhood of the optimal solution of the LQR problem. Based on this result, an off-policy data-driven …
We study robust output regulation for parabolic partial differential equations and other infinite-dimensional linear systems with analytic semigroups. As our main results we show that robust output tracking and disturbance rejection for our class of systems can be achieved using a finite-dimensional controller and present algorithms for construction of two different internal model based robust controllers. The controller parameters are chosen based on a Galerkin approximation of the original PDE system and employ balanced truncation to reduce the orders of the controllers. In the second part o…
Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying manifold leads to the so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian sys…
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…
The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact that system operator generates a strongly continuous group. Moreover, in this situation the spectrum consists of eigenvalues only, located in a strip parallel to the imaginary axis and they can decomposed into finitely many sets having each a uniform gap.
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 …
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.
The class of port-Hamiltonian systems incorporates many physical models, such as mechanical systems in the finite-dimensional case and wave and beam equations in the infinite-dimensional case. In this paper we study a subclass of linear first order port-Hamiltonian systems. These systems are exactly observable when the energy is not dissipated internally and when sufficient observations are made at the boundary. In this article we study the observability properties for these systems when internal dissipation of energy is possible. We cannot show the exact observability, but we do show that the…
