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George Weiss

Publications and source records attributed to George Weiss.

18 recordsLinked to original sources
arXiv2026

Output Regulation for Linear Parabolic Systems Using Finite-Dimensional Tracking-Error-Based Control

This paper addresses the output regulation problem for 1-D diffusion-reaction system, where both the disturbance and reference signals are generated by an unstable exosystem. We propose a constructive approach to the design of a finite-dimensional tracking error-based regulator for this class of possibly unstable systems. Based on the regulator equations, a combined plant is derived, converting the output regulation problem into a partial stabilization problem for the combined system. Using the modal decomposition method, the observability of the truncated modes is characterized under appropriate transmission zeros and observability conditions. For the controller design, unlike in the standard stabilization case, where the observer gain is dependent on the unstable modes only, the observer gain here has full order due to the coupling introduced by the exosystem. We prove that the observer gain can be designed such that its norm remains uniformly bounded with respect to the dimension of the observer. LMI-based conditions are provided for determining the observer dimension, and it is shown that the LMI is feasible for a sufficiently large dimension. Finally, the output regulation problem in the presence of unknown time-varying measurement delays is analyzed. Numerical examples are provided to validate the theoretical results.

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arXiv2026

Aclass of incrementally scattering-passive nonlinear systems

We investigate a special class of nonlinear infinite dimensional systems. These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator M from the semigroup generator of a scattering passive linear system. While the linear system may have unbounded linear damping (for instance, boundary damping) which is only densely defined, the nonlinear damping operator M is assumed to be defined on the whole state space. We show that this new class of nonlinear infinite dimensional systems is well-posed and incrementally scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system.

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arXiv2026

Projected incrementally scattering passive systems on closed convex sets

In this article we show that the projected dynamical system obtained by restricting the state of an incrementally scattering passive system to a closed and convex subset K of the state space (a real Hilbert space), is also an incrementally scattering passive system. First we show that the projection of a maximal dissipative operator to the tangent cones of K is again maximal dissipative, hence, it determines a contraction semigroup.

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arXiv2026

Second order systems on Hilbert spaces with nonlinear damping

We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems out of thin air introduced by M. Tucsnak and G. Weiss in 2003. The modified differential equation contains a new nonlinear damping term, that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.

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arXiv2026

The strong stability of the Perron-Frobenius semigroup and almost global attractivity

We discuss some useful properties of the solution map (flow) of a nonlinear dynamical system with a finite-dimensional state space. Then, we introduce the Perron-Frobenius semigroup, and we prove that it is a positive strongly continuous semigroup of contractions. We show that, given a nonlinear system and an invariant set, this set is an almost global attractor if and only if certain Perron-Frobenius semigroups associated to the nonlinear system are strongly stable. Unlike other works on the Perron-Frobenius semigroup from the literature, we do not require the existence of a compact and invariant state-space for the dynamical system, we allow trajectories with finite escape time, and we do not require the attractor to be locally (Lyapunov) stable. Two simple examples are used throughout the paper to illustrate the theory.

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arXiv2025

Projected integral control of impedance passive nonlinear systems

We propose an abstract framework for solving the constrained set-point tracking problem for impedance passive infinite-dimensional nonlinear systems. The class of systems considered is governed by monotone differential inclusions and allows us to exploit the theory of contraction semigroups. To account for possible operational constraints, e.g., bounds on the input, we replace a classical integral controller with a projected integral controller. This guarantees that the integrator state remains in a given closed convex set, where said constraints are satisfied. We showcase our results through three case studies.

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arXiv2022

PI control of stable nonlinear plants using projected dynamical systems

This paper presents a novel anti-windup proportional-integral controller for stable multi-input multi-output nonlinear plants. We use tools from projected dynamical systems theory to force the integrator state to remain in a desired (compact and convex) region, such that the plant input steady-state values satisfy the operational constraints of the problem. Under suitable monotonicity assumptions on the plant steady-state input-output map, we use singular perturbation theory results to prove the existence of a sufficiently small controller gain ensuring closed-loop (local) exponential stability and reference tracking for a feasible set of constant references. We suggest a particular controller design, which embeds (when possible) the right inverse of the plant steady-state input-output map. The relevance of the proposed controller scheme is validated through an application in the power systems domain, namely, the output (active and reactive) power regulation for a grid-connected synchronverter.

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arXiv2021

Strong stabilization of (almost) impedance passive systems by static output feedback

The plant to be stabilized is a system node $\Sigma$ with generating triple $(A,B,C)$ and transfer function $\bf G$, where $A$ generates a contraction semigroup on the Hilbert space $X$. The control and observation operators $B$ and $C$ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $E$ such that, if we replace ${\bf G}(s)$ by ${\bf G}(s)+E$, the new system $\Sigma_E$ becomes impedance passive. An easier case is when $\bf G$ is already impedance passive and a special case is when \mm $\Sigma$ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $u=-\kappa y+v$, where $u$ is the input of the plant and $\kappa>0$, stabilizes $\Sigma$, strongly or even exponentially. Here, $y$ is the output of \m $\Sigma$ and $v$ is the new input. Our main result is that if for some $E\in{\mathcal L}(U)$, $\Sigma_E$ is impedance passive, and \m $\Sigma$ is approximately observable or approximately controllable in infinite time, then for sufficiently small $\kappa$ the closed-loop system is weakly stable. If, moreover, $\sigma(A)\cap i{\mathbb R}$ is countable, then the closed-loop semigroup and its dual are both strongly stable.

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arXiv2021

The equilibrium points and stability of grid-connected synchronverters

Virtual synchronous machines are inverters with a control algorithm that causes them to behave towards the power grid like synchronous generators. A popular way to realize such inverters are synchronverters. Their control algorithm has evolved over time, but all the different formulations in the literature share the same "basic control algorithm". We investigate the equilibrium points and the stability of a synchronverter described by this basic algorithm, when connected to an infinite bus. We formulate a fifth order model for a grid-connected synchronverter and derive a necessary and sufficient condition for the existence of equilibrium points. We show that the set of equilibrium points with positive field current is a two-dimensional manifold that can be parametrized by the corresponding pair $(P,Q)$, where $P$ is the active power and $Q$ is the reactive power. This parametrization has several surprizing geometric properties, for instance, the prime mover torque, the power angle and the field current can be seen directly as distances or angles in the $(P,Q)$ plane. In addition, the stable equilibrium points correspond to a subset of a certain angular sector in the $(P,Q)$ plane. Thus, we can predict the stable operating range of a synchronverter from its parameters and from the grid voltage and frequency. Our stability result is based on the intrinsic two time scales property of the system, using tools from singular perturbation theory. We illustrate our theoretical results with two numerical examples.

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arXiv2021

Saturating PI control of stable nonlinear systems using singular perturbations

This paper presents an anti-windup PI controller, using a saturating integrator, for a single-input single-output stable nonlinear plant, whose steady-state input-output map is increasing. We prove that, under reasonable assumptions, there exists an upper bound on the controller gain such that for any constant reference input, the corresponding equilibrium point of the closed-loop system is exponentially stable, with a "large" region of attraction. When the state of the closed-loop system converges to this equilibrium point, then the tracking error tends to zero. The closed-loop stability analysis employs Lyapunov methods in the framework of singular perturbations theory. Finally, we show that if the plant satisfies the asymptotic gain property, then the closed-loop system is globally asymptotically stable for any sufficiently small controller gain. The effectiveness of the proposed PI controller is proved by showing how it performs as part of the control algorithm of a synchronverter (a special type of DC to AC power converter).

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arXiv2020

Stabilizability properties of a linearized water waves system

We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function $u$, times a given function $h$ of the height along the active boundary. The state $z$ of the system consists of two functions: the water level $\zeta$ along the top boundary, and its time derivative $\dot\zeta$. We prove that for suitable functions $h$, there exists a bounded feedback functional $F$ such that the feedback $u=Fz$ renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like $(1+t)^{-\frac{1}{6}}$. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill, Paunonen, Seifert, Stahn and Tomilov (2019).

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arXiv2019

Solving the regulator problem for the one-dimensional Schrodinger equation via backstepping

We investigate the regulator problem (tracking and disturbance rejection) for a system (plant) described by a boundary controlled anti-stable linear one-dimensional Schrodinger equation, using the backstepping approach. The output to be controlled is not required to be measurable and its observation operator is assumed to be admissible for a certain operator semigroup that is related to the operator semigroup of the original plant. We consider both the state feedback and the output feedback regulator problem. In the latter case, the measurement from the Schrodinger equation is taken at the boundary. First we show that the open-loop system is well-posed. We design a state feedback control law that solves the regulator problem by the backstepping method. Then, a finite-dimensional reference observer and an infinite-dimensional disturbance observer are designed. Putting these together, we obtain an output feedback controller with internal loop that achieves output regulation.

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arXiv2019

Minimal order controllers for output regulation of nonlinear systems

This paper is about the nonlinear local error feedback regulator problem. The plant is a nonlinear finite-dimensional system with a single control input and a single output and it is locally exponentially stable around the origin. The plant is driven, via a separate disturbance input, by a Lyapunov stable exosystem whose states are nonwandering. The reference signal that the plant output must track is a nonlinear function of the exosystem state. The local error feedback regulator problem is to design a dynamic feedback controller, with the tracking error as its input, such that (i) the closed-loop system of the plant and the controller is locally exponentially stable, and (ii) the tracking error tends to zero for all sufficiently small initial conditions of the plant, the controller and the exosystem. Under the assumption that the above regulator problem is solvable, we propose a nonlinear controller whose order is relatively small - typically equal to the order of the exosystem, and which solves the regulator problem. The emphasis is on the low order of the controller. The stability assumption on the plant (which can be relaxed to some extent) is crucial for making it possible to design a low order controller. We will show, under certain assumptions, that our proposed controller is of minimal order. Three examples are presented - the first illustrates our controller design procedure using an exosystem whose trajectories are periodic even though the state operator of the linearized exosystem contains a nontrivial Jordan block. The second example is more involved, and shows that sometimes a nontrivial immersion of the exosystem is needed in the design. The third example, based on output voltage regulation for a boost power converter, shows how the regulator equations may reduce to a first order PDE with no given boundary conditions, but which nevertheless has a locally unique solution.

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arXiv2017

Output feedback exponential stabilization of a nonlinear 1-D wave equation with boundary input

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in $L^2(0,\infty)$ but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded.

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arXiv2017

Output feedback exponential stabilization for 1-D unstable wave equations with boundary control matched disturbance

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in $L^2[0,\infty)$. Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded.

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arXiv2016

Almost global asymptotic stability of a grid-connected synchronous generator

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an "infinite bus", i.e. a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $2\pi$). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the {\em exact swing equation}, which resembles a forced pendulum equation and is equivalent to our fourth order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500 kW synchronverter.

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arXiv2016

Stability of the integral control of stable nonlinear systems

PI controllers are the most widespread type of controllers and there is an intuitive understanding that if their gains are sufficiently small and of the correct sign, then they always work. In this paper we try to give some rigorous backing to this claim, under specific assumptions. Let $\bf P$ be a nonlinear system described by $\dot x=f(x,u)$, $y=g(x)$, where the state trajectory $x$ takes values in $R^n$, $u$ and $y$ are scalar and $f,g$ are of class $C^1$. We assume that there is a Lipschitz function $\Xi:[u_{min},u_{max}]\rightarrow R^n$ such that for every constant input $u_0\in[u_{min},u_{max}]$, $\Xi(u_0)$ is an exponentially stable equilibrium point of $\bf P$. We also assume that $G(u)=g(\Xi(u))$, which is the steady state input-output map of $\bf P$, is strictly increasing. Denoting $y_{min}=G(u_{min})$ and $y_{max}=G(u_{max})$, we assume that the reference value $r$ is in $(y_{min},y_{max})$. Our aim is that $y$ should track $r$, i.e., $y\rightarrow r$ as $t\rightarrow\infty$, while the input of $P$ is only allowed to be in $[u_{min},u_{max}]$. For this, we introduce a variation of the integrator, called the saturating integrator, and connect it in feedback with $\bf P$ in the standard way, with gain $k>0$. We show that for any small enough $k$, the closed-loop system is (locally) exponentially stable around an equilibrium point $(Xi(u_r),u_r)$, with a large region of attraction $X_T\subset R^n\times[u_{min},u_{max}]$. When the state $(x(t),u(t))$ of the closed-loop system converges to $(\Xi(u_r),u_r)$, then the tracking error $r-y$ tends to zero. The compact set $X_T$ can be made larger by choosing a larger parameter $T>0$, resulting in smaller $k$.

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