Input-to-state stability (ISS) allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this habilitation thesis, we provide a broad picture of infinite-dimensional input-to-state stability theory.
This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard L p functional spaces, with p $\in$ [2, $\infty$], and relies mostly on some results collected in Haraux (2009). The asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant system.
Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficien…
A class of distributed systems with a cyclic interconnection structure is considered. These systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns. In this paper, a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified. For these systems global stability results hold if the "secant" criterion is satisfied. In the linear case, it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic Lya…
We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies norm-to-integral input-to-state stability. This property in turn is equivalent to input-to-state stability, if the system satisfies certain mild regularity assumptions. For a particular class of linear systems with unbounded admissible input operators, explicit constructions of noncoercive Lyapunov functions are provided. The theory is applied to a heat equation with…
Stability is a critical feature of distributed linear multi-input-multi-output systems. Global asymptotic stability usually can be guaranteed when using decentralised or distributed control architectures, if: (i) conservative controllers are designed, (ii) collective stability conditions are satisfied, or (iii) interaction terms are neutral. This paper extends the collective stability method to incorporate adaptive controllers, and shows that this method is insufficient for systems with large-gain interconnections. Subsequently, we show that global asymptotic stability can be systematically en…
For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.
Time distributed optimization is an implementation strategy that can significantly reduce the computational burden of model predictive control by exploiting its robustness to incomplete optimization. When using this strategy, optimization iterations are distributed over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. This paper presents a general systems theoretic analysis framework for time distributed optimiza…
The paper deals with the optimal sizing and allocation of dispersed generation, and distributed storage systems. The optimization aims at minimizing the sum of the costs sustained by the distributor for the power losses, for network upgrading, for carrying out the reactive power service and the costs of storage and capacitor installation, over a planning period of several years. Also, analysis of the effect of distributed generation capacity and location on voltage stability enhancement of distribution networks are presented in this paper. The analysis is performed using a steady state voltage…
Distributed cooperative droop control consisting of the primary decentralized droop control and the {secondary} distributed correction control is studied in this paper, which aims to achieve an exact current sharing between generators, worked in the voltage control mode, of DC microgrids. For the DC microgrids with the distributed cooperative droop control, the dynamic stability has not been well investigated although its steady performance has been widely reported. This paper focuses on the stability problem and shows it is equivalent to the semistability problem of a class of second-order ma…
We consider the problem of characterizing and assessing the voltage stability in power distribution networks. Different from previous formulations, we consider the branch-flow parametrization of the power system state, which is particularly effective for radial networks. Our approach to the voltage stability problem is based on a local, approximate, yet highly accurate characterization of the determinant of the power flow Jacobian. Our determinant approximation allows us to construct a voltage stability index that can be computed in a fully scalable and distributed fashion. We provide an upper…
We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result…
We revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the well-known Gradient Tracking and with distributed integral control. Ov…
We give sufficient conditions for Input-to-State Stability in $C^{1}$ norm of general quasilinear hyperbolic systems with boundary input disturbances. In particular the derivation of explicit Input-to-State Stability conditions is discussed for the special case of $2\times 2$ systems.
We consider the problem of quantifying and assessing the steady-state voltage stability in radial distribution networks. Our approach to the voltage stability problem is based on a local, approximate, and yet highly accurate characterization of the determinant of the Jacobian of the power flow equations parameterized according to the branch-flow model. The proposed determinant approximation allows us to construct a voltage stability index that can be computed in a fully distributed or in a hierarchical fashion, resulting in a scalable approach to the assessment of steady-state voltage stabilit…
Distributed consensus-based controllers for optimal secondary frequency regulation of microgrids and power systems have received substantial attention in recent years. This paper provides a Lyapunov-based proof that, under a time-scale separation, these control schemes are stabilizing for a wide class of nonlinear power system models, and under weak assumptions on (i) the objective functions used for resource allocation, and (ii) the graph topology describing communication between agents in the consensus protocol. The results are illustrated via simulation on a detailed test system.
This paper deals with several related notions of output stability with respect to inputs. The inputs may be thought of as disturbances; when there are no inputs, one obtains generalizations of the classical concepts of partial stability. The main notion studied is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions. A companion paper establishes necessary and suffi…
The death spiral hypothesis in electric utility represents a positive feedback phenomenon in which a regulated utility is driven to financial instability by rising prices and declining demand. We establish conditions for the existence of death spiral and conditions of stable adoption of distributed energy resources. We show in particular that linear tariffs always induce death spiral when the fixed operating cost of the utility rises beyond a certain threshold. For two-part tariffs with connection and volumetric charges, the Ramsey pricing that optimizes myopically social welfare subject to th…
For a broad class of infinite-dimensional systems, we characterize input-to-state practical stability (ISpS) using the uniform limit property and in terms of input-to-state stability. We specialize our results to the systems with Lipschitz continuous flows and evolution equations in Banach spaces. Even for the special case of ordinary differential equations our results are novel and improve existing criteria for ISpS.
This paper considers robust stability analysis of a large network of interconnected uncertain systems. To avoid analyzing the entire network as a single large, lumped system, we model the network interconnections with integral quadratic constraints. This approach yields a sparse linear matrix inequality which can be decomposed into a set of smaller, coupled linear matrix inequalities. This allows us to solve the analysis problem efficiently and in a distributed manner. We also show that the decomposed problem is equivalent to the original robustness analysis problem, and hence our method does …
This work studies the problem of distributed optimization in heterogeneous linear multi-agent systems. Instead of relying on a perfect communication network as in many existing distributed optimization approaches, we considered two important issues related to communication networks. First, assumed that communication delays exist when each agent receives information from its neighbors. Further, since communications networks are generally unreliable, we assumed each agent interacted with the other agents through random digraphs. Finally, to prove convergence to optimal solutions, delay-dependent…
We establish the local input-to-state stability of a large class of disturbed nonlinear reaction-diffusion equations w.r.t. the global attractor of the respective undisturbed system.
In this paper we study the global exponential stability in the $L^{2}$ norm of semilinear $1$-$d$ hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the $L^{2}$ norm with respect to both interior and boundary disturbances.
This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated. The work is related to partial stability of differential equations (the particular case which arises when there are no external inputs).
In distributed model predictive control (DMPC), where a centralized optimization problem is solved in distributed fashion using dual decomposition, it is important to keep the number of iterations in the solution algorithm, i.e. the amount of communication between subsystems, as small as possible. At the same time, the number of iterations must be enough to give a feasible solution to the optimization problem and to guarantee stability of the closed loop system. In this paper, a stopping condition to the distributed optimization algorithm that guarantees these properties, is presented. The sto…
Time-distributed Optimization (TDO) is an approach for reducing the computational burden of Model Predictive Control (MPC). When using TDO, optimization iterations are distributed over time by maintaining a running solution estimate and updating it at each sampling instant. In this paper, TDO applied to input constrained linear MPC is studied in detail, and analytic expressions for the system gains and a bound on the number of optimization iterations per sampling instant required to guarantee closed-loop stability is derived. Further, it is shown that the closed-loop stability of TDO-based MPC…
In distributed model predictive control (MPC), the control input at each sampling time is computed by solving a large-scale optimal control problem (OCP) over a finite horizon using distributed algorithms. Typically, such algorithms require several (virtually, infinite) communication rounds between the subsystems to converge, which is a major drawback both computationally and from an energetic perspective (for wireless systems). Motivated by these challenges, we propose a suboptimal distributed MPC scheme in which the total communication burden is distributed also in time, by maintaining a run…
In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the $L^2$-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the $L^2$-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respective…
This paper continues the study of the integral input-to-state stability (IISS) property. It is shown that the IISS property is equivalent to one which arises from the consideration of mixed norms on states and inputs, as well as to the superposition of a ``bounded energy bounded state'' requirement and the global asymptotic stability of the unforced system. A semiglobal version of IISS is shown to imply the global version, though a counterexample shows that the analogous fact fails for input to state stability (ISS). The results in this note complete the basic theoretical picture regarding IIS…
This note deals with Bounded-Input-Bounded-Output (BIBO) stability for semilinear infinite-dimensional dynamical systems allowing for boundary control and boundary observation. We give sufficient conditions that guarantee BIBO stability based on Lipschitz conditions with respect to interpolation spaces. Our results can be applied to guarantee feasibility of funnel control for coupled ODE-PDE systems, as shown by means of an example from chemical engineering.
This paper discusses the effect of sequential conflict resolution maneuvers of an infinite aircraft flow through a finite control volume. Aircraft flow models are utilized to simulate traffic flows and determine stability. Pseudo-random flow geometry is considered to determine airspace stability in a more random airspace, where aircraft flows are spread over a given positive width. The use of this aircraft flow model generates a more realistic flow geometry. A set of upper bounds on the maximal aircraft deviation during conflict resolution is derived. Also with this flow geometry it is proven …
We consider the global consensus problem for multi-agent systems with input saturation over digraphs. Under a mild connectivity condition that the underlying digraph has a directed spanning tree, we use Lyapunov methods to show that the widely used distributed consensus protocol, which solves the consensus problem for the case without input saturation constraints, also solves the global consensus problem for the case with input saturation constraints. In order to reduce the overall need of communication and system updates, we then propose a distributed event-triggered control law. Global conse…
This paper deals with strong versions of input-to-state stability and integral input-to-state stability of infinite-dimensional linear systems with an unbounded input operator. We show that infinite-time admissibility with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable but, unlike in the case of exponentially stable systems, not a necessary one.
In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability not…
In their paper "Stability to weak dissipative bresse system", Alabau et al. studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. Our goal is to extend their results, by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we sh…
Input-to-state stability (ISS) and $\mathcal{L}_2$-gain are well-known robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and $\mathcal{L}_2$-gain properties, demonstrating several qualitative equivalences between these two approaches. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting these qualitative equivalences.
In this work, we study stability of distributed filtering of Markov chains with finite state space, partially observed in conditionally Gaussian noise. We consider a nonlinear filtering scheme over a Distributed Network of Agents (DNA), which relies on the distributed evaluation of the likelihood part of the centralized nonlinear filter and is based on a particular specialization of the Alternating Direction Method of Multipliers (ADMM) for fast average consensus. Assuming the same number of consensus steps between any two consecutive noisy measurements for each sensor in the network, we fully…
We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also answer, by means of suitable counterexamples, two open questions concerning weak input-to-state stability (and its relation to other common stability concepts) raised in the aforementioned paper.
In this work, we address the distributed optimization problem with event-triggered communication by the notion of input feedforward passivity (IFP). First, we analyze the distributed continuous-time algorithm over uniformly jointly strongly connected balanced digraphs in an IFP-based framework. Then, we propose a distributed event-triggered communication mechanism for this algorithm. Next, we discretize the continuous-time algorithm by the forward Euler method with a constant stepsize irrelevant to network size, and show that the discretization can be seen as a stepsize-dependent passivity deg…
Input-to-state stability (ISS) for systems described by partial differential equations has seen intensified research activity recently, and in particular the class of boundary control systems, for which truly infinite-dimensional effects enter the situation. This note reviews input-to-state stability for parabolic equations with respect to general $L^{p}$-input-norms in the linear case and includes extensions of recent results on semilinear equations.
We study the recently introduced notion of output-input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. The subject of this paper is developing the theory of output-input stability in the multi-input, multi-output setting. We show that output-input stability can be viewed as a combination of two system properties, one related to detectability and the other to left-invertibility. For systems affine in controls, we provide a necessary and sufficient condition for output-input stability, which relies on Hirschorn's nonlinear structu…
Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input-output performance becomes more important. We analyze the input-output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit exp…
In this paper, a distributed optimization problem is investigated via input feedforward passivity. First, an input-feedforward-passivity-based continuous-time distributed algorithm is proposed. It is shown that the error system of the proposed algorithm can be decomposed into a group of individual input feedforward passive (IFP) systems that interact with each other using output feedback information. Based on this IFP framework, convergence conditions of a suitable coupling gain are derived over weight-balanced and uniformly jointly strongly connected (UJSC) topologies. It is also shown that t…
Distributed Output Regulation for a Class of Nonlinear Multi-Agent Systems with Unknown-Input Leaders
School of Automation, Beijing University of Posts and Telecommunications（北京邮电大学自动化学院）；Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences（中国科学院系统与控制重点实验室，系统科学研究所）
A notion of detectability for nonlinear systems is discussed. Within the framework of ``input to state stability'' (ISS), a dual notion of ``output to state stability'' (OSS), and a more complete detectability notion, ``input-output to state stability'' (IOSS) have appeared in the literature. This note addresses a variant of the IOSS property, using an integral norm to measure signals, as opposed to the standard supremum norm that appears in ISS theory.
Suitable continuity and boundedness assumptions on the function f defining the dynamics of a time-varying nonimpulsive system with inputs are known to make the system inherit stability properties from the zero-input system. Whether this type of robustness holds or not for impulsive systems was still an open question. By means of suitable (counter)examples, we show that such stability robustness with respect to the inclusion of inputs cannot hold in general, not even for impulsive systems with time-invariant flow and jump maps. In particular, we show that zero-input global uniform asymptotic st…
