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In this paper, we analyze the behavior of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee. We prove that after a large number of iterations, if the stochastic approximation process enters the domain of attraction of an attracting set, it gets locked into the attracting set with high probability. We demonstrate that the above-mentioned result is an effective instrument for analyzing stochastic approximation schemes in the absence of a stability guarantee, by using it to obtain an alternate criterion for convergence in the presence of a locally attracting set for the mean field and by using it to show that a feedback mechanism, which involves resetting the iterates at regular time intervals, stabilizes the scheme when the mean field possesses a globally attracting set, thereby guaranteeing convergence. The results in this paper build on the works of Borkar, Andrieu et al., and Chen et al., by allowing for the presence of set-valued drift functions. [ABSTRACT FROM AUTHOR]

In this paper, the locally fixed-time and globally fixed-time $\mathcal {H}_{\infty }$ control problems for the port-controlled Hamiltonian (PCH) systems are investigated via the interconnection and damping assignment passivity-based control (IDA-PBC) technique. Compared with finite-time stabilization, where the convergence time of the closed-loop system's states relies on the initial values, the settling time of fixed-time stabilization can be adjusted to achieve desired equilibrium point regardless of initial conditions. The concepts of fixed-time $\mathcal {H}_{\infty }$ control, fixed-time stability region (or region of attraction), and fixed-time stability boundary are presented in this paper, and the criterions of globally fixed-time attractivity of a prespecified locally fixed-time stability region are obtained. Combining the locally fixed-time stability of an equilibrium point and the globally fixed-time attractivity of a prespecified fixed-time stability region, the globally fixed-time $\mathcal {H}_{\infty }$ control problem of PCH system is effectively solved. Two novel control laws are designed to deal with the globally fixed-time $\mathcal {H}_{\infty }$ control problem, and the conservativeness in estimating the settling time is also briefly discussed. An illustrative example shows that the theoretical results obtained in this paper work very well in the fixed-time $\mathcal {H}_{\infty }$ control design for PCH systems. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems In which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most. important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarily equations, the boundary and continuity, conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper. [ABSTRACT FROM AUTHOR]

This paper examines the long-standing problem of linear quadratic regulation and stabilization for Itô stochastic systems with input delay. This problem remains a primary challenge because the separation principle does not hold for Itô stochastic systems. This paper presents a complete solution to the problem: 1) The (sufficient and necessary) solvability condition of the optimal control and the analytical controller are given based on the modified Riccati differential equation defined herein. 2) The sufficient and necessary stabilization condition in mean square sense is explored. We show that the Itô stochastic system with input delay is stabilized if and only if the modified algebraic Riccati equation developed in this paper has a unique positive-definite solution. The essential obstacle encountered in this paper concerns a Delayed Forward-Backward Stochastic Differential Equation (D-FBSDE), which is mathematical challenging. [ABSTRACT FROM PUBLISHER]

A production system is characterized by both its steady state and transient properties. While extensive research efforts have been spent in the analysis of the steady state of production systems, very few results, especially analytical ones, have been reported regarding their transient behavior. Indeed, transient behavior of production systems has significant practical and theoretical implications. A better understanding of the transient properties of production systems is critical to effective utilization of real-time production data for efficient factory floor operation and management. In the framework of serial production lines with geometric machines and finite buffers, this paper develops mathematical models for transient analysis and derives closed-form expressions for evaluating the production rate, consumption rate, work-in-process, and probabilities of machine starvation and blockage, during transients. In addition, a computationally efficient algorithm based on aggregation is developed to approximate the transient performance measures with high accuracy. Numerical experiments show that the methods developed can be applied to systems with time-varying machine parameters as well. [ABSTRACT FROM AUTHOR]

In this paper, we study the cooperative global robust output regulation problem for a class of lower triangular nonlinear uncertain multi-agent systems. We first introduce a type of distributed internal model to convert the cooperative global robust output regulation problem into a global robust stabilization problem of a so-called augmented multi-agent system which is in the block lower triangular form. We then further show that, under a set of standard assumptions, the augmented multi-agent system can be globally stabilized via a distributed state feedback control law by developing a block backstepping technique. A special case of the main result of this paper leads to the solution of the leader-following global robust consensus problem for a class of nonlinear uncertain multi-agent systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how insight obtained from a mean-field model can be used to create an architecture for approximate dynamic programming (ADP) for a certain class of games comprising of a large number of agents. The general technique is illustrated with the aid of a mean-field oscillator game model introduced in our prior work. The states of the model are interpreted as the phase angles for a collection of nonhomogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. The paper introduces ADP techniques for design and adaptation (learning) of approximately optimal control laws for this model. For this purpose, a parameterization is proposed, based on an analysis of the mean-field PDE model for the game. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters while in an online setting, a steepest descent algorithm is proposed. The paper provides a detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control penalty parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments. [ABSTRACT FROM AUTHOR]

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied primarily on the conditions of homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems by deriving convergence results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results shows that if the undelayed system has a convergent trajectory that is unbounded in all components as $t\rightarrow-\infty$, then the system is globally asymptotically stable for arbitrary bounded time-varying delays. This follows from a more general result derived in the paper that allows to quantify delay-independent regions of attraction, which can be used to prove global asymptotic stability for various classes of systems. These also recover various known delay-independent stability results that are discussed within the paper. [ABSTRACT FROM PUBLISHER]

This paper extends the result of an earlier paper on constructive robust nonlinear output feedback controller design, which is based on the use of IQCs and minimax LQG control. In this paper, in addition to copies of the nonlinearities, we include copies of the inverses of the nonlinearities in the controller. Consequently more IQCs are exploited in the design procedure, which gives a smaller bound on the closed-loop cost function than the earlier result. [ABSTRACT FROM AUTHOR]

The article (Bazanella et al., 2019) presented the first results on generic identifiability of dynamic networks with partial excitation and partial measurements. All previous papers assumed that either all nodes are excited or all nodes are measured. One key contribution of that paper was to establish a set of necessary conditions on the excitation and measurement pattern (EMP) that guarantee generic identifiability: all sources must be excited and all sinks measured, and all other nodes must be either excited or measured. In this article, we show that two other types of nodes, which are defined by the local topology of the network, play an essential rôle in the search for a valid EMP, i.e., one that guarantees generic identifiability. We have called these nodes dources and dinks. We show that a network is generically identifiable only if, in addition to the abovementioned conditions, all dources are excited and all dinks are measured. We also show that sources and dources are the only nodes in a network that always need to be excited, and that sinks and dinks are the only nodes that need to be measured for an EMP to be valid.

This work is concerned with a compositional data-driven approach for formal safety verification of large-scale continuous-time dynamical systems with unknown models. The proposed framework enjoys the interconnection matrix and joint dissipativity-type properties of subsystems, described by the notion of stochastic storage certificates. In the first part of the paper, we cast the required conditions for constructing storage certificates as a robust optimization program (ROP). Since the proposed ROP is not tractable due to the unknown model appearing in one of its constraints, we propose a scenario optimization program (SOP) corresponding to the original ROP by collecting finite numbers of data from trajectories of each subsystem. By establishing a probabilistic relation between the optimal value of SOP and that of ROP, we construct a storage certificate for each unknown subsystem based on the number of data and a required level of confidence. We accordingly propose a compositional technique based on dissipativity reasoning to construct stochastic barrier certificates of interconnected systems based on storage certificates of individual subsystems. By leveraging the acquired barrier certificate, we quantify a lower bound on the probability that an interconnected system never reaches a certain unsafe region in finite-time horizons with an a-priori guaranteed confidence. We also propose an auxiliary compositional approach without requiring any compositionality condition but at the cost of providing a potentially conservative safety guarantee. In the second part of the paper, we propose our approaches for deterministic continuous-time systems with unknown dynamics. We verify our results over an unknown room temperature network containing 100 rooms.

The paper studies the model reduction problem using the notion of least squares moment matching. For linear systems, the main idea is to approximate a transfer function by ensuring that the interpolation conditions imposed by moment matching are satisfied in a least squares sense. The paper revisits this idea using tools from output regulation theory to develop a nonlinear enhancement of the notion of least squares moment matching and a unifying model reduction framework both for linear and nonlinear systems. The proposed framework allows for the direct computation of models through optimization, the use of weights to modulate the quality of approximation, and the possibility of enforcing prescribed properties, such as stability and passivity, via additional constraints. Parameterized families of models achieving least squares moment matching are also determined and shown to admit natural geometric and system-theoretic interpretations. The theory is illustrated by numerical examples.

In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected convergence time of the unbiased quantized consensus over general but fixed networks. It is shown that the maximum expected convergence time lies within a constant factor of the maximum hitting time of an appropriate lazy random walk, using the theory of harmonic functions for reversible Markov chains. Following this, and using electric resistance analogy of the reversible Markov chains, the paper provides an upper bound of $O(nmD\log n)$ for the expected convergence time to consensus, where $n$, $m$ and $D$, denote, respectively, the number of nodes, the number of edges, and the diameter of the network. Moreover, the paper shows that the above upper bound is tight up to a factor of $\log n$ for some simple graphs such as line graph and cycle. Finally, the results are extended to bound the expected convergence time of the underlying dynamics in time-varying networks. Modeling such dynamics as the evolution of a time inhomogeneous Markov chain, the paper derives an upper bound for the expected convergence time of the dynamics using the spectral representation of the networks. This upper bound is significantly better than earlier ones given for different quantized consensus protocols over time-varying graphs. [ABSTRACT FROM PUBLISHER]