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The topic of this paper is the backstepping-based solution of the robust state feedback regulation problem for general linear heterodirectional hyperbolic systems with spatially varying coefficients. Thereby, a large class of outputs is considered, i.e., mixed distributed and pointwise in-domain or boundary outputs. The state feedback regulator design is based on the $\boldsymbol {p}$ -copy internal model principle. This results in a stabilization problem for a cascade of $\boldsymbol {m+p}$ boundary controlled transport partial differential equations (PDEs) with $\boldsymbol {p}$ ordinary differential equations (ODEs). For this, a systematic backstepping approach is presented, in which all design equations are explicitly solvable in backstepping coordinates. Furthermore, the solvability conditions for the considered output regulation problem can be evaluated in closed form utilizing the plant transfer behavior. Robust output regulation is verified for nondestabilizing model uncertainties and unknown disturbance input locations. A $\boldsymbol {4 \times 4}$ hyperbolic system with uncertain transport velocities and couplings in the transport PDEs illustrates the results of this paper. [ABSTRACT FROM AUTHOR]

This paper introduces the internally positive representation (IPR) of linear coupled differential-difference systems with time-varying delays. Then, a stability analysis is conducted on this class of systems: Relying on results established under positivity assumptions, the IPR technique leads to a novel sufficient condition for the delay-independent asymptotic stability of coupled differential-difference systems that holds for not necessarily positive systems. As a consequence, an analogous stability condition for neutral-type equations with time-varying delays is also obtained. The results are discussed and favorably compared to similar conditions proposed in the literature, and finally validated by means of numerical examples. [ABSTRACT FROM AUTHOR]

This paper considers a singular problem in optimal control of continuous-time bimodal switched linear systems. A relaxed switched system with a continuous-valued switching signal is considered and the representations of singular control and singular arcs are derived. The similarity in the structure between the singular control and a stabilizing switching law is revealed and an approximation of the singular control by a well-defined switching signal is addressed. The results are demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

We consider a class of singularly perturbed stochastic differential equations with linear drift terms, and present a reduced-order model that approximates both the slow and fast variable dynamics when the time-scale separation is large. We show that, on a finite time interval, moments of all orders of the slow variables for the original system become closer to those of the reduced-order model as time-scale separation is increased. A similar result holds for the first and second moments of the fast variable approximation. Biomolecular systems with linear propensity functions, modeled by the chemical Langevin equation fit the class of systems considered in this paper. Thus, as an application example, we analyze the tradeoffs between noise and information transmission in a typical gene regulatory network motif, for which both the slow and fast variables are required. [ABSTRACT FROM AUTHOR]

This paper is concerned with variational optimal control constructions whose solution yields a sampling algorithm. The particular form of the sampling algorithm considered here is a particle filter, designed to numerically approximate the solution to the global optimization problem. The theoretical significance of this study comes from its variational aspects. Specifically, the control input represents the solution of a mean-field-type optimal control problem. Its parametric counterpart, obtained when a parametric form of density is known a priori, is shown to be equivalent to the natural gradient algorithm. Explicit formulae for the filter are derived when the objective function is quadratic and the density is Gaussian. The optimal control construction of the particle filter is a significant departure from the classical importance sampling–resampling-based approaches. [ABSTRACT FROM AUTHOR]

We consider both finite-horizon and infinite-horizon versions of a dynamic game with $N$ selfish players who observe their types privately and take actions that are publicly observed. Players’ types evolve as conditionally independent Markov processes, conditioned on their current actions. Their actions and types jointly determine their instantaneous rewards. In dynamic games with asymmetric information, a widely used concept of equilibrium is perfect Bayesian equilibrium (PBE) which consists of a strategy and belief pair that simultaneously satisfy sequential rationality and belief consistency. In general, there does not exist a universal algorithm that decouples the interdependence of strategies and beliefs over time in calculating PBE. In this paper, for the finite-horizon game with independent types, we develop a two-step backward–forward recursive algorithm that sequentially decomposes the problem (w.r.t. time) to obtain a subset of PBEs, which we refer to as structured Bayesian perfect equilibria (SPBE). In such equilibria, a player's strategy depends on his/her history only through a common public belief and its current private type. The backward recursive part of this algorithm defines an equilibrium generating function. Each period in the backward recursion involves solving a fixed-point equation on the space of probability simplexes for every possible belief on types. Using this function, equilibrium strategies and beliefs are generated through a forward recursion. We then extend this methodology to the infinite-horizon model, where we propose a time-invariant single-shot fixed-point equation, which in conjunction with a forward recursive step, generates the SPBE. Sufficient conditions for the existence of SPBE are provided. With our proposed method, we find equilibria that exhibit signaling behavior. This is illustrated with the help of a concrete public goods example. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delay differential inclusion $\dot{x}(t)\in F(t,x_t)$ , where $x_t$ denotes the history function of $x(\cdot)$ along an interval of time. We extend the celebrated Filippov's theorem to this case. Then, we further generalize this theorem to the case when the state variable $x$ is constrained to the closure of an open subset $K\subset \mathbb {R}^n$. Under a new “inward pointing condition,” we give a relaxation result stating that the set of trajectories lying in the interior of the state constraint is dense in the set of constrained trajectories of the convexified inclusion ${\dot{x}(t)\in \mbox{co}\,F(t,x_t)}$. [ABSTRACT FROM AUTHOR]

This paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations–hyperbolic equations with boundary, trace, and integral nonlocal terms. Using the backstepping approach, the controller is designed by formulating an integral operator, whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation. By applying the method of successive approximations, the kernel's well-posedness is given. We prove the exponential stability of the origin of the system in a suitable Hilbert space. Moreover, a wave system with nonlocal terms is stabilized by applying the above result. [ABSTRACT FROM AUTHOR]

In this paper, source localization by a network of primitive binary sensors under various imperfections are studied. Detailed analysis and mathematical modeling of imperfect binary sensors are presented. Imperfections include sensor failures of two types, drifting, uncertainty, and heterogeneity in binary sensor trigger thresholds, presence of noise, and nonradial symmetry of sensing ranges. Theoretical results, including asymptotical convergence, are established, in particular in the presence of substantial outliers due to sensor failure and large noise. Efficient numerical algorithms are proposed and simulated supporting the theoretical analysis. [ABSTRACT FROM PUBLISHER]

In this paper, we present an adaptive optimal control approach applicable to a wide class of large-scale nonlinear systems. The proposed approach avoids the so-called loss-of-stabilizability problem and the problem of poor transient performance that are typically associated with adaptive control designs. Moreover, it does not require the system model to be in a certain parameterized form, and most importantly, it is able to efficiently handle systems of large dimensions. Theoretical analysis establishes that the proposed methodology guarantees stability and exponential convergence to state trajectories that can be made as close as desired to the optimal ones. A numerical example demonstrates the capability of the proposed approach to overcome loss-of-stabilizability problems. Moreover, simulation experiments for energy-efficient climate control performed on a ten-office building demonstrate the effectiveness of the proposed approach in large-scale nonlinear applications. [ABSTRACT FROM PUBLISHER]

This paper puts forward a simple mean-field game that captures some of the key dynamic features of crowd and pedestrian flows in multilevel building evacuations. It considers both microscopic and macroscopic route choice by strategic agents. To achieve this, we use mean-field differential game with local congestion measure based on the location of the agent in the building. Including the local mean-field term and its evolution along the path causes a sort of dispersion of the flow: the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing costs at the stairs and exits. Each agent state is represented by a center of a box that follows a simple first-order dynamical system in an Euclidean space. Each agent will move to one of the closest exits that is safer and with less congested path. First, we formulate the problem and derive optimality equations using maximum principle and dynamic programming with boundary conditions. Second, well posedness and existence results are provided. Numerics and simulations are carried out to illustrate mean-field equilibria of a safer evacuation process. Finally, the methodology is shown to be flexible enough to include movement noises and stochastic structural component of the building. [ABSTRACT FROM AUTHOR]

This paper deals with the observer problem for dynamical systems in a behavioral context. We are given a dynamical system together with a partition of the system variables into a set of known or measured variables and a set of unknown, to be estimated variables. The observer problem is to find a system that produces an estimate of the unknown variables on the basis of the known or measured variables. For a given plant and partition, we establish a characterization of all error behaviors that can be achieved by interconnecting the plant with some observer. The main result of this paper is a very general, behavioral formulation of an internal model principle for observers. We will show that a nonintrusive observer achieves a stable error behavior if and only if, in addition to a detectability condition on the observer, the observer behavior contains the anti-stabilizable part of the plant behavior. [ABSTRACT FROM AUTHOR]

Unlike many complex networks studied in the literature, social networks rarely exhibit unanimous behavior, or consensus. This requires a development of mathematical models that are sufficiently simple to be examined and capture, at the same time, the complex behavior of real social groups, where opinions and actions related to them may form clusters of different size. One such model, proposed by Friedkin and Johnsen, extends the idea of conventional consensus algorithm (also referred to as the iterative opinion pooling) to take into account the actors’ prejudices, caused by some exogenous factors and leading to disagreement in the final opinions. In this paper, we offer a novel multidimensional extension, describing the evolution of the agents’ opinions on several topics. Unlike the existing models, these topics are interdependent, and hence the opinions being formed on these topics are also mutually dependent. We rigorously examine stability properties of the proposed model, in particular, convergence of the agents’ opinions. Although our model assumes synchronous communication among the agents, we show that the same final opinions may be reached “on average” via asynchronous gossip-based protocols. [ABSTRACT FROM PUBLISHER]

For linear discrete state-space models, under certain conditions, the linear least-mean-squares filter estimate has a convenient recursive predictor/corrector format, aka the Kalman filter. The purpose of this paper is to show that the linear minimum variance distortionless response (MVDR) filter shares exactly the same recursion, except for the initialization which is based on a weighted least-squares estimator. If the MVDR filter is suboptimal in mean-squared error sense, it is an infinite impulse response distortionless filter (a deconvolver) which does not depend on the prior knowledge (first- and second-order statistics) on the initial state. In other words, the MVDR filter can be pre-computed and its behaviour can be assessed in advance independently of the prior knowledge on the initial state. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following problem: what passivity properties can be inferred for a system by studying only an approximate mathematical model for it. Our results show that an excess of passivity (whether in the form of input strictly passive, output strictly passive or very strictly passive) in the approximate model guarantees a certain passivity index for the system, provided that the norm of the error between the approximate and the true models is sufficiently small in a suitably defined sense. Further, we consider $(Q,S,R)$-dissipative systems and show that $(Q,S,R)$- dissipativity has a similar robustness property, even though the supply rates for the system and its approximation may be different. These results may be particularly useful if either the approximate model is much easier to analyze, or if the precise system model is unknown. We illustrate the results by considering particular approximation methods, e.g., model reduction, discretization, quantization, and linearization around an equilibrium point. [ABSTRACT FROM PUBLISHER]

This paper proposes a new iterative solution method for nonlinear stochastic optimal control problems based on path integral analysis. First, we provide an iteration law for solving a stochastic Hamilton-Jacobi-Bellman (SHJB) equation associated to this problem, which is a nonlinear partial differential equation (PDE) of second order. Each iteration procedure of the proposed method is represented by a Cauchy problem for a linear parabolic PDE, and its explicit solution is given by the Feynman-Kac formula. Second, we derive a suboptimal feedback controller at each iteration by using the path integral analysis. Third, the convergence property of the proposed method is investigated. Here, some conditions are provided so that the sequence of solutions for the proposed iteration converges, and the SHJB equation is satisfied. Finally, numerical simulations demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system. We also define a new notion of moment matching and present a family of (nonlinear) parametrized reduced order models that achieve moment matching. We establish relations with existing notions of moment for nonlinear systems, showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching, respectively, are equivalent. Furthermore, we compute the reduced order model that matches the moments at two chosen signal generators (one exciting the input of the system and another driven by the system), simultaneously. We also present a family of models computed on the basis of a nonlinear extension of the Petrov-Galerkin projection that achieve moment matching. Finally, we specialize the results to the case of nonlinear, input-affine systems. [ABSTRACT FROM AUTHOR]

Port-Hamiltonian (PH) theory is a novel, but well established modeling framework for nonlinear physical systems. Due to the emphasis on the physical structure and modular framework, PH modeling has become a prime focus in system theory. This has led to a considerable research interest in the control of PH systems, resulting in numerous nonlinear control techniques. General nonlinear control methodologies are classified in a spectrum from model-based to model-free, where adaptation and learning typically lie close to the end of the range. Various articles and monographs have provided a detailed overview of model-based control techniques on PH models, but no survey is specifically dedicated to the learning and adaptive control methods that can benefit from the PH structure. To this end, we provide a comprehensive review of the current learning and adaptive control methodologies that have been adapted specifically to PH systems. After establishing the required theoretical background, we elaborate on various general machine learning, iterative learning, and adaptive control techniques and their application to PH systems. For each method we highlight the changes from the general setting due to PH model, followed by a detailed presentation of the respective control algorithm. In general, the advantages of using PH models in learning and adaptive controllers are: i) Prior knowledge in the form of PH model speeds up the learning. ii) In some instances new stability or convergence guarantees are obtained by having a PH model. iii) The resulting control laws can be interpreted in the context of physical systems. We conclude the paper with notes on open research issues. [ABSTRACT FROM AUTHOR]

Stochastic Trust-Region Response-Surface method (STRONG) is a new response-surface-based framework for simulation optimization. The appeal of STRONG lies in that it preserves the advantages, yet eliminates the disadvantages, of traditional response surface methodology (RSM) that has been used for more than 50 years. Specifically, STRONG does not require human involvement in the search process and can guarantee to converge to the true optimum with probability one (w.p.1). In this paper, we propose an improved framework, called STRONG-X, that enhances the efficiency and efficacy of STRONG to widen its applicability to more practical problems. For efficiency improvement, STRONG-X includes a newly-developed experimental scheme that consists of construction of optimal simulation designs and an assignment strategy for random number streams to obtain computational gains. For efficacy improvement, a new variant, called STRONG-XG, is developed to achieve convergence under generally-distributed responses, as opposed to STRONG and STRONG-X where convergence is guaranteed only when the response is normal. An extensive numerical study is conducted to evaluate the efficiency and efficacy of STRONG-X and STRONG-XG. Moreover, two illustrative examples are provided to show the viability of STRONG-X and STRONG-XG in practical settings. [ABSTRACT FROM AUTHOR]

Nonconvex feasible parameter sets are encountered in set membership identification whenever the regressor vector is affected by bounded uncertainty. This occurs for example when considering standard output error models, or when the available measurements are provided by binary or quantized sensors. In this paper, a unifying framework is proposed to deal with several identification problems involving a nonconvex feasible parameter set and a procedure is proposed for approximating the minimum volume orthotope containing the feasible set. The procedure exploits different relaxations for autoregressive and input parameters, based on the solution of a sequence of linear programming problems. The proposed technique is shown to provide tight bounds in some special cases. Moreover, it is extended to cope with bounds not aligned with the parameter coordinates, in order to obtain polytopic approximations of the feasible set. A number of numerical tests on randomly generated models and data sets demonstrates the accuracy of the computed set approximations. [ABSTRACT FROM AUTHOR]

While determining the order as well as the matrices of a black-box linear state-space model is now an easy problem to solve, it is well-known that the estimated (fully parameterized) state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. By assuming that the system has been identified consistently in a fully parameterized form, the question addressed in this paper then is how to recover the physical parameters from this initially estimated black-box form. Two solutions to solve such a parameterization problem are more precisely introduced. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to nonsmooth optimization is presented to transform the initial fully parameterized model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. By assuming that the physical state-space form is identifiable and the initial fully parameterized model is consistent, it is proved that the global solutions of these two optimization problems are unique. The proposed algorithms are presented, along with an example of a physical system. [ABSTRACT FROM AUTHOR]

When a dynamic optimization problem is not decomposable by a stage-wise backward recursion, it is nonseparable in the sense of dynamic programming. The classical dynamic programming-based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies. Among these notorious nonseparable problems, the dynamic mean-variance portfolio selection formulation had posed a great challenge to our research community until recently. Different from the existing literature that invokes embedding schemes and auxiliary parametric formulations to solve the dynamic mean-variance portfolio selection formulation, we propose in this paper a novel mean-field framework that offers a more efficient modeling tool and a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the multi-period mean-variance-type portfolio selection problems. [ABSTRACT FROM AUTHOR]