We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.
Flexible manufacturing has been the trend in the area of the modern chemical process nowadays. One of the essential characteristics of flexible manufacturing is to track time-varying target trajectories (e.g. diversity and quantity of products). A possible tool to achieve time-varying targets is contraction theory. However, the contraction theory was developed for continuous time systems and there lacks analysis and synthesis tools for discrete-time systems. This article develops a systematic approach to discrete-time contraction analysis and control synthesis using Discrete-time Control Contr…
We present quadratically convergent algorithms to compute the extremal value of a real parameter for which a given rational transfer function of a linear time-invariant system is passive. This problem is formulated for both continuous-time and discrete-time systems and is linked to the problem of finding a realization of a rational transfer function such that its passivity radius is maximized. Our new methods make use of the Hybrid Expansion-Contraction algorithm, which we extend and generalize to the setting of what we call root-max problems.
In this paper, we develop a novel contraction framework for stability analysis of discrete-time nonlinear systems with parameters following stochastic processes. For general stochastic processes, we first provide a sufficient condition for uniform incremental exponential stability (UIES) in the first moment with respect to a Riemannian metric. Then, focusing on the Euclidean distance, we present a necessary and sufficient condition for UIES in the second moment. By virtue of studying general stochastic processes, we can readily derive UIES conditions for special classes of processes, e.g., i.i…
Stochastic contraction analysis is a recently developed tool for studying the global stability properties of nonlinear stochastic systems, based on a differential analysis of convergence in an appropriate metric. To date, stochastic contraction results and sharp associated performance bounds have been established only in the specialized context of state-independent metrics, which restricts their applicability. This paper extends stochastic contraction analysis to the case of general time- and state-dependent Riemannian metrics, in both discrete-time and continuous-time settings, thus extending…
This paper develops a distributed model predictive control (DMPC) strategy for a class of discrete-time linear systems with consideration of globally coupled constraints. The DMPC under study is based on the dual problem concerning all subsystems, which is solved by means of the primal-dual gradient optimization in a distributed manner using Laplacian consensus. To reduce the computational burden, the constraint tightening method is utilized to provide a capability of premature termination with guaranteeing the convergence of the DMPC optimization. The contraction theory is first adopted in th…
We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for $L_1$-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global a…
We derive an exact deterministic nonlinear observer to compute the continuous state of an inertial navigation system based on partial discrete measurements, the so-called strapdown problem. Nonlinear contraction is used as the main analysis tool, and the hierarchical structure of the system physics is sytematically exploited. The paper also discusses the use of nonlinear measurements, such as distances to time-varying reference points.
We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov's Theorem. We use our methods to build strict Lyapunov functions for time-varying hybr…
Reachability analysis of hybrid systems has been used as a safety verification tool to assess offline whether the state of a system is capable of remaining within a designated safe region for a given time horizon. Although it has been applied to stochastic hybrid systems, little work has been done on the equally important problem of reachability under incomplete or noisy measurements of the state. Further, there are currently no computational methods or results for reachability analysis of partially observable discrete time stochastic hybrid systems. We provide the first numerical results for …
Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices. Although this provides the potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on controllers but on their partial derivatives, which makes control design challenging. In this paper, we illustrate this so-called integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers and then establish an LMI framework of contraction-based control design for nonlinear discrete-tim…
In the first part of the paper, we consider a discrete-time stochastic control system. We show that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of the system as well as the set of their mathematical expectations converge (as the time horizon tends to infinity) to a convex and compact (non-random) set, which is shown to coincide with the set of stationary probabilities of the system. In the second part, we apply the results obtained in the first part to deal with a hybrid system that evolves in continuous time and is subjected t…
For discrete-time systems, governed by Kraus maps, the work of D. Petz has characterized the set of universal contraction metrics. In the present paper, we use this characterization to derive a set of quadratic Lyapunov functions for continuous-time systems, governed by Lindblad differential equations, that have a steady-state with full rank. An extremity of this set is given by the Bures metric, for which the quadratic Lyapunov function is obtained by inverting a Sylvester equation. We illustrate the method by providing a strict Lyapunov function for a Lindblad equation designed to stabilize …
Stabilization of highly nonlinear hybrid stochastic differential delay equations by periodically intermittent feedback controls based on discrete-time observations with asynchronous switching
We present a numerically tractable formulation for computing the optimal control of the class of hybrid dynamical systems whose trajectories are continuous. Our formulation, an extension of existing relaxed-control techniques for switched dynamical systems, incorporates the domain information of each discrete mode as part of the constraints in the optimization problem. Moreover, our numerical results are consistent with phenomena that are particular to hybrid systems, such as the creation of sliding trajectories between discrete modes.
The emergence of Big Data has enabled new research perspectives in the discrete choice community. While the techniques to estimate Machine Learning models on a massive amount of data are well established, these have not yet been fully explored for the estimation of statistical Discrete Choice Models based on the random utility framework. In this article, we provide new ways of dealing with large datasets in the context of Discrete Choice Models. We achieve this by proposing new efficient stochastic optimization algorithms and extensively testing them alongside existing approaches. We develop t…
In the discrete modeling approach for hybrid control systems, the continuous plant is reduced to a discrete event approximation, called the DES-plant, that is governed by a discrete event system, representing the controller. The observability of the DES-plant model is crucial for the synthesis of the controller and for the proper closed loop evolution of the hybrid control system. Based on a version of the framework for hybrid control systems proposed by Antsaklis, the paper analysis the relation between the properties of the cellular space of the continuous plant and a mechanism of plant-symb…
We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincare map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via…
Input-to-State Stability Implications in Contraction Theory
Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）；Department of Mechanical Engineering and the Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara（加州大学圣塔芭芭拉分校机械工程系及控制、动力学系统与计算中心）
This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary o…
Three similar convergence notions are considered. Two of them are the long established notions of convergent dynamics and incremental stability. The other is the more recent notion of contraction analysis. All three convergence notions require that all solutions of a system converge to each other. In this paper, we investigate the differences between these convergence properties for discrete-time, time-varying nonlinear systems by comparing the properties in pairs and using examples. We also demonstrate a time-varying smooth Lyapunov function characterization for each of these convergence noti…
We consider the problem of optimizing hybrid structures (mixture of discrete and continuous input variables) via expensive black-box function evaluations. This problem arises in many real-world applications. For example, in materials design optimization via lab experiments, discrete and continuous variables correspond to the presence/absence of primitive elements and their relative concentrations respectively. The key challenge is to accurately model the complex interactions between discrete and continuous variables. In this paper, we propose a novel approach referred as Hybrid Bayesian Optimi…
This article aims to study coupled mean-field equation and ODEs with discrete events motivated by vehicular traffic flow. Precisely, multi-lane traffic flow in presence of human-driven and autonomous vehicles is considered, with the autonomous vehicles possibly influenced by external policy makers. First a finite-dimensional hybrid system is developed based on the continuous Bando-Follow-the-Leader dynamics coupled with discrete events due to lane-change maneuvers. Then the mean-field limit of the finite-dimensional hybrid system is rigorously derived for the dynamics of the human-driven vehic…
