Partition Of State Space Research Articles

Optimal control of partitioned hybrid systems via discrete-time Hamilton–Jacobi theory

Computational framework for optimal control of hybrid systems with a partitioned state space is presented. It is shown that necessary conditions for optimality for a discrete-time dynamic system can be solved concurrently for various boundary conditions, according to the recent development of discrete-time Hamilton–Jacobi theory. This unique property is utilized to construct computationally efficient numerical optimization of hybrid systems where discrete switching dynamics occurs at the boundary between partitions of the configuration space. A benchmark example shows that the proposed approach has substantial computational advantages compared with the existing ones.

Partitioned event graph: formalizing LP-based modelling of parallel discrete-event simulation

Logical process (LP) is a modelling paradigm widely used in parallel discrete-event simulation (PDES). However, effective methods for formalizing LP-based modelling of PDES are lacking. This prevents an unambiguous, platform-independent description of LP-based models. We present a formalism named partitioned event graph (PEG) as a solution. PEG extends classical event graph formalism towards a formal specification for LP-based PDES models. We map between PEG and LP-based models, define the structural operational semantics (SOS) in a timed-labelled transition system, and discuss the Wallclock time-based execution. We propose a PEG-based model transformation framework for PDES, which has three model representation phases and distinguishes amongst four kinds of personnel roles. Finally, we present a domain-specific language (DSL) for the PDES of a Lotka–Volterra system and obtain preliminary parallel simulation results using YinHe Simulation Utilities for Parallel Environment (YHSUPE). The case study shows that the PEG-based framework not only effectively transforms a DSL into the LP paradigm, but will also result in efficient parallel simulation on a specific platform. In summary, by setting out specific characteristics for event scheduling and state space partition in the LP paradigm, PEG provides a formal method for model behaviour analysis and cross-platform model transformation.

Observer-Based Control for DC–DC Converters

The paper proposes a computational technique for the synthesis of a suitable observer-based control for DC–DC switching converters. The work will be carried out directly on the Piece-Wise Affine convincing representation obtained from the Bonds Graph modeling method. The aim was to regulate by switching the converter outputs to an average (not common) equilibrium reference. Based on the Lyapunov theory, a systematic state feedback control is derived from a tractable Bilinear Matrix Inequality formulation of the problem. The design control searches for a single Lyapunov-like function that satisfies practical quadratic stabilization properties in an appropriate continuous state space partition and provides a way to drive the system states into a quantifiable small ball around the non-equilibrium desired reference. To complete the method, a simple state estimation procedure is introduced to avoid state measurements, the technique is based on Luenberger-like observer structure. The approach can be implemented as an embedded software system to generate several continuous supply levels. Satisfactory simulation results are obtained for several examples, the illustrative theoretical demonstrations for some of these switching devices are reported.

Approximation of Model Predictive Control Laws for Polynomial Systems

AbstractA fast implementation of a given predictive controller for polynomial systems is introduced by approximating the optimal control law with a piecewise constant function defined over a hyper‐cube partition of the system state space. Such a state‐space partition is computed in order to guarantee stability, an a priori fixed trajectory error as well as input and state constraints fulfilment. The presented approximation procedure is achieved by solving a set of nonconvex polynomial optimization problems, whose approximate solutions are computed by means of semidefinite relaxation techniques for semialgebraic problems.

Mapping densities in a noisy state space

Weak noise smooths out fractals in a chaotic state space and introduces a maximum attainable resolution to its structure. The balance of noise and deterministic stretching/contraction in each neighborhood introduces local invariants of the dynamics that can be used to partition the state space. We study the local discrete-time evolution of a density in a two-dimensional hyperbolic state space, and use the asymptotic eigenfunctions for the noisy dynamics to formulate a new state space partition algorithm. 1. Motivation and outline Chaotic systems’ main feature is their high sensitivity to initial conditions. That makes direct numerical integration of the equations difficult and often calls for alternative methods for the evaluation of long-time averages of observables, such as decay of correlations, diffusion coefficients, energy spectra, or escape rates [7]. To properly weigh these averages, one needs to understand which regions of the state space are more or less relevant for the dynamics, in other words make a partition [3]. Invariants of the dynamics such as unstable periodic oribts have been successfully used to partition the state space [1]. However, noise, modelled by stochastic variables, erases periodic orbits. One has to look for new invariants. For that reason, we previously have studied [4, 2] the evolution of densities of trajectories and determined eigenfunctions of the local Fokker-Planck operator in the vicinity of the deterministic periodic orbits. The eigenfunctions are then used to partition the state space. All that was done in discrete time in one dimension. In order to develop a similar algorithm in higher dimensions, the first step is again to study the evolution of densities in the neighborhood of the periodic points of the deterministic system. In the present contribution we focus on the asymptotic evolution in two dimensions, forward and backward in time of a noiseless hyperbolic map (sect. 2), to which we successively add weak, uncorrelated, isotropic noise (sect. 3). In both cases the densities asymptotically align with the unstable (stable) direction of the monodromy matrix when iterated forward (backward) in time. We finally use our results to propose a definition of a neighborhood for an optimal partition of the state space. 2. Deterministic evolution We start by reviewing the deterministic evolution of densities and observables in the neighborhood of a fixed point x0 of the two-dimensional map x′ = f(x). We assume that the fixed point is hyperbolic, i.e., that the Jacobian matrix evaluated at the fixed point, Mi j(x0) = ∂ fi(x) ∂x j ∣∣∣∣∣ x=x0 , (1) has eigenvalues |Λs| 1. Consider the simplest example, a map f(x) = (Λsx,Λuy) (2) that is contracting along the x-axis and expanding along the y-axis. Now consider a density of trajectories ρ(x), for instance a Gaussian placed around the fixed point of f(x), and apply the Perron-Frobenius operator [1] to it: L ρ(x) = ∫ dz δ (x − f(z)) ρ(z) = 1 |ΛuΛs| ρ ( x Λs , y Λu ) , (3) so that, after n iterations, Lnρ(x) = 1 |Λu| ρ( x Λ n s , y Λ n u ) |Λs| . (4) One can see this as a density, which is losing mass by a factor of |Λu| at each iteration. This expression can be renormalized by a factor of |Λu| , when taking the limit n → ∞. If the initial density is a normalized Gaussian ρ(x) ∝ exp [ −(x2 + y2)/2σ2 ] , we obtain lim n→∞ |Λu|L = lim n→∞ exp [ − x2 2(σΛs )2 ] √ 2πσ2Λ s = δ(x) (5) meaning the limiting density is supported on the y-axis, the unstable manifold of the fixed point of the map. Of course this was the simplest possible example, given that the contracting and expanding directions of the fixed point are already separated by the coordinates. This is not the case in general, and one needs to do something different from what we just described. We will follow Rugh’s formalism [8] for a general two-dimensional map f(x) with a hyperbolic fixed point x0: the equation fy(xi, yi) = y f has ar X iv :1 30 3. 09 51 v1 [ nl in .C D ] 5 M ar 2 01 3 a unique solution, which we can call φs(xi, y f ), which is analytic and a contraction. On the other hand, one can define φu(xi, y f ) = fx ( xi, φs(xi, y f ) ) (6)

Optimal Estimation of Free Energies and Stationary Densities from Multiple Biased Simulations

When studying high-dimensional dynamical systems such as macromolecules, quantum systems, and polymers, a prime concern is the identification of the most probable states and their stationary probabilities or free energies. Often, these systems have metastable regions or phases, prohibiting the estimation of the stationary probabilities by direct simulation. Efficient sampling methods such as umbrella sampling, metadynamics, and conformational flooding have been developed that perform a number of simulations where the system's potential is biased so as to accelerate the rare barrier crossing events. A joint free energy profile or stationary density can then be obtained from these biased simulations with the weighted histogram analysis method. This approach (a) requires a few essential order parameters to be defined in which the histogram is set up, and (b) assumes that each simulation is in global equilibrium. Both assumptions make the investigation of high-dimensional systems with previously unknown energy landscape difficult. Here, we introduce the transition matrix based unbiasing method (TMU), a simple and efficient estimation method which dismisses both assumptions. The configuration space is discretized into sets, but these sets are not only restricted to a preselected slow coordinate but can be clusters that form a partition of high-dimensional state space. The assumption of global equilibrium is replaced by requiring only local equilibrium within the discrete sets, and the stationary density or free energy is extracted from the transitions between clusters. We prove the asymptotic convergence and normality of TMU, give an efficient approximate version of it, and demonstrate its usefulness in numerical examples.

State-Partition-Based Control of Discrete Event Systems for Enforcement of Regular Language Specifications

We consider the solution of supervisory control problems for discrete event systems modeled by automata or bounded Petri nets where the specification is expressed as a regular sublanguage of the system language and where the supervisor is restricted to be state-partition-based with respect to original system state space, i.e., the state space of the automaton or the set of reachable markings of the Petri net. State-partition-based supervisors are completely characterized by a partition of the original system state space into legal and illegal states: transitions between legal states are always enabled while transitions from legal to illegal states are always disabled. We present a general algorithm that calculates all state-partition-based supervisors that result in safe and non-blocking controlled languages. The algorithm uses a vertex-cover-type algorithm on the representation of the supremal controllable sublanguage in order to obtain the desired partitions. This work is motivated by the application of discrete event control techniques to the avoidance of classes of concurrency bugs in multithreaded programs. State-partition-based supervisors are especially advantageous in that application as they allow more concurrency at run-time. More generally, this class of supervisors is required when the representation of the supervisor must be based on the system's original state space; this occurs for memoryless supervisors in automaton-based control or in supervision based on place invariants in Petri-net-based control, for instance.

Nonsynchronized Robust Filtering Design for Continuous-Time T–S Fuzzy Affine Dynamic Systems Based on Piecewise Lyapunov Functions

This paper investigates the problem of robust H(∞) state estimation for a class of continuous-time nonlinear systems via Takagi-Sugeno (T-S) fuzzy affine dynamic models. Attention is focused on the analysis and design of an admissible full-order filter such that the resulting filtering error system is asymptotically stable with a guaranteed H(∞) disturbance attenuation level. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the filter implementation with state-space partition may not be synchronous with the state trajectories of the plant. Based on piecewise quadratic Lyapunov functions combined with S-procedure and some matrix inequality linearization techniques, some new results are presented for the filtering design of the underlying continuous-time T-S fuzzy affine systems. Illustrative examples are given to validate the effectiveness and application of the proposed design approaches.

Finite Abstractions of Max-Plus-Linear Systems

This work puts forward a novel technique to generate finite abstractions of autonomous and nonautonomous Max-Plus-Linear (MPL) models, a class of discrete-event systems used to characterize the dynamics of the timing related to successive events that synchronize autonomously. Nonautonomous versions of MPL models embed within their dynamics nondeterminism, namely a signal choice that is usually regarded as an exogenous control or schedule. In this paper, abstractions of MPL models are characterized as finite-state Labeled Transition Systems (LTS). LTS are obtained first by partitioning the state space (and, for the nonautonomous model, by covering the input space) of the MPL model and by associating states of the LTS to the introduced partitions, then by defining relations among the states of the LTS based on dynamical transitions between the corresponding partitions of the MPL state space, and finally by labeling the LTS edges according to the one-step timing properties of the events of the original MPL model. In order to establish formal equivalences, the finite abstractions are proven to either simulate or to bisimulate the original MPL model. This approach enables the study of general properties of the original MPL model by verifying (via model checking) equivalent logical specifications over the finite LTS abstraction. The computational aspects related to the abstraction procedure are thoroughly discussed and its performance is tested on a numerical benchmark.

Limit Order Books with Stochastic Market Depth

We propose a partitioning algorithm to solve a class of linear-quadratic Markov decision processes with inequality constraints and non-convex stagewise cost; within each region of the partitioned state space, the value function and the optimal policy have analytical quadratic and linear forms, respectively. The algorithm is applied to two applications. In the main application, we present a model for limit order books with stochastic market depth to study the optimal order execution problem. Stochastic market depth is consistent with empirical studies and necessary to accommodate various order activities, such as limit order submission at and outside the best quotes and order cancellation, which may account for a large proportion of limit order activities. The optimal execution policy obtained by the algorithm significantly outperforms that of a deterministic market depth model in numerical examples. In the second application, we use the algorithm to compute the exact optimal solution to the renewable electricity management problem, for which previously only an approximate solution is known. As a comparison, we show that the approximate solution can be quite inaccurate for some initial states and thus demonstrate the advantage of the exact solution in this application.

A proposition of adaptive state space partition in reinforcement learning with Voronoi tessellation

This paper presents a new adaptive segmentation of continuous state space based on vector quantization algorithm such as Linde---Buzo---Gray for high-dimensional continuous state spaces. The objective of adaptive state space partitioning is to develop the efficiency of learning reward values with an accumulation of state transition vector in a single-agent environment. We constructed our single-agent model in continuous state and discrete actions spaces using Q-learning function. Moreover, the study of the resulting state space partition reveals a Voronoi tessellation. In addition, the experimental results show that this proposed method can partition the continuous state space appropriately into Voronoi regions according to not only the number of actions, but also achieve a good performance of reward-based learning tasks compared with other approaches such as square partition lattice on discrete state space.

Self-organized topology of recurrence-based complex networks

With the rapid technological advancement, network is almost everywhere in our daily life. Network theory leads to a new way to investigate the dynamics of complex systems. As a result, many methods are proposed to construct a network from nonlinear time series, including the partition of state space, visibility graph, nearest neighbors, and recurrence approaches. However, most previous works focus on deriving the adjacency matrix to represent the complex network and extract new network-theoretic measures. Although the adjacency matrix provides connectivity information of nodes and edges, the network geometry can take variable forms. The research objective of this article is to develop a self-organizing approach to derive the steady geometric structure of a network from the adjacency matrix. We simulate the recurrence network as a physical system by treating the edges as springs and the nodes as electrically charged particles. Then, force-directed algorithms are developed to automatically organize the network geometry by minimizing the system energy. Further, a set of experiments were designed to investigate important factors (i.e., dynamical systems, network construction methods, force-model parameter, nonhomogeneous distribution) affecting this self-organizing process. Interestingly, experimental results show that the self-organized geometry recovers the attractor of a dynamical system that produced the adjacency matrix. This research addresses a question, i.e., "what is the self-organizing geometry of a recurrence network?" and provides a new way to reproduce the attractor or time series from the recurrence plot. As a result, novel network-theoretic measures (e.g., average path length and proximity ratio) can be achieved based on actual node-to-node distances in the self-organized network topology. The paper brings the physical models into the recurrence analysis and discloses the spatial geometry of recurrence networks.

Preference for Flexibility and Dynamic Consistency

Abstract Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency , is linked to a comparative theory of preference for flexibility . When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.

Optimal Control for Hybrid Systems With Partitioned State Space

For hybrid systems where the continuous state space is partitioned by switching manifolds, the discrete state changes autonomously if the continuous state hits a manifold. Recently, we introduced a version of the minimum principle for the optimal control of such systems, where necessary optimality conditions were provided for the case that a trajectory passes through the intersection of switching manifolds . Further, we introduced an algorithm based on these conditions for computing optimal controls by varying not only the continuous state on switching manifolds based on gradient information, but also the sequence of discrete states . It was shown that, the combinatorial complexity of former computational schemes based on the minimum principle can be avoided, since not all possible discrete state sequences need to be analyzed separately. In this note, theoretical aspects of the algorithm and practical considerations for a successful implementation are discussed in detail. Further, a comparison to existing algorithms for the solution of hybrid optimal control problems with autonomous switching is presented. The efficiency of the proposed algorithm is demonstrated by a novel numerical example.

A computationally efficient state-space partitioning approach to pricing high-dimensional American options via dimension reduction

This paper studies the problem of pricing high-dimensional American options. We propose a method based on the state-space partitioning algorithm developed by Jin et al. (2007) and a dimension-reduction approach introduced by Li and Wu (2006). By applying the approach in the present paper, the computational efficiency of pricing high-dimensional American options is significantly improved, compared to the extant approaches in the literature, without sacrificing the estimation precision. Various numerical examples are provided to illustrate the accuracy and efficiency of the proposed method. Pseudcode for an implementation of the proposed approach is also included.

Fault detection of discrete-time T–S fuzzy affine systems based on piecewise Lyapunov functions

This paper investigates the problem of robust fault detection for a class of discrete-time nonlinear systems, which are represented by Takagi–Sugeno (T–S) fuzzy affine dynamic models with norm-bounded uncertainties. The objective is to design an admissible fault detection filter guaranteeing the asymptotic stability of the resulting residual system with prescribed performances. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the fault detection filter implementation with state-space partition may not be synchronized with the state trajectories of the plant. Based on a piecewise quadratic Lyapunov function combined with S-procedure and some matrix inequality convexification techniques, the results are formulated in the form of linear matrix inequalities. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach.

Variable-resolution Velocity Roadmap Generation Considering Safety Constraints for Mobile Robots

This research develops a new roadmap method for autonomous mobile robots based on variable-resolution partitioning of continuous state space. Unlike conventional roadmaps which include position information only, the proposed roadmap also includes velocity information. Each node of the proposed roadmap consists of a fixed position and a range of velocity values, where velocity ranges are determined by variable-resolution partitioning of velocity space. An ordered pair of nodes are connected by a directed link if any combination of velocity values within the ranges represented by the nodes produce a trajectory satisfying a set of safety constraints. In this manner, an executable trajectory connecting arbitrary starting node and destination node is obtained by applying a graph search technique on the proposed roadmap. The proposed method is evaluated through simulations.

A patchy approximation of explicit model predictive control

The explicit solution of multi-parametric optimisation problems (MPOP) has been used to construct an off-line solution to relatively small- and medium-sized constrained control problems. The control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics. In this context, it can be shown that the optimal control law is a piecewise linear (PWL) state feedback defined over polytopic cells of the state space. However, as the complexity of the related optimisation problems increases, the memory footprint and implementation of such explicit optimal solution may be burdensome for the available hardware, principally due to the high number of polytopic cells in the state-space partition. In this article we provide a solution to this problem by proposing a patchy PWL feedback control law, which intend to approximate the optimal control law. The construction is based on the linear interpolation of the exact solution at the vertices of a feasible set and the solution of an unconstrained linear quadratic regulator (LQR) problem. With a hybrid patchy control implementation, we show that closed-loop stability is preserved in the presence of additive measurement noise despite the existence of discontinuities at the switch between the overlapping regions in the state-space partition.

Hysteresis Switching Design for Stabilization of a Class of Switched Neutral Systems

AbstractStabilization of a class of switched neutral systems with discrete time‐varying delays is investigated in this paper. The upper bound of the derivative of the discrete time‐varying delay is an arbitrary given unknown constant. Each subsystem is not assumed stable. In order to avoid sliding modes and chattering, a hysteresis switching law is designed based on state space partition, which considers the information of both discrete delays and neutral delays sufficiently. The obtained delay‐dependent stabilization criterion is given in terms of linear matrix inequality by introducing free‐weighting matrices. Finally, a simulation example is given to motivate and exemplify the proposed strategy.

On formalism and stability of switched systems

In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the theory of differential inclusions, we devise a Lyapunov stability theorem suitable for this class of switched systems. With this, we prove a Lyapunov stability theorem for piecewise linear switched systems by means of a concrete class of Lyapunov functions. Contrary to existing results on the subject, the stability theorems in this paper include Filippov (or relaxed) solutions and allow infinite switching in finite time. Finally, we show that for a class of piecewise linear switched systems, the inertia of the system is not sufficient to determine its stability. A number of examples are provided to illustrate the concepts discussed in this paper.