Partition Of State Space Research Articles

A mutual information statistic for assessing state space partitions of dynamical systems.

We propose a mutual information statistic to quantify the information encoded by a partition of the state space of a dynamical system. We measure the mutual information between each point's symbolic trajectory history under a coarse partition (one with few unique symbols) and its partition assignment under a fine partition (one with many unique symbols). When applied to a set of test cases, this statistic demonstrates predictable and consistent behavior. Empirical results and the statistic's formulation suggest that partitions based on trajectory history, such as the ordinal partition, perform best. As an application, we introduce the weighted ordinal partition, an extension of the popular ordinal partition with parameters that can be optimized using the mutual information statistic, and demonstrate improvements over the ordinal partition in time series analysis. We also demonstrate the weighted ordinal partition's applicability to real experimental datasets.

State-Space Basins for Monopedal Jumping with Stable Landing

Abstract Maintaining stability during jumping remains a challenge due to its hybrid dynamics. Despite recent advancements in the control of jumping robots, existing research lacks an accurate and comprehensive criterion for the stability of jumping. This study proposes a general criterion for jumping that incorporates the system’s nonlinear dynamics and environmental constraints. The stability basins are formulated as a state-space partition based on a rigorous definition extending the balanced state basin (BSB) concept, which has been validated as a stability criterion for biped robots in legged stance. A hybrid approach, using a combination of analytical and optimization methods, solves the flight and stance phases of jumping as independent sub-problems. The basins are computed for a monopedal jumping robot, Salto-1P, and validated with simulations. To demonstrate its general applicability, two distinct flight-to-stance tasks are considered: targeted jumping and cat-like righting. Analysis of each task’s stability basins and the landing BSB reveals the inherent trade-offs between stability and the task requirements for variations in the initial state variables such as speed, orientation, and height. The stability basins represent the system’s fundamental controller-independent stability characteristics and can be applied to the mechanical design, control, and fall prediction of jumping robots.

Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

Representing turbulent statistics with partitions of state space. Part 1. Theory and methodology

This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive–response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Generalized-Type Multistability of Almost Periodic Solutions for Memristive Cohen-Grossberg Neural Networks.

This article investigates a generalized type of multistability about almost periodic solutions for memristive Cohen-Grossberg neural networks (MCGNNs). As the inevitable disturbances in biological neurons, almost periodic solutions are more common in nature than equilibrium points (EPs). They are also generalizations of EPs in mathematics. According to the concepts of almost periodic solutions and Ψ -type stability, this article presents a generalized-type multistability definition of almost periodic solutions. The results show that (K+1)n generalized stable almost periodic solutions can coexist in a MCGNN with n neurons, where K is a parameter of the activation functions. The enlarged attraction basins are also estimated based on the original state space partition method. Some comparisons and convincing simulations are given to verify the theoretical results at the end of this article.

Information Bottleneck Approach for Markov Model Construction.

Markov state models (MSMs) have proven valuable in studying the dynamics of protein conformational changes via statistical analysis of molecular dynamics simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multiresolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multiresolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on a specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSM construction.

Secure state estimation for affine T-S fuzzy systems under sparse sensor attacks

This paper is concerned with the secure state estimation problem of affine T-S fuzzy systems under sparse sensor attacks. The objective is to construct a bank of fuzzy state estimators with an adaptive switching mechanism to ensure the attacked sensor measurements be isolated, and the estimation error system is asymptotic stable with a guaranteed H∞ performance. The case of unmeasurable plant premise variables is considered, which leads to the estimator implementation according to estimators’ state-space partitions may not be always synchronous with the plant. By constructing piecewise Lyapunov functions and employing the S-procedure and matrices decoupled techniques, the constructed state estimators synthesis conditions are derived in linear matrix inequalities (LMIs) form. Compared with existing methods, the proposed approach avoids restrictions on the structure of Lyapunov matrices and derives less conservative estimator design conditions. Finally, the effectiveness and superiorities of the proposed method is verified with a numerical example.

A deep learning method for pricing high-dimensional American-style options via state-space partition

A deep learning method for pricing high-dimensional American-style options via state-space partition

Goal-conditioned offline reinforcement learning through state space partitioning

AbstractOffline reinforcement learning (RL) aims to create policies for sequential decision-making using exclusively offline datasets. This presents a significant challenge, especially when attempting to accomplish multiple distinct goals or outcomes within a given scenario while receiving sparse rewards. Prior methods using advantage weighting for offline goal-conditioned learning improve policies monotonically. However, they still face challenges from distribution shift and multi-modality that arise due to conflicting ways to reach a goal. This issue is especially challenging in long-horizon tasks, where the presence of multiple, often conflicting, solutions makes it hard to identify a single optimal policy for transitioning from a state to a desired goal. To address these challenges, we introduce a complementary advantage-based weighting scheme that incorporates an additional source of inductive bias. Given a value-based partitioning of the state space, the contribution of actions expected to lead to target regions that are easier to reach, compared to the final goal, is further increased. Our proposed approach, Dual-Advantage Weighted Offline Goal-conditioned RL, outperforms several competing offline algorithms in widely used benchmarks. Furthermore, we provide a theoretical guarantee that the learned policy will not be inferior to the underlying behavior policy.

Multiple Mittag-Leffler Stability of Fractional-Order Complex-Valued Memristive Neural Networks With Delays.

This article discusses the coexistence and dynamical behaviors of multiple equilibrium points (Eps) for fractional-order complex-valued memristive neural networks (FCVMNNs) with delays. First, based on the state space partition method, some sufficient conditions are proposed to guarantee that there are multiple Eps in one FCVMNN. Then, the Mittag-Leffler stability of those multiple Eps is proved by using the Lyapunov function. Simultaneously, the enlarged attraction basins are obtained to improve and extend the existing theoretical results in the previous literature. In addition, some existing stability results in the literature are special cases of a new result herein. Finally, two illustrative examples with computer simulations are presented to verify the effectiveness of theoretical analysis.

Data-driven control for discrete-time piecewise affine systems

This paper studies the problem of data-driven control for discrete-time piecewise affine (PWA) systems. Based on a sequel of sampled control inputs and states with satisfactory rank conditions, the closed loop dynamics of PWA systems can be represented and therefore the explicit information of system matrices is not needed in system stability analysis and controller synthesis processes. First, via S-procedure, system stability analysis results have been obtained, and then, two novel data-driven controller design methods in the form of linear matrix inequalities and equations have been proposed such that the resulting closed loop system is asymptotically stable. For PWA systems with unknown state space partition information, an online data-driven control method has been further proposed to guarantee the asymptotic stability of the resulting closed loop systems. Finally, simulation studies of two examples have been given to validate the effectiveness of the proposed data-driven controller design approaches.

Safe control of logical control networks with random impulses

Under the framework of a hybrid-index model, this paper investigates safe control problems of state-dependent random impulsive logical control networks (RILCNs) on both finite and infinite horizons, respectively. By using the ξ-domain method and the constructed transition probability matrix, the necessary and sufficient conditions for the solvability of safe control problems have been established. Further, based on the technique of state-space partition, two algorithms are proposed to design feedback controllers such that RILCNs can achieve the goal of safe control. Finally, two examples are shared to demonstrate the main results.

Robust stability analysis of continuous PWA systems

This paper studies the robust stability of the origin of continuous-time Piecewise Affine (PWA) systems. We consider an implicit representation based on vector-valued ramp functions to model PWA continuous-time systems. From this representation, we derive Linear Matrix Inequality (LMI) stability conditions suitable to deal with polytopic uncertainties that can modify both the dynamics and the regions of the state-space partition of the PWA system. Two numerical examples illustrate the effectiveness of the proposed conditions.

A Branch-independence-based Reliability Assessment Approach for Transmission Systems

This paper proposes a branch-independence-based reliability assessment approach for transmission systems. The approach consists of branch decoupling and state-space partition techniques. By integrating an impact-increment-based reliability index calculation model and the proposed branch decoupling technique, a proportion of sampled contingency states no longer need to be analyzed using the time-consuming optimal power flow (OPF) algorithm. In this way, the technique speeds up the calculation of reliability indices. Since first-order contingency states have a high probability of being sampled, we propose a state-space partition technique to replace first-order contingency state simulation with first-order contingency state enumeration. Consequently, the calculation of reliability indices is further accelerated by avoiding a large amount of repetitive OPF analyses during simulation process without affecting reliability index accuracy. The validity and applicability of our approach are verified using the IEEE 118-bus and IEEE 145-bus systems. Numerical results indicate that the proposed approach can improve computational efficiency without decreasing accuracy.

Multiple asymptotical [formula omitted]-periodicity of fractional-order delayed neural networks under state-dependent switching

This paper presents theoretical results on multiple asymptotical ω-periodicity of a state-dependent switching fractional-order neural network with time delays and sigmoidal activation functions. Firstly, by combining the geometrical properties of activation functions with the range of switching threshold, a partition of state space is given. Then, the conditions guaranteeing that the solutions can approach each other infinitely in each positive invariant set are derived. Furthermore, the S-asymptotical ω-periodicity and the convergence of solutions in positive invariant sets are discussed. It is worth noting that the number of attractors increases to 3n from 2n in a neural network without switching. Finally, three numerical examples are given to substantiate the theoretical results.

Rapid Feedback Stabilization of Quantum Systems With Application to Preparation of Multiqubit Entangled States.

For stochastic quantum systems with measurement feedback, this article proposes a rapid switching control scheme based on state space partition and realizes the rapid stabilization of an eigenstate of an observable operator. Meanwhile, we apply the proposed scheme to the preparation of typical entangled states in multiqubit systems. In view of the convergence obstacle caused by the symmetric structure of the state space, especially in the case with degenerate observable operators, we first partition the state space into a subset containing the target state and its complement to distinguish the target state from its antipodal points, and then design the corresponding control laws in these two subsets, respectively, by using different Lyapunov functions. The interaction Hamiltonians are also constructed to drive the system state to the desired subset first, and further to the target state. In particular, the control law designed in the undesired subset guarantees the strictly monotonic descent of the corresponding Lyapunov function, which makes the system trajectory switch between the two subsets at most twice and has the potential to speed up the convergence process. We also prove the stability of the closed-loop system with the proposed switching control law based on the stochastic Lyapunov stability theory. By applying the proposed switching control scheme to a three-qubit system, we achieve the preparation of a GHZ state and a W state.

State space partitioning based on constrained spectral clustering for block particle filtering

To overcome the curse of dimensionality of particle filter (PF), the block PF (BPF) partitions the state space into several blocks of smaller dimension so that the correction and resampling steps can be performed independently on each block. Despite its potential performance, this approach has a practical limitation. BPF requires an additional input compared to classical PF: the partition from which blocks are defined. A poor choice will not offer the expected performance gain.In this paper, we formulate the partitioning problem as a clustering problem and propose a data-driven partitioning method based on constrained spectral clustering (CSC) to automatically provide an appropriate partition.We design a generalized BPF that contains two new steps: (i) estimation of the state vector correlation matrix from particles, (ii) CSC using this estimate to determine blocks. The proposed method succeeds in providing an online partition into blocks restricted in size and grouping the most correlated state variables. This partition allows the BPF to escape the curse of dimensionality by reducing the variance of the filtering distribution estimate while limiting the level of bias. Since our approach relies on particles that are already necessary to generate as part of BPF, the computation overhead is limited.

Protocol-based filtering for fuzzy Markov affine systems with switching chain

This paper investigates the reduced-order filter design method for fuzzy Markov jump affine systems with dynamic event-triggered protocol and uncertain packet dropouts. Aiming at approximating the nonlinearities with high accuracy and broadening practical application, the fuzzy Markov jump affine systems associated with mode-dependent operating regions are developed. A novel nonhomogeneous Markov process is proposed to describe the generalized stochastic jumping among subsystems, whose time-varying transition probabilities are orchestrated by a higher-level deterministic switching signal. Compared with the conventional Markov process with arbitrary switchings, the average dwell-time strategy is adopted in the novel nonhomogeneous Markov process to improve the dynamic performance. A dynamic event-triggered protocol is provided to alleviate the network transmission burden by elongating the interval between any two consecutive events. Based on the variation of state-space partitions, nonsynchronous reduced-order generalized filters are considered. Under this framework, sufficient criteria are attained to ensure stochastic stability and prescribed l2−l∞ performance for the filtering error systems. Eventually, the theoretical findings are verified by two simulation examples.

Guaranteed performance control of switched positive systems: A switching policy design method

In this article, a state-dependent switching policy for handling the guaranteed performance control problem, by means of the partition of the nonnegative state space, for switched positive systems, is devised. The mixed performance, induced by the linear guaranteed cost and L1-gain performance for disturbance attenuation, is investigated. A corresponding optimization problem devoting to the mixed performance is considered, including an upper bound of linear guaranteed cost and an L1-gain performance. Moreover, sufficient conditions are also established by the joint design of switching rules and controllers to ensure the positivity and mixed performance of the closed-loop systems. Eventually, two examples are used to illustrate the validness and practicability of the developed results.