Dimension Of State Space Research Articles

Hydropower Aggregation by Spatial Decomposition—An SDDP Approach

The balance between detailed technical description, representation of uncertainty and computational complexity is central in long-term scheduling models applied to hydro-dominated power system. The aggregation of complex hydropower systems into equivalent energy representations (EER) is a commonly used technique to reduce dimensionality and computation time in scheduling models. This work presents a method for coordinating the EERs with their detailed hydropower system representation within a model based on stochastic dual dynamic programming (SDDP). SDDP is applied to an EER representation of the hydropower system, where feasibility cuts derived from optimization of the detailed hydropower are used to constrain the flexibility of the EERs. These cuts can be computed either before or during the execution of the SDDP algorithm and allow system details to be captured within the SDDP strategies without compromising the convergence rate and state-space dimensionality. Results in terms of computational performance and system operation are reported from a test system comprising realistic hydropower watercourses.

A New Exact Algorithm for Single-Commodity Vehicle Routing with Split Pickups and Deliveries

We present a new exact algorithm to solve a challenging vehicle routing problem with split pickups and deliveries, named as the single-commodity split-pickup and split-delivery vehicle routing problem (SPDVRP). In the SPDVRP, any amount of a product collected from a pickup customer can be supplied to any delivery customer, and the demand of each customer can be collected or delivered multiple times by the same or different vehicles. The vehicle fleet is homogeneous with limited capacity and maximum route duration. This problem arises regularly in inventory and routing rebalancing applications, such as in bike-sharing systems, where bikes must be rebalanced over time such that the appropriate number of bikes and open docks are available to users. The solution of the SPDVRP requires determining the number of visits to each customer, the relevant portions of the demands to be collected from or delivered to the customers, and the routing of the vehicles. These three decisions are intertwined, contributing to the hardness of the problem. Our new exact algorithm for the SPDVRP is a branch-price-and-cut algorithm based on a pattern-based mathematical formulation. The SPDVRP relies on a novel label-setting algorithm used to solve the pricing problem associated with the pattern-based formulation, where the label components embed reduced cost functions, unlike those classical components that embed delivered or collected quantities, thus significantly reducing the dimension of the corresponding state space. Extensive computational results on different classes of benchmark instances illustrate that the newly proposed exact algorithm solves several open SPDVRP instances and significantly improves the running times of state-of-the-art algorithms. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72222011, 71971090, 71821001, 72171112], by the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], and by the Research Grants Council of Hong Kong SAR, China [Grant 15221619]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1249 .

PnP Operations to Maintain Persistent Feasibility in Linear Constrained Systems

Multi-agent systems often have high dimensional state and input spaces that change rapidly as agents join and leave the system. The plug-and-play (PnP) design paradigm provides a framework to design controllers in this context by shifting most computations for individual agents to design-time. Smaller, numerically tractable PnP algorithms are then used to allow the agents to join and leave the network while maintaining persistent feasibility and stability. In constrained systems, persistent feasibility is assured within feasible control invariant (CI) sets. This work develops a PnP framework to compute these sets for a class of distributed, non-cooperative linear systems. The agents are allowed a preview of the cross-disturbance but no other restrictions are placed on the agents’ controllers. At design-time, undisturbed and maximally disturbed CI sets are found for each agent individually. As the network changes, the agents’ current CI sets are computed using linear combinations of these extreme CI sets. With this, on-line constraint adjustments can be accomplished through distributed optimization problems that scale well for large networks. The overall solution is agnostic to the local controller design and accounts for the local states at the time of the PnP operations.

Golay codes and quantum contextuality

It is shown that the codewords of the binary and ternary Golay codes can be converted into rays in ${\mathbb{RP}}^{23}$ and ${\mathbb{RP}}^{11}$ that provide proofs of the Kochen-Specker theorem in real state spaces of dimensions 24 and 12, respectively. Some implications of these results are discussed.

Using photonic reservoirs as preprocessors for deep neural networks

Artificial neural networks are very time consuming and energy intensive to train, especially when increasing the size of the neural network in an attempt to improve the performance. In this paper, we propose to preprocess the input data of a deep neural network using a reservoir, which has originally been introduced in the framework of reservoir computing. The key idea of this paper is to use such a reservoir to transform the input data into a state in a higher dimensional state-space, which allows the deep neural network to process the data with improved performance. We focus on photonic reservoirs because of their fast computation times and low-energy consumption. Based on numerical simulations of delay-based reservoirs using a semiconductor laser, we show that using such preprocessed data results in an improved performance of deep neural networks. Furthermore, we show that we do not need to carefully fine-tune the parameters of the preprocessing reservoir.

Autonomous robot navigation based on a hierarchical cognitive model

AbstractWe propose a hierarchical cognitive navigation model (HCNM) to improve the self-learning and self-adaptive ability of mobile robots in unknown and complex environments. The HCNM model adopts the divide and conquers approach by dividing the path planning task into different levels of sub-tasks in complex environments and solves each sub-task in a smaller state subspace to decrease the state space dimensions. The HCNM model imitates animal asymptotic properties through the study of thermodynamic processes and designs a cognitive learning algorithm to achieve online optimum search strategies. We prove that the learning algorithm designed ensures that the cognitive model can converge to the optimal behavior path with probability one. Robot navigation is studied on the basis of the cognitive process. The experimental results show that the HCNM model has strong adaptability in unknown and environment, and the navigation path is clearer and the convergence time is better. Among them, the convergence time of HCNM model is 25 s, which is 86.5% lower than that of HRLM model. The HCNM model studied in this paper adopts a hierarchical structure, which reduces the learning difficulty and accelerates the learning speed in the unknown environment.

Markovian policy network for efficient robot learning

What poses a challenge for robot learning is the fact that considerable training samples and time are required to find an optimal policy in the high-dimensional robot state space. To mitigate this problem of robot learning, there is an incentive in incorporating appropriate prior knowledge. In this paper, a novel framework named Markovian Policy Network (MPN) is presented to take full advantage of the structural prior knowledge for robots to achieve efficient robot learning. Specifically, we first make efforts to reveal the Markov Property of the robot, which characterizes the intrinsic prior knowledge for robotic physical structure. Next, the proposed framework is constructed by employing a modified graph neural network to incorporate the Markov Property of the robot into policy networks. By exploiting the structural prior knowledge of robots in our framework, the high-dimensional original state space could be decomposed into several low-dimensional state spaces with conditional independence to realize substantial reductions in the dimensionality of state space. Empirical results on diverse robotic systems including the challenging dexterous manipulation task, demonstrate not only the effectiveness but the robustness of the proposed MPN for robotic skill learning, delivering advancement substantially over traditional policy learning methods.

Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method

In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.

Cost function for low-dimensional manifold topology assessment

In reduced-order modeling, complex systems that exhibit high state-space dimensionality are described and evolved using a small number of parameters. These parameters can be obtained in a data-driven way, where a high-dimensional dataset is projected onto a lower-dimensional basis. A complex system is then restricted to states on a low-dimensional manifold where it can be efficiently modeled. While this approach brings computational benefits, obtaining a good quality of the manifold topology becomes a crucial aspect when models, such as nonlinear regression, are built on top of the manifold. Here, we present a quantitative metric for characterizing manifold topologies. Our metric pays attention to non-uniqueness and spatial gradients in physical quantities of interest, and can be applied to manifolds of arbitrary dimensionality. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional projections can be found. We delineate a few applications of the cost function to datasets representing argon plasma, reacting flows and atmospheric pollutant dispersion. We demonstrate how the cost function can assess various dimensionality reduction and manifold learning techniques as well as data preprocessing strategies in their capacity to yield quality low-dimensional projections. We show that improved manifold topologies can facilitate building nonlinear regression models.

Study on leaky Lamb waves in functionally graded composites loaded by asymmetric fluids

The present work aims at contributing to the investigation of a numerical method to describe the leaky Lamb waves, which propagate along the functionally graded composite plates when undergoing asymmetric fluids. The motions are considered under the two-dimensional plane strain assumption. In some respects, the proposed numerical scheme is an extension of the spectral collection method calculating the Lamb waves in a vacuum plate (stress-free on the surfaces). However, the proposed scheme implements differently because the displacement field here is influenced by both vertical heterogeneous and loaded fluids. The solution procedure relies on a linearization algorithm using a high dimensional state space, which permits the transformation of the nonlinear eigenvalue problem to the polynomial eigenvalue problem. Two comparisons are given to validate the effectiveness and support the method. Finally, results on specific examples (functionally graded composite plate immersed in the same and different fluids) are discussed to study leaky Lamb waves' propagating properties.

Toward black hole entropy in chiral loop quantum supergravity

Recently, many geometric aspects of $\mathcal{N}$-extended AdS supergravity in chiral variables have been encountered and clarified. In particular, if the theory is supposed to be invariant under SUSY transformations also on boundaries, the boundary term has to be the action of a $\mathrm{OSp}(\mathcal{N}|2)_{\mathbb{C}}$ super Chern-Simons theory, and particular boundary conditions must be met. Based on this, we propose a way to calculate an entropy $S$ for surfaces, presumably including black hole horizons, in the supersymmetric version of loop quantum gravity for the minimal case $\mathcal{N}=1$. It proceeds in analogy to the non-supersymmetric theory, by calculating dimensions of quantum state spaces of the super Chern-Simons theory with punctures, for fixed quantum (super) area of the surface. We find $S = a_H/4$ for large areas and determine the subleading correction. Due to the non-compactness of $\mathrm{OSp}(1|2)_{\mathbb{C}}$ and the corresponding difficulties with the Chern-Simons quantum theory, we use analytic continuation from the Verlinde formula for a compact real form, $\mathrm{UOSp}(1|2)$, in analogy to work by Noui et al. This also entails studying some properties of $\mathrm{OSp}(1|2)_{\mathbb{C}}$ representations that we have not found elsewhere in the literature.

Stochastic Planner-Actor-Critic for Unsupervised Deformable Image Registration

Large deformations of organs, caused by diverse shapes and nonlinear shape changes, pose a significant challenge for medical image registration. Traditional registration methods need to iteratively optimize an objective function via a specific deformation model along with meticulous parameter tuning, but which have limited capabilities in registering images with large deformations. While deep learning-based methods can learn the complex mapping from input images to their respective deformation field, it is regression-based and is prone to be stuck at local minima, particularly when large deformations are involved. To this end, we present Stochastic Planner-Actor-Critic (spac), a novel reinforcement learning-based framework that performs step-wise registration. The key notion is warping a moving image successively by each time step to finally align to a fixed image. Considering that it is challenging to handle high dimensional continuous action and state spaces in the conventional reinforcement learning (RL) framework, we introduce a new concept `Plan' to the standard Actor-Critic model, which is of low dimension and can facilitate the actor to generate a tractable high dimensional action. The entire framework is based on unsupervised training and operates in an end-to-end manner. We evaluate our method on several 2D and 3D medical image datasets, some of which contain large deformations. Our empirical results highlight that our work achieves consistent, significant gains and outperforms state-of-the-art methods.

Relativizing State Space: Deciphering China’s New Dynamics of Urban Transformation Engineered through the Creation of National New Areas

Abandonment of the taken-for-granted attitude of territoriality in the studies of state space has been followed by diverse concerns and competing interpretations with different focuses notably on state–market relations, spatiality of social life, and relativization of scale. Inspired by Lefebvre’s spatial triad, applied textual analysis, and in-depth interview, this study critically evaluates the Chinese practices of setting up National New Areas to reshape the trajectory of urban and regional development using the Liangjiang National New Area (LNNA). Our research foregrounds the relativizing dimensions of state space in which the creation of the LNNA was conceived, perceived, and lived by key stakeholders holding different positions and vested interests. The LNNA is the spatial manifestation of the special vision and mission of China’s national developmental agenda and the adjustment of power relations within an authoritarian Party-state. However, it is a controversial project which requires negotiation, contestation, and reconciliation among grassroots people. This study shows that even in China, there existed pervasive negotiation and resistance from diverse stakeholders from the bottom up. Our research suggests an alternative perspective that goes beyond the popular dichotomy of state–market or society–space relations in the studies of state space and takes seriously the dialectical relations among the forces at work in the (re)production of state space.

Scalable Controllability Analysis of Structured Networks

This article deals with strong structural controllability of structured networks. A structured network is a family of heterogeneous structured systems (called node systems) that are interconnected by means of a structured interconnection law, all given by pattern matrices. Here, we consider structured networks with single-input–single-output node systems. It is shown that such a structured network is strongly structurally controllable if and only if an associated structured system is. This structured system will, in general, have a very large state space dimension and, therefore, existing tests for verifying strong structural controllability are not tractable. The main result of this article circumvents this problem. We show that controllability can be tested by replacing the original network by a new network in which all original node systems have been replaced by (auxiliary) node systems with state space dimensions either 1 or 2. Hence, controllability of the original network can be verified by testing controllability of a structured system with state space dimension at most twice the number of node systems, regardless of the state space dimensions of the original node systems.

Data-Driven Power Flow Calculation Method: A Lifting Dimension Linear Regression Approach

The high-precision parameters in distribution networks are difficult to obtain, which brings difficulties to the model-based methods and analysis. With the widespread deployment of high-precision measurement units, data-driven methods have greater advantages in practice. In addition, with massive integration of distributed renewable generation and fast electric vehicle chargers, the fluctuations of net load increase significantly. The data-driven power flow calculation method based on the linear model becomes difficult to obtain accurate results for the low nonlinear adaptability. To improve the data-driven power flow calculation accuracy under high penetration of renewable distribution generation, this paper proposed an approach with high adaptability to the nonlinearity of power flow. Based on the thought of Koopman operator theory, the nonlinear relationship in power flow calculation is converted into a linear mapping in a higher dimension state space, which can significantly improve the calculation accuracy. Case studies on different IEEE cases have demonstrated that the proposed method can realize higher accuracy in power flow calculation with large-scale power fluctuations, compared to the existing data-driven method, in both mesh and radial networks. Finally, measurement data of a practical 10kV distribution network has been further used to verify the effectiveness of the proposed method in practical applications.

Stochastic reduced order modelling and analysis of rotating bladed discs

A computational finite-element (FE)-based technique is proposed for developing a stochastic reduced order model for rotating bladed disc with spatial random inhomogeneities. The spatial inhomogeneities imply the system to be randomly mistuned. The formulation assumes the availability of a high fidelity FE model for the tuned system. The corresponding FE matrices are antisymmetric on account of the Coriolis forces due to rotation. The spatial inhomogeneities, available from limited point measurements on the blades, are modelled as non-Gaussian random fields with arbitrary distributions. A low order stochastic computational model is developed by projecting the FE model onto a reduced dimensional state space defined in terms of specified observable nodal points and expressing the stochasticity through an arbitrary polynomial chaos (aPC) basis. This model enables probabilistic quantification of the variabilities in the system response and estimating failure probabilities. The methodology enables drastic reduction in the state space and stochastic dimensions, addresses the practical difficulties with having limited measurable data points, antisymmetric FE matrices, aPC representation in complex irregular geometries and carrying out probabilistic analyses on industrial systems, at significantly reduced computational costs. The methodology is illustrated through an academic rotor and an industrial rotor blade.

A new approach to modeling and simulation of the nonlinear, fractional behavior of Li-ion battery cells

Two nonlinear infinite dimensional models of a Lithium-ion battery are introduced. The proposed models interpolate linear fractional models which are obtained from a widely spread equivalent circuit model. The linear models are parameterized by means of so-called dynamic impedance measurements, i.e. impedance measurements in several operating points. In each operating point, the fractional input output behavior is recovered by an infinite dimensional state space description. For numerical implementation the introduced models require for an appropriate approximation. Therefore, a Krylov subspace method and a finite element approach are proposed. The nonlinear lumped parameter models are validated on the basis of experimental data.

A Reinforcement Learning Approach for Optimizing the Age-of-Computing-Enabled IoT

Age of Information (AoI) is a newly rising metric for measuring the freshness of information. In this article, we consider a multidevice computing-enabled Internet of Things (IoT) system with a common destination, in which the status update sampled by the device can be offloaded directly to the destination for computing or computed by the device and then delivered to the destination, and jointly design offloading and scheduling policies to minimize the average weighted sum of AoI and energy consumption. The challenge lies in computing mode selection and its strong coupling with scheduling decisions. To address this issue, we formulate the optimization problem as a bilevel discrete-time Markov decision process (MDP) and approximate the optimal solution by relative value iteration. Furthermore, the threshold structure of the MDP policy is shown. However, with the expansion of the system scale, the MDP policy will suffer from the curse of dimensionality. In light of this, we develop a learning-based algorithm based on emerging deep reinforcement learning (DRL) to reduce the dimensionality of state space and utilize a late experience storage method to train two heterogeneous artificial neural networks (ANNs) synchronously during the training process. Simulation results show the structure of the MDP policy and verify the performance of the DRL policy is near-optimal.

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

In this paper we investigate realizability of discrete time linear dynamical systems (LDSs) in fixed state space dimension. We examine whether there exist different Θ = (A,B,C,D) state space realizations of a given Markov parameter sequence Y with fixed B, C and D state space realization matrices. Full observation is assumed in terms of the invertibility of output mapping matrix C. We prove that the set of feasible state transition matrices associated to a Markov parameter sequence Y is convex, provided that the state space realization matrices B, C and D are known and fixed. Under the same conditions we also show that the set of feasible Metzler-type state transition matrices forms a convex subset. Regarding the set of Metzler-type state transition matrices we prove the existence of a structurally unique realization having maximal number of non-zero off-diagonal entries. Using an eigenvalue assignment procedure we propose linear programming based algorithms capable of computing different state space realizations. By using the convexity of the feasible set of Metzler-type state transition matrices and results from the theory of non-negative polynomial systems, we provide algorithms to determine structurally different realization. Computational examples are provided to illustrate structural non-uniqueness of network-based LDSs.

On d-Collision-Free Dynamical Systems

We present several recent results and generalizations that have been obtained in the theory of collision-freeness studied in Zerz and Herty (2019). A nonlinear ODE system ẋ(t) = f(x(t)) is called collision-free if the solution to the initial value problem with x(0) = x0 has distinct components for all times t whenever the initial state x0 has distinct components. This is an important structural property of particle systems. Here, we address the case where the state of the i-th particle has d components xi1, ..., xid, and a collision occurs if there exists some time t and i ≠ j such that xik(t) = xjk(t) for all 1 ≤ k ≤ d. Thus, the already known case corresponds to d = 1. In this paper, we characterize d-collision-freeness for linear, polynomially nonlinear, and general C1-systems. Moreover, we show that d-collision-freeness implies D-collision-freeness for all multiples D of d that divide the overall state space dimension. In particular, 1-collision-freeness implies d-collision-freeness. Finally, we address the connection between permutation symmetry and d-collision-freeness.