Extended State Space Research Articles

Control of probability flow in Markov chain Monte Carlo-Nonreversibility and lifting.

The Markov chain Monte Carlo (MCMC) method is widely used in various fields as a powerful numerical integration technique for systems with many degrees of freedom. In MCMC methods, probabilistic state transitions can be considered as a random walk in state space, and random walks allow for sampling from complex distributions. However, paradoxically, it is necessary to carefully suppress the randomness of the random walk to improve computational efficiency. By breaking detailed balance, we can create a probability flow in the state space and perform more efficient sampling along this flow. Motivated by this idea, practical and efficient nonreversible MCMC methods have been developed over the past ten years. In particular, the lifting technique, which introduces probability flows in an extended state space, has been applied to various systems and has proven more efficient than conventional reversible updates. We review and discuss several practical approaches to implementing nonreversible MCMC methods, including the shift method in the cumulative distribution and the directed-worm algorithm.

A model of task-level human stepping regulation yields semistable walking.

A simple lateral dynamic walker, with swing leg dynamics and three adjustable input parameters, is used to study how motor regulation affects frontal-plane stepping. Motivated by experimental observations and phenomenological models, we imposed task-level multi-objective regulation targeting the walker's optimal lateral foot placement at each step. The regulator prioritizes achieving step width and lateral body position goals to varying degrees by choosing a mixture parameter. Our model thus integrates a lateral mechanical template, which captures the fundamental mechanics of frontal-plane walking, with a lateral motor regulation template, an empirically verified model of how humans manipulate lateral foot placements in a goal-directed manner. The model captures experimentally observed stepping fluctuation statistics and demonstrates how linear empirical models of stepping dynamics can emerge from first-principles nonlinear mechanics. We find that task-level regulation gives rise to a goal-equivalent manifold in the system's extended state space of mechanical states and inputs, a subset of which contains a continuum of period-1 gaits forming a semistable set: perturbations off of any of its gaits result in transients that return to the set, though typically to different gaits.

Environment-mediated long-ranged correlations in many-body system

Quantum states in complex aggregates are unavoidably affected by environmental effects, which typically cannot be accurately modeled by simple Markovian processes. As system sizes scale up, nonperturbative simulation becomes thus unavoidable, but they are extremely challenging due to the intimate interplay of intrinsic many-body interaction and time-retarded feedback from environmental degrees of freedom. In this work, we utilize the recently developed quantum dissipation with minimally extended state space approach to address reservoir induced long-ranged temporal correlations in finite size Ising-type spin chains. For thermal reservoirs with ohmic and subohmic spectral density, we simulate the quantum time evolution from finite to zero temperature. The competition between thermal fluctuations, quantum fluctuations, and anti-/ferromagnetic interactions reveals a rich pattern of dynamical phases, including dissipative induced phase transitions and spatiotemporal correlations.

Extended state space for describing renormalized Fock spaces in QFT

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.

A model of task-level human stepping regulation yields semistable walking.

A simple lateral dynamic walker, with swing leg dynamics and three adjustable input parameters, is used to study how motor regulation affects frontal plane stepping. Motivated by experimental observations and phenomenological models, we imposed task-level multiobjective regulation targeting the walker's optimal lateral foot placement at each step. The regulator prioritizes achieving step width and lateral body position goals to varying degrees by choosing a mixture parameter. Our model thus integrates a lateral mechanical template, which captures fundamental mechanics of frontal-plane walking, with a lateral motor regulation template, an empirically verified model of how humans manipulate lateral foot placements in a goal-directed manner. The model captures experimentally observed stepping fluctuation statistics and demonstrates how linear empirical models of stepping dynamics can emerge from first-principles nonlinear mechanics. We find that task-level regulation gives rise to a goal equivalent manifold in the system's extended state space of mechanical states and inputs, a subset of which contains a continuum of period-1 gaits forming a semistable set: perturbations off of any of its gaits result in transients that return to the set, though typically to different gaits.

Trajectory tracking control of wearable upper limb rehabilitation robot based on Laguerre model predictive control

Wearable rehabilitation robots have become an important auxiliary tool in rehabilitation therapy, providing effective rehabilitation training and helping to recover damaged muscles and joints. In response to the difficulty of traditional control methods in solving various constraints in the trajectory tracking process of the Upper Limb Rehabilitation Robot (ULRR), this study uses model predictive control to study the trajectory tracking problem of the upper limb rehabilitation robot. Firstly, based on the Lagrangian dynamic model of wearable rehabilitation robots, an extended state space model with pseudo linearization of the system was established. Given the performance indicators and various constraints of the system, a corresponding model predictive controller is designed based on the Laguerre model to ensure system performance while greatly reducing the computational complexity of predictive control. Secondly, the stability of the model predictive controller is demonstrated, and a disturbance observer is introduced into the controller to achieve compensation for slow-varying perturbations; a joint space sliding mode variable is also introduced to achieve simultaneous tracking of the joint’s desired position and desired velocity. Finally, taking a planar two bar robot as an example, comparative simulation verification was conducted on unconstrained joint trajectory tracking and constrained joint trajectory tracking. The simulation results show that the model predictive controller can achieve simultaneous tracking of joint expected trajectory and expected speed while meeting various constraints. It has good effects in improving patient motion control ability and reducing patient fatigue, providing new research ideas and methods for the field of rehabilitation therapy.

An improved state space model predictive control for linear systems with input disturbance

This paper presents an improved model predictive control (MPC) algorithm for linear systems with input disturbance. Based on the developed extended non-minimum state space input disturbance (ENMSS-ID) model, the input disturbance model structure is incorporated into the MPC framework and the objective function of the MPC optimization problem is improved to weigh the system output increments. This enables the algorithm simultaneously to achieve good input disturbance rejection performance for systems with known input disturbances and reduce the controllers’ sensitivity to model mismatch. An existing optimal estimation method is introduced to estimate the input disturbance, together with the proposed strategy to improve estimation convergence. Offset-free property is also proven to show the steady-state performance of the designed control scheme. Finally, two benchmark plants are studied to illustrate the effectiveness and advantages of the proposed algorithm.

Initialisation from lattice Boltzmann to multi-step Finite Difference methods: Modified equations and discrete observability

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes—seen as dynamical systems on a commutative ring—can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods.

Parameter estimation of biased harmonic signal

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9, 10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].

Physics-informed deep 1D CNN compiled in extended state space fusion for seismic response modeling

Artificial neural networks have been proven promisingly powerful in developing a data-driven surrogate model for rapid seismic response modeling, while very few of them embody a physical mechanism with explicit interpretability. Herewith, this study proposes a novel physics-informed deep 1D convolutional neural network compiled in extended state space fusion (SSM-CNN) for enhanced seismic response modeling. In the SSM-CNN, an innovative parameter-free physics-constrained mechanism is designed and embedded for performance enhancement by construing the differential nexus of state variables derived from the state-space representation of initial structural response. Both the designing philosophy and mechanism translating of the SSM-CNN are elaborated exhaustively. To fully demonstrate the capability and superiority of proposed model compared to normal CNN, two groups of filed records of earthquake events with different sources from PEER and CESMD dataset were allocated for numerical and experimental validations, respectively. The validation results confirmed the effectiveness and superiority of physics-informed SSM-CNN in seismic response prediction. The sensitivity analysis and generalization analysis were also performed on the models for better understanding their interpretability. Involved algorithms and models in present study were integrated and developed into a web program, which affirmed a promising engineering applicability for rapid seismic response prediction.

Foundations of Nonequilibrium Statistical Mechanics in Extended State Space

The review provides a pedagogical but comprehensive introduction to the foundations of a recently proposed statistical mechanics (μNEQT) of a stable nonequilibrium thermodynamic body, which may be either isolated or interacting. It is an extension of the well-established equilibrium statistical mechanics by considering microstates mk in an extended state space in which macrostates (obtained by ensemble averaging A^) are uniquely specified so they share many properties of stable equilibrium macrostates. The extension requires an appropriate extended state space, three distinct infinitessimals dα=(d,de,di) operating on various quantities q during a process, and the concept of reduction. The mechanical process quantities (no stochasticity) like macrowork are given by A^dαq, but the stochastic quantities C^αq like macroheat emerge from the commutator C^α of dα and A^. Under the very common assumptions of quasi-additivity and quasi-independence, exchange microquantities deqk such as exchange microwork and microheat become nonfluctuating over mk as will be explained, a fact that does not seem to have been appreciated so far in diverse branches of modern statistical thermodynamics (fluctuation theorems, quantum thermodynamics, stochastic thermodynamics, etc.) that all use exchange quantities. In contrast, dqk and diqk are always fluctuating. There is no analog of the first law for a microstate as the latter is a purely mechanical construct. The second law emerges as a consequence of the stability of the system, and cannot be violated unless stability is abandoned. There is also an important thermodynamic identity diQ≡diW ≥ 0 with important physical implications as it generalizes the well-known result of Count Rumford and the Gouy-Stodola theorem of classical thermodynamics. The μNEQT has far-reaching consequences with new results, and presents a new understanding of thermodynamics even of an isolated system at the microstate level, which has been an unsolved problem. We end the review by applying it to three different problems of fundamental interest.

MPC-ESO Position Control Strategy for a Miniature Double-Cylinder Actuator Considering Hose Effects.

Miniature hydraulic actuators are especially suitable for narrow-space and harsh environment arrangement. However, when using thin and long hoses to connect components, the volume expansion caused by pressurized oil inside can have significant adverse effects on the performance of the miniature system. Moreover, the volumetric variation relates to many uncertain factors that are difficult to describe quantitatively. This paper conducted an experiment to test the hose deformation characteristics and presents the Generalized Regression Neural Network (GRNN) to describe the hose behavior. On this basis, a system model of a miniature double-cylinder hydraulic actuation system was established. To decrease the impact of nonlinearity and uncertainty on the system, this paper proposes a Model Predictive Control (MPC) based on Augmented Minimal State-Space (AMSS) model and Extended State Observer (ESO). The extended state space acts as the prediction module model for the MPC, and the disturbance of the ESO estimates is fed to the controller to improve the anti-disturbance capability. The full system model is validated by comparison between the experiment and the simulation. For a miniature double-cylinder hydraulic actuation system, the proposed MPC-ESO control strategy contributes to a better dynamic than conventional MPC and fuzzy-PID. In addition, the position response time can be reduced by 0.5 s and achieves a 4.2% reduction in steady-state error, especially for high-frequency motion. Moreover, the actuation system with MPC-ESO exhibits better performance in suppressing the influence of the load disturbance.

An Extended State Space Model for Aggregation of Large-Scale EVs Considering Fast Charging

This article presents an extended state space model for aggregation of large-scale electric vehicles (EVs) for frequency regulation and peak load shaving in power systems. The proposed model systematically deals with the fast charging of EVs as an effective solution for immediate charging requirements. Furthermore, the proposed extended state space model increases the flexibility of the EV aggregator (EVA) by enabling the EVs to participate in ancillary services with both regular and fast charging/discharging rates. This will help the EVA to provide a prompt and efficient response to severe generation-consumption imbalances. A probabilistic control approach is developed which reduces the communication burden of the EVA. Furthermore, the uncertainties related to EV users' behavior are modeled in real-time. The simulations are conducted for a typical power system including a large population of EVs, a conventional generator (CG), and a wind generation system. It is shown that the proposed aggregation model can accurately describe the aggregated behavior of a large population of EVs enabling them to efficiently participate in frequency regulation and peak load shaving services. Finally, the performance of EVA is evaluated for different driving behaviors and state of charge (SOC) levels of the EV population.

Observer of harmonic oscillator parameters

This paper deals with the problem of asymptotically estimating amplitude, frequency and phase of a sinusoidal signal by adopting the theory of invariant relations proposed in analytical mechanics [8] and further investigated in [9], [10] (see also [11]). The problem of frequency, amplitude and phase estimation of a sinusoidal signal has attracted a remarkable research attention in the past and current literature. The reasons of this interest rely on several engineering applications where an effective and robust solution to this problem is crucial. To mention few, it is worth mentioning problems of harmonic disturbance compensation in automatic control, design of phase-looked loop circuits in telecommunication, adaptive filtering in signal processing, etc. In principle, the method of least squares, Fourier analysis, Laplace transform provide a potential solution to the corresponding problems. However, these methods may not be suitable, for example, for control algorithms with real-time data processing. The goal of this paper is to suggest a further contribution to this task by showing how to solve the problem at hand through the observer’s theory. The method of invariant relations is used for the asymptotically observation scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns variables can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. The asymptotic convergence of the estimates of the sought phase vector components to their true value is proved. The simulation results demonstrate the effectiveness of the proposed method of solving the state observation problem of the harmonic oscillator. It should be noted that a more general approach, which forms an appropriate method for solving observation problems for nonlinear dynamical systems due to the synthesis of invariant manifold, was proposed as a modification of the I\&I method (Input and Invariance) of stabilization of nonlinear systems in [12, 13].

On geometry of multiscale mass action law and its fluctuations

The classical mass action law in chemical kinetics is put into the context of geometric multiscale thermodynamics, which allows for description of chemical reactions with inertial effects. The kinetics is extended to an enlarged state space with reaction rates as new state variables, and it has the structure a Lie-algebroid dual. Subsequently, the dynamics is lifted to the Liouville description within kinetic theory of the enlarged state space, so that we can include also dynamics of fluctuations. The lifted kinematics has the geometric structure of a matched pair, which allows for reduction to moments by a Lie-algebra homomorphism, as in the Grad hierarchy. In particular, the first and second moments then lead to evolution equations for chemical kinetics with inertia and for correlations between composition and reaction rates. Finally, dissipation is added in the extended state space which leads to the classical mass action law when the moments relax to their respective quasi-equilibria. We demonstrate, for instance, the possibility of oscillating homogeneous chemical reactions, and how correlations between composition and reaction rates contribute to the chemical kinetics.

Complex Langevin approach to interacting Bose gases

Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to standard Monte Carlo techniques, simulations beyond their scope almost exclusively rely on quantum-mechanical techniques, which are inherently limited to small particle numbers. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional non-relativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent/-species models.

Improved constrained min-max optimization for MPC tracking control in industrial networked process systems

Based on a modified min-max optimization strategy, an improved design of model predictive tracking control (MPC) is proposed to guarantee the performance of industrial process control systems when the networked system suffers from communication faults. Packet losses and uncertainties exist in the networked control system, which may deteriorate the performance of the systems and even cause a safety accident. To handle with this problem, an extended state space model is utilized in the proposed MPC design for enhancing control performance firstly. Then an improved min-max optimization approach is adopted for obtaining the optimal solution of the corresponding MPC method, where two separate optimization steps are proposed to enhance the control system operation. Different from MPC strategies that employ traditional state space models, the state variables can be tuned additionally for the modified MPC design because the tracking error information and the state variables are united in the improved model. As a consequence, extra degree of freedom is acquired for the relevant MPC scheme, and then better control performance is expected. Simulations on the temperature regulation process under model/plant mismatches and packet losses are simulated to evaluate the validity of the proposed MPC method finally.

Exact Bayesian Inference for Discretely Observed Markov Jump Processes Using Finite Rate Matrices

ABSTRACT We present new methodologies for Bayesian inference on the rate parameters of a discretely observed continuous-time Markov jump process with a countably infinite statespace. The usual method of choice for inference, particle Markov chain Monte Carlo (particle MCMC), struggles when the observation noise is small. We consider the most challenging regime of exact observations and provide two new methodologies for inference in this case: the minimal extended statespace algorithm (MESA) and the nearly minimal extended statespace algorithm (nMESA). By extending the Markov chain Monte Carlo statespace, both MESA and nMESA use the exponentiation of finite rate matrices to perform exact Bayesian inference on the Markov jump process even though its statespace is countably infinite. Numerical experiments show improvements over particle MCMC of between a factor of three and several orders of magnitude. Supplementary materials for this article are available online.

Robust particle-Kalman filtering phase unwrapping algorithm for wrapped fringe patterns.

This paper presents a robust phase unwrapping algorithm based on a particle-Kalman filter for wrapped fringe patterns by combining a particle filter and an extended Kalman filter, which formulates the phase unwrapping problem of wrapped fringe patterns as an optimal state estimation problem under the frame of the particle-Kalman filter. First, a state space equation for state variables is extended to the second order of Taylor series, and a local phase gradient estimator based on a modified matrix pencil model is used to obtain the first-order and second-order phase gradient information required by the extended state space equation, which is conducive to enhancing the phase unwrapping accuracy of the proposed procedure. Second, the initial estimate of unwrapped phase is obtained through applying an efficient phase unwrapping program based on a particle filter to unwrap noisy wrapped pixels. Finally, the initial estimate of unwrapped phase obtained by the particle filter is taken as the predicted estimate of state variables and further processed by the extended Kalman filter to obtain the final estimate of unwrapped phase. In addition, an efficient quality-guided strategy that has been demonstrated well is used to guarantee that the particle-Kalman filter efficiently and accurately unwraps wrapped pixels along a suitable path. Results obtained with synthetic data and experimental data demonstrate the effectiveness of the proposed method and show that this new approach can obtain more acceptable solutions from noisy wrapped fringe patterns, with respect to some of the most commonly used methods.

Improved Constrained Predictive Functional Control Using Extended Non-Minimal State Space Formulation for the Cement Production Process

In this article, an improved constraint dealing method and an extended state space model-based constrained predictive functional control (PFC) approach is developed for the denitration process in cement production. To improve the control effect of the presented strategy, first, an enhanced form-based state space model in which the changes of the output predictions can be regulated to achieve smoother system dynamics is constructed. Then, a modified constraint dealing scheme, which integrates the relaxation factors to improve the success probability of solving the relevant optimization, is also introduced for the proposed method. On the basis of the two merits, better ensemble control performance is anticipated. Simulations on the regulation of the NOx concentration in the denitration process prove the validity of the proposed constrained PFC strategy.