Extended State Space Research Articles

On the structure of uniformly hyperbolic chain control sets

The following theorem is proved: Let Q be an isolated compact chain control set of a control-affine system on a smooth manifold M. If Q is uniformly hyperbolic without center bundle, then the lift of Q to the extended state space U×M, where U is the space of control functions, is a graph over U. In other words, for every control u∈U there is a unique x∈Q such that the corresponding state trajectory φ(t,x,u) evolves in Q.

Guide on set invariance for delay difference equations

This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs).Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. The notion of D-invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model.The present paper contains a sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D-invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D-invariant sets is not necessarily D-invariant, while the convex hull of a non-convex D-invariant set is D-invariant.

Extended state space recursive least squares

A new extended state space recursive least squares (ESSRLS) algorithm is proposed for state estimation of nonlinear systems. It is based on state space recursive least squares (SSRLS) approach and uses first order linearization of the system. It inherits the capability of obtaining state estimate without knowledge of process and measurement noise covariance matrices (Q and R respectively). The proposed approach is considered to provide new design option for scenarios where noise statistics and system dynamics vary. ESSRLS is initialized using delayed recursion method and a forgetting factor λ is employed to optimize the performance. The selection of λ can be problem specific as shown through experimental validations. However a value closer to and less than unity is generally recommended. Theoretical bases are validated by applying this algorithm to problems of tracking a non-conservative oscillator, a damped system with amplitude death and a signal modeled by mixture of Gaussian kernels. Simulation results show an MSE performance gain of 20 dB and 23 dB over extended Kalman filter (EKF) and unscented Kalman filter (UKF) while tracking van der Pol oscillator without knowledge about noise variances. The computational complexity of ESSRLS falls within that of EKF and UKF.

A Fuzzy Networked Control System Following Frequency Transmission Strategy

At present, network control systems employ a common approximation to solve the connectivity issue due to time delays coupled with external factors . However, this approach tends to be complex in terms of time delays, and the inherent local phase is missing. Therefore, it is necessary to study the behavior of the delays as well as the integration of the differential equations of these bounded delays. The related time delays need to be known a priori, but from a dynamic real-time perspective in order to understand the dynamic phase behavior. The objective of this paper is to demonstrate the inclusion of the data frequency transmission and time delays that are bounded as parameters of the dynamic response from a real-time scheduling approximation, considering the local phase situation. The related control law is designed considering a fuzzy logic approximation for nonlinear time delays coupling. The main advantage is the integration of this behavior through extended state space representation. This keeps certain linear and bounded behavior leading to a stable situation during an events presentation, based on an accurate data transmission rate. An expected result is that the basics of the local phase missing as a result of the local bounded time delays from the lack of tide synchronization conforms to the modeling approximation.

Systematic coarse-graining in nucleation theory.

In this work, we show that the standard method to obtain nucleation rate-predictions with the aid of atomistic Monte Carlo simulations leads to nucleation rate predictions that deviate 3 - 5 orders of magnitude from the recent brute-force molecular dynamics simulations [Diemand et al., J. Chem. Phys. 139, 074309 (2013)] conducted in the experimental accessible supersaturation regime for Lennard-Jones argon. We argue that this is due to the truncated state space the literature mostly relies on, where the number of atoms in a nucleus is considered the only relevant order parameter. We here formulate the nonequilibrium statistical mechanics of nucleation in an extended state space, where the internal energy and momentum of the nuclei are additionally incorporated. We show that the extended model explains the lack in agreement between the molecular dynamics simulations by Diemand et al. and the truncated state space. We demonstrate additional benefits of using the extended state space; in particular, the definition of a nucleus temperature arises very naturally and can be shown without further approximation to obey the fluctuation law of McGraw and LaViolette. In addition, we illustrate that our theory conveniently allows to extend existing theories to richer sets of order parameters.

Immersion and invariance observers with time-delayed output measurements

We address the problem of constructing (globally) convergent, (reduced-order) observers for general nonlinear systems when the output measurements are subject to constant time-delays. Immersion and invariance (I&I) techniques are used to derive a general tool for constructing I&I observers in the presence of time-delays. We show that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. In this manuscript, the observer may play two different roles. On the one hand, it may be used to reconstruct a delayed version of the unmeasured state from measurements of the available delayed outputs. We show that if the time-delay is known, standard I&I techniques can be directly applied to estimate the delayed unmeasured states. In this case, we refer to the observer as a retarded immersion and invariance observer. On the other hand, the observer may be used to reconstruct both the delay-free unmeasured states and the delay-free output from measurements of the delayed output. In this case, we refer to it as an immersion and invariance predictor. Two examples with chaotic oscillators are presented to show the performance of the observers.

Time and quantum theory: A history and a prospectus

The historical part of this paper analyzes in detail how ideas and expectations regarding the role of time in quantum theory arose and evolved in the early years of quantum mechanics (from 1925 to 1927). The general theme is that expectations which seemed reasonable from the point of view of matrix mechanics and Dirac׳s q-number formalism became implausible in light of Dirac–Jordan transformation theory, and were dashed by von Neumann׳s Hilbert space formalism which came to replace it. Nonetheless, I will identify two concerns that remain relevant today, and which blunt the force of Hilgevoord׳s (2005) claim that the demand that time feature as an observable arose as the result of a simple conceptual error. First, I advocate the need for event time observables, which provide a temporal probability distribution for the occurrence of a particular event. Second, I claim that Dirac׳s use of the extended phase space to define time and (minus the) energy as conjugates is not subject to ‘Pauli׳s Theorem,’ the result that rules out time observables in von Neumann׳s formalism. I also claim that the need to define these event time observables leads to a novel motivation for considering Dirac׳s extended state space.

Experimental study of wireless structural vibration control considering different time delays

With the development of wireless communication technology, active structural vibration control based on a wireless sensor network has tended to replace the traditional wired control method. However, the problem of time delay in a wireless control system is inevitable and requires serious attention. In this study, a wireless active vibration control scheme consisting of a cantilever beam with a piezoelectric actuator is proposed and implemented. Experimental results indicate that wireless control gives good control performance; however, because of the influence of time delay, the performance of wireless control is slightly worse than that of wired control. Therefore, a novel method for time delay compensation is presented in this study to resolve this problem. This approach takes advantage of the finite difference method to extend the state space of the cantilever beam. Additional time delay states are used to form the extended state space model for time delay compensation. Simulation and experimental results demonstrate that this method can effectively compensate for time delay and enables the wireless control system to exhibit excellent control performance that can be favorably compared with that of wired control.

Study on the MOESP Subspace Identification Method and its Simulation

A detailed description of the basic principles of MOESP subspace system identification methods is introduced. The extended state space equation which is formed by Hankel matrix is used in subspace identification, and then the augmented observation matrix and state space vector estimation are all got by SVD decomposition. The simulation results on a multi input/output system demonstrate the effectiveness and feasibility of the method.

Optimal Control with Budget Constraints and Resets

We consider both discrete and continuous control problems constrained by a fixed budget of some resource, which may be renewed upon entering a preferred subset of the state space. In the discrete case, we consider deterministic shortest path problems on graphs with a full budget reset in all preferred nodes. In the continuous case, we derive augmented PDEs of optimal control, which are then solved numerically on the extended state space with a full/instantaneous budget reset on the preferred subset. We introduce an iterative algorithm for solving these problems efficiently. The method's performance is demonstrated on a range of computational examples, including optimal path planning with constraints on prolonged visibility by a static enemy observer.

Approximate Solution of Linear-Quadratic Problem in Discrete-Time Multi-Delay Systems: Derivation and Quality Assessment

The paper addresses infinite-horizon linear-quadratic optimal control problem in discrete-time systems with multiple input-output delays. An extended state-space description is introduced which allows one to translate the original, complex problem into a standard solution framework. In order to circumvent the issues related to increased problem dimension in the extended state space, a modification of the solution procedure is proposed. The computational effort to determine the gains of thus obtained suboptimal controller is reduced to the undelayed system part. The proposed suboptimal strategy shows improved performance over the optimal one in certain classes of applications.

An improved decoupling structure based state space MPC design with improved performance

In this paper, an improved decoupling structure is first proposed for multivariable processes, where the resulted decoupler can avoid zero-pole cancelation and simultaneously guarantee feasibility by tuning of additional parameters. A subsequent single-input–single-output design of model predictive control (MPC) is then proposed using a new state space realization of the independent single loops, where we adopt a new extended non-minimal state space structure to enhance control performance. Through this new design model, dynamics of processes’ state variables can be directly taken into account. The decoupler property, closed-loop control performance, the relationship with traditional state space MPC and robust stability issues are fully analyzed. The efficiency of the proposed is evaluated through a typical process in the recent literature, in comparison with a typical non-minimal state space MPC recently developed.

Model Predictive Control with Feedforward Strategy for Gas Collectors of Coke Ovens

In coking process, the production quality, equipment life, energy consumption, and process safety are all influenced by the pressure in gas collector pipe of coke oven, which is frequently influenced by disturbances. The main control objectives for the gas collector pressure system are keeping the pressures in collector pipes at appropriate operating point. In this paper, model predictive control (MPC) strategy is introduced to control the collector pressure system due to its ability to handle constraint and good control performance. Based on a method proposed to simplify the system model, an extended state space model predictive control is designed, which combines the feedforward strategy to eliminate the disturbance. The simulation results in a system with two coke ovens show the feasibility and effectiveness of the control scheme.

Stabilisation of Markov jump linear systems subject to both state and mode detection delays

This study addresses the stabilisation problem of a class of discrete-time Markov jump linear systems subject to time delays which occur in both the system state and mode detection. The mode detection delay means that there exists a lag between the switching of the system modes and the switching of the corresponding controllers. The state delays are considered to be random but with upper and lower bounds, and the mode detection delay is assumed to be constant. With an extended state space, the underlying system can be remodelled as a Markov jump linear system in which the system complexity and the number of system modes are both increased to a certain extent, but remain unchanged when the mode detection delay becomes larger. A sufficient condition guaranteeing the stochastic stability of the system is obtained via a modified stochastic Lyapunov function. The effectiveness of the proposed design approach is demonstrated by a numerical example.

More Risk-Sensitive Markov Decision Processes

We investigate the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite horizon that is generated by a Markov decision process (MDP). In contrast to a risk-neutral decision maker this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility. We show that this optimization problem can be solved by an ordinary MDP with extended state space and give conditions under which an optimal policy exists. In the case of an infinite time horizon we show that the minimal discounted cost can be obtained by value iteration and can be characterized as the unique solution of a fixed-point equation using a “sandwich” argument. Interestingly, it turns out that in the case of a power utility, the problem simplifies and is of similar complexity than the exponential utility case, however has not been treated in the literature so far. We also establish the validity (and convergence) of the policy improvement method. A simple numerical example, namely, the classical repeated casino game, is considered to illustrate the influence of the certainty equivalent and its parameters. Finally, the average cost problem is also investigated. Surprisingly, it turns out that under suitable recurrence conditions on the MDP for convex power utility, the minimal average cost does not depend on the parameter of the utility function and is equal to the risk-neutral average cost. This is in contrast to the classical risk-sensitive criterion with exponential utility.

Anisotropic universe with anisotropic sources

We analyze the state space of a Bianchi-I universe with anisotropic sources. Here we consider an extended state space which includes null geodesics in this background. The evolution equations for all the state observables are derived. Dynamical systems approach is used to study the evolution of these equations. The asymptotic stable fixed points for all the evolution equations are found. We also check our analytic results with numerical analysis of these dynamical equations. The evolution of the state observables are studied both in cosmic time and using a dimensionless time variable. Then we repeat the same analysis with a more realistic scenario, adding the isotropic (dust like dark) matter and a cosmological constant (dark energy) to our anisotropic sources, to study their co-evolution. The universe now approaches a de Sitter space asymptotically dominated by the cosmological constant. The cosmic microwave background anisotropy maps due to shear are also generated in this scenario, assuming that the universe contains anisotropic matter along with the usual (dark) matter and vacuum (dark) energy since decoupling. We find that they contribute dominantly to the CMB quadrupole. We also constrain the current level of anisotropy and also search for any cosmic preferred axis present in the data. We use the Union 2 Supernovae data to this extent. An anisotropy axis close to the mirror symmetry axis seen in the cosmic microwave background data from Planck probe is found.

Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials

Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials

Milestoning with coarse memory

Milestoning is a method used to calculate the kinetics of molecular processes occurring on timescales inaccessible to traditional molecular dynamics (MD) simulations. In the method, the phase space of the system is partitioned by milestones (hypersurfaces), trajectories are initialized on each milestone, and short MD simulations are performed to calculate transitions between neighboring milestones. Long trajectories of the system are then reconstructed with a semi-Markov process from the observed statistics of transition. The procedure is typically justified by the assumption that trajectories lose memory between crossing successive milestones. Here we present Milestoning with Coarse Memory (MCM), a generalization of Milestoning that relaxes the memory loss assumption of conventional Milestoning. In the method, milestones are defined and sample transitions are calculated in the standard Milestoning way. Then, after it is clear where trajectories sample milestones, the milestones are broken up into distinct neighborhoods (clusters), and each sample transition is associated with two clusters: the cluster containing the coordinates the trajectory was initialized in, and the cluster (on the terminal milestone) containing trajectory's final coordinates. Long trajectories of the system are then reconstructed with a semi-Markov process in an extended state space built from milestone and cluster indices. To test the method, we apply it to a process that is particularly ill suited for Milestoning: the dynamics of a polymer confined to a narrow cylinder. We show that Milestoning calculations of both the mean first passage time and the mean transit time of reversal-which occurs when the end-to-end vector reverses direction-are significantly improved when MCM is applied. Finally, we note the overhead of performing MCM on top of conventional Milestoning is negligible.

State Space Model Predictive Control Using Partial Decoupling and Output Weighting for Improved Model/Plant Mismatch Performance

Focusing on multivariable control, this paper presents a design method of model predictive control that enjoys the benefits of both the partial decoupling and the state space design. The multivariable process is first decoupled into a set of multi-input single-output (MISO) structures and then transformed into an extended state space model (partial decoupling extended state space model, PD-ESS). Consequently, a systematic design of model predictive control is proposed. The proposed controller is tested on three typical cases for comparison with previous controllers. Results show that control performance is improved. In addition, a closed-form of transfer function representation that facilitates frequency analysis of the control system is provided for further insight into the proposed method.

Extended designs algebraically

Extended designs distinguish non-terminating and aborting executions of sequential, non-deterministic programs. We show how to treat them algebraically based on techniques we have previously applied to total and general correctness approaches. In particular, we propose modifications to the definition of an extended design which make the theory more clear and simplify calculations, and an approximation order for recursion. We derive explicit formulas for operators on extended designs including non-deterministic choice, sequential composition, while-loops and full recursion. We show how to represent extended designs as designs or prescriptions over an extended state space. The new theory generalises our previous algebraic theory of general correctness by weakening its axioms. It also integrates with partial, total and general correctness into a common foundation which gives a unified semantics of while-programs. Program transformations derived using this semantics are valid in all four correctness approaches.