Finite-dimensional State-space Model Research Articles

Generating series for networks of Chen–Fliess series

Consider a set of single-input, single-output nonlinear systems whose input-output maps are described only in terms of convergent Chen-Fliess series without any assumption that finite dimensional state space models are available. It is shown that any additive or multiplicative interconnection of such systems always has a Chen-Fliess series representation that can be computed explicitly in terms of iterated formal Lie derivatives.

Iterative Learning Control for a Class of Multivariable Distributed Systems With Experimental Validation

This article develops an iterative learning control (ILC) design for a class of multiple-input-multiple-output systems where a distributed heating system is used as a particular example to experimentally validate the design. The class of systems considered is described by a parabolic partial differential equation, which, for control design, is approximated by a finite-dimensional state-space model obtained by applying the method of integro-differential relations combined with a projection approach. In some cases, including the distributed heating system, this approximation may result in a nonminimum phase system and, hence, pose an additional design challenge. In this work, the ILC law is computed in the frequency domain by solving a convex optimization problem, and its performance is evaluated in both simulation and experiment.

Load reduction of a monopile wind turbine tower using optimal tuned mass dampers

ABSTRACTWe investigate to apply tuned mass dampers (TMDs) (one in the fore–aft direction, one in the side–side direction) to suppress the vibration of a monopile wind turbine tower. Using the spectral element method, we derive a finite-dimensional state-space model Σd from an infinite-dimensional model Σ of a monopile wind turbine tower stabilised by a TMD located in the nacelle. Σ and Σd can be used to represent the dynamics of the tower and TMD in either the fore–aft direction or the side–side direction. The wind turbine tower subsystem of Σ is modelled as a non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) system consisting of an Euler–Bernoulli beam equation describing the dynamics of the flexible tower and the Newton–Euler rigid body equations describing the dynamics of the heavy rotor-nacelle assembly (RNA) by neglecting any coupling with blade motions. Σd can be used for fast and accurate simulation for the dynamics of the wind turbine tower as well as for optimal TMD designs. We show that Σd agrees very well with the FAST (fatigue, aerodynamics, structures and turbulence) simulation of the NREL 5-MW wind turbine model. We optimise the parameters of the TMD by minimising the frequency-limited -norm of the transfer function matrix of Σd which has input of force and torque acting on the RNA, and output of tower-top displacement. The performances of the optimal TMDs in the fore–aft and side–side directions are tested through FAST simulations, which achieve substantial fatigue load reductions. This research also demonstrates how to optimally tune TMDs to reduce vibrations of flexible structures described by partial differential equations.

Network dynamics of the brain and influence of the epileptic seizure onset zone.

The human brain is a dynamic networked system. Patients with partial epileptic seizures have focal regions that periodically diverge from normal brain network dynamics during seizures. We studied the evolution of brain connectivity before, during, and after seizures with graph-theoretic techniques on continuous electrocorticographic (ECoG) recordings (5.4 ± 1.7 d per patient, mean ± SD) from 12 patients with temporal, occipital, or frontal lobe partial onset seizures. Each electrode was considered a node in a graph, and edges between pairs of nodes were weighted by their coherence within a frequency band. The leading eigenvector of the connectivity matrix, which captures network structure, was tracked over time and clustered to uncover a finite set of brain network states. Across patients, we found that (i) the network connectivity is structured and defines a finite set of brain states, (ii) seizures are characterized by a consistent sequence of states, (iii) a subset of nodes is isolated from the network at seizure onset and becomes more connected with the network toward seizure termination, and (iv) the isolated nodes may identify the seizure onset zone with high specificity and sensitivity. To localize a seizure, clinicians visually inspect seizures recorded from multiple intracranial electrode contacts, a time-consuming process that may not always result in definitive localization. We show that network metrics computed from all ECoG channels capture the dynamics of the seizure onset zone as it diverges from normal overall network structure. This suggests that a state space model can be used to help localize the seizure onset zone in ECoG recordings.

Dynamic modeling and simulation for the rigid flexible coupling system with a non-tip payload in non-inertial coordinate system

The rigid flexible coupling system with a payload at non-tip position of the flexible beam is studied in this paper. Using the theory about mechanics problems in a non-inertial coordinate system and the subsystem modeling principle, the dynamic equations of the rigid flexible coupling system with dynamic stiffening are established via material mechanics and the angular momentum principle. This paper elucidates that dynamic stiffening is produced by the coupling effect of the centrifugal inertial load distributed on the beam and the transverse vibration deformation of the beam. First, the payload at non-tip position is incorporated into the continuous dynamic equations of the system by means of the Dirac function and the Heaviside function. Then, the constrained modes of the flexible beam are analyzed. According to the boundary conditions of the constrained modes, a reasonable position interval of the payload is proposed. It corrects the inaccurate understandings in some papers. Finally, based on the conclusions of orthogonalization about the normal constrained modes, the finite dimensional state space model suitable for controller design is obtained. In addition, in order to make the finite dimensional dynamic model established more practical, based on the common constrained modes, the dimensionless variables and parameters are introduced, and the dimensionless finite dimensional dynamic model of the system is derived. The numerical simulation results show that: dynamic stiffening is included in the first-order dynamic model established in this paper, therefore, it can indicate the dynamic responses of the rigid flexible coupling system with large overall motion accurately; the payload has a softening effect on the dynamic behaviors of the flexible beam, and the effect would be more obvious when the payload has a larger rotary inertia or a larger mass, or lies closer to the tip of the beam.

Nonlinear Estimation of Fluid Properties using the Time Domain Response of a Vibrating Microcantilever

A novel approach is presented for the modelling and analysis of the dynamic response of a microcantilever beam vibrating in a viscous fluid to determine the properties of the fluid. This new approach is based on modelling the time response of the deflection of the free end of the microcantilever, whereas other methods developed in the field of rheological measurements using vibrating microcantilevers are generally based on the analysis of the frequency response of the deflection of the cantilever. The proposed approach consists of spatially discretizing the beam along its length, approximately representing it with a finite-dimensional state-space model and determining the fluid properties using nonlinear state/parameter estimation techniques. In this work, the discrete-time extended Kalman filter is employed to find estimates of the fluid’s viscosity and density using temporal measurements of the deflection at the free end of the beam. Simulation studies illustrating the validity of the proposed approach are presented.

A data-driven framework for neural field modeling

This paper presents a framework for creating neural field models from electrophysiological data. The Wilson and Cowan or Amari style neural field equations are used to form a parametric model, where the parameters are estimated from data. To illustrate the estimation framework, data is generated using the neural field equations incorporating modeled sensors enabling a comparison between the estimated and true parameters. To facilitate state and parameter estimation, we introduce a method to reduce the continuum neural field model using a basis function decomposition to form a finite-dimensional state-space model. Spatial frequency analysis methods are introduced that systematically specify the basis function configuration required to capture the dominant characteristics of the neural field. The estimation procedure consists of a two-stage iterative algorithm incorporating the unscented Rauch–Tung–Striebel smoother for state estimation and a least squares algorithm for parameter estimation. The results show that it is theoretically possible to reconstruct the neural field and estimate intracortical connectivity structure and synaptic dynamics with the proposed framework.

Optimal control of diffusion-convection-reaction processes using reduced-order models

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process.

OPTIMAL TRANSITION CONTROL OF DIFFUSION-CONVECTION-REACTION PROCESSES

An approach for the optimal transition control of diffusion-convection- reaction processes based on finite-dimensional models is presented. The finite-dimensional state-space models are constructed directly from the process PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. The optimal controller is designed based on the finite-dimensional state-space model using continuous-time linear quadratic regulator (LQR) techniques.

Model building and verification for active control of microvibrations with probabilistic assessment of the effects of uncertainties

Microvibrations, generally defined as low-amplitude vibrations at frequencies up to 1 kHz, are of critical importance in a number of areas. It is now well known that, in general, the suppression of such microvibrations to acceptable levels requires the use of active control techniques which, in turn, require sufficiently accurate and tractable models of the underlying dynamics on which to base controller design and initial performance evaluation. Previous work has developed a systematic procedure for obtaining a finite-dimensional state-space model approximation of the underlying dynamics from the defining equations of motion, which has then been shown to be a suitable basis for robust controller design. In this paper, the experimental validation of this model prior to experimental studies is described in order to determine the effectiveness of the designed controllers. This includes details of the experimental rig and also the use of methods for assessing the safety of the resulting structure against uncertain parameters.

ON COMPARTMENTAL MODELLING OF MIXING PHENOMENA

In this paper the problem of modelling partial mixing phenomena, mostly relevant in environmental and reactors modelling practice, is considered. The ultimate modelling goal is to find identifiable, finite-dimensional state-space models, which are physically interpretable, realisable and which describe partial mixing. Hence, realization theory will be linked to prior physical systems knowledge to answer the question which mixing models are good candidates in environmental/reactor systems modelling. The starting point is compartmental systems modelling with backflows. It appears however that only a limited set of low-dimensional structures is identifiable. From the real world example given in this paper it appears that an appropriate, physically interpretable and realisable model within this class of models cannot be easily found.

Fixed memory filter for real-time estimation of noise-corrupted signals

Introduction I N many engineering applications, such as guidance and navigation of aeronautical vehicles, the signals need to be estimated from noisy measurements in real time. The (minimum-variance) Kalman-Bucy filter' that relies on a finite-dimensional state-space model of the plant dynamics and the measurement history is extensively used for this purpose. Techniques of adaptive filtering' that are based on the principle of recursive least squares algorithm and rely on an inputoutput relationship of the plant dynamics are also used. Given a linear (or linearized) time-invariant finite-dimensional model of plant dynamics in the autoregressive moving average (ARMA) setting, we propose a fixed-memory, nonrecursive filter for real-time signal estimation. The key idea is to construct a simple, nonrecursive filter based on weighted averaging of a finite array of past values of measured inputs and outputs of the plant. A fixed memory filter is very useful in the applications of smart sensors and/or active sensors where memory constraints are tight and computationally efficient algorithms are required for real-time implementation on a microchip collocated with the sensor. The filter memory length depends on the plant dynamics and the allowable bound of estimation error and, for real-time applications, is often selected as a tradeoff between execution time and accuracy of the filter. This Engineering Note presents the concept and formulation of a fixed-memory filter based on an ARMA model of plant dynamics. The criteria for selection of the filter memory length are not included in this Note. Although the filter algorithm is derived for single-input single-output (SISO) systems, the proposed filter can be extended to multi-input multioutput (MIMO) systems.

Identification of the deterministic part of MIMO state space models given in innovations form from input-output data

In this paper we describe two algorithms to identify a linear, time-invariant, finite dimensional state space model from input-output data. The system to be identified is assumed to be excited by a measurable input and an unknown process noise and the measurements are disturbed by unknown measurement noise. Both noise sequences are discrete zero-mean white noise. The first algorithm gives consistent estimates only for the case where the input also is zero-mean white noise, while the same result is obtained with the second algorithm without this constraint. For the special case where the input signal is discrete zero-mean white noise, it is explicitly shown that this second algorithm is a special case of the recently developed Multivariable Output-Error State Space ( moesp) class of algorithms based on instrumental variables. The usefulness of the presented schemes is highlighted in a realistic simulation study.

Sufficient sequences and state-space models for random processes

Let ( U_{1}, U_{2} , ...) be a sequence of observed random variables whose probability distributions are described by a parameterized family of density functions \{p_{k}(u_{1}, ..., u_{k}; \theta)\} . If there exists a sequence of sufficient statistics for \theta(T_{1}(U_{1}) , T_{2}(U_{1}, U_{2}) , ...), and if a realizability assumption holds, then there is a finite-dimensional state-space model whose output process agrees with ( U_{1}, U_{2} , ...) in distribution.

Recent Developments in Statistical Signal Processing

Publisher Summary Determining optimal algorithms and implementing the optimal algorithms are the two major aspects of statistical signal processing. There is considerable scope for application of ideas from linear system theory. One way to restrict complexity is to make further assumptions on the problem. A very common one in the last two decades has been to assume that there is an underlying finite-dimensional state-space model for the problem, in which case the memory can be fixed at the size of the state vector; however, the resulting filters will still be time variant. In any case, the important point is that these new results make for a much more direct link between signal processing theory and digital circuit implementations. This offers some promise in significantly reducing the software bottlenecks encountered in fitting all algorithms into simple general purpose architecture.

State-Space Models of Gravity Disturbance Gradients

State-space models of gravity disturbance gradients are needed to design the Kalman filters used for gradiometer-aided inertial navigation. However, the physics of the gravity field guarantee that many of the cross-spectral densities of gravity disturbance gradients are nonrational. A procedure is demonstrated for fitting gravity disturbance correlation functions with finite-dimensional state-space models.