Discrete-time State-space Representation Research Articles

Real-Time Speed Estimation for an Induction Motor: An Automated Tuning of an Extended Kalman Filter Using Voltage-Current Sensors.

This paper aims at achieving real-time optimal speed estimation for an induction motor using the Extended Kalman filter (EKF). Speed estimation is essential for fault diagnosis in Motor Current Signature Analysis (MCSA). The estimation accuracy is obtained by exploring the noise covariance matrices estimation of the EKF algorithm. The noise covariance matrices are determined using a modified subspace model identification approach. In order to reach this goal, this method compares an estimated model of a deterministic system, derived from available input-output datasets (using voltage-current sensors), with the discrete-time state-space representation used in the Kalman filter equations. This comparison leads to the determination of model uncertainties, which are subsequently represented as noise covariance matrices. Based on the fifth-order nonlinear model of the induction motor, the rotor speed is estimated with the optimized EKF algorithm, and the algorithm is tested experimentally.

Noise covariance matrix estimation with subspace model identification for Kalman filtering

SummaryA problem frequently encountered in Kalman filtering is the tuning of the noise covariance matrices. Indeed, misspecifying their values can drastically reduce the performance of the Kalman filter. Unfortunately, in most practical cases, noise statistics are not known a priori. This paper focuses on a method relying on subspace model identification theory to determine them accurately. This solution is developed for linear time invariant systems with stationary random disturbances having constant covariance matrices. Practically, these noise covariance matrices are determined from the comparison between an estimated state space representation and the discrete time state space representation involved in the Kalman filter equations. The method developed in this paper departs from most of the solutions available in the literature by the fact that it does not need any tuning parameter to be chosen by the user. After discussing theoretical results, several numerical examples are given to demonstrate the efficiency of the approach.

Robust Control of Three-Phase VSI with LCL Filter for Distributed Generation Power Quality Improvement

This paper proposes a methodology to control a three-phase voltage source inverter with LCL filter for distributed generation (DG) in microgrid scenario. Thus, the presented control strategy allows the DG system to operate in both grid-connected and islanded modes with a smooth transition between them. Therefore, the DG control system strategy is performed by splitting the control input into two control-stages; the first one is in charge of reproducing sinusoidal voltages at the filter capacitors; meanwhile, the second is dedicated to modulate these voltages in order to achieve proper amplitude and phase for correct output current injection. By means of discrete-time state-space representation of the system, state-feedback controllers are obtained employing linear matrix inequalities considering uncertainties on grid inductances and optimization of the H2 norm for each operation mode. Finally, the feasibility of the proposed DG system is demonstrated and validated through computational simulations, as well as experimental evaluations.

A particle filter-based framework for real-time state estimation of a non-linear hyperbolic PDE system describing transient flows in CO2 pipelines

In this work, pressure, temperature and mass flow observations are assimilated into a fluid flow model, describing the dynamics of CO2 flow in liquid or dense-phase during normal pipeline operations. The goal is to improve its state estimates using two Sequential Monte Carlo (SMC) methods, namely the sequential importance resampling (SIR) filter and the auxiliary sampling importance resampling filter (ASIR). After constructing a state-space model of the nonisothermal, single-phase flow model, the SIR and ASIR filters are implemented. The state variables are updated with simulated measurements. The partial differential equations are semi-discretized via a fourth order, five-point approximation and solved with the fourth-order Runge–Kutta method to obtain finite-dimensional discrete-time state-space representations. The system states are then combined into an augmented state vector. The resulting nonlinear state-space model is used for the design of the particle filter that provides real-time estimations of the system states. Numerical experiments were carried out in order to (1) examine the impact of ensemble size on accuracy and computational cost, (2) analyze the effects of different model and observation error covariance structures on the filters performance and (3) investigate the robustness of the filters by adding a Student’s t-distributed noise to the model.

State-space estimation with a Bayesian filter in a coupled PDE system for transient gas flows

The accuracy of the first-principle models describing the evolution of gas dynamics in pipelines is sometimes limited by the lack of understanding of the gas transport phenomena. In this paper, a stochastic filtering approach is proposed based on a sequential Monte Carlo method to provide real-time estimates of the state in gas pipelines. After constructing a state-space model of the compressible single-phase flow based on the laws of conservation of mass and momentum, the optimal sequential importance resampling filter (SIR) is implemented. The state variables are updated with simulated measurements. The two-step Lax–Wendroff method is used for the discretization of the partial differential equations describing the gas model in both space and time to obtain finite-dimensional discrete-time state-space representations. The system states are then combined into an augmented state vector. The resulting nonlinear state-space model is used for the design of the particle filter that provides real-time estimations of the system states. Simulation results for a coupled PDE system describing an unsteady isothermal gas flow demonstrate the effectiveness of the proposed method. A sensitivity analysis is conducted to examine the performance of the filter for different model and observation error covariances and observation intervals.

An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions.

Discrete state space modeling and control of nonlinear unknown systems

A novel procedure for integrating neural networks (NNs) with conventional techniques is proposed to design industrial modeling and control systems for nonlinear unknown systems. In the proposed approach, a new recurrent NN with a special architecture is constructed to obtain discrete-time state-space representations of nonlinear dynamical systems. It is referred as the discrete state-space neural network (DSSNN). In the DSSNN, the outputs of the hidden layer neurons of the DSSNN represent the system's (pseudo) state. The inputs are fed to output neurons and the delayed outputs of the hidden layer neurons are fed to their inputs via adjustable weights. The discrete state space model of the actual system is directly obtained by training the DSSNN with the input–output data. A training procedure based on the back-propagation through time (BPTT) algorithm is developed. The Levenberg–Marquardt (LM) method with a trust region approach is used to update the DSSNN weights. Linear state space models enable to use well developed conventional analysis and design techniques. Thus, building a linear model of a system has primary importance in industrial applications. Thus, a suitable linearization procedure is proposed to derive the linear state space model from the nonlinear DSSNN representation. The controllability, observability and stability properties are examined. The state feedback controllers are designed with both the linear quadratic regulator (LQR) and the pole placement techniques. The regulator and servo control problems are both addressed. A full order observer is also designed to estimate the state variables. The performance of the proposed procedure is demonstrated by applying for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) nonlinear control problems.

Discrete-time Modelling and Robust Analysis of a Buck Converter

A discrete-time state-space representation of a Buck converter is presented. This analytical state-space model provides explicit insight on how circuit parameters influence the stability and transient performance for this converter type. Variations in the power rail components and parasitic resistances are accommodated, which are typically neglected in compensator synthesis for switched mode power supplies. With this representation, robust stability and feedback performance can be readily assessed. Moreover, the model can be employed directly for the design of digital robust compensators, and it provides a basis for the development of compensators in analytical form rather than the use of numerical discretisation techniques.

On the State-Space Realization of LPV Input-Output Models: Practical Approaches

A common problem in the context of linear parameter-varying (LPV) systems is how input-output (IO) models can be efficiently realized in terms of state-space (SS) representations. The problem originates from the fact that in the LPV literature discrete-time identification and modeling of LPV systems is often accomplished via IO model structures. However, to utilize these LPV-IO models for control synthesis, commonly it is required to transform them into an equivalent SS form. In general, such a transformation is complicated due to the phenomenon of dynamic dependence (dependence of the resulting representation on time-shifted versions of the scheduling signal). This conversion problem is revisited and practically applicable approaches are suggested which result in discrete-time SS representations that have only static dependence (dependence on the instantaneous value of the scheduling signal). To circumvent complexity, a criterion is also established to decide when an linear-time invariant (LTI)-type of realization approach can be used without introducing significant approximation error. To reduce the order of the resulting SS realization, an LPV Ho-Kalman-type of model reduction approach is introduced, which, besides its simplicity, is capable of reducing even non-stable plants. The proposed approaches are illustrated by application oriented examples.

Step-by-Step Eigenvalue Analysis With EMTP discrete-time Solutions

This paper presents a methodology to obtain a discrete-time state-space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. Compared with conventional state-space solutions, the nodal EMTP solution is computationally very efficient. Compared with the phasor solutions used in transient stability analysis, the proposed approach may capture a wider range of eigenvalues and system operating states. An important advantage of extracting the system eigenvalues from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time-scale solutions. The proposed technique can be implemented as an extension of an EMTP-based simulator. Within our research group at UBC, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator.

Transport Parameter Estimations of Plasma Transport Dynamics Using the Extended Kalman Filter

The accuracy of first-principle predictive models for the evolution of plasma profiles is sometimes limited by the lack of understanding of the plasma transport phenomena. It is possible then to develop approximate transport models for the prediction of plasma dynamics, which are consistent with the available diagnostic data. This data-driven approach, usually referred to as phenomenological modeling, arises as an alternative to the more classical theory-driven approach. In this paper, we propose a stochastic filtering approach based on an extended Kalman filter to provide real-time estimates of poorly known or totally unknown transport coefficients. We first assume that plasma dynamics is governed by tractable models obtained by first principles. However, the transport parameters are considered unknown and to be estimated. These estimates will be based solely on input/output diagnostic data and limited understanding of the transport physics. Numerical methods (e.g., finite differences) can be used to discretize the partial differential equation models both in space and time to obtain finite-dimensional discrete-time state-space representations. The system states and to-be-estimated parameters are then combined into an augmented state vector. The resulting nonlinear state-space model is used for the design of an extended Kalman filter that provides real-time estimations not only of the system states but also of the unknown transport coefficients. Simulation results demonstrate the effectiveness of the proposed method for a benchmark transport model in cylindrical coordinates.

Control of Hydraulic Manipulator System Using Disturbance Observer.

This paper proposes an alternative control algorithm for a hydraulic manipulator using disturbance observers. The hydraulic actuators of the manipulator are velocity-controlled by a proportional type flow-control valve. We first model the actuator, which has a disturbance and a time delay in its input, using a discrete-time state space representation. Then we design a disturbance observer for the model with a time delay of n sampling periods in its input. The total control system is synthesized using the observer so that the disturbance to the system is eliminated. The resultant control system is robust against the disturbances to the system, such as mechanical friction and variation of the load of the actuator. Simulation results illustrate the effectiveness of the observer. Finally the proposed method is applied to a hydraulic manipulator with three degrees of freedom and the experimental results demonstrate the validity of the proposed control system.

Coherence of Distributed Applications Under Critical Time Constraints

The increasing complexity of control systems, as well as the large dimensions of the controlled processes, lead to the development of distributed control. New control architectures are mainly based on communication and on intelligent instrumentation concepts. Thus, field buses, as well as smart sensors and actuators, constitute a new field for investigation, in terms of their own definition and working mechanisms, as well as in terms of their use within the framework of distributed applications design. Field bus protocols like FIP (Factory Instrumentation Protocol), are based on a producer-consumers diffusion process, and introduce critical time concepts, which should be very usefull when designing automatic control systems, whose algorithms arc most frequently based on discrete-time state-space representations of the process to control or to supervise. However, the communication protocol gives some tools in order to handle the critical time constraints in real time operation, but does not provide any means for the designers to settle the numerical values of these constraints. This paper describes a real time application as a collection of software and hardware modules, each of them being characterized by the variables it produces and consumes. The consumed variables are associated to critical time constraints, since they have to represent the state of the process at times which arc coherent with the execution instants of the modules which consume them. Logical and temporal coherence arc defined on such an application : logical coherence ensures that each consumed variable has been produced, and that its producer is unique ; temporal coherence checks whether all the consumers of a produced variable are able to fulfill their critical time requirement.

Digital modelling, ideal state reconstructor, and control for time-delay sampled-data systems

The cascaded discrete-time state-space representation of a cascaded continuous-time system with fractional input delays is established. Based on the time-delay digital modelling, a practically imple-mentable ideal state reconstructor is also established such that system states are exactly reconstructed via the measurement histories of inputs and outputs without a state observer. By utilizing the block-pulse function approximation the digital modelling of cascaded continuous-time systems with fractional input delays can be carried out, and an artificial input design method is proposed to determine the state feedback gain. Thus the practically implementable digital control law can be established for digital control of time-delay sampled-data systems. An illustrative example is shown to demonstrate the effectiveness of the proposed method.

Constrained eigensystem realization algorithm for lightly damped distributed structures

The temporal correlation method for modal identification of lightly damped distributed structures is extended to include system realization and is shown to be a constrained version of the eigensystem realization algorithm. The method relies on the temporal and spatial orthogonality of the modes of vibration, which are properties of lightly damped or nearly self-adjoint distributed systems. Using these properties, the system realization and modal identification can be performed in the configuration space, as opposed to the state space. In this manner, the computational requirements of the eigensystem realization algorithm are decreased, in addition to constrain- ing the algorithm to take advantage of the lightly damped behavior of the system so that the technique may be less sensitive to sensor noise in the system measurements and provide improved realization capabilities. Analytical and experimental examples illustrate and verify the method. The results are compared with the standard version of the eigensystem realization algorithm. HE eigensystem realization algorithm (ERA) has been gaining popularity for use as a tool for identifying modal parameters of structures. Developed and tested by J.-N. Juang and R. S. Pappa at NASA Langley Research Center, the ERA has been used successfully for minimum-order system realiza- tion and modal identification of structures. The ERA was first introduced as an extension of the Ho-Kalman algorithm for constructing a discrete-time state-space representation of a linear system from noisy sensor measurements.1 In the time domain, the ERA uses the discrete-time state equations x(k +1) = Ax(k) + Bu(k) and the output equation y = Cx(k) and consists of constructing the coefficient matrices A, B, and C using the impulse response. The method is particularly well suited for systems with closely spaced or repeated eigenvalues, as it allows more than a single set of output data generated by different inputs to be used in the technique. This makes it possible to identify eigenfunctions belonging to repeated ei- genvalues. The minimum-order system realization consists of determining the system order and then constructing the mini- mum-order discrete-time state-transition matrix A. The solu- tion of the eigenvalue problem associated withvl renders the identified modal parameters. Several recent works have been published by Juang and Pappa1-6 and Longman and Juang.7 In addition, Pappa has developed a very useful public domain ERA code for implementation on VAX/VMS systems. In many ongoing structures/controls experiments, the struc- ture possesses light levels of damping and demands active and/or passive control for vibration suppression. Usually, many sensors are used for system identification and state estimation of the structure to ensure adequate performance of the control system. The degree of noise contamination in the sensors is generally unknown and can make the system realiza- tion process especially difficult. Modal survey tests typically use a large number of sensors to adequately define mode shapes, and they can be greater in number than the number of modes to be identified. Using a large number of measurement stations enhances the noise filtering capabilities of the identifi- cation as the sensor measurements can use spatial filters (e.g.,