Reduced State Space Model Research Articles

Robust leader–follower synchronization of electric power generators

We consider the problem of synchronizing two electric power generators, one of which (the leader) is serving a time-varying electrical load, so that they can ultimately be connected to form a single power system. Each generator is described by a second-order reduced state–space model. We assume that the generator not serving an external load initially (the follower) has access to measurements of the leader’s phase angle, corrupted by some additive disturbances. By using these measurements, and leveraging results on reduced-order observers with ISS-type robustness, we propose a procedure that drives (i) the angular velocity of the follower close enough to that of the leader, and (ii) the phase angle of the follower close enough to that of the point at which both systems will be electrically connected. An explicit bound on the synchronization error in terms of the measurement disturbance and the variations in the electrical load served by the leader is computed. We illustrate the procedure via numerical simulations.

Solid Element Rotordynamic Modeling of a Rotor on a Flexible Support Structure Utilizing Multiple-Input and Multiple-Output Support Transfer Functions

The authors present an improved modeling approach to analyze the coupled rotor-support dynamics by modeling the rotor with solid finite elements (FEs) and utilizing multiple-input and multiple-output transfer functions (TFs) to represent the flexible support. A state-space model is then employed to perform general rotordynamic analyses. Transfer functions are used to simulate dynamic characteristics of the support structure, including cross-coupling between degrees-of-freedom. These TFs are derived by curve-fitting the frequency response functions of the support model at bearing locations. The impact of the polynomial degree of the TF on the response analysis is discussed, and a general rule is proposed to select an adequate polynomial degree. To validate the proposed approach, a comprehensive comparison between the complete solid FE rotor-support model (CSRSM) and the reduced state-space model (RSSM) is presented. Comparisons are made between natural frequencies, critical speeds, unbalance response, logarithmic decrement, and computation time. The results show that the RSSM provides a dynamically accurate approximation of the solid FE model in terms of rotordynamic analyses. Moreover, the computation time for the RSSM is reduced to less than 20% of the time required for the CSRSM. In addition, the modes up to 100,000 cpm are compared among the super-element, beam element, and RSSM. The results show that the RSSM is more accurate in predicting high-frequency modes than the other two approaches. Further, the proposed RSSM is useful for applications in vibration control and active magnetic bearing systems.

A receptor state space model of the insulin signalling system in glucose transport.

Insulin is a potent peptide hormone that regulates glucose levels in the blood. Insulin-sensitive cells respond to insulin stimulation with the translocation of glucose transporter 4 (GLUT4) to the plasma membrane (PM), enabling the clearance of glucose from the blood. Defects in this process can give rise to insulin resistance and ultimately diabetes. One widely cited model of insulin signalling leading to glucose transport is that of Sedaghat et al. (2002) Am. J. Physiol. Endocrinol. Metab. 283, E1084-E1101. Consisting of 20 deterministic ordinary differential equations (ODEs), it is the most comprehensive model of insulin signalling to date. However, the model possesses some major limitations, including the non-conservation of key components. In the current work, we detail mathematical and sensitivity analyses of the Sedaghat model. Based on the results of these analyses, we propose a reduced state space model of the insulin receptor subsystem. This reduced model maintains the input-output relation of the original model but is computationally more efficient, analytically tractable and resolves some of the limitations of the Sedaghat model.

Robust control of resistive wall modes using pseudospectra

A novel methodology to design robust feedback controllers for the stabilization of resistive wall modes (RWMs) in tokamaks is presented. A linear state space model is used that describes the system composed of the plasma, the resistive wall, the active coils and the magnetic field sensors. The full three-dimensional (3D) geometry of the wall and the coils is taken into account. The control system is represented by a parametrized matrix whose robust stability properties are optimized under variations of the parameters. To make the optimization process feasible, a reduced state space model is constructed by means of a novel technique involving an orthogonal projection. The orthogonality is essential to the robust stability concept used. The latter is based upon the idea of matrix pseudospectra and therefore accounts for the sensitivity of eigenvalues. Furthermore, the transient growth of perturbations is investigated. Namely, in some theoretically stable linear systems, initial perturbations can grow by large factors before they eventually decay, rendering them practically unstable.A detailed analysis of a simple, ITER-like test case provides the following general, conceptual insights into the RWM stabilization problem: (i) it is important to consider not only the eigenvalues themselves, but also their sensitivity, (ii) transient amplification might be an issue requiring consideration and (iii) the transient peak can be substantially reduced by a careful choice of the optimization objective and the sensor configuration.

Vehicle Parameter Estimation and Stability Enhancement

In this paper, high order sliding mode observers are used to estimate tires longitudinal forces, vehicle side slip angle and velocities. Two types of high order sliding mode observers are used. A third order one is used to estimate the longitudinal forces, while for the estimation of the vehicle side slip and velocities, an observer based on the super twisting algorithm is proposed. Validations with the simulator ve-dyna pointed out the good performance and the robustness of the proposed observers. Controller design for the braking is accomplished using a reduced state space model representing the movement of the vehicle center of gravity in the (X, Y ) plane. Driver's reactions are taken into account. The performance of the closed loop system is carried out by means of simulation tests.

Vehicle parameter estimation and stability enhancement using sliding modes techniques

In this paper, tyres longitudinal forces, vehicle side slip angle and velocity are identified and estimated using sliding modes observers. Longitudinal forces are identified using higher order sliding mode observers. In the estimation of the vehicle side slip angle and vehicle velocity, an observer based on the broken super-twisting algorithm is proposed. Validations with the simulator VE-DYNA pointed out the good performance and the robustness of the proposed observers. After validating these observers, controller design for the braking is accomplished using a reduced state space model representing the movement of the vehicle centre of gravity in the (X, Y) plane. Driver's reactions are taken into account. The performance of the closed loop system is carried out by means of simulation tests.

On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so-called singular value functions of a system describe the relative importance of each state component from an input–output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, distinct sets of singular value functions and balanced realizations are possible.

Model Reduction of Large Space Structures Using Approximate Component Cost Analysis

State-space model reduction methods, such as the component cost analysis (CCA), requires the solution of Lyapunov equations of order equal to the order of the system. However, for many engineering applications, the storage requirements and computational time needed for the solution of such large Lyapunov equations is often prohibitive.Inthiswork,theuseofKrylov-subspaceiterativemethodsisexaminedtoobtain low-rankapproximate solutionsofLyapunov equationsforusein CCAmodelreduction oflargespacestructures.Themethodsareapplied to obtain reduced-order models of the International Space Station multibody assembly stages for simulation and control purposes. In addition, closed-form expressions for cost-equivalent and cost-decoupled realizations are derived based on the approximate Lyapunov solutions. It is shown that using the proposed methods, approximate CCA reduced-order models can be obtained with a signie cant reduction in the computational effort and time.

Discrete system dominance index for model reduction

This paper will use a saccadic eye movement model to present a discrete system state-space model reduction method using an eigenspace dominance index. Also a relationship between the discrete-time dominance indices and similar continuous-time dominance indices will be established for a continuous system that is discretized by a zero-order hold. Numerical advantages to using the discrete-time system when the continuous-time system contains small near zero eigenvalues are discussed.

Combining var estimation and state space model reduction for simple good predictions

AbstractA combination of VAR estimation and state space model reduction techniques are examined by Monte Carlo methods in order to find good, simple to use, procedures for determining models which have reasonable prediction properties. The presentation is largely graphical. This helps focus attention on the aspects of the model determination problem which are relatively important for forecasting. One surprising result is that, for prediction purposes, knowledge of the true structure of the model generating the data is not particularly useful unless parameter values are also known. This is because the difficulty in estimating parameters of the true model causes more prediction error than results from a more parsimonious approximate model.

Fundamental approach to equivalent systems analysis

CRITICAL review is presented, and an alternate extended procedure is offered, for the equivalent system technique1 for evaluating aircraft handling qualities. In the equivalent systems approach, a numerical search algorithm has been employed to find the reduced-order model, of classical aircraft form, such that the frequency response of the higher-order system (aircraft) is well approximated over a specified frequency range. Questions have been raised, however especially when a good approximation is not obtained with the method, and are related to the following2: 1) the nonuniqueness of solutions, 2) the interpretation of the matching cost, 3) the goodness of fit required, 4) the uncertainty as to whether to fix or free parameters in the lower order transfer functions, 5) the appropriate treatment of the multi-input/multi-output case, and 6) the concept of effective system dynamic order. The origins of some of these problems may be generated by the reduction procedure, while others may result from not considering certain aspects of closed-loop system analysis or from the inability to accurately define the task. And still others may arise because each system possesses an effective order, often times higher than that desired. Because of some of these fundamental difficulties, the reduced-order modeling objective of approximating the aircraft's frequency response is re-examined, and when and how to match multiple frequency responses will be reviewed. An alternate state-space model-reduction approach will be proposed. The original transfer function (matrix) G(s) of dynamic order n is to be reduced, via a state-space transformation T. The construction of T involves no numerical search algorithm. Furthermore, the resulting model Gr is unique for the selected dynamic order >. The least effective dynamic order is determined a priori by evaluating a set of frequency-domain matching error bounds. These error bounds, furthermore, are applicable to each i-j element of [G(s) — Gr(s)]s=J(0 over all CD. These error bounds may naturally be interpreted on a Bode plot. Finally, the approach is applicable to multi-input/multioutput systems.

Identification of essential states for reduced-order models using a modal analysis

The problem of state-space model reduction and, in particular, the identification of essential states is considered. The method of a state/output energy index is reviewed and shown, in some cases, to yield insufficient information to determine the essential states. The modal indices of Litz are reassessed, exposing their explicit link with controllability. A new method of essential-state selection is proposed. First the dominant modes are identified, and then these are used to indicate which states contain a significant contribution from these dominant modes. The algorithm can be used to select the essential states in either the state vector or the output. The method is demonstrated on a model having the structure of a rolling mill stand model.