Approximation Of Linear Systems Research Articles

A New Scheme for the Approximation of Linear Dynamical Systems and Its Application to Controller Design

A New Scheme for the Approximation of Linear Dynamical Systems and Its Application to Controller Design

Stable approximation of SISO and MIMO linear dynamic systems

Stable approximation of SISO and MIMO linear dynamic systems

Computationally Efficient Hierarchical Model Predictive Control Via Koopman Operator

Abstract Combined powertrain and velocity optimization can achieve significant energy efficiency improvements. However, due to the multitime scales in the system, the optimization is performed hierarchically and by separating time scales. To enforce state constraints, iteration between controller is introduced, for example, using Lagrange multipliers as metric for constraint violation. In this paper, an extension of the Koopman operator theory is presented with to obtain a data-driven approximation of the multipliers' behavior hence eliminating the need for iterations. Because the evolution of the Lagrange multipliers is the result of a fast dynamics optimization problem, and not the response of a nonlinear dynamical system, a novel technique in which the Lagrange multipliers are interpreted as a dynamic system is presented here. The approximate Koopman linear system is then derived using extended dynamic mode decomposition and it is integrated with the slow dynamic optimization. Results show that the Koopman augmented controller, which is solved as one single optimization, meets state and input constraints and achieves similar energy savings compared to an iterative approach.

Space-Time Discontinuous Galerkin Methods for Weak Solutions of Hyperbolic Linear Symmetric Friedrichs Systems

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the {textrm{L}}_2 norm at final time. Numerical experiments confirm that the a priori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator.

Robust regulation for superheated steam temperature control based on data-driven feedback compensation

Control of the superheated steam temperature is a significant technical challenge to coal-fired power plants due to the strong nonlinearity, large inertia and parameter uncertainties with external disturbances. This paper proposes a robust proportion–integration–differentiation (PID) control of the superheated steam temperature with proven stability and the external disturbance rejection. The design is based on the proposed data-driven feedback compensator (DFC) which is a neural network (NN) trained to isolate the nonlinearity and inertia from the PID controller, allowing the system for tuning the PID controller can be simplified as an approximate linear system with the modeling error based-feedback and feed-forward compensations. Then, the PID controller is tuned by the Nyquist stability criterion to place every closed-loop pole to a specific region in the left-half s-plane to guarantee the stability with considering the modeling error. Besides, a NN based feed-forward compensator is trained as the inverse model of the feedback compensator to further smooth the temperature fluctuation caused by the disturbance. The simulations and engineering implementation of the superheated steam temperature controls for a coal-fired power plant in Australia show the effectiveness.

Variational Inference and Learning of Piecewise Linear Dynamical Systems.

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because their computational complexity increases exponentially with time. In this article, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms: a filter and a smoother. We show that the model parameters can be split into two sets: static and dynamic parameters, and that the former parameters can be estimated offline together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to the head-pose tracking, and we thoroughly compare our algorithms with several state of the art trackers.

Metric entropy limits on recurrent neural network learning of linear dynamical systems

One of the most influential results in neural network theory is the universal approximation theorem [1], [2], [3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems—including time-varying systems—by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn—or identify in system-theory parlance—stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.

Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems

Abstract This article is dedicated to the long time behavior of a finite volume approximation of general symmetrizable linear hyperbolic system on a bounded domain. In the continuous case this problem is very difficult, and the $\omega $–limit set (namely the set of all the possible long time limits) may be large and complicated to depict if no dissipation is introduced. In this article we prove that in general, with a stable finite volume scheme, the discrete solution converges to a steady state when the time goes to infinity. This property is a direct consequence of the numerical dissipation mechanisms used for stabilizing the discretization. We apply this result for determining the long time limit for several stabilizations of the wave system, and perform a formal link with the low Mach number problem of the nonlinear Euler system. Numerical experiments with the wave system are performed for confirming the theoretical results obtained.

A Novel Reduction Approach for Linear System Approximation

A Novel Reduction Approach for Linear System Approximation

STABILITY OF NON-LINEAR DYNAMICAL SYSTEM

The mainobjective of this research is to study the stability of thenon-lineardynamical system by using the linearization technique of three dimension systems toobtain an approximate linear system and find its stability. We apply this technique to reaches to the stability of the public non linear dynamical systems of dimension. Finally, some proposed examples (example (1) and example (2)) are given to explain this technique and used the corollary.

Internal model control design based on approximation of linear discrete dynamical systems

In this paper, a new direct discrete approximation based internal model control design is proposed to the linear discrete dynamical systems. The approximation method is used to determine an accurate and stable reduced-order model for the considered original higher-order discrete-time system. The method involves an enhanced Differential Evolution algorithm to ascertain the stable denominator polynomial coefficients, and the preferable reduced numerator polynomial coefficients are evaluated by using the improved discrete multi-point Padé approximation approach. The method deploys on discrete step integral square error minimization between the original dynamical system and the approximated model, together with retaining their discrete impulse response energy values. The approximated model has been considered an internal (predictive) model and proceeds with an optimal internal model controller design to improve the discrete dynamical system behaviour according to the reference input/the set point. The controller's best performance is attained by tuning the single filter parameter ′λ′ by minimizing the integral square error between the reference input and the actual output of the dynamical system using the enhanced differential evolution algorithm. The acceptability and applicability of the proposed process reduction-based controller design have been validated on a single-input single-output supersonic jet engine inlet dynamical model. The controller robust study is conducted by inserting 10% disruption uncertainty in the system dynamical model poles and zeros. The method has also been extended to the discrete multi-input multi-output dynamical model of the single machine infinite bus power system to develop an optimal internal model control-based power system stabilizer. The simulation results showing better reference input tracking, comparison of performance indices, and also highlight the efficacy of the proposed controller design.

Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method.

In this paper we propose a data driven realization and model order reduction (MOR) for linear fractional-order system (FoS) by applying the Loewner-matrix method. Given the interpolation data which obtained by sampling the transfer function of a FoS, the minimal fractional-order state space descriptor model that matching the interpolation data is constructed with low computational cost. Based on the framework, the commensurate order α of the fractional-order system is estimated by solving a least squares optimization in terms of sample data in case of unknown order-α. In addition, we present an integer-order approximation model using the interpolation method in the Loewner framework for FoS with delay. Finally, several numerical examples demonstrate the validity of our approach.

Takagi–Sugeno fuzzy sampled‐data model‐based less conservative stability criterion for a Francis hydropower unit's governing system

In this study, the stability analysis and controller design of a hydropower unit's governing system (HUGS) were studied based on a Takagi–Sugeno (T–S) fuzzy model and constant sampled-data control. First, according to the T–S fuzzy theory, the non-linear Francis hydro-turbine governing system under rigid water hammer is linearised and then the approximate linear system was obtained. Second, a fuzzy sampled-data controller was designed in the case of periodic sampling, and the closed-loop sampling system was discretised. After constructing the Lyapunov function and using the forward difference method, the stabilisation condition was given with less conservatism in a symmetrical linear matrix inequality form. Finally, the numerical simulation results showed that under four different sampling periods, the HUGS can quickly achieve stability and that it has different stability performance. In addition, the superiority of the controller was verified in comparison with traditional proportional–integral–derivative control and fuzzy control techniques.

Tele-operation of Autonomous Vehicles over AWGN Channel

This paper is concerned with the tele-operation of autonomous vehicles over analog Additive White Gaussian Noise (AWGN) channel, which is subject to transmission noise and power constraint. The nonlinear dynamic of autonomous vehicle is described by the unicycle model and is cascaded with a bandpass filter acting as encoder. Using the describing function method, the nonlinear dynamic of autonomous vehicle is represented by an approximate linear system. Then, the available results for linear control over analog AWGN channel are extended to account for linear continuous time systems with non - real valued and multiple real valued eigenvalues and for tracking a non-zero reference signal. Subsequently, by applying the extended results on the describing function of autonomous vehicles, a mean square control technique including an encoder, decoder and a controller is presented for reference tracking of the tele-operation of autonomous vehicles over AWGN channel. The satisfactory performance of the proposed control technique is illustrated by computer simulations.

Portable deep learning singlet microscope.

Having the least lenses, the significant feature of the singlet imaging system, helps the development of the portable and cost-effective microscopes. A novel method of monochromatic/color singlet microscopy, which is combined with only one aspheric lens and deep learning computational imaging technology, is proposed in this article. The designed singlet aspheric lens is an approximate linear signal system, which means modulation-transfer-function curves on all field-of-views (5 mm diagonally) are almost coincident with each other. The purpose of the designed linear signal system is to further improve the resolution of our microscope by using deep learning algorithm. As a proof of concept, we designed a singlet microscopy based on our method, which weighs only 400 g. The experimental data and results of the sample USAF-1951 target and bio-sample (the Equisetum-arvense Strobile L.S), prove that the performance of the proposed singlet microscope is competitive to a commercial microscope with the 4X/NA0.1 objective lens. We believe that our idea and method would guide to design more cost-effective and powerful singlet imaging system.

On the Correctness of Finite-Rank Approximations by Series of Shifted Gaussians

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite-rank approximations by linear systems. The main result for this approach is to establish correctness of the finite-rank linear system under consideration. And the main result of the paper is to prove correctness of the finite-rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non-zero.

Signal approximations by shifted Gaussians: a direct approach by finite linear systems

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite–rank approximations by linear systems. The main result for this approach is to establish correctness of the finite–rank linear system under consideration. And the main result of the paper is to prove correctness of the finite–rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non–zero.

Invariants in Optimal Control: An Exact Solution of the Optimal Stabilization Problem

The stabilizing optimal feedback is a function onthe separatrix of stable points of the associated Hamiltoniansystem. Three geometric objects - the symplectic form, Hamiltonianvector field, and Lyapunov function, generating the separatrix - are %intrinsic to the optimal control system. They areinvariantly attached to the optimal control system under canonicaltransformations of the phase space. The separatrix equations can be writtenin terms of these invariants through invariant operations.There is a computable representative of the equivalence class,containing the original system. It is its linear approximation system at the stable point.

Optimal approximation of fractional‐order systems with model validation using CFOA

Optimal integer-order transfer function approximation of linear time-invariant fractional-order systems (FOS) using a straight-forward approach is proposed here. The error between the rational approximant and the FOS (commensurate/non-commensurate/oscillatory types etc.) is minimised directly in the frequency domain, subject to design stability constraints, by using the colliding bodies optimisation (CBO) algorithm. Comparisons with other well-known metaheuristics regarding the computational time and the design effort expended in parameter tuning justify the efficiency of the proposed method. The CBO-optimised models achieve a superior solution quality regarding both the magnitude and phase responses in comparison to the reported literature. Analogue circuit implementations of the models using the current feedback operational amplifier are also presented. Simulations carried out in OrCAD PSPICE environment validate the practical efficacy of the proposed models.

BIFURCATIONS OF STEADY STATES OF THE BIAXIAL CREW MODEL

Many transport systems (car, airplane, ship) have symmetry property. For such systems symmetrical deviations of some control parameter (steering wheel) from the neutral position lead to symmetry of the dynamic behaviour of the object (equivalence of left and right turns). In this case the straight-line motion of an object with constant velocity corresponds to a trivial solution of the corresponding dynamic system (symmetric solution). Conditions for dangerous-safe loss of symmetric junction stability (the case of one zero proper linear approximation system) can be obtained from the analysis of the true birth-fusion of stationary states with a stationary state corresponding to a symmetric solution. This approach makes it possible to obtain conditions of dangerous and safe loss of symmetric junction stability equivalent to the conditions of MM. Bautin [3] with the lowest possible computational cost. The proposed approach is illustrated by the example of the stability analysis of a nonlinear model of a double-axle crew with excessive rotation. It is shown that the condition of hazardous and safe loss of stability of straight-line motion (symmetric junction) is determined by the ratio between dimensionless coefficients of axle deflection and coupling coefficients on transaxle axes. However, clutch coefficients are not included in the linear system of equations of perturbed motion but refer to a substantially nonlinear characteristic of the forces of withdrawal.