Control-Theoretic Forward Error Analysis of Iterative Numerical Algorithms

Reducing plug flow effect on measured rheological results of cement-based materials: Image analysis and numerical iteration approach

On coupled FEM-BEM simulation of the magneto-mechanical behavior of single-crystalline Ni-Mn-Ga alloys

A self-adaptive relaxed primal-dual iterative algorithm for solving the split feasibility and the fixed point problem

The dynamical systems theory developed in [4, 7, 8, 9] is applied to the stability analysis of control systems in which the feedback control law requires in real time the solution of a set of nonlinear algebraic equations. Since small sampling period is assumed, the stability and performance of the controlled process can be studied with a continuous-time formulation. A singularly perturbed system is used to model the combined dynamics of the system being controlled and a numerical iterative algorithm required to compute the control law. An updating control procedure has been proposed based on the iterative nature of the control algorithm. The results obtained in [9] regarding the behavior of a dynamical system that models the numerical algorithms lead to a considerable simplification in the analysis. For the case of control problem involving inverse kinematics, the numerical algorithm which solves for inverse kinematics can be considered as an observer (or an estimator) of the state space variables. The study provides an estimate of the required speed of computations to preserve the stability of the controller.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

We propose a hybrid dynamical system as a continuous analog to the block-iterative multiplicative algebraic reconstruction technique (BI-MART), which is a well-known iterative image reconstruction algorithm for computed tomography. The hybrid system is described by a switched nonlinear system with a piecewise smooth vector field or differential equation and, for consistent inverse problems, the convergence of non-negatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem. Namely, we can prove theoretically that a weighted Kullback-Leibler divergence measure can be a common Lyapunov function for the switched system. We show that discretizing the differential equation by using the first-order approximation (Euler's method) based on the geometric multiplicative calculus leads to the same iterative formula of the BI-MART with the scaling parameter as a time-step of numerical discretization. The present paper is the first to reveal that a kind of iterative image reconstruction algorithm is constructed by the discretization of a continuous-time dynamical system for solving tomographic inverse problems. Iterative algorithms with not only the Euler method but also the Runge-Kutta methods of lower-orders applied for discretizing the continuous-time system can be used for image reconstruction. A numerical example showing the characteristics of the discretized iterative methods is presented.

In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system, J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.

An iterative numerical algorithm for solving multi-parameter inverse problems of evolutional partial differential equations

The problem of uplink communications is considered where multiple mobile terminals equipped with multiple antennas are communicating with one base station also having multiple antennas. A theory for jointly optimizing the transmitter and receiver finite impulse response (FIR) multiple-input multiple-output (MIMO) filters is developed. The signals from the individual users are assumed to be uncorrelated with each other and the additive channel noise is uncorrelated with the original signals. The FIR MIMO channel filters are assumed to be known for both the mobile terminals and the base station. All the input signals to the FIR MIMO channels are assumed to be power-constrained. For given FIR MIMO channels with maximum allowable average input powers, the transmitter and receiver FIR MIMO filters are jointly optimized such that the mean square error (MSE) between the desired and reconstructed signals is minimized. An iterative numerical optimization algorithm is proposed. Numerical simulation results show that the proposed method has substantially better performance than a code division multiple access (CDMA) based system using minimum MSE receiver filters.

A novel second-order partial differential equation (PDE) — based image restoration technique is proposed here. The considered smoothing method is based on a nonlinear hyperbolic differential model combined to a two-dimension filter kernel. The considered PDE model is well-posed and it is solved numerically by developing an explicit iterative finite difference method-based numerical approximation algorithm that is consistent to the combined PDE-based model and is converging fast to its weak solution. Our successful restoration tests and method comparison are also discussed.

In the area of the numerical analysis of stochastic Petri Nets we present an algorithm which enables the reduction of storage requirements for generator matrices of the underlying Markov chain. We show that neither the generator matrix nor “parts” of them need be generated and stored. The solvable model class contains the superposed stochastic automatas defined by Donatelli [2] as a special case. The state spaces of the underlying Markov chains in the examples range from about 107 up to 108 states with up to 109 matrix entries and we show that for such models a solution is possible. Further, this algorithm can be easily integrated in tools which contain iterative numerical solution techniques.KeywordsGenerator MatrixMatrix EntryProcedure CallColumn IndexInput PlaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A parametric study on geometrically nonlinear behavior of curved beams with single and double link rods, and supported on moving boundary

As robots become more capable in terms of hardware, and more complex tasks are considered, optimality starts playing a more important role in the design of algorithms implemented in these systems. Optimality is a guiding principle that directs the computation of feasible and efficient solutions to different robotics tasks. In control theory, this principle is implemented online as a set of efficient numerical optimization algorithms, that in addition to solving the task, purports to save a suitably defined effort or energy term. This thesis investigates trajectory generation, learning and control for dynamic tasks from the unifying point of view of optimization. As an application, we focus on Table Tennis, a chal- lenging task where robots are yet to outperform humans. We believe that the required dexterity and accuracy for this dynamical task hinges on the developments in online optimization and efficient learning algorithms. We consider trajectory generation for table tennis in the first part of the thesis. In highly dynamic tasks like table tennis that involve moving targets, planning is necessary to figure out when, where and how to intercept the target. Motion planning can be very challenging in robotic table tennis in particular, due to time constraints, dimension of the search space and joint limits. Conventional planning algorithms often rely on a fixed virtual hitting plane to construct robot striking trajectories. These algorithms, however, generate restrictive strokes and can result in unnatural strategies when compared with human playing. In this thesis, we introduce a new trajectory generation framework for robotic table tennis that does not involve a fixed hitting plane. A free-time optimal control approach is used to derive two different trajectory optimizers. The resulting two algorithms, Focused Player and Defensive Player, encode two different play- styles. We evaluate their performance in simulation and in our robot table tennis platform with a high speed cable-driven seven DOF robot arm. The algorithms return the balls with a higher probability to the opponent’s court when compared with a virtual hitting plane based method. Moreover, both can be run online and the trajectories can be corrected with new ball observations. In the second part of the thesis, we look at how such trajectories, computed on the kine- matics level, can be tracked accurately with learning control based approaches. Highly dynamic tasks like table tennis require large accelerations and precise tracking for successful perfor- mance. To track desired trajectories well, such tasks usually rely on accurate models and/or high gain feedback. While kinematic optimization allows for efficient representation and online generation of hitting trajectories, learning to track such dynamic movements with inaccurate models remains an open problem. In particular, stability issues surrounding the learning per- formance, in the iteration domain, can prevent the successful implementation of model based learning approaches. To achieve accurate tracking for these tasks in a stable and efficient way, we propose a new adaptive Iterative Learning Control algorithm that is implemented efficiently using a recursive approach. Moreover, covariance estimates of model matrices are used to ex- ercise caution during learning. We evaluate the performance of the proposed approach in our robotic table tennis platform, where we show how the performance of two Barrett WAMs can be optimized. Our implementation on the table tennis platform compares favorably with two state-of-the-art approaches. Finally, we discuss an alternative learning from demonstrations approach, where we learn sparse representations from demonstrated movements. Learning from demonstrations is an easy and intuitive way to show examples of successful behavior to a robot. However, the fact that humans optimize or take advantage of their body and not of the robot, usually called the embodiment problem in robotics, often prevents industrial robots from executing the task in a straightforward way. The shown movements often do not or cannot utilize the degrees of freedom of the robot efficiently, and typically suffer from excessive execution errors. In the last chapter, we show a new approach that can alleviate some of these difficulties by learning sparse representations of movement. Moreover, the number of learned parameters are independent of the degrees of freedom of the robot. Sparsity is a desirable feature for policy search Rein- forcement Learning algorithms that adapt the parameters of these movement primitives. By ranking the learned parameters on the Elastic Net path in terms of importance, we note that our approach could be potentially useful to combat the curse of dimensionality in robot learning applications. We show preliminary results on the real robot setup, including a successful table tennis serve using our new movement primitive representation. Throughout the thesis, we present and analyze in detail new control and learning algo- rithms. Efficient online optimization approaches are presented that can be used to solve not just table tennis problems, but they can be adapted to solve different dynamic tasks.

Hyperspectral imaging is beneficial in a diverse range of applications from diagnostic medicine, to agriculture, to surveillance to name a few. However, hyperspectral images often suffer from degradation such as noise and low resolution. In this paper, we propose an effective model for hyperspectral image (HSI) restoration, specifically image denoising and super-resolution. Our model considers three underlying characteristics of HSIs: sparsity across the spatial-spectral domain, high correlation across spectra, and non-local self-similarity over space. We first exploit high correlation across spectra and non-local self-similarity over space in the degraded HSI to learn an adaptive spatial-spectral dictionary. Then, we employ the local and non-local sparsity of the HSI under the learned spatial-spectral dictionary to design an HSI restoration model, which can be effectively solved by an iterative numerical algorithm with parameters that are adaptively adjusted for different clusters and different noise levels. In experiments on HSI denoising, we show that the proposed method outperforms many state-of-the-art methods under several comprehensive quantitative assessments. We also show that our method performs well on HSI super-resolution.

A computer‐aided method for solving the problem of partial or one‐parameter synthesis of a lossless non‐uniform transmission line of finite length is presented. This method is essentially the same as the pulse‐spectrum technique for solving the inverse problems of determining the velocity coefficients of the wave equation from known information of the solution on the boundary. The basic idea of this method is that the original synthesis problem in the time‐domain is first reduced to a CW synthesis problem in the frequency‐domain, by either Fourier transform or Laplace transform, and then the synthesis is carried out in the frequency‐domain by using an iterative numerical algorithm. Numerical simulations are carried out to test the feasibility and to study the general performance of this computer‐aided synthesis procedure without the real design data. It is found that this computer‐aided method does give excellent results.