scholarly journals Double Tracking Control for the Directed Complex Dynamic Network via the Observer of Outgoing Links

2021 ◽  
Author(s):  
Peitao Gao ◽  
Yinhe Wang ◽  
Lizhi Liu ◽  
Lili Zhang ◽  
Xiao Tang
Keyword(s):  
Dynamic Model ◽  
Tracking Control ◽  
Dynamic Network ◽  
Complex Dynamic ◽  
System Perspective ◽  
Tracking Controllers ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Vector Differential Equation

Abstract From the large system perspective, the directed complex dynamic network is considered as being composed of the nodes subsystem (NS) and the links subsystem (LS), which are coupled with together. Different from the previous studies which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behavior of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Since the vectors possess the flexible mathematical operational properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problems of NS and OLS. Under the state of OLS can be unavailable, the asymptotical state observer of OLS is designed in this paper, by which the tracking controllers of NS and OLS are synthesized to ensure achieving the double tracking goals. Finally, the example simulations for supporting the theoretical results are also provided.

scholarly journals Perturbation analysis of a matrix differential equation ẋ = ABx

2018 ◽  
Vol 3 (1) ◽  
pp. 97-104 ◽  
Author(s):  
M. Isabel García-Planas ◽  
Tetiana Klymchuk
Keyword(s):  
Normal Form ◽  
Equivalence Transformations ◽  
First Order ◽  
Similarity Transformations ◽  
Canonical Pair ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Contragredient Equivalence ◽  
Canonical Matrix

AbstractTwo complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

On matrix differential equations with several unbounded delays

2006 ◽  
Vol 17 (4) ◽  
pp. 417-433 ◽  
Author(s):  
J. ĈERMÁK
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Equation ◽  
Matrix Differential Equation ◽  
Continuous Vector ◽  
Asymptotic Bounds ◽  
The Matrix ◽  
Exponentially Stable ◽  
Matrix Differential Equations ◽  
Matrix Differential

The paper focuses on the matrix differential equation \[ \dot y(t)=A(t)y(t)+\sum_{j=1}^{m}B_j(t)y(\tau_j(t))+f(t),\quad t\in I=[t_0,\infty)\vspace*{-3pt} \] with continuous matrices $A$, $B_j$, a continuous vector $f$ and continuous delays $\tau_j$ satisfying $\tau_k\circ\tau_l =\tau_l\circ\tau_k$ on $I$ for any pair $\tau_k,\tau_l$. Assuming that the equation \[ \dot y(t)=A(t)y(t)\] is uniformly exponentially stable, we present some asymptotic bounds of solutions $y$ of the considered delay equation. A system of simultaneous Schröder equations is used to formulate these asymptotic bounds.

scholarly journals Calibrating linear continuous-time dynamical systems via perturbation analysis

Filomat ◽  
2018 ◽  
Vol 32 (5) ◽  
pp. 1909-1915
Author(s):  
Peter Weng ◽  
Frederick Phoa
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Perturbation Analysis ◽  
Continuous Time ◽  
Riccati Equations ◽  
Linear Dynamical Systems ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Time Linear

This work considered the continuous-time linear dynamical systems described by the matrix differential equations, and aimed at studying the perturbation analysis via solving perturbed linear dynamical systems. In specific, we solved Riccati differential equations and continuous-time algebraic Riccati equations with finite and infinite times respectively. Moreover, we stated some assumptions on the existence and uniqueness of the solutions of the perturbed Riccati equations. Similar techniques were applied to the discrete-time linear dynamical systems. Two numerical examples illustrated the efficiency and accuracy.

Oscillation Criteria for Matrix Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 184-199 ◽  
Author(s):  
H. C. Howard
Keyword(s):  
Existence Theorems ◽  
Symmetric Matrix ◽  
Order System ◽  
Matrix Differential Equation ◽  
First Order ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

Dynamic Analysis of Elastic Link Mechanisms by Reduction of Coordinates

1972 ◽  
Vol 94 (2) ◽  
pp. 577-581 ◽  
Author(s):  
R. C. Winfrey
Keyword(s):  
Dynamic Analysis ◽  
Complete Solution ◽  
Practical Interest ◽  
Elastic Behavior ◽  
Computational Time ◽  
Linear Matrix ◽  
Considerable Saving ◽  
Matrix Differential Equations ◽  
Elastic Link ◽  
Matrix Differential

Techniques for the solution of linear matrix differential equations have previously been applied to the dynamic analysis of a mechanism. However, because the mechanism changes geometry as it rotates, a large number of solutions are necessary to predict the mechanism’s elastic behavior for even a few revolutions. Also, a designer is frequently concerned with the elastic behavior of only one point on the mechanism and has no practical interest in a complete solution. For these reasons, a method is given here for reducing the total number of coordinates to one coordinate at the point of design interest. A considerable saving in computational time is obtained since the dynamic solution involves one degree of freedom instead of many. Further, since any solution will make use of some limiting assumptions, results here indicate that, for design purposes, reducing the coordinates does not significantly affect comparable accuracy.

Real-time trajectory tracking control of Stewart platform using fractional order fuzzy PID controller optimized by particle swarm algorithm

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zafer Bingul ◽  
Oguzhan Karahan
Keyword(s):  
Dynamic Model ◽  
Fractional Order ◽  
Trajectory Tracking ◽  
Tracking Control ◽  
Fuzzy Pid ◽  
Simulation Environment ◽  
Trajectory Tracking Control ◽  
Content Type ◽  
Control Approach ◽  
Coupling Dynamic

Purpose The purpose of this paper is to address a fractional order fuzzy PID (FOFPID) control approach for solving the problem of enhancing high precision tracking performance and robustness against to different reference trajectories of a 6-DOF Stewart Platform (SP) in joint space. Design/methodology/approach For the optimal design of the proposed control approach, tuning of the controller parameters including membership functions and input-output scaling factors along with the fractional order rate of error and fractional order integral of control signal is tuned with off-line by using particle swarm optimization (PSO) algorithm. For achieving this off-line optimization in the simulation environment, very accurate dynamic model of SP which has more complicated dynamical characteristics is required. Therefore, the coupling dynamic model of multi-rigid-body system is developed by Lagrange-Euler approach. For completeness, the mathematical model of the actuators is established and integrated with the dynamic model of SP mechanical system to state electromechanical coupling dynamic model. To study the validness of the proposed FOFPID controller, using this accurate dynamic model of the SP, other published control approaches such as the PID control, FOPID control and fuzzy PID control are also optimized with PSO in simulation environment. To compare trajectory tracking performance and effectiveness of the tuned controllers, the real time validation trajectory tracking experiments are conducted using the experimental setup of the SP by applying the optimum parameters of the controllers. The credibility of the results obtained with the controllers tuned in simulation environment is examined using statistical analysis. Findings The experimental results clearly demonstrate that the proposed optimal FOFPID controller can improve the control performance and reduce reference trajectory tracking errors of the SP. Also, the proposed PSO optimized FOFPID control strategy outperforms other control schemes in terms of the different difficulty levels of the given trajectories. Originality/value To the best of the authors’ knowledge, such a motion controller incorporating the fractional order approach to the fuzzy is first time applied in trajectory tracking control of SP.

scholarly journals A Gradient System for Low Rank Matrix Completion

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 51 ◽  
Author(s):  
Carmela Scalone ◽  
Nicola Guglielmi
Keyword(s):  
Sparse Matrix ◽  
Matrix Completion ◽  
Low Rank ◽  
Frobenius Norm ◽  
Gradient System ◽  
Large Matrix ◽  
Rank Matrix ◽  
Matrix Differential Equations ◽  
Low Rank Matrix Completion ◽  
Matrix Differential

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

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