Control Of Linear Dynamic Systems Research Articles

Universality and Control of Fat Tails

Motivated by applications in hydrodynamics and networks of thermostatically-control loads in buildings we study control of linear dynamical systems driven by additive and also multiplicative noise of a general position. Utilizing mathematical theory of stochastic multiplicative processes we present a universal way to estimate fat, algebraic tails of the state vector probability distributions. This prompts us to introduce and analyze mean-q-power stability criterion, generalizing the mean-square stability criterion, and then juxtapose it to other tools in control.

Дослідження плавності ходу багатоосних технологічних систем

Goal. To improve the smooth running of the modular energy unit by creating an elastic-dissipative connection of the energy module with the technological one. Methods. Mathematical formalization of the studied process was based on the provisions of theoretical mechanics, tractor theory, statistical dynamics, frequency methods of the theory of automatic control of linear dynamical systems for testing statistically random perturbing effects. The experimental studies were based on both generally accepted standards and developed original methods using tensometric, vibrographic equipment with the recording of signals recorded on a PC using an analog-to-digital converter. Processing of experimental results was performed on a PC using probability theory and correlation spectral analysis. Results. According to the results of the study, an adequate dynamic model of a modular power tool in the plow unit as a multi-axis dynamic system is obtained, which allows studying its motion in the longitudinal-vertical plane. The values were obtained of the amplitude characteristics of the vertical displacements of the rear axle of the power module during the reproduction of the oscillations of the path profile and traction resistance of the plow at different values of the coefficient of dissipative resistance of the two modules of the modular power vehicle. Based on these experimental studies, the dependences of the normalized correlation functions and the spectral densities of the accelerations of the vertical oscillations of the rear axle of the energy module at different values of the coefficient of resistance of the dissipative connection of the two modules were obtained. Their analysis made it possible to substantiate the value of the rational rigidity of the elastic-dissipative connection of the energy and technological modules of the modular energy unit of universal use. Conclusions. Studies proved that with increasing the coefficient of resistance of the dissipative connection of the two modules to 1.80 kN·s·m–1, the amplitude of oscillations of the rear axle of the power module decreased 10 times, and resonant peaks of frequency characteristics were shifted to low frequencies from 16.5 s–1 to 6.0 s–1 with simultaneous reduction of their maximum values by 4 times. The peaks of the amplitude-frequency characteristics for the reproduction of the modular energy unit of oscillations of the traction resistance of the plow shifted from 13 to 10 s–1 with a simultaneous decrease in their maximum values by 3 times. It is proved that the rational stiffness of the elastic-dissipative connection of two modules falls on the value of the resistance coefficient 1.65 kN·s·m–1. The dispersion of the accelerations of the vertical oscillations of the modular energy unit was reduced by 3 times. It is established that the oscillations of the traction resistance of agricultural tools have 10 times less effect on the smoothness of the modular power tool in the plowing unit than the oscillations of the longitudinal profile of the track.

Quadratic Constrained Periodic Optimization for Bandlimited Linear Systems Via the Fourier-Based Method

AbstractMotivated by engineering applications, we address bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimization can be cast as a minimization problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimized. With this parametrization, we address linear quadratic problems involving periodic bandlimited dynamics by using quadratic minimization with parametric time-dependent constraints. We hence investigate the implications of a discretization of linear continuous time constraints and propose an algorithm that provides a feasible suboptimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal control of DC/AC power converters, and optimal energy harvesting from pulsating mechanical energy sources.

Синтез H-inf регуляторов на конечном промежутке времени

The article considers a problem of stabilization of an aircraft, which can be solved by finding output control. Sufficient conditions for H-inf- output control for linear dynamic systems are obtained. Two approximate methods of closed-loop systems synthesis are proposed. The first one is based on finding a pseudo-inverse matrix of the linear measuring system model. The second one based on synthesizing an asymptotic observer, which allows to formulate an estimate of a state vector, giving a possibility of further use of the obtained estimate in the control law. The paper contains a step-by-step algorithm of finding an H-inf -control of a closed-loop system.

On Linear Quadratic Regulation of Linear Port-Hamiltonian Systems

The linear quadratic regulator is a widely used and studied optimal control technique for the control of linear dynamical systems. It consists in minimizing a quadratic cost functional of the states and the control inputs by the means of solving a linear Riccati equation. The effectiveness of the linear quadratic regulator relies on the cost function parameters hence, an appropriate selection of these parameters is of mayor importance in the control design. Port-Hamiltonian system modelling arise from balance equations, interconnection laws and the conservation of energy. These systems encode the physical properties in their structure matrices, energy function and definition of input and output ports. This paper establishes a relation between two classical passivity based control tools for port-Hamiltonian systems, namely control by interconnection and damping injection, with the linear quadratic regulator. These relations allow then to select the weights of the quadratic cost functional on the base of physical considerations. A simple RLC circuit has been used to illustrate the approach.

Controllability of Linear Dynamical Systems Under Input Sparsity Constraints

In this article, we consider the controllability of a discrete-time linear dynamical system with sparse control inputs. Sparsity constraints on the input arises naturally in networked systems, where activating each input variable adds to the cost of control. We derive algebraic necessary and sufficient conditions for ensuring controllability of a system with an arbitrary transfer matrix. The derived conditions can be verified in polynomial time complexity, unlike the more traditional Kalman-type rank tests. Further, we characterize the minimum number of input vectors required to satisfy the derived conditions for controllability. Finally, we present a generalized Kalman decomposition-like procedure that separates the state-space into subspaces corresponding to sparse-controllable and sparse-uncontrollable parts. These results form a theoretical basis for designing networked linear control systems with sparse inputs.

Adaptive Iterative Learning Control for Linear Dynamic Systems with Large Errors in Unknown Parameters

Adaptive Iterative Learning Control for Linear Dynamic Systems with Large Errors in Unknown Parameters

Integrated system identification and modal state estimation for control of flexible space structures

A novel approach of integrated system identification and modal state estimation is proposed for control of linear dynamical systems including flexible space structures. There are four steps involved in this approach. First, the relation between a stochastic state space model of a dynamical system and the coefficients of its autoregressive model with exogenous input is derived. Second, an adaptive least-squares transversal predictor is used to estimate the coefficients of the model. Third, a state space model and a steady state Kalman filter gain of the dynamical system are then identified from the coefficients of the model by using the eigensystem realization algorithm. Fourth, a modal state estimator is constructed using the modal parameters of the identified model. On-line implementation of this algorithm can continually improve the modal parameters and the filter gain. It can also gradually update the system model when the system characteristics are slowly changing. A numerical example is used to illustrate the feasibility of the new approach.

The Study of Movement of the Wide Span Tractor-Based Field Machine Unit with Power Method of its Control

Abstract Recently, rapid development of controlled traffic farming convincingly demonstrates the broad prospects of using the wide span tractor-based field machines throughout the world. The issue at hand lies in the fact that having a wide span tractor-based machine with steering-wheel system moving along the tracks of a constant technological track is not a rational option. According to the concept of bridge and rutting agriculture, it is advisable to subordinate the automated means of mechanization to the principles of the functioning of the coordinate transport system in which machines can move only in two mutually perpendicular directions and for implementation of which the field must have strictly defined dimensions. Therefore, a method of power steering similar to a crawler tractor (track-type tractor) can be more efficient in the automatic driving of bridge means of mechanization within the coordinate-transport system than a kinematic control. Furthermore, methodology utilized for selection of design schemes, parameters and operating modes of machine-tractor aggregates can not be used for study of dynamics of the wide span tractor-based field machine. Theoretical study is based on theoretical mechanics, theory of mobile energy facilities, statistical dynamics and theory of automatic control of linear dynamical systems with reproduction of statistically random control and disturbing input effects. Purpose of the research lies in development of a dynamic model of plane parallel motion in the horizontal plane of a wide span tractor-based field machine unit using a power (onboard) method of control, which would allow investigation of impacts of the control parameters and disturbing influences on the controllability and stability of its motion. Mathematical models have been developed and new regularities of the straightforward parallel movement of the wide span tractor-based field machine unit for controlled traffic farming have been obtained. The results obtained allow the validation of new schemes, design parameters and modes of operation with acceptable controllability and stability of movement in the horizontal plane with a power control method of the chassis.

Optimal variable-order fractional PID controllers for dynamical systems

This paper studies the design of variable-order fractional proportional–integral–derivative (VFPID) controllers for linear dynamical systems. For this purpose, a technique to discretize fractional differential equations with variable-order operators is proposed. The resulting model and a particle swarm optimization algorithm are used to search for the optimal parameters of the VFPID controllers. Three examples illustrate the performance of the closed-loop system under the action of the VFPID and compare it with the results obtained by means of the PID and fractional PID (FPID). Furthermore, time-dependent parameters are also tested, showing that the VFPID yields a superior performance.

Exact controllability of linear dynamical systems: a geometrical approach

In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems it is of interest to study the exact controllability; this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, we focus the study on the obtention of the set of all B making the system (A, B) exact controllable

Dropout feedback parametrized policies for stochastic predictive controller

Abstract: This article presents a novel control policy class for the networked control of linear dynamical systems with bounded inputs. The control channel is assumed to have i.i.d. Bernoulli packet dropouts. Our proposed class of policies is parametrized relative to the past dropouts. We show how to augment the underlying optimization problem with a constant negative drift constraint in order to ensure mean-square boundedness of the closed-loop states. The resulting convex quadratic program can be solved periodically online. The states of the closed loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control.

CONTROLLABILITY AND ALGORITHMS OF COMPLEX NETWORKS

In the controllability study of complex networks,there are basically two core issues:The minimal number of inputs and the minimal number of controlled nodes needed to achieve full control.Liu and Barabasi applied the modern control theory to the investigation of network structural controllability of linear dynamical systems and proposed a method to calculate the minimal number of inputs.This solves the computable problem of the complex network structural controllability.As to undirected or weighted networks in real networks,the paper introduces the concept of node control capacity for them.By using the method of adding inputs or edges,we propose algorithms to identify the minimal set of driver nodes as well as the approximately minimal set of controlled nodes to ensure the full control of network.

Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

Constrained Control Design for Linear Systems with Geometric Adversary Constraints

Inspired by some practical applications concerning collision avoidance topics, this paper focuses on the optimal control of linear dynamical systems in the presence of a set of adversary constraints. One of the novelties is the type of constraints introduced in the receding horizon optimization problem. These constraints can be considered as “adversary” by their non convex characteristics which make the convergence of the system dynamics towards the “zero” equilibrium position an impossible task. The present paper proposes a dual-mode control strategy which builds on an optimization based controller and a fixed constrained control law whenever the adversary constraints are activated. In order to illustrate the benefits of the proposed method, typical applications involving the control of multi-agent systems are considered.

Network controllability based on node overloaded failure

Liu and Barabasi applied the modern control theory to the network controllability of linear dynamical systems and proposed a method to calculate the minimal set of driver node which controls the states of all nodes in a linear time invariant complex network with any topology. The network controllability model solves the computable problems of the network controllability. Facing the problem of node overloaded failure in real networks, in the paper we investigate the model of network controllability based on node overloaded failure. Through the simulation of betweenness and Weibull failure model, the results demonstrate that the difficulty in maintaining the controllability of SF network is significantly greater than that of ER network. In the target failure mechanism, even if the failure signals input rarely to the networks, they can greatly increase the difficulty of network controllability. Besides, the node failure based high betweenness centrality is more efficient than failure based high degree on damaging network controllability, which indicates the nodes with high betweenness centrality play an important role in maintaining the network controllability. Furthermore, taking the reasonable measures for different load failure model can prevent the networks from inducing a step uncontrollable phenomenon.

Control of linear dynamical systems by time transformations

Necessary and sufficient conditions for the controllability of solutions of linear inhomogeneous integral equations are obtained.

Nonsmooth optimization via quasi-Newton methods

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

Formula omitted] optimal semistable control for linear dynamical systems: An LMI approach

In this paper, we develop H 2 semistability theory for linear dynamical systems. Using this theory, we design H 2 optimal semistable controllers for linear dynamical systems. Unlike the standard H 2 optimal control problem, a complicating feature of the H 2 optimal semistable stabilization problem is that the closed-loop Lyapunov equation guaranteeing semistability can admit multiple solutions. An interesting feature of the proposed approach, however, is that a least squares solution over all possible semistabilizing solutions corresponds to the H 2 optimal solution. It is shown that this least squares solution can be characterized by a linear matrix inequality minimization problem.

Application of a parametrization method to problem of optimal control

A new approach to the problem of optimal control of linear dynamic systems is proposed that makes use of a method of input and state parametrization to transform the original problem into a problem of the Calculus of Variations. In contrast to the standard approaches for this class of problems, two salient features of the new approach are that no Lagrange multiplier functions need to be invoked and that the class of inputs can be restricted to the – relatively small – class of continuous functions, even for problems with fixed end-states. The resulting necessary conditions of optimality, i.e., the Euler–Lagrange equation and the boundary conditions for the transformed problem, are proved to be equivalent to those resulting from the standard method of First Variations. In case of quadratic cost functionals, the new approach provides a simpler alternative to the well known, but equally difficult, Riccati differential equation approach and results in a simple dynamic state-feedback implementation of the optimal control.