Technical stability of stationary automatic control systems of variable structure with switched filters

Sufficient conditions for the technical stability in measure of a nonstationary control system with variable structure are established. The controller of the system has feedback-switched filters functioning together with shaper and actuator. It is assumed that the nonstationary parameters of the system vary within given ranges, at a finite rate, with appropriate control laws, with adjustment against mismatch signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach for analysis of technical stability does not involve sliding mode conditions. Criteria of technical instability in measure for the control system under consideration are formulated using the properties of systems of comparison from below. The general criteria of technical stability and instability are applied to nonstationary filtered-control systems of variable structure of the third order. The comparison method based on normalized Lyapunov functions is used __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 110–127, June 2006.     

Technical Stability of Forced Motion in Nonstationary Automatic-Control Systems of Variable Structure

Sufficient conditions of technical stability are established for nonstationary automatic-control systems of variable structure. The case is considered where for appropriate controls and for all admissible initial perturbations from a predefined measurable set of initial states, the nonstationary parameters of a process under external actions vary within given ranges. The control process depends on the mismatch signal and its derivatives up to an order that is less by one than the order of the equations of motion. The associated system of differential equations contains discontinuous time-dependent coefficients. A relationship is established between the eigenvalues of the quadratic forms of the given Lyapunov functions and the technical-stability criteria. It is shown that the technical-stability conditions do not depend on the sliding-mode conditions for variable-structure control processes     

Stability of Nonstationary Automatic-Control Systems of Variable Structure in Forced Motion

Conditions of stability in sliding mode are obtained for nonstationary dynamic automatic-control processes of variable structure. The nonstationary parameters of a control process subjected to external perturbations vary in given ranges, whereas the parameters of the switching plane remain constant for the appropriately chosen parameters of discontinuous logic control laws. The corresponding system of differential equations contains coefficients that vary stepwise in time together with the parameters of the controlled process. The general stability criteria for the solutions of variable-structure systems are used to analyze the stability of motion in sliding mode of an automatic variable-structure control system of the second order     

Technical Stability Conditions for Systems of Variable Structure with Derivatives of the Control Function

We derive sufficient conditions for the technical stability of forced states of automatic control systems with a variable structure depending on the derivatives of the control function. For arbitrary admissible initial perturbations from a measurable set of initial states of a control system with filtering, the technical stability conditions do not depend on the sliding conditions in the given domain of the system parameters. We find how the properties of the eigenvalues of the quadratic forms of the Lyapunov functions are related to the technical stability conditions for automatic control systems of variable structure with filtering. The technical stability conditions are analyzed for automatic control systems with third-order filtering where the forming signal creates eight structures     

Technical Stability of Forced Automatic-Control Variable-Structure Systems

Sufficient conditions for technical stability of functional states of controlled systems with a variable structure are derived. Allowance is made for external perturbations acting on the given process for all possible initial distributions from the set of the process' initial states predetermined in a quadratic measure. The differential equations characterizing the system under consideration include coefficients that vary stepwise with stepwise change in the parameters of the discontinuous control function. It is shown that the conditions of technical stability obtained do not necessarily depend on the existence of sliding modes in variable-structure systems. The eigenvalues of the quadratic forms of the corresponding Lyapunov functions are found to relate to the criteria of technical stability of automatic-control variable-structure processes     

Stability in sliding mode of nonstationary automatic-control systems of variable structure with switched filters

Stability conditions for a nonstationary automatic-control system of variable structure in sliding mode are established. The controller of the system has feedback-switched filters functioning together with the shaper and actuator. The nonstationary parameters of the system vary within given ranges, at a finite rate, under appropriate control laws, with adjustment for the error signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach to stability analysis is based on the existence conditions for the sliding mode at the switching boundary in the phase space. The general stability and instability criteria are applied to nonstationary automatic filtered-control systems of variable structure of the third order __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 116–134, October 2006.     

Technical Stability of Discontinuous Control Systems with a Continuous Set of Initial Perturbations

Conditions of technical stability in a given measure are derived for autonomous dynamic discontinuous-control systems for all possible initial states in a given domain of admissible initial perturbations of the output processes. The criteria formulated depend on the properties of the roots of the secular equation of a given quadratic form assigned to the control system under investigation     

A single adaptive controller with one variable for synchronizing two identical time delay hyperchaotic Lorenz systems with mismatched parameters

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameter identification and synchronization of uncertain hyperchaotic time-delayed systems. Based on the Lyapunov stability theory and the adaptive control theory, a single adaptive controller with one variable for synchronizing two identical time-delay hyperchaotic Lorenz systems with mismatch parameters is proposed. The parameter update laws and sufficient conditions of the scheme are obtained for both linear feedback and adaptive control approaches. Numerical simulations are also given to show the effectiveness of the proposed method.     

Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs

Gyroscopes are one of the most interesting and everlasting nonlinear nonautonomous dynamical systems that exhibit very complex dynamical behavior such as chaos.In this paper,the problem of robust stabi...     

Adaptive finite-time stabilization of uncertain non-autonomous chaotic electromechanical gyrostat systems with unknown parameters

This paper deals with the problem of robust finite-time stabilization of non-autonomous chaotic gyrostat systems. It is assumed that the parameters of the gyrostat system are completely unknown in advance and the system is perturbed by unknown uncertainties and disturbances. Some update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and the update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in a finite time. The finite-time stability and convergence of the closed-loop system are analytically proved. A numerical simulation is given to demonstrate the applicability and robustness of the proposed finite-time controller and to verify the theoretical results.     

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

We derive a flow formulation of differential-algebraic equations (DAEs), implicit differential equations whose dynamics are restricted by algebraic constraints. Using the framework of derivatives arrays and the strangeness-index, we identify the systems that are uniquely solvable on a particular set of initial values and thus possess a flow, the mapping that uniquely relates a given initial value with the solution through this point. The flow allows to study system properties like invariant sets, stability, monotonicity or positivity. For DAEs, the flow further provides insights into the manifold onto which the system is bound to and into the dynamics on this manifold. Using a projection approach to decouple the differential and algebraic components, we give an explicit representation of the flow that is stated in the original coordinate space. This concept allows to study DAEs whose dynamics are restricted to special subsets in the variable space, like a cone or the nonnegative orthant.     

Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives

The positivity of quadratic integrals involving variable coefficients and derivatives of any order is studied. The result is determined by the solution of an initial value problem for a system of first order nonlinear differential equations. The system is identified as the matrix Riccati differential equation in control theory. A complete conclusion is reached by considering the cases when the solution is bounded and when the solution is unbounded.     

时间发展平面混合流的三维演化

采用高精度差分方法和群速度控制方法，求解三维可压缩Ｎ Ｓ方程，直接数值模拟了时间发展的平面混合流．研究了平面混合流三维拟序结构的形成及发展．给出了流动失稳后涡的卷起，相邻两涡的对并，激波的形成及发展．指出，涡对并所诱导产生的激波对三维拟序结构的形成及发展过程是重要的．     

Forced sliding mode control for chaotic systems synchronization

Synchronization of chaotic systems is considered to be a common engineering problem. However, the proposed laws of synchronization control do not always provide robustness toward the parametric perturbations. The purpose of this article is to show the use of synergy-cybernetic approach for the construction of robust law for Arneodo chaotic systems synchronization. As the main method of design of robust control, the method of design of control with forced sliding mode of the synergetic control theory is considered. To illustrate the effectiveness of the proposed law, in this article it is compared with the classical sliding mode control and adaptive backstepping. The distinctive features of suggested robust control law are the more good compensation of parametric perturbations (better performance indexes—the root-mean-square error (RMSE), average absolute value (AVG) of error) without designing perturbation observers, the ability to exclude the chattering effect, less energy consuming and a simpler analysis of the stability of a closed-loop system. The study of the proposed control law and the change of its parameters and the place of parametric perturbation’s application is carried out. It is possible to significantly reduce the synchronization error and RMSE, as well as AVG of error by reducing some parameters, but that leads to an increase in control signal amplitude. The place of application of parametric disturbances (slave or master system) has no effect on the RMSE and AVG of error. Offered approach will allow a new consideration for the design of robust control laws for chaotic systems, taking into account the ideas of directed self-organization and robust control. It can be used for synchronization other chaotic systems.

     

Gas Flows with Several Thermal Nonequilibrium Modes

We study gas flows with any finite number of thermal nonequilibrium modes. The equations describing such flows are a hyperbolic system with several relaxation equations. An important feature is entropy increase dictated by physics for any irreversible process. Under physical assumptions we obtain properties of thermodynamic variables relevant to stability. By the energy method we prove global existence and uniqueness for the Cauchy problem when the initial data are small perturbations of constant equilibrium states. We give a precise formulation of the fundamental solution for the linearized system, and use it to obtain large time behavior of solutions to the nonlinear system. In particular, we show that the entropy increases but stays bounded. The resulting asymptotic picture of nonequilibrium flows is in a pointwise sense both in space and in time.     

用具有负定矩阵的梯度系统构造稳定的变质量力学系统

随着科学技术的发展,对喷气飞机、火箭等变质量系统动力学的研究显得越来越重要, 并且总是希望变质量系统的解是稳定的或渐近稳定的. 而通用的研究稳定性的Lyapunov直接法有很大难度, 因为直接从微分方程出发构造Lyapunov函数往往很难实现. 本文给出一种研究稳定性的间接方法, 即梯度系统方法. 该方法不但能揭示动力学系统的内在结构, 而且有助于探索系统的稳定性、渐进性和分岔等动力学行为. 梯度系统的函数V通常取为Lyapunov函数, 因此梯度系统比较适合用Lyapunov函数来研究. 列写出变质量完整力学系统的运动方程,在系统非奇异情形下,求得所有广义加速度. 提出一类具有负定矩阵的梯度系统, 并研究该梯度系统解的稳定性. 把这类梯度系统和变质量力学系统有机结合,给出变质量力学系统的解可以是稳定的或渐近稳定的条件, 进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的变质量力学系统. 通过具体例子,研究了变质量系统的单自由度运动,在怎样的质量变化规律、微粒分离速度和加力下,其解是稳定的或渐近稳定的. 本文的构造方法也适合其它类型的动力学系统.      

DYNAMICAL CONTROL OF STABILITY FOR BIRKHOFFIAN SYSTEM$^{\bf 1)}$

本文研究 Birkhoff 系统和广义 Birkhoff 系统平衡稳定性的动力学控制. 首先建立系统的运动方程和平衡方程. 其次,研究 Birkhoff 系统中控制参数出现在 Birkhoff 函数中平衡稳 定性的动力学控制. 方法是通过选取控制参数使得 Birkhoff 函数 $B$ 成为定号函数,而其时间导数 $\dot {B}$ 为与 $B$ 反号的常号函数. 再次,研究广义 Birkhoff 系统平衡稳定性的动力学控制,通过选取 Birkhoff 函数或附加项中包含控制参数的方法,使得 Birkhoff 函数是定号函数,而其时间导数为反号的常号函数,从而控制系统的平衡稳定性. 最后举例说明结果的应用.     

New methods to find solutions and analyze stability of equilibrium of nonholonomic mechanical systems

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives \({\ddot{q}}_0 ,{{\dddot{q}{} }}_{\!0} ,\ldots \) is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples.     

Finite time stability analysis of PD fractional control of robotic time-delay systems

A finite time stability test procedure is proposed for robotic system where it appears a time delay in PD^α fractional control system. Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time-delay systems given in state-space form. Sufficient conditions for this kind of stability, for particular class of fractional time-delay systems are derived.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.