Stabilization of the second-order linear time-invariant control systems by a delayed feedback

We consider a static stabilization problem for a two-dimensional linear time-invariant control system with a delayed feedback. We obtain the necessary and sufficient conditions for the stabilizability of the system under consideration. The theorems proved in this paper show that such a delayed feedback approach is efficient in stabilizing the second-order linear systems.     

On periodic solutions of nonlinear second order differential systems

The paper deals with the second order differential systems with periodic boundary conditions. In the first part of the paper four different methods are employed to find the unique solution of the linear systems. One of the methods is a shooting type which converts the periodic boundary value problem into its equivalent initial value problem. In the second part of the paper several sufficient conditions are provided for the existence and uniqueness for the nonlinear systems. The technique developed for the linear systems to convert into its equivalent initial value problem is used in an iterative way for the nonlinear systems. It is shown that the iterations converge quadratically.     

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper considers the problem of positive real control for two-dimensional (2-D) discrete delayed systems in the Fornasini–Marchesini second local state-space model. Attention is focused on the design of dynamic output feedback controllers, which guarantee that the closed-loop system is asymptotically stable and the closed-loop transfer function is extended strictly positive real. We first present a sufficient condition for extended strictly positive realness of 2-D discrete delayed systems. Based on this, a sufficient condition for the solvability of the positive real control problem is obtained in terms of a linear matrix inequality (LMI). When the LMI is feasible, an explicit parametrization of a desired output feedback controller is presented. Finally, we provide a numerical example to demonstrate the application of the proposed method.     

Delayed feedback stabilization and the Huijberts–Michiels–Nijmeijer problem

A short survey on delayed feedback stabilization is given. The Huijberts–Michiels–Nijmeijer problem on the delayed feedback stabilization of unstable equilibria of two- and three-dimensional dynamical systems is considered. It is shown that the methods of delayed feedback stabilization of unstable periodic orbits can be used with advantage for the stabilization of unstable equilibria. An analytical study based on the D-decomposition method is given. Efficient necessary and/or sufficient conditions for the stabilizability of the systems in question are obtained in the form of explicit analytic expressions. These conditions define the boundaries of stabilizability domains in terms of system parameters. It follows from these conditions that the introduction of a delayed feedback control generally extends the possibilities of stationary stabilization of linear systems with delay-free feedback.     

Stabilization of stationary affine control systems with discrete time

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.     

Spectrum control problem for strongly irregular periodic vibrations of linear systems with zero mean of the coefficient matrix

We consider a linear periodic control system with zero mean of the coefficient matrix and with linear state feedback control periodic with the same period. We obtain necessary and sufficient conditions for the solvability of the frequency spectrum control problem with a given goal set for strongly irregular periodic vibrations. In this problem, one should find a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.     

Asynchronous pole assignment for linear systems with blocks of incomplete rank

We consider a linear periodic control system with constant matrix multiplying the control in the critical case under the assumption that the mean of the coefficient matrix is block triangular and some blocks of its nonstationary part have incomplete column rank. A linear state feedback control periodic with the same period as the system itself is considered. We obtain necessary and sufficient conditions for the solvability of the asynchronous pole assignment problem, that is, the problem of finding a feedback factor such that the closed-loop system has a strongly irregular periodic solution with desired frequencies.     

参数不确定性广义周期时变系统的鲁棒H_∞控制

研究状态矩阵和控制输入矩阵均具不确定性广义周期时变系统的鲁棒H_∞控制问题.提出参数不确定性广义周期时变系统广义可镇定和广义二次可镇定且具有H_∞性能指标的概念,利用线性矩阵不等式(LMI)方法,得到了参数不确定性广义周期时变系统广义二次可镇定且具有H_∞性能指标γ的充要条件,给出了相应的鲁棒H_∞状态反馈控制律的设计方法.最后,通过数值算例说明了设计方法的有效性.     

Approximate controllability of impulsive fractional evolution equations of order 1<α<2$$ 1&amp;lt;\alpha &amp;lt;2 $$ with state-dependent delay in Banach spaces

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state-dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear-regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results.     

一类不确定广义周期时变系统的鲁棒H_∞控制

本文研究了状态矩阵具不确定性广义周期时变系统的鲁棒H∞控制问题.利用线性矩阵不等式(LMI)方法,在给出不确定广义周期时变系统广义可镇定和广义二次可镇定且具有H∞性能指标概念的基础上,得到了该系统广义二次可镇定且具有H∞性能指标γ的充要条件,并给出了相应的鲁棒H∞状态反馈控制律的设计方法,推广了周期系统的鲁棒控制理论结果.最后,通过数值算例说明了设计方法的有效性.     

Control synthesis for a three-dimensional nilpotent system

We consider the control problem for a system described by ordinary differential equations with linear controls. We present sufficient conditions for finding an exact solution of the control problem for a three-dimensional nilpotent system with a two-dimensional linear control in the form of programmed controls and feedback controls. We consider two examples of the computation of controls with the use of linear vector fields on the plane.     

Synthesis of minimal linear stabilizers

We consider the stabilizer synthesis problem for linear stationary control dynamical systems; attention is mainly focused on the investigation of possibilities to reduce the order of such stabilizers. The problem is considered in two settings. The first setting deals with the construction of stabilizers of given order (in particular, the minimum possible order); no conditions are imposed on the spectrum of the closed system. For scalar (single-input-single-output) control systems, we suggest an approach to the solution of this problem and obtain necessary and sufficient conditions for the existence of a stabilizer of a given order. The second problem is that of synthesizing a stabilizer of minimum order with given dynamic properties (a given spectrum or a given distribution of the spectrum of the closed system, in particular, with a guaranteed convergence rate of the closed system). For this problem, we suggest two approaches that permit one to obtain an upper bound for the dimension of such stabilizers.     

Asynchronous pole assignment for linear systems with independent blocks

We consider a linear periodic control system with linearly independent column bases of the blocks of the coefficient matrix, which has zero mean value and admits a block triangular representation. For the case of a linear state feedback control periodic with the same period as the system itself, we obtain necessary and sufficient conditions for the solvability of the asynchronous pole assignment problem, i.e., the problem of finding a feedback coefficient such that the closed-loop system has a strongly irregular periodic solution with the desired frequencies.     

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired nth order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix

We consider a linear periodic control system such that the ranks of the matrix multiplying the control and the extended matrix consisting of the averaged coefficient matrix and the matrix multiplying the control are the same. We assume that the control has the form of feedback linear in the state variables and is periodic with the same period as the system itself. We pose the problem of control of the frequency spectrum of strongly irregular periodic oscillations with an objective set, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies. We obtain necessary and sufficient conditions for the solvability of this problem.     

Control of asynchronous spectrum of linear systems with a two-sided dependence of blocks of complete column rank

We consider a linear periodic control system with a two-sided dependence of blocks of complete column rank in the nonstationary component of the coefficient matrix in the critical case. In this case, the nontrivial intersection of vector subspaces formed by linear spans of the columns in the blocks can be arbitrary. We assume that the control is given in the form of feedback linear in the state variables and is periodic with the period of the system. We derive necessary and sufficient conditions for the solvability of the control problem for the asynchronous spectrum, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.     

Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.