Approximate controllability of stochastic delay differential systems driven by Poisson jumps with instantaneous and noninstantaneous impulses

Control systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed-point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results.     

Guaranteed roll-off in a class of high-gain feedback design problems

A number of synthesis problems associated with (almost) disturbance decoupling by state or measurement feedback is considered. Starting from a mathematical definition of the notion of high-frequency roll-off, known results on the solvability of these problems are generalized to the situation in which we require their solvability together with a certain guaranteed roll-off between disturbance and control. The conditions are formulated in terms of the solvability and approximate solvability of certain matrix equations in rational functions.     

一类可对称化矩阵反问题的最小二乘解

1.引言 用Rn×m,ORn×n,SRn×n及ASRn×n分别表示n×m实矩阵,n阶实正交矩阵,n阶实对称矩阵和n阶实反对称矩阵的全体组成的集合.用S⊥表示集合S的正交补,A(?)B表示A和B的正交直和.设A,B∈Rn×m,定义A与B的内积为     

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.     

一类非线性奇异控制系统的块解耦

讨论了在正则状态下反馈作用下，一类非线性奇异控制系统的输入输出块解耦问题，首先给一种算法，然后利用该算法导出系统输入块解耦问题可解的充分必要条件。     

Semi-global robust output regulation for a class of singular nonlinear systems via nonlinear internal model

Most of existing results on robust output regulation problem of singular nonlinear systems are limited to local solutions. In this paper, the semi-global robust output regulation problem for a class of singular nonlinear systems is investigated by using a nonlinear internal model. Attaching a nonlinear internal model to the singular nonlinear system yields an augmented singular nonlinear system whose semi-global robust stabilisation solution leads to the solution of the semi-global robust output regulation problem of the original singular nonlinear system. The solvability conditions of the semi-global output regulation problem are established by addressing the solvability of the robust stabilisation problem of augmented singular nonlinear system. Finally, a numerical simulation example is used to illustrate the design of the semi-global regulator for the singular nonlinear systems.     

Discrete‐time mixed ℋ︁2/ℋ︁∞ nonlinear filtering

In this paper, we consider the discrete‐time mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Necessary and sufficient conditions for the solvability of this problem with a finite‐dimensional filter are given in terms of a pair of coupled discrete‐time Hamilton–Jacobi‐Isaac's equations (DHJIE) with some side‐conditions. For linear systems, it is shown that these conditions reduce to a pair of coupled discrete‐time algebraic‐Riccati‐equations (DAREs) or a system of linear matrix inequalities (LMIs) similar to the ones for the control case. Both the finite‐horizon and infinite‐horizon problems are discussed. Moreover, sufficient conditions for approximate solvability of the problem are also derived. These solutions are especially useful for computational purposes, considering the difficulty of solving the coupled DHJIEs. An example is also presented to demonstrate the approach. Copyright © 2010 John Wiley & Sons, Ltd.     

GLOBAL ROBUST SERVOMECHANISM OF A CLASS OF NONLINEAR SYSTEMS USING OUTPUT FEEDBACK

The robust servomechanism problem (alternatively, output regulation problem) of the class of nonlinear systems in lower triangular form has been extensively studied in recent years. The semi‐global solution was first given by either state feedback or output feedback. The global solution by state feedback was given very recently. However, the global solution by output feedback has long been an open problem. In this paper, we present a set of solvability conditions of the global robust servomechanism problem for this class of nonlinear systems by output feedback.     

Simultaneous decoupling and disturbance rejection—a structural approach

The combined problem of disturbance rejection and decoupling for linear systems is considered. New necessary and sufficient conditions are derived for the existence of a state feedback controller that rejects the disturbances and simultaneously decouples the closed-loop system. These conditions are simple, intuitively appealing and are expressed in terms of system structural invariants such as infinite zero orders. Then we consider the disturbance rejection with simultaneous decoupling problem for structured linear systems. Necessary and sufficient conditions for the generic solvability of this problem are provided. They are expressed in terms of input-output path length comparisons on the system associated graph.     

Finite settling time stabilisation for multivariable discrete-time systems: a polynomial equation approach

The multivariable case of Finite Settling Time Stabilisation (FSTS) of linear discrete-time systems is considered in this paper. An algebraic approach is adopted which leads to the solution of a polynomial matrix Diophantine equation. This gives rise to the parametrisation of all FSTS controllers in a Ku?era–Youla–Bonjiorno sense and the FSTS problem is further reduced to a linear algebra problem over the real numbers. Subsequently, the family of all causal FSTS controllers is parametrised, and necessary and sufficient conditions for strong FSTS (stable controllers) are derived. The minimal McMillan degree solution and minimal complexity controllers are examined and new bounds are given. The analysis provides the means for the parametrisation of families of FSTS controllers with certain complexity. Finally the problems of tracking, disturbance rejection and partially assigned dynamics in FST sense are considered and conditions for their solvability are given.     

线性时滞系统基于观测器的最优输出跟踪控制器近似设计

研究线性时滞系统的最优输出跟踪控制器近似设计过程.首先根据原系统最优输出跟踪控制问题构造了两个分别具有已知初始条件和终端条件的微分方程迭代序列,并证明它们一致收敛于原问题的最优解.然后通过对该解序列的有限次迭代,得到最优输出跟踪控制问题的一个近似解,进一步给出一个计算近似最优输出跟踪控制律的算法.最后通过构造降维参考输入观测器解决了最优输出跟踪控制器中前馈项的物理可实现问题.仿真结果表明该方法是有效的,且易于实现.     

Solution and stability of the damped Mathieu equation by the Wentzel-Kramers-Brillouin approach

An approximate solution for a linear, second-order, time-varying differential equation, which specializes to the damped Mathieu equation, is derived by using the Wentzel-Kramers-Brillouin approach. The conditions for the approximate solution to be valid are used as a stability criterion. The damped Mathieu equation is considered as a special case.     

The solution set of the mixed ls-tls problem

Although the least squares (LS) problem and the total least squares (TLS) problem have received much attention, the mixed LS-TLS problem still remains to be investigated. In this paper, we study the necessary and sufficient solvability conditions for the mixed LS-TLS problem in Frobenius norm. The solution set, the minimum Frobenius norm solution and an algorithm to obtain the solution are also given. Further we propose the mixed LS-TLS problem in spectral norm and present the solvability condition.     

Sensor Location for Diagnosis in Linear Systems: A Structural Analysis

We consider here the fault detection and isolation (FDI) problem for linear systems. We are interested in designing a set of observer-based residuals, in such a way that the transfer from the faults to the residuals is diagonal and the transfer from the disturbances to the residuals is zero. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. This problem can be solved in terms of the graph that can be associated in a natural way with a structured system. When the FDI solvability conditions are not satisfied, we assume that internal variables can be measured at a cost and look into the question of wether the problem is solvable with these new measurements. We give solvability conditions for a solution with a minimal number of additional sensors and among such solutions provide a minimal cost solution for the sensor location problem under consideration. We pay particular attention to the internal analysis of the system, and we propose a structural decomposition of the system associated graph based on some particular separators. This analysis leads to the definition of a reduced system. We prove that some potential additional sensors are inefficient for solving our FDI problem and that the FDI problem can be solved using only measurements on the reduced system     

Sufficient and necessary conditions for the solvability of the state feedback regulation problem

In this paper, we discuss the state feedback output regulation problem (SFRP) for infinite‐dimensional linear control systems with infinite‐dimensional exosystems. Under the polynomial stabilizability assumption, sufficient and necessary conditions are given for the solvability of the SFRP. The solvability of this problem is characterized in terms of the solvability of a pair of linear regulator equations. An application of the solvability of the SFRP for polynomial stable SISO system is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Solving Initial Value Problems for Ordinary Differential Equations by two approaches: BDF and piecewise-linearized methods

Many scientific and engineering problems are described using Ordinary Differential Equations (ODEs), where the analytic solution is unknown. Much research has been done by the scientific community on developing numerical methods which can provide an approximate solution of the original ODE. In this work, two approaches have been considered based on BDF and Piecewise-linearized Methods. The approach based on BDF methods uses a Chord-Shamanskii iteration for computing the nonlinear system which is obtained when the BDF schema is used. Two approaches based on piecewise-linearized methods have also been considered. These approaches are based on a theorem proved in this paper which allows to compute the approximate solution at each time step by means of a block-oriented method based on diagonal Padé approximations. The difference between these implementations is in using or not using the scale and squaring technique.Five algorithms based on these approaches have been developed. MATLAB and Fortran versions of the above algorithms have been developed, comparing both precision and computational costs. BLAS and LAPACK libraries have been used in Fortran implementations. In order to compare in equality of conditions all implementations, algorithms with fixed step have been considered. Four of the five case studies analyzed come from biology and chemical kinetics stiff problems. Experimental results show the advantages of the proposed algorithms, especially when they are integrating stiff problems.     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstract second-order infinite-dimensional dynamical systems is considered. It is proved using the frequency-domain method, that approximate controllability of second-order system can be verified by the approximate controllability conditions for the corresponding simplified first-order system. General results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends earlier results on approximate controllability of second-order abstract dynamical systems.     

Relaxed fault detection and isolation: An application to a nonlinear case study

Given a number of possibly concurrent faults (and disturbances) that may affect a nonlinear dynamic system, it may not be possible to solve the standard fault detection and isolation (FDI) problem, i.e., to detect and isolate each single fault from all other, possibly concurrent faults and disturbances, due to the violation of the available necessary conditions of geometric nature. Motivated by a robotic application where this negative situation structurally occurs, we propose some relaxed formulations of the FDI problem and show how necessary and sufficient conditions for their solution can be derived from those available for standard FDI. The design of a hybrid residual generator follows directly from the fulfillment of the corresponding solvability conditions. In the considered nonlinear case study, a robotic system affected by possible actuator and/or force sensor faults, we detail the application of these results and present experimental tests for validation.     

Simple Group Pursuit Subject to Phase Constraints and Data Delay

The problem of the simple pursuit of a single evader by a group of pursuers subject to phase constraints is considered. It is assumed that the pursuers receive information about the evader’s control and affect the control system with a time delay. Sufficient conditions for the solvability of the pursuit problem are obtained.