Interval frequency negative imaginary systems

This paper studies interval frequency negative imaginary (IFNI) systems based on minimal state-space realisations. Firstly, the concept of the IFNI transfer functions is introduced, and the relationship between the IFNI transfer functions and the interval frequency positive real transfer functions is established. Secondly, based on the generalised KYP lemma, a necessary and sufficient condition is derived for IFNI transfer functions with minimal state-space realisation in terms of linear matrix inequalities. Our results coincide with the existing negative imaginary lemma when the interval frequency set becomes the positive frequency set. Also, a time domain interpretation of the IFNI property is provided in terms of the system input, output and state. Finally, two examples are used to illustrate the developed theory.     

N-D representation and generalised Kalman–Yakubivich–Popov lemma of spatially interconnected systems with interconnected chains

This paper presents a multidimensional (N-D) hybrid Roesser model for a class of spatially interconnected systems (SISs) with interconnected chains. A generalised Kalman–Yakubivich–Popov (KYP) lemma for N-D hybrid Roesser model is developed, which plays a significant role in the system control and synthesis. As the application of the developed generalised KYP lemma, robust finite-frequency H_∞ filter problem for SISs with interconnected chains is studied. Moreover, it is shown that the developed finite-frequency filter design methods are generally less conservative than the entire frequency ones. Finally, a practical example is provided to validate the effectiveness of the proposed method.     

Nonlinear stochastic passivity,feedback equivalence and global stabilization

This paper addresses several important issues including stochastic passivity, feedback equivalence and global stabilization for a class of nonlinear stochastic systems. Based on a nonlinear stochastic Kalman–Yacubovitch–Popov (KYP) lemma, we investigate the relationship between a stochastic passive system and the corresponding zero‐output system. Different from the deterministic case, it is shown for the first time that feedback equivalence to a stochastic passive system requires a strong minimum‐phase condition, not the minimum‐phase one. Following the stochastic passivity theory, global stabilization results are established for a class of nonlinear stochastic systems with relative degree 1≤r<n. An example is presented to illustrate the effectiveness of our results. Copyright © 2011 John Wiley & Sons, Ltd.     

Global state and feedback equivalence of nonlinear systems

The problem of finding global state space transformations and global feedback of the form u(t)= α(x) + ν(t) to transform a given nonlinear system to a controllable linear system on Rⁿ or on an open subset of Rⁿ, is considered here. We give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem.     

H ∞ model reduction for discrete‐time positive systems with inhomogeneous initial conditions

This paper addresses the H _∞ model reduction problem of discrete‐time positive linear systems with inhomogeneous initial conditions. For an asymptotically stable positive system with non‐zero initial condition, our goal is to approximate it by a reduced‐order initial‐valued positive system without introducing significant error. We establish a necessary and sufficient condition for the existence of a desired reduced‐order model such that the output error between the original system and the reduced‐order one is bounded by a weighted sum of the magnitude of the input and that of the initial condition. Moreover, based on congruent transformation and the dual form of bounded real lemma, several equivalent conditions are derived in terms of LMIs and an iterative convex optimization algorithm is developed accordingly. Finally, an illustrative example is presented to show the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

H∞ Control for Continuous‐Time Mean‐Field Stochastic Systems

An H_∞‐type control is considered for mean‐field stochastic differential equations (SDEs) in this paper. A stochastic bounded real lemma (SBRL) is proved for mean‐field stochastic continuous‐time systems with state‐ and disturbance‐dependent noise. Based on SBRL, a sufficient condition is given for the existence of a stabilizing H_∞ controller in terms of coupled nonlinear matrix inequalities.     

Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests

We present an algorithm which robustly computes the intersection curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n+1 unknowns. The solution of such a system is typically a curve in Rⁿ⁺¹. This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in Rⁿ⁺¹. Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible applications of the solver, namely solving the surface–surface intersection problem, computing 3D trisector curves, flecnodal curves or kinematic simulations in 3D are also discussed.     

Output Feedback H∞ Control for Discrete‐time Mean‐field Stochastic Systems

This paper is to consider dynamic output feedback H_∞ control of mean‐field type for stochastic discrete‐time systems with state‐ and disturbance‐dependent noise. A stochastic bounded real lemma (SBRL) of mean‐field type is derived. Based on the SBRL, a sufficient condition with the form of coupled nonlinear matrix inequalities is derived for the existence of a stabilizing H_∞ controller. Moreover, a numerical example is given to examine the effectiveness of the theoretical results.     

An improved delay‐dependent bounded real lemma (BRL) and H∞ controller synthesis for linear neutral systems

In this paper, a novel delay‐dependent bounded real criterion and an improved sufficient condition are derived for the design of an H_∞ state‐feedback controller for linear neutral time‐delay systems. On the basis of an augmented Lyapunov‐Krasovskii functional, a new bounded real lemma is introduced in terms of a convex linear matrix inequality (LMI) condition that can be solved using interior point algorithms. The bounded real lemma is extended to obtain a sufficient condition for the existence of a delay‐dependent H_∞ memoryless state‐feedback controller. Neither any model transformation nor bounding of any of the cross terms are utilized while deriving the bounded real lemma. Moreover, the use of any free slack matrix variable approach is avoided to a certain extent in order not to increase the complexity of the synthesis problem. A cone complementary nonlinear minimization algorithm is employed to achieve a feasible solution set for the synthesis conditions. Finally, seven numerical examples are given to illustrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A NEW INTEGRAL INEQUALITY APPROACH TO DELAY‐DEPENDENT ROBUST H∞ CONTROL

A design method for the robust H_∞ control of an uncertain linear system with a time‐varying state delay is proposed. First, an integral inequality that we recently obtained is employed to establish a new delay‐dependent bounded real lemma for a system with a time‐varying delay. The lemma uses neither a model transformation nor a bounding technique for cross terms. Then, the lemma is used in combination with a matrix decomposition method to derive delay‐dependent conditions for the existence of robust H_∞ control based on linear matrix inequalities. Finally, some numerical examples are given to demonstrate the validity of the method.     

有限频域分析与设计的广义KYP引理方法综述

频域方法是控制理论与工程领域的一种基本研究手段,许多控制问题都可归结为有限频域性能指标的分析与综合问题.广义Kalman-Yakubovich-Popov（KYP）引理建立了频域方法（传递函数）与时域方法（状态空间）之间的一座桥梁,成为近年来系统与控制理论领域的研究热点之一.本文首先从信号和系统两个角度阐明有限频域分析与设计的背景和意义,并依次讨论三种主要研究方法（经典控制理论方法、频率加权法和广义性能指标法）各自的优缺点.然后简单介绍广义KYP引理的主体内容,并详细总结当前基于广义KYP引理的有限频域分析与设计的主要方向及研究进展.最后给出在使用广义KYP引理时很重要但容易忽视的几点注记,同时指明该领域目前存在并值得未来进一步研究的关键问题.     

On the Kalman–Yakubovich–Popov lemma for discrete-time positive linear systems: a novel simple proof and some related results

Theorems of alternatives on the feasibility of linear matrix inequalities (LMIs) are used in order to provide novel simple proofs for two considered versions of the Kalman–Yakubovich–Popov (KYP) lemma for discrete-time positive linear systems. Two different and novel recursive methods, to determine whether a positive matrix is or is not Schur, are obtained as an application of an existing connection between the strict inequality version of the KYP lemma for single-input single-output (SISO) discrete-time positive linear systems and a Schur matrix condition. Examples are included which provide illustration on these recursive methods.     

H∞-type control for discrete-time stochastic systems

In this paper we consider discrete-time, linear stochastic systems with random state and input matrices which are subjected to stochastic disturbances and controlled by dynamic output feedback. The aim is to develop an H^∞-type theory for such systems. For this class of systems a stochastic bounded real lemma is derived which provides the basis for a linear matrix inequality (LMI) approach similar to, but more general than the one presented in Reference 1 for stochastic differential systems. Necessary and sufficient conditions are derived for the existence of a stabilizing controller which reduces the norm of the closed-loop perturbation operator to a level below a given threshold γ. These conditions take the form of coupled nonlinear matrix inequalities. In the absence of the stochastic terms they get reduced to the linear matrix inequalities of deterministic H^∞-theory for discrete time systems. Copyright © 1999 John Wiley & Sons, Ltd.     

H∞ filtering for singular Markovian jump systems with time delay

The problem of H_∞ filtering is considered for singular Markovian jump systems with time delay. In terms of linear matrix inequality (LMI) approach, a delay‐dependent bounded real lemma (BRL) is proposed for the considered system to be stochastically admissible while achieving the prescribed H_∞ performance condition. Based on the BRL and under partial knowledge of the jump rates of the Markov process, both delay‐dependent and delay‐independent sufficient conditions that guarantee the existence of the desired filter are presented. The explicit expression of the desired filter gains is also characterized by solving a set of strict LMIs. Some numerical examples are given to demonstrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley & Sons, Ltd.     

H ∞ control for non-linear stochastic systems: the output-feedback case

The H _∞ output-feedback control problem for non-linear stochastic systems is considered. A solution for a large class of non-linear stochastic systems is introduced (including non-linear diffusion systems as a subclass). This solution is based on a bounded real lemma for non-linear stochastic systems that was previously established via a stochastic dissipativity concept. The theory yields sufficient conditions for the closed-loop system to possess a prescribed L ₂-gain bound in terms of two Hamilton Jacobi inequalities: one that is associated with the state feedback part of the problem is n-dimensional (where n is the underlying system's state dimension) and the other inequality that stems from the estimation part is 2n-dimensional. Both stationary and non–stationary systems are considered. Stability of the closed-loop system is established, both in the mean-square and the in-probability senses. As the solution to the Hamilton Jacobi inequalities may, in general, lead to a non–realisable state estimator, a modification of the associated 2n-dimensional Hamilton Jacobi inequality is made in order to circumvent this realisation problem, while preserving the system's L ₂-gain bound. For time-invariant systems, the problem of robust output-feedback is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear state-dependent matrix inequalities.     

Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems

For the general multidimensional perturbed oscillators y^″+Ky=f(y,y^′) with K∈Rd×d, the order conditions for the ARKN methods are presented by X. Wu et al. [Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250–2257]. These methods integrate exactly the multidimensional unperturbed oscillators and are highly efficient when the perturbing function is small. In this paper, we are concerned with the analysis of the concrete multidimensional ARKN methods based on the order conditions for the multidimensional ARKN methods. We extent the matrix K to a general case, in which K is not required to be symmetric. Numerical experiments demonstrate that the novel multidimensional ARKN methods presented in this paper are more efficient compared with some well-know RKN methods in the scientific literature.     

Robust H∞ filtering of stochastic time‐delay systems with state dependent noise

This paper is concerned with the H_∞ filtering design for discrete‐time stochastic time‐delay systems with state dependent noise. A sufficient condition for the existence of H_∞ filter design is presented via linear matrix inequalities. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

广义时滞系统时滞依赖H∞控制的改进结果

讨论广义时滞系统的时滞依赖控制问题. 基于线性矩阵不等式(LMI)方法和增广Lyapunov泛函, 给出保证系统正则、无脉冲、稳定且满足性能指标的时滞依赖有界实引理. 在此引理的基础上, 给出基于严格LMI的状态反馈控制器存在的时滞依赖条件. 同时给出所需状态反馈控制器的明确表示. 数值例子表明本文的结果改进了已有结论的保守性.     

Efficiently computing geodesic offsets on triangle meshes by the extended Xin–Wang algorithm

Geodesic offset curves are important for many industrial applications, such as solid modeling, robot-path planning, the generation of tool paths for NC machining, etc. Although the offset problem is well studied in classical differential geometry and computer-aided design, where the underlying surface is sufficiently smooth, very few algorithms are available for computing geodesic offsets on discrete representation, in which the input is typically a polyline curve restricted on a piecewise linear mesh. In this paper, we propose an efficient and exact algorithm to compute the geodesic offsets on triangle meshes by extending the Xin–Wang algorithm of discrete geodesics. We define a new data structure called parallel-source windows, and extend both the “one angle one split” and the filtering theorem to maintain the window tree. Similar to the original Xin–Wang algorithm, our extended algorithm has an O(n) space complexity and an O(n²logn) asymptotic time complexity, where n is the number of vertices on the input mesh. We tested our algorithm on numerous real-world models and showed that our algorithm is exact, efficient and robust, and can be applied to large scale models with complicated geometry and topology.     

H disturbance attenuation for state delayed systems

In this note, the differential game and dissipation inequality are applied to the disturbance attenuation or H^∞-control for linear systems with delayed state. Firstly, a simple sufficient condition on the existence of a γ-suboptimal H^∞ state feedback controller is given, which is independent of delay, and an observer-based dynamic output feedback solution is presented in terms of Riccati inequalities (or Riccati equations). Secondly, a sufficient condition on the existence of a delay-dependent state feedback is presented and the criterion is presented by a matrix inequality which can be solved by numerical methods.