On lossless negative imaginary systems

The paper is concerned with the notion of lossless negative imaginary systems and their stabilization using strictly negative imaginary controllers through positive feedback. Firstly, the concept of lossless negative imaginary transfer functions is introduced and some properties of such transfer functions are studied. Secondly, a Lossless Negative Imaginary Lemma is given which establishes conditions on matrices appearing in a minimal state-space realization that are necessary and sufficient for a transfer function to be lossless negative imaginary. Thirdly, a necessary and sufficient condition is provided for the stabilization of a lossless negative imaginary system by a strictly negative imaginary controller. Finally, a flexible structure example is presented to illustrate the theory.     

A generalized negative imaginary lemma and Riccati-based static state-feedback negative imaginary synthesis

In this paper, we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition. Then, an algebraic Riccati equation method is given to determine if a system is negative imaginary. Also, a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property. The controller synthesis procedure is based on the proposed negative imaginary lemma. Using this procedure, the closed-loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty, such as in the case of unmodeled spill-over dynamics in a lightly damped flexible structure. A numerical example is presented to illustrate the usefulness of the results.     

Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response

A necessary and sufficient condition, expressed simply as the dc loop gain (i.e., the loop gain at zero frequency) being less than unity, is given in this note to guarantee the internal stability of a feedback interconnection of linear time-invariant (LTI) multiple-input multiple-output systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise, for example, when considering transfer functions from force actuators to colocated position sensors, and are commonly important in, for example, lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem, which refers to the stability of feedback interconnections of positive real (or passive) systems. A complete state-space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided.     

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.     

Strictly positive real matrices and the Lefschetz-Kalman-Yakubovichlemma

The authors give necessary and sufficient conditions in the frequency domain for rational matrices to be strictly positive real. Based on this result, the matrix form of the Lefschetz-Kalman-Yakubovich lemma is proved, which gives necessary and sufficient conditions for strictly positive real transfer matrices in the state-space realization form     

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state-space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. By a particular transform of these conditions, a direct algorithm for testing positive realness is determined that requires only checking a set of simple algebraic conditions. This provides an alternative procedure to the positive real lemma and to the s-domain inequalities. Based on this algorithm, a synthesis of a positive real system via output feedback is presented     

SIMO和MISO系统绝对稳定性问题的一个猜想和几个推论

首先分析了具有多个非线性特性的 SIMO 和 MISO 系统的绝对稳定性问题, 指出应用已知的频域判据来解决上述问题很难奏效. 然后, 基于所有孤立部分传递函数都正实的充分必要条件给出了上述系统为稳定的一个猜想, 当传递函数的零极点都位于虚轴上时, 由这一猜想得到了一个已知的结论; 当传递函数的零极点都位于实轴上时, 由这一猜想得到了一个新的结论, 本文证明该结论是正确的; 最后, 根据这一猜想, 给出了传递函数极点位于复平面的一个例子, 它涉及到一类系数矩阵为时变正定矩阵的振动方程的稳定性问题, 值得去深入研究.     

New versions of Barbalat’s lemma with applications

This note presents a set of new versions of Barbalat’s lemma combining with positive (negative) definite functions. Based on these results, a set of new formulations of Lyapunov-like lemma are established. A simple example shows the usefulness of our results.     

On Kalman-Yakubovich-Popov lemma for stabilizable systems

The Kalman-Yakubovich-Popov (KYP) lemma has been a cornerstone in system theory and network analysis and synthesis. It relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain. This note proves that the KYP lemma is also valid for realizations which are stabilizable and observable     

Low frequency positive real control for delta operator systems

A strictly positive real control problem for delta operator systems in a low frequency range is presented by using the generalized Kalman–Yakubovic?–Popov lemma. The objective of the strictly positive real control problem is to design a controller such that the transfer function is strictly positive real and the resulting closed-loop system is stable. Sufficient conditions for the low frequency strictly positive real controller of the closed-loop delta operator systems are presented in terms of solutions to a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness and potential for the developed techniques.     

The positive real lemma and construction of all realizations of generalized positive rational functions

We here extend the well known positive real lemma (also known as the Kalman-Yakubovich-Popov lemma) to a complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through a static output feedback, to be generalized positive.     

On the bounded real lemma

In this paper we deal with special generalizations of the well-known bounded real lemma. We develop general statements for comparison of rational transfer functions in the sense of quadratic forms on the imaginary axis. The main results can be expressed by means of Hamiltonian matrices.     

Eigenvalue-based characterization and test for positive realness ofscalar transfer functions

An eigenvalue-based characterization of positive realness of transfer functions of a single-input single output time-invariant linear system is derived. Based on this characterization, we propose an efficient computational procedure to determine if a given transfer function is positive real. The input for this eigenvalue-based test is any given, not necessarily minimal, state-space representation of the linear system. The test only involves standard matrix computations, such as computing eigenvalues of a matrix or a matrix pencil. Results of numerical experiments are reported     

Least squares based iterative identification for a class of multirate systems

This paper studies modeling and identification problems for multi-input multirate systems with colored noises. The state-space models are derived for the systems with different input updating periods and furthermore the corresponding transfer functions are obtained. To solve the difficulty of identification models with unmeasurable noises terms, the least squares based iterative algorithm is presented by replacing the unmeasurable variables with their iterative estimates. Finally, the simulation results indicate that the proposed iterative algorithm has advantages over the recursive algorithms.     

Reciprocally convex approach to stability of systems with time-varying delays

Whereas the upper bound lemma for matrix cross-product, introduced by Park (1999) and modified by Moon, Park, Kwon, and Lee (2001), plays a key role in guiding various delay-dependent criteria for delayed systems, the Jensen inequality has become an alternative as a way of reducing the number of decision variables. It directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities, resulting in a linear combination of positive functions weighted by the inverses of convex parameters. This paper suggests the lower bound lemma for such a combination, which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.     

Robust strict positive realness: new results for intervalplant-controller families

A frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions. Based on simple frequency domain properties of transfer functions, the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness. The main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 Kharitonov segments of the interval plant. The argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions, such as robust stability of H_∞-norm computation, can be deduced from a fixed number of segments of transfer functions of the family     

Minimality and local state decompositions of a nonlinear state space realization using energy functions

In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state-space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies minimality. Using the nonlinear analogue of the Kalman decomposition, connections are then established between minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.