A Lyapunov Variable-Free KYP Lemma for SISO Continuous Systems

This technical note proposes a novel frequency-selective Kalman-Yakubovich-Popov (FS–KYP) lemma for analysis of single-input single-output (SISO) continuous systems. In contrast to existing approaches, the proposed method only uses a minimal number of variables due to the absence of Lyapunov variables in semidefinite programming (SDP) formulation. The SDP formulation is extended to polytopic uncertain systems also without any additional variable and yields an efficient method for computation of the $H_{infty} $ gain for polytopic systems. The viability of the proposed method are demonstrated through several numerical examples.      

Sum-of-Squares Decomposition via Generalized KYP Lemma

The Kalman–Yakubovich–Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) of a proper rational function and a linear matrix inequality (LMI). A recent result generalized the KYP lemma to characterize an FDI of a possibly nonproper rational function on a portion of a curve on the complex plane. This note examines implications of the generalized KYP result to sum-of-squares (SOS) decompositions of matrix-valued nonnegative polynomials of a single complex variable on a curve in the complex plane. Our result generalizes and unifies some existing SOS results, and also establishes equivalences among FDI, LMI, and SOS.      

New Condition for FD,All Filters,and New KYP Lemma

This paper presents a necessary and sufficient condition for the fault detection (FD) problem, under a quite general and natural assumption. The condition is actually a constant inertia on the extended imaginary axis (EIA) of a para‐hermitian rational matrix (RM). An observer solution is given. Another two main results are the presentation of all problem solutions, and a necessary and sufficient condition for constant inertia on the EIA or in a given frequency region, for a class of para‐hermitian RMs, in terms of linear matrix inequalities. The latter result can be regarded as a generalization of the Kalman‐Yakubovich‐Popov (KYP) lemma.     

Experimentally verified generalized KYP Lemma based iterative learning control design

This paper considers iterative learning control law design for plants modeled by discrete linear dynamics using repetitive process stability theory. The resulting one step linear matrix inequality based design produces a stabilizing feedback controller in the time domain and a feedforward controller that guarantees convergence in the trial-to-trial domain. Additionally, application of the generalized Kalman–Yakubovich–Popov (KYP) lemma allows a direct treatment of differing finite frequency range performance specifications. The results are also extended to plants with relative degree greater than unity. To support the algorithm development, the results from an experimental implementation are given, where the performance requirements include specifications over various finite frequency ranges.     

Integrated servo-mechanical design of high-performance mechatronics using generalized KYP Lemma

High-performance mechatronics have specifications which are difficult to achieve when the mechanical plant is non-minimum phase and a low-order controller is used. In this paper, an integrated servo-mechanical design algorithm is proposed for systematic finite frequency redesign of a mechanical plant using the generalized Kalman–Yakubovic–Popov Lemma. The synthesis of a minimum phase plant is carried out based on a predesigned low-order controller, positive realness constraints, and performance specifications of the overall control system. Our simulation results using the proposed algorithm achieve a high-bandwidth control system with disturbance attenuation capabilities at the phase-stabilized resonant modes of the plant with the low-order controller.     

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.     

A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities

Absolute stability criteria for systems with multiple hysteresis non-linearities are given in this paper. It is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem. If the linear subsystem satisfies a particular linear matrix inequality and a simple residue condition, then, as is proven, the non-linear system will be asymptotically stable. The main stability theorem is developed using a combination of passivity, Lyapunov and Popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non-linear closed loop system. The invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses. The class of non-linearities covered by the analysis is very general and includes multiple slope-restricted memoryless non-linearities as a special case. Simple numerical examples are used to demonstrate the effectiveness of the new analysis in comparison to other recent results, and graphically illustrate state asymptotic stability.     

A generalized KYP lemma based approach for disturbance rejection in data storage systems

This paper is concerned with disturbance rejection for data storage systems which applies microactuator to expand its servo bandwidth. An approach based on the generalized Kalman–Yakubovic–Popov (KYP) lemma together with the Youla parameterization is proposed which allows us to suppress disturbances of particular frequencies within or beyond the servo bandwidth via convex optimization. The rejection of narrowband high-frequency disturbances is achieved by using this approach. Simulation together with implementation results demonstrate the simplicity and effectiveness of the design method as compared with the traditional frequency weighting method.     

A note on Reusken's Lemma

A modification of the Lemma of Reusken is given. It allows us to improve the estimate of the smoothing property in cases where the contraction number of the iteration is small. This is of importance for robust multi-grid methods. Moreover, we describe a simple semi-iterative smoother with better asymptotic behaviour than for the stationary iterative smoother.     

The Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems

In this paper we generalize the Kalman-Yakubovich-Popov Lemma to the Pritchard-Salamon class of infinite-dimensional systems, i.e. systems determined by semigroups of operators on a Hilbert space with unbounded input and output operators. At the same time we prove a comparison theorem for Riccati equations of the same class.     

An anti-windup technique for LMI regions

The anti-windup problem seeks to minimize closed loop performance deterioration due to input nonlinearities, such as saturation, for a given linear time-invariant plant and controller. This paper presents a linear matrix inequality (LMI) based method that attempts to minimize performance deterioration while explicitly restricting the anti-windup closed loop dynamics. The restriction placed on the dynamics is described via LMI regions, which is a form of regional pole placement. Finally, the techniques discussed in this paper are demonstrated on an electro-hydraulic testbed.     

A new LMI condition for robust stability of discrete-time uncertain systems

In this paper, a new robust stability condition is derived for uncertain discrete-time linear systems with convex polytopic uncertainties. The condition is expressed in terms of a set of linear matrix inequalities (LMIs) involving only the vertices of the polytope domain. It enables us to determine robust stability of uncertain systems easily by solving some LMIs. A rigorous proof is given to show that an interesting result appeared recently is a special case of the proposed condition. Numerical examples also demonstrate the merit of the present condition in the aspect of conservativeness over other results in the literature.     

LMI designmethod for networked-based PID control

In this paper, we propose a methodology for the design of networked PID controllers for second-order delayed processes using linear matrix inequalities. The proposed procedure takes into account time-varying delay on the plant, time-varying delays induced by the network and packed dropouts. The design is carried on entirely using a continuous-time model of the closed-loop system where time-varying delays are used to represent sampling and holding occurring in a discrete-time digital PID controller.     

L分析的线性矩阵不等式方法及其优化算法

通过采用S—过程和投影引理，得到了结构奇异值上界的LMI判据，该判据是基于状态空间描述的，从而消除了频率扫描过程和频率响应曲线拟合过程，并具有较好的数值性态，以该判据为基础，给出了计算结构奇异值上界的优化投影迭代算法，并将该方法应用于基准测试系统和典型电力系统，以验证其有效性，数值结果表明，该方法与经典频域方法和状态空间方法相比具有更好的求解效率。     

LMI tools for eventually periodic systems

This paper is focused on the concept of an eventually periodic linear discrete-time system. We derive a necessary and sufficient analysis condition for checking open-loop stability and performance of such systems, and use this to derive exact controller synthesis conditions given eventually periodic plants. All the conditions derived are provided in terms of semi-definite programming problems. The motivation for this work is controlling nonlinear systems along prespecified trajectories, notably those which eventually settle down into periodic orbits and those with uncertain initial states.     

A Mechanized Proof of the Basic Perturbation Lemma

We present a complete mechanized proof of the result in homological algebra known as basic perturbation lemma. The proof has been carried out in the proof assistant Isabelle, more concretely, in the implementation of higher-order logic (HOL) available in the system. We report on the difficulties found when dealing with abstract algebra in HOL, and also on the ongoing stages of our project to give a certified version of some of the algorithms present in the Kenzo symbolic computation system. J. Aransay was partially supported by Ministerio de Educación y Ciencia, MTM2006/06513, and by Gobierno de La Rioja ANGI2005/19 and J. Rubio was partially supported by Ministerio de Educación y Ciencia, MTM2006/06513, and by Gobierno de La Rioja ANGI2005/19.     

A Structure Exploiting Preprocessor for Semidefinite Programs Derived From the Kalman-Yakubovich-Popov Lemma

Semidefinite programs derived from the Kalman-Yakubovich-Popov (KYP) lemma are quite common in control and signal processing applications. The programs are often of high dimension which makes them hard or impossible to solve with general-purpose solvers. Here we present a customized preprocessor, KYPD, that utilizes the inherent structure of this particular optimization problem. The key to an efficient implementation is to transform the optimization problem into an equivalent semidefinite program. This equivalent problem has much fewer variables and the matrices in the linear matrix inequality constraints are of low rank. KYPD can use any primal-dual solver for semidefinite programs as an underlying solver.     

Generalized KYP lemma: unified frequency domain inequalities with design applications

The celebrated Kalman-Yakubovic/spl caron/-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an S-procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects-the frequency range and the class of systems-and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers.     

A novel criterion for nonlinear time-delay systems using LMI fuzzy Lyapunov method

This study presents a kind of fuzzy robustness design for nonlinear time-delay systems based on the fuzzy Lyapunov method, which is defined in terms of fuzzy blending quadratic Lyapunov functions. The basic idea of the proposed approach is to construct a fuzzy controller for nonlinear dynamic systems with disturbances in which the delay-independent robust stability criterion is derived in terms of the fuzzy Lyapunov method. Based on the robustness design and parallel distributed compensation (PDC) scheme, the problems of modeling errors between nonlinear dynamic systems and Takagi–Sugeno (T–S) fuzzy models are solved. Furthermore, the presented delay-independent condition is transformed into linear matrix inequalities (LMIs) so that the fuzzy state feedback gain and common solutions are numerically feasible with swarm intelligence algorithms. The proposed method is illustrated on a nonlinear inverted pendulum system and the simulation results show that the robustness controller cannot only stabilize the nonlinear inverted pendulum system, but has the robustness against external disturbance.     

LMI Consensus Condition for Discrete-time Multi-agent Systems

This paper examines a consensus problem in multi-agent discrete-time systems, where each agent can exchange information only from its neighbor agents. A decentralized protocol is designed for each agent to steer all agents to the same vector. The design condition is expressed in the form of a linear matrix inequality. Finally, a simulation example is presented and a comparison is made to demonstrate the effectiveness of the developed methodology.