Finite Frequency Analysis And Synthesis Based On Generalized KYP Lemma

It is known that all the components of practical control systems, such as sensors,actuators and plants, usually have limited response frequency bandwidth, and that manypractical signals in engineering, such as seismic waves and gust disturbances, have theirmost energy within one or a few frequency ranges. For considering and making use of thefinite frequency properties of systems and signals to improve control system performance,many frequency-domain performance specifications in control system design are usuallyrestricted within some finite frequency range(s). Traditionally, a main approach to deal-ing with restricted frequency-domain specifications is frequency weighting, but this ap-proach has several drawbacks, for instance, the weighted plant has an increased order andthere is no systematic method for selecting weighting functions, etc. To overcome thesedrawbacks, a milestone development in the past ten years is the generalized Kalman–Yakubovich–Popov(KYP) lemma proposed by Iwasaki and Hara, which provides moredirect and effective method for handling restricted frequency-domain specifications. Inview of the fundamental role of the standard KYP lemma in ??∞ control theory—themost rapidly developing branch of robust control theory in the past three decades, thegeneralized KYP lemma has drawn extensive attention since it was proposed, and manyscholars have obtained a huge number of results in the framework of the generalizedKYP lemma. However, there still are some basic but challenging control problems thathave not be solved in this framework, and the existing results are also much conservative.Based on previous works of others, this thesis, in the framework of the generalized KYPlemma, systematically investigates the problems of performance analysis, controller de-sign, filtering and estimation, and model approximation with restricted frequency-domainspecifications, and presents generalized KYP lemma based new methods for system anal-ysis and synthesis. Meanwhile, part of the developed works in this thesis is applied tosome engineering problems, for instance, active structural control of wind turbines.Chapter 1 introduces the background of finite frequency analysis and synthesis prob-lems, focusing on analyzing the limitations of the existing works in the framework ofthe generalized KYP lemma. Chapter 2 and Chapter 3, based on the generalized KYPlemma, investigate controller design problems and further extend the generalized KYPlemma to two-dimensional systems. For the static output-feedback control problem withrestricted frequency-domain specifications, Chapter 2 proposes a “two-stage” controllerdesign method based on the generalized KYP lemma, and constructs two heuristic iter-ative algorithms for controller design and initial value optimization, respectively, whichare further extended to the robust output-feedback control problem. The developed de-sign method fills the gap that there is no static output-feedback controller design resultbased on the generalized KYP lemma, and it also provides a useful guide for developingnew methods for computing reduced-order filters and reduced-order models. Chapter 3presents a novel two-dimensional generalized KYP lemma for the two-dimensional FMmodel, which is further applied to the generalized positive real control problem, lead-ing to two controller design methods for the state-feedback law and the dynamic output-feedback control law, respectively. The developed new generalized KYP lemma unifiesthe existing bounded real lemma and positive real lemma for the FM model, and can serveas a fundamental tool for other synthesis problems of two-dimensional systems.Chapter 4 is concerned with generalized ??∞ filtering problems of linear systems.For the reduced-order filter design problem, by combining a matrix separation techniqueand the “two-stage” idea, iterative algorithms are first constructed for filter design andauxiliary matrix optimization, respectively, based on which, a generalized ??∞ strategyis further developed for solving the channel equalization problem. For the generalized??∞ filtering problem of two-dimensional systems, it is first revealed that the existingtwo-dimensional bounded real lemmas are rather conservative because they neglect thestructural property of the FM model, and then using the two-dimensional generalizedKYP lemma developed in Chapter 3, two filter design methods in terms of linear matrixinequality are derived. Compared with the traditional ??∞ filter design method, the de-veloped approaches can not only make use of the finite frequency properties of signalsfor improving filter performance, but even for standard ??∞ filtering problems, can alsoprovide less conservative filter designs.Chapter 5 focuses on studying another basic problem in control theory, that is, modelapproximation. First, for original linear systems that are passive, by applying the general-ized KYP lemma and the “two-stage” idea, an iterative algorithm is proposed for comput-ing reduced-order models, which not only optimizes the generalized ??∞ error level, andalso preserves the passivity property of the original systems. Second, for single-input-single-output transfer functions, based on the polynomial version of the generalized KYPlemma, a matrix inequality condition with fewer variables is obtained for the existence ofan approximation transfer function, and an effective algorithm is further constructed foroptimizing the approximation transfer function. In addition, using the developed approxi-mation algorithm, a class of digital filter design problems are formulated and solved in thetransfer function approximation framework. As a typical application of the generalizedKYP lemma, the presented design results of digital filters not only demonstrate the ef-fectiveness of the developed model approximation approaches, but also extend the designmethod presented by Iwasaki and Hara from digital filters with finite impulsive responseto those with infinite impulse response.Chapter 6 applies the generalized KYP lemma to stability analysis and stabilizationproblems of time-delay systems. First, by incorporating a frequency-partitioning tech-nique and the generalized KYP lemma, two linear matrix inequality criteria are proposedfor delay-independent stability of time-delay systems, which, for the cases of a singledelay and multiple commensurate delays, are necessary and sufficient. With the stabil-ity analysis results in hand, sufficient conditions are further derived for the existence ofstate-feedback stabilization controllers for time-delay systems. On one hand, the pro-posed stability criteria overcome or relax the conservatism of the existing sufficient sta-bility conditions; on the other hand, they also features the merit of extendability to othersynthesis problems of time-delay systems.Chapter 7 presents a study of the active structural control problem of floating windturbines, which applies the controller design method developed in Chapter 2 to optimiza-tion of structural controller parameters. For the NREL 5-MW wind turbine model in-stalled on a barge platform, a linear design method with three degrees of freedom(DOFs)are first identified from the input-output data that are generated from a high-fidelity sim-ulator. Based on the obtained design model, two ??∞ active structural controllers withdifferent gains are designed by the method presented in Chapter 2. Through modal analy-sis for the open-loop and closed-loop design models, the qualitative relationship betweenthe local magnitude of the design model and the controller gain is revealed, which in-terprets the possibility of achieving a compromise between the controller gain and thewave disturbance attenuation performance by the generalized ??∞ design method. Fi-nally, the effectiveness of the designed controllers are verified by simulation experimentson the full-DOF nonlinear wind turbine model. The work in this part is an exploratoryapplication of the generalized KYP lemma, especially the developed controller designapproach, in engineering practice, which not only further demonstrates the flexibility andeffectiveness of the generalized KYP lemma in dealing with restricted frequency-domainspecifications, but also provides useful theoretical evidences that can be referred to forstructural design and control of wind turbines.