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Frequency-dependent Switching Control for Disturbance Attenuation of Linear Systems
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Abstract
The generalized Kalman-Yakubovich-Popov lemma as established by Iwasaki and Hara in 2005 marks a milestone in the analysis and synthesis of linear systems from a finite-frequency perspective. Given a pre-specified frequency band, it allows us to produce passive controllers with excellent in-band disturbance attenuation performance at the expense of some of the out-of-band performance. This paper focuses on control design of linear systems in the presence of disturbances with non-strictly or non-stationary limited frequency spectrum. We first propose a class of frequency-dependent excited energy functions (FD-EEF) as well as frequency-dependent excited power functions (FD-EPF), which possess a desirable frequency-selectiveness property with regard to the in-band and out-of-band excited energy as well as excited power of the system. Based upon a group of frequency-selective passive controllers, we then develop a frequency-dependent switching control (FDSC) scheme that selects the most appropriate controller at runtime. We show that our FDSC scheme is capable to approximate the solid in-band performance while maintaining acceptable out-of-band performance with regard to global time horizons as well as localized time horizons. The method is illustrated by a commonly used benchmark model.
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This paper deals with the development of a mixed active‐passive microvibration mitigation solution capable of attenuating the transmitted vibrations generated by reaction wheels to a satellite structure. A dedicated simulation environment, provided by the European Space Agency and Airbus Defence and Space industries, serves as a support for testing the proposed solution at satellite level. This paper covers modeling, control system design, and worst‐case analysis for a typical satellite observation mission that requires high pointing stability. Combined with a novel disturbance model for the reaction wheel perturbations, the pointing performance and stability requirements are reformulated as bounds on the worst‐case system gains. Subsequently, the active microvibration controller is tuned to manage the conflicting design goals and optimize different trade‐offs between robustness and performance. Finally, robust stability margins and worst‐case performance bounds with respect to various system uncertainties, time‐varying reaction wheel spin rates, actuator saturation, and time delays are obtained using the structured singular value, integral quadratic constraints, and time‐domain nonlinear simulations.
This paper investigates a sliding mode control for a wind turbine in finite frequency (FFSMC). The sliding mode control (SMC) method can be designed for wind energy conversion systems. However, the fluctuations of wind speed perhaps reduce the robustness of the SMC. A dynamic compensator is introduced to design the sliding surface in order to overcome the difficulty of dealing with a fluctuation of wind speed, but the fast change of this fluctuation in certain finite ranges can lead to degrade the compensator performance. In order to solve this problem, a finite frequency approach based on the generalised Kalman-Yakubovich-Popov (KYP) lemma is proposed, and the compensator parameters are obtained in terms of linear matrix inequality (LMI) which can be solved efficiently using existing numerical tools. On the other hand, the reaching law is used to reduce the chattering that is produced by the traditional approach of sliding mode. Finally, the simulation results illustrate the effectiveness of the proposed control strategy compared to a non-singular terminal sliding mode control.
The output power of wind farms has significant randomness and variability, which results in adverse impacts on power system frequency stability. This paper extracts wind power fluctuation feature with the HHT (Hilbert-Huang Transform) method. Firstly, the original wind power data was decomposed into several IMFs (Intrinsic Mode Functions) and a tendency component by using the EMD (Empirical Mode Decomposition) method. Secondly, the instantaneous frequency of each IMF was calculated. On this basis, taking a WSCC 9-bus power system as benchmark, the impact on power system frequency caused by wind power fluctuation was simulated in a real-time simulation platform, and the key component which results in the frequency deviation was found. The simulation results validate the wind power fluctuation impacts on frequency deviation, underlying the following study on power system frequency stability under the situation of large-scale intermittent generation access into the grid.
Response of vehicle suspension system has traditionally been studied under the assumption of stationary road excitation. However, road excitation becomes a non-stationary random input in the time domain when the vehicle travels at a varying speed. In this paper, we investigate the non-stationary response of vehicle suspension under random road excitation. Despite its importance and relevant applications, this problem has not been well addressed in the literature. The non-stationary random road excitations in the time domain are computed using a high precision Gaussian-Legendre integration method. The evolutionary spectral theory is then applied to evaluate the response in the time-frequency domain. Numerical results for a quarter-car model are presented and effects of vehicle acceleration and road roughness on the ride comfort, energy dissipation from the suspension, and time-variant half-power frequency band for the constant speed and various constant acceleration cases are examined. The consideration of energy dissipation provides new insights on the relationships between ride comfort and fuel consumption, which is important for designs of optimized energy-ride comfort suspension systems under non-stationary road excitations.
This book focuses on most recent theoretical findings on control issues for active suspension systems. The authors first introduce the theoretical background of active suspension control, then present constrained H∞ control approaches of active suspension systems in the entire frequency domain, focusing on the state feedback and dynamic output feedback controller in the finite frequency domain which people are most sensitive to. The book also contains nonlinear constrained tracking control via terminal sliding-mode control and adaptive robust theory, presenting controller design of active suspensions as well as the reliability control of active suspension systems. The target audience primarily comprises research experts in control theory, but the book may also be beneficial for graduate students alike.
An event-triggered integral sliding mode controller design problem for linear systems with disturbance is investigated in this paper. A time-varying triggering threshold is employed in the event-triggered condition to guarantee that the trajectory of system will enter a bounded region. Two types of integral sliding mode switching functions are established. One is the continuous-state-dependent sliding mode switching function, which is used to analyze the reachability of sliding mode surface. The other is the triggered-state-dependent sliding mode switching function, which is applied to design the event-triggered integral sliding mode controller. Moreover, a sufficient condition of ultimate boundedness for sliding motion is established and a positive lower bound of the internal execution time is ensured. Finally, two examples are presented to illustrate the effectiveness of obtained theoretical results.
This paper presents a reliable control strategy for quadrotors to achieve a satisfactory tracking performance and provide a stable yaw angle control in the presence of wind disturbances. Based on the prelinearizing transformation and singular perturbation techniques, a quadrotor flight system is decomposed into two interconnected subsystems in different time-scales: 1) the angular rotations and 2) the linear translations. For the fast subsystem running as the inner loop, finite frequency H∞ control is used to enhance robustness and provide adequate decoupling such that each translational dynamic is controlled independently. For the slow subsystem running as the outer loop, H∞ loop shaping control via proportion-integration-differentiation control is used to achieve a desired servo performance even in the presence of wind disturbance. Simulations are performed under different wind conditions to validate the effectiveness of the new control strategy.
Manual flight control system design for fighter aircraft is one of the most demanding problems in automatic control. Fighter aircraft dynamics generally have highly coupled uncertain and nonlinear dynamics. Multivariable control design techniques offer a solution to this problem. Robust Multivariable Flight Control provides the background, theory and examples for full envelope manual flight control system design. It gives a versatile framework for the application of advanced multivariable control theory to aircraft control problems. Two design case studies are presented for the manual flight control of lateral/directional axes of the VISTA-F-16 test vehicle and an F-18 trust vectoring system. They demonstrate the interplay between theory and the physical features of the systems.
Because of unpredictable fluctuations in wind power output caused by sudden changes in weather conditions, operations that balance supply and demand in power systems will gradually become more difficult with the massive deployment of wind power generation. Therefore, it is necessary to quantitatively evaluate wind power output fluctuations in a way that corresponds to frequency controls in power system operations. In this study, we analyzed data for the fluctuations of actual wind power output at 20 wind farms, as designated by three basic definitions: the changes in time-averaged values, the maximum power-fluctuation range within a time window, and the deviations from a centered moving average value. The results indicated relevant differences between the definitions and demonstrated the importance of percentile analysis. We proposed a novel quantitative definition of the output fluctuations associated with the frequency controls of power systems: primary turbine-governor control, secondary load-frequency control, and tertiary economic load-dispatch control. Using these definitions to quantify fluctuations demonstrates how each smoothing technique can contribute to reducing the reserve capacity necessary for frequency control in the power system operations. Our proposed definitions for dividing the frequency ranges of the fluctuations were confirmed as a convenient and practical method for quantitatively evaluating the fluctuations and determining the reserve capacity required for stable power system operations.
Simple state-space formulas are presented for a controller solving a standard H_∞-problem. The controller has the same state-dimension as the plant, its computation involves only two Riccati equations, and it has a separation structure reminiscent of classical LQG (i.e., H_2) theory. This paper is also intended to be of tutorial value, so a standard H_2-solution is developed in parallel.
This paper deals with the problem of vibration suppression for seismic-excited buildings in finite frequency domain, and the possible faults caused by actuators are also considered in controller׳s design to improve the stability of the system. Firstly, with the consideration of seismic wave׳s effect, a mathematical model of building structure system is established. Then the finite frequency theory is introduced to the process of controller design in order to reduce seismic-excited building vibration over a certain frequency band and enhance disturbance suppression performance. Furthermore, taking actuators׳ faults into account, the active fault tolerance control method is also added to the finite frequency domain controller to compensate for the faults effect and remain the performance of building system at an acceptable level when faults occur. Finally, simulation of a three-degree-of-freedom linear building structure under earthquake excitation is given to illustrate the effect of the proposed approach.
This expository paper sets out the principal results in H// infinity control theory in the context of continuous-time linear systems. It treats a general regulator problem: a controller is to be designed to regulate the output of a plant subjected to exogenous inputs, such as disturbances, sensor noises and reference signals. A theory for the regulator problem begins by specifying a model of the plant (the model may be a set, to reflect uncertainty), a model of the exogenous inputs, the performance requirements of the controlled system and the allowable class of controllers. The focus is on the mathematical theory rather than computational methods.
A state-dependent switching law that obeys a dwell time constraint and guarantees the stability of a switched linear system is designed. Sufficient conditions are obtained for the stability of the switched systems when the switching law is applied in presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each subsystem such that it is non-increasing at the switching instants. During the dwell time, this function varies piecewise linearly in time. After the dwell, the system switches if the switching results in a decrease in the value of the LF. The method proposed is also applicable to robust stabilization via state-feedback. It is further extended to guarantee a bound on the L2-gain of the switching system; it is also used in deriving state-feedback control law that robustly achieves a prescribed L2 -gain bound.
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.
Background mathematics: Function spaces.- The standard problem.- Stability theory.- Background mathematics: Operators.- Model-matching theory: Part I.- Factorization theory.- Model-matching theory: Part II.- Performance bounds.
In this paper, we consider sensor and actuator fault detection and estimation issues for large scale wind turbine systems where individual pitch control (IPC) is used for load reduction. The faults considered are the blade root bending moment sensor faults and blade pitch actuator faults. In the first part, with the aid of a dynamical model of the wind turbine system, a so-called H∞/H− observer in the finite frequency range, is applied to generate the residual for fault detection. The observer is designed to be sensitive to faults but insensitive to disturbances, such as wind turbulence. When there is a detectable fault, the observer sends an alarm signal if the residual evaluation is larger than a predefined threshold. In addition to the fault detection, we also consider the fault estimation problem, where a dynamic filter is used to estimate the fault magnitude. The effectiveness of the proposed approach is demonstrated by simulation results for several fault scenarios. Copyright © 2010 John Wiley & Sons, Ltd.
This paper considers a control synthesis problem for linear systems to meet design specifications in terms of multiple frequency domain inequalities in (semi)finite ranges. Our approach is based on the generalized Kalman–Yakubovich–Popov (GKYP) lemma, and dynamic output feedback controllers of order equal to the plant are considered. A new multiplier expansion is proposed to convert the synthesis condition to a linear matrix inequality (LMI) condition through a standard linearizing change of variables. In a single objective setting, the LMI condition may or may not be conservative, depending on the choice of the basis for the multiplier expansion. We provide a qualification for the basis matrix to yield non-conservative LMI conditions. It is difficult to determine the basis matrix meeting such a qualification in general. However, it is shown that qualified bases can be found for some cases, and that the qualification condition can be used to find reasonable choices of the basis for other cases. The synthesis method is then extended to the multiple objective case where a sufficient condition is given for the existence of a controller to meet all the prescribed specifications. Finally, design examples for an active magnetic bearing are given to illustrate the effectiveness of the proposed design method. Copyright © 2006 John Wiley & Sons, Ltd.
The recently developed generalized Kalman-Yakubovic-Popov (KYP) Lemma establishes the equivalence between a frequency domain property and a linear matrix inequality (LMI), and allows to solve a certain class of system design problems with multiple specifications on gain/phase properties over several frequency ranges. This paper applies the KYP lemma in a control design for microactuator to suppress the narrow-band disturbances through shaping sensitivity functions in data storage systems such as servo track writers for hard disk drives (HDDs). The KYP Lemma is applied together with the Youla parameterization approach to allow a convex optimization for achieving the desired specifications of sensitivity functions. Both LDV based experiment and implementation in a servo track writer demonstrate the effectiveness of the design method in rejecting specific frequency disturbances. In particular, the design method improves the track density of the servo track writer with 425k track-per-inch (TPI)
This technical note studies the problem of designing reliable H ∞ controllers with adaptive mechanism for linear systems. A new method for designing indirect adaptive reliable controller via state feedback is presented for actuator fault compensations. Based on the on-line estimation of eventual faults, the proposed reliable controller parameters are updated automatically to compensate the fault effects on systems. A notion of adaptive H ∞ performance index is proposed to describe the disturbance attenuation performances of closed-loop systems. The design conditions are given in terms of solutions to a set of linear matrix inequalities (LMIs). The resultant designs can guarantee the asymptotic stability and adaptive H ∞ performances of closed-loop systems even in the cases of actuator failures. The effectiveness of the proposed design method is illustrated via a numerical example.
Many of the significant results in systems and control literature rely on characterizations of system properties in terms of frequency domain inequalities (FDIs) and/or time domain inequalities (TDIs). Classical FDIs, required to hold on the entire frequency range, have been interpreted by equivalent TDIs so that satisfaction of one implies that of the other. Recent developments have addressed FDIs within (semi)finite frequency ranges to increase flexibility in the system analysis and synthesis. This paper provides necessary and sufficient conditions for a general FDI to hold within a restricted frequency range in terms of TDIs to be satisfied by every inputs within a certain class specified by a matrix-valued integral quadratic constraint.
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.
This paper deals with the problem of H∞ control for seismic-excited buildings in finite frequency domain. The objective of designing controllers is to guarantee the asymptotic stability of the closed-loop system and attenuate the disturbance from earthquake excitation. After analyzing the frequency spectrum of some typical earthquakes, a frequency range in which earthquakes have a higher strength and lead to more serious damage to buildings, 0.3–8.8 Hz, is confirmed. Based on H∞ control theory and linear matrix inequality techniques, a new approach for building vibration control over finite frequency range is presented. In comparison with the entire frequency control, the finite frequency approach proves much better performance in building vibration control over a certain frequency band in which strong earthquakes happen. A three-degree-of-freedom linear building structure under earthquake excitation is considered and simulations are employed to validate the effectiveness of the proposed approach in reducing seismic-excited building vibration.
This paper presents all controllers for the general ∞ control problem (with no assumptions on the plant matrices). Necessary and sufficient conditions for the existence of an ∞ controller of any order are given in terms of three Linear Matrix Inequalities (LMIs). Our existence conditions are equivalent to Scherer's results, but with a more elementary derivation. Furthermore, we provide the set of all ∞ controllers explicitly parametrized in the state space using the positive definite solutions to the LMIs. Even under standard assumptions (full rank, etc.), our controller parametrization has an advantage over the Q-parametrization. The freedom Q (a real-rational stable transfer matrix with the ∞ norm bounded above by a specified number) is replaced by a constant matrix L of fixed dimension with a norm bound, and the solutions (X, Y) to the LMIs. The inequality formulation converts the existence conditions to a convex feasibility problem, and also the free matrix L and the pair (X, Y) define a finite dimensional design space, as opposed to the infinite dimensional space associated with the Q-parametrization.
The synthesis problem on how to select the linear vector fields among the possible ones such that a switched system becomes (exponentially) stable is treated in this paper. Earlier synthesis results guarantee stability under the restriction that sliding motions do not occur. In this paper, we extend the design procedure allowing sliding motions to occur by introducing additional conditions. Furthermore, if desirable sliding motions can be avoided by introducing hysteresis in the switching strategy. One example is given to illustrate the synthesis procedure
This paper considers a robust control synthesis problem for uncertain linear systems to meet design specifications given in terms of multiple frequency domain inequalities in (semi)finite ranges. We restrict our attention to static gain feedback controllers. We develop a new multiplier method that allows for reduction of synthesis conditions to linear matrix inequality problems. We study conditions under which the reduction is exact (nonconservative) in the single-objective nominal setting. Although the multiplier method is conservative in the general setting of multi-objective robust control, numerical design examples demonstrate the utility of the method for the state feedback case.
In this paper, a method for designing PID controllers to meet multiple frequency-domain inequalities (FDI) specifications on the open-loop transfer function in finite and semi-infinite frequency ranges is presented. The method, which is based on the generalized Kalman-Yakubovich-Popov (GKYP) lemma, enables direct loop shaping through LMI optimization without frequency gridding or weights. The resulting synthesis condition is nonconservative; its infeasibility implies that no PID controller exists to meet the specifications. The design method extends to systems with parametric uncertainties using Lyapunov function, which depends on the parameter in a linear fractional manner. The robust PID control design is also reduced to an LMI optimization problem, albeit with potential conservatism. The effectiveness of the GKYP synthesis has been demonstrated through several numerical examples. In addition to PID control design, robust GKYP synthesis applies to open-loop shaping and filtering problems, where the poles of the controller or filter are fixed.
The celebrated Kalman-Yakubovicˇ-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.
In this paper, the problem of sensitivity, reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability, of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n times n matrices of H^{infty} frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility, of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity, to output disturbances can be reduced to an optimal value approaching the singularity, measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitivity resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.
Analysis and design methodologies for robust aeroservoelastic structures
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An innovative finite frequency H∞
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Since 2003, he has been a Professor of mathematics in industry and technology with the TU Chemnitz. In 2010, he was appointed as one of the four Directors of the Max Planck Institute for Dynamics of Complex Technical Systems
- K S Lawrence
- Usa Chemnitz-Zwickau
Peter Benner received the Diploma degree in mathematics from the RWTH Aachen University, Aachen, Germany, in 1993, the Ph.D. degree in mathematics from the University of Kansas, Lawrence, KS, USA, and the TU
Chemnitz-Zwickau, Germany, in February 1997, and the Habilitation (Venia Legendi) degree in mathematics
from the University of Bremen, Germany, in 2001. After spending a term as a Visiting Associate Professor with
the TU Hamburg-Hamburg, Germany, he was a Lecturer in Mathematics with the TU Berlin, Germany, from
2001 to 2003. Since 2003, he has been a Professor of mathematics in industry and technology with the TU
Chemnitz. In 2010, he was appointed as one of the four Directors of the Max Planck Institute for Dynamics
of Complex Technical Systems, Magdeburg, Germany. Since 2011, he has been an Honorary Professor with
the Otto-von-Guericke University of Magdeburg, Germany. His research interests include scientific computing,
numerical mathematics, systems theory, and optimal control. He is a SIAM Fellow (Class of 2017).