Finite Frequency Range Research Articles

A Gramian matrix approach to synthesizing finite-frequency H2 controller

This paper studies the H2 control problem for linear continuous-time systems over a finite-frequency range. Using the finite-frequency Gramian matrix approach, a necessary and sufficient condition is obtained for the characterization of the finite-frequency H2 performance of a Hurwitz stability system. With such a characterization, a sufficient condition for the solvability of the finite-frequency H2 control problem is derived. An iterative algorithm is then constructed to solve the corresponding controller gain numerically. Finally, a numerical example of a two-cart-one-spring system is given to illustrate the effectiveness of the proposed scheme.

H∞ observer-based controller synthesis for fractional order systems over finite frequency range

This paper focuses on the H∞ observer-based control problem for fractional order systems (FOSs) over finite frequency range. Firstly, as an extension to the KYP lemma, the structure singular value–μ and its upper bound are analyzed and determined. The necessary and sufficient conditions of the H∞ performance for FOSs over finite frequency range are given through the generalized μ analysis method. This method is different from the traditional μ-analysis method, which is only suitable for integer order systems. Secondly, according to these obtained LMI-based criteria for H∞ performance index and a key projection lemma, we design an H∞ observer-based controller to stabilize the estimation error and decrease the H∞ norm for the FOS simultaneously. With the help of the matrix congruence transformation, the feedback gain matrix is parameterized by a scalar matrix. Thirdly, to deal with the nonlinear terms in matrix inequalities, an iterative linear matrix inequality (ILMI) algorithm is developed. Finally, two numerical examples are provided to explain the correctness of the established results.

On the behaviour of the upstream-travelling waves in merging twin-jet systems

There is currently considerable interest in the guided-jet mode, as a result of recent works demonstrating it being the upstream component of various resonant systems in high-speed flows. For given jet operating conditions, the mode is known to exist over only a finite-frequency range that, for a twin-jet system, has been observed to vary with both jet separation and solution symmetry. Vortex-sheet and finite-thickness linear stability models are here employed to consider the behaviour of the guided-jet mode as the two jets are brought together, for both a planar and round twin-jet system. It is demonstrated that in both cases as the twin-jet system merges it forms a higher-order mode of an equivalent single-jet geometry. This then imposes a constraint on the guided-jet mode as the finite-frequency range must change to meet that of the equivalent geometry the system merges to, explaining the previously observed dependence on jet separation.

Robust Feedforward Control of Discrete Uncertain Systems Over a Finite Frequency Range

Robust Feedforward Control of Discrete Uncertain Systems Over a Finite Frequency Range

Robust Fixed-Order Controller Design for Uncertain Systems With Generalized Common Lyapunov Strictly Positive Realness Characterization

This article investigates the design of a robust fixed-order controller for single-input–single-output (SISO) polytopic systems with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the performance specifications on sensitivity shaping are conformed in a specific finite frequency range. Utilizing the notion of generalized common Lyapunov strictly positive realness (CL-SPRness), the equivalence between strictly positive realness (SPRness) and strictly bounded realness (SBRness) is established; and then, the specifications on robust stability and performance are transformed into the SPRness of newly constructed systems and further characterized in the framework of linear matrix inequality (LMI) conditions. The proposed methodology avoids the tedious yet mandatory evaluations of the specifications on all vertices of the uncertain polytopic system in an explicit form. Instead, solving five LMIs exclusively suffices for ensuring the robust stability and performance regardless of the number of vertices, and thus, the typically heavy computational burden is considerably alleviated. It is also noteworthy that the proposed methodology additionally provides the necessary and sufficient conditions for this robust controller design with the consideration of a prescribed finite frequency range, and therefore, significantly less conservatism is attained in the system performance.

Sensor fault detection and minimum detectable fault analysis for dynamic point-the-bit rotary steerable system

In this paper, the sensor fault detection problem considering the drilling disturbances is studied for the dynamic point-the-bit rotary steerable system. Firstly, the DPRSS is modeled as a linear system with the drilling disturbances, including unknown inputs, measurement noises, and model perturbations. Then, a finite-frequency zonotopic fault detection observer is proposed. The finite-frequency range H− performance and the P-radius criterion are considered to design the observer gains such that the residuals are sensitive to sensor faults and robust against the drilling disturbances simultaneously. Subsequently, the calculation method of minimum detectable faults is presented for the proposed sensor fault detection mechanism. Finally, simulations and experiments are presented to illustrate the effectiveness of the proposed methods.

On the development of a virtual test bench to assess sound absorption of materials in an alpha chamber

A description is given of a virtual test-bench (VTB) designed by engineering center of Peter the Great St. Petersburg Polytechnic University to calculate a sound absorption coefficient of various materials. Developed VTB differs from known programs in that it allows a sound absorption coefficient of various materials to be determined with minimum involvement of an engineer. This VTB differs from other programs also by using infinite elements jointly with finite elements, which increases adequacy of the discrete model being used, and also the configuration of the boundary of an alpha chamber being used. The known programs use various phenomenological mathematical models of porous materials such as Johnson-Champoux-Allard (JCA) model. The VTB is based on fundamental mathematical models and statistical energy analysis (SEA) that allow describing adequately the established or transitional processes of sound absorption and reflection by a porous material the properties of which are not homogenized. The value of this VTB, which is created on the basis of a VA One software complex supplemented by a set of files of boundary conditions, files of solvers’ settings, secondary finite element (FE) models, is that VTB allows standardized calculations to be performed to determine the sound absorption coefficient of a material with minimum involvement of an engineer and the obtained result to be submitted for detailed processing. Developed virtual test-bench enables determination of a sound absorption coefficient for various materials within the entire finite frequency range. The result of the calculation is displayed as a graph of dependency of the material sound absorption coefficient on frequency.

Self-triggered finite-time [formula omitted] control for Markov jump systems with multiple frequency ranges performance

In the present study, a self-triggered finite-time control policy is investigated for Markov jump systems (MJSs) under multiple frequency ranges restrictions. A self-triggered scheme is developed to realize the system in a resource-aware pattern which will fully analyze the effect of transition probability on the characteristics of the finite frequency range. The developed scheme updates the control input of the system through the history of measurements. Then, taking into account several factors such as transient performance, frequency domain specification, and computational burden, a self-triggered finite-time controller is constructed. The developed controller concurrently ensures the predefined disturbance suppression level and the finite-time boundedness of the considered system while decreasing the computational and communication cost. The simulation example of a cart-spring system illustrates the effectiveness and superiority of the proposed approach over existing methods.

Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllers

In this paper, the robust stability of interval fractional order plants with one time delay controlled by fractional order controllers is investigated in a general form. For robust stability analysis of the closed loop system by the zero exclusion principle, the distance between the origin and the value set of the characteristic function needs to be checked. It is known that the outer vertices of this value set may change at some switching frequencies and the repetitive calculation of these vertices at switching frequencies leads to additional calculations. In this study initially, new necessary and sufficient conditions are proposed to check the robust stability of a delayed fractional order closed loop system. Then, a novel robust stability testing function is presented based on some vertices, which are fixed for all positive frequencies. Therefore, no additional calculation is needed to obtain the outer vertices of the characteristic function value set for any pair of the switching frequencies. Also, a finite frequency range is presented to reduce the computational cost noticeably. Eventually, three numerical examples are given to verify the efficiency of the results of this paper.

Antiresonance in switched systems with only unstable modes

The authors report the emergence of antiresonance in a variation of the Stuart-Landau model, which alternates between two unstable modes but displays a stable response in a finite range of switching frequencies.

Iterative Learning Control for Fractional Order Linear Systems with Time Delay Based on Frequency Analysis

To overcome the deficiencies of time delay in the repetitive control of fractional-order linear systems, PDα-type iterative learning control (ILC) law and P & convolution-type ILC law are designed for input and state time delay, respectively. Convergence conditions are derived in frequency domain via contraction mapping principle. Besides, the convergence frequency domain of proposed feedback controllers is obtained over a finite frequency range to design the controllers effectively. Then, the effectiveness of the proposed theoretical schemes is demonstrated using two numerical examples. The influence of time delay is eliminated, and output trajectory convergence to the desired one is guaranteed. Moreover, the Nyquist diagram of transfer function G(s) and time delay variation are analyzed in frequency domain to reveal the influence of convergence on the system.

Cyber-Physical Attacks Against State Estimators Based on a Finite Frequency Approach

This paper studies the design problem of the attack strategy for cyber-physical systems (CPSs). In this paper, a finite frequency attack strategy is designed for CPSs such that the performance of the state estimator can be reduced with a low attack detection probability. Different from the existing methods with a predefined finite frequency range, the attack strategy is designed for the system to satisfy H_/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance index in different finite frequency ranges in the proposed method, and a suitable frequency interval can be selected as that of the attack signal. Since the property of the finite frequency of the attack signal could be made full use of such that a good attack effect can be obtained in the proposed method. Finally, an example is given to illustrate the effectiveness of the developed method.

Optimal Absorption of Acoustic Waves by a Boundary

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the energy of a sound wave, we consider a frequency model (the Helmholtz equation) with a damping on the boundary. The damping on the boundary is firstly related with the damping in the volume, knowing the macroscopic parameters of a fixed porous medium. Once the well-posedness results are proved for the time-dependent and the frequency models in the class of bounded $n$-sets (for instance, locally uniform domains with a $d$-set boundary, containing self-similar fractals or Lipschitz domains as examples), the shape optimization problem of minimizing the acoustical energy for a fixed frequency is considered. To obtain an efficient wall shape for a large range of frequencies, we define the notion of $\epsilon$-optimal shapes and prove their existence in a class of multiscale Lipschitz boundaries when we consider energy dissipation on a finite range of frequencies, and in a class of fractals for an infinite frequency range. The theory is illustrated by numerical results.

Validation of high-fidelity ion cyclotron range of frequencies antenna coupling simulations in full 3D geometry against experiments in the ASDEX Upgrade tokamak

We address the validation of finite elements ion cyclotron range of frequencies (ICRF) antenna coupling simulation against experiments performed in the ASDEX Upgrade tokamak. Measurements of the loading resistance in ICRF-heated, magnetically-perturbed 3D plasma discharges are compared against numerical predictions of the RAPLICASOL code. To this end, the 3D induction field and the 3D density profile are modeled by concatenating the PARVMEC, BMW and EMC3-EIRENE codes. The 3D density is input to RAPLICASOL, where full-wave simulations are performed on a finite element mesh retaining full 3D geometry in the ICRF antenna model and the plasma description. The results are further compared with RAPLICASOL simulations using a 1D density profile as measured at the outboard midplane in the same experiments. We find that simulations using a 1D density profile overestimate the change in loading resistance by a factor of ∼197 − 248%, while simulations using the full 3D density profile are in agreement with experiments within a factor ∼16%.

Flexural waves in a periodic non-uniform Euler-Bernoulli beam: Analysis for arbitrary contour profiles and applications to wave control

The flexural wave in a periodic non-uniform Euler-Bernoulli beam with arbitrarily contoured profiles is studied by utilizing the power series expansion method. The convergence criterion that makes the power series expansion method applicable is also illustrated. The validation is carried out by comparing the theoretical results with that from the finite element analysis when the beam thickness varies in different forms. For a quadratic thickness variation, the first band gap evolution versus the structural parameter is investigated, based on which a flexural-wave-based low-pass filter for frequency shunting and a rectangular lens for energy focusing are designed. It is revealed in the frequency domain analysis that the flexural wave with a lower frequency can propagate further when it travels into the wave filter. The lens designed exhibits a good focusing phenomenon with the focusing size smaller than one wavelength, and has a good performance at a certain finite frequency range. The theoretical method and design scheme can provide effective guidance for the flexural wave control.

Robust finite‐frequency filter design for linear uncertain systems using polynomially parameter‐dependent approach

This study investigates the problem of robust filter design for continuous-time linear uncertain systems affected by external noises in finite-frequency (FF) ranges. The objective is to design an admissible filter guaranteeing asymptotic stability of the filtering error system with a prescribed FF noise attenuation level. To this end, homogenous polynomially parameter-dependent Lyapunov functions are used to assess the stability of the filtering error system, and robust FF bounded realness lemma is used to describe the noise attenuation specifications. Polynomially parameter-dependent matrices and some scalars are introduced to facilitate filter design and reduce conservatism. Moreover, a sequential convex approximation method is used to replace the non-convex constrained set with a convex feasible one. On this basis, an iterative algorithm is developed to calculate the filter parameters. It is demonstrated that the proposed robust FF filter can achieve a better noise attenuation performance in the FF range than the existing ones in the literature. An example about F-18 aircraft model is given to illustrate the effectiveness of the proposed method.

Structure preserving model reduction of second‐order time‐delay systems via approximate Gramians

In this study, a model-order reduction approach based on approximate Gramians for second-order time-delay systems is proposed. By treating the second-order time-delay system as a special case of the integro-differential system, it exhibits a connection with the time-delay Gramians of the corresponding first-order system. After applying the approximate Gramians and balanced truncation procedures, the resulting reduced system is expected to well approximate the original system. Besides, the e -embedding technique is applied to the case when the damping matrix of the second-order time-delay system is singular. Finally, numerical examples show the good performance in the time domain and the finite frequency range.

Iterative Learning Control for Linear Differential Systems With Additional Performance Requirements

This paper develops the systematic procedure for designing of iterative learning control (ILC) algorithms through the differential repetitive process setting. This means that the proposed approach can be directly applied to plants with differential dynamics and allows to satisfy the additional requirements on the resulting dynamics. In particular, the proposed design procedure enforces a required frequency attenuation over a finite frequency range and includes regional pole constraints. Additionally, an important result extension to the plants with relative degree greater than unity is presented. The sufficient conditions for the existence of the controllers are derived in terms of linear matrix inequalities, which are immediately extended to deal with time varying uncertainties. Finally, the simulations for a typical actuator of tracking servo system prove that the design is effective and brings some advantages when compared to the existing alternatives.

A Design Method of Optimal PID-Based Repetitive Control Systems

This paper is concerned with the design of a PID-based repetitive control strategy for a strictly proper plant with requirements of high-precision tracking performance for periodic signals and good rejecting performance for an external disturbance with a finite frequency range. The PID-based controller consists of a PID controller and a repetitive controller connected in series, and its control parameters are optimized to meet the performance requirements. First, utilizing Youla–Kucera Parameterization, the necessary and sufficient conditions for the existence of the PID-based repetitive controller are derived. Second, from the requirements of the control performance and the analysis of the sensitivity function, the description of the PID-based repetitive controller is deduced. Third, the design of PID control parameters depends on the stability of closed-loop system and the attenuation performance. By introducing an arbitrarily small relaxation variable and applying the $H_\infty $ control method, the design problem is transformed into a generalized eigenvalue minimization problem (GVEP) under linear matrix inequality (LMI) constraints. In addition, the largest cutoff frequency in the repetitive controller is also achieved by solving a GVEP. Finally, simulations are applied to examine the validity and efficacy of the proposed method.

Output Feedback Based Iterative Learning Control with Finite Frequency Range Specifications via a Heuristic Approach for Batch Processes with Polytopic Uncertainties

For robust control and iterative optimization of industrial batch processes with polytopic uncertainties, this paper proposes a robust output feedback based iterative learning control (ILC) design in terms of finite frequency range stability specifications. Robust stability conditions for the closed-loop ILC system along both time and batch directions are first established based on the generalized Kalman-Yakubovich-Popov lemma and linear repetitive system theory. To facilitate the ILC controller design with respect to process uncertainties described in a polytopic form, extended sufficient conditions for the system stability are then derived in terms of matrix inequalities. Correspondingly, a two-stage heuristic approach is developed to iteratively compute feasible ILC controller gains for implementation. An illustrative example is given to demonstrate the effectiveness of the proposed control design.