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Separated periodic/aperiodic state feedback control using periodic/aperiodic separation filter based on lifting
Abstract
Automatic machines have recently advanced into diverse environments. A periodic state and an aperiodic state of the system usually have different physical meanings and require different control objectives in the diverse environments. To enable a new control approach based on the periodic and aperiodic states for the different control objectives, a periodic/aperiodic separation filter (PASF) that separates a state into a periodic state and an aperiodic state is proposed in this paper. Furthermore, using the PASF, the periodic/aperiodic state feedback control and the separation principle are established.
... Classically, in the motion control field, impedance control has been studied (24)- (27) . This study aims at imparting presence to the current sound with virtual wall generation based on the impedance control framework. ...
... According to (27), a virtual wall whose acoustic impedance is valuable is generated at desired position. Figure 6 shows the block diagram of acoustic impedance control system. ...
... In this section, the virtual acoustic impedance at a desired position is made equal to the acoustic impedance of the desired medium by controlling the sound pressure emitted by the secondary sound source represented by (27). Subsequently, the generation of a virtual wall is confirmed through simulation. ...
Realistic systems have received increasing attention auditory sense plays a significant role, sound-field reproduction that generates high-presence sound-field is important. Key factor affecting the presence of sound-field is the influence of echo. Therefore, this study focuses on the effect of echo for a more realistic sound-field reproduction. In this study, the sound wave from a sound source is exposed to the echo effect through an acoustic impedance control performed by constructing a virtual wall. The virtual acoustic impedance is controlled by the sound wave generated from a secondary sound source the virtual wall acoustic impedance varies at a certain distance from the sound source. Therefore, an echo effect is produced through the control of the acoustic impedance the listener feel as if direct sound was reflected at the position of the virtual wall.
... Accordingly, the P/A motion control using P/A velocity and P/A force is proposed. A periodic/aperiodic separation filter (PASF) [22] separates velocity and force into the P/A velocity and P/A force, respectively. The separation of objectives and motion into P/A objectives and P/A motions, respectively, enables the simultaneous realization of two control objectives. ...
... However, these methods focus only on the periodicity. Moreover, although the previous study [22] proposed the PASF and P/A state feedback control, it is impractical because disturbance compensation and practical simultaneous control are not addressed. In this article, six types of practical P/A motion controls, that correspond to different combinations of the P/A velocity, P/A force, and P/A impedance controls, are constructed. ...
... Then, the robust stability condition based on the small-gain theorem is W (s)T (s) ∞ < 1. T (s) is a complementary sensitivity function, which is an index to evaluate the robust stability. The complementary sensitivity functions of the P/A motion control systems are given by (12), (17), (22), (27), (32), and (37). To satisfy the robust stability condition, the cutoff frequency g of the DOB and parameters for the P/A motion control: K i , K d , K f , M v , D v , and K v can adjust the complementary sensitivity functions. ...
Motion control is a fundamental technique used in automated mechanical systems. Classically, velocity, force, and impedance are controlled in motion control systems, but simultaneous control is difficult. This study proposes periodic/aperiodic (P/A) motion control based on periodicity and aperiodicity of motion. The P/A motion control separately applies different control methods to P/A motions using P/A velocity and P/A force, which are extracted using a periodic/aperiodic separation filter (PASF) from velocity and force. Accordingly, six types of P/A motion controls are constructed in this study, which correspond to different combinations of the P/A velocity, P/A force, and P/A impedance controls. To construct the P/A motion control systems, acceleration control based on a disturbance observer is used. The acceleration control system, which rejects disturbances, enables the P/A motion control design to ignore disturbances. The experiments were conducted to validate the six P/A motion controls, which simultaneously realized two P/A motion control objectives.
... This article proposes a quasi-periodic disturbance observer (QDOB) based on an internal model of a quasi-periodic disturbance for simultaneously realizing them. The internal model is realized using a periodic/aperiodic separation filter [26], [27], where each time delay is integrated with a zero-phase low-pass filter. This internal model leads to the wideband harmonic suppression, which is robust against perturbations in each cycle, slight changes over cycles, and/or slight variations in the period of the quasi-periodic disturbance. ...
... For the Q-filter of the QDOB (5), this article uses a periodicpass filter of a first-order periodic/aperiodic separation filter proposed in [26], [27]. Since the lifted quasi-periodic disturbance D τ (c) satisfies (9), it consists of low-frequency signals at frequencies less than or equal to the separation frequency ρ. ...
Periodic disturbances composed of harmonics typically occur during periodic operations, impairing performance of mechanical and electrical systems. To improve the performance, control of periodic-disturbance suppression has been studied, such as repetitive control and periodic-disturbance observers. However, actual periodic disturbances are typically quasiperiodic owing to perturbations in each cycle, identification errors of the period, variations in the period, and/or aperiodic disturbances. For robustness against quasiperiodicity, although wideband harmonic suppression is expected, conventional methods have trade-offs among harmonic suppression bandwidth, amplification of aperiodic disturbances, and deviation of harmonic suppression frequencies. This paper proposes a quasiperiodic disturbance observer to compensate for quasiperiodic disturbances while simultaneously achieving the wideband harmonic suppression, non-amplification of aperiodic disturbances, and proper harmonic suppression frequencies. A quasiperiodic disturbance is defined as comprising harmonics and surrounding signals. On the basis of this definition, the quasiperiodic disturbance observer is designed using a periodic-pass filter of a first-order periodic/aperiodic separation filter for its Q-filter, time delay integrated with a zero-phase low-pass filter, and an inverse plant model with a first-order low-pass filter. The periodic-pass filter achieves the wideband harmonic suppression while the zero-phase and first-order low-pass filters prevent the amplification of aperiodic disturbances and deviation of harmonic suppression frequencies. For the implementation, the Q-filter is discretized by an exact mapping of the s-plane to the z-plane, and the inverse plant model is discretized by the backward Euler method. The experiments validated the frequency response and position-control precision of the proposed method.
... Their frequency characteristics have many band-stop characteristics, but they are difficult to design their cutoff frequencies, convergence speed, and high-order filters. Similar to the comb filter, a periodic/aperiodic separation filter (PASF) was proposed [9], which separates a signal into quasi-periodic and quasi-aperiodic signals defined in [8]. The first-order PASF can easily adjust its cutoff frequency and convergence speed [7], and the high-order PASF can perform more rigid harmonics elimination [8]. ...
... Lastly, the moving standard deviation (MSD) of the quasi-aperiodic force is computed and compared to a threshold for the detection. Moreover, for the PASF, we investigated a separation frequency that realizes a desired cutoff frequency for the firstorder PASF, the time constant of the first-order PASF on the basis of the previous studies [7,9], and highorder PASF based on [8]. They enable direct design of the cutoff frequency and time constant, and the high-order filter improves the harmonics elimination performance without deteriorating chatter signal extraction performance. ...
Process harmonics often need to be eliminated for accurate regenerative chatter diagnosis. To eliminate the harmonics with many band-stop characteristics, this paper uses a periodic/aperiodic separation filter (PASF) and proposes a chatter detection algorithm based on the PASF. The PASF can design the cutoff frequency and time constant, directly, and the high-order PASF improves the harmonics elimination performance without deteriorating chatter signal extraction performance. The experiments validated the proposed algorithm under several conditions of spindle speeds, cutting types, thresholds, and cutoff frequencies. Their results showed the robustness of the proposed chatter detection algorithm against the threshold selection and change of cutting conditions.
... The periodic/Aperiodic separation control was proposed to achieve quasi-periodic and quasi-aperiodic controls on a single-degree-of-freedom [17]- [19]. By using a periodic/aperiodic separation filter (PASF), a state is separated into quasi-periodic and quasi-aperiodic states, and the separated quasi-periodic and quasi-aperiodic tasks are simultaneously achieved. ...
Robots are typically expected to perform multiple tasks. For realizing multiple tasks using multi-degrees-of-freedom of a robot, priority control was developed. The priority control prioritizes conflicting tasks by projecting lower-priority task velocity into a null space of higher-priority task spaces. Similarly, for realizing two tasks on a single-degree-of-freedom of a robot, periodic/aperiodic separation control was developed. The periodic/aperiodic separation control separates a state into quasi-periodic and quasi-aperiodic states and feedback controls them separately. Thus, the priority control and periodic/aperiodic separation control realize multiple tasks in a different manner, and this paper proposes an integration of them to increase the number of simultaneously realized multiple tasks. Furthermore, this study analyzed the stability and conducted experiments to validate the proposed method. In the experiments, the proposed method achieved to increase the number of realizable tasks from six to nine by performing the quasi-periodic and quasi-aperiodic tasks in addition to prioritized tasks.
... In order to solve this probl em, periodic signals were handled by previous studies. Repet itive cont rol is a well-known method to eliminate a periodic disturb ance [12]- [ 15]. [16) not only focused on a periodic signal but also on an aperiodic signal. With the increasing demand of industry for high-performance control systems, as a two-degree-offreedom controller, disturbance observers (DOB s) were studied [17]- [23]. ...
This research tries to stabilise a three degree-of-freedom series-type double inverted pendulum system by employing an adaptive state feedback control approach optimised by a particle swarm optimisation (PSO) algorithm. To this end, the dynamical equations are derived via Lagrangian method and linearised by utilising approximation approaches. Then, the state feedback scheme is implemented to control the system states, that is, the angle and angular velocity of the first and second poles as well as the position and velocity of the cart. In order to timely regulate the control gains, the adaptation rules based on the gradient descent method and sliding surface relations are applied. The particle swarm optimisation algorithm is implemented to optimally tune the design parameters of the controller based on a proper objective function defined as summation of integrals of absolute values of the system errors. The time responses of the angles of the poles and the position of the cart clearly depict the feasibility and efficiency of the introduced control strategy to stabilise the system from different initial conditions to the final desired values.
Periodicity and aperiodicity can exist in a state simultaneously and typically become quasi-periodicity and quasi-aperiodicity in a dynamically changing state. The quasi-periodic and quasi-aperiodic states existing in the periodic/aperiodic state mostly correspond to different phenomena and require different controls. For separation control of these states, this paper defines the periodic/aperiodic, quasi-periodic, and quasi-aperiodic states to construct a periodic/aperiodic separation filter that separates the periodic/aperiodic state into the quasi-periodic and quasi-aperiodic states. Based on these definitions, the linearity of periodic-pass and aperiodic-pass functions and the orthogonality of quasi-periodic and quasi-aperiodic state functions are proved. Subsequently, the periodic/aperiodic separation filter composed of periodic-pass and aperiodic-pass filters that realize the periodic-pass and aperiodic-pass functions is designed and integrated with a Kalman filter for estimation of the quasi-periodic and quasi-aperiodic states.
Periodicity and aperiodicity can exist in a state simultaneously and typically become quasi-periodicity and quasi-aperiodicity in a dynamically changing state. The quasi-periodic and quasi-aperiodic states existing in the periodic/aperiodic state mostly correspond to different phenomena and require different controls. For separation control of these states, this paper defines the periodic/aperiodic, quasi-periodic, and quasi-aperiodic states to construct a periodic/aperiodic separation filter that separates the periodic/aperiodic state into the quasi-periodic and quasi-aperiodic states. Based on these definitions, the linearity of periodic-pass and aperiodic-pass functions and the orthogonality of quasi-periodic and quasi-aperiodic state functions are proved. Subsequently, the periodic/aperiodic separation filter composed of periodic-pass and aperiodic-pass filters that realize the periodic-pass and aperiodic-pass functions is designed and integrated with a Kalman filter for estimation of the quasi-periodic and quasi-aperiodic states.
Foods are essential for our society, and they are transported and delivered after packaging to avoid contamination and deterioration of freshness. Nevertheless, there is a package leak problem in automatic packaging processes that would risk impairing human health. In order to protect human health and guarantee the reliability of the industrial automatic food production, automatic leak testing is necessary for all packages. This study developed a package leak tester and proposed an aperiodic anomaly test using the tester to detect the package leaks automatically. The developed package leak tester realized accurate reaction force measurement enabling accurate leak detection, and the proposed aperiodic anomaly test using an aperiodic derivative of the reaction force based on a leaking package model achieved robust leak detection against disturbances. The package leak tester and the aperiodic anomaly test were validated by testing 10,000 package data randomly generated by 388 experimental package data.
An analytical and experimental study of the pulsed ultrasonic comb filtering effect is presented in this work intending to provide a fundamental tool for data analysis and phenomenon understanding in pulsed ultrasonics. The basic types of comb filter, feedforward and feedback filters, are numerically simulated and demonstrated. The characteristic features of comb filters, which include the formula for determining the locations of the spectral peaks or notches and the relationship between its temporal characteristics (relative time delay between constituent pulses) and its spectral characteristics (frequency interval between peaks or notches), are theoretically derived. To demonstrate the applicability of the comb filtering effect, it is applied to measuring the sound velocities and thickness of a thin plate sample. It is proven that the comb filtering effect based method not only is capable of accurate measurements, but also has advantages over the conventional time-of-flight based method in thin plate measurements. Furthermore, the principles developed in this study have potential applications in any pulsed ultrasonic cases where the output signal shows comb filter features.
In this paper, we propose a design method of digital repetitive controllers to track continuous-time periodic references based on sampled-data delayed signal reconstruction. It is well known that perfect tracking is not possible for arbitrary periodic signals with a fixed period if the plant to be controlled is strictly proper.
Human mobility is affected by different types of pathologies and also decreases gradually with age. In this context, Smart Walkers may offer important benefits for human assisted-gait in rehabilitation and functional compensation scenarios. This paper proposes a new interaction strategy for human–walker cooperation. The presented strategy is based on the acquisition of human gait parameters by means of data fusion from inertial measurement units and a laser range finder. This paper includes the mathematical formulation of the controller, simulations, and practical experimentation of the interaction strategy, in order to show the performance of the control system, including the param-eter detection methodology. In the experimental study, despite the continuous oscillation during the walking, the parameter estima-tion was suitable for assisted ambulation, showing an appropriate adaptive behavior with changes in human linear velocity. Finally, the controller keeps the walker continuously following in front of the human gait, and it is shown how the walker orientation follows the human orientation during the real experiments. Index Terms—Human–robot interaction, inertial measurement units (IMUs) sensor, laser range finder (LRF), multimodal inter-face, walker-assisted gait.
In this paper, we propose novel LMI conditions for the stability and l2 gain performance analysis of discrete-time linear periodically time-varying (LPTV) systems. These LMIs are convex with respect to all of the coefficient matrices of the LPTV systems and this property is expected to be promising when dealing with several control system analysis and synthesis problems. For example, we can apply those LMIs straightforwadly to robust performance analysis problems of LPTV systems that are affected by polytopic-type uncertainties. Even though our approach for robust performance analysis is conservative in general, we can reduce the conservatism gradually by artificially regarding the original N-eriodic system as pN-periodic and increasing p. In addition, thanks to the simple structure of the LMI conditions, we can readily derive a viable test to verify the exactness of the computation results.
Repetitive operations are widely conducted by automatic machines in industry. Periodic disturbances induced by the repetitive operations must be compensated to achieve precise functioning. In this paper, a periodic-disturbance observer (PDOB) based on the disturbance observer (DOB) structure is proposed. The PDOB compensates a periodic disturbance including the fundamental wave and harmonics by using a time-delay element. Furthermore, an adaptive PDOB is proposed for the compensation of frequency-varying periodic disturbances. An adaptive notch filter is used in the adaptive PDOB to estimate the fundamental frequency of the periodic disturbance. Simulations compare the proposed methods with a repetitive controller and the DOB. Practical performances are validated in experiments using a multiaxis manipulator. The proposal provides a new framework based on the DOB structure to design controllers using a time-delay element.
A novel, cable-driven soft joint is presented for use in robotic rehabilitation exoskeletons to provide intrinsic, comfortable human-robot interaction. The torque-displacement characteristics of the soft elastomeric core contained within the joint are modelled. This knowledge is used in conjunction with a dynamic system model to derive a sliding mode controller (SMC) to implement low-level torque control of the joint. The SMC controller is experimentally compared to a baseline feedback-linearised PD controller across a range of conditions and shown to be robust to un-modelled disturbances. The torque controller is then tested with six healthy subjects while they perform a selection of ADLs, which has validated its range of performance. Finally, a case study with a participant with spastic cerebral palsy is presented to illustrate the potential of both the joint and controller to be used in a physiotherapy setting to assist clinical populations.
Rejection of unknown periodic disturbances in multi-channel systems has several industrial applications that include aerospace, consumer electronics, and many other industries. This paper presents a design and analysis of an output-feedback robust adaptive controller for multi-input multi-output continuous-time systems in the presence of modeling errors and broadband output noise. The trade-off between robust stability and performance improvement as well as practical design considerations for performance improvements are presented. It is demonstrated that proper shaping of the open-loop plant singular values as well as over-parameterizing the controller parametric model can significantly improve performance. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme. Copyright
Several continuous-time frequency estimators for a measured sinusoidal signal, which have been proposed in the literature, are reviewed, reinterpreted and compared both theoretically and by simulations. It is argued that adaptive notch filters are feedback algorithms that contain a local adaptive observer in the feedback loop. It is shown that the adaptive notch filter (ANF), which was originally conceived as a discrete-time ANF, is basically equivalent to the recently proposed adapted frequency locked loop called orthogonal signal generator. They both require a sufficiently slow frequency estimation and can be interpreted as third-order adaptive observers. They exhibit local convergence properties for the estimation errors, that is, the convergence to zero is guaranteed provided that their initial error is sufficiently small. Three adaptive observers, which were independently proposed in 2002, are third-order frequency estimators whose estimation errors are exponentially convergent to zero from any initial condition and for any value of frequency, amplitude and phase in the measured sinusoidal signal. They have the additional advantage of not requiring the frequency estimation dynamics to be sufficiently slow. Conversely, they may be interpreted as adaptive notch filters. Second-order frequency estimators have been proposed as well: they may be interpreted as adaptive reduced-order observers. Copyright © 2015 John Wiley & Sons, Ltd.
Harmonic extraction and control is one of the main elements in active power filter design. The advances in the digital signal processor (DSP)-based control have facilitated the use of sophisticated filters and control techniques to achieve better dynamic performance, which was difficult earlier with the use of analog design. To improve the dynamic response of the active harmonic controller, we propose a comb-filter-based solution, which can be implemented on a DSP platform. Performance evaluation results for the load transient show a substantial improvement in the dynamic performance of the harmonic compensator compared with the conventional low-pass-filter-based approaches.
An intelligent human-robot interaction (HRI) system with adjustable robot behavior is presented. The proposed HRI system assists the human operator to perform a given task with minimum workload demands and optimizes the overall human-robot system performance. Motivated by human factor studies, the presented control structure consists of two control loops. First, a robot-specific neuro-adaptive controller is designed in the inner loop to make the unknown nonlinear robot behave like a prescribed robot impedance model as perceived by a human operator. In contrast to existing neural network and adaptive impedance based control methods, no information of the task performance or the prescribed robot impedance model parameters is required in the inner loop. Then, a task-specific outer-loop controller is designed to find the optimal parameters of the prescribed robot impedance model to adjust the robot’s dynamics to the operator skills and minimize the tracking error. The outer loop includes the human operator, the robot and the task performance details. The problem of finding the optimal parameters of the prescribed robot impedance model is transformed into a linear quadratic regulator (LQR) problem which minimizes the human effort and optimizes the closed-loop behavior of the HRI system for a given task. To obviate the requirement of the knowledge of the human model, integral reinforcement learning is used to solve the given LQR problem. Simulation results on an x-y table and a robot arm, and experimental implementation results on a PR2 robot confirm the suitability of the proposed method.
The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as “lifting”, is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data.
A repetitive control method is presented that is implemented in real-time for periodic wind disturbance rejection for linear systems with multiple inputs and multiple outputs and with both repetitive and non-repetitive disturbance components. The novel repetitive controller can reject the periodic wind disturbances for fixed-speed wind turbines and variable-speed wind turbines operating above-rated and we will demonstrate this on an experimental “smart” rotor test section. The “smart” rotor is a rotor where the blades are equipped with a number of control devices that locally change the lift profile on the blade, combined with appropriate sensors and controllers. The rotational speed of wind turbines operating above-rated will vary around a defined reference speed, therefore methods are given to robustify the repetitive controllers for a mismatch in the period. The design of the repetitive controller is formulated as a lifted linear stochastic output-feedback problem on which the mature techniques of discrete linear control may be applied. For real-time implementation, the computational complexity can be reduced by exploiting the structure in the lifted state-space matrices. With relatively slow changing periodic disturbances it is shown that this repetitive control method can significantly reduce the structural vibrations of the “smart” rotor test section. The cost of additional wear and tear of the “smart” actuators are kept small, because a smooth control action is generated as the controller mainly focuses on the reduction of periodic disturbances.
This paper presents a trigonometrical approach to design a simple second order wideband compensator of a comb decimation filter. The method is based on the similarity of the inverse passband comb characteristic and the squared sine function. The design parameter B, which is the amplitude of the squared sine function, depends only on the number K of the cascaded comb filters. Considering that the parameter B can be expressed in terms of only addition and shift operations, we may obtain a multiplierless filter. The proposed filter performs wideband compensation efficiently using a maximum of four additions/subtractions, and for a given K, can be applied to any value of the decimation factor M.
The problem of both causal and noncausal identification of linear stochastic systems with quasi-harmonically varying parameters is considered. The quasi-harmonic description allows one to model nonsinusoidal quasi-periodic parameter changes. The proposed identification algorithms are called generalized adaptive comb filters/smoothers because in the special signal case they reduce down to adaptive comb algorithms used to enhance or suppress nonstationary harmonic signals embedded in noise. The paper presents a thorough statistical analysis of generalized adaptive comb algorithms, and demonstrates their statistical efficiency in the case where the fundamental frequency of parameter changes varies slowly with time according to the integrated random-walk model.
The paper addresses the problem of attenuation (rejection) of unknown and time varying multiple narrow band disturbances without measuring them. In this context the disturbance model is unknown and time varying while the model of the plant is known (obtained by system identification) and almost invariant. This requires to use an adaptive feedback approach. The term “adaptive regulation” has been coined to characterize this control paradigm. Application domains include: mechanical and mechatronics systems, active vibration and noise suppression systems, some types of bio-chemical reactors.The paper reviews the various techniques proposed for solving this problem. It will focus on the presentation of the direct and indirect adaptive regulation strategies using the internal model principle and the Youla–Kucera parametrization which have been extensively used in applications.The paper also reviews a number of applications including: active suspension systems, active vibration control systems, active noise control, bio-chemical reactors, distributed flexible mechanical structures and Blu-ray disc drives. Real time results obtained on various applications will illustrate the methodology.
Adaptive regulation is an important issue with a lot of potential for applications in active suspension, active vibration control, disc drives control and active noise control. One of the basic problems from the “control system” point of view is the rejection of multiple unknown and time varying narrow band disturbances without using an additional transducer for getting information upon the disturbances. An adaptive feedback approach has to be considered for this problem. Industry needs to know the state of the art in the field based on a solid experimental verification on a representative system using currently available technology. The paper presents a benchmark problem for suppression of multiple unknown and/or time-varying vibrations and an associated active vibration control system using an inertial actuator with which the experimental verifications have been done. The objective is to minimize the residual force by applying an appropriate control effort through the inertial actuator. The system does not use any additional transducer for getting real-time information about the disturbances.
The benchmark has three levels of difficulty and the associated control performance specifications are presented. A simulator of the system has been used by the various contributors to the benchmark to test their methodology. The procedure for real-time experiments is briefly described. The performance measurement methods used will be presented as well as an extensive comparison of the results obtained by various approaches.
In this paper we overview, compare and elaborate on the invariant representations of periodic systems. Precisely, with reference to discrete-time systems, we first introduce the concept of periodic transfer function from which a notion of generalized frequency response can be worked out. Then we discuss the following four reformulations: (i) time lifted, (ii) cyclic, (iii) frequency lifted and (iv) Fourier. A number of interesting links will be established, and many theoretical aspects somewhat overlooked in the existing literature will be clarified. All reformulations are first worked out from the input–output description and then elaborated in a state-space formalism.
In this note, an approach based on lifting is introduced for the observer based fault detection (FD) of linear discrete-time periodic systems. The main contribution of this note is to give an analytic expression to transform the linear time-invariant (LTI) observer based residual generator designed based on the lifted LTI reformulation of the periodic system into a periodic observer based residual generator to reduce detection delay. The result enables a systematic transfer of the existing observer based FD methods for discrete LTI systems to discrete-time periodic systems.
Online frequency estimation of a pure sinusoidal signal is a classical problem that has many practical applications. Recently an ANF with global convergence property has been developed for this purpose. There exist some practical applications in which signals are not pure sinusoidal and contain harmonics. Therefore, online frequency estimation of periodic but not necessarily sinusoidal signals espoused by such applications becomes quite important. This note presents an alternative stability analysis for a modified ANF that permits the presence of harmonics in the incoming signal. Also, this stability analysis is simpler and alleviates the problem complexity even in the case of pure sinusoidal signal. Simulation results confirm theoretical issues.