Frequency-domain analysis of linear time-periodic systems

Sandberg, Henrik; Möllerstedt, Erik; Bernhardsson,

doi:10.1109/tac.2005.860294

Cited by 63 publications

(36 citation statements)

References 31 publications

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“…A fuller description with equations and expanded motivation can be found in Kiemel et al (2016, pre-print available at http://arxiv.org/abs/1607.01746). Our method is based on existing theory for linear time-periodic systems (e.g., Wereley and Hall, 1990; Möllerstedt and Bernhardsson, 2000; Sandberg et al, 2005) extended for general limit-cycle systems in which perturbations can reset the phase of the oscillator. Our method assumes that the system has smooth dynamics (see Ankarali and Cowan, 2014 for a method designed for hybrid LTP systems).…”

Section: Methodsmentioning

confidence: 99%

“…However, methods used to compute the ϕIRF of an LTP system are not necessarily valid for limit-cycle systems, because perturbations can reset the phase of the oscillator, violating the assumption of periodicity. Much of the theory for LTP systems assumes that a transient perturbation produces a transient response (Sandberg et al, 2005), which is not true when the perturbation resets phase. The novelty of the method used in this study is that it accounts for phase resetting and, thus, can be applied to walking.…”

Section: Introductionmentioning

confidence: 99%

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Using a System Identification Approach to Investigate Subtask Control during Human Locomotion

Logan

Kiemel

Jeka

2017

Front. Comput. Neurosci.

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Here we apply a control theoretic view of movement to the behavior of human locomotion with the goal of using perturbations to learn about subtask control. Controlling one's speed and maintaining upright posture are two critical subtasks, or underlying functions, of human locomotion. How the nervous system simultaneously controls these two subtasks was investigated in this study. Continuous visual and mechanical perturbations were applied concurrently to subjects (n = 20) as probes to investigate these two subtasks during treadmill walking. Novel application of harmonic transfer function (HTF) analysis to human motor behavior was used, and these HTFs were converted to the time-domain based representation of phase-dependent impulse response functions (ϕIRFs). These ϕIRFs were used to identify the mapping from perturbation inputs to kinematic and electromyographic (EMG) outputs throughout the phases of the gait cycle. Mechanical perturbations caused an initial, passive change in trunk orientation and, at some phases of stimulus presentation, a corrective trunk EMG and orientation response. Visual perturbations elicited a trunk EMG response prior to a trunk orientation response, which was subsequently followed by an anterior-posterior displacement response. This finding supports the notion that there is a temporal hierarchy of functional subtasks during locomotion in which the control of upper-body posture precedes other subtasks. Moreover, the novel analysis we apply has the potential to probe a broad range of rhythmic behaviors to better understand their neural control.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Using a System Identification Approach to Investigate Subtask Control during Human Locomotion

Logan

Kiemel

Jeka

2017

Front. Comput. Neurosci.

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“…Semi-global stabilization of discrete-time periodic systems with bounded controls has been analyzed in [22]. The convergence of truncated representations of the frequency-response operator of a linear time-periodic system has been studied in [23]. Several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients have been investigated in [24].…”

Section: Introductionmentioning

confidence: 99%

Control of periodic sampling systems subject to actuator saturation

Yang

Shi

et al. 2014

Intl J Robust & Nonlinear

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Summary In this paper, the control problem of linear systems with periodic sampling period subject to actuator saturation is considered via delta operator approach. Using periodic Lyapunov function, sufficient conditions of local stabilization for periodic sampling systems are given. By solving an optimization problem, we derive the periodic feedback control laws and the estimate of the domain of attraction. As the saturation function sat(·) belongs to the sector [0,1], sufficient conditions are derived by constructing saturation‐dependent Lyapunov functions to ensure that the periodic sampling system is globally asymptotically stable. A numerical example is given to illustrate the theoretical results proposed in this paper. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Using the Fourier decomposition of the impulse response operator, the structure of the HTF and the elements fG n g n2Z can be derived leading to the same results as with the EMP approach [28].…”

Section: A Harmonic Transfer Functionmentioning

confidence: 99%

Robust Stability Analysis of a Linear Time-Periodic Active Helicopter Rotor

Maurice

Farolfi

Saupe

et al. 2012

Journal of Guidance, Control, and Dynamics

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Robust stability of a helicopter hingeless rotor model used for air resonance alleviation is investigated in this paper. This aeroelastic phenomenon can be described naturally considering the rotor as a whole through a time-periodic change of coordinates called multiblade coordinates transformation. To assess robust stability of the resulting continuous linear time-periodic system, this paper defines a set of uncertainties specific to the helicopter rotor problem. The uncertain system is written in the form of a linear fractional representation and transformed using a frequency-lifting technique. This leads to a time-invariant representation of the uncertain system, the so-called harmonic transfer function. The extension of the standard -analysis to such lifted systems is proposed in the paper. Compared with a similar approach from literature based on an elementary example, the technique requires a moderate increase of the number of inputs and outputs, thus reducing the numerical effort involved in the computation of the -bounds. The methodology, applied to the uncertain closed-loop helicopter rotor, shows robust stability for the defined set of uncertainty. The results are compared to an analysis performed on the base of a linear time-invariant system to quantify how strongly robust stability bounds are underestimated when the periodicity is not accounted for. Nomenclature A, B, C, D = system, control, measurement, and feedthrough matrices of a linear time-periodic system A, B, C, D = Toeplitz form of time-periodic matrices A, B, C, and D G = harmonic transfer function Y GU G ibc = nominal single input/single output individual blade control model used in the weight definitioñ G i max ibc = admissible single input/single output system allowing the maximal variation from nominal model G ibc G k o ;k i = truncation of the harmonic transfer function G to the k i th input's and k o th output's harmonics G nom = nominal plant with additional inputs/outputs for the parametric uncertainty fG n g n2Z = elements of the harmonic transfer function G H mix = uncertain closed-loop augmented plant in -structure M = blade lagging moments defined in multiblade coordinates, Nm s = complex frequency T = fundamental period of a linear time-periodic system, s Tt = multiblade coordinates transformation from multiblade coordinates to individual blade control u, x, y = input, state, and output signal vectors of a state-space representation U, X , Y = frequency-lifted form of vectors u, x, and y u ibc = trailing-edge flaps command transformed in individual blade control u ibc Tt u unc = multiplicative input of the lumped uncertainty, defined in multiblade coordinates u 1 , u 2 = augmented plant inputs, used for the parametric uncertainty u 3 = multiplicative input of the lumped uncertainty, defined in individual blade control V, W = frequency-lifted form of the uncertainty output and input vectors v and w v, w = uncertainty output and input vectors of a linear fractional representation W = single input/single output individual blade con...

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Frequency-domain analysis of linear time-periodic systems

Cited by 63 publications

References 31 publications

Using a System Identification Approach to Investigate Subtask Control during Human Locomotion

Using a System Identification Approach to Investigate Subtask Control during Human Locomotion

Control of periodic sampling systems subject to actuator saturation

Robust Stability Analysis of a Linear Time-Periodic Active Helicopter Rotor

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