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In this paper we consider the problem of disturbance response for a simple system of coupled harmonic oscillators. We suppose that the oscillators are connected in a string in which each oscillator tries to track its predecessor. Motivated by terminology from the problem of vehicle platooning, we say that the system is string unstable if the effect...
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... the series connection, or string, of n single-loop feedback systems depicted in Figure 1. We assume that these systems are all identical, with each plant described by a proper rational transfer function of the form ...
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... that we desire each oscillator in the string to track the position of its immediate predecessor. Following the terminology used in vehicle platooning, we refer to the system in Figure 1 as a predecessor-following control architecture. Denote the commanded position to the lead oscillator by r 1 (t), and the positions and tracking errors of the ith oscillator as y i (t) and e i (t), respectively. ...
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... (6) implies that any disturbance to the lead oscillator at this frequency will be amplified as it propagates to successive oscillators. As the number of oscillators increases, the error will be amplified without bound, and the string in Figure 1 will be string unstable. In studies of string instability in vehicle platooning, one may derive sufficient conditions for string instability using the following integral relation, dual to the Bode sensitivity integral, that must be satisfied by the complementary sensitivity function [8, Theorem 3.1.5]. ...
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... that the right hand side of (13) is nonnegative, and that W (ω, α) > 0 for all frequencies except ω = α. It follows that if L(s) has at least two pairs of poles at ±jα, then there must exist a frequency for which |T (jω)| > 1, and thus that the string of oscillators in Figure 1 is string unstable. Suppose that L(s) contains only a single pair of poles at ±jα, namely, those due to the plant (1). ...
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... 3 (Interpretation of K α ): (a) Consider the se- ries connection of feedback systems in Figure 1, with plant (1) and stabilizing compensator C(s). Assume that r 1 (t) = t sin αt, and define the steady state error for the first system as the response that persists after the transient response decays, denoted by e ss 1 (t). ...
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... next result uses Theorem 3, together with the fact that all the subsystems in Figure 1 are identical, to show that the steady state tracking errors for each subsystem are identical. ...
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... 1: (a) Let e ss k (t) denote the steady state track- ing error of the k'th subsystem in Figure 1 in response to the input r 1 (t) = t sin αt. Then ...
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... goal in the present section is to derive a lower bound on sup ω |T (jω)| that holds whenever the system is assumed to satisfy appropriate performance specifications. If this lower bound exceeds unity, then we may conclude that the system in Figure 1 is string unstable. We will be interested in the case for which neither sufficient condition for string instability derived in Section III is satisfied; however, our methods will also yield a lower bound for the case in which L(s) has at least two pairs of poles at ±jα. ...
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... bounds (21)- (23) constrain the rate at which |T (jω)| converges to one as ω approaches α, and are a consequence of the requirement (20) that the transient response converges rapidly to zero. The following assumption implies that the system in Figure 1 has the ability to track low frequency commands with a specified error. For example, we may wish to apply a command that "steers" the entire string of coupled oscillators to a new position while oscillating. ...
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Citations
... We then extend our string instability analysis to consider the heterogeneous controller design and a more general communication range between oscillators. Preliminary versions of this paper appear in two conference publications [18,19]. ...
In this paper, we consider the problem of disturbance response and error amplification for a simple system of coupled harmonic oscillators. We first suppose that identical oscillators are connected in a string in which each oscillator attempts to track its predecessor by using the same control law that depends on the relative position information from its immediate predecessor. Such an oscillator string is called a homogeneous oscillator string with predecessor-following architecture. Motivated by terminology from the problem of vehicle platooning, we say that the synchronized oscillator system is string unstable if the effect of a disturbance to the lead oscillator is amplified as it propagates along the string. With the use of a new Bode-like integral relation that must be satisfied by the complementary sensitivity function, we provide sufficient conditions for string instability. The sufficient conditions show that any string of oscillators that satisfies certain time domain performance specifications and bandwidth limitations must necessarily be string unstable. We further introduce a concept of time headway for the oscillator system and extend our analysis of string instability to consider the heterogeneous oscillator string and a more general communication range. Copyright © 2014 John Wiley & Sons, Ltd.
... Assuming instead that all side channels have zero capacity, our result recovers a previous bound derived in [30] under the predecessor-following control strategy. Unlike the bounds presented in [25,26], our results do not assume specific plant dynamics. ...
This paper studies the problem of disturbance propagation in a string of vehicles aiming to proceed along a given trajectory while keeping a constant distance between each vehicle and its successor. It is assumed that each vehicle can control its position based on the spacing error with respect to the preceding vehicle in the string, as well as on coded information transmitted by the lead vehicle. Using information-theoretic techniques, this paper establishes a lower bound to the integral of the sensitivity function of spacing errors with respect to a stochastic disturbance acting on the lead vehicle. The derived bound depends on the open-loop poles and zeros of the vehicles’ dynamics as well as on the (possibly nonlinear) controller used at each vehicle. The lower bound is shown to be tight for a specific class of systems and controllers.
In this paper we consider the disturbance response for a heterogeneous string of harmonic oscillators. We assume the individual controllers can communicate with a limited range of neighbors and be tuned independently. Motivated by the speed-dependent spacing policy in vehicle platooning, we develop an extension of the concept of time headway for oscillator systems. Using fundamental limitations theory, we derive a lower bound on the peak magnitude response of the transfer function that describes how a disturbance applied to the lead oscillator propagates along the string to produce a response in the last oscillator. We then give sufficient conditions for string instability in terms of the separation policy, communication ranges, certain time domain performance specifications, and bandwidth limitations.
