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Disturbance propagation in large interconnected systems
Abstract
Focuses on control of vehicle strings and the theoretical issues arising from this problem. The key result is that a control structure where each vehicle uses only information about its predecessor is fundamentally sensitive to disturbances. Specifically, small disturbances acting on one vehicle can propagate and have a large effect on another vehicle. A similar, though less general, result is derived for a control structure where each vehicle looks at both neighbors.
... Other authors have discussed the decentralized control of nested systems in the past [14]- [17], however, considering different performance criteria. In addition, Seiler et al. [18] analyzed potential disturbance propagations that can arise from interconnected systems, demonstrating that one needs to have access to the leader's information to prevent disturbances from amplifying downstream. ...
... We now want to apply Lemma 5; however, this can only be done when r = 0. To this end, note Remark 3, and we augment the controller memory to L = L +r . From Lemma 5, we know that (18) implies the existence of L, H ∈ N 0 and positivedefinite {X C } ∈A L+H such that the following is satisfied for all i ∈J and ∈ A L+H +1 : ...
In this article, we develop synthesis methods for decentralized control of switched systems with mode-dependent (more generally, path-dependent) performance specifications and apply them to an experimental vehicular testbed. This specification flexibility is important when achievable system performance varies greatly between modes, as a mode-independent specification will lead to designs that do not extract all the system performance available in each mode. More specifically, under these specifications, we derive exact conditions for the existence of block lower triangular path-dependent controllers with ℓ₂-induced-norm performance. The synthesis conditions are given in the form of a semidefinite program (SDP) for both uniform and path-dependent performance bounds. Building on previous results, we also introduce a basis-based approach that allows computational complexity to be more carefully controlled. We apply the resulting design methods to a group of miniature quadcopters, illustrating different features of the presented methods, along with some practical engineering considerations.
... Canudas de Wit and Brogliato [2] provided a detailed overview of string stability and how various control polices and inter-vehicle spacing strategies affect string stability. Seiler et al [10] analyzed various classes of linear controllers with regard to string stability. In the area of non-holonomic mobile robot control, Aguiar et al [1] used Lyapunov functions to design a nonlinear controller that produces smooth trajectories. ...
... This result has been proved for systems that employ more complicated dynamics by many others (e.g. Peppard [9], Seiler et al[10], etc). Leader-centric Controllers: Leader-centric controllers implement two constraints. ...
This paper analyzes the stability properties of a decentralized hybrid control system for maintaining formations. Utilizing
only local sensing, the system assembles strings or “platoons” of robots that has each robot maintaining a fixed bearing to
its nearest neighbor. Using these platoons, the system is able to construct more complicated geometries. A piecewise linear
controller based on bidirectional controller design is utilized to ensure the stability of the system. The system is demonstrated
in simulation as well as on a physical set on non-holonomic mobile robots.
... The study of stability and disturbance propagation in vehicle formations is increasingly receiving deserved attention from the controls community. The range of applications which have motivated such research is wide -from car platoons and formation of robots to unmanned air vehicles (UAV) [1], [2], [3], [4]. ...
Recent works have combined tools from graph theory and control theory to derive stability criteria for vehicle formations. The formulation typi-cally models information exchange between vehicles in a formation by using graph-theoretic tools. This work builds upon these ideas by studying the link between topology of these graphs and the dynamical behavior of a formation subject to additive disturbances. The problem is re-formulated by using Kronecker products. It is then shown that if the eigenvalues of the graph Laplacian matrix are real and distinct, then the eigenvectors of the graph Laplacian are singular disturbance directions for the formation. In the general case, the Schur decompo-sition of the graph Laplacian is used to decompose the study of the disturbance propagation in the formation into a number of studies on smaller subsystems. Illus-trative examples demonstrate the scheme.
... There has been a significant amount of research in to distance based vehicle self-separation in different domains such as highways and military formations of unmanned aerial vehicles. Seiler et al, 2002 [16] summarise these results by concluding "that 'string stability' cannot be obtained when vehicles use only relative spacing information to maintain a constant distance behind their Copyright © 2007by EUROCONTROL predecessor." Slater, 2002 showed using a linear aircraft model and linear controller that "controls based only on the neighbouring vehicle result in unacceptable performance, but good string behaviour can be achieved if lead vehicle state is included in the feedback law". ...
Research results from domains such as road highway traffic and military vehicle platoons conclude that string stability cannot be obtained when vehicles use only relative spacing information to maintain constant distance behind a predecessor. Air traffic studies confirm these results but such instability has not been shown for constant time delay based airborne spacing. Unlike constant distance based, constant time delay based spacing has the potential to enhance stability by anticipating changes in spacing using the preceding aircraft’s history. This simulation based study analysed the merging of aircraft by constant time delay based spacing over the period of the order of an hour to observe any build up of error propagation effects. Aircraft descended from 12,000 feet to 4,000 feet, each trying to achieve ninety seconds spacing with respect to its predecessor. The spacing anticipation time for each trailing aircraft to react to the preceding aircraft’s time history was varied from 0 to 20s. Without anticipation, a time spacing error was observed to propagate at about 20 knots groundspeed in a forwards direction (towards runway) growing to about -3.5s (trail aircraft too early and too close). This compression wave was avoided by increasing spacing anticipation to 10s. Values larger than 10s reversed the error and moved it upstream. A tuned scenario was repeated for 5,400 random values of initial time spacing error and top of descent with automatic and manual airborne spacing modes. Time spacing accuracy and pilot activity were measured to be within required values previously derived for a pair of aircraft but the corresponding cost in speed variation was higher.
... Recent article [15] addressed some fundamental design limitations in vehicular platoons. In particular, it was shown that string stability of a finite platoon with linear dynamics cannot be achieved with any linear controller that uses only information about relative distance between the vehicle on which it acts and its immediate predecessor. ...
We investigate some fundamental limitations and tradeoffs in the control of vehicular platoons. These systems are of considerable practical importance as they represent an example of systems on lattices in which different units are dynamically coupled only through feedback controls. We demonstrate that in very long platoons, to avoid large position, velocity, and control amplitudes, one needs to explicitly account for the initial deviations of vehicles from their desired trajectories. We further derive explicit constraints on feedback gains - for any given set of initial conditions - to achieve desired position transients without magnitude and rate saturation. These constraints are used to generate the trajectories around which the states of the platoon system are driven towards their desired values without the excessive use of control effort. All results are illustrated using computer simulations of platoons containing a large number of vehicles.
We investigate the effects of actuator and sensor bandwidths on robust stability of intelligent cruise control of autonomous vehicles with different feedback biasing strategies for two types of boundary conditions. The first one is a platoon of vehicles on an open stretch of highway, and the other one with periodic boundary conditions. The goal of the feedback is to maintain the same relative velocity and keep a certain safe distance between adjacent vehicles. We obtain in closed form the stability regions in the parameter space of the feedback gains for different biasing strategies of controllers. If there is biasing, it is shown that the two types of boundary conditions engender qualitative different stability characteristics: when the number of vehicles n is large, the robust stability in the case of periodic boundary conditions is much weaker than open boundary conditions. The spatially discrete model is approximated by a continuum model described by partial differential equations. The n-mode truncation of the continuum model and the spatially discrete model predict approximately the same region of stability for lower order modes, as well as the same qualitative features of overall stability region in the parameter space of feedback gains.
A passive decomposition framework for the formation and maneuver controls for multiple rigid bodies is proposed. In this approach, the group dynamics of the multiple agents is decomposed into two decoupled systems: The shape system representing internal group formation shape (formation, in short), and the locked system abstracting the overall group maneuver as a whole (maneuver, in short). The decomposition is natural in that the shape and locked systems have dynamics similar to the mechanical systems, and the total energy is preserved. The shape and locked system can be decoupled without the use of net energy. The decoupled shape and locked systems can be controlled individually to achieve the desired formation and maneuver tasks. Since all agents are given equal status, the proposed scheme enforces a group coherence among the agents. By abstract-ing a group maneuver by its locked system whose dynamics is similar to that of a single agent, a hierarchical control structure for the multiple agents can be easily imposed in the proposed framework. A decentralized version of the controller is also proposed, which requires only undirected line communication (or sensing) graph topology.
Zusammenfassung
Funkbasierte Kommunikationsnetze bieten die Möglichkeit, Informationen zwischen sich bewegenden Objekten zu übertragen und die Informationsverbindung immer dann zu nutzen, wenn eine Annäherung der Objekte untereinander eine Koordination ihrer Regelungen erfordert. Die wichtigste Frage bei derartigen Regelungen lautet: Welche Informationskopplungen sind nützlich und welche Verbesserungen der Regelgüte können durch diese Informationskopplungen erreicht werden? Diese Frage wird für Systeme beantwortet, die aus einer Reihenschaltung von Teilsystemen mit gleichartiger Dynamik bestehen und deren Regelung bei schwacher Kopplung eine hinreichende Autonomie der Teilsysteme und bei starker Kopplung ein kooperatives Verhalten der Teilsysteme im Sinne der Güteforderungen an das Gesamtsystem gewährleistet.
This note focuses on disturbance propagation in vehicle strings. It is known that using only relative spacing information to follow a constant distance behind the preceding vehicle leads to string instability. Specifically, small disturbances acting on one vehicle can propagate and have a large effect on another vehicle. We show that this limitation is due to a complementary sensitivity integral constraint. We also examine how the disturbance to error gain for an entire platoon scales with the number of vehicles. This analysis is done for the predecessor following strategy as well as a control structure where each vehicle looks at both neighbors.
An important aspect of an Automated Highway System is automatic vehicle following. Automatic Vehicle follower systems must address the problem of string stability, i.e., the problem of spacing error propagation, and in some cases, amplification upstream from one vehicle to another, due to some disturbance at the head of the platoon. An automatic vehicle following controller design that is (asymptotically) stable for one vehicle following another is not necessarily (asymptotically) stable for a string of vehicles. The dynamic coupling between vehicles in such close-formation platoons is a function of the available information (communicated as well as sensed), decentralized feedback control laws and the vehicle spacing policy in use. In the first half of this paper, we develop a framework for establishing conditions for stability of the string in the presence of such dynamic interactions. We then develop a metric for analyzing the performance of a platoon resulting from different vehicle following control algorithms. This metric is the guaranteed rate of attenuation/non-amplification of spacing errors from one vehicle to another. In the latter half of this paper, we outline and analyze various constant spacing vehicle follower algorithms. All these algorithms are analyzed for sensing/actuation lags.
"April 1997." Thesis (Ph. D. in Mechanical Engineering)--University of California, Berkeley, 1994.
In this paper we introduce the notion of string stability of a
countably infinite interconnection of a class of nonlinear systems.
Intuitively, string stability implies uniform boundedness of all the
states of the interconnected system for all time if the initial states
of the interconnected system are uniformly bounded. It is well known
that the I/O gain of all the subsystems less than unity guarantees that
the interconnected system is I/O stable. We derive sufficient
(“weak coupling”) conditions which guarantee asymptotic
string stability of a class of interconnected systems. Under the same
“weak coupling” conditions, string stable interconnected
systems remain string stable in the presence of small
structural/singular perturbations. In the presence of parameter
mismatch, these “weak coupling” conditions ensure that the
states of all the subsystems are all uniformly bounded when gradient
based parameter adaptation law is used and that they go to zero
asymptotically
This paper presents a systematic analysis of a longitudinal control law for a platoon of non-identical vehicles using a non-linear model to represent the vehicle dynamics of each vehicle within the platoon. The basic idea is to take full advantage of recent advances in communication and measurement and using these advances in longitudinal control of a platoon of vehicles: in particular, we assume that for i = 1,2,... vehicle i knows at all times vl and al (the velocity and acceleration of the lead vehicle) in addition to the distance between vehicle i and the preceding vehicle, i - 1.
The automatic control of a string of high-speed, densely-packed vehicles is studied in a unified approach. Various input structures for the control schemes and their stability are studied and compared. The optimal decentralized regulation theory is applied to seek the best feedback control gains for various input structures. Value of information is defined to evaluate the relative merit of these inputs. The results can be applied for on-board computer designs in developing new transportation technology such as 'personal rapid transit' systems.
This paper presents a preliminary system study of a longitudinal control law for a platoon of nonidentical vehicles using a simplified nonlinear model for the vehicle dynamics. This study advances the art of automatic longitudinal control for a platoon of vehicles in the sense that it considers longer platoons composed of nonidentical vehicles, furthermore, the longitudinal control laws presented in this study take advantage of communication possibilities not available in the recent past. We assume that for i = 1, 2, . . . vehicle i knows at all times v1 and a1 (the velocity and acceleration of the lead vehicle) in addition to the distance between vehicle i and the preceding vehicle, i - 1. A control law is developed and is tested on a simulation of a platoon of 16 vehicles where the lead vehicle increases its velocity at a rate of 3 m.s-2; it is shown that the distance between successive vehicles does not change by more than 0.12 m in spite of variations in the masses of the vehicles (from the nominal), of communication delay and of noise in measurements.
Automated vehicles traveling in platoons must exhibit stability both individually and as a group, a property referred to as "string stability". We propose a new framework for evaluating the longitudinal string stability properties of platoons of automated vehicles. In this framework, the platoon is considered to be a mass-spring-damper system with linear characteristics. The resulting closed-loop representation yields transfer functions and impulse responses that can be analyzed to determine the string stability properties of the platoon. This framework facilitates qualitative comparisons of the effects of various controller characteristics, such as time headway and intervehicle communication, on string stability.
This paper discussed the conditions under which longitudinal disturbances in strings of coupled vehicles can be made to attenuate as they travel upstream. The “slinky” effect is defined as the condition where these disturbances are amplified, a condition that can lead to collisions among upstream vehicles. Input/output linearization is applied to linearize the spacing error to throttle dynamics. Well known linear norm relationships are then applied to investigate the “slinky” effect It is shown that broadcasting the velocity and acceleration of the lead vehicle to all vehicles in die string is sufficient to eliminate the slinky effect.