Communication Requirements for the Stability of Platoons

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Communication Requirements for the Stability of
Platoons

• Pete Seiler
• Professor J.K. Hedrick
• Vehicle Dynamics and Control Lab
• University of California at Berkeley
• October 18, 2001

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Automated Highway Systems (AHS)

• AHS are motivated by traffic congestion
• and the desire to increase capacity.
• PATH uses a platooning strategy
• Closely spaced vehicles travel in small
• platoons with large separation
• between the platoons
• Research on the control of vehicle
• strings dates back to the 1960s.

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Simple Vehicle Following

• A reasonable strategy Use a radar to follow a
  fixed distance behind the preceding vehicle.
• Plant
• Error
• Controller
• Error Dynamics

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Amplification of Errors

• Theorem Middleton and Goodwin, 1990 If H(s)
  is a rational transfer function with at least two
  poles at the origin and if the feedback loop is
  stable, then the complementary sensitivity
  function must satisfy
• A simple consequence is that for any
• stabilizing controller there exists an w
• such that T(jw)gt1.
• Interpretation Errors with significant
• energy at this frequency will be amplified.
• This is commonly called string instability.

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Error Amplification Example
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Implications for Control

• Communication of leader information
• solves this problem
• Controller
• Error Dynamics
• Similar results on disturbance amplification
  exist for a large class of bidirectional
  controllers
• If communication of leader information is not
  possible, then performance requirements must be
  weakened.

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Communications and Control

• Motivation Wireless networks can be used to
  solve the problem of error amplification.
• Communication/Control Questions
• What information should be communicated?
• What are the limits of network performance for
  acceptable control?
• Develop tractable conditions for the following
• Analyze degradation in closed loop performance
  due to network
• Given a level of network performance, synthesize
  a controller

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Problem Setup

• Plant
• Network Model
• Augmented open loop model

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Tools for Networked Systems

• The augmented open loop plant falls into the
  general class of Markov Jump Linear Systems.
• We have used filtering theory to determine
  theoretical bounds on required network
  performance for simple problems.
• We have set up an optimization problem that
  allows us to analyze the performance and/or
  synthesize controllers for more complicated
  networked control problems.

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Platoon Example

• We will analyze the effect of the wireless
  network on the performance of a 4 car platoon.
• The vehicle model is discretized at a 20
  msec sample rate.
• Demo controller
• Lead and preceding vehicle information must be
  communicated to each vehicle.

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Platoon Performance vs. Packet Loss
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Conclusions

• Local decentralized controllers lack
  scalability since they are sensitive to
  disturbances. This motivates the need for
  communication between vehicles.
• We developed a computationally efficient method
  to analyze the effect of a network. This
  algorithm can also be used for control design.
• We used this algorithm to analyze the effect of
  the network in a platoon of 4 cars.

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Markovian Jump Linear Systems

• MJLS
• There are many definitions of stability
• Stochastic Stability
• Mean Square Stability
• Exponential Mean Square Stability
• Almost Sure Stability
• Theorem Ji and Chizeck, 1991 For a MJLS,
  conditions 1-3 are equivalent and any one of them
  implies 4.

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H? Conditions

• Plant (P)
• If the plant is stable, define the H? norm as
• Theorem If pijpj for i,j1,,N then the MJLS is
  stable and satisfies iff there exists
  a matrix Ggt0 such that

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Optimal Controllers

• Goal Find a dynamic output feedback controller
  (or show that one fails to exist) such that the
  closed loop system is stable and . We
  assume the controller has the following form
• Closed Loop State Matrices
• Applying the Theorem, the closed loop is stable
  and iff there exists a matrix Ggt0
  such that

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Disturbance Propagation Theorem

• Theorem Given any Mgt0 there exists a platoon
• length such that for any
  KLA(s) or for
• any KBD(s) with no poles at the origin.
• Proof Outline
• For KLA, disturbances are propagated through
  T(S). For example, the effect of d1(s) on eN(s)
  is
• The result follows since T(jw)gt1 at some
  frequency.
• For KBD, disturbances propagate in both
  directions. The closed loop map simplifies to
• First we show that has an
  eigenvalue tending to zero as the platoon length
  tends to infinity. We then construct the bad
  disturbance from the corresponding eigenvector.

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Conditions for Mean-Square Stability

• Theorem Costa and Fragoso (1993) The stability
  of the MJLS is equivalent to the existence of
  matrices Gigt0 for i1,,N satisfying any of the
  following
• Corollary If pijpj for i,j1,,N then the MJLS
  is stable iff there exists a matrix Ggt0 such
  that

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H? Conditions Proof

• Theorem If pijpj for i,j1,,N then the MJLS is
  stable and satisfies iff there exists
  a matrix Ggt0 such that
• Proof Outline
• Sufficiency follows from a stochastic Lyapunov
  argument.
• For necessity, we show that
  implies that a related Riccati equation has a
  solution. This is shown via contraposition.
  This Riccati equality can easily be turned into a
  Riccati inequality which leads to the specified
  matrix inequality.