Control and the Synchronous Model of Computation

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Control and the Synchronous Model of Computation

• Raja Sengupta
• Systems Program, Civil Environmental
  Engineering
• University of California, Berkeley
• http//path.berkeley.edu/raja
• Joint Work with
• S. Dickey, J. Misener, D. Nelson, S. Tan, S.
  Rezaei, P. Seiler, Q.Xu, M. Zennaro, UC Berkeley
• H. Krishnan, General Motors

2
Relevant Papers

• Xu Q., Sengupta R. Real-time Estimation of a
  Markov Process over a Noisy Digital Communication
  Channel. Submitted to the IEEE Transactions on
  Automatic Control, June 2005.
• Zennaro M. Sengupta R. Distributing Synchronous
  Programs Using Bounded Queues. Accepted by EMSOFT
  2005.
• Seiler P., Sengupta R. An H_infinity Approach to
  Networked Control. IEEE Transactions on Automatic
  Control, vol.50, no.3, March 2005, pp.356-64.
• Ergen M., Lee D., Sengupta R., Varaiya P.
  Wireless Token Ring Protocol. IEEE transactions
  on Vehicular Technology, vol.53, no.6, Nov. 2004,
  pp.1863-81.
• Seiler P., Sengupta R. A Bounded Real Lemma For
  Jump Systems. IEEE Transactions on Automatic
  Control, vol.48, no.9, Sept. 2003, pp.1651-4.

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With the advent of Digital Computing Control
linked to the Synchronous Model of Computation
Verified (stability,safety, fairness,optimality)
Syntax
Semantics
Execution Platform
compiled
relevance
P. Varaiya. A question about hierarchical
systems. System Theory modeling, analysis and
control, Kluwer, 2000.
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With the advent of Digital Computing Control
linked to the Synchronous Model of Computation
Caspi P. Embedded Control From Asynchrony to
Synchrony and Back, EMSOFT 2001
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With the advent of Digital Computing Control
linked to the Synchronous Model of Computation
Caspi P. Embedded Control From Asynchrony to
Synchrony and Back, EMSOFT 2001
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Extend Control to Networked Environments?
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Extend Control to Networked Environments?
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Distributing the Synchronous Model
Cascade Composition
Berry 91
yf(u)
ug(y)
Feedback Composition Fixpoint Semantics

• Both order and timing would have to be enforced
  across networks

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Tension Execution time in each round could
become highly variable
IEEE 802.11b Synchronous Broadcast Performance
Data (DCF)
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The Logical Order can be enforced Without
Scheduling! Zennaro, Sengupta EMSOFT05

• Implementation problem given a map ? from RA to
  STS traces we want an implementation map ? such
  that, for all STS s and RA r the following
  holds
• (r ?(s) ? r?t) ? s ? ?(t)
• Modularity preservation we seek a composition
  operator xRA with respect to which ? is a
  monomorphism between (STS, xSTS) and (RA, xRA).
  We want this operator to be implementable across
  a network

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BDSP library contd
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In certain environments the timing can be
enforced as well

• Time Triggered Architecture (Kopetz etal.) is a
  wired example
• Simulink to Lustre to TTA - Tripakis etal. 04
• The token ring protocol is a wireless example
  (Sengupta etal. 04)

Token Rotation Time
Rotation number
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The DES Supervisory Control Theory Response
Control in a CSP-style Model of Computation
Plant
Supervisor
!c1
?c1
!c2
X
?u1
!u1
?u2
!u2

• The sender blocks till the receiver receives
• Ramadge 87

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The DES Supervisory Control Theory Response
Control in a CSP-style Model of Computation
Plant
Supervisor
!c1
Network
?c1
!c2
X
?u1
!u1
?u2
Network
!u2

• The rendezvous time could be variable
• Not the right model for real-time computation
• A real-time computation should not block for
  non-deterministic time

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The Hybrid Systems ResponseA non-blocking
Synchronous thread with a blocking Asynchronous
thread
x gt 1 ? !output
xf(x)
xg(x)

• The hybrid system paradigm responded to this by
  making each component have two concurrent threads
  of computation
• Deshpande SHIFT IEEETAC, Sifakis rtss 99
• All dataflows between the asynchronous process
  and the synchronous process are local

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The Hybrid Systems ResponseA non-blocking
Synchronous thread with a blocking Asynchronous
thread

• The hybrid model semantics and execution might
  differ significantly if one tried to send data
  over a network from the synchronous thread

?output (z)
x gt 1 ? !output (x)
xf(x)
xg(x)
Rcv Rndzvs Ack
Req rndzvs
Read x
Rndzvs Ack
Rcv x
comm delay
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Networked Control Systems

• DES/Hybrid Systems is generalizing into NCS
• DES/Hybrid Model of Computation is mostly
  CSP-style (RPC)
• Sender blocks message by message
• Generalization
• Kahn-style models of computation enabling a
  sender to stream data
• Semantics explicitly accounting for network
  transport performance
• Loss/Erasure, delay, distortion

send
receive

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Networked Control Systems

• DES/Hybrid Systems is generalizing into NCS
• DES/Hybrid Model of Computation is mostly
  CSP-style (RPC)
• Sender blocks message by message
• Generalization
• Kahn-style models of computation enabling a
  sender to stream data
• Partially blocking sender Obtained by TCP,
  messages flow FIFO, blocks when buffers full
  (Tilbury etal. 01)
• Synchronous non-blocking Unreliable periodic
  FIFO message flow (Seiler Sengupta etal. 01)
• Asynchronous non-blocking Aperiodic FIFO message
  flow (Hespanha etal. 04)

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NCS Non-blocking synchronous senderMarkovian
Jump Linear Semantics
Sync state estimate
Sync state output
Intermittent output
Network
Estimator
Controller
Vehicle 1
Vehicle 2
System 2

• Estimator An asynchronous to synchronous
  converter in the interior of an end-to-end
  synchronous system

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Problem Setup

• Plant
• Network Model
• Augmented Plant

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The Optimal Estimator in the LQG case(Seiler,
PhD thesis 2001)

• Problem Given all past measurements,
  y(0),,y(k), and network observations,
  q(0),,q(k), find the state estimate,
  which minimizes
• There is a separation theorem
• Theorem The optimal state estimate is given by
  the Kalman Filter

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Stability of Kalman Filter

• Typically (A,C) detectable ensures that the
  Kalman filter is stable M(k) stays bounded as
  k??. For this problem, M(k) is a stochastic
  process
• If EM(k) grows unbounded as k??, then the
  infinite time LQG cost is infinite and plant
  cannot be stabilized by any controller.
• What conditions must the network satisfy to
  ensure that EM(k) stays bounded?

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Stability of Kalman Filter

• Theorem Assume (A,Bw) is stabilizable. EM(k)
  grows unbounded if p r(A)2 ? 1. If C is
  nonsingular then p r(A)2 lt 1 is sufficient for
  EM(k) to stay bounded.
• Proof (1)Use the following matrix inequality and
  induction
• (2) If C is nonsingular,
  . Use this to derive an upper bound

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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
  mean square stable and satisfies iff
  there exists a matrix Ggt0 such that

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Platoon Example

• Use feedback linearization and model each vehicle
  as
• Spacing error
• State space vehicle dynamics
• Demo controller

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4-car Platoon Example
e
a0
u
gu
gn
n
P
e
u
Network
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Vehicle Following Performance vs. Packet Loss
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Classical Information Theory

• If the source is ergodic the minimal
  communication rate to transmit the source with an
  arbitrarily small probability of error is set by
  its entropy
• Shannon 1948
• Rate-Distortion function (Shannon 1959)
• The minimal bit rate R to represent a continuous
  alphabet source to achieve a given distortion D
• Distortion is the limit of average error
• D
• These rates are achieved as communication delays
  go to infinity
• We care about finite time performance

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Literature

• Classical rate-distortion theory
• Infimum of the rate required to achieve given
  distortion
• Distortion is defined in asymptotic way
• Channel is not considered
• Neuhoff and Gilbert, 82
• Problem Causal source code
• Difference
• Source code, dont consider channel
• Study asymptotic property
• Varaiya and Walrand 83
• Problem Optimal causal coding/decoding over
  memoryless channel
• Difference
• Discrete alphabet and Hamming distance
• Channel with noiseless feedback
• Farvardin, Vaishampayan, 87, 91
• Problem Quantization over noisy channel
• Difference Consider memoryless quantization,
  algorithm solution to reach local optimality
• Brockett and Wang, 97
• Problem Design coder and estimator that have
  stable mean square error

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Literature

• Sahai, 00
• The highest achievable data rate (anytime
  capacity) of a channel such that the bit error
  probability decays with delay at given rate.
• Tatikonda, 01
• Problem The lower bound of rate required for
  controllability, observability and stability
• Difference Bit-rate constraint without channel
  error
• Seiler, Sengupta, 01
• Problem Control and estimation over wireless
  channel
• Difference Real erasure channel, no quantization
• Tunc and Varaiya, 02, 03
• Problem State estimation when the measurement is
  transmitted over binary symmetric channel
• Difference
• Look for the coder/decoder to stabilize
  estimation error
• Noiseless channel feedback
• Gastpar, Rimoldi, and Vetterli 03
• Problem Optimal causal source/channel coding
• Difference Consider asymptotic property
• Tenekietzis, 04
• Problem Real-time optimal coding and decoding
  over noisy channel
• Difference Discrete alphabet source

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The MMSE Problem Statement
Given and design and to
minimize the mean square error
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Our Problem Statement
Markovian System
Encoder
Memoryless Digital Channel
Receiver Memory Update
Decoder
Note perfect memory case is included with
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Structure of the optimal encoder and decoder

• Theorem 1 For fixed decoder, there is no loss of
  optimality if one restricts attention to an
  encoder of the form
• Encoder is a function of the current state and
  the probability density function of the state of
  the receiver
• Generalization of Teneketzis04 to continuous
  valued case
• Theorem 2 For fixed encoder, the optimal decoder
  is the expectation of conditioned on all
  received symbols, i.e.
• and when receiver has perfect memory
• Standard result

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The optimal encoder has a threshold structure

• Theorem 3 The optimal encoder
  partitions with hyperplanes
• For scalar case, there is an optimal encoder that
  is a threshold encoder
• Optimal encoder design is a finite dimensional
  optimization problem of dimension no greater than
  the channel alphabet
• Extends Farvardin 87 result from IID process to
  Markov process

1
0


X
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Iterative algorithm for a locally optimal
threshold Scalar system, Binary symmetric
channel

• The best threshold T, for fixed reconstruction
  levels, is the mid-point of R0 and R1.
• The best reconstruction levels R0 and R1, for
  fixed threshold T, are the expectation of X
  conditioned on received symbol and receiver
  memory.
• Iteratively optimize T and R0 and R1.
• The MSE decreases with each iteration and
  therefore the algorithm converges.
• Algorithm is similar to Farvardin 87,
  Blahut-Arimoto 72 for rate-distortion functions,
  Lloyd-Max quantization without noisy channel

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Conjecture

• We think it may be a good idea to encode the
  innovation, i.e.,
• We examined the problem of optimal encoding of a
  memoryless Gaussian source

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Memoryless Gaussian source over binary symmetric
channel Problem
Gaussian Random vector to be transmitted
Encoder
Binary Symmetric Channel
Decoder
For any given Q(), the optimal decoder is
conditional expectation
Objective function
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Memoryless Gaussian source over binary symmetric
channel Results

• Theorem 4
• To transmit a zero-mean non-singular n-dimension
  Gaussian random vector X over the binary
  symmetric channel, the optimal encoder is a
  hyperplane through the origin and orthogonal to
  the principal component
• Encode the principal component

X1
0
k
1
X2
1
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Example 2-D Vector Case Result
X1
0
k
1
X2
1
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Control Is Still Exploring New Models Of
Computation
Computer Sciences
Control Sciences
Operations Research
Petri Nets/ Stroboscope/ CSIM
Simulink/ Esterel/ S/R, SDF
Asynchronous Product
CCS
CSP
Strongest composition and mathematical properties
Hard to enforce at short time scales over large
distances
Weak composition and mathematical properties.
Easy to realize over distances even within
milliseconds

• Procedures of component integration are well
  defined for compilation
• Mathematically derive system behavior from
  component behavior