Near Optimal Rate Selection for Wireless Control Systems
Abusayeed Saifullah, Chengjie Wu, Paras Tiwari, You Xu, Yong Fu,
Chenyang Lu, Yixin Chen
Slide 2
Wireless Control System 2 Wireless control network Employs
sensor-actuator control loops In process monitoring and control
WirelessHART Standard for industrial process control Control
performance Depends not only on controller design But also on
real-time communication in shared network Optimizing control
performance under limited bandwidth Requires a scheduling-control
co-design
Slide 3
Rate Selection as Co-Design Effects of low sampling rates
Degraded control performance Effects of high sampling rates
Congestion in the network Long communication delays imply degraded
performance Choice of sampling rates Must balance between control
and real-time communication We address near optimal rate selection
As scheduling-control co-design 3
Slide 4
Control cost of i under rate f i [Seto, RTSS, 96] Performance
deviation from continuous counterpart Approximated as under
sensitivity coefficients Performance Index in Terms of Rate 4
Digital implementation of control loop i Periodic sampling at rate
f i Performance deviates from continuous counterpart Continuous
sensing Ideal scenario Impractical under resource constraints
Performance index Overall control cost of n loops
Slide 5
System Model 5 Control network model A WirelessHART network
Control loops are numbered as 1,2, , n To maintain stability,
sampling rate f i of each loop i Must be at least f i min Cannot
exceed f i max Transmission scheduling Rate monotonic Real-time
requirement: end-to-end delay sampling period
Slide 6
Formulation of Rate Selection 6 Minimize control cost subject
to for every control loop i Formulated as a constrained non-linear
optimization Determine sampling rates to
Slide 7
Delay Bounds 7 We derived delay bounds in a previous work
[RTAS, 11] Iterative fixed-point algorithm needs pseudo polynomial
time Not very practical for expensive non-linear optimization We
extend the results to a polynomial time method We use the
polynomial time delay bounds in our optimization
Slide 8
Polynomial Time Delay Bounds 8 Delay bound of control loop 5
Rate of Control Loop 5 Rate of Control Loop 6 In terms of decision
variables (rates), the delay bounds are Non-linear Non-convex
Non-differentiable Our optimization is thus non-convex,
non-differentiable, not in closed form
Slide 9
Optimization Space 9 Lagrangian Dual function Rate of Control
Loop 5 Rate of Control Loop 6 The dual surface under 2 rate changes
among 12 loops The dual surface indicates The existence of an
excessive number of local extrema The difficulty of the
optimization problem
Slide 10
Solution Approaches Subgradient method A standard non-linear
optimization approach 10 Gradient method upon convex relaxation
Based on new delay bounds that are convex and smooth Simulated
annealing based penalty approach A global optimization framework
Greedy heuristic The simplest and straightforward approach For a
quick solution
Slide 11
Greedy Heuristic A simple and intuitive greedy heuristic To get
a faster solution With a reasonable control cost The approach
Starts by selecting the minimum rate for each control loop
Increases rate of the loop that causes maximum decrease in cost The
procedure is continued as long as all loops are schedulable
Performance observation Very fast in execution Easily gets trapped
into local minima 11
Slide 12
Subgradient Method Traditionally effective to escape from local
extrema Handles non-differentiability and non-convexity Guided by
the subgradients when gradient cannot be determined Gradient method
is unsuitable for our optimization Performance observation
Convergence is extremely slow Quality of solution is extremely bad
Reasons Existence of an excessive number of local minima
Complicated and ineffective subgradient direction 12
Slide 13
Simulation Using Testbed Topology 13 Our sensor network testbed
topology as the control network 74 TelosB motes Spread over Brayn
Hall and Jolley Hall of Washington University The Gateway is
colored in blue
Slide 14
Evaluation: Greedy and Subgradient 14 The control cost in
subgradient method is higher Execution time in subgradient method
is significantly higher
Slide 15
Simulated Annealing (SA) A global unconstrained optimization
framework Requires no gradient information Can easily escape from
local minima Particularly suitable for our problem Subgradient
directions have been seen to be less informative SA-based penalty
approach SA extention for constrained optimization [Chen, Com. Opt.
Vol 47] Constraint violations are penalized with a non-negative
penalty Uses a new objective function: 15
Slide 16
Evaluation: SA-based Penalty Method 16 Subgradient: the highest
cost Greedy: better than subgradient SA: the least control cost
Subgradient: longest exec. time SA: faster than subgradient Greedy:
the fastest
Slide 17
Convex and Smooth Delay Bound We derive convex and smooth delay
bounds 17 Exploration of three methods suggests a balance between
execution time and control cost Rate of Control Loop 5 Rate of
Control Loop 6 Control cost Renders smooth solution surface
Slide 18
New Delay Bounds Derived through convex relaxation of pseudo
polynomial time bounds in our prev. work [RTAS, 11] 18 Competitive
against polynomial time bounds
Slide 19
Gradient Descent Method 19 Gradient based steepest descent
method Follows the (unique) gradient at current position New convex
and smooth delay bounds reduce our problem to a convex optimization
problem Gradient based approach can be applied Performance Fast in
execution Quality solution since the new delay bounds are not
overly pessimistic
Slide 20
Evaluation: Gradient Descent Method 20 Subgradient method is
both inefficient and ineffective Greedy heuristic is very fast but
incurs higher control cost SA incurs the least cost, but takes long
time Gradient method hits the balance between the two metrics
Slide 21
Conclusion Scheduling-control co-design is critical To optimize
control performance in a wireless control system We address the
co-design of optimal rate selection For control networks based on
WirelessHART We study four methods for this difficult optimization
Greedy heuristic: very fast at the cost of higher control cost
Subgradient method: ineffective due to many local minima SA: low
control cost at the cost of long execution time Elegant approach: a
convex relaxation with smooth delay bounds hits balance between the
two 21