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Near Optimal Rate Selection for Wireless Control Systems Abusayeed Saifullah, Chengjie Wu, Paras Tiwari, You Xu, Yong Fu, Chenyang Lu, Yixin Chen

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Evaluation: Greedy and Subgradient 14 The control cost in subgradient method is higher Execution time in subgradient method is significantly higher

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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

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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

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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

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New Delay Bounds Derived through convex relaxation of pseudo polynomial time bounds in our prev. work [RTAS, 11] 18 Competitive against polynomial time bounds

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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

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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

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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