Near Optimal Rate Selection for Wireless Control Systems

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

2. Wireless Control System • 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-controlco-design

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

4. Performance Index in Terms of Rate • Continuous sensing • Ideal scenario • Impractical under resource constraints • Digital implementation of control loop i • Periodic sampling at rate fi • Performance deviates from continuous counterpart • Control cost of i under rate fi [Seto, RTSS ,96] • Performance deviation from continuous counterpart • Approximated as under sensitivity coefficients • Performance index • Overall control cost of nloops

5. System Model • Control network model • A WirelessHART network • Control loops are numbered as 1,2, …, n • To maintain stability, sampling rate fi of each loop i • Must be at least fimin • Cannot exceed fimax • Transmission scheduling • Rate monotonic • Real-time requirement: end-to-end delay ≤ sampling period

6. Formulation of Rate Selection • Formulated as a constrained non-linear optimization • Determine sampling rates to Minimize control cost subject to for every control loop i

7. Delay Bounds • 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

8. Polynomial Time Delay Bounds • 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 Delay bound of control loop 5 Rate of Control Loop 6 Rate of Control Loop 5

9. Optimization Space • The dual surface under 2 rate changes among 12 loops Lagrangian Dual function Rate of Control Loop 5 Rate of Control Loop 6 • The dual surface indicates • The existence of an excessive number of local extrema • The difficulty of the optimization problem

10. Solution Approaches • Subgradient method • A standard non-linear optimization approach • Greedy heuristic • The simplest and straightforward approach • For a quick solution • Simulated annealing based penalty approach • A global optimization framework • Gradient method upon convex relaxation • Based on new delay bounds that are convex and smooth

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

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

13. Simulation Using Testbed Topology • 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

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

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:

16. Evaluation: SA-based Penalty Method • Subgradient: the highest cost • Greedy: better than subgradient • SA: the least control cost • Subgradient: longest exec. time • SA: faster than subgradient • Greedy: the fastest

17. Convex and Smooth Delay Bound • Exploration of three methods suggests a balance between execution time and control cost • We derive convex and smooth delay bounds • Renders smooth solution surface Control cost Rate of Control Loop 5 Rate of Control Loop 6

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

19. Gradient Descent Method • New convex and smooth delay bounds reduce our problem to a convex optimization problem • Gradient based approach can be applied • Gradient based steepest descent method • Follows the (unique) gradient at current position • Performance • Fast in execution • Quality solution since the new delay bounds are not overly pessimistic

20. Evaluation: Gradient Descent Method • 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

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