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

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

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

 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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