1. Optimization-based Cross-Layer Design in Networked Control Systems Jia Bai, Emeka P. Eyisi Yuan Xue and Xenofon D. Koutsoukos

2. Outline Dynamic Tuning Retransmission Limit of IEEE 802.11 MAC Protocol for Networked Control Systems – From controller to network –Essential information from controller to network Performance –Network support Optimal retransmission scheme at MAC layer Distributed Sampling Rate Adaptation for Networked Control Systems –From network to controller –Essential information from network to controller Congestion signal (in terms of price) –controller support Sampling rate adaptation Plantcontroller network Application Data/ Performance/ Congestion

3. Overview Goal –Optimize the overall control system performance Approach –Decompose the design concern into two factors Control layer: ensure stability using a passivity-based architecture Communication layer: optimize performance by adjusting the network operation parameters –Focus on the design of IEEE 802.11 wireless network Studying the impact of MAC retransmission on NCS performance

4. System Model (1/3) Networked control system –Control layer: a controller controls the plant, a robotic arm, to follow –Communication layer: UDP connection over IEEE 802.11 wireless network Stability vs. performance –Stable: system output stay bounded for any bounded input –Performance: how fast and accurate the plant can track the control signal within bound

5. System Model (2/3) Control Layer –Passive control architecture for the digital control of a plant over wireless network Plant: continuous linear time-invariant system Controller: discrete-time linear time-invariant system Block b: transformation of power variables and wave variables, preserving system passivity

6. System Model (3/3) Communication layer –A wireless channel using IEEE 802.11 MAC protocol with random errors –A corrupted frame got retransmitted –Given packet loss probability, a larger retransmission limit increases successful transmission rate, but with longer delay.

7. Observations (1/4) Goal –Understand how network loss and delay affect the NCS performance –How retransmission strategy can be designed to minimize the effect Metrics –The velocity of the plant tracks a sinusoidal reference input –Performance evaluated by instantaneous tracking error

8. Observations (2/4) Effect of packet loss –Disable MAC retransmission mechanism –(a): average tracking error with standard deviation –(b): maximum tracking error

9. Observations (3/4) Effect of network delay –Loss free network –D is the amount of delay introduced before packet transmission –Performance degrades when D exceeding certain a sampling time

10. Observations (4/4) Effect of MAC retransmission

11. Observation summary NCS performance based on passivity is negatively affected by network factors MAC retransmission limit and the NCS performance follows a convex relation The optimal value depends on control system properties as well as network factors Fixed retransmission limit in standard MAC is not optimal for NCS performance Retransmission limit needs dynamic adjustments

12. MAC control design (1/2) Dynamically adjust the retransmission limit based on the feedback from the control system –Keep the performance error within threshold Piggyback the update on data packets Form a time-scale-decomposed system

13. MAC control design (2/2) Indicate whether NCS is within the error threshold The change of the retransmission limit follows the change of the tracking error

14. Simulation results Impact of initial value of retransmission limit –With no background traffic –Wireless channel error probability 83%

15. Simulation results (2/3) Impact of background traffic –Fixed packet error probability of 50% –Varying background traffic with Poisson distribution

16. Simulation results (3/3) Impact of channel error probability –One pair of Poisson background traffic –Packet error probability of 40% and 80%

17. Concluding Remarks NCS design is decomposed into two spaces: –Stability guaranteed by a passive control structure –Performance optimized by adjusting networked protocol parameters Convex relation between retransmission limit and NCS performance exists MAC-layer control algorithm designed to dynamically tune the retransmission limit

18. Outline Dynamic Tuning Retransmission Limit of IEEE 802.11 MAC Protocol for Networked Control Systems – From controller to network –Essential information from controller to network Performance –Network support Optimal retransmission scheme at MAC layer Distributed Sampling Rate Adaptation for Networked Control Systems –From network to controller –Essential information from network to controller Congestion signal (in terms of price) –controller support Sampling rate adaptation Plantcontroller network Application Data/ Performance/ Congestion

19. Overview Goal –Sampling rate adaptation for dynamic resource management –Maximize the aggregated NCS system performance –Fully exploit the varying available bandwidth Approach –Quantify the relationship between the NCS performance and the sampling rate using a utility function –Formulate NCS sampling rate adaptation as an optimal resource allocation problem where aggregated system utility is maximized subject to the bandwidth constraint

20. System Model

21. Control System Model Continuous time system Discrete time system Zero-mean white noise

22. Utility Function When the system is discretized to implement a digital controller over a network or computer, the control system response to disturbances degrades compared to the continuous closed loop case. To characterize the impact of sampling rate on the disturbance rejection ability of a digital controller, we consider the ratio of the discrete-time system plant state and its continuous-time counterpart. Root mean square of the noise covariance matrix

23. Network Model Multihop Wireless Network –Topology graph –Node set –Wireless link set End-to-End Flow –Flow set –Assumption: Flow route known a priori –Flow passes a set of wireless links –Routing Matrix 12345 6 7 f1f1 f2f2

24. Wireless Transmission Model Physical Channel –Single wireless channel with capacity Contention Model –IEEE 802.11 style protocol model [Gupta00] –Two transmissions contend if their sources or destinations are within the communication range 1234 ack data ack data

25. Contention Model Contention Graph [Luo00][Nand00][Jain03] – –Vertex: wireless links in the topology graph –Edge: two links contend with each other 12345 6 7 {1,2} {2,3} {3,6} {4,5} {6,7} {3,4}

26. Resource Model {1,2} {2,3} {3,6} {4,5} {6,7} {3,4} Resource Element in Multihop Wireless Network Facet of independent set polytope of contention graph Approximation: Maximal clique in contention graph Capacity under ideal MAC scheduling Constraint on Wireless Link Rate Allocation –Wireless link aggregated rate –

27. Constraint on End-to-End Flow Rate Allocation –Clique-flow matrix –Resource constraint Resource Constraint {1,2} {2,3} {3,6} {4,5} {6,7} {3,4} 12345 6 7 f1f1 f2f2 f3f3 f4f4 f 1 f 2 f 3 f 4 q1q1 q2q2 q3q3 +  Routing matrix

28. Problem Formulation Optimization Objective –Optimal resource allocation Optimization Constraint –Resource constraint Putting things together

29. Optimization Solution Lagrangian Form Dual Problem Gradient Projection

30. Sampling Rate Adaptation Iterative Algorithm Convergence and Stability The algorithm converges to the unique optimal point

31. Economical Interpretation Price serves as a signal to distributed resource allocation price Resource price User price User choice: maximizing net benefit User demand Relation between demand and supply User price: depending on usage pattern

32. Simulation results Convergence Performance Comparison With Rate Adaptation Fixed Sampling Rate

33. Concluding Remarks Distributed sampling rate adaptation enables dynamic resource management –Improved NCS performance –Better wireless resource utilization Utility function characterizes the performance in terms of robustness of NCS Utility-based optimization framework leads to distributed solution