Problem Statement Given a control system where components, i.e. plant, sensors, controllers, actuators, are connected via a communication network, design an optimal controller for the system Example Pursuit Evasion Games (PEGs) Team of pursuers needs to catch evaders within the field of interest Sensor network provides pursuers’ team sensing data about evaders using a multihop network. A central controller uses this data to generate control inputs for the pursuers (cooperative strategy). Control inputs are sent wirelessly to each pursuer. Observation and control packets maybe lost or randomly delayed. Control Issues Random data loss and delay cannot be neglected; they must accounted for in the control design. Classical control theory does not take into account packet losses. Existing estimation and control algorithms need to be modified accordingly. Packet loss, delay depend on the specific communication infrastructure. Our Approach For the sake of generality, we want to keep our analysis at high level of abstraction, extracting the necessary network parameters. We will not consider delay; in other words a packet delayed over its usable time is considered lost. A simple loss model: Bernoulli processes of parameters to model packet losses between sensors and estimator and between controller and actuators respectively Protocol Types: TCP-like: Packet acknowledgement is provided UDP-like: Packet acknowledgement is absent We want to address and solve the optimal estimation and control problems. Plant is linear time invariant and noise is gaussian Model is discrete time and synchronous November 18, 2004 TCP-like protocols: Separation Principle applies Estimation: Kalman Filter is the optimal estimator Error covariance P t diverges if the arrival probability is lower than the critical value, which is a function of the fastest system eigenvalue LQG Control Optimal Controller is linear for both finite and infinite horizon case Infinite horizon: The optimal LQG controller bounds the state if the following condition on the arrival probability holds: On the Control of Networked Embedded Systems (NESs) Bruno Sinopoli Shankar Sastry Plant Aggregate Sensor Controller State estimator Communication Network Communication Network System Estimator Plant z -1 Model z Controller UDP-like protocols: In general Separation Principle does not hold anymore Estimation and control are coupled Optimal control problem is nonlinear, non-convex (sum of squares) An approximate linear static feedback can be derived Performance comparison  max  min       c i cPt ct t PtMPE PPE ||max 1 1 0condition initialany and 1for ][ 0condition initial some and 0for ][lim Conclusions: Issues of delay and packet loss cannot be neglected in the analysis and control of NESs We introduce a simple packet loss model and provide analytical tools to design estimators and controllers for two types of protocols: TCP-like: this protocol is more difficult to implement but it yields a linear controller, easier to compute UDP-like: easier protocols design, more complex controller NO FREE LUNCH: where do you place you complexity?