# NCS: Packet-based Control: the UDP case

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In this lecture we consider the Linear Quadratic Gaussian (LQG) optimal control problem in the discrete time setting and when data loss may occur between the sensors and the estimation-control unit and between the latter and the actuation points. We focus on the case where the arrival of the control packet is acknowledged at the receiving actuator, as it happens with the common Transfer Control Protocol (TCP). We start by showing that the separation principle holds. Additionally, we can prove that the optimal LQG control is a linear function of the state. Finally, building upon the results shown in the previous lecture on estimation with unreliable communication, we show the existence of critical arrival probabilities below which the optimal controller fails to stabilize the system. This is done by providing analytic upper and lower bounds on the cost functional. In the previous lectures we showed that, for protocols where packets are acknowledged at the receiver (e.g.\ TCP type protocols), the separation principle holds. Moreover, the optimal LQG control is a linear function of the state. Finally, we showed the existence of critical arrival probabilities below which the optimal controller fails to stabilize the system. In this lecture we focus on UDP-like protocols. It turns out that when there is no feedback on whether a control packet has been delivered or not (e.g. UDP type protocols), the LQG optimal controller is in general nonlinear function of the information state. In the particular case where there is no measurement noise and the observation matrix C is invertible, we are able to show that the optimal controller is again linear, even if the separation principle still doesn't hold. Necessary conditions on the arrival probabilities for state boundedness are provided.

## Lecture Materials

For this lecture consider pages 71-88.

## Reading

LQG Control with Missing Observation and Control Packets, B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla and S. Sastry. Ifac 05

An LQG Optimal Linear Controller for Control Systems with Packet Losses, B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla and S. Sastry. CDC 05

## Additional Resources

- Real-Time Control Systems with Delays, by Johan Nilsson, PhD Thesis.

## Books

Stochastic Systems: Estimation, Identification and Adaptive Control, by P.R. Kumar, P. Varaiya, Prentice Hall, 1986. Difficult to find (Richard has a copy though). Even if it is not the most user friendly reading, chapters 6 to 8 contain a good reference for dynamic programming and LQG control.

Dynamic Programming and Optimal Control, by D. Bertsekas.

Neuro-Dynamic Programming, by D. Bertsekas and J. Tsitsiklis.