#### PhD Thesis

## Real-Time Control Systems with Delays

### Johan Nilsson

##### Abstract

Control loops that are closed over a communication network get more and more common. A problem with such systems is that the transfer delays will be varying with different characteristics depending on the network hardware and software. The network delays are typically varying due to varying network load, scheduling policies in the network and the nodes, and due to network failures. Two network models of different complexity are studied:

Random delays that are independent from transfer to transfer,

Random delays with probability distribution functions governed by an
underlying Markov chain.

The delay models are verified by experimental measurements of network delays.

In the thesis it is shown how to analyze stability and expected performance of linear controllers where the network delays are described by one of the two network models above. Methods to evaluate quadratic cost functions are developed. Through the same analysis we find criteria for mean square stability of the closed loop for the different network models.

The Linear Quadratic Gaussian (LQG) optimal controller is developed for the two delay models. The derived controller uses knowledge of old time delays. These can be calculated using ``timestamping'' of messages in the network. ``Timestamping'' means that every transfered signal is marked with the time of generation. The receiving node can then calculate how long the transfer delay was by comparing the timestamp with the node's internal clock.

##### Keywords

Clock synchronization, Delay compensation, Distributed computer control systems, Jump linear systems, Linear quadratic control, Real-time systems, Stochastic control, Stochastic parameters, Timing jitter

PhD Thesis ISRN LUTFD2/TFRT--1049--SE, Department of Automatic Control, Lund University, Sweden, January 1998.