Research Summary:
Joint Design of Control and Communication
Building a distributed control system over a wireless network offers a lot of flexibility in terms of installation, mobility and maintenance. Yet this is a very challenging problem.
Control systems and communication networks are typically designed using very different principles. Traditional control theory requires the feedback data to be accurate, timely and lossless. Conversely, random delay and packet loss are generally accepted in communication network design. Moreover, this delay and loss is much more pronounced in wireless networks than in wired networks due to limited spectrum and power, time-varying channel gains, and interference. Therefore, a joint design is necessary and we need a new approach for this joint design.
Joint design of control and communication is two-fold: the controller design needs to be robust and adaptive to the communication faults such as random delays and packet losses, while the network should be designed with the goal of optimizing the end-to-end control performance. The Kalman filtering in the presence of random packet losses [1] [2] is our first step in designing controllers that adapt to communication faults. In the effort to design the communication network to optimize the control performance, we first separately studied the link layer design tradeoffs [3] and the MAC layer design tradeoffs [4] [5]. Recently we propose a cross-layer framework to jointly design all the layers of the network to deliver the best end-to-end control performance [6].
Kalman Filtering in the Presence of Random Packet Losses
Background:
Partial observation losses can occur in a distributed control system where measurements are taken at different sensors that are at different physical locations or one sensor needs to send its data in multiple packets.
Classical Kalman filter is the optimal minimum mean square estimator when the observations are continuously available. In distributed control applications over wireless networks, packet losses are inevitable. Thus it is important to know the optimal state estimate in the presence of packet losses. Particularly, we allow partial observation losses and we are interested in the network resource allocation among different sensors.
We formulate the Kalman filtering problem with partial observation losses and derive the Kalman filter updates with partial observation measurements. We show that with these partial measurements the Kalman filter and its error covariance matrix iteration become stochastic, since they now depend on the random packet arrivals of the sensor measurements, which can be lost or delayed when transmitted over a communication network. The communication network needs to provide a sufficient throughput for each of the sensor measurements in order to guarantee the stability of the Kalman filter, where the throughput captures the rate of the sensor measurements correctly received.
We investigate the statistical convergence properties of the error covariance matrix iteration as a function of the throughput of the sensor measurements. A throughput region that guarantees the convergence of the error covariance matrix is found by solving a feasibility problem of a linear matrix inequality. We also find an unstable throughput region such that the state estimation error of the Kalman filter is unbounded. When the Kalman filter is stable, the expected error covariance matrix is bounded both from above and from below. We also solve the optimal throughput allocation problem with respect to the upper bound using a semi-definite program.
Publications:
[1] X. Liu and A. J. Goldsmith, Kalman Filtering with Partial Observation Losses, Proc. IEEE Conference on Decision and Control, 2004.
[2] X. Liu and A. J. Goldsmith, Kalman Filtering with Partial Observation Losses, Submitted to IEEE Transactions on Automatic Control, May 2004.
Optimize the Network Design with respect to the Control Performance
Background:
In a distributed control application, the network design objective is to optimize the control performance. This control performance is a complex function of the network parameters, such as throughput, packet delay and packet loss probabilities. The goal of optimizing the control performance imposes implicit tradeoffs on the wireless network design as opposed to the explicit tradeoffs typical in wireless data and voice applications. The implicit design tradeoffs make the design optimization difficult.
Research Findings:
We first study the link layer design tradeoffs in a TDMA based network [3]. The trade-offs of data rate, time delay and packet loss in the communication link layer design are intricate and implicit. We uncover some surprising insights. In particular, an uncoded link design, which is often undesirable due to its unreliability, can be optimal under certain circumstances since it can achieve the optimal tradeoff among data resolution, time delay and packet loss probability. We also find that an analog soft decoding link can perform better than a digital communication link under low channel gains. This suggests fundamental changes in wireless link designs may be necessary to support robust distributed control systems.
When multiple transmitter/receiver pairs share a wireless network, the medium access control (MAC) protocol greatly affects the packet delay and packet losses, which in turn affect the control performance. We study the effects of different MAC protocols on the control performance in [4] [5]. We show that Polling gives little performance improvement over TDMA. We also show that distributed medium access control can cause significant performance degradation. This performance degradation can be reduced if we choose the control parameters, for example, the sample period, properly.
Most recently, we present a cross-layer framework for the joint design of wireless networks and distributed controllers [6]. We first present this framework for a broad class of distributed control applications. We then illustrate this framework by a cross-layer optimization of the link layer, MAC layer, and sample period selection in an inverted pendulum system. Our results indicate that cross-layer design significantly improves the performance and stability of the controller.
Publications:
[3] X. Liu and A. J. Goldsmith, Wireless Communication Tradeoffs in Distributed Control, Proc. IEEE Conference on Decision and Control, Dec. 2003.
[4] X. Liu and A. J. Goldsmith, Wireless Medium Access Control in Distributed Control Systems, Proc. Allerton Conference on Communication, Control and Computing, Oct. 2003.
[5] X. Liu and A. J. Goldsmith, Wireless Medium Access Control in Networked Control Systems, Proc. IEEE American Control Conference, June 2004.
[6] X. Liu and A. J. Goldsmith, Wireless Network Design for Distributed Control, Proc. IEEE Conference on Decision and Control, Dec. 2004. (Invited Paper)