## Joint Optimization of Communication Rates and Linear SystemsL. Xiao, M. Johansson, H. Hindi, S. Boyd, and A. Goldsmith
We consider a linear system, such as a controller or estimator, in which several signals are transmitted over communication channels with bit rate limitations. We focus on finding the allocation of communication resources such as transmission powers, bandwidths, or time-slot fractions, that yields optimal system performance. Assuming conventional uniform quantization and a standard white-noise model for quantization errors, we consider two specific problems. In the first, we assume that the linear system is fixed and address the problem of allocating communication resources to optimize system performance. We observe that this problem is often convex (at least, when we ignore the constraint that individual quantizers have an integral number of bits), hence readily solved. We describe a general dual decomposition method for solving these problems that exploits the special structure often found in network resource allocation problems. This method reduces to the standard waterfilling techniques used in problems with only one coupling constraint. We briefly describe how the integer bit constraints can be handled, and give a bound on how suboptimal these heuristics can be. The second problem we consider is that of jointly allocating communication resources and designing the linear system in order to optimize system performance. This problem is in general not convex, but can be solved heuristically in a way that exploits the special structure of the communication resource allocation problems, and appears to work well in practice. We demonstrate these ideas and methods on two numerical examples. In the first, we consider a networked estimator in which sensors transmit measurements over a multiple access channel, and we optimize bandwidth, power allocation, and bit rates to the sensors. In the second example, we consider a networked LQG controller, in which the sensor signals are transmitted over a multiple access channel and the actuator signals are transmitted over a broadcast channel. The sensor and actuator channels have separate power limits, but share a common bandwidth constraint. Here we allocate power and bandwidth to each actuator and sensor channel, as well as the total bandwidth available to the sensors and actuators, and in addition optimize the controller itself. |