Optimal Power Management in Wireless Control Systems

Abstract

This paper considers the control of a linear plant when plant state information is being transmitted from a sensor to the controller over a wireless fading channel. The power allocated to these transmissions determines the probability of successful packet reception and is allowed to adapt online to both channel conditions and plant state. The goal is to design plant input and transmit power policies that minimize an infinite horizon cost combining power expenses and the conventional linear quadratic regulator control cost. Since plant inputs and transmit powers are in general coupled, a restricted information structure is imposed allowing them to be designed separately. Under this information structure the standard LQR controller becomes the optimal plant input policy, while the optimal communication policy follows a Markov decision process minimizing transmit power at the sensor and state estimation error at the controller. The optimal power adaptation to channel and plant states is examined qualitatively for general forward error correcting codes. In the particular case of capacity achieving codes event-triggered policies are recovered, where the sensor decides whether to transmit or not based on plant and channel conditions. Approximate dynamic programming is employed to derive a family of tractable suboptimal communication policies exhibiting the same qualitative features as the optimal one. The performance of our suboptimal policies is shown in simulations and is contrasted to other simple transmission policies.

Opportunistic Control Over Shared Wireless Channels

“…Subtracting g(µ) = L(α(µ), p(µ), µ) from both sides of this inequality and expanding the terms of the Lagrangian as in (22) we get…”

“…Hence we turn our focus to problem (24). A more convenient expression for the Lagrangian defined in (22) can be obtained by rearranging terms to get…”

“…The SNR is proportional to the received power level expressed by the product h · p of channel fading and the allocated transmit power. Overall we express the probability of success by a given relationship of the form q(h ij,k · p ij,k ) -for more details on this model, the reader is referred to [22]. An illustration of this relationship is given in Fig.…”

“…Depending on the measured channel states the access point decides which plant is scheduled to close the loop during the time slot. We note that perfect state information is not required, as h can denote in practice an estimate of the channel state -see also [22]. We also point out that even though the pilot signals for the channel estimation incur some power consumption, we assume that practically this is much lower than the power necessary for transmitting the packets of the control systems, especially for large packet lengths (e.g.…”

“…Such a channel-aware mechanism can opportunistically exploit channel information to, e.g., grant channel access to control loops experiencing favorable channel conditions, or equivalently avoid closing the loop under adverse conditions. Similar to our previous work for single-loop systems in [22], [23] we allow for the selection of transmit power when a system is scheduled. Transmit power and channel fading determine the probability of successful message delivery at the receiver.…”