## 2 Related Work
### Packet Loss Compensation  
Several previous works have considered the compensation of omitted samples and variable delays in networked systems, both wired and wireless. This work ranges from relatively simple first-order hold interpolators to be used in intelligent actuators (e.g., [22]) through to advanced techniques based upon \(H_{\infty}\) filtering of the data stream, application of adaptive sampling controls, and the use of buffering schemes within a robust control framework (e.g., [23, 24]). In [23], the use of playback buffers in conjunction with information redundancy is suggested to overcome packet losses, packet disordering and variable latency of UDP/IP communication links in a closed loop. Buffer sizes should be selected to cover worst-case latencies of the links and in the absence of other methods can be estimated based upon gathered statistical data. A frequency-domain robust control method is proposed to overcome the additional link delays and provide additional resilience to occasions in which buffer sizes are violated at run-time. In [24], [PERSON] _et al._ consider an adaptive sampling rate control scheme for a DCS subject to stochastic packet disordering, transmission delays and packet losses of the type induced by the UDP/IP protocol. They propose an augmented closed-loop DCS and develop an adaptive tracking controller to stabilize the resulting stochastic system. The implementation of such techniques can be non-trivial, and both [23] and [24]--plus closely related methods--require a high-fidelity process model plus knowledge of the network characteristics such as delays and losses. The work in [23] requires the solution to a large non-convex optimization problem, obtained by sweeping the frequency parameter over a specified range using a given search granularity. Although principally aimed at off-line control design, it may be adapted for on-line use. In [24], multiple consecutive losses are included in the model, and a non-linear search is required to be carried out on-line. No discussion of complexity is provided.  
For simple PID-based networked control systems, some conceptually simple techniques (which are related to the Send-On-Delta approaches previously discussed) make use of timestamps and gain scheduling (e.g., [25]). In such techniques, the timestamps are employed to automatically adjust (schedule) the integral and derivative PID gains based on the actual time--as opposed to the ideal time--that has elapsed between successively received samples. Although principally aimed at jitter compensation, a similar technique has been employed for dropout compensation in WSANs; it is now adapted for use in the higher layers of WirelessHART [17] by disabling the dynamic (I and D) elements when packets are omitted. Although such a technique provides for much improved performance over regular PID upon channel recovery [17], the control signal is still effectively \"frozen\", with the last known state of the plant used to apply proportional-only control during interference bursts [17].  
As with the wired case, more advanced classes of method have been developed specifically for the industrial wireless domain. In particular, the use of predictive filtering at information sink nodes in a network has been found useful in overcoming many of the issues related to imprecise sampling and omissions. A variety of methods have been proposed, ranging from the use of state-space dynamic models and Kalman filters (e.g., [26]) through to simpler transfer function dynamic models and ARMA filters (e.g., [14] and the references therein). Although Kalman filtering approaches such as [26] give resilience to unmeasured disturbances, their implementation complexity is high; and as mentioned, a major drawback across all of these methods lies in the assumption that a time-invariant mathematical model of the underlying process and measurement dynamics is available. In the following section, a technique is presented that may provide a more appropriate solution for many industrial control applications.