## 5 Controller Packet Loss Compensation  
### Principle  
Considering Figure 1, the other source of potential information losses in the control loop lies in the wireless link between the controller and actuator nodes. Again, under the assumption that a large proportion of samples will be successfully delivered in a well-designed wireless network, one possible way to add resilience against such information losses in the loop is to use buffering. Suppose that the controller is implemented as the causal, discrete control law \(D(z)=U(z)/E(z)\), where \(U(z)\) and \(E(z)\) are the z-transformed representations of the sampled-data control signal \(u(k)\) and feedback error \(e(k)=r(k)-y(k)\), with \(r(k)\) the reference (setpoint) signal at step \(k\). Then upon receipt of the new actual/estimated process value \(y(k)\) or \(\hat{y}(k)\), the controller will first form the new error \(e(k)\) and then the new control signal \(u(k)\) via \(D(z)\).  
Since the control \(u(k)\) (and hence \(\Delta u(k)\)) are available, the prediction equation (3) allows the calculation of \(\hat{y}(k+1)\). Under the assumption that \(r(k+1)=r(k)\), \(\hat{e}(k+1)\) can also be calculated and hence \(\hat{u}(k+1|k)\) determined through the control law \(D(z)\). This procedure may be applied recursively to determine the future controls to apply over a horizon of \(H\) samples, _i.e._, the signals \(u(k)\), \(\hat{u}(k+1|k)\), \(\hat{u}(k+2|k)\),..., \(\hat{u}(k+H|k)\). This sequence effectively represents the \"best guess\" of future open-loop controls that would be generated by controller \(D(z)\) to drive the process Model (1) towards the current reference \(r(k)\), and can be calculated with a complexity dependent upon the number of terms in the model and controller, and the length of \(H\). It is the \"best guess\" in the sense that disturbances entering the system at or before the time that the last sensor packet was successfully delivered are taken into account, but those affecting the system after this time are not. Since the process model parameters are accurately tracked by RLS, and assuming the controller \(D(z)\) achieves the desired closed-loop performance, then this sequence of open-loop controls should achieve a reasonable quality of control. In the buffering approach, at each sample time the controller node re-calculates this sequence and places it in the packet to be forwarded to the actuator.  
The controller-sensor data packet transfer (DT2 in Figure 2) has an effective deadline of \(T\) time units after the start of the previous actuation event. A packet omission for the \(k^{th}\) iteration of the control loop can easily be detected in the actuator node by the absence of a new packet at time \(t=kT\). In this case,supposing that packet \(k-1\) arrived successfully, the estimated control \(\hat{u}(k|k-1)\) can be applied instead of \(u(k)\). In the case of multiple successive controller-actuator packet omissions, then if the last successful packet was received at step \(j\) (with \(j<k\leq j+H\)), there is an estimated control \(\hat{u}(k|j)\) available to apply to the process. Under the assumption that the horizon \(H\geq\beta\), good resilience to burst errors is achievable. Since a DCS control signal is often encoded as a 16-bit integer, the overhead of such an approach is typically only \(2H\) extra bytes per packet which is relatively small.