# Energy-Efficient Wireless Communication Strategy for Precision Agriculture Irrigation Control
## 3 Energy-Efficient Communication
### Self-Triggered Control  
Closed-loop control algorithms have been traditionally implemented using periodical sampling. Although periodicity simplifies the control system design and analysis, it also  
Figure 4: Validation results for process identification.  
Figure 5: Precision irrigation control loop.  
leads to the unnecessary use of communication, energy, and computation resources in the system devices. Aperiodic sampling avoids this situation by conducting measurements only when a relevant event is about to happen, and it can be implemented using either a reactive (event-based) or a proactive approach (self-triggered). In the first one, the sampling instant is determined when a monitored output variable reaches a specific threshold, while, in the latter one, the next update time is predicted based on the evolution of the process state variables.  
Self-triggered control (STC) has an advantage over event-triggered control (ETC) since it does not require dedicated additional energy-consuming hardware to conduct continuous measurements. Instead, it uses process dynamics knowledge to implement the model-based algorithm to estimate the next sampling event. We considered two different self-triggered strategies: the first one calculates the measured error to estimate the next sampling instant [17], while the second one uses the Lyapunov function to trigger the control updates [20].  
In the error-based triggering condition \(\gamma_{1}\), the process states were forced to stay within a given error defined as a function of the norm of the measured states; thereby, the next sampling time \(h+1\) was obtained when the measured error exceeded the limit that was proportional to the current process state. Thus,  
\[\gamma_{1}:\;h+1\;=\;min\;\{\;h\leq max\_limit\;\;\mid\;\;\;\;\mid e(h)|> \sigma\mid x(kh)\mid\;\;\}, \tag{7}\]  
where \(0<\sigma\leq 1\) denotes proportionality, \(max\_limit\) is the maximum allowable sampling period, and the measured error is defined as  
\[e(h)=\;x(kh+h)-x(kh), \tag{8}\]  
which represents the difference between the estimated next state value with respect to the current one.  
In the Lyapunov-based triggering condition \(\gamma_{2}\), the estimation of the next sampling instant is determined by evaluating the Lyapunov decaying rate over time \(\Delta V\). Given a linear discrete control model defined by Equation (4), the system is asymptotically stable in the Lyapunov sense if there is a function \(V\) that satisfies  
\[V(x(kh))=\;x(kh)^{T}\;P\;x(kh)>0, \tag{9}\]  
and  
\[\Delta V(x(kh))=\;x(kh)^{T}\;\Big{(}\Phi^{T}P\Phi-P\Big{)}\;x(kh)<0, \tag{10}\]  
where  
\[\Delta V(x(kh))=\;V(x(kh+h))-V(x(kh)), \tag{11}\]  
the discrete-time Lyapunov matrix \(P\) must be a positive definite matrix, and its existence certifies the system stability since \(V\) is always decreasing. To ensure that the Lyapunov function rate is proportional to the current value, then the next sampling time \(h+1\) is obtained when the estimated Lyapunov function decays under a limit proportional to the current function value. Thus,  
\[\gamma_{2}:\;h+1\;=\;min\;\{\;h\leq max\_limit\;\;\mid\;\;V(x(h))<\sigma V( x(kh))\;\;\}, \tag{12}\]  
where \(0<\sigma\leq 1\) also denotes proportionality.  
In any case, if the plant dynamics for the crop irrigation process are known, then the evolution of the state variables can be estimated to predict the next update time from the last measurement.