# Protocol for Energy-Efficiency in Networked Control Systems Based on WSN
## 1 Introduction
The MJLS class allows to model the success or failure transmission simply changing the values of the input and output matrices of the system [18]. As previously mentioned, the performance criterion adopted to synthesize the controler is the \(\mathcal{H}_{\infty}\) norm of the closed-loop system [19]. Another important contribution of this paper is the proposition of a systematic procedure to determine the transition probability matrix and the modeling of the Markov chain operation modes. Since several measurement signals can be read by a single sensor (or control signals can be sent to the same actuator), the operation modes are determined by fail or success in the transmission of packets associated to the total amount of those devices (product between the number of sensors and actuators). Additionally, from the viewpoint of the remote controller, only the packet loss of the measurement signals can be recognized, indicating that some of the modes (associated with the control input) are not observable. To the best of the authors' knowledge, there are only sub-optimal conditions for the design of dynamic output-feedback controllers in the partially mode-dependent context. On the other hand, concerning the design of filters (full-order or state-observer) and state-feedback controllers, there exist necessary and sufficient conditions even for the scenario regarding partial availability of the chain modes in the particular case of MJLS with generalized Bernoulli distribution [20; 21], a problem investigated in this paper. For this reason, it was chosen to perform the design of the controller in two steps: synthesis of a state-observer filter and synthesis of a state-feedback gain. Although each part of the design is optimal, the joint design (filter + state-feedback controller) lost its optimality because the separation principle is not valid for \(\mathcal{H}_{\infty}\) control of MJLS. Even though, if all the system states are accessible for feedback or if the packet loss does not occur in the transmission of the measurement or control signals, the same design conditions provide the optimal performance. The remainder of this paper is briefly summarized in what follows. Section 2 presents different strategies available in the literature to model the packet loss in WSN scenario. Section 3 introduces the mathematical background necessary for the presentation of the results proposed in the paper, such as the definition of the Markov model for dynamical systems controlled through communication network; the methods to design the \(\mathcal{H}_{\infty}\) output-feedback controller; the formulas for the calculation of the transmission success probability and mathematical expectation of the global number of transmissions using the Hop-by-Hop transport scheme; and the energy consumption model for WSN. Section 4 presents the modeling of the Markov operation modes and the computation of the probabilities of transition among them for problems of control through semi-reliable communication networks with multiple information sources. Section 5 describes in details the three heuristics of implementation of the proposed energy saving protocol. In Section 6 the energy saving protocol is applied to a standard control problem: the coupled water tank, a multiple-input multiple-output (MIMO) system with two inputs and two outputs. The numerical results in terms of the \(\mathcal{H}_{\infty}\) performance degradation and the reduction of the global energy consumption when compared with a classical control system (implemented in a full-reliable network) are also presented. Section 7 provides a critical assessment of the results and presents final considerations and some future perspectives. Appendix A presents a tutorial for the use of the R-Package 'Hopbyhop', which facilitates the computation of the transmission success probabilities and the mathematical expectation of the global number of transmissions in a Hop-by-Hop network and also provides a validation of the calculations by means of a Monte Carlo simulation.