Protocol-Based Control for Cyber-Physical Systems with Environment-Dependent Energy Harvesting Sensors

In this paper, a protocol-based controller is designed for Cyber-Physical Systems (CPSs) with multiple sensors, which are powered by environment-dependent energy harvesting (EDEH) devices, respectively. The Round-Robin (RR) protocol is adopted to coordinate the data transmission of sensors. The protocol-based transmission can be realized only when the energy harvested by EDEH devices is sufficient. The aim of this paper is to design the protocol-based controller to ensure the stochastic finite-time boundedness (FTB) with EDEH and RR protocol. Firstly, modeling the EDEH by a switching sequence with varying sojourn probabilities, assuming a finite battery capacity constraint, and associating protocol-based transmission with a given energy cost, we propose a new recursive model to depict the dynamic of energy levels for each sensor. Then, combining with stochastic analysis and the dynamic of energy levels, the explicit expressions of the controller for each environment with average dwell time (ADT) are obtained. Finally, an example is provided to demonstrate the effectiveness of the designed controllers.


Introduction
Due to the rapid development of computation, networking and communication, CPSs have attracted significant research interest [1,2,3], which combine the cyber and physical process quite tightly. So far, CPSs have been applied in many fields, i.e. autonomous vehicles [4], industrial automation [5]. In these applications, a hot research topic is to study the control performance under the resources constraint (energy, bandwidth, computation and so on) [6]. In [7] and [8], the eventtriggering control scheme has been proposed for SCPs to decrease the data transmission burden with the limited communication resource. In [9], the resource allocation and control problem have been jointly studied for industrial CPSs under the power and subcarrier constraints.
As a vitally important branch of the resources constraint, energy problem often exists in CPSs [10,11,12,13]. In some applications of CPSs, the communication devices are battery powered vie a fixed power or a replaceable battery [14]. However, for some application scenarios, it is neither affordable nor feasible to always use a fixed supply or rechargeable battery, like Forest Fire where x(k) ∈ R nx represents the state variable, u(k) ∈ R nu and y(k) ∈ R ny denote the input and the output, w(k) ∈ R nx and v(k) ∈ R ny are the noise, which are unknown but satisfy where s ∈ Z + , π l,k and Γ l,k are known matrices. The information exchange between sensors and controllers takes place over a communication network with limited resources. In the following, we will establish the measurement model under RR protocol and the environment-dependent energy harvesting.

Round Robin Protocol
In this paper, RR protocol is used to schedule the transmission, under which only one sensor can transmit information in turns. By defining r(k) = mod(k − 1, n y ) + 1 as the label of the active sensor, the measurement can be denoted as .., δ(n y − j)}, j = 1, 2, ..., n y , δ(·) ∈ {0, 1} is the Kronecker delta function.

Environment-dependent energy harvesting
We denote the energy level for i-th sensor as z i k ∈ {0, 1, 2, ..., S} where S is the maximum energy level. The harvested energy at the i-th environment is represented by h i k which is an environmentdependent independently identically distributed stochastic process with the probability distribution where σ k ∈ {1, 2, ..., M } is the environment factor, p i γ,β satisfies 0 ≤ p i γ,β ≤ 1 and +∞ γ=1 p i γ,β = 1. When z k = 0, the sensor can transmit the measurement to the controller consuming 1 unit of energy. Thus, the indicator function and the measurement are further expressed by . Moreover, the dynamics of the energy level for i-th sensor is with z i 0 ≤ S. Then, the controller can be described as where K β (k) is the controller gain to be designed. where 1} is the Kronecker delta function, we can rewrite the closed-loop nonlinear system as where c 1 > c 0 > 0 are real scalars and R is a weight coefficient matrix.
The purpose is to design the controller such that system (4) stochastic FTB under RR protocol and EDEH.

Main Results
, the recursion of the probability ρ i k,β under the β-th environment is given by where Proof. According to (2), the probability z i k = t (0 ≤ t < S) under the β-th environment can be denoted by Afterwards, the probability of z i k = S can be expressed as Combining (6) and (9), we can obtain the recursion (8). The proof is complete.

Numerical simulation
A numerical example is proposed in this section. For a nonlinear system with three sensors and the following parameters According to the given recursive algorithm, the controller gain can be obtained recursively by Matlab. Table 2 lists the desired parameters of controller K 1 (k) or K 2 (k) from the time k = 1 to k = 12. The bound is shown in Fig. 2. The switched environments are shown in Fig. 3. The energy level and measurement for i-th sensor are shown in Fig. 4-5. The simulation has confirmed that the designed controllers perform very well.

Conclusions
In this paper, the protocol-based control problem has been investigated for CPSs with EDEH devices. Firstly, an iterative algorithm of varying probabilities under EDEH and the RR protocol has been employed to describe the dynamics of the energy level for each sensor. Then, by utilizing ADT method and the varying probability of energy harvesting, the environment-dependent control problem case has been equivalently reduced to a general switched system with time-varying probability. A sufficient condition has been derived to guarantee the SFTB of the closed-loop system. Subsequently, the desired controller has been designed in terms of the proposed algorithm that can be adopted effectively by using available software. At last, an example has been given to proof the availability of the proposed design method.