Distributed State Estimation of a Non-linear process system with interconnected subsystems

In this paper authors propose a distributed state estimation scheme for the hybrid system with interconnected subsystems to estimate the states. The system considered in this work has different subsystems which can interact with each other via their states over a communication network. The objective is to implement the distributed state estimation scheme for interconnected subsystems in which each subsystem sensors are connected to the communication network, the estimator has been used for each subsystem to estimate the current states by utilizing the states of the neighboring subsystems over the communication network. The proposed state estimation framework utilizes unscented Kalman filter algorithm. Unscented Kalman filter has been designed for each subsystem to estimate the current state estimates which corrupted by state and measurement noise. The benchmark system taken to implement distributed state estimation is continuous stirred tank reactor units which are strongly interconnected via their neighboring states. The reactors temperature is maintained at unstable operating point by a decentralized proportional integral controller for each subsystem by utilizing the measurements of the sensors from the network. The estimation framework has been verified in the absence of the network failure to stabilize the plant at unstable operating point. The estimate of the corresponding subsystems is used to compute the controller output in the absence of the network failure with minimal sharing over the network.


INTRODUCTION
In [1] the authors have proposed a robust unknown input observer for state estimation and fault detection using linear parameter varying model is proposed. The parameters of the Unknown Input Observer (UIO) are obtained by solving the linear matrix inequalities (LMIs) and linear matrix equalities (LMEs) and also convergence of the UIO is analyzed through Lyapunov theory. The state of a nonlinear dynamical system is estimated through consensus-based networked estimation. It is mainly focus on a family of distributed state estimation algorithms which relies on the extended Kalman filter linearization paradigm. The effectiveness of the nonlinear consensus filter is analyzed with target tracking applications [2]. A mathematical model of distributed state estimation is constructed for nonlinear networked systems against denial-of-service attacks. The feasibility of the distributed state estimation is confirmed and a sufficient condition of the proposed estimation method is tested [3]. An ICMSMT 2020 IOP Conf. Series: Materials Science and Engineering 872 (2020) 012050 IOP Publishing doi:10.1088/1757-899X/872/1/012050 2 estimation of a state of discrete nonlinear systems with uncertainties and sensor delays is examined in [4]. A distributed state estimation method is applied for power system applications [5], continuoustime stochastic process [6] and stochastic non-linear systems with multi-step transmission delays [7]. Industrial systems consists of complex systems which contains Distributed Model Predictive Control (DMPC) scheme is emerging as one of the most effective control schemes for the control of interconnected subsystems. The distributed state estimation scheme [8 -11] has attracted the attention of the researchers and distributed Kalman filter for sensor networks are available in the literature [12,13]. Recently a series of work, related to DMHE have been proposed by Zhang and Liu [11].
The non-linear dynamic system can be decomposed into 'm' interconnected subsystems, where th i subsystem is described by the non-linear state (Eq. 1) and measurement equations (Eq. 2) as shown below:

PROCESS DESCRIPTION
The two interconnected continuous stirred tank reactor (CSTR's) [14] with recycle has been taken as the benchmark system, to demonstrate the proposed distributed state estimation scheme. The well mixed non-isothermal interconnected CSTR is shown in Figure1.Three irreversible exothermic chemical reactions has been taking place inside the chemical reactors of the form A k 1 B, A k 2 U and A k 3 R takes place, where A is the fresh reactant species, B the desired product , and U and R undesired byproducts as shown in Figure 1. The CSTR1 has two input streams one is having fresh reactant species with flow rateF 0 , molar concentration C A0 and temperature T 0 , and the second input stream recycled from the CSTR2 with flow rate F r, molar concentration C A2 and temperature T 2 . The CSTR 2 has another input stream having fresh species A with flow rate of F 3, molar concentration C A03 and temperature T 03 . The output of the CSTR 2 is recycled to CSTR1. Both the reactors are provided with a outer jacket to remove or to provide heat to the reactors as the chemical reaction is exothermic. The mass and energy balance governing the reactors is given below. The dynamics of the reactors has been treated as two separate subsystems.

Subsystem -1
The dynamics of the reactor 1 is given by the mathematical model given below

Subsystem -2
The dynamics of the reactor 2 is given by the mathematical model given below Where,

IMPLEMENTATION OF DISTRIBUTED STATE ESTIMATION
The dynamics of the interconnected system is decomposed into subsystems, the measurements from each subsystem is passed to the network. The objective of the distributed estimation scheme is to share the measurements of the each subsystem over the network so the local controller i.e decentralized controller is implemented by utilizing the measurements over the network. The control action to each subsystem is computed based on the sensor information shared over the network, each decentralized controller is provided with model based estimation of its own subsystem as shown in Figure 2.

SIMULATION RESULTS
From the simulation results of the implementation of distributed state estimation of interconnected subsystems temperature T1 of the CSTR1 is presented in Figure 3, the temperature is initially maintained at unstable steady state upto 400 samplings instants at 400 th sampling instant a network failure is introduced the estimator is switched automatically to provide the estimate of the current subsystem.     The Concentration profile of both the reactors is shown in Figure 4 and Figure 6. The temperature profile of the CSTR 2 is shown in Figure 5.

CONCLUSION
The authors have proposed a distributed state estimation scheme of an interconnected subsystem. It has been implemented in an interacting CSTR process which is able to provide the state estimate fairly by utilizing the subsystem estimate.
Where  is a secondary scaling parameter,  is a factor determining the spread of sigma points around (k 1| k 1) x  and is usually set between 1e-4 to 1. The parameter  is used to incorporate prior knowledge of distribution of x and for Gaussian distribution its optimum value is 2. The 2L+1 sigma points have been derived from the state (k | k 1) x and covariance of the state vector P(k 1 | k 1)  , where L is the dimension of the state vector.