Identification of Spatially Interconnected Systems Using Neural Network Mukhtar Ali, Hossam Abbas, Saulat S. Chughtai and Herbert Werner Abstract— This paper presents an identification technique based on linear recurrent neural network to identify spatially interconnected systems both in open and closed-loop form. The latter has not been addressed in the literature for the systems under consideration. The paper considers identification of two- dimensional (time and space) systems; the method can be easily extended to have more than one dimension in space. In this paper we consider a semi-causal (causal in time and non-causal in space) two-dimensional (2-D) system, which may be separable or non-separable but the method can also be used for 2-D systems which are causal in both dimensions. Furthermore the algorithm can handle boundary conditions. The effectiveness of the method is shown with application to simulation examples. I. INTRODUCTION In the last decade there has been a renewed interest in distributed control of complex engineering systems that are multidimensional and composed of similar subsystems, which interact with their closest neighbors. Such systems include unmanned aerial vehicles and satellites flying in formations, vehicle platoons and automated highway systems as well as flexible structures, fluid flow and systems that are also characterized by the same class of partial differential equations. The control synthesis of two-dimensional or multidimensional systems is a rich field and a number of methods have been proposed to design controllers for such systems. Distributed controllers are designed for spatially invariant system in Fourier domain in [1]; a method that can be used to design simple controllers for distributed interconnected systems by using the separability condition between the temporal and spatial dimension of the plant is presented in [2]. LMI-based control design methods have been developed for spatially invariant distributed systems using multidimensional optimization, where the implementation of the controller is almost decentralized in nature like in [3], [4], [5], [6], [7]. To synthesize optimal and/or robust control schemes for spatially interconnected systems, it is important to have a sufficiently accurate model of the system. One possibility to obtain such a model is to construct it based on the governing partial differential equations (PDE), followed by the experimental identification of the physical parameters, M.Ali, S.S.Chughtai and H.Werner are with the Insti- tute of Control Systems, Hamburg University of Tech- nology, Eissendorfer Str. 40, 21073 Hamburg, Germany, {mukhtar,saulat.chughtai,h.werner}@tu-harburg.de, H.Abbas is with the Electrical Engineering Department, Faculty of Engineering, Assiut University, 71515 Assiut, Egypt, {h abbas}@aun.edu.eg. see e.g. [8], but such a model may be less accurate due to unmodeled dynamics. Few results are available in the literature for identification of 2-D and multidimensional distributed systems. Identifica- tion of transfer function models of 2-D causal systems is presented in [9], the method is based on the 2-D Hankel theory. Methods to identify 2-D and multidimensional non- causal rational transfer functions are given in [10], [11] but are less practical due to their dependence on the impulse response of the system. Identification of 2-D state-space models for separable-in- denominator systems based on the impulse response of the system is discussed in [12] while identification from input- output data is presented in [13]. The latter has also the advantage that it gives a state-space model in balanced form. A state-space based model identification method for spatially distributed interconnected systems is proposed in [14], which we can call decentralized subspace identification of spatially interconnected systems. Recently a systems identification technique to identify 2-D transfer function models for 2-D systems has been proposed in [15]. This method can be used to identify 2-D transfer function models for separable or non-separable systems and is applicable to causal, semi-causal (spatially interconnected systems) and non-causal systems. Furthermore, it can handle various boundary conditions. The method is based on least squares estimation. The method can only identify models in open loop (stable models). For identification of unstable plant models closed-loop identification has to be used. There are two methods for closed-loop identification: direct closed- loop identification and indirect closed-loop identification [16]. The latter has the advantage that the method does not suffer from bias due to noise correlated with the input signal, as the input signal for identification is taken to be an external reference signal. The indirect closed-loop iden- tification method for linear time-invariant one-dimensional systems has the problem of the separation of the plant from the controller. The method of separation is usually based on assumptions on closed-loop order and controller structure as in [17] and [18] and therefore is less practical. The use of the stated methods for separation of controller and plant for spatially distributed systems will be more complex. This paper presents methods to identify models for a spatially interconnected system in both open and closed-loop form. The main motivation of this work is to identify unstable spatially interconnected systems in closed-loop and also to see the effect of additive output noise. In this work we