Can state space equations be represented as x(k)=Ax(k-1) and y(k)=Cx(k) ?
Yes, it is an autonomous system (without external input) in discrete time.
(1) Quote "I want to estimate states using neural networks"
The papers [RP95], [NP90] and [PI91] estimate system states using neural networks.
[RP95] I. RIVALS and L. PERSONNAZ, "BLACK-BOX MODELING WITH STATE-SPACE NEURAL NETWORKS," Neural Adaptive Control Technology, 1995.
https://www.neurones.espci.fr/Articles_PS/NACT1.pdf
[NP90] K.S. Narendra and K. Parthasarathy, "Identification and Control of Dynamical Systems Using Neural Networks", IEEE Transactions on Neural Networks, 1(1), March 1990.
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=80202
[PI91] M.M. Polycarpou and P.A. Ioannou, "Identification and Control of Nonlinear Systems Using Neural Network Models: Design and Stability Analysis", Technical Report, 1991.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.1816
(2) Quote "Can state space equations be represented as x(k)=Ax(k-1) and y(k)=Cx(k)?"
The experimental result shown by Figure 23 in the paper [Hong99] has validated the concise answer provided by Esmaeil Jahanshahi.
Figure 23: The response of the system with observer when the input is zero and the disturbance input is a unit step
[Hong99] Y. Hong, “The Controller Design For Linear System: A State Space Approach,” Technical Report, National University of Singapore, November 1999. DOI: 10.13140/RG.2.2.21548.21120/1 Available from the following RG link.
Preprint The Controller Design For Linear System: A State Space Approach
[HY07] Y. Hong and O.W.W. Yang, “Self-Tuning Optimal PI Rate Controller for End-to-End Congestion With LQR Approach,” Proceedings of 20th International Teletraffic Congress (ITC-20), Ottawa, Canada, June 2007, pp.829-840. Available from the following RG link.
https://www.researchgate.net/publication/221584579_Self-tuning_Optimal_PI_Rate_Controller_for_End-to-End_Congestion_With_LQR_Approach
(3) Reference books on state space representation
"What are good books for control of linear and nonlinear systems?"
https://www.researchgate.net/post/What_are_good_books_for_control_of_linear_and_nonlinear_systems
(4) Discussions on control system design
"What are trends in control theory and its applications in physical systems (from a research point of view)?"
https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2
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Conference Paper Self-tuning Optimal PI Rate Controller for End-to-End Conges...
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