Fully symbolic-based technique for solving complex state-space control systems

Amera Mahmoud Abd-Alrahem, Hala Elhadidy, Kamel Elserafi, Hassen Taher Dorrah


Despite the superiority of symbolic approaches over the purely numerical approaches in many aspects, it does not receive the proper attention due to its significant complexity, high resources requirement and long drawn time which even grows significantly with the increase of model dimensions. However, its merits deserve every attempt to overcome the difficulties being faced. In this paper, a fully generic symbolic-based technique is proposed to deal with complex state space control problems. In this technique, depending on the model dimension if exceeds a predefined limit, the state space is solved using the partitioned matrices theory and blockwise inversion formula. Experimental results demonstrate that the proposed technique overcomes all the previously mentioned barriers and gives the same results when compared to numerical methods (Simulink). Moreover, it can be used to gain useful information about the system itself, provides an indication of which parameters are more important and reveals the sensitivity of system model to single parameter variations.


Symbolic Algebra; State space solution; Partition matrix theory; Ship autopilot; Liquid container; Wind turbine


A. M. Abd-Alrahem, et al., “Toward Symbolic Representation and Analysis of Parameter Varying Control Systems,” in Proc. 20th International Middle East Power Systems Conference MEPCON 2018, Dec. 2018, pp. 894–899.

J. R. Leigh, Control theory : a guided tour. Institution of Engineering and Technology, 3rd ed., London, 2012.

N. Munro, The Symbolic Methods in Control System Analysis and Design. Stevenage, UK: Institution of Electrical Engineers, 1999.

B. Palancz, et al., “Product review-Control system professional suite,” IEEE Control Systems Magazine, vol. 25, no. 2, pp. 67–75, 2005.

M. T. Söylemez and İ. Üstoğlu, ‘‘Block Diagram Reduction Toolbox’’, 2006.

G. Pola, et al., “Symbolic control design of nonlinear systems with outputs,” Automatica, vol. 109, p. 108511, 2019.

M. Fakhroleslam, et al., “Time-optimal symbolic control of a changeover process based on an approximately bisimilar symbolic model,” Journal of Process Control, vol. 81, pp. 126–135, 2019.

A. Girard, et al., “Safety controller synthesis for incrementally stable switched systems using multiscale symbolic models,” IEEE Transactions on Automatic Control, vol. 61, no. 6, pp. 1537–1549, 2016.

N. M. Radaydeh and M. R. D. Al-Mothafar, “Small-signal modeling of current-mode controlled modular DC-DC converters using the state-space algebraic approach,” International Journal of Electrical & Computer Engineering (IJECE), vol. 10, no. 1, pp. 139–150, 2020.

M. Mizoguchi and T. Ushio, “Deadlock-free output feedback controller design based on approximately abstracted observers,” Nonlinear Analysis: Hybrid Systems, vol. 30, pp. 58–71, 2018.

A. Borri, et al., “Design of symbolic controllers for networked control systems,” IEEE Transactions on Automatic Control, vol. 64, no. 3, pp. 1034–1046, 2019.

L. Shamgah, et al., “A Symbolic approach for multi-target dynamic reach-avoid problem a symbolic approach for multi-target dynamic reach-avoid problem,” in 2018 IEEE 14th International Conference on Control and Automation, Alaska, USA, pp. 1022–1027, 2018.

H. T. Dorrah, et al., “Derivation of symbolic-based embedded feedback control stabilization expressions with experimentation,” Journal of Electrical Systems and Information Technology, vol. 5, no. 3, pp. 427–441, 2018.

V. Mladenović, et al., “Symbolic analysis as universal tool for deriving properties of symbolic analysis as universal tool for deriving properties of non-linear algorithms – case study of EM algorithm,” Acta Polytechnica Hungarica, vol. 11, no. 2, pp. 117-136, 2014.

E. Setiawan, et al., “Accurate symbolic steady state modeling of buck converter,” International Journal of Electrical & Computer Engineering (IJECE), vol. 7, no. 5, pp. 2374-2381, Oct 2017.

K. Ogata, Modern Control Engineering, 5th ed., Prentice Hall, 2010.

R. Fateman. (2003). Manipulation of matrices symbolically. Available: http//www. cs. berkeley. edu/~ fateman/papers/symmat2. pdf.

T. -T. Lu and S. -H. Shiou, “Inverses of 2×2 block matrices,” Computers & Mathematics with Applications, vol. 43, no. 1, pp. 119–129, 2002.

I. C. S. Cosme, et al., “Memory-usage advantageous block recursive matrix inverse,” Applied Mathematics and Computation, vol. 328, pp. 125–136, 2018.

Matlab, Symbolic Math Toolbox, User’s Guide, Ver. 2016b. The MathWorks Inc., Natick, MA, USA, 2016.

R. Fraga and L. Sheng, “An effective state-space feedback autopilot for ship motion control,” Journal of Control Engineering and Technology, vol. 2, no. 2, pp. 62–69, 2012.

K. Yano, et al., “Motion control of liquid container considering an inclined transfer path,” Control Engineering Practice, vol. 10, no. 4, pp. 465–472, 2002.

R. Yang, “Control of wind turbines using takagi-sugeno approach,” M.S. thesis, Politècnica de Catalunya University, Barcelona, Spain, 2017.

DOI: http://doi.org/10.11591/ijece.v11i1.pp%25p
Total views : 0 times

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

ISSN 2088-8708, e-ISSN 2722-2578