Synthesis of Guaranteed Stability Regions of a Nonstationary Nonlinear System with a Fuzzy Controller

Vasiliy Berdnikov, Valeriy Lokhin

Abstract


The paper proposes a method for constructing guaranteed regions of stability of nonstationary nonlinear systems on the plane of parameters of a fuzzy PID controller. It is shown that this method allows to determine the full stability areas, which are significantly larger than the areas determined by classical methods (frequency circle criterion, quadratic Lyapunov functions). This improvement is achieved by using the algorithm for constructing spline Lyapunov functions. This type of Lyapunov functions is based on the necessary and sufficient conditions of stability, while the classical methods are only sufficient conditions of stability. In this regard, on the basis of the proposed method, it is possible to calculate the maximum sizes of the sectors in which the nonlinear characteristics in the channels of the fuzzy PID controller should be located. Examples of the synthesis of fuzzy P, PI, PID controllers for a nonstationary control object of the third order are given. Numerical experiments show that the expansion of the boundaries of nonlinear characteristics allows to improve the accuracy in the steady state, and also to almost double the speed of the automatic control system with a nonstationary object. The advantages over linear controllers are demonstrated. The proposed method guarantees the stability inside the calculated stability regions for any character of the change in the nonstationary parameter, as well as for any character of the change in the nonlinear characteristics in the corresponding sectors.


Keywords


Nonstationary Nonlinear System; Stability Regions; Lyapunov Functions; Circle Criteria; Spine Functions; PID Controllers; Fuzzy Logic; Adaptive Systems.

References


K. A. Pupkov, N. D. Egupov, Nonstationary Automatic Control Systems: analyze, synthesis and optimization, Moscow: Izdatelstvo MGTU im. N.E. Baumana, 2007.

N. Siddique, “Fuzzy control”, Studies in Computational Intelligence, vol. 517, pp. 95-135, 2014. doi: 10.1007/978-3-319-02135-5.

R. Guclu, “Sliding mode and PID control of a structural system against earthquake”, Mathematical and Computer Modelling, vol. 44, Iss. 1–2, pp. 210-217, 2006. doi: 10.1016/j.mcm.2006.01.014.

X. Liao, P. Yu. Absolute Stability of Nonlinear Control Systems, Dordrecht: Springer, 2008. doi: 10.1007/978-1-4020-8482-9.

V.P Berdnikov, “Algorithm of determination of non-stationary nonlinear systems full stability areas”, Russian Technological Journal, vol. 5, no. 6, pp. 55-72, 2017.

V.P. Berdnikov, “Modified algorithm of determination of non-stationary nonlinear systems full stability areas”, Russian Technological Journal, vol. 6, no 3, pp. 39-53, 2018.

V.P. Berdnikov, “Improving efficiency of the procedure of Lyapunov spline-functions construction for nonlinear nonstationary systems”, Russian Technological Journal, vol. 6, no. 5, pp. 25-44, 2018. doi: 10.32362/2500-316X-2018-6-5-25-44.S.

A.P Molchanov, E.S. Pyatnickij, “Lyapunov functions defining the necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems, I”, Automation and Remote Control, no. 47, pp. 344–354, 1986.

A.P Molchanov, E.S. Pyatnickij, “Lyapunov functions defining the necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems, II”, Automation and Remote Control, no. 47, pp. 443–451, 1986.

A.P Molchanov, E.S. Pyatnickij, “Lyapunov functions defining the necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems, III”, Automation and Remote Control, no. 47, pp. 620–630, 1986.

J. Jantzen, Foundations of Fuzzy Control: A Practical Approach. Chichester, England: John Wiley and Sons, 2013. doi: 10.1002/9781118535608.

D. Altshuller, Frequency domain criteria for absolute stability: a delay-integral-quadratic constraints approach, London: Springer, 2013. doi: 10.1007/978-1-4471-4234-8.

A.T. Barabanov, A. S. Lisogurskiy, “Algebraic approach to the formation of fast algorithms for the study of absolute stability”, Dynamic Systems, vol. 4, no. 2, pp. 121-134, 2014.

V. A. Kamenetskiy, “Frequency-domain stability conditions for hybrid systems”, Automation and Remote Control, vol. 78, no. 12, pp. 2101–2119, 2017. doi: 10.1134/S0005117917120013.

M. Lipkovich, A. Fradkov, “Equivalence of MIMO Circle Criterion to Existence of Quadratic Lyapunov Function”, IEEE Transactions on Automatic Control, vol. 61, no. 7, pp. 1895 – 1899, 2016. doi: 10.1109/TAC.2015.2487479.

S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA: Society for Industrial Mathematics, 1994.

S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.

I. M. Makarov, V. M. Lokhin, Intellectual Automatic Control Systems. Moscow: Fizmatlit, 2001.

N. A. Kazachek, “The use of intelligent algorithms based on fuzzy logic in management of industrial facilities”, Science Review, no. 20, pp. 165-170, 2015.

V. M. Lokhin, N. A. Kazachek, V. A. Ryabcov. Complex research of dynamics of control systems with fuzzy P–controller. International Journal of Materials, Mechanics and Manufacturing. vol.4, no. 2 pp. 140-144, 2016. doi: 10.7763/IJMMM.2016.V4.242.

Y. Boutalis, D. Theodoridis, T. Kottas, M. A. Christodoulou, System Identification and Adaptive Control: Theory and Applications of the Neurofuzzy and Fuzzy Cognitive Network Models. Berlin: Springer, 2014. doi: 10.1007/978-3-319-06364-5.

F. Manenti, F. Rossi a, A. G. Goryunov, V. F. Dyadik, K. A. Kozin, I. S. Nadezhdin, S. S. Mikhalevich, “Fuzzy adaptive control system of a non-stationary plant with closed-loop passive identifier”, Resource-Efficient Technologies, vol. 9, no. 1, pp. 10-18, 2015. doi: 10.1016/j.reffit.2015.07.001.

I. S. Nadezhdin, A. G. Goryunov, F. Manenti, “Fuzzy Adaptive Control System of a Non-Stationary Plant”, IOP Conference Series: Materials Science and Engineering, vol. 142, no. 1, pp. 1–8, 2016. doi: 10.1088/1757-899X/142/1/012048.


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DOI: 10.28991/cej-2019-03091229

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