Abstract
A simplified neural network model is proposed to solve a class of linear matrix inequality problems. The stability and solvability of the proposed neural network are analyzed and discussed theoretically. In comparison with the previous neural network models (Lin and Huang, Neural Process Lett 11:153–169, 2000; Lin et al., IEEE Trans Neural Netw 11:1078–1092, 2000), the simplified one is composed of two layers rather than three layers, and the neuron array in each layer is triangular rather than square. The proposed approach can therefore reduce the complexity of the neural network architecture. In addition, the simplified neural network can also be extended to solve multiple linear matrix inequalities with specific constraints, which enlarges the application domain of the proposed approach. Finally, examples are given to illustrate the effectiveness and efficiency of the simplified neural network.
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References
Gahinet P, Apkarian P (1994) A linear matrix inequality approach to H ∞ control. Int J Robust Nonlin Contr 4: 421–448
Li H, Fu M (1997) A linear matrix inequality approach to Robust H ∞ Filtering. IEEE Trans Signal Process 45: 2338–2350
Molchanov AP, Pyatnitskii ES (1989) Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst Contr Lett 13: 59–64
Liao X, Chen G, Sanchez EN (2002) Delay-dependent exponential stability analysis of delayed neural networks: a LMI approach. Neural Netw 15: 855–866
Poolla K, Khargonekar P, Tikku A et al (1994) A time-domain approach to model validation. IEEE Trans Automat Contr 39: 951–959
Anderson B, Moore JB (1990) Optimal control: linear quadratic methods. Prentice-Hall, Englewood Cliffs, New Jersey
Megretsky A (1993) Necessary and sufficient conditions of stability: a multiloop generalization of the circle criterion. IEEE Trans Automat Contr 38: 753–756
Cheng L, Hou ZG et al (2007) Constrained multi-variable generalized predictive control using a dual neural network. Neural Comput Appl 16: 505–512
Hou ZG, Gupta MM et al (2007) A recurrent neural network for hierarchical control of interconnected dynamic systems. IEEE Trans Neural Netw 18: 466–481
Cheng L, Hou ZG et al (2008) A neutral-type delayed projection neural network for solving nonlinear variational inequalities. IEEE Trans Circuits Syst II 55: 806–810
Cheng L, Hou ZG et al (2009) Solving linear variational inequalities by projection neural network with time-varying delays. Phys Lett A 373: 1739–1743
Cichocki A, Unbehauen R (1992) Neural network for computing eigenvalue and eigenvectors. Biol Cybern 68: 155–164
Wang J, Wu G (1996) A multilayer recurrent neural network for on-line synthesis of minimum-norm linear feedback control systems via pole assignment. Automatica 32: 435–442
Zhang Y, Ge SS (2005) Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans Neural Netw 16: 1477–1490
Wang J, Wu G (1998) A multilayer recurrent neural network for solving continuous-time algebraic Riccati equations. Neural Netw 11: 939–950
Cichocki A, Unbehauen R (1993) Neural networks for optimization and signal processing. John Wiley, New York
Xia Y, Wang J, Hung DL (1999) Recurrent neural networks for solving linear inequalities and equations. IEEE Trans Circuits Syst I 46: 452–462
Jiang DC, Wang J (1999) A recurrent neural network for real-time semidefinite programming. IEEE Trans Neural Netw 10: 81–93
Boyd S, Ghaoui LE, Feron E et al (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia
Slotine J-JE, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs, New Jersey
Gahinet P, Nemirovski A, Laub NJ et al (1995) LMI control toolbox for use with Matlab. The MathWorks, Natick
Lin C-L, Huang T-H (2000) A novel approach solving for linear matrix inequalities using neural networks. Neural Process Lett 11: 153–169
Lin C-L, Lai C-C, Huang T-H (2000) A neural network for linear matrix inequality problems. IEEE Trans Neural Netw 11: 1078–1092
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Cheng, L., Hou, ZG. & Tan, M. A Simplified Neural Network for Linear Matrix Inequality Problems. Neural Process Lett 29, 213–230 (2009). https://doi.org/10.1007/s11063-009-9105-5
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DOI: https://doi.org/10.1007/s11063-009-9105-5