# Numerical Solution for Nonlinear-quadratic Switching Control Systems with Time Delay

Volume 9, Issue 3, pp 175-187
• 1098 Views

### Abstract

This paper contributes an efficient numerical approach for optimal control of switched system with time delay via Bezier curves. A simple transformation is first used to map the optimal control problem with varying switching times into a new optimal control problem with fixed switching times. Then, the Bezier curves is used to approximate the optimal control problem a NLP. The NLP could be solved by using known algorithms.

### Keywords

• switched systems
• Bezier control points
• time delay systems
• dynamical system.

•  34E20
•  34A45
•  93C70
•  90C20

### References

• [1] J. V. Beltran, J. Monterde, Bezier solutions of the wave equation, Lecture notes in Computational Sciences, Computational Science and its Applications-ICCSA (A. Laganà, M. L. Gavrilova, V. Kumar, Y. Mun, C. J. K.Tan & O. Gervasi eds). Lecture Notes in Computer Science, Berlin: Springer, 3044 (2004), 631–640

• [2] A. Bemporad, A. Giua, C. Seatzu, Synthesis of state-feedback optimal controllers for continuous-time switched linear systems, In Proc. 41st IEEE conference on decision and control, Las Vegas, Nevada USA (2002)

• [3] R. Cholewa, A. J. Nowak, R. A. Bialecki, L. C. Wrobel, Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems, Computational Mechanics, 28 (2002), 282-290

• [4] C. H. Chu, C. C. L. Wang, C. R. Tsai, Computer aided geometric design of strip using developable Bezier patches, Computers in Industry, 59 (2008), 601-611

• [5] M. Egerstedt, Y. Wardi, F. Delmotte, Optimal control of switching times in switched dynamical systems, In Proc. IEEE conference on decision and control, 3 (2003), 2138-2143

• [6] M. Evrenosoglu, S. Somali, Least squares methods for solving singularly perturbed two-point boundary value problems using Bézier control points, Applied Mathematics Letters, 21 (2008), 1029-1032

• [7] G. E. Farin, Curve and surfaces for computer aided geometric design, First ed, Academic Press, New York (1988)

• [8] B. Farhadinia, K. L. Teo, R. C. Loxton, A computational method for a class of non-standard time optimal control problems involving multiple time horizons, Mathematical Computer Modelling, 49 (2009), 1682-1691

• [9] M. Gachpazan, Solving of time varying quadratic optimal control problems by using Bezier control points, Computational and Applied Mathematics, 30 (2011), 367-379

• [10] J. Geromela, P. Colaneri, P. Bolzern, Passivity of switched linear systems: Analysis and control design, Systems and Control Letters, 61 (2012), 549-554

• [11] F. Ghomanjani, M. H. Farahi, The Bezier control points method for solving delay differential equation, Intelligent Control and Automation, 3 (2012), 188-196

• [12] F. Ghomanjani, M. H. Farahi, M. Gachpazan, Bezier control points method to solve constrained quadratic optimal control of time varying linear systems, Computational and Applied Mathematics, 31 (2012), 1-24

• [13] A. Giua, C. Seatzu, C. Van Der Me, Optimal conrol of switched autonomous linear systems, In Proc. IEEE conference on decision and control, (2001), 2472-2477

• [14] K. Harada, E. Nakamae, Application of the Bezier curve to data interpolation, Computer-Aided Design, International Journal of Computer Mathematics, 14 (1982), 55-59

• [15] M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems, Applied Mathematics and Computation, 173 (2005), 169-198

• [16] I. Hwang, J. Li, D. Du, A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach, International Journal of Control, 81 (2008), 277-293

• [17] H. Juddu, Spectral method for constrained linear-quadratic optimal control, Mathematics Computers In simulation, 58 (2002), 159-169

• [18] B. Lang, The synthesis of wave forms using Bezier curves with control point modulation, In: The Second CEMS Research Student Conference, 1st edn. Morgan kaufamann, San Francisco (2004)

• [19] A. T. Layton, M. Van de Panne, A numerically evident and stable algorithm for animating water waves, The visual Computer, 18 (2002), 41-53

• [20] M. Kamgarpour, C. Tomlin, On optimal control of non-autonomous switched systems with a fixed mode sequence, Automatica, 48 (2012), 1177-1181

• [21] M. Margaliot, Stability analysis of switched systems using variational principles: An introduction, Automatica, 42 (2006), 2059-2077

• [22] B. Niua, J. Zhaoa, Stabilization and $L_2$-gain analysis for a class of cascade switched nonlinear systems: An average dwell-time method, Nonlinear Analysis: Hybrid Systems, 5 (2001), 671-680

• [23] G. Nürnberger, F. Zeilfelder, Developments in bivariate spline interpolation, Journal of Computational and Applied Mathematics, Computers and Mathematics with Applications, 121 (2000), 125-152

• [24] H. Prautzsch, W. Boehm, M. Paluszny, Bezier and B-Spline Techniques, Springer, (2001)

• [25] Y. Q. Shi, H. Sun, Image and video compression for multimedia engineering, CRC Press LLc, (2000)

• [26] R. Winkel, Generalized Bernstein Polynomials and Bezier Curves: An Application of Umbral Calculus to Computer Aided Geometric Design, Advances in Applied Mathematics, 27 (2001), 51-81

• [27] C. Wu, K. L. Teo, R. Li, Y. Zhao, Optimal control of switched systems with time delay, Appl, Math. Letters, 19 (2006), 1062-1067

• [28] X. Xu, G. Zhai, S. He, Some results on practical stabilizability of discrete-time switched affine systems, Nonlinear Analysis: Hybrid Systems, 4 (2010), 113-121

• [29] X. Xu, P. J. Antsakalis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16

• [30] J. Zheng, T. W. Sederberg, R. W. Johnson, Least squares methods for solving differential equations using Bezier control points, Applied Numerical Mathematics, 48 (2004), 237-252