Abstract
This chapter deals with the development of a state estimator and adaptive controller based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities as well as external perturbations. The set of stabilizing feedback matrices is given by a specific matrix inequality including the characteristic matrix of the attractive ellipsoid that contains all possible bounded trajectories around the origin. Here we present two modifications of the AEM that allow us to use online information obtained during the process and to adjust matrix parameters participating in constraints that characterize the class of adaptive stabilizing feedbacks. The proposed method guarantees that under a specific persistent excitation condition, the controlled system trajectories converge to an ellipsoid of “minimal size” having a minimal trace of the corresponding inverse ellipsoidal matrix.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The solution X ∗ of the optimization problem
$$\displaystyle{ \left\Vert A - XB\right\Vert ^{2} \rightarrow \mathop{\min }\limits_{ X} }$$(12.21)satisfies the identity
$$\displaystyle{ \begin{array}{c} \frac{\partial } {\partial X}\left\Vert A - XB\right\Vert ^{2} = \frac{\partial } {\partial X}\mathrm{tr}\left\{\left(A - XB\right)\left(A^{\intercal }- B^{\intercal }X^{\intercal }\right)\right\} \\ = -2\left(AB^{\intercal }- X^{+}BB^{\intercal }\right) = 0, \end{array} }$$or equivalently, \(X^{{\ast}}BB^{\intercal } = AB^{\intercal }\), whose solution (in the case \(BB^{\intercal } > 0\)) is
$$\displaystyle{ X^{{\ast}} = AB^{+} + Y \left(I - BB^{+}\right),\text{ }B^{+}:= B^{\intercal }\left(BB^{\intercal }\right)^{-1}, }$$where Y is any matrix of the corresponding size. So one has
$$\displaystyle{ \begin{array}{c} \left\Vert X^{{\ast}}\right\Vert ^{2} = \left\Vert AB^{+}\right\Vert ^{2} + \left\Vert Y \left(I - BB^{+}\right)\right\Vert ^{2} \\ + 2\mathrm{tr}\left\{AB^{+}\left(I -\left(\left(BB^{\intercal }\right)^{-1}B\right)B^{\intercal }\right)Y ^{\intercal }\right\} = \\ \left\Vert AB^{+}\right\Vert ^{2} + \left\Vert Y \left(I - BB^{+}\right)\right\Vert ^{2} \geq \left\Vert AB^{+}\right\Vert ^{2}. \end{array} }$$This means that the solution of the optimization problem (12.21) of minimal norm is X ∗ = AB +.
- 2.
We use the variable coordinate change x 1 = q 1, x 2 = q 2, \(x_{3} =\dot{ q}_{1}\), and \(x_{4} =\dot{ q}_{2}\). The Pendubot top position is (x 1, x 2, x 3, x 4) = (π∕2, 0, 0, 0).
- 3.
H + is the pseudoinverse matrix to H defined in the Moore–Penrose sense (Poznyak 2008).
Bibliography
Anderson, B. D. O. (1985). Adaptive systems, lack of persistency of excitation and bursting phenomena. Automatica, 21, 247–258.
Barabanov, A., & Granichin, O. (1984). Optimal controller for linear plants with bounded noise. Automation and Remote Control, 45, 39–46.
Davila, J., & Poznyak, A. (2011). Dynamic sliding mode control design using attracting ellipsoid method. Automatica, 47, 1467–1472.
Duncan, G., & Schweppe, F. (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Transactions on Automatic Control, 16, 411–423.
Francis, B., Helton, J., & Zames, G. (1984). H∞-optimal feedback controllers for linear multivariable systems. IEEE Transactions on Automatic Control, 29, 888–900.
Gonzalez-Garcia, S., Polyakov, A., & Poznyak, A. (2011). Using the method of invariant ellipsoids for linear robust output stabilization of spacecraft. Automation and Remote Control, 72, 540–555.
Holmstrom, K., Goran, A., & Edvall, M. (2009). User’s Guide for TOMLAB/CPLEX v12.1.
Ioannou, P., & Sun, J. (1996). Robust adaptive control. Upper Saddle River, NJ: Prentice Hall.
Kurzhanski, A., & Veliov, V. (1994). Modeling techniques and uncertain systems. New York: Birkhäuser.
Lakshmikantham, V., Leela, S., & Martynyuk, A. (1990). Practical stability of nonlinear systems. New Jersey, London, Singapore, Hong Kong: World Scientific.
Ljung, L. (1999). Sistem identification: Theory for the user. New Jersey: Prentice Hall PTR.
Narendra, K. S., & Annaswamy, A. M. (2005). Stable adaptive systems. New York: Dover Publications Inc.
Nazin, S., Polyak, B., & Topunov, M. (2007). Rejection of bounded exogenous disturbances by the method of invariant ellipsoids. Automation and Remote Control, 68, 467–486.
Ordaz, P., & Poznyak, A. (2012). The Furuta’s pendulum stabilization without the use of a mathematical model: Attractive ellipsoid method with KL-adaptation. In 51-th IEEE Conference on Decision and Control, Maui, Hawaii.
Ordaz, P., & Poznyak, A. (2014). “KL”-gain adaptation for attractive ellipsoid method. IMA Journal of Mathematical Control and Information. doi:10.1093/imamci/dnt046.
Peaucelle, D., Henrion, D., Labit, Y., & Taitz, K. (2002). User’s guide for SeDuMi Interface 1.04.
Polyakov, A., & Poznyak, A. (2011). Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control. Automatica, 47, 1450–1454.
Poznyak, A. (2004). Variable structure systems: From Principles to Implementation, chapter 3. IEE Control Series (Vol. 66, pp. 45–78). London, UK: The IET.
Poznyak, A. (2008). Advanced mathematical tools for automatic control engineers: Deterministic techniques. Amsterdam: Elsevier.
Sastry, S. (1999). Nonlinear systems. New York: Springer.
Sastry, S., & Bodson, M. (1994). Adaptive control: stability, convergence, and robustness. New York: Prentice-Hall.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Poznyak, A., Polyakov, A., Azhmyakov, V. (2014). Attractive Ellipsoid Method with Adaptation. In: Attractive Ellipsoids in Robust Control. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-09210-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-09210-2_12
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-09209-6
Online ISBN: 978-3-319-09210-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)