0
Research Papers

Optimizing a Class of Nonlinear Singularly Perturbed Systems Using SDRE Technique

[+] Author and Article Information
Seyed Mostafa Ghadami

Department of Electrical Engineering,
Science and Research Branch,
Islamic Azad University,
Ashrafi Esfehani Freeway,
Hesarak Street, Poonak,
Tehran 1477893855, Iran
e-mail: m.gadami@srbiau.ac.ir

Roya Amjadifard

Assistant Professor
Kharazmi University,
Department of Computer Engineering,
Tehran 37551-31979, Iran
e-mail: amjadifard@khu.ac.ir

Hamid Khaloozadeh

Professor
Industrial Control Center of Excellence,
K. N. Toosi University of Technology,
Tehran 16314, Iran
e-mail: h_khaloozadeh@kntu.ac.ir

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 13, 2011; final manuscript received May 14, 2013; published online September 4, 2013. Assoc. Editor: Warren E. Dixon.

J. Dyn. Sys., Meas., Control 136(1), 011003 (Sep 04, 2013) (13 pages) Paper No: DS-11-1351; doi: 10.1115/1.4024602 History: Received November 13, 2011; Revised May 14, 2013

In this paper, we address the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the state-dependent Riccati equation (SDRE) technique and singular perturbation theory. In such systems, both slow and fast variables are nonlinear. Moreover, the performance index for the system states is nonlinearly quadratic. In this study, unlike conventional methods, linearization does not occur around the equilibrium point, and it provides a description of the system as state-dependent coefficients (SDCs) in the form f(x) = A(x)x. One of the advantages of the state-dependent Riccati equation method is that no information about the Jacobian of the nonlinear system, just like the Hamilton–Jacobi–Belman (HJB) equation, is required. Thus, the state-dependent Riccati equation has simplicity of the linear quadratic method. On the other hand, one of the advantages of the singular perturbation theory is that it reduces high-order systems into two lower order subsystems due to the interaction between slow and fast variables. In the proposed method, the singularly perturbed state-dependent Riccati equations are first derived for the system under study. Using the singular perturbation theory, the singularly perturbed state and state-dependent Riccati equations are separated into outer layer, initial, and final layer correction equations. These equations are then solved to obtain the optimal control law. Simulation results in comparison with the previous methods indicate the desirable performance and efficiency of the proposed method. However, it should be noted that due to the dependence of the proposed method on the choice of state-dependent matrices and the presence of a nonlinear optimal control problem, the results are generally suboptimal.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

The design procedure stages in the proposed method

Grahic Jump Location
Fig. 2

Performance index for different values of ε

Grahic Jump Location
Fig. 3

The optimal control u for various values of ε in time interval of 0–10 s

Grahic Jump Location
Fig. 4

The optimal control u for various values of ε in initial-layer correction

Grahic Jump Location
Fig. 5

The slow variable x1

Grahic Jump Location
Fig. 6

The fast variable x2 for various values of ε in time interval of 0–10 s

Grahic Jump Location
Fig. 7

The fast variable x2 for various values of ε in initial-layer correction

Grahic Jump Location
Fig. 8

The Riccati gain of P11 for different values of ε

Grahic Jump Location
Fig. 9

The Riccati gain P21 for different values of ε in time interval of 0–10 s

Grahic Jump Location
Fig. 10

The Riccati gain P21 for different values of ε in final layer correction

Grahic Jump Location
Fig. 11

The Riccati gain P22 for different values of ε in time interval of 0–10 s

Grahic Jump Location
Fig. 12

The Riccati gain P22 for different values of ε in final layer correction

Grahic Jump Location
Fig. 13

Single link flexible joint robot manipulator

Grahic Jump Location
Fig. 14

The optimal control u in time interval of 0–2 s

Grahic Jump Location
Fig. 15

The slow and fast variables in time interval of 0–2 s

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In