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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.

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References

Çimen, T., 2010, “Systematic and Effective Design of Nonlinear Feedback Controllers Via the State-Dependent Riccati Equation (SDRE) Method,” Annu. Rev. Control, 34, pp. 32–51. [CrossRef]
Pearson, J. D., 1962, “Approximation Methods in Optimal Control,” J. Electron. Control, 13, pp. 453–469. [CrossRef]
Wernli, A., and Cook, G., 1975, “Suboptimal Control for the Nonlinear Quadratic Regulator Problem,” Automatica, 11, pp. 75–84. [CrossRef]
Mracek, C. P., and Cloutier, J. R., 1998, “Control Designs for the Nonlinear Benchmark Problem Via the State-Dependent Riccati Equation Method,” Int. J. Robust Nonlinear Control, 8, pp. 401–433. [CrossRef]
Hammett, K. D., Hall, C. D., and Ridgely, D. B., 1998, “Controllability Issues in Nonlinear State-Dependent Riccati Equation Control,” AIAA J. Guid. Control Dyn., 21, pp. 767–773. [CrossRef]
Shamma, J. S., and Cloutier, J. R., 2003, “Existence of SDRE Stabilizing Feedback,” IEEE Trans. Autom. Control, 48, pp. 513–517. [CrossRef]
Khaloozadeh, H., and Abdolahi, A., 2002, “An Iterative Procedure for Optimal Nonlinear Tracking Problem,” Proceedings of the 7th International Conference on Control, Automation, Robotics and Vision (ICARCV), pp. 1508–1512.
Khaloozadeh, H., and Abdolahi, A., 2004, “A New Iterative Procedure for Optimal Nonlinear Regulation Problem,” Proceedings of the 3rd International Conference on System Identification and Control Problems (SICPRO), pp. 1256–1266.
Beeler, S. C., 2004, “State-Dependent Riccati Equation Regulation of Systems With State and Control Nonlinearities,” NASA and National Institute of Aerospace, Hampton, VA, Technical Report NASA/CR-2004-213245, NIA-2004-08.
Banks, H. T., Lewis, B. M., and Tran, H. T., 2007, “Nonlinear Feedback Controllers and Compensators, A State-Dependent Riccati Equation Approach,” Comput. Optim. Appl., 37, pp. 177–218. [CrossRef]
Çimen, T., 2012, “Survey of State-Dependent Riccati Equation in Nonlinear Optimal Feedback Control Synthesis,” J. Guid. Control Dyn., 35(4), pp. 1025–1047. [CrossRef]
Pukdeboon, C., 2012, “Optimal Output Feedback Controllers for Spacecraft Attitude Tracking,” Asian J. Control. (in press). [CrossRef]
Naidu, D. S., and Calise, A. J., 2001, “Singular Perturbations and Time Scales in Guidance and Control of Aerospace Systems: A Survey,” AIAA J. Guid. Control Dyn., 24, pp. 1057–1078. [CrossRef]
Kokotovic, P., and Sannuti, P., 1968, “Singular Perturbation Method for Reducing the Model Order in Optimal Control Design,” IEEE Trans. Autom. Control, 13, pp. 377–384. [CrossRef]
Fridman, E., 2000, “Exact Slow-Fast Decomposition of Nonlinear Singularly Perturbed Optimal Control Problem,” Syst. Control Lett., 40, pp. 121–131. [CrossRef]
Fridman, E., 2001, “A Descriptor System Approach to Nonlinear Singularly Perturbed Optimal Control Problem,” Automatica, 37, pp. 543–549. [CrossRef]
Amjadifard, R., Beheshti, M. T. H., and Yazdanpanah, M. J., 2011, “Robust Stabilization for a Class of Nonlinear Singularly Perturbed Systems,” ASME J. Dyn. Syst. Meas. Control., 133(5), p. 051004. [CrossRef]
Amjadifard, R., Yazdanpanah, M. J., and Beheshti, M. T. H., 2005, “Robust Regulation of a Class of Nonlinear Singularly Perturbed Systems,” 16th IFAC World Congress, July 4-8, Prague, Czech Republic, pp. 1009-1009. [CrossRef]
Lewis, F. L., and Syrmos, V. L., 1995, Optimal Control, 2nd ed., Wiley Interscience, New York.
O'Malley, R., 1974, Introduction to Singular Perturbations, Academic Press, New York.
Kokotovic, P., and Yackel, R., 1972, “Singular Perturbation of Linear Regulators: Basic Theorems,” IEEE Trans. Autom. Control, 17, pp. 29–37. [CrossRef]
Amjadifard, R., Khadem, S. E., and Khaloozadeh, H., 2001, “Position and Velocity Control of a Flexible Joint Robot Manipulator Via Fuzzy Controller Based on Singular Perturbation Analysis,” Proceedings of the 10th IEEE International Conference on Fuzzy Systems, pp. 348–351.

Figures

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Fig. 1

The design procedure stages in the proposed method

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Fig. 2

Performance index for different values of ε

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Fig. 3

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

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Fig. 4

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

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Fig. 5

The slow variable x1

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Fig. 6

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

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Fig. 7

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

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Fig. 8

The Riccati gain of P11 for different values of ε

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Fig. 9

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

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Fig. 10

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

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Fig. 11

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

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Fig. 12

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

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Fig. 13

Single link flexible joint robot manipulator

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Fig. 14

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

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Fig. 15

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

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