Non-Affine Nonlinear Systems Adaptive Optimal
Trajectory Tracking Controller Design and Application
Haoping WANG1,*, Yang TIAN1, Christian VASSEUR2
1 Sino-French International Joint Laboratory of Automatic Control and Signal Processing (LaFCAS),
Nanjing University of Science & Technology (NUST),
Nanjing 210094, China email@example.com
2 LAGIS- CNRS UMR 8219, LaFCAS,
University Lille Nord de France
Lille France, 59600
* Corresponding author
Abstract: For considering a general non-affine in control nonlinear systems, this present paper introduces a new Adaptive Nonlinear Optimal Recursive Control (ANORC) used for trajectory tracking. This referred ANORC controller which is developed on a L2 norm prescribed to minimize a trajectory tracking error has advantages of adaptive turning and optimal features. Furthermore, under the feedback of the systems output, this ANORC which adopts a control structure in recursive scheme and employs no knowledge of internal dynamics ensures the realization of the systems output trajectory tracking. Then to demonstrate the performances and effectiveness of this proposed ANORC controller, three different academic numerical examples (in the form of Single Input Single Output (SISO), Multi-Input Single Output (MISO) and Multi-Input Multi-Output (MIMO)) are implemented.
Keywords: Adaptive nonlinear optimal control, non-affine in control nonlinear systems, model free recursive control; trajectory tracking
CITE THIS PAPER AS:
Haoping WANG*, Yang TIAN, Christian VASSEUR, Non-Affine Nonlinear Systems Adaptive Optimal Trajectory Tracking Controller Design and Application, Studies in Informatics and Control, ISSN 1220-1766, vol. 24 (1), pp. 05-12, 2015. https://doi.org/10.24846/v24i1y201501
During the last decades, the control of nonlinear systems which concerns normally the stabilization or trajectory tracking control is generally considered difficult and becomes worldly active and hot research subjects [1-5]. The combined consideration of twofold for a controlled systems, which means the realization not only the control (trajectory tracking or stabilization), but also the optimization of a prescribed performance index (such as minimization the trajectory tracking error, the fuel consumption, or particles emissions, etc) are not easy. Currently there exists no adequate work relating the optimal control of nonlinear systems, especially for the non-affine in control nonlinear systems, since their corresponding Hamilton-Jacobi- Bellman (HJB) equation are difficult to resolve [6-7].
Compared to linear or nonlinear affine in control systems to solve their corresponding Riccati Equation (RE), Algebraic Riccati Equation (ARE) and HJB, the nonlinear non-affine in control systems are more complex and has no feasible solutions when their systems dynamics are not explicit. Recently, online adaptive approximation- based optimal control which refers to an online approximator based Adaptive Critic Designs (ACD) is gradually proposed [8-9]. While in reference , a relative new optimal adaptive control (OAC) is proposed for MIMO nonlinear systems in the form of strict feedback. The optimal adaptive feedback scheme is introduced for the affine systems to estimate the solution of HJB equation online which becomes the optimal feedback control input for the closed-loop system.
In adaptive control literature, various nonlinear systems and methods are appeared and discussed . Nonlinear systems in form of strict feedback in a variety of ways and their stability are studied with the application of the normal Backstepping scheme without any optimality. One of the most popular approaches is called “dynamic inversion and feedback linearization”. Recently, the inverse optimal control for strict feedback systems which consist on an associated cost function based control law is introduced in . Furthermore, in many applications, the inverse function for nonlinear systems is difficult or even impossible to find. In many practical cases, systems full states are not fully accessible. Then output feedback nonlinear controllers which use model-based observers were developed when their systems dynamics are known. While in many cases where their systems dynamics are unknown, the control of unknown strict feedback systems using adaptive neural-network based scheme is given to approximate their uncertain dynamics .
Therefore in our previous paper, a composed adaptive controller which is based on a recursive model free stabilization sub-controller (RMFSSC) and a recursive uncertain dynamic compensation sub-controller (RUDCSC) are proposed for nonlinear systems in . The RMFSSC is a recursive model free controller based on the theory of Piecewise-Continuous Systems (PCS) which are a particular class of hybrid systems with autonomous switching and controlled impulses [13-17].
Using PCS theory, piecewise-continuous controllers were firstly developed to enable sampled trajectory tracking of linear systems [13-17]. Then for improving trajectory tracking performance, a derived piecewise continuous controller and recursive model free controller were proposed in [15-16]. Unfortunately, the referred proposed RMFSSC is defined without any prescribed cost function to minimize and not considered for the case of non-affine in control nonlinear systems. Thus in this present paper for non-affine in control nonlinear systems, a new Adaptive Nonlinear Optimal trajectory tracking Recursive Controller (ANORC) is proposed. This referred ANORC controller which employs only the output of the non-affine in control nonlinear systems and a L2 norm to minimize the trajectory tracking error has a relative simple architecture and can be implemented to the case where the internal dynamics and parameters of the controlled non-affine in control nonlinear systems are unknown.
The following paper is organized as: in section 2, the research problem of the considered trajectory tracking of non-affine in control nonlinear systems is introduced. Then in Section 3, a two steps design approach of adaptive nonlinear optimal trajectory tracking recursive controller is developed. After that in Section 4 based on the proposed method, three academic numerical examples are implemented to validate the proposed method performance and robustness. Finally it is followed by some conclusion remarks in Section 5.
- SLOTINE, J. E., W. P. LI, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey, 1991.
- ISIDORI, A, Nonlinear Control Systems. 3rd edition, Springer-Verlag, London, 1995.
- KHALIL, H. K., Nonlinear Systems. 3rd edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.
- DINH, H., S. BHASIN, D. KIM, W. E. DIXON, Dynamic Neural Network-based Global Output Feedback Tracking Control for Uncertain Second-order Nonlinear Systems, In American Control Conference, pages June 2012, pp. 6418-6423.
- ZARGARZADEH, H., T. DIERKS, S. JAGANNATHAN, Optimal Adaptive Control of Nonlinear Continuous-time Systems in Strict Feedback Form with Unknown Internal Dynamics. In 51st IEEE Conference on Decision and Control, December 2012. Maui, Hawaii, USA , pp. 4127-4132.
- PEREIRA DA SILVA, P. S., BATISTA, On State Representations of Nonlinear Implicit Systems. vol. 83, no. 3, 2010, pp. 441-456.
- SONG, Y. D., Q. SONG, Survey of the Latest Developments in Control of Non-affine Systems. In Proceedings of the 30th Chinese Control Conference, Yantai, China, 2011.
- VRABIE, D., K. VAMVOUDAKIS, F. L. LEWIS, Adaptive Optimal Controllers based on Generalized Policy Iteration in a Continuous-time Framework, In Proceedings of the IEEE Mediterranean Conference on Control and Automation, 2009.
- VRABIE, D., F. L. LEWIS, Neural Network Spproach to Continuous-time Direct Adaptive Optimal Control for Partially Unknown Nonlinear Systems, Neural Networks, vol. 22, 2009.
- TAO, J. E., P. KOKOTOVIC, Adaptive Control of Systems, John Wiley & Sons, 1996.
- LI, Z. H., M. KRSTIC, Optimal Design of Adaptive Tracking Controllers for Nonlinear Continuous Systems, vol. 33, no. 8, 1997, pp. 1459-1473.
- WANG, H. P., B. KALCHEV, Y. TIAN, I. SIMEONOV, N. CHRISTOV, Modelling and Composed Recursive Model Free Control for the Anaerobic Digestion Process, In Advances in Intelligent Control Systems and Computer Science, Springer, January 2013, pp. 265-278.
- KONCAR, V., C. VASSEUR, Control of Linear Systems using Piecewise Continuous Systems, vol. 150, no. 6, November 2003, pp. 565-576.
- WANG, H. P., C. VASSEUR, V. KONCAR, A. CHAMROO, N. CHRISTOV, Sampled Tracking for Delayed Systems using Two Time-scales Sampled-data Controllers, Studies in Informatics and Control, vol. 19, no. 4, 2010, 339-346.
- WANG, H. P., C. VASSEUR, N. CHRISTOV, V. KONCAR, Vision Servoing of Robot Systems using Piecewise Continuous Controllers and Observers, Mechanical Systems and Signal Processing, vol. 33, December 2012, pp. 132–141.
- LI, S. Z., H. P. WANG, Y. TIAN, N. CHRISTOV, A. AITOUCHE, NNPID based Stator Voltage Oriented Vector Control for DFIG, Studies in Informatics and Control, vol. 23, no. 1, 2014, pp. 5-12.
- WANG, H. P., Y. TIAN, N. CHRISTOV. Piecewise Continuous Observers for Linear Systems with Sampled and Delayed Output. International Journal of System Science.
- LIN, W., Time-varying Feedback Control of Nonaffine Nonlinear Systems without Drift, 29, no. 2, 1996, pp. 101-110.