\(H_\infty\) output tracking control for polynomial parameter-varying systems via homogeneous Lyapunov methods. (English) Zbl 1541.93084
Summary: This paper investigates the \(H_\infty\) output tracking control for polynomial parameter-varying systems – a class of nonlinear parameter-varying systems. Based on the homogeneous polynomial Lyapunov function (HPLF), a state feedback controller is presented to make the augmented system (composed of the original system and the tracking error system) robust exponentially stable, so that the output of the original system can track the reference signal and satisfy the \(H_\infty\) output tracking performance. The conservativeness of the theoretical results is reduced due to the following facts: (1) the HPLF can depend on the full states and time-varying parameters; and (2) there are no constraints on the input coefficient matrix and the inverse of the Lyapunov matrix. In particular, when the system states and time-varying parameters are confined to a compact set, local results are obtained based on the generalized S-procedure. The solvable conditions are given in terms of state-and-parameter-dependent linear matrix inequalities, which can be solved by sum of squares (SOS) techniques. Finally, the feasibility and effectiveness of the proposed method are verified by a numerical example and the heading control of unmanned surface vehicle.
MSC:
| 93B36 | \(H^\infty\)-control |
| 93C10 | Nonlinear systems in control theory |
| 93D30 | Lyapunov and storage functions |
| 93B52 | Feedback control |
Keywords:
\(H_\infty\) output tracking control; homogeneous polynomial Lyapunov function; polynomial parameter-varying systems; nonlinear parameter-varying systems; sum of squaresReferences:
| [1] | Yang, X.; Huang, J., Output regulation of time-varying nonlinear systems, Asian J. Control, 14, 5, 1387-1396, 2012 · Zbl 1303.93073 |
| [2] | Ngoc, P. H.A.; Hieu, L. T., New criteria for exponential stability of nonlinear difference systems with time-varying delay, Internat. J. Control, 86, 9, 1646-1651, 2013 · Zbl 1278.93218 |
| [3] | Li, Z. X.; Ji, H. B., Distributed consensus and tracking control of second-order time-varying nonlinear multi-agent systems, Internat. J. Robust Nonlinear Control, 27, 17, 3549-3563, 2017 · Zbl 1386.93010 |
| [4] | Long, L. J., Integral ISS for switched nonlinear time-varying systems using indefinite multiple Lyapunov functions, IEEE Trans. Autom. Control, 64, 1, 404-411, 2019 · Zbl 1423.93345 |
| [5] | F. Previdi, M. Lovera, Identification of a class of linear models with nonlinearly varying parameters, in: European Control Conference, Karlsruhe, Germany, 31 Aug.-03 Sept., 1999, pp. 2813-2817. |
| [6] | Zemouche, A.; Rajamani, R.; Kheloufi, H., Robust observer-based stabilization of Lipschitz nonlinear uncertain systems via LMIs - discussions and new design procedure, Internat. J. Robust Nonlinear Control, 27, 11, 1915-1939, 2017 · Zbl 1366.93596 |
| [7] | Pham, T. P.; Sename, O.; Dugard, L., A nonlinear parameter varying observer for real-time damper force estimation of an automotive electro-rheological suspension system, Internat. J. Robust Nonlinear Control, 31, 17, 8183-8205, 2021 · Zbl 1527.93154 |
| [8] | Zhou, X.; Xie, X. P.; Yue, D., Further studies on state estimation of discrete-time nonlinear parameter varying systems based on a new multiinstant switching observer, Internat. J. Robust Nonlinear Control, 31, 17, 8206-8217, 2021 · Zbl 1527.93440 |
| [9] | Lan, J. L.; Zhao, D. Z., Robust model predictive control for nonlinear parameter varying systems without computational delay, Internat. J. Robust Nonlinear Control, 31, 17, 8273-8294, 2021 · Zbl 1527.93056 |
| [10] | Fu, R.; Sun, H. F.; Zeng, J. P., Exponential stabilisation of nonlinear parameter-varying systems with applications to conversion flight control of a tilt rotor aircraft, Internat. J. Control, 92, 11, 2473-2483, 2019 · Zbl 1425.93233 |
| [11] | Zhang, Y. X.; Wang, Q.; Dong, C. Y., \( H_\infty\) Output tracking control for flight control systems with time-varying delay, Chin. J. Aeronaut., 16, 5, 1251-1258, 2019 |
| [12] | Liu, M. Q.; Zhang, S. L.; Chen, H. Y., \( H_\infty\) Output tracking control of discrete-time nonlinear systems via standard neural network models, IEEE Trans. Neural Netw. Learn. Syst., 25, 10, 1928-1935, 2014 |
| [13] | Yan, H. C.; Hu, C. Y.; Zhang, H., \( H_\infty\) Output tracking control for networked systems with adaptively adjusted event-triggered scheme, IEEE Trans. Syst. Man Cybern. A, 49, 10, 2050-2058, 2019 |
| [14] | Yang, D.; Zhao, J., \( H_\infty\) Output tracking control for a class of switched LPV systems and its application to an aero-engine model, Internat. J. Robust Nonlinear Control, 21, 12, 2102-2120, 2017 · Zbl 1370.93110 |
| [15] | Zhao, Y.; Zhao, J., \( H_\infty\) Output tracking bumpless transfer control for switched linear systems, IMA J. Math. Control Inform., 38, 1, 159-176, 2021 · Zbl 1475.93037 |
| [16] | Galeani, S.; Possieri, C.; Sassano, M., Asymptotic tracking for nonminimum phase linear systems via steady-state compensation, IEEE Trans. Autom. Control, 66, 9, 4176-4183, 2021 · Zbl 1471.93077 |
| [17] | Lian, J.; Ge, Y. L., Robust \(H_\infty\) output tracking control for switched systems under asynchronous switching, Nonlinear Anal. Hybrid Syst., 8, 57-68, 2013 · Zbl 1257.93031 |
| [18] | Pakkhesal, S.; Mohammadzaman, I., Less conservative output-feedback tracking control design for polynomial fuzzy systems, IET Control Theory Appl., 12, 7, 1843-1852, 2018 |
| [19] | Zhao, Y. B.; Lam, H. K.; Song, G., Relaxed stability conditions for polynomial-fuzzy-model-based control system with membership function information, IET Control Theory Appl., 11, 10, 1493-1503, 2017 |
| [20] | Pang, A. P.; He, Z.; Zhao, M. H., Sum of squares approach for nonlinear \(H_\infty\) control, Complexity, 2018, Article 8325609 pp., 2018, 7pages · Zbl 1407.93115 |
| [21] | C. Ebenbauer, J. Renz, F. Allgower, Polynomial feedback and observer design using nonquadratic Lyapunov functions, in: The 44th IEEE Conference on Decision Control, Seville, Spain, 15 Dec., 2005, pp. 7587-7592. |
| [22] | Ding, B. C., Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi-Sugeno systems via nonparallel distributed compensation law, IEEE Trans. Fuzzy Syst., 18, 5, 994-1000, 2010 |
| [23] | Tsai, S. H.; Jen, C. Y., \( H_\infty\) Stabilization for polynomial fuzzy time-delay system: a sum-of-squares approach, IEEE Trans. Fuzzy Syst., 26, 6, 3630-3644, 2018 |
| [24] | Li, Y.; Zeng, J. P.; Duan, Z. S., Robust observer-based \(H_\infty\) tracking control for polynomial uncertain systems via a homogeneous Lyapunov approach, Internat. J. Control, 2023 |
| [25] | Lo, J. C.; Lin, C. W., Polynomial fuzzy observed-state feedback stabilization via homogeneous Lyapunov methods, IEEE Trans. Fuzzy Syst., 26, 4, 2873-2885, 2018 |
| [26] | Lo, J. C.; Liu, J. W., Polynomial static output feedback \(H_\infty\) control via homogeneous Lyapunov functions, Internat. J. Robust Nonlinear Control, 29, 5, 1639-1659, 2019 · Zbl 1416.93068 |
| [27] | Huang, Y. W.; Lin, T.; Huang, W. C., Extended bounded real lemma based sum of squares for static output feedback \(H_\infty\) heading control, Internat. J. Robust Nonlinear Control, 32, 14, 7879-7895, 2022 · Zbl 1528.93057 |
| [28] | Huang, Y. W.; Lin, F., Heading control based on extended homogeneous polynomial Lyapunov function, Internat. J. Robust Nonlinear Control, 2023 |
| [29] | Sabbaghian-Bidgoli, F.; Farrokhi, M., Robust fuzzy observer-based fault-tolerant control: A homogeneous polynomial Lyapunov function approach, IET Control Theory Appl., 17, 1, 74-91, 2023 |
| [30] | Ding, J. Y.; Liu, Y.; Yu, J. Y., Interval type-2 polynomial fuzzy fault detection scheme with a multi-order homogeneous polynomial Lyapunov functions considering unmeasurable premise variables, Inform. Sci., 626, 559-571, 2023 · Zbl 1536.93497 |
| [31] | Fu, R.; Zeng, J. P.; Duan, Z. S., \( H_\infty\) Mixed stabilization of nonlinear parameter-varying systems, Internat. J. Robust Nonlinear Control, 28, 17, 5232-5246, 2018 · Zbl 1408.93118 |
| [32] | Zhang, B.; Jia, Y. M.; Du, J. P., A new class of strict Lyapunov functions for nonlinear time-varying systems, Automatica, 112, Article 108697 pp., 2020 · Zbl 1430.93185 |
| [33] | Papachristodoulou, A.; Anderson, J.; Valmorbida, G., SOSTOOLS: Sum of squares optimisation toolbox for MATLAB (version 3.01), 2016 |
| [34] | Reznick, B., Some concrete aspects of Hilbert’s 17th problem, Contemp. Math., 253, 51-72, 2000 |
| [35] | Jarvis-Wloszek, Z. W., Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization, 2003, University of California: University of California Berkeley, CA, (Ph.D. thesis) |
| [36] | Zhu, P. F.; Zeng, J. P., Observer-based control for nonlinear parameter-varying systems: A sum-of-squares approach, ISA Trans., 111, 121-131, 2021 |
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