American Institute of Mathematical Sciences

September  2015, 5(3): 585-611. doi: 10.3934/mcrf.2015.5.585

Generalized homogeneous systems with applications to nonlinear control: A survey

 1 Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, United States 2 Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, United States 3 School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

Received  November 2014 Revised  May 2015 Published  July 2015

This survey provides a unified homogeneous perspective on recent advances in the global stabilization of various nonlinear systems with uncertainty. We first review definitions and properties of homogeneous systems and illustrate how the homogeneous system theory can yield elegant feedback stabilizers for certain homogeneous systems. By taking advantage of homogeneity, we then present the so-called Adding a Power Integrator (AAPI) technique and discuss how it can be employed to recursively construct smooth state feedback stabilizers for uncertain nonlinear systems with uncontrollable linearizations. Based on the AAPI technique, a non-smooth version as well as a generalized version of AAPI approaches can be further developed from a homogeneous viewpoint, resulting in solutions to the global stabilization of genuinely nonlinear systems that may not be controlled, even locally, by any smooth state feedback. In the case of output feedback control, we demonstrate in this survey why the homogeneity is the key in developing a homogeneous domination approach, which has been successful in solving some difficult nonlinear control problems including, for instance, the global stabilization of systems with higher-order nonlinearities via output feedback. Finally, we show how the notion of Homogeneity with Monotone Degrees (HWMD) plays a decisive role in unifying smooth and non-smooth AAPI methods under one framework. Other applications of HWMD will be also summarized and discussed in this paper, along the directions of constructing smooth stabilizers for nonlinear systems in special forms and low-gain'' controllers for a class of general upper-triangular systems.
Citation: Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control & Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585
References:
 [1] V. Andrieu, L. Praly and A. Astofi, Homogeneous approximation and recursive observer design and output feedback,, SIAM J. Control Optim., 47 (2008), 1814. doi: 10.1137/060675861. [2] V. Andrieu, L. Praly and A. Astolfi, High gain observers with updated gain and homogeneous correction terms,, Automatica, 45 (2009), 422. doi: 10.1016/j.automatica.2008.07.015. [3] A. Bacciotti, Local Stabilizability of Nonlinear Control Systems,, World Scientific, (1992). [4] A. Bacciotti and L. Roiser, Liapunov Functions and Stability in Control Theory,, volume 267 of Lecture Notes in Control and Information Sciences, (2001). [5] D. Bestle and M. Zeitz, Canonical form observer design for nonlinear time-variable systems,, Internat. J. Control, 38 (1983), 419. [6] S. Celikovsky and E. Aranda-Bricaire, Constructive nonsmooth stabilization of triangular systems,, Systems Control Lett., 36 (1999), 21. doi: 10.1016/S0167-6911(98)00062-0. [7] J. M. Coron and L. Praly, Adding an integrator for the stabilization problem,, Systems Control Lett., 17 (1991), 89. doi: 10.1016/0167-6911(91)90034-C. [8] W. P. Dayawansa, Recent advances in the stabilization problem for low dimensional systems,, In Proc. of 1992 IFAC NOLCOS, (1993), 1. doi: 10.1016/B978-0-08-041901-5.50006-3. [9] W. P. Dayawansa, C. F. Martin and G. Knowles, Asymptotic stabilization of a class of smooth two-dimensional systems,, SIAM J. Control Optim., 28 (1990), 1321. doi: 10.1137/0328070. [10] S. Ding, C. Qian and S. Li, Global stabilization of a class of feedforward systems with lower-order nonlinearities,, IEEE Trans. Autom. Control, 55 (2010), 691. doi: 10.1109/TAC.2009.2037455. [11] S. Ding, C. Qian, S. Li and Q. Li, Global stabilization of a class of upper-triangular systems with unbounded or uncontrollabel linearizations,, Internat. J. Robust Nonlinear Control, 21 (2011), 271. doi: 10.1002/rnc.1591. [12] J. Franz, Control Design for a Class of Nonlinear Systems Using Limited Information and Its Application to robotics,, Master's thesis, (2009). [13] J. P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems applications to bioreactors,, IEEE Trans. Autom. Control, 37 (1992), 875. doi: 10.1109/9.256352. [14] W. Hahn, Stability of Motion,, Springer-Verlag, (1967). [15] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,, In Differential Equations: Stability and Control, 127 (1991), 249. [16] X. Huang, W. Lin and B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems,, Automatica, 41 (2005), 881. doi: 10.1016/j.automatica.2004.11.036. [17] A. Isidori, Nonlinear Control Systems,, Springer-Verlag, (1995). doi: 10.1007/978-1-84628-615-5. [18] M. Kawski, Stabilization of nonlinear systems in the plane,, Systems Control Lett., 12 (1989), 169. doi: 10.1016/0167-6911(89)90010-8. [19] M. Kawski, Homogeneous stabilizing feedback laws,, Control Theory and Advanced Technology, 6 (1990), 497. [20] H. K. Khalil and A. Saberi, Adaptive stabilization of a class of nonlinear systems using high-gain feedback,, IEEE Trans. Autom. Control, 32 (1987), 1031. doi: 10.1109/TAC.1987.1104481. [21] P. V. Kokotovic and R. Freeman, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques,, Springer, (1996). doi: 10.1007/978-0-8176-4759-9. [22] A. J. Krener and A. Isidori, Linearization by output injection and nonlinear observers,, Systems Control Lett., 3 (1983), 47. doi: 10.1016/0167-6911(83)90037-3. [23] H. Lei and W. Lin, Robust control of uncertain systems with polynomial nonlinearity by output feedback,, Internat. J. Robust Nonlinear Control, 19 (2009), 692. doi: 10.1002/rnc.1349. [24] J. Li, C. Qian and M. Frye, A dual observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities,, Internat. J. Robust Nonlinear Control, 19 (2009), 1697. doi: 10.1002/rnc.1401. [25] W. Lin and C. Qian, New results on global stabilization of feedforward systems via small feedback,, In Proc. of the 37th IEEE Conference on Decision and Control, (1998), 873. [26] W. Lin and X. Li, Synthesis of upper-triangular nonlinear systems with marginally unstable free dynamics using state-dependent Saturation,, Internat. J. Control, 72 (1999), 1078. doi: 10.1080/002071799220434. [27] W. Lin and H. Lei, Taking advantage of homogeneity: a unified framework for output feedback control of nonlinear systems (plenary paper),, In Proc. of the 7th IFAC Nonlinear Control Systems Symposium, (2007), 27. [28] W. Lin and C. Qian, Adding one power integrator: A tool for global stabilization of high-order lower-triangular systems,, Systems Control Lett., 39 (2000), 339. doi: 10.1016/S0167-6911(99)00115-2. [29] W. Lin and C. Qian, Robust regulation of a chain of power integrators perturbed by a lower-triangular vector field,, Internat. J. Robust Nonlinear Control, 10 (2000), 397. doi: 10.1002/(SICI)1099-1239(20000430)10:5<397::AID-RNC477>3.0.CO;2-N. [30] R. Marino and P. Tomei, Dynamic output feedback linearization and global stabilization,, Systems Control Lett., 17 (1991), 115. doi: 10.1016/0167-6911(91)90036-E. [31] F. Mazenc, Stabilization of feedforward systems approximated by a non-linear chain of integrators,, Systems Control Lett., 32 (1997), 223. doi: 10.1016/S0167-6911(97)00091-1. [32] F. Mazenc and L. Praly, Adding integrations, saturated controls, and stabilization for feedforward systems,, IEEE Trans. Autom. Control, 41 (1996), 1559. doi: 10.1109/9.543995. [33] F. Mazenc, L. Praly and W. P. Dayawansa, Global stabilization by output feedback: Examples and counterexamples,, Systems Control Lett., 23 (1994), 119. doi: 10.1016/0167-6911(94)90041-8. [34] J. Polendo, Global Synthesis of Highly Nonlinear Dynamic Systems with Limited and Uncertain Information,, PhD thesis, (2006). [35] J. Polendo and C. Qian, A universal method for robust stabilization of nonlinear systems: unification and extension of smooth and nonsmooth approaches,, In Proc. of the 2006 American Control Conference, (2006), 4285. doi: 10.1109/ACC.2006.1657392. [36] J. Polendo and C. Qian, A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback,, Internat. J. Robust Nonlinear Control, 17 (2007), 605. doi: 10.1002/rnc.1139. [37] J. Polendo and C. Qian, An expanded method to robustly stabilize uncertain nonlinear systems,, Communications in Information and Systems, 8 (2008), 55. doi: 10.4310/CIS.2008.v8.n1.a4. [38] J. Polendo, C. Qian and C. Schrader, Homogeneous domination and decentralized control problem for nonlinear system stabilization,, In Advances in Statistical Control, (2008), 257. doi: 10.1007/978-0-8176-4795-7_13. [39] C. Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear system,, In Proc. of the 2005 American Control Conference, (2005), 4708. [40] C. Qian and W. Lin, Using small feedback to stabilize a wider class of feedforward systems,, In Proc. of IFAC World Congress, (1999), 309. [41] C. Qian and W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 46 (2001), 1061. doi: 10.1109/9.935058. [42] C. Qian and W. Lin, Output feedback control of a class of nonlinear systems: A nonseparation principle paradigm,, IEEE Trans. Autom. Control, 47 (2002), 1710. doi: 10.1109/TAC.2002.803542. [43] C. Qian and W. Lin, Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 51 (2006), 1457. doi: 10.1109/TAC.2006.880955. [44] C. Qian and W. Lin, Homogeneity with incremental degrees and global stabilisation of a class of high-order upper-triangular systems,, Internat. J. Control, 85 (2012), 1851. doi: 10.1080/00207179.2012.706713. [45] L. Roiser, Homogeneous lyapunov function for homogeneous continuous vector field,, Systems Control Lett., 19 (1992), 467. doi: 10.1016/0167-6911(92)90078-7. [46] C. Rui, M. Reyhangolu, I. Kolmanovsky, S. Cho and H. N. McClamroch, Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system,, In Proc. of the 36th IEEE Conference on Control and Decision, 4 (1997), 3998. doi: 10.1109/CDC.1997.652490. [47] A. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls,, Systems Control Lett., 18 (1992), 165. doi: 10.1016/0167-6911(92)90001-9. [48] A. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation,, IEEE Trans. Autom. Control, 41 (1996), 1256. doi: 10.1109/9.536496. [49] W. Tian, C. Qian and H. Du, A generalised homogeneous solution for global stabilisation of a class of non-smooth upper-triangular systems,, Internat. J. Control, 87 (2014), 951. doi: 10.1080/00207179.2013.862347. [50] W. Tian, C. Zhang, C. Qian and S. Li, Global stabilization of inherently non-linear systems using continuously differentiable controllers,, Nonlinear Dynamics, 77 (2014), 739. doi: 10.1007/s11071-014-1336-y. [51] J. Tsinias, A theorem on global stabilization of nonlinear systems by linear feedback,, Systems Control Lett., 17 (1991), 357. doi: 10.1016/0167-6911(91)90074-O. [52] J. Tsinias and M. P. Tzamtzi, An explicit formula of bounded feedback stabilizers for feedforward systems,, Systems Control Lett., 43 (2001), 247. doi: 10.1016/S0167-6911(01)00107-4. [53] B. Yang and W. Lin, Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization,, IEEE Trans. Autom. Control, 50 (2005), 619. doi: 10.1109/TAC.2005.847084. [54] V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems,, Groningen: Noordhoff, (1964).

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References:
 [1] V. Andrieu, L. Praly and A. Astofi, Homogeneous approximation and recursive observer design and output feedback,, SIAM J. Control Optim., 47 (2008), 1814. doi: 10.1137/060675861. [2] V. Andrieu, L. Praly and A. Astolfi, High gain observers with updated gain and homogeneous correction terms,, Automatica, 45 (2009), 422. doi: 10.1016/j.automatica.2008.07.015. [3] A. Bacciotti, Local Stabilizability of Nonlinear Control Systems,, World Scientific, (1992). [4] A. Bacciotti and L. Roiser, Liapunov Functions and Stability in Control Theory,, volume 267 of Lecture Notes in Control and Information Sciences, (2001). [5] D. Bestle and M. Zeitz, Canonical form observer design for nonlinear time-variable systems,, Internat. J. Control, 38 (1983), 419. [6] S. Celikovsky and E. Aranda-Bricaire, Constructive nonsmooth stabilization of triangular systems,, Systems Control Lett., 36 (1999), 21. doi: 10.1016/S0167-6911(98)00062-0. [7] J. M. Coron and L. Praly, Adding an integrator for the stabilization problem,, Systems Control Lett., 17 (1991), 89. doi: 10.1016/0167-6911(91)90034-C. [8] W. P. Dayawansa, Recent advances in the stabilization problem for low dimensional systems,, In Proc. of 1992 IFAC NOLCOS, (1993), 1. doi: 10.1016/B978-0-08-041901-5.50006-3. [9] W. P. Dayawansa, C. F. Martin and G. Knowles, Asymptotic stabilization of a class of smooth two-dimensional systems,, SIAM J. Control Optim., 28 (1990), 1321. doi: 10.1137/0328070. [10] S. Ding, C. Qian and S. Li, Global stabilization of a class of feedforward systems with lower-order nonlinearities,, IEEE Trans. Autom. Control, 55 (2010), 691. doi: 10.1109/TAC.2009.2037455. [11] S. Ding, C. Qian, S. Li and Q. Li, Global stabilization of a class of upper-triangular systems with unbounded or uncontrollabel linearizations,, Internat. J. Robust Nonlinear Control, 21 (2011), 271. doi: 10.1002/rnc.1591. [12] J. Franz, Control Design for a Class of Nonlinear Systems Using Limited Information and Its Application to robotics,, Master's thesis, (2009). [13] J. P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems applications to bioreactors,, IEEE Trans. Autom. Control, 37 (1992), 875. doi: 10.1109/9.256352. [14] W. Hahn, Stability of Motion,, Springer-Verlag, (1967). [15] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,, In Differential Equations: Stability and Control, 127 (1991), 249. [16] X. Huang, W. Lin and B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems,, Automatica, 41 (2005), 881. doi: 10.1016/j.automatica.2004.11.036. [17] A. Isidori, Nonlinear Control Systems,, Springer-Verlag, (1995). doi: 10.1007/978-1-84628-615-5. [18] M. Kawski, Stabilization of nonlinear systems in the plane,, Systems Control Lett., 12 (1989), 169. doi: 10.1016/0167-6911(89)90010-8. [19] M. Kawski, Homogeneous stabilizing feedback laws,, Control Theory and Advanced Technology, 6 (1990), 497. [20] H. K. Khalil and A. Saberi, Adaptive stabilization of a class of nonlinear systems using high-gain feedback,, IEEE Trans. Autom. Control, 32 (1987), 1031. doi: 10.1109/TAC.1987.1104481. [21] P. V. Kokotovic and R. Freeman, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques,, Springer, (1996). doi: 10.1007/978-0-8176-4759-9. [22] A. J. Krener and A. Isidori, Linearization by output injection and nonlinear observers,, Systems Control Lett., 3 (1983), 47. doi: 10.1016/0167-6911(83)90037-3. [23] H. Lei and W. Lin, Robust control of uncertain systems with polynomial nonlinearity by output feedback,, Internat. J. Robust Nonlinear Control, 19 (2009), 692. doi: 10.1002/rnc.1349. [24] J. Li, C. Qian and M. Frye, A dual observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities,, Internat. J. Robust Nonlinear Control, 19 (2009), 1697. doi: 10.1002/rnc.1401. [25] W. Lin and C. Qian, New results on global stabilization of feedforward systems via small feedback,, In Proc. of the 37th IEEE Conference on Decision and Control, (1998), 873. [26] W. Lin and X. Li, Synthesis of upper-triangular nonlinear systems with marginally unstable free dynamics using state-dependent Saturation,, Internat. J. Control, 72 (1999), 1078. doi: 10.1080/002071799220434. [27] W. Lin and H. Lei, Taking advantage of homogeneity: a unified framework for output feedback control of nonlinear systems (plenary paper),, In Proc. of the 7th IFAC Nonlinear Control Systems Symposium, (2007), 27. [28] W. Lin and C. Qian, Adding one power integrator: A tool for global stabilization of high-order lower-triangular systems,, Systems Control Lett., 39 (2000), 339. doi: 10.1016/S0167-6911(99)00115-2. [29] W. Lin and C. Qian, Robust regulation of a chain of power integrators perturbed by a lower-triangular vector field,, Internat. J. Robust Nonlinear Control, 10 (2000), 397. doi: 10.1002/(SICI)1099-1239(20000430)10:5<397::AID-RNC477>3.0.CO;2-N. [30] R. Marino and P. Tomei, Dynamic output feedback linearization and global stabilization,, Systems Control Lett., 17 (1991), 115. doi: 10.1016/0167-6911(91)90036-E. [31] F. Mazenc, Stabilization of feedforward systems approximated by a non-linear chain of integrators,, Systems Control Lett., 32 (1997), 223. doi: 10.1016/S0167-6911(97)00091-1. [32] F. Mazenc and L. Praly, Adding integrations, saturated controls, and stabilization for feedforward systems,, IEEE Trans. Autom. Control, 41 (1996), 1559. doi: 10.1109/9.543995. [33] F. Mazenc, L. Praly and W. P. Dayawansa, Global stabilization by output feedback: Examples and counterexamples,, Systems Control Lett., 23 (1994), 119. doi: 10.1016/0167-6911(94)90041-8. [34] J. Polendo, Global Synthesis of Highly Nonlinear Dynamic Systems with Limited and Uncertain Information,, PhD thesis, (2006). [35] J. Polendo and C. Qian, A universal method for robust stabilization of nonlinear systems: unification and extension of smooth and nonsmooth approaches,, In Proc. of the 2006 American Control Conference, (2006), 4285. doi: 10.1109/ACC.2006.1657392. [36] J. Polendo and C. Qian, A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback,, Internat. J. Robust Nonlinear Control, 17 (2007), 605. doi: 10.1002/rnc.1139. [37] J. Polendo and C. Qian, An expanded method to robustly stabilize uncertain nonlinear systems,, Communications in Information and Systems, 8 (2008), 55. doi: 10.4310/CIS.2008.v8.n1.a4. [38] J. Polendo, C. Qian and C. Schrader, Homogeneous domination and decentralized control problem for nonlinear system stabilization,, In Advances in Statistical Control, (2008), 257. doi: 10.1007/978-0-8176-4795-7_13. [39] C. Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear system,, In Proc. of the 2005 American Control Conference, (2005), 4708. [40] C. Qian and W. Lin, Using small feedback to stabilize a wider class of feedforward systems,, In Proc. of IFAC World Congress, (1999), 309. [41] C. Qian and W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 46 (2001), 1061. doi: 10.1109/9.935058. [42] C. Qian and W. Lin, Output feedback control of a class of nonlinear systems: A nonseparation principle paradigm,, IEEE Trans. Autom. Control, 47 (2002), 1710. doi: 10.1109/TAC.2002.803542. [43] C. Qian and W. Lin, Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems,, IEEE Trans. Autom. Control, 51 (2006), 1457. doi: 10.1109/TAC.2006.880955. [44] C. Qian and W. Lin, Homogeneity with incremental degrees and global stabilisation of a class of high-order upper-triangular systems,, Internat. J. Control, 85 (2012), 1851. doi: 10.1080/00207179.2012.706713. [45] L. Roiser, Homogeneous lyapunov function for homogeneous continuous vector field,, Systems Control Lett., 19 (1992), 467. doi: 10.1016/0167-6911(92)90078-7. [46] C. Rui, M. Reyhangolu, I. Kolmanovsky, S. Cho and H. N. McClamroch, Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system,, In Proc. of the 36th IEEE Conference on Control and Decision, 4 (1997), 3998. doi: 10.1109/CDC.1997.652490. [47] A. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls,, Systems Control Lett., 18 (1992), 165. doi: 10.1016/0167-6911(92)90001-9. [48] A. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation,, IEEE Trans. Autom. Control, 41 (1996), 1256. doi: 10.1109/9.536496. [49] W. Tian, C. Qian and H. Du, A generalised homogeneous solution for global stabilisation of a class of non-smooth upper-triangular systems,, Internat. J. Control, 87 (2014), 951. doi: 10.1080/00207179.2013.862347. [50] W. Tian, C. Zhang, C. Qian and S. Li, Global stabilization of inherently non-linear systems using continuously differentiable controllers,, Nonlinear Dynamics, 77 (2014), 739. doi: 10.1007/s11071-014-1336-y. [51] J. Tsinias, A theorem on global stabilization of nonlinear systems by linear feedback,, Systems Control Lett., 17 (1991), 357. doi: 10.1016/0167-6911(91)90074-O. [52] J. Tsinias and M. P. Tzamtzi, An explicit formula of bounded feedback stabilizers for feedforward systems,, Systems Control Lett., 43 (2001), 247. doi: 10.1016/S0167-6911(01)00107-4. [53] B. Yang and W. Lin, Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization,, IEEE Trans. Autom. Control, 50 (2005), 619. doi: 10.1109/TAC.2005.847084. [54] V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems,, Groningen: Noordhoff, (1964).
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