Control of MIMO nonlinear systems: a data-driven model inversion approach. (English) Zbl 1415.93138
Summary: A data-driven control design approach for multiple input multiple output nonlinear systems is presented in this paper. The approach, called Nonlinear Inversion Control (NIC), is based on the identification of a polynomial prediction model of the system to control and the on-line inversion of this model. The main features of the NIC approach can be summarized as follows: it does not require a physical model of the plant to control which, in many real-world situations, may be difficult to derive; it can guarantee a-priori properties such as closed-loop stability and tracking error accuracy; it is general, numerically efficient and relatively simple. Extensive simulations are carried out to test the numerical efficiency of the NIC approach. A simulated example of industrial interest is also presented, concerned with control of a robotic manipulator.
MSC:
| 93C35 | Multivariable systems, multidimensional control systems |
| 93B51 | Design techniques (robust design, computer-aided design, etc.) |
| 93C55 | Discrete-time control/observation systems |
| 93C10 | Nonlinear systems in control theory |
References:
| [1] | Abreu, G. C. M. De, Teixeira, R. L., & Ribeiro, J. F. (2000). A neural network-based direct inverse control for active control of vibrations of mechanical systems. In Proceedings of the sixth Brazilian symposium on neural networks; Abreu, G. C. M. De, Teixeira, R. L., & Ribeiro, J. F. (2000). A neural network-based direct inverse control for active control of vibrations of mechanical systems. In Proceedings of the sixth Brazilian symposium on neural networks |
| [2] | Anuradha, D. B., Prabhaker Reddy, G., & Murthy, J. S. N. (2009). Direct inverse neural network control of a continuous stirred tank reactor, cstr. In Proceedings of the international multiconference of engineers and computer scientists; Anuradha, D. B., Prabhaker Reddy, G., & Murthy, J. S. N. (2009). Direct inverse neural network control of a continuous stirred tank reactor, cstr. In Proceedings of the international multiconference of engineers and computer scientists |
| [3] | Astrom, K. J.; Wittenmark, B., Adaptive control (1995), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0217.57903 |
| [4] | Bader, Michael, How to construct space-filling curves, (Space-filling curves: An introduction with applications in scientific computing (2013), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 15-30 · Zbl 1283.68012 |
| [5] | Battilotti, S., Robust stabilization of nonlinear systems with pointwise norm-bounded uncertainties: a control Lyapunov function approach, IEEE Transactions on Automatic Control, 44, 1, 3-17 (1999) · Zbl 0965.93094 |
| [6] | Bellman, R., Dynamic programming, P (Rand Corporation) (1957), Princeton University Press · Zbl 0077.13605 |
| [7] | Brown, M. D.; Lightbody, G.; Irwin, G. W., Nonlinear internal model control using local model networks, IEE Proceedings - Control Theory and Applications, 144, 6, 505-514 (1997) |
| [8] | Cabrera, J. B.D.; Narendra, K. S., Issues in the application of neural networks for tracking based on inverse control, IEEE Transactions on Automatic Control, 44, 11, 2007-2027 (1999) · Zbl 0955.93022 |
| [9] | Calafiore, G. C.; El Ghaoui, L., Optimization models (2014), Cambridge University Press · Zbl 1342.90001 |
| [10] | Campi, M. C.; Savaresi, S. M., Direct nonlinear control design: The virtual reference feedback tuning (VRFT) approach, IEEE Transactions on Automatic Control, 51, 1, 14-27 (2006) · Zbl 1366.93251 |
| [11] | Chen, J.; Gu, G., Control-oriented system identification: An \(H_\infty\) approach (2000), John Wiley & Sons: John Wiley & Sons New York |
| [12] | Chen, Fu-Chuang; Khalil, H. K., Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE Transactions on Automatic Control, 40, 5, 791-801 (1995) · Zbl 0925.93461 |
| [13] | Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Homogeneous polynomial forms for robustness analysis of uncertain systems (2009), Springer · Zbl 1218.93002 |
| [14] | Dreesen, P., Batselier, K., & De Moor, B. (2012). Back to the roots: Polynomial system solving, linear algebra, systems theory. In 16th IFAC symposium on system identification; Dreesen, P., Batselier, K., & De Moor, B. (2012). Back to the roots: Polynomial system solving, linear algebra, systems theory. In 16th IFAC symposium on system identification · Zbl 1269.65026 |
| [15] | Dreesen, P.; Moor, B. De, Polynomial optimization problems are eigenvalue problems, (Model-Based Control (2009), Springer), 49-68 |
| [16] | Fagiano, L.; Novara, C., Learning a nonlinear controller from data: theory, computation and experimental results, IEEE Transactions on Automatic Control, 61, 7, 1854-1868 (2016) · Zbl 1359.93181 |
| [17] | Findeisen, R.; Allgower, F.; Biegel, L. T., Assessment and future directions of nonlinear model predictive control, (Lecture Notes in Control and Information Sciences (2007), Springer) · Zbl 1116.93009 |
| [18] | Formentin, S.; De Filippi, P.; Corno, M.; Tanelli, M.; Savaresi, S. M., Data-driven design of braking control systems, IEEE Transactions on Control Systems Technology, 21, 1, 186-193 (2013) |
| [19] | Formentin, S.; Novara, C.; Savaresi, S. M.; Milanese, M., Active braking control system design: the D2-IBC approach, IEEE/ASME Transactions on Mechatronics, 20, 4, 1573-1584 (2015) |
| [20] | Franzè, G., A nonlinear sum-of-squares model predictive control approach, IEEE Transactions on Automatic Control, 55, 6, 1466-1471 (2010) · Zbl 1368.93170 |
| [21] | Freeman, A.; Kokotovic, V., Robust nonlinear control design (1996), Birkhäuser: Birkhäuser Boston · Zbl 0857.93001 |
| [22] | Freidovich, L. B.; Khalil, H. K., Performance recovery of feedback-linearization-based designs, IEEE Transactions on Automatic Control, 53, 10, 2324-2334 (2008) · Zbl 1367.93498 |
| [23] | Grune, L.; Pannek, J., Nonlinear model predictive control - theory and algorithms, (Communications and Control Engineering (2011), Springer) · Zbl 1220.93001 |
| [24] | Hou, Z.; Gao, H.; Lewis, F. L., Data-driven control and learning systems, IEEE Transactions on Industrial Electronics, 64, 5, 4070-4075 (2017) |
| [25] | Hou, Z.; Jin, S., Data-driven model-free adaptive control for a class of mimo nonlinear discrete-time systems, IEEE Transactions on Neural Networks, 22, 12, 2173-2188 (2011) |
| [26] | Hou, Z.; Jin, S., A novel data-driven control approach for a class of discrete-time nonlinear systems, IEEE Transactions on Control Systems Technology, 19, 6, 1549-1558 (2011) |
| [27] | Hou, Z.; Wang, Z., From model-based control to data-driven control: Survey, classification and perspective, Information Sciences, 235, 3-35 (2013), Data-based Control, Decision, Scheduling and Fault Diagnostics · Zbl 1284.93010 |
| [28] | Hsu, K.; Novara, C.; Vincent, T.; Milanese, M.; Poolla, K., Parametric and nonparametric curve fitting, Automatica, 42/11, 1869-1873 (2006) · Zbl 1120.65013 |
| [29] | Hunt, K. J.; Sbarbaro, D., Neural networks for nonlinear internal model control, IEE Proceedings D Control Theory and Applications, 138, 5, 431-438 (1991) · Zbl 0755.93034 |
| [30] | Isidori, A., Nonlinear control systems (1995), Springer · Zbl 0569.93034 |
| [31] | Khalil, H. K., Nonlinear systems (1996), Prentice Hall |
| [32] | Kwiatkowski, A., & Werner, H. (2005). LPV control of a 2-DOF robot using parameter reduction. In Proceedings of the IEEE conference on decision and control and european control conference; Kwiatkowski, A., & Werner, H. (2005). LPV control of a 2-DOF robot using parameter reduction. In Proceedings of the IEEE conference on decision and control and european control conference |
| [33] | Lasserre, J. B., Moments, positive polynomials and their applications (2009), Imperial College Press |
| [34] | Levin, A. U.; Narendra, K. S., Control of nonlinear dynamical systems using neural networks. II. Observability, identification, and control, IEEE Transactions on Neural Networks, 7, 1, 30-42 (1996) |
| [35] | Lewis, F. L.; Vrabie, D.; Vamvoudakis, K. G., Reinforcement learning and feedback control: Using natural decision methods to design optimal adaptive controllers, IEEE Control Systems, 32, 6, 76-105 (2012) · Zbl 1395.93584 |
| [36] | Magni, L.; Raimondo, D. M.; Allgower, F., Nonlinear model predictive control - towards new challenging applications, (Lecture notes in control and information sciences (2009), Springer) |
| [37] | Maier, C., Bohm, C., Deroo, F., & Allgower, F. (2010). Predictive control for polynomial systems subject to constraints using sum of squares. In 49th IEEE conference on decision and control.; Maier, C., Bohm, C., Deroo, F., & Allgower, F. (2010). Predictive control for polynomial systems subject to constraints using sum of squares. In 49th IEEE conference on decision and control. |
| [38] | Milanese, M., Gerlero, I., Novara, C., Conte, G., Cisternino, M., & Pedicini, C., et al. (2015). Nonlinear mimo data-driven control design for the air and charging systems of diesel engines. In SAE - ICE2015 - 12th international conference on engines and vehicles; Milanese, M., Gerlero, I., Novara, C., Conte, G., Cisternino, M., & Pedicini, C., et al. (2015). Nonlinear mimo data-driven control design for the air and charging systems of diesel engines. In SAE - ICE2015 - 12th international conference on engines and vehicles |
| [39] | Milanese, M.; Novara, C., Set membership identification of nonlinear systems, Automatica, 40/6, 957-975 (2004) · Zbl 1109.93020 |
| [40] | Milanese, M.; Novara, C., Model quality in identification of nonlinear systems, IEEE Transactions on Automatic Control, 50, 10, 1606-1611 (2005) · Zbl 1365.93105 |
| [41] | Milanese, M.; Novara, C., Computation of local radius of information in SM-IBC identification of nonlinear systems, Journal of Complexity, 23, 937-951 (2007) · Zbl 1128.93014 |
| [42] | Milanese, M.; Novara, C., Unified set membership theory for identification, prediction and filtering of nonlinear systems, Automatica, 47, 10, 2141-2151 (2011) · Zbl 1228.93115 |
| [43] | Narendra, K. S.; Parthasarathy, K., Identification and control of dynamical systems using neural networks, IEEE Transactions on Neural Networks, 1, 1, 4-27 (1990) |
| [44] | Nersesov, S. G.; Haddad, M. M., On the stability and control of nonlinear dynamical systems via vector Lyapunov functions, IEEE Transactions on Automatic Control, 51, 2, 203-215 (2006) · Zbl 1366.93553 |
| [45] | Norgaard, M.; Ravn, O.; Poulsen, N.; Hansen, L. K., Neural networks for modeling and control of dynamic systems (2000), Springer · Zbl 0953.93003 |
| [46] | Novara, C., Set membership identification of state-space LPV systems, (dos Santos, P. Lopes; Perdicoúlis, T. P. Azevedo; Novara, C.; Ramos, J. A.; Rivera, D. E., Linear parameter-varying system identification - new developments and trends, advanced series in electrical and computer engineering (Vol. 14) (2011), World Scientific), 65-93 · Zbl 1297.93074 |
| [47] | Novara, C., Sparse identification of nonlinear functions and parametric set membership optimality analysis, IEEE Transactions on Automatic Control, 57, 12, 3236-3241 (2012) · Zbl 1369.41030 |
| [48] | Novara, C., Polynomial model inversion control: numerical tests and applications (2015) |
| [49] | Novara, C.; Canale, M.; Milanese, M.; Signorile, M. C., Set membership inversion and robust control from data of nonlinear systems, International Journal of Robust and Nonlinear Control, 24, 18, 3170-3195 (2014) · Zbl 1302.93084 |
| [50] | Novara, C.; Fagiano, L.; Milanese, M., Direct feedback control design for nonlinear systems, Automatica, 49, 4, 849-860 (2013) · Zbl 1285.93041 |
| [51] | Novara, C.; Formentin, S., Data-driven inversion-based control of nonlinear systems with guaranteed closed-loop stability, IEEE Transactions on Automatic Control, 63, 4, 1147-1154 (2018) · Zbl 1390.93373 |
| [52] | Novara, C.; Formentin, S.; Savaresi, S. M.; Milanese, M., Data-driven design of two degree-of-freedom nonlinear controllers: the D2-IBC approach, Automatica, 72, 19-27 (2016) · Zbl 1344.93054 |
| [53] | Novara, C., & Karimshoushtari, M. (2016). A data-driven model inversion approach to cancer immunotherapy control. In 55th IEEE conference on decision and control; Novara, C., & Karimshoushtari, M. (2016). A data-driven model inversion approach to cancer immunotherapy control. In 55th IEEE conference on decision and control |
| [54] | Novara, C., Rabbone, I., & Tinti, D. (2018). Data-driven polynomial MPC and application to blood glucose regulation in a diabetic patient. In Proc. of the European control conference; Novara, C., Rabbone, I., & Tinti, D. (2018). Data-driven polynomial MPC and application to blood glucose regulation in a diabetic patient. In Proc. of the European control conference |
| [55] | Novara, C.; Vincent, T.; Hsu, K.; Milanese, M.; Poolla, K., Parametric identification of structured nonlinear systems, Automatica, 47, 4, 711-721 (2011) · Zbl 1215.93145 |
| [56] | Parker, R. S. (2002). Nonlinear model predictive control of a continuous bioreactor using approximate data-driven models. In American control conference; Parker, R. S. (2002). Nonlinear model predictive control of a continuous bioreactor using approximate data-driven models. In American control conference |
| [57] | Parrilo, P. A., Structured semidefinite programs and semi-algebraic geometry methods in robustness and optimization (2000), California Institute of Technology, (Ph.D. thesis) |
| [58] | Polycarpou, M. M., Stable adaptive neural control scheme for nonlinear systems, IEEE Transactions on Automatic Control, 41, 3, 447-451 (1996) · Zbl 0846.93060 |
| [59] | Qu, Z., (Robust control of nonlinear uncertain systems. Robust control of nonlinear uncertain systems, Wiley series in nonlinear science (1998)) · Zbl 0938.93004 |
| [60] | Ravi Sriniwas, G.; Arkun, Y., A global solution to the nonlinear model predictive control algorithms using polynomial ARX models, Computers and Chemical Engineering, 21, 4, 431-439 (1997) |
| [61] | Sjöberg, J.; Zhang, Q.; Ljung, L.; Benveniste, A.; Delyon, B.; Glorennec, P., Nonlinear black-box modeling in system identification: a unified overview, Automatica, 31, 1691-1723 (1995) · Zbl 0846.93018 |
| [62] | Tibshirani, R., Regression shrinkage and selection via the Lasso, Royal Statistical Society. Series B, 58, 1, 267-288 (1996) · Zbl 0850.62538 |
| [63] | Tropp, J. A., Just relax: convex programming methods for identifying sparse signals in noise, IEEE Transactions on Information Theory, 52, 3, 1030-1051 (2006) · Zbl 1288.94025 |
| [64] | Tseng, P., Convergence of a block coordinate descent method for nondifferentiable minimization, Journal of Optimization Theory and Applications, 109, 3, 475-494 (2001) · Zbl 1006.65062 |
| [65] | Yao, B.; Tomizuka, M., Adaptive robust control of MIMO nonlinear systems in semi-strict feedback forms, Automatica, 37 (2001) · Zbl 0983.93035 |
| [66] | Yesildirek, A.; Lewis, L., Feedback linearization using neural networks, Automatica, 31, 11, 1659-1664 (1995) · Zbl 0847.93032 |
| [67] | Yin, Shen; Luo, Hao; Ding, S. X., Real-time implementation of fault-tolerant control systems with performance optimization, IEEE Transactions on Industrial Electronics, 61, 5, 2402-2411 (2014) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
