# American Institute of Mathematical Sciences

April  1995, 1(2): 223-236. doi: 10.3934/dcds.1995.1.223

## Suboptimal feedback control for a class of nonlinear systems using spline interpolation

 1 Department of Mathematics, University of Western Australia, Nedlands, WA, 6009, Australia, Australia, Australia

Received  September 1994 Published  February 1995

We consider a class of nonlinear quadratic regulator problems where the system dynamics are affine in the control. It has been shown recently that an optimal feedback control law for this class of problems can be given in terms of the solution of a state dependent algebraic Ricatti equation (ARE) at each instance of time. However, in most practical problems it is not possible to find an analytic solution to the ARE and hence numerical schemes to calculate suboptimal controls are required. In this paper, we consider one such scheme based on cubic basis spline interpolation. It is shown that if the chosen partitioning of the state space is sufficiently small, the resulting suboptimal controller leads to a stable closed loop system.
Citation: V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223
 [1] Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355 [2] Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117 [3] Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019008 [4] Anne-Sophie de Suzzoni. Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space. Communications on Pure & Applied Analysis, 2014, 13 (3) : 991-1015. doi: 10.3934/cpaa.2014.13.991 [5] Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115 [6] Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions. Conference Publications, 2011, 2011 (Special) : 941-952. doi: 10.3934/proc.2011.2011.941 [7] Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931 [8] Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151 [9] Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267 [10] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 [11] Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101 [12] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89 [13] Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147 [14] Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058 [15] Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455 [16] Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037 [17] Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039 [18] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [19] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [20] Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307

2018 Impact Factor: 1.143