Successive Galerkin approximation algorithms for nonlinear optimal and robust control

Nonlinear optimal control and nonlinear H infinity control are two of the most significant paradigms in nonlinear systems theory. Unfortunately, these problems require the solution of Hamilton-Jacobi equations, which are extremely difficult to solve in practice. To make matters worse, approximation techniques for these equations are inherently prone to the so-called 'curse of dimensionality'. While there have been many attempts to approximate these equations, solutions resulting in closed-loop control with well-defined stability and robustness have remained elusive. This paper describes a recent breakthrough in approximating the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. Successive approximation and Galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing, closed-loop control laws with well-defined stability regions. In addition, we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po...

[1]  Z. Rekasius,et al.  Suboptimal design of intentionally nonlinear controllers , 1964 .

[2]  Ruey-Wen Liu,et al.  Construction of Suboptimal Control Sequences , 1967 .

[3]  Donald E. Kirk,et al.  Optimal Control Theory , 1970 .

[4]  K. Park.  An approximate method for the synthesis of optimal control of distributed systems , 1972, CDC 1972.

[5]  William L. Garrard,et al.  Design of nonlinear automatic flight control systems , 1977, Autom..

[6]  G. Gopalakrishnan Nair,et al.  Suboptimal control of nonlinear systems , 1978, Autom..

[7]  G. Siouris,et al.  Optimum Systems Control , 1979, IEEE Transactions on Systems, Man and Cybernetics.

[8]  George N. Saridis,et al.  An Approximation Theory of Optimal Control for Trainable Manipulators , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[9]  S. Glad.  On the gain margin of nonlinear and optimal regulators , 1984, 1982 21st IEEE Conference on Decision and Control.

[10]  J. Tsitsiklis,et al.  Guaranteed robustness properties of multivariable, nonlinear, stochastic optimal regulators , 1983, The 22nd IEEE Conference on Decision and Control.

[11]  C. Fletcher.  Computational Galerkin Methods , 1983 .

[12]  E. Ryan.  Optimal feedback control of bilinear systems , 1984 .

[13]  S. Tzafestas,et al.  Stabilizing optimal control of bilinear systems with a generalized cost , 1984 .

[14]  T. Glad.  Robust Nonlinear Regulators based on Hamilton-Jacobi Theory and Lyapunov Functions , 1985 .

[15]  S. Torkel Glad.  Robustness of nonlinear state feedback - A survey , 1987, Autom..

[16]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[17]  R.T. Reichert,et al.  Robust autopilot design using μ-synthesis , 1990, 1990 American Control Conference.

[18]  S. Bittanti,et al.  The Riccati equation , 1991 .

[19]  Thor I. Fossen,et al.  Adaptive control of nonlinear systems: A case study of underwater robotic systems , 1991, J. Field Robotics.

[20]  A. Schaft.  L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control , 1992 .

[21]  G. Saridis,et al.  Suboptimal control for nonlinear stochastic systems , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[22]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[23]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[24]  C. J. Goh,et al.  On the nonlinear optimal regulator problem , 1993, Autom..

[25]  J. L. Sedwick,et al.  Successive approximation solution of the HJI equation , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[26]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[27]  Jie Huang,et al.  Numerical approach to computing nonlinear H-infinity control laws , 1995 .

[28]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[29]  R.A. Freeman,et al.  Optimal nonlinear controllers for feedback linearizable systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[30]  John T. Wen,et al.  Improving the performance of stabilizing controls for nonlinear systems , 1996 .

[31]  Randal W. Beard,et al.  Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation , 1997, Autom..

[32]  Timothy W. McLain,et al.  Nonlinear Optimal Control of a Hydraulically Actuated Positioning System , 1997 .

[33]  G. Saridis,et al.  Approximate Solutions to the Time-Invariant Hamilton–Jacobi–Bellman Equation , 1998 .

[34]  Timothy W. McLain,et al.  Successive Galerkin approximations to the nonlinear optimal control of an underwater robotic vehicle , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[35]  R. Beard,et al.  Numerically efficient approximations to the Hamilton-Jacobi-Bellman equation , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[36]  Timothy W. McLain,et al.  NONLINEAR OPTIMAL CONTROL DESIGN OF A MISSILE AUTOPILOT , 1998 .