Numerical method for the solution of the regulator equation with application to nonlinear tracking

A numerical method to solve the so-called regulator equation is presented here. This equation consists of partial differential equations combined with algebraic ones and arises when solving the output-regulation problem. Solving the regulator equation is becoming difficult especially for the nonminimum phase systems where reducing variables against algebraic part leads to a potentially unsolvable differential part. The proposed numerical method is based on the successive approximation of the differential part of the regulator equation by the finite-element method while trying to minimize a functional expressing the error of its algebraical part. The method is analyzed to obtain theoretical estimates of its convergence and it is tested on an example of the ''two-carts with an inverted pendulum'' system. Simulations are included to illustrate the suggested approach.

[1]  W. Wonham,et al.  Error feedback and internal models on differentiable manifolds , 1982, 1982 21st IEEE Conference on Decision and Control.

[2]  Dan Wang,et al.  A neural network-based approximation method for discrete-time nonlinear servomechanism problem , 2001, IEEE Trans. Neural Networks.

[3]  Jie Huang,et al.  Output regulation of nonlinear systems with nonhyperbolic zero dynamics , 1995, IEEE Trans. Autom. Control..

[4]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[5]  B. Rehak,et al.  Real-Time Error-Feedback Output Regulation of Nonhyperbolically Nonminimum Phase System , 2007, 2007 American Control Conference.

[6]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[7]  B. Paden,et al.  Nonlinear inversion-based output tracking , 1996, IEEE Trans. Autom. Control..

[8]  Jie Huang,et al.  On the solvability of the regulator equations for a class of nonlinear systems , 2003, IEEE Trans. Autom. Control..

[10]  A. Isidori.  Nonlinear Control Systems , 1985 .

[11]  Christopher I. Byrnes,et al.  Output regulation for nonlinear systems: an overview , 2000 .

[12]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[13]  Jie Huang,et al.  Neural network enhanced output regulation in nonlinear systems , 2001, Autom..

[14]  Jie Huang,et al.  On a nonlinear multivariable servomechanism problem , 1990, Autom..

[15]  S. Čelikovský,et al.  Universal algorithm for the solution of the regulator equation , 2004 .

[16]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[17]  Jie Huang,et al.  Remarks on the robust output regulation problem for nonlinear systems , 2001, IEEE Trans. Autom. Control..

[18]  Jie Huang.  Asymptotic tracking of a nonminimum phase nonlinear system with nonhyperbolic zero dynamics , 2000, IEEE Trans. Autom. Control..

[19]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[20]  A. Isidori,et al.  Output regulation of nonlinear systems , 1990 .

[21]  Jie Huang,et al.  Nonlinear Output Regulation: Theory and Applications , 2004 .