Input-constrained closed-loop systems with grazing bifurcations in optimal robust design

In this work, the normal vector method for robust design is considered to account for actuator saturation effects when unknown time-varying disturbances are present, and desired dynamic properties have to be guaranteed. The normal vector method ensures that desired dynamic properties hold despite uncertain parameters by maintaining a minimal distance between the operating point and so-called critical manifolds where the process behavior changes qualitatively. In this paper input saturation is considered for the first time in the normal vector framework. In order to solve the resulting optimization problem, first and second order derivatives of the flow of a dynamical system has to be computed efficiently. For this purpose, a new platform for source-level manipulation of mathematical models, currently under development at RWTH Aachen University, is proposed to solve the technical difficulties arising when the event of actuator saturation takes place.

[1]  Wolfgang Marquardt,et al.  Discrete first- and second-order adjoints and automatic differentiation for the sensitivity analysis of dynamic models , 2010, ICCS.

[2]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[3]  Jesus Alvarez,et al.  The Global Stabilization of a Two-Input Three-State Polymerization Reactor with Saturated Feedback , 1993, 1993 American Control Conference.

[4]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[5]  Martin Mönnigmann,et al.  Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems , 2002, J. Nonlinear Sci..

[6]  Christopher L. E. Swartz,et al.  Input saturation effects in optimizing control - inclusion within a simultaneous optimization framework , 2004, Comput. Chem. Eng..

[7]  Doraiswami Ramkrishna,et al.  Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors , 1982 .

[8]  Martin Mönnigmann,et al.  Steady-State Process Optimization with Guaranteed Robust Stability and Feasibility , 2003 .

[9]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[10]  Martin Mönnigmann,et al.  Normal Vectors on Critical Manifolds for Robust Design of Transient Processes in the Presence of Fast Disturbances , 2008, SIAM J. Appl. Dyn. Syst..

[11]  R. Aguilar,et al.  Dynamic behavior of a continuous stirred bioreactor under control input saturation , 1998 .

[12]  Christopher L.E. Swartz,et al.  Rigorous Handling of Input Saturation in the Design of Dynamically Operable Plants , 2004 .

[13]  Wolfgang Marquardt,et al.  Robust Design of Closed-Loop Nonlinear Systems with Input and State Constraints , 2006 .

[14]  Paul I. Barton,et al.  Modeling, simulation, sensitivity analysis, and optimization of hybrid systems , 2002, TOMC.

[15]  P. I. Barton,et al.  Parametric sensitivity functions for hybrid discrete/continuous systems , 1999 .

[16]  Eduardo F. Camacho,et al.  On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach , 2003, ECC.

[17]  Paul I. Barton,et al.  State event location in differential-algebraic models , 1996, TOMC.

[18]  P. Gill,et al.  Fortran package for nonlinear programming. User's Guide for NPSOL (Version 4. 0) , 1986 .