Johannes Gerhard

Normal Vectors on Critical Manifolds for Robust Design of Transient Processes in the Presence of Fast Disturbances

Information on steady-state bifurcations, most notably stability boundaries, is frequently used for the analysis and design of nonlinear systems. The bifurcation points separate regions with different dynamic behavior and thus give valuable information about nonlinear systems. They cannot, however, reflect the impact of fast disturbances on the transient behavior of nonlinear systems. The influence of fast disturbances can be addressed by bifurcation points that are defined as critical points during the transient behavior of a dynamic system in the presence of fast disturbances. Specifically, we consider two types of points—grazing points and end-points. At a grazing point the trajectory of a nonlinear system tangentially touches a hypersurface spanned by a state or output constraint. At an end-point the trajectory crosses the hypersurface at a specified final time. These critical points unfold to manifolds in the parameter space of the nonlinear system separating parts of the parameter space that admit t...

Robust Stable Nonlinear Control and Design of a CSTR in a Large Operating Range

Abstract In this work an approach based on nonlinear dynamics is used for the integrated design and controller tuning of a CSTR. The approach enables integrated design optimization and robust tuning of a linearizing feedback controller. The controller setting found by the approach guarantees robust stability of the process over a large range of set point variations even in the presence of parameter uncertainty. The method enforces robust stability by introducing lower bounds on the parametric distance of the operating point to critical boundaries in the space of process and controller parameters. For stabilization of a large range of operations, a lower bound on the distance to a nontransversal Hopf bifurcation has to be considered in this particular case.

ROBUST YAW CONTROL DESIGN WITH ACTIVE DIFFERENTIAL AND ACTIVE ROLL CONTROL SYSTEMS

Abstract A simple robust yaw controller for the nonlinear single-track model is designed, making use of active differential and active roll control systems. Robustness is studied for uncertainties in several model parameters, namely the vehicle longitudinal velocity, the road adherence coefficients and the hand wheel angle. Constructive nonlinear dynamics are employed for the controller design. The controller parameters are selected by solving an optimization problem. Stability of the solution is guaranteed by constraints that ensure a minimal distance between the nominal operating point and a stability boundary in the space of uncertain parameters.

TOWARDS ROBUST DESIGN OF CLOSED-LOOP NONLINEAR SYSTEMS WITH INPUT AND STATE CONSTRAINTS

Abstract In this paper we address the task of finding a robust process and control design for nonlinear systems with uncertainties and disturbances such that bounds on inputs and outputs are not violated. The solution of this task is approached by Constructive Nonlinear Dynamics (CNLD), an optimization based method developed by the authors in recent years. CNLD guarantees robustness by backing off a nominal point of operation from critical manifolds. Critical manifolds are boundaries in the space of system and controller parameters that separate regions with qualitatively different system behavior, such as a region with stable operating points from a region with unstable system behavior. In this work, CNLD is adopted and extended to account for bounds and constraints on trajectories of inputs and states. Critical boundaries in the parameter space are presented that separate a region where all trajectories stay within the bounds from a region where trajectories violate the constraints. Constraints ensuring a minimal back off from this new type of critical manifold are derived. Application to an illustrative case study demonstrates the feasibility of the approach.

Integrated process and control design by the normal vector approach: Application to the Tennessee-Eastman process

Abstract This paper presents a large-scale application of the normal vector approach to demonstrate that the complexity of robust dynamic optimization with application to the integration of process and control design can be treated successfully for complex nonlinear systems. The case study further demonstrates that our approach can deal with a multi-dimensional uncertainty space. The normal vector approach is able to automatically identify the worst-case scenarios and find a solution that is optimal with respect to the cost function and robust with respect to path constraints on inputs and states in the presence of parameterized disturbances. The tedious analysis of a large number of different disturbance realizations is not required.

Stability-Preserving Optimization in the Presence of Fast Disturbances

We present algebraic conditions on the trajectory of a dynamical system to approximately describe a certain type of system robustness. The corresponding equations can be used as constraints in a robust optimization procedure to select a set of optimal design parameters for a dynamical system which is subject to fast disturbances. Robustness is ensured by requiring the disturbance parameters to stay sufficiently far away from critical manifolds in the disturbance parameter space, at which the system would lose stability. The closest distance to the critical manifolds is measured along their normal vectors.