Michael Zeitz

Canonical form observer design for non-linear time-variable systems

An observer of canonical (phase-variable) form for non-linear time-variable systems is introduced. The development of this non-linear time-variable form requires regularity of the non-linear time-variable- observability matrix of the system. From the relationships derived during the development, it follows that a non-linear time-variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non-linearities about the reconstructed state trajectories is permissible. This is an assumption similar to that of the extended Kalman filter based on a linearization about the current estimate.

The extended Luenberger observer for nonlinear systems

Abstract For nonlinear single-input single-output systems dot x = f(x, u), y = h(x, u) , the relationships for a state transformation into the nonlinear observer canonical form are developed. It is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation, if the transformed nonlinearities are linearized about the reconstructed state. With reference to the extended Kalman filter algorithm, this nonlinear observer design is called the extended Luenberger observer.

Flatness based control of a nonlinear chemical reactor model

The nonlinear model of a continuous stirred tank reactor is shown to be flat. The flatness permits the design of suitable trajectories on the basis of the explicit stationary solution and the tracking of these trajectories asymptotically using quasi-static state feedback linearization. A nonlinear local observer with a time-varying gain is designed allowing the realisation of the state feedback in the case of partial measurement of the state. Simulation results illustrate the tracking behavior of the closed loop with the observer.

A new approach to inversion-based feedforward control design for nonlinear systems

The finite-time transition between stationary setpoints of nonlinear SISO systems is considered as a scenario for the presentation of a new design approach for inversion-based feedforward control. Design techniques which are based on a stable system inversion result in input trajectories with pre- and/or post-actuation intervals. The presented approach treats the considered transition task as a two-point boundary value problem (BVP) and yields causal feedforward trajectories, which are constant outside the transition interval. The main idea of this approach is to provide free parameters in the desired output trajectory to solve the BVP of the internal dynamics. Thereby, a standard MATLAB function can be used for the numerical solution of the BVP. Feedforward control design techniques are illustrated by simulation results for a simple example.

A block triangular nonlinear observer normal form

Abstract The nonlinear observer normal form is generalized to a block triangular observer normal form, which can be interpreted as a series connection of subsystems in observer form. Based on this new form, exponential observers can be designed using methods from the linear case. Necessary and sufficient conditions for the existence of state transformations into this normal form are given.

Brief paper: Swing-up of the double pendulum on a cart by feedforward and feedback control with experimental validation

The swing-up maneuver of the double pendulum on a cart serves to demonstrate a new approach of inversion-based feedforward control design introduced recently. The concept treats the transition task as a nonlinear two-point boundary value problem of the internal dynamics by providing free parameters in the desired output trajectory for the cart position. A feedback control is designed with linear methods to stabilize the swing-up maneuver. The emphasis of the paper is on the experimental realization of the double pendulum swing-up, which reveals the accuracy of the feedforward/feedback control scheme.

Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme

Das 1992 von Fliess, Levine, Martin und Rouchon eingeführte Konzept der Flachheit eröffnet einen neuen Zugang zur Analyse und zum Entwurf nichtlinearer Systeme. Flache nichtlineare Systeme sind eine Verallgemeinerung der linearen steuerbaren Systeme und ermöglichen einen systematischen Entwurf von Steuerungen und Regelungen zur Trajektorienfolge. Zur Realisierung der flachheitsbasierten Folgeregelung können nichtlineare Beobachter mit zeitvarianter Verstärkung verwendet werden. Die Flachheitsanalyse und der flachheitsbasierte Entwurf einer Steuerung, einer asymptotischen Folgeregelung und eines nichtlinearen Folgebeobachters werden erläutert. Dazu wird das Beispiel eines kinematischen Fahrzeugmodells betrachtet.

Observability canonical (phase-variable) form for non-linear time-variable systems

Abstract An observability canonical form for non-linear time-variable systems, [xdot]=f(x,u,t), y=h(x,u,t), is introduced by analogy with the corresponding linear phase-variable forms. The transformation into observability canonical form follows from the nonlinear observability map, whose jacobian must be assumed to be a regular matrix in the considered domains of state x, input u and time t. If this observability matrix can be inverted analytically or numerically, the transformation into the observability canonical coordinates can be achieved directly. As opposed to linear systems, the non-linear observability canonical form with input depends, additionally, on the time derivatives of the input. This restricts a practical implementation.

Feedforward Control Design for Finite-Time Transition Problems of Nonlinear Systems With Input and Output Constraints

The article extends a recently presented approach to feedforward control design for nonlinear systems to additionally account for input and output constraints. The inversion-based design treats a finite-time transition problem as a two-point boundary value problem (BVP) in the coordinates of the input-output normal form. To account for constraints on the output and its time derivatives, the input-output dynamics is replaced by a new system, which is systematically constructed by means of saturation functions. The solvability of the BVP requires a sufficient number of free parameters in an ansatz function. The resulting BVP with free parameters can be solved in a straightforward manner (e.g., with the Matlab function bvp4c). Input constraints can additionally be considered as constraints on the highest output derivative. The approach is applicable to nonlinear and nonminimum-phase systems, which is illustrated for the side-stepping of an inverted pendulum on a cart.

PROMOT: A Modeling Tool for Chemical Processes

The novel process modeling tool PROMOT supports the object-oriented modeling of chemical processes for the simulation environment DIVA. In PROMOT, differential-algebraic process models can be built by aggregating structural and behavioral modeling entities that represent the topological structure or the dynamic and steady-state behavior, respectively, of the investigated chemical processes. Process models and their modeling entities may be defined either in an object-oriented modeling language or with a graphical user interface. This paper discusses the modeling concept, the modeling language, the knowledge representation aspects, and the implementation of PROMOT.