Determination of Non-Minimum Phase Systems by the Structural Approach
Svaricek, Ferdinand, Prof. Dr.-Ing.
Svaricek, Ferdinand, Prof. Dr.-Ing.; Röbenack, Klaus, Prof. Dr.-Ing. habil.
Date oral examination:
Dynamisches System ; Regelkreis ; Strukturanalyse ; Nichtlineares System ; Strukturelle Stabilität ; Hochschulschrift
control engineering, dynamical systems, non-minumum phase, structural stability, structural propertiesy,
Many dynamic processes and systems that we encounter regularly are characterized by the system-theoretical property “non-minimum phase”. A system is called non-minimum phase if it has unstable zero dynamics or, in the linear case, zeros with non-negative real parts. Non-minimum phase systems are generally challenging to control. The closed control loop tends to become unstable, which results in the fact that only small control gains can be chosen. Furthermore, non-minimum phase systems do not have a stable inverse, so that in particular many nonlinear methods of controller design are not applicable. Both properties lead to a poor trajectory tracking behavior. For this reason, it is necessary to check a priori if a system is non-minimum phase in order to select a suitable control method. There are usually two approaches to model a plant. A dynamic model can be derived from the physical relations in the system or measurements of the inputs and outputs can be used for system identification methods. In the first case, a system with parameters, which have to be determined, is obtained and, in the second case, a set of numerical systems is generated. For technical systems, the obtained models are usually very complex. Nevertheless, many parameters are known only imprecisely. This leads to the fact that the application of common numerical methods for the analysis of system-theoretical properties is difficult. In these cases, the structural analysis may be helpful. The structural approach only considers the dependencies in the system, which means, whether certain system states, inputs or outputs depend on one another. In this thesis, it is investigated whether it is possible to determine only by the structure of a system whether it is non-minimum phase. Therefore, the following approach is taken. The term (non-)minimum phase was used in 1940 by Bode for stable linear systems with at least one zero with positive real part and in the eighties by Isidori for systems with unstable zero dynamics. It is examined how these two concepts can be unified. The result is that essentially the definition of non-minimum phase systems according to Isidori applies also to systems that are non-minimum phase according to Bode. Further, the representation of dynamical systems as a graph is used to determine the zeros by the graph-theoretic approach. For this purpose, known methods are extended to the general case of non-square and degenerated MIMO systems. These extended methods are then applied to structural systems. Structural properties apply to almost all systems of the same structure in the numerical sense. By the non-existence of certain subgraphs - the feedback cycle families - in the graph of a system, a sufficient criterion for the non-minimum phase property of a numerical system can be given. This means that almost all numerical realizations of a structurally non-minimum phase system are also non-minimum phase in the numerical sense. This also raises the question whether there exists a structural non-minimum phase property that is valid numerically for all realizations. This leads to the well-known concept of strong-structural properties that hold numerically for all realizations of a structural system. In this context, strong-structurally non-minimum phase systems are defined. Finally, three extensions are presented. First, the developed methods are applied to analyze the stability of structural systems. This leads to strong-structurally not asymptotically stable systems, whose numerical realizations cannot be asymptotically stable under any choice of numerical non-zero parameters. Often, in addition to the existence of a dependency in the system, the direction, i. e. the sign, of the dependency is also known. It is shown that for systems, for which no non-minimum phase property can be determined structurally, a non-minimum phase property can be found by considering the signs of the dependencies in the system. Finally, the application of the methods to nonlinear systems is discussed. It turns out that in general the structurally non-minimum phase property does not exist for nonlinear systems. However, a structural criterion for not asymptotically stable nonlinear systems can be found. «
Many dynamic processes and systems that we encounter regularly are characterized by the system-theoretical property “non-minimum phase”. A system is called non-minimum phase if it has unstable zero dynamics or, in the linear case, zeros with non-negative real parts. Non-minimum phase systems are generally challenging to control. The closed control loop tends to become unstable, which results in the fact that only small control gains can be chosen. Furthermore, non-minimum phase systems do not ha... »