Alexander Weinmann

STABILITY MARGIN AND SPHERICAL UNCERTAINTY

The article addresses the stability margin of multivariable dynamic systems as a comprehensive study of the influence of the uncertainties on the stability margin. Spherical uncertainties of a given norm are considered, minimal in Frobenius norm sense, producing a maximum influence on the stability margin of the system. The resulting procedure nonconservatively calculates the stability margin change incorporating given spherically bounded uncertainties. The extremal uncertainties are derived in their structural details. Both large-scale and small uncertainties are taken into consideration. The calculations also determine how much the uncertainties may be increased without violating a lower limit of the stability margin or before instability would occur. Entirely and partially perturbed coefficient matrices of the dynamic system representation are looked into.

A DIALOG-ORIENTED AND GRADIENT-BASED STABILITY MARGIN IN UNCERTAIN SYSTEMS

ABSTRACT Gradients and matricial gradients for optimally increasing the stability margin and the admissible uncertainty of a dynamic system are the targets of this presented article. To design a dynamic system, the gradients are used in a dialog between a system scientist and gradient-based computer support. The stability margin is derived for output state controllers, including regular state controllers. The resonant frequency and the damping factors are investigated as a direct function of the maximum admissible uncertainty. The resulting gradients are extended to observer-assisted controllers, to minimum-order observers, and to dynamic-output state controllers.

ROBUSTNESS GRADIENTS FOR DISCRETE-TIME CONTROL SYSTEMS

ABSTRACT This article addresses gradients for discrete-time control system design. First, the stability margin is investigated based on the minimum distance of the Mikhailov hodograph to the origin or on the minimum distance of the closed-loop poles to the unit circle. The gradients with respect to the output state controller matrix are derived and applied for stepwise, improving the dynamics of the closed-loop system. Second, the maximum admissible uncertainty of the plant is considered based on the unstructured type of uncertainty. For robustification, gradients to attain maximum admissible uncertainty are introduced.

CONTROL SYSTEMS DESIGN DRIVEN BY WORST-INITIAL-ENERGY SENSITIVITY

For automatic control purposes, various time-weighted indices of performance are considered. The sensitivity with respect to the controller matrix is derived and included for stepwise system design. The dependence on initial conditions is excluded by using a selected worst-case per-unit condition. A phobia effect is observed and used for preserving stability during the system design process. The method is advisable for improving linear system behavior based on eigenvalue assignment with imperfect eigenvalue assumptions. The sensitivity expressions are also utilized for adaptation in order to smooth chaotic motion in nonlinear control.

Application oriented control system design based on the fuzziness of parameter uncertainties

The parameter uncertainties of the plant are included into the design process of closed-loop control. For this purpose the membership functions of the uncertain parameters are used. The design methods are based on individual closed-loop parameters, on differential equations, eigenvalue assignment and Riccati controllers. Weighting the closed-loop parameter membership function can help to include particular sensitivity considerations concerning stability. There is a remarkable difference in performance between results of control systems designed with respect to nominal or center-of-gravity data.

Matrix assignment and restriction of the actuator peaks by modified Widrow-Hoff learning

Generalizing the pole assignement method leads to matrix assignment. An additional constraint is included concerning the actuating signal magnitude in time domain. Widrow-Hoff search is applied to the associated controller norm in order to match the magnitude of the actuator signals.ZusammenfassungDie Polvorgabe von Regelkreisen, auf Matrizenvorgabe erweitert, wird mit der Einschränkung versehen, dass die Maximalamplitude der Stellsignale im Zeitbereich begrenzt wird. Die Begrenzung bedient sich der Zustandsreglermatrix-Norm als Hilfsgröße. Bewerkstelligt wird die Begrenzung mit Widrow-Hoff-Lernalgorithmen. Für jeden Vektor der Zustandsreglermatrix wird ein Neuron verwendet.

Eigenvalue cycles for robust control systems

ZusammenfassungIn der vorliegenden Arbeit wird ein Zyklus von Unsicherheiten um die nominellen Parameter des linearen oder linearisierten Regelungssystems angesetzt. Vorzugsweise werden zwei unsichere Parameter verwendet. Die Abbildung des Unsicherheitszyklus der Regelstrecke auf die Eigenwerte des geschlossenen Regelkreises liefert den Eigenwertzyklus. Operatorgestützte oder automatische Abänderung des Unsicherheitszyklus ist gut geeignet, die zulässigen Unsicherheiten und die Reglerparameter gegeneinander abzuwägen; so sind die Grenzen einer vorgegebenen Stabilitätsgüte auszuschöpfen und weitere konstruktive Ingenieurleistung einzubringen möglich. Bevorzugt werden die Unsicherheiten in einer unstrukturierten Version, begrenzt durch eine vorgewählte spezielle Norm. Die Methode vermeidet die Einbeziehung jeglicher überzähliger Annahmen. Unter ausreichender Computerunterstützung ist die Methode für Systeme von beliebiger Komplexheit anwendbar.SummaryIn this article, a cycle of uncertainties around the nominal parameters of the linear or linearized control system is established. Preferably two uncertain parameters are used. Mapping the uncertainty cycle of the control plant on the eigenvalues of the closed-loop control system yields eigenvalue cycles. Operator-guided or automatic modifying the uncertainty cycle is well-suited to balance the admissible uncertainty and the controller parameters; in order to exhaust the limits of a given stability degree and to perform detailled engineering. Using the uncertainties in an unstructured version limited by a preselected specific norm is preferred. The method is nonconservative. Under sufficient computer assistance it is applicable for systems of arbitrary complexity.

Robustification of control systems under uncertainty proportions

ZusammenfassungBetrachtet werden lineare zeitkontinuierliche Systeme mit unstrukturierten Unsicherheiten innerhalb einer speziellen direkten Matrixnorm. Die höchstzulässigen Unsicherheiten werden auf analytischem Weg hergeleitet und an gegebene Entwurfskriterien angeglichen. Keine überflüssigen Reserveannahmen werden benötigt. Ein Gradientenalgorithmus für den Regler wird vorgeschlagen, um die zulässigen Unsicherheiten in Bezug auf ihre Norm erhöhen zu können und so die gesamte Regelung zu robustifizieren.SummaryThe uncertainty of a linear continuous-time system is chosen unstructured by means of a specific direct matrix norm with given proportion interrelation. The maximum admissible uncertainties are derived analytically and the control system is tuned for a given design performance. No conservatism takes place. A gradient algorithm for the controller is suggested in order to increase the admissible uncertainty in norm sense and to robustify the control system.

THE ADMISSIBLE SPHERICAL UNCERTAINTY FOR DYNAMIC SYSTEMS IN STATE SPACE

Among the family of additive-type uncertainties, the uncertainty with the exactly largest admissible Frobenius norm is derived which does not affect stability of a continuous-time linear dynamic system. Special emphasis is put on second-order systems. Selecting the monotonic and oscillatoric stability border, the admissible uncertainty is derived in explicit detail. Closed derivation for the uncertainties is feasible for linear and bilinear ones. The matrix-based results are also given for arbitrary order of the dynamic system. Minimum degree of stability and nice stability are also addressed.

Adhesion control in railroad traction

This article presents a new approach to adhesion control for railroad traction. Based on a combined torque and speed observation the controller detects the actual adhesion utilization at the wheel and applies an adequate torque limitation. Hence the traction unit optimally takes use of the available adhesion. No measurement of the absolute train velocity is needed. The effectiveness of the proposed method has been investigated by railroad tests with a subway electric traction unit. The results and their good conformity with computer simulation are presented in detail.ZusammenfassungEin neuer Ansatz zur Adhäsionsregelung für elektrische Triebfahrzeuge wird vorgestellt: Basierend auf der Beobachtung des übertragbaren Drehmoments und der Fahrzeuggeschwindigkeit ermittelt der Regler die aktuelle Kraftschlussausnutzung am Rad und bewirkt eine geeignete Begrenzung des Antriebsmoments. Damit wird eine optimale Ausnutzung des verfügbaren Kraftschlusses erzielt, ohne auf eine Messung der absoluten Fahrzeuggeschwindigkeit angewiesen zu sein. Die Wirksamkeit des vorgestellten Verfahrens wurde durch Messungen an einem U-Bahn-Triebfahrzeug geprüft. Die Messergebnisse und ihre gute Übereinstimmung mit Simulationsergebnissen werden im Einzelnen dargelegt.