738 research outputs found
Output-input stability and minimum-phase nonlinear systems
This paper introduces and studies the notion of output-input stability, which
represents a variant of the minimum-phase property for general smooth nonlinear
control systems. The definition of output-input stability does not rely on a
particular choice of coordinates in which the system takes a normal form or on
the computation of zero dynamics. In the spirit of the ``input-to-state
stability'' philosophy, it requires the state and the input of the system to be
bounded by a suitable function of the output and derivatives of the output,
modulo a decaying term depending on initial conditions. The class of
output-input stable systems thus defined includes all affine systems in global
normal form whose internal dynamics are input-to-state stable and also all
left-invertible linear systems whose transmission zeros have negative real
parts. As an application, we explain how the new concept enables one to develop
a natural extension to nonlinear systems of a basic result from linear adaptive
control.Comment: Revised version, to appear in IEEE Transactions on Automatic Control.
See related work in http://www.math.rutgers.edu/~sontag and
http://black.csl.uiuc.edu/~liberzo
Passivity based stabilization of non-minimum phase nonlinear systems
summary:A cascade scheme for passivity-based stabilization of a wide class of nonlinear systems is proposed in this paper. Starting from the definitions and basic concepts of passivity-based stabilization via feedback (which are applicable to minimum phase nonlinear systems expressed in their normal forms) a cascade stabilization scheme is proposed for minimum and non-minimum phase nonlinear systems where the constraint of stable zero dynamics imposed by previous stabilization approaches is abandoned. Simulation results of the proposed algorithm are presented to demonstrate its performance
Output feedback control and robustness in the gap metric
Zusammenfassung
Mueller, Markus:
Output feedback control and robustness in the gap metric
Ilmenau : Univ.-Verl. Ilmenau, 2009. - 254 S.
ISBN 978-3-939473-60-2
Die vorgelegte Arbeit behandelt den Entwurf und die Robustheit von drei
verschiedenen Regelstrategien für lineare Differentialgleichungssysteme mit
mehrdimensionalen Ein- und Ausgangssignalen (MIMO): Stabilisierung durch
Ausgangs-Ableitungs-Rückführung, Lambda-tracking und Funnel-Regelung.
Damit diese Regler bei der Anwendung auf ein lineares System die gewünschten
Stabilisierung/Regelung erbringen, ist eine explizite Kenntnis der
Systemmatrizen nicht notwendig. Es müssen nur strukturelle Eigenschaften des
Systems bekannt sein: der Relativgrad, dass das System minimalphasig ist, und
dass die sogenannte "high-frequency gain" Matrix positiv definit ist. Diese
stukturellen Eigenschaften werden für MIMO-Systeme in den ersten Kapiteln der
Arbeit ausführlich behandelt. Für MIMO-Systeme mit nicht striktem Relativgrad
wird eine Normalform hergeleitet, die die gleichen Eigenschaften wie die
bekannte Normalform für SISO-Systeme oder MIMO-Systeme mit striktem Relativgrad
aufweist.
Die Normalform sowie Minimalphasigkeit und Positivität der "high-frequency
gain" Matrix bilden die Grundlage dafür, dass die oben genannten
Regelstrategien Systeme mit diesen Eigenschaften im jeweiligen Sinn
stabilisieren.
Robustheit bzw. robuste Stabilisierung beschreibt folgendes Prinzip: falls
ein geschlossener Kreis aus einem linearen System und einem Regler in gewissem
Sinne stabil ist und die Gap-Metrik (der Abstand) zwischen dem im geschlossenen
Kreis betrachteten System und einem anderen "neuen" System hinreichend klein
ist, so ist der geschlossene Kreis aus dem "neuen" System und dem gleichen
Regler wieder stabil. Die gleiche Aussage stimmt auch für den Fall, dass man
den Regler und nicht das System austauscht.
Für Ausgangs-Ableitungs-Rückführung wird gezeigt, dass, falls diese ein System
stabilisiert, die auftretenden Ableitungen des Ausgangs durch
Euler-Approximationen der Ableitungen ersetzt werden können, falls diese
hinreichend genau sind.
Für Lambda-tracking und Funnel-Regelung wird gezeigt, dass beide Regler auch
für die Stabilisierung linearer Systeme verwendet werden können, die einen
geringen Abstand zu einem System haben, dass die o.g. Voraussetzungen erfüllt,
selbst diese Voraussetzungen aber nicht erfüllen.Abstract:
This dissertation considers the design and robustness analysis of three different control strategies for linear systems of differential equations with multidimensional input and output signals (MIMO): high-gain output derivative feedback control, lambda-tracking and funnel control. To apply these control strategies to linear systems and achieve the desired control objectives (stabilization or tracking), the explicit system's data needs not to be known, but certain structural properties of the systems are required. The system's relative degree must be known, the system must be minimum phase and the so-called "high-frequency gain" matrix must be positive definite.
These properties are considered in detail for linear MIMO-systems with non-strict relative degree. A normal form is developed which has the same properties as the well-known normal form for SISO-systems or MIMO-systems with strict relative degree.
Normal form, minimum phase property and positivity of the high-frequency gain matrix are the crucial assumptions for the application of the control strategies mentioned above. It is shown that each controller achieves certain control objectives when applied to any system which satisfies these assumptions.
The result on robustness and robust stability are as follows: if a closed-loop system represented by the application of a controller to a linear plant is stable (in some sense), and the gap metric (i.e. the distance) between the stabilised system and a different "new" system is sufficiently small, then the closed-loop system represented by the application of the controller to the "new" system is again stable. This conclusion holds also true when changing the roles of system and controller.
For high-gain output derivative feedback control it is shown that the controller still stabilizes a system when the derivatives of the output are replaced by Euler approximations of the derivatives, provided the approximation is sufficiently precise.
For lambda-tracking and funnel control it is shown that both controllers may be applied to systems which are "close" (in terms of a small gap) to any system from the class of minimum phase systems, with relative degree one and positive definite high-frequency gain matrix, but not necessarily satisfy any of these assumptions
H∞ control of nonlinear systems: a convex characterization
The nonlinear H∞-control problem is considered with an emphasis on developing machinery with promising computational properties. The solutions to H∞-control problems for a class of nonlinear systems are characterized in terms of nonlinear matrix inequalities which result in convex problems. The computational implications for the characterization are discussed
- …