Feedback control : systems with higher unknown relative degree, input constraints and positivity [[Elektronische Ressource]] / Norman Hopfe. Gutachter: Fabian Wirth ; Armin Hoffmann. Betreuer: Achim Ilchmann

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• Publié par: technische_universitat_ilmenau
• Publié le: 01 janvier 2010
• Nombre de lectures: 37
• Poids de l'ouvrage: 5 Mo

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Norman Hopfe

Feedback control

Systems with higher unknown relative degree, input
constraints and positivity




Feedback control

Systems with higher unknown relative degree,
input constraints and positivity



Norman Hopfe




Universitätsverlag Ilmenau
2010 Impressum

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Diese Arbeit hat der Fakultät für Mathematik und Naturwissenschaften der
Technischen Universität Ilmenau als Dissertation vorgelegen.
Tag der Einreichung: 29. Januar 2010
1. Gutachter: Prof. Dr. Achim Ilchmann
(Technische Universität Ilmenau)
2. Gutachter: Prof. Dr. Fabian Wirth
(Universität Würzburg)
3. Gutachter: Prof. Dr. Armin Hoffmann
(Technische Universität Ilmenau)
Tag der Verteidigung: 23. April 2010



Technische Universität Ilmenau/Universitätsbibliothek
Universitätsverlag Ilmenau
Postfach 10 05 65
98684 Ilmenau
www.tu-ilmenau.de/universitaetsverlag


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ISBN 978-3-939473-89-3 (Druckausgabe)
urn:nbn:de:gbv:ilm1-2010000306


Titelfoto: photocase.com | Lily Abstract
Thethesisdealswiththecontroloflinearsingleinput,singleoutputandmultiinput,
multi output systems with unknown but bounded relative degree and linear multi in-
put, multi output Volterra-Stieltjes systems. The following two control strategies are
considered: adaptivehigh-gainoutputderivativefeedbackcontrolandfunnelcontrol.
Eachcontrolstrategyrequiresthestructuralpropertiesofstrictrelativedegree,stable
zero dynamics andpositive high-frequency gain.
For many control applications their parameters are not precisely known. In gen-
eral, it cannot be expected to have complete information about a system, but instead
only structural properties are known. One possible control strategy is an adaptive
controller. Theaim of Chapter 2 is a universal adaptive controller whichlearns from
the behaviour of the system and achieves a prespecified control objective. Possible
objectives arestabilization of the systemand λ-tracking.
For example, for systems with relative degree one the λ-tracking controller means
that the output of the system should stay close to a given reference signal, where a
prespecified small tracking error of sizeλ>0 is tolerated. For systems with higher
relativedegreethe λ-trackingcontrollerusestheoutputanditsderivatives. Thedraw-
backofthederivativescanbesolvedifanobserverisusedwhichestimatestheoutput
of the system and its first derivatives. It has to be noted that this controller stabilize
or track any system if the relative degree is known, provided the system has stable
zero dynamics andthe high-frequency gain matrix is positive definite.
In the thesis the adaptive λ-tracking controller is extended to systems with unknown
relative degree, where an upper bound of the relative degree is known. This is
achieved if ahigh-gain outputderivative feedbackisallowed. It isproven λ-tracking
and stabilization are guaranteed. An advantage of the proposed controller is its rela-
tive simple structure which is helpful for the implementation and the understanding
how the controller works. In this thesis, the adaptive λ-tracking controller is applied
to a serially connected mass-spring damper system with unknown relative degree 1,
2 or 3.
The main drawback is that the gain k(·) increases. In Chapter 3, the well known
concept of funnel control for systems with relative degree one is introduced which
5overcomes this drawback. It is shown that the classical funnel controller applied to
linear multi input, multi output systems achieves in presence of input saturation the
control objectives of funnel control. The presence of explicit input constraints is a
distinguishingfeatureofthisthesis. Afeasibilityrelationshipisderivedunderwhich
theefficiencyoffunnelcontrolinthepresenceofinputsaturationisestablished. The
drawback is that sufficient a priori system information is required in order to check
thefeasibility condition.
Chapter 5 generalizes the classical results of funnel control and the new results of
input constraints to linear multi input, multi output Volterra-Stieltjes systems with
relative degree one.
It has to be noted that the system in Chapter 3 has strict relative degree one which
is important. The aim of Chapter 4 is to generalize the results for funnel control to
linear single input, single output systems with relative degree two. It is known that
the funnel controller can be extended to systems with higher relative degree, where
the controller involves a filter, the feedback strategy dynamic and a backstepping
construction of the feedback strategy. A drawback is that the controller is no longer
simple.
It is shown that the simplicity of the control strategy can be preserved if derivative
feedback is allowed. The thesis designs a simple feedback structure which relies on
two funnels; one for the output and the other one for its derivative. This new funnel
controller is robust for systems of unknown relative degree, i.e. the new funnel con-
troller can be achieved to linear single input, single output systems with unknown
relative degree one or two.
If the system has relative degree one, then the application of the new funnel con-
troller yields a closed-loop system which is an implicit differential equation. An
existence and uniqueness result for a maximal solution of an implicit ordinary dif-
ferential equation is proven. Moreover, the results of Chapter 3 are generalized to
systemswith relative degree two.
Chapter 5 considers time-varying and time-invariant linear multi input, multi output
Volterra-Stieltjes systems with regard to positivity, various stability concepts, zero
dynamicsand funnel control.
Positive systems, i.e., loosely speaking, for any non-negative input and any non-ne-
gativeinitialcondition,thecorrespondingsolutionofthesystemisalsonon-negative,
are of great practical importance which occurs quite often in numerous applications
and in nature. An existence and uniqueness result for Volterra-Stieltjes systems is
proven and in this case the concept of positivity is characterized. Thereafter,various
stability concepts for linear time-invariant systems are generalized to time-invariant
6Volterra-Stieltjessystemsandthedifferencesarepresented. Explicitcriteriaforvari-
ousstabilityconceptsarederivedforpositive Volterra-Stieltjesequations. Inviewof
the concept of (stable) zero dynamics, Byrnes-Isidori form and Appendix 1.1, these
conceptsaregeneralizedtotime-invariantVsystems. Itisproventhat
positive Volterra-Stieltjes systems with stable zero dynamics and a special structure
oftheinputoutputmatrices(inparticular,relativedegreeone)arehigh-gainstabiliz-
able while preservingpositivity.
These results are exploited to generalize funnel controller to Volterra-Stieltjes sys-
tems in this thesis. In case of stable zero dynamics and suitable assumptions on the
high-frequency gain matrix funnel control is guaranteed and also positivity of the
trajectory of the closed-loop system. Under a suitable feasibility assumption, funnel
control is possible in the presence of input constraints which generalizes the results
of Chapter 3 to Volterra-Stieltjes systems. A further modification of the proposed
funnel controller is presented which guarantees non-negative input. These results
are applied to a control problem in anesthesia. The control objective is to keep the
concentration of anestheticgas close to a target value chosen by theanesthetist.
7Zusammenfassung
Diese Dissertation behandelt die Regelung von linearen Systemen mit mehrdimen-
sionalen Eingangen¨ und Ausgangen¨ und unbekanntem, aber beschranktem,¨ Relativ-
gradundlineareVolterra-StieltjesSystememitmehrerenEingangen¨ undAusgangen.¨
Die vorgelegte Arbeit behandelt die folgenden zwei Regler: adaptive Ruckf¨ uhrung¨
des Ausgangssignals und dessen Ableitung und Funnel Regelung. Fur¨ alle Regler
werden bestimmte strukturelle Voraussetzungen an die Systeme gestellt, auf die der
Regler angewendet werden soll.
Fur¨ vieleRegelungsanwendungensindkeinegutenModellevorhandenoderdieMod-
elle sind nur ungenau bekannt. Gewohnlich¨ kann nicht erwartet werden, dass voll-
standige¨ Informationen eines Systems vorhanden sind. Stattdessen sind nur struk-
turelle Eigenschaften (z.B. stabile Nulldynamik, Relativgrad) bekannt. Bei der Re-
gelung solcher Systeme kann ein adaptiver Regler angewendet werden. Das Ziel
von Kapitel 2 ist, einen universellen adaptiven Regler zu entwerfen, der vom Sys-
¨ ¨temverhaltenlerntundeinvorabfestgelegtesRegelungszielgewahrleistet. Mogliche
Zielsetzungen sind Stabilisierungdes Systems und λ-tracking.
Die meisten Regler, die λ-tracking benutzen, konnen¨ nur fur¨ Systeme mit Relativ-
grad eins angewendet werden. Das bedeutet, dass der Ausgang des Sy