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Control and Analysis of Pulse-Modulated Systems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

The thesis consists of an introduction and four appended papers. In the introduction we give an overview of pulse-modulated systems and provide a few examples of such systems. Furthermore, we introduce the so-called dynamic phasor model which is used as a basis for analysis in two of the appended papers. We also introduce the harmonic transfer function and finally we provide a summary of the appended papers.

The first paper considers stability analysis of a class of pulse-width modulated systems based on a discrete time model. The systems considered typically have periodic solutions. Stability of a periodic solution is equivalent to stability of a fixed point of a discrete time model of the system dynamics.

Conditions for global and local exponential stability of the discrete time model are derived using quadratic and piecewise quadratic Lyapunov functions. A griding procedure is used to develop a systematic method to search for the Lyapunov functions.

The second paper considers the dynamic phasor model as a tool for stability analysis of a general class of pulse-modulated systems. The analysis covers both linear time periodic systems and systems where the pulse modulation is controlled by feedback. The dynamic phasor model provides an $\textbf{L}_2$-equivalent description of the system dynamics in terms of an infinite dimensional dynamic system. The infinite dimensional phasor system is approximated via a skew truncation. The truncated system is used to derive a systematic method to compute time periodic quadratic Lyapunov functions.

The third paper considers the dynamic phasor model as a tool for harmonic analysis of a class of pulse-width modulated systems. The analysis covers both linear time periodic systems and non-periodic systems where the switching is controlled by feedback. As in the second paper of the thesis, we represent the switching system using the L_2-equivalent infinite dimensional system provided by the phasor model. It is shown that there is a connection between the dynamic phasor model and the harmonic transfer function of a linear time periodic system and this connection is used to extend the notion of harmonic transfer function to describe periodic solutions of non-periodic systems. The infinite dimensional phasor system is approximated via a square truncation. We assume that the response of the truncated system to a periodic disturbance is also periodic and we consider the corresponding harmonic balance equations. An approximate solution of these equations is stated in terms of a harmonic transfer function which is analogous to the harmonic transfer function of a linear time periodic system. The aforementioned assumption is proved to hold for small disturbances by proving the existence of a solution to a fixed point equation. The proof implies that for small disturbances, the approximation is good.

Finally, the fourth paper considers control synthesis for switched mode DC-DC converters. The synthesis is based on a sampled data model of the system dynamics. The sampled data model gives an exact description of the converter state at the switching instances, but also includes a lifted signal which represents the inter-sampling behavior. Within the sampled data framework we consider H-infinity control design to achieve robustness to disturbances and load variations. The suggested controller is applied to two benchmark examples; a step-down and a step-up converter. Performance is verified in both simulations and in experiments.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , xiii, 25 p.
Series
Trita-MAT. OS, ISSN 1401-2294 ; 08/OS/04
Keyword [en]
Pulse-width modulation, Periodic systems, Stability analysis, Harmonic analysis, Lyapunov methods, Dynamic phasors, Harmonic transfer function, Switched mode power converters, Sampled data modeling, H-infinity synthesis
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4713ISBN: 978-91-7178-959-4 (print)OAI: oai:DiVA.org:kth-4713DiVA: diva2:13578
Public defence
2008-05-23, D3, D, Lindstedsvägen 5, KTH, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100628Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2010-06-30Bibliographically approved
List of papers
1. Stability analysis of a class of PWM systems
Open this publication in new window or tab >>Stability analysis of a class of PWM systems
2007 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 52, no 6, 1072-1078 p.Article in journal (Refereed) Published
Abstract [en]

This note considers stability analysis of a class of pulsewidth modulated (PWM) systems that incorporates several different switched mode dc-de- converters. The systems of the class typically have periodic solutions. A sampled data model is developed and used to prove stability of these solutions. Conditions for global and local exponential stability are derived using quadratic and piecewise quadratic Lyapunov functions. The state space is partitioned and the stability conditions are verified by checking a set of coupled linear matrix inequalities (LMIs).

Keyword
dc-dc converter, Lyapunov methods, pulsewidth modulated (PWM) systems, sampled data modeling, stability analysis, LYAPUNOV FUNCTIONS, HYBRID SYSTEMS, PIECEWISE, CONVERTERS
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-13746 (URN)10.1109/TAC.2007.899082 (DOI)000247353300011 ()2-s2.0-34447115060 (Scopus ID)
Note
QC 20100628Available from: 2010-06-28 Created: 2010-06-28 Last updated: 2017-12-12Bibliographically approved
2. Dynamic phasor analysis of pulse-modulated systems
Open this publication in new window or tab >>Dynamic phasor analysis of pulse-modulated systems
2007 (English)In: Proceedings Of The 46th IEEE Conference On Decision And Control, Vols 1-14, 2007, 3938-3945 p.Conference paper, Published paper (Refereed)
Abstract [en]

The paper considers stability analysis of a general class of pulse modulated systems in a phasor dynamic framework. The dynamic phasor model exploits the cyclic nature of the modulation functions by representing the system dynamics in terms of a Fourier series expansion defined over a moving time-window. The contribution of the paper is to show that a special type of periodic Lyapunov function can be used to analyze the system and that the analysis conditions become tractable for computation after truncation. The approach provides a trade-off between complexity and accuracy that includes standard state space averaged models as a special case.

Series
IEEE Conference On Decision And Control - Proceedings, ISSN 0191-2216
Keyword
Pulse-modulated systems; Stability analysis; Lyapunov methods; Dynamic phasors; Periodic systems; Harmonic transfer function
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13763 (URN)000255181702044 ()2-s2.0-62749086376 (Scopus ID)978-1-4244-1497-0 (ISBN)
Conference
46th IEEE Conference on Decision and Control Location: New Orleans, LA Date: DEC 12-14, 2007
Note
QC 20100628Available from: 2010-06-28 Created: 2010-06-28 Last updated: 2011-09-30Bibliographically approved
3. Harmonic analysis of pulse-width modulated systems
Open this publication in new window or tab >>Harmonic analysis of pulse-width modulated systems
2009 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 45, no 4, 851-862 p.Article in journal (Refereed) Published
Abstract [en]

The paper considers the so-called dynamic phasor model as a basis for harmonic analysis of a class switching systems. The analysis covers both periodically switched systems and non-periodic systems where the switching is controlled by feedback. The dynamic phasor model is a powerful tool for exploring cyclic properties of dynamic systems. It is shown that there is a connection between the dynamic phasor model and the harmonic transfer function of a linear time periodic system and this connection is used to extend the notion of harmonic transfer function to describe periodic solutions of non-periodic systems.

Keyword
Pulse-width modulation, Harmonic analysis, Dynamic phasors, Periodic systems, Switched-mode circuits, TIME-PERIODIC-SYSTEMS, STABILITY ANALYSIS, CONVERTERS
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-13748 (URN)10.1016/j.automatica.2008.10.029 (DOI)000265155700001 ()2-s2.0-61849182595 (Scopus ID)
Note
QC 20100628Available from: 2010-06-28 Created: 2010-06-28 Last updated: 2017-12-12Bibliographically approved
4. Sampled data control of DC-DC converters
Open this publication in new window or tab >>Sampled data control of DC-DC converters
(English)Article in journal (Other academic) Submitted
Keyword
Switched mode DC-DC converters; Sampled data modeling; H-infinity control synthesis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-13782 (URN)
Note
QC 20100628Available from: 2010-06-28 Created: 2010-06-28 Last updated: 2010-06-28Bibliographically approved

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