Model Order Reduction Based on Semidefinite Programming

The main topic of this PhD thesis is complexity reduction of linear time-invariant models. The complexity in such systems is measured by the number of differential equations forming the dynamical system. This number is called the order of the system. Order reduction is typically used as a tool to model complex systems, the simulation of which takes considerable time and/or has overwhelming memory requirements. Any model reflects an approximation of a real world system. Therefore, it is reasonable to sacrifice some model accuracy in order to obtain a simpler representation. Once a low-order model is obtained, the simulation becomes computationally cheaper, which saves time and resources. A low-order model still has to be "similar" to the full order one in some sense. There are many ways of measuring "similarity" and, typically, such a measure is chosen depending on the application. Three different settings of model order reduction were investigated in the thesis. The first one is H infinity model order reduction, i.e., the distance between two models is measured by the H infinity norm. Although, the problem has been tackled by many researchers, all the optimal solutions are yet to be found. However, there are a large number of methods, which solve suboptimal problems and deliver accurate approximations. Recently, research community has devoted more attention to large-scale systems and computationally scalable extensions of existing model reduction techniques. The algorithm developed in the thesis is based on the frequency response samples matching. For a large class of systems the computation of the frequency response samples can be done very efficiently. Therefore, the developed algorithm is relatively computationally cheap. The proposed algorithm can be seen as a computationally scalable extension to the well-known Hankel model reduction, which is known to deliver very accurate solutions. One of the reasons for such an assessment is that the relaxation employed in the proposed algorithm is tightly related to the one used in Hankel model reduction. Numerical simulations also show that the accuracy of the method is comparable to the Hankel model reduction one. The second part of the thesis is devoted to parameterized model order reduction. A parameterized model is essentially a family of models which depend on certain design parameters. The model reduction goal in this setting is to approximate the whole family of models for all values of parameters. The main motivation for such a model reduction setting is design of a model with an appropriate set of parameters. In order to make a good choice of parameters, the models need to be simulated for a large set of parameters. After inspecting the simulation results a model can be picked with suitable frequency or step responses. Parameterized model reduction significantly simplifies this procedure. The proposed algorithm for parameterized model reduction is a straightforward extension of the one described above. The proposed algorithm is applicable to linear parameter-varying systems modeling as well. Finally, the third topic is modeling interconnections of systems. In this thesis an interconnection is a collection of systems (or subsystems) connected in a typical block-diagram. In order to avoid confusion, throughout the thesis the entire model is called a supersystem, as opposed to subsystems, which a supersystem consists of. One of the specific cases of structured model reduction is controller reduction. In this problem there are two subsystems: the plant and the controller. Two directions of model reduction of interconnected systems are considered: model reduction in the nu-gap metric and structured model reduction. To some extent, using the nu-gap metric makes it possible to model subsystems without considering the supersystem at all. This property can be exploited for extremely large supersystems for which some forms of analysis (evaluating stability, computing step response, etc.) are intractable. However, a more systematic way of modeling is structured model reduction. There, the objective is to approximate certain subsystems in such a way that crucial characteristics of the given supersystem, such as stability, structure of interconnections, frequency response, are preserved. In structured model reduction all subsystems are taken into account, not only the approximated ones. In order to address structured model reduction, the supersystem is represented in a coprime factor form, where its structure also appears in coprime factors. Using this representation the problem is reduced to H infinity model reduction, which is addressed by the presented framework. All the presented methods are validated on academic or known benchmark problems. Since all the methods are based on semidefinite programming, adding new constraints is a matter of formulating a constraint as a semidefinite one. A number of extensions are presented, which illustrate the power of the approach. Properties of the methods are discussed throughout the thesis while some remaining problems conclude the manuscript

Robust Regulation for Infinite-Dimensional Systems and Signals in the Frequency Domain

In this thesis, the robust output regulation problem is studied both in the time domain and in the frequency domain. The problem to be addressed is to find a stabilizing controller for a given plant so that every signal generated by an exogenous system, or shortly exosystem, is asymptotically tracked despite perturbations in the plant or some external disturbances. The exosystem generating the reference and disturbance signals is assumed to be infinite-dimensional. The main contribution of this thesis is to develop the robust regulation theory for an infinite-dimensional exosystem in the frequency domain framework. In order to do that, the time domain theory is studied in some detail and new results that emphasize the smoothness requirement on the reference and disturbance signals due to infinite-dimensionality of the exosystem are presented. Two types of controllers are studied, the feedforward controllers and the error feedback controllers, the latter of which facilitate robust regulation. These results exploit the structure at infinity of tha plant transfer function. In this thesis, a new definition of the structure at infinity suitable for infinite-dimensional systems is developed and its properties are studied. The frequency domain theory developed is based on the insights into the corresponding time domain theory. By following some recent time domain ideas the type of robustness and stability types are chosen so that they facilitate the use of an infinite-dimensional exosystem. The robustness is understood in the sense that stability should imply regulation. The chosen stability types resemble the time domain polynomial and strong stabilities and allow robust regulation of signals that have an infinite number of unstable dynamics along with transfer functions vanishing at infinity. The main contribution of this thesis is the formulation of the celebrated internal model principle in the frequency domain terms in a rather abstract algebraic setting. Unlike in the existing literature, no topological aspect of the problem is needed because of the adopted definition of robustness. The plant transfer function is only assumed to have a right or a left coprime factorization but not necessarily both. The internal model principle leads to a necessary and sufficient condition for the solvability of the robust regulation problem. The second main contribution of the thesis is to design frequency domain controllers for infinite-dimensional systems and exosystems. In this thesis, the Davison’s simple controller design for stable plants is extended to infinite-dimensional systems and exosystems. Then a controller design procedure for unstable plants containing two phases is proposed. In the first phase, a stabilizing controller is constructed for a given plant. The second phase is to design a robustly regulating controller for a stable part of the plant. This design procedure nicely combines with the Davison’s type controllers and is especially suitable for infinite-dimensional plants with transfer functions in the Callier-Desoer class of transfer functions

Youla–Kučera Parametrization with no Coprime Factorization—Single-Input Single-Output Case

We present a generalization of the Youla—Kučera parametrization to obtain all stabilizing controllers for single-input and single-output plants. This uses three parameters and can be applied to plants that may not admit coprime factorizations. In this generalization, at most two rational expressions of plants are required, while the Youla–Kučera parametrization requires precisely one rational expression

Homogeneous finite-gain Lp-stability analysis on homogeneous systems

In dieser Arbeit wird gezeigt, dass die klassische Lp-Stabilität und Lp-Verstärkung für beliebige stetige, gewichtete homogene Systeme nicht wohldefiniert ist. Indem die klassische Lp-Norm von Signalen zu einer homogenen Lp-Norm so angepasst wird, dass diese bezüglich der Gewichtsvektoren homogen ist, ist es möglich zu zeigen, dass jedes intern stabile homogene System für hinreichend große p eine global definierte endliche homogene Lp-Verstärkung besitzt. Mit Hilfe einer homogenen Lyapunov-Funktion kann die homogene Lp-Stabilität durch eine homogene partielle Differentialungleichung charakterisiert werden, die sich im eingangsaffinen Fall in eine homogene Hamilton-Jacobi-Ungleichung transformieren lässt. Des Weiteren werden in dieser Arbeit detaillierte Methoden zur Abschätzung von oberen Schranken für homogene Lp-Verstärkungen aus diesen Ungleichungen abgeleitet. Dies schließt die homogene L∞-Verstärkung und die homogene Eingangs-Zustands-Verstärkung ebenfalls ein. Bei rückgekoppelten homogenen Systemen, bei denen die Gewichtsvektoren zwischen den Systemen zueinander passend sind, erlaubt die additive Ungleichung für die homogene Lp-Norm die Einführung des homogenen Small-Gain Theorems für beliebige p, wodurch eine Stabilitätsanalyse des geschlossenen Regelkreises ermöglicht wird. Weiterhin können homogene H∞-Regler entworfen werden, wenn das System eingangsaffin ist. Da die konventionellen Werkzeuge der linearen Systemtheorie nicht zur Verfügung stehen, können solche homogenen H∞-Regler nur garantieren, dass der geschlossene Regelkreis eine homogene Lp-Verstärkung hat, die kleiner als ein bestimmbarer Wert ist. Ihre Optimalität kann hingegen nicht garantiert werden. In jedem Kapitel werden mehrere kurze Beispiele vorgestellt, um zu veranschaulichen, wie eine solche homogene Lp-Verstärkung berechnet werden kann. Insbesondere ist eine detaillierte Analyse des “Continuous Super-Twisting Like Algorithm" mit tieferen Einblicken für interessierte Leser enthalten.In this thesis, it is shown that the classical Lp-stability and Lp-gain is not well-defined for arbitrary continuous weighted homogeneous systems. By modifying the classical Lp-norm of signals to be homogeneous w.r.t. some weight vectors, which is called homogeneous Lp-norm, it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous Lp-gain, for p sufficiently large. With the help of a homogeneous Lyapunov function, homogeneous Lp-stability can be characterized by a homogeneous partial differential inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. Furthermore, in this thesis some detailed methods to calculate upper estimates for the homogeneous Lp-gain are provided from theses inequalities. This also includes the homogeneous L∞-gain and homogeneous Input-to-State gain. For feedback interconnected systems, if the weight vectors between plants are matched, the additive inequality for homogeneous Lp-norm allows the introduction of the homogeneous small gain theorem for each p, enabling stability analysis on the closed loop system. Finally, some homogeneous H∞-controllers can be designed, if the system is affine in the control input. Without the convenient tools for the linear systems, such homogeneous H∞-controllers can only guarantee that the closed loop system has homogeneous Lp-gain less than some derivable numbers, its optimality can not be guaranteed. Several short examples are presented within each chapter to illustrate how such homogeneous Lp-gain can be calculated. In particular a detailed analysis on the Continuous Super-Twisting Like Algorithm is included with deeper insight for interested readers