Hankel-type Model Reduction Based on Frequency Response Matching

In this paper, a stability preserving model reduction algorithm for single-input single-output linear time invariant systems is presented. It performs a data fitting in the frequency domain using semidefinite programming methods. Computing the frequency response of a model can be done efficiently even for large scale models making this approach applicable to those. The relaxation used to obtain a semidefinite program is similar to one used in Hankel model reduction. Therefore accuracy of approximation is also similar to Hankel model reduction one. The approach can be easily extended to frequency-weighted and parameter dependent model reduction problems

Active Vibration Control of Structures using an Impedance Matching Control Technique

Active vibration control of structures has gained a lot of interest in recent years. This paper presents an active vibration control methodology of a structure using piezoelectric actuators. The proposed methodology is useful in practical applications where the system to be controlled is difficult to model due to the presence of complex boundary conditions. The impedance matching control technique uses a power flow approach wherein the controller is designed such that the power flow into the structure is minimized. The system transfer function is obtained from the experimental collocated actuator/sensor pair data using Eigen Realisation Algorithm (ERA). The controller is designed for the system transfer function according to impedance matching theory. The above approach is targeted towards the vibration control of wind tunnel stings, which suffer from flow-induced vibration. A wind tunnel sting model is designed and fabricated for this study. The real time implementation of the impedance matching controller has been carried out using dSPACE® Digital Signal Processor (DSP) card. The results are encouraging and demonstrate the feasibility of applying this technique in the wind tunne

Linear Control Theory with an ℋ∞ Optimality Criterion

This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods

Passivity-preserving parameterized model order reduction using singular values and matrix interpolation

We present a parameterized model order reduction method based on singular values and matrix interpolation. First, a fast technique using grammians is utilized to estimate the reduced order, and then common projection matrices are used to build parameterized reduced order models (ROMs). The design space is divided into cells, and a Krylov subspace is computed for each cell vertex model. The truncation of the singular values of the merged Krylov subspaces from the models located at the vertices of each cell yields a common projection matrix per design space cell. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parameterized ROM. Pertinent numerical results validate the proposed technique

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

Model Reduction Using Semidefinite Programming

In this thesis model reduction methods for linear time invariant systems are investigated. The reduced models are computed using semidefinite programming. Two ways of imposing the stability constraint are considered. However, both approaches add a positivity constraint to the program. The input to the algorithms is a number of frequency response samples of the original model. This makes the computational complexity relatively low for large-scale models. Extra properties on a reduced model can also be enforced, as long as the properties can be expressed as convex conditions. Semidefinite program are solved using the interior point methods which are well developed, making the implementation simpler. A number of extensions to the proposed methods were studied, for example, passive model reduction, frequency-weighted model reduction. An interesting extension is reduction of parameterized linear time invariant models, i.e. models with state-space matrices dependent on parameters. It is assumed, that parameters do not depend on state variables nor time. This extension is valuable in modeling, when a set of parameters has to be chosen to fit the required specifications. A good illustration of such a problem is modeling of a spiral radio frequency inductor. The physical model depends nonlinearly on two parameters: wire width and wire separation. To chose optimally both parameters a low-order model is usually created. The inductor modeling is considered as a case study in this thesis

Model Reduction Techniques and Their Application to Helicopter Models

Techniques of model reduction are important not only because a reduced order model may be needed for control system design or for model validation from measured response data, but also because in many applications accurate information about the plant dynamics may only be available, or required, for a restricted range of frequencies. In the content of aircraft flight mechanics models reduced order descriptions are also of considerable importance for handling qualities studies. In this document, various model reduction techniques are reviewed. An 'equivalent system approximation' approach has been selected and applied to the reduction of a helicopter flight dynamics model. The adequacy and degree of accuracy of this 'equivalent system approximation' reduced order model was verified through comparison with a high order model using the Puma helicopter as an example. Excellent agreement between the results from the reduced order model and the original high order system model were obtained over selected range of frequency. Another approximate method --- extended Levy's complex-curve fitting method using a modified least-squares approach has been extended to the multi-input multi-output and has also applied to the reduction of the helicopter flight dynamics model for a Puma helicopter. Very good agreement between the results from the reduced order models and the original system model were again obtained. Comparison of Levy's method with the 'equivalent system' approach showed that in the latter physical insight can be used in the reduction process whereas Levy's method is purely a curve fitting technique. Both techniques can, however, provide useful reduced-order descriptions for given frequency ranges. The extended Levy approach and the 'equivalent systems' approach have both been implemented using the MATLAB software package

Scattering Suppression from Arbitrary Objects in Spatially-Dispersive Layered Metamaterials

Concealing objects by making them invisible to an external electromagnetic probe is coined by the term cloaking. Cloaking devices, having numerous potential applications, are still face challenges in realization, especially in the visible spectral range. In particular, inherent losses and extreme parameters of metamaterials required for the cloak implementation are the limiting factors. Here, we numerically demonstrate nearly perfect suppression of scattering from arbitrary shaped objects in spatially dispersive metamaterial acting as an alignment-free concealing cover. We consider a realization of a metamaterial as a metal-dielectric multilayer and demonstrate suppression of scattering from an arbitrary object in forward and backward directions with perfectly preserved wavefronts and less than 10% absolute intensity change, despite spatial dispersion effects present in the composite metamaterial. Beyond the usual scattering suppression applications, the proposed configuration may serve as a simple realisation of scattering-free detectors and sensors