Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

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

Phase Retrieval via Matrix Completion

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover

Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis

Reduced cost of sensors and increased computing power is enabling the development and implementation of control systems that can simultaneously regulate multiple variables and handle conflicting objectives while maintaining stringent performance objectives. To make this a reality, practical analysis and design tools must be developed that allow the designer to trade-off conflicting objectives and guarantee performance in the presence of uncertain system dynamics, an uncertain environment, and over a wide range of operating conditions. As a first step towards this goal, we organize and streamline a promising robust control approach, Robust Linear Parameter Varying control, which integrates three fields of control theory: Integral Quadratic Constraints (IQC) to characterize uncertainty and nonlinearities, Linear Parameter Varying systems (LPV) that formalizes gain-scheduling, and convex optimization to solve the resulting robust control Linear Matrix Inequalities (LMI). To demonstrate the potential of this approach, it was applied to the design of a robust linear parametrically varying controller for an ecosystem with nonlinear predator-prey-hunter dynamics

Control oriented modelling of an integrated attitude and vibration suppression architecture for large space structures

This thesis is divided into two parts. The main focus of the research, namely active vibration control for large flexible spacecraft, is exposed in Part I and, in parallel, the topic of machine learning techniques for modern space applications is described in Part II. In particular, this thesis aims at proposing an end-to-end general architecture for an integrated attitude-vibration control system, starting from the design of structural models to the synthesis of the control laws. To this purpose, large space structures based on realistic missions are investigated as study cases, in accordance with the tendency of increasing the size of the scientific instruments to improve their sensitivity, being the drawback an increase of its overall flexibility. An active control method is therefore investigated to guarantee satisfactory pointing and maximum deformation by avoiding classical stiffening methods. Therefore, the instrument is designed to be supported by an active deployable frame hosting an optimal minimum set of collocated smart actuators and sensors. Different spatial configurations for the placement of the distributed network of active devices are investigated, both at closed-loop and open-loop levels. Concerning closed-loop techniques, a method to optimally place the poles of the system via a Direct Velocity Feedback (DVF) controller is proposed to identify simultaneously the location and number of active devices for vibration control with an in-cascade optimization technique. Then, two general and computationally efficient open-loop placement techniques, namely Gramian and Modal Strain Energy (MSE)-based methods, are adopted as opposed to heuristic algorithms, which imply high computational costs and are generally not suitable for high-dimensional systems, to propose a placement architecture for generically shaped tridimensional space structures. Then, an integrated robust control architecture for the spacecraft is presented as composed of both an attitude control scheme and a vibration control system. To conclude the study, attitude manoeuvres are performed to excite main flexible modes and prove the efficacy of both attitude and vibration control architectures. Moreover, Part II is dedicated to address the problem of improving autonomy and self-awareness of modern spacecraft, by using machine-learning based techniques to carry out Failure Identification for large space structures and improving the pointing performance of spacecraft (both flexible satellite with sloshing models and small rigid platforms) when performing repetitive Earth Observation manoeuvres

Robustness analysis for power systems based on the structured singular value tools and the [nu] gap metric

Modern power systems are operated more stressed than ever because of the advent of deregulation and competition. One of the important issues in the design of controllers for a stressed system is to evaluate the stability of the controlled system over a range of operating conditions.;The conventional controllers are designed to make the system stable under certain conditions of operation. The time consuming time domain simulation is then used to evaluate the controllers for a few selected operating conditions around which the controllers are designed. Such a design and evaluation procedure cannot guarantee robustness of the controller over the whole range of operating conditions.;In this dissertation, practical algorithms to perform robustness analysis based on two tools, structured singular value and the nu gap metric, are investigated. The power system stabilizer is used as the controller and small signal stability is of interest.;The robustness problem in the SSV framework is set up for the multimachine power system. In this formulation, an improved uncertainty characterization has been used to capture the effect of parameter variations in terms of the varying elements of the linearized system matries, which are derived from the component differential equations and the network algebraic equations separately. SVD decomposition is used to reduce the size of the problem. Based on the resulting framework, a branch and bound scheme is proposed to intelligently select frequency intervals on which the frequency sweep test can be performed further to find the peak of mu. Instead of blindly choosing frequency intervals to sweep, which could ignore important frequency points on the mu plots, this scheme provides searching under guidance. The analysis procedure accurately predicts the range of stable operating conditions which are verified by repeated eigenvalue analysis.;Fir the robustness in terms of nu gap metric, we set up the feedback configuration for multimachine power system. The frequency response of the nu gap metric is plotted and its relationship with that of the stability margin is used to determine the stability of the perturbed systems. A weighted nu gap metric is defined and its frequency domain interpretation is explored to further reduce the conservatism of the results.;Finally, a feedback configuration is carefully developed to carry out the McFarlane and Glover Hinfinity loop shaping design procedure. The effect of the damping controller on improving system dynamic performance is also examined.;Comparisons are made between the two major analysis tools via the results on the same test systems with the same scenarios