Arbitrary pole placement with the extended Kautsky-Nichols-van Dooren parametric form with minimum gain

We consider the classic problem of pole placement by state feedback. We revisit the well-known eigenstructure assignment algorithm of Kautsky, Nichols and van Dooren [1] and extend it to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then employed to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with minimum gain

Robust arbitrary pole placement with the extended Kautsky-Nichols-van Dooren parametric form

We consider the classic problem of pole placement by state feedback. Our recent work [1] offered an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing gain matrix to deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. The method was adapted from the classic eigenstructure assignment algorithm of Kautsky, Nichols and van Dooren [2]. In this paper we employ this parametric formula to introduce an unconstrained nonlinear optimisation algorithm to obtain a gain matrix that delivers any desired pole placement with optimal robustness

Robust polynomial controller design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The work presented in this thesis was motivated by the desire to establish an alternative approach to the design of robust polynomial controllers. The procedure of pole-placement forms the basis of the design and for polynomial systems this generally involves the solution of a diophantine equation. This equation has many possible solutions which leads directly to the idea of determining the most appropriate solution for improved performance robustness. A thorough review of many of the aspects of the diophantine equation is presented, which helps to gain an understanding of this extremely important equation. A basic investigation into selecting a more robust solution is carried out but it is shown that, in the polynomial framework, it is difficult to relate decisions in the design procedure to the effect on performance robustness. This leads to the approach of using a state space based design and transforming the resulting output feedback controller to polynomial form. The state space design is centred around parametric output feedback which explicitly represents a set of possible feedback controllers in terms of arbitrary free parameters. The aim is then to select these free parameters such that the closed-loop system has improved performance robustness. Two parametric methods are considered and compared, one being well established and the other a recently proposed scheme. Although the well established method performs slightly better for general systems it is shown to fail when applied to this type of problem. For performance robustness, the shape of the transient response in the presence of model uncertainty is of interest. It is well known that the eigenvalues and eigenvectors play an important role in determining the transient behaviour and as such the sensitivities of these factors to model uncertainty forms the basis on which the free parameters are selected. Numerical optimisation is used to select the free parameters such that the sensitivities are at a minimum. It is shown both in a simple example and in a more realistic application that a significant improvement in the transient behaviour in the presence of model uncertainty can be achieved using the proposed design procedure.UK Science and Engineering Research Counci

Robust control of redundantly actuated dynamical systems

The eigenstructure assignment scheme for robust multivariable feedback control is extended to redundantly actuated dynamical systems. It is shown that an orthonormal set of close loop eigenvectors is always exactly assignable in the case of redundant actuation proving the inherent robustness in the control design methodology. A choice of close loop eigenvector set to minimize the feedback gain matrix is suggested. Partial Eigenstructure Assignment methodology is proposed for second order mechanical systems. A methodology for coordinated actuation of redundant actuator sets by a trained weighted minimum norm solution is presented. To apply the methodology to hyper-redundant actuator arrays, for application to smart actuator arrays, a novel adaptive discretization algorithm is proposed. The adaptive aggregation strategy, based on the physics of the system, introduces nodes, to optimize a performance index of the overall plant model. The dimensionality of the inputs thus reduces to a finite number, making it a candidate plant for control by the robust redundant control scheme. The adaptive aggregation together with robust redundant control methodology is demonstrated on a finite element model of a novel morphing wing. This schema unifies the traditionally disparate methods of modeling and controller design

Robust eigenstructure assignment in geometric control theory

In this paper we employ the Rosenbrock system matrix pencil for the computation of output-nulling subspaces of linear time-invariant systems which appear in the solution of a large number of control and estimation problems. We also consider the problem of finding friends of these output-nulling subspaces, i.e., the feedback matrices that render such subspaces invariant with respect to the closed-loop map and output-nulling with respect to the output map, and which at the same time deliver a robust closed-loop eigenstructure. We show that the methods presented in this paper offer considerably more robust eigenstructure assignment than the other commonly used methods and algorithms

Lead pursuit control of multiphase drives

Los accionamientos multifásicos, compuestos por una máquina eléctrica de más de tres fases alimentada por un convertidor de potencia, han atraído recientemente un importante interés en la comunidad investigadora debido a las ventajas que presentan frente a las máquinas trifásicas convencionales. Este es el caso de la mejor distribución de potencia por fase, la menor producción de armónicos en el convertidor de potencia y, la más importante, la tolerancia a fallos, lo cual significa que la máquina multifásica puede seguir funcionando cuando una o varias fases se pierden, siempre que el número restante de fases sea igual o mayor que tres. Debido a esta alta fiabilidad, los accionamientos multifásicos son especialmente adecuados para aplicaciones relacionadas con los vehículos eléctricos (terrestres, marítimos y aéreos) y las energías renovables por razones de seguridad y/o económicas. El uso de controladores avanzados y de alto rendimiento en accionamientos multifásicos es particularmente relevante, ya que las estrategias de control convencionalmente aplicadas a los accionamientos trifásicos no terminan de alcanzar un estándar en su extensión al caso multifásico. La razón es la mayor complejidad y número de variables a controlar. En este contexto, los controladores predictivos han encontrado un interesante nicho de aplicación en convertidores de potencia y accionamientos multifásicos debido a su formulación intuitiva y flexible: un modelo del sistema es usado para calcular las predicciones de las variables controladas, que luego se comparan con las referencias impuestas dentro de una función de coste. Esta estrategia permite incorporar varios objetivos de control y restricciones en el proceso de control a través de la función de coste. Sin embargo, es bien sabido que este tipo de controlador sufre de un alto coste computacional y contenido armónico de corriente que limita su aplicación en los accionamientos multifásicos. La investigación desarrollada en esta Tesis se centra en la mitigación de las limitaciones citadas siguiendo dos objetivos principales: • La incorporación de observadores de corrientes rotóricas en el controlador predictivo para mejorar así la precisión del modelo predictivo y, consecuentemente, el rendimiento del sistema de control, principalmente en términos de contenido armónico y pérdidas por conmutación en el convertidor de potencia. Un observador de Luenberger es construido para este propósito utilizando una estrategia innovadora de posicionamiento de polos en su diseño. • La introducción de un grado de libertad adicional en el controlador predictivo basado en tiempos de muestreo variables e implementado usando el concepto de lead pursuit. El resultado es un controlador novedoso que conduce a una resolución en los tiempos de conmutación más fina en comparación con las técnicas predictivas más convencionales, lo que proporciona una reducción importante en el contenido armónico. Las estrategias de control propuestas son validadas mediante simulación y experimentación utilizando un accionamiento compuesto por una máquina de inducción de cinco fases como caso de ejemplo. Los resultados y conclusiones derivadas de esta investigación han sido presentados en cinco trabajos principales publicados en revistas internacionales de alto impacto, los cuales constituyen las contribuciones de esta Tesis por compendio de artículos. Sin embargo, otros trabajos relacionados con la línea de investigación han sido también publicados en artículos de revista y conferencia y en un capítulo de libro.Multiphase drives, constituted by an electric machine with more than three phases fed by a power converter, have recently attracted an important interest in the research community due to the advantages that they present over the conventional three-phase ones. This is the case of the better power distribution per phase, the lower harmonic production in the power converter, and the most important one, the fault-tolerant capability, which means that the multiphase machine can still be operated when one or several phases are missing, provided that the number of remaining phases is equal or greater than three. Due to this high reliability, multiphase drives are specially well suited for applications related to electric vehicles (terrestrial, maritime and aerial) and renewable energies for safety and/or economical reasons. The use of advanced and high-performance controllers in multiphase drives is particularly relevant, since the control strategies conventionally applied to three-phase drives do not reach a standard in their extension to the multiphase case. The reason is the greater complexity and number of variables that must be controlled. In this context, predictive controllers have found an interesting niche of application in power converters and multiphase drives due to their intuitive and flexible formulation: a model of the system is used to compute predictions of the controlled variables, which are later compared with the imposed references in a cost function. This strategy permits incorporating several control objectives and constraints in the control process through the cost function. However, it is well known that this type of controller suffers from a high computational cost and current harmonic content that limit its application in multiphase drives. The research developed in this Thesis work is focused on the mitigation of the cited limitations following two main goals: • The incorporation of rotor current observers in the predictive controller in order to improve the accuracy of the predictive model and, consequently, the control system performance, principally in terms of harmonic content and commutation losses in the power converter. A Luenberger observer is constructed for that purpose using an innovative pole-placement strategy in its design. • The introduction of an additional degree of freedom in the predictive controller based on variable sampling times and implemented using the lead-pursuit concept. The result is a novel controller that leads to a finer resolution in the commuting times in comparison with more conventional predictive techniques, which provides an important reduction in the harmonic content. The proposed control strategies are validated by simulation and experimentation using a five-phase induction machine drive as case example. The results and conclusions derived from this research have been presented in five main works published in high-impact international journals, which constitute the contributions of this article compendium Thesis. Nevertheless, other related works have also been published in journal and conference papers and a book chapter

Optimal control with structure constraints and its application to the design of passive mechanical systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering; and, (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2002.Page 214 blank.Includes bibliographical references.Structured control (static output feedback, reduced-order control, and decentralized feedback) is one of the most important open problems in control theory and practice. In this thesis, various techniques for synthesis of structured controllers are surveyed and investigated, including H2 optimization, H[infinity] optimization, L1 control, eigenvalue and eigenstructure treatment, and multiobjective control. Unstructured control-full- state feedback and full-order control-is also discussed. Riccati-based synthesis, linear matrix inequalities (LMI), homotopy methods, gradient- and subgradientbased optimization are used. Some new algorithms and extensions are proposed, such as a subgradient-based method to maximize the minimal damping with structured feedback, a multiplier method for structured optimal H2 control with pole regional placement, and the LMI-based H2/H[infinity]/pole suboptimal synthesis with static output feedback. Recent advances in related areas are comprehensively surveyed and future research directions are suggested. In this thesis we cast the parameter optimization of passive mechanical systems as a decentralized control problem in state space, so that we can apply various decentralized control techniques to the parameter design which might be very hard traditionally. More practical constraints for mechanical system design are considered; for example, the parameters are restricted to be nonnegative, symmetric, or within some physically-achievable ranges. Marginally statable systems and hysterically damped systems are also discussed. Numerical examples and experimental results are given to illustrate the successful application of decentralized control techniques to the design of passive mechanical systems, such as multi-degree-of-freedom tuned-mass dampers, passive vehicle suspensions, and others.by Lei Zuo.S.M

Optimisation of condition number for eigenstructure problems

This thesis deals with a new solution for the problem of eigenstructure assignment in control systems design. A wide range of challenging issues is examined involving the problem of eigenstructure assignment and associated system properties through different forms of complexity which are strongly related to control system design. In this thesis, specific attention has been given to the issue of skewness of the closedloop eigenframe of the state matrix. In fact, the aim is to develop a new methodology for determining the best angle between closed-loop eigenvectors by optimising the minimal condition number of the closed-loop eigenvector matrix. This problem is strongly linked to sensitivity of eigenvalues to parameter uncertainty, perturbations to model parameter uncertainty. The importance of this methodology can be expressed in terms of results related to the Sensitivity of eigenvalues, Relative measures of controllability and observability and also deviations from strong stability to overshooting behaviour. Among this variety of eigenstructure assignment methods, special consideration has been paid to Geometric Theory, which introduces an alternative solution to the assignability of spectrum of controllability subspaces (cs) based on an eigenvector approach and then develops a new pole assignment algorithm based on openloop/ closed-loop spectra as a practical application of this approach. In order to tackle the problem of measuring the skewness of angles between closed-loop eigenvalues, some measures for Eigenframe skewness have been defined in general and so the necessary and efficient conditions have been derived for the angle between some subspaces in a direct sum decomposition to be maximized. This has been done via three metrics; Condition Number, Determinant of Gram Matrix and Singular Values. The thesis presents the parametrisation of closed-loop eigenframes result by the method generated in [4]. Within this thesis, a non-smooth algorithm has been developed in order to select the most orthogonal closed-loop eigenframe and so the influence of selected closed-loop spectra. Also, the parametrisation of controllability subspaces in a standard direct sum decomposition using matrix fraction description (MFD) has been derived. Within this thesis the construction and the existence of controllability subspaces connected to (A,B)-invariant subspaces, has also been studied. In addition, an algebraic description of the total system behaviour which leads to an algebraic characterisation of the total input, state and output behaviour in an implicit formulation is given based on properties of MFD descriptions, a topic which remains open for future studies