Stability analysis and vibration control of a class of negative imaginary systems

This paper presents stability analysis and vibration control of a class of negative imaginary systems. A flexible manipulator that moves in a horizontal plane is considered and is modelled using the finite element method. The system with two poles at the origin is shown to possess negative imaginary properties. Subsequently, an integral resonant controller (IRC) which is a strictly negative imaginary controller is designed for the position and vibration control of the system. Using the IRC, the closed-loop system is observed to be internally stable and simuation results show that satisfactory hub angle response is achieved. Furthermore, vibration magnitudes at the resonance modes are suppressed by 48 dB

A local stability condition for dc grids with constant power loads

Currently, there are an increasing number of power electronics converters in electrical grids, performing the most diverse tasks, but most of them, work as constant power loads (CPLs). This work presents a sufficient condition for the local stability of dc linear time-invariant circuits with constant power loads for all the possible equilibria (depending on the drained power) of the systems. The condition is shown as a method with successive steps that should be met. Its main step is expressed as a linear matrix inequality test which is important for easiness of verification reasons. The method is illustrated with two examples: a single-port RLC circuit connected to a CPL and a two-port linear dc circuit connected to two CPLs.Postprint (published version

Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

In this paper, we propose a notion of dissipativity with dynamic supply rates for nonlinear differential input-state-output equations via the use of auxiliary systems. This extends the classical dissipativity with static supply rates and miscellaneous dynamic quadratic forms. The main results of this paper concern Lyapunov (asymptotic/exponential) stability analysis for nonlinear feedback dissipative systems that are characterised by dissipation inequalities with respect to compatible dynamic supply rates. Importantly, dissipativity conditions guaranteeing partial stability of the state of the feedback systems without concerning that of the state of the auxiliary systems are provided. They are shown to recover several existing results in the literature. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made

New Results on Negative Imaginary Systems Theory with Application to Flexible Structures and Nano-Positioning

Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure control systems, and are often very sensitive to changes in environmental variables. These can lead to vibrational effects which limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extend by using force actuators combined with colocated measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received a great attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory. In this thesis, we provide a generalization and development of negative imaginary systems theory to include a wider class of systems. In the generalization of NI systems theory, we provide a new negative imaginary definition that allows for flexible systems with free body motion. Also, we provide a new stability condition for a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative imaginary (SNI). This general stability result captures all previous NI stability results which have been developed. This thesis also presents analytical tools for negative imaginary systems theory, which can be useful in the practical applications of the theory. Two methods that can be used for checking the negative imaginary property for a given system are presented. Also, methods for enforcing NI dynamics on mathematical system models to satisfy an NI Property are explored. A systematic method to design controllers for NI systems with guaranteed robust stability also is presented. A practical application of control system design for a three-mirror cavity locking system is presented in the end of the thesis

Técnicas de control para la mejora de la estabilidad en redes eléctricas DC con convertidores operando a potencia constante

Some electrical devices in direct current (DC) networks, under certain operating characteristics, behave as constant power loads (CPLs). This behavior is defined by presenting negative incremental resistance, compromising the stability of the supply networks. The current evolution of electrical networks favors the proliferation and increase of this unstable behavior. In this research, a stability analysis method is presented that allows verifying a sufficient condition for the local stability of linear and time invariant DC circuits with CPLs, for all possible equilibria of the system that arise when varying the power consumed by the CPLs. Furthermore, this method is expressed by linear matrix inequalities that can be verified by convex programming. It is also proposed to connect a stabilizing device in parallel, given an accessible connection port in the electrical network, which through an adequate control law, allows ensuring the local stability of linear and time-invariant DC networks with CPLs. The techniques used to obtain the control law have been H8 control, self-tuning control, and non-linear control. In addition, it is shown that the proposed solution meets the objectives established a priori, based on the worst-case regarding the system stability, through numerical results in simulation and experimental results obtained in a prototype plant designed and built within the framework of this research.Algunos dispositivos eléctricos de las redes de corriente continua (DC), bajo determinadas características de funcionamiento, se comportan como cargas de potencia constante (CPLs). Este comportamiento, se define por presentar resistencia incremental negativa, comprometiendo la estabilidad de las redes de suministro. La evolución actual de las redes eléctricas, favorece la proliferación y aumento de este comportamiento inestable. En esta investigación, se presenta un método de análisis de estabilidad que permite comprobar una condición suficiente para la estabilidad local de circuitos de DC lineales e invariantes en el tiempo con CPLs, para todos los posibles equilibrios del sistema que surgen al variar la potencia consumida por las CPLs. Además, este método se expresa en forma de desigualdades matriciales lineales que pueden ser verificadas mediante programación convexa. También se propone la conexión de un dispositivo estabilizador en paralelo, dado un puerto de conexión accesible en la red eléctrica, que mediante una ley de control adecuada, permita asegurar la estabilidad local de redes de DC lineales e invariantes en el tiempo con CPLs. Las técnicas utilizadas en la obtención de la ley de control han sido control H8, control autosintonizado y control no lineal. Además, se demuestra que la solución propuesta cumple los objetivos establecidos a priori, basados en el peor caso respecto a la estabilidad del sistema, mediante resultados numéricos en simulación y resultados experimentales obtenidos en una planta prototipo diseñada y construida en el marco de este trabajo de investigación.Postprint (published version