Stability of closed-loop fractional-order systems and definition of damping contours for the design of controllers

Fractional complex order integrator has been used since 1991 for the design of robust control-systems. In the CRONE control methodology, it permits the parameterization of open loop transfer function which is optimized in a robustness context. Sets of fractional order integrators that lead to a given damping factor have also been used to build iso-damping contours on the Nichols plane. These iso-damping contours can also be used to optimize the third CRONE generation open-loop transfer function. However, these contours have been built using non band-limited integrators, even if such integrators reveal to lead to unstable closed loop systems. One objective of this paper is to show how the band-limitation modifies the left half-plane dominant poles of the closed loop system and removes the right half-plane ones. It is also presented how to obtain a fractional order open loop transfer function with a high phase slope and a useful frequency response. It is presented how the damping contours can be used to design robust controllers, not only CRONE controllers but also PD and QFT controllers

Fractional robust control with iso-damping property

This article deals with the problem of the reduction of structural vibrations with isodamping property. The proposed methodology is based on: - a contour defined in the Nichols plane and significant of the damping ratio of the closed-loop response - a robust control method that uses fractional order integration. The methodology is applied to an aircraft wing model made with a beam and a tank whose different levels of fillings are considered as uncertainties

Multi-objective LQR with Optimum Weight Selection to Design FOPID Controllers for Delayed Fractional Order Processes

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.An optimal trade-off design for fractional order (FO)-PID controller is proposed in this paper with a Linear Quadratic Regulator (LQR) based technique using two conflicting time domain control objectives. The deviation of the state trajectories and control signal are automatically enforced by the LQR. A class of delayed FO systems with single non-integer order element has been controlled here which exhibit both sluggish and oscillatory open loop responses. The FO time delay processes are controlled within a multi-objective optimization (MOO) formulation of LQR based FOPID design. The time delays in the FO models are handled by two analytical formulations of designing optimal quadratic regulator for delayed systems. A comparison is made between the two approaches of LQR design for the stabilization of time-delay systems in the context of FOPID controller tuning. The MOO control design methodology yields the Pareto optimal trade-off solutions between the tracking performance for unit set-point change and total variation (TV) of the control signal. Numerical simulations are provided to compare the merits of the two delay handling techniques in the multi-objective LQR-FOPID design, while also showing the capability of load disturbance suppression using these optimal controllers. Tuning rules are then formed for the optimal LQR-FOPID controller knobs, using the median of the non-dominated Pareto solution to handle delays FO processes

Stability Analysis and Decentralized Control of Coupled Oscillators with Feedback Delays

Most dynamic systems do not react instantaneously to actuation signals. The temporal evolution of some others is based on retarded communications or depends on information from the past. In such cases, the mathematical models used to describe these systems must include information about the past dynamics of the states. These models are often referred to as delay or retarded systems. Delays could channel energy in and out of a system at incorrect time intervals producing instabilities and rendering controllers\u27 performance ineffective. The purpose of this research is two folds. The first investigates the effect of inherent system delays on the stability of coupled oscillators subjected to decentralized control and the second studies the prospectus of augmenting the delay into a larger delay period that could actually stabilize the coupled system and enhance its damping characteristics. Towards these ends, a system of two linearly-coupled oscillators with decentralized delayed-proportional feedback is considered. A comprehensive linear stability analysis is utilized to generate maps that divide the controllers\u27 gain and delay domain into regions of stability for different coupling values. These maps are then used to draw definite conclusions about the effect of coupling on the stability of the closed-loop in the presence of delay. Once the stability maps are generated, the Lambert-W function approach is utilized to find the stability exponents of the coupled system which, in turn, is used to generate damping contours within the pockets of stability. These contours are used to choose gain-delay combinations that could augment the inherent feedback delays into a larger delay period which can enhance the damping characteristics and reduce the system settling time significantly. An experimental plant comprised of two mass-spring-damper trios coupled with a spring is installed to validate the theoretical results and the proposed control hypothesis. Different scenarios consisting of different gains and delays are considered and compared with theoretical findings demonstrating very good agreement. Furthermore, the proposed delayed-proportional feedback decentralized controller is tested and its ability to dampen external oscillations is verified through different experiments. Such a research endeavor could prove very beneficial to many vital areas in our life. A good example is that of the coupled system of the natural and artificial cardiac pacemakers where the natural pacemaker represents a rhythmic oscillating system and the coupled artificial pacemaker provides a stabilizing signal through a feedback mechanism that senses the loss in rhythm. In this system, even the minute amount of delay in the sensing-actuating could prove very detrimental. The result of this research contributes to the solution of this and similar problems

Implementation of Delayed-Feedback Controllers on Continuous Systems and Analysis of their Response under Primary Resonance Excitations

During the last three decades, a considerable amount of research has been directed toward understanding the influence of time delays on the stability and stabilization of dynamical systems. From a control perspective, these delays can either have a compounding and destabilizing effect, or can actually improve controllers\u27 performance. In the latter case, additional time delay is carefully and deliberately introduced into the feedback loop so as to augment inherent system delays and produce larger damping for smaller control efforts. While delayed-feedback algorithms have been successfully implemented on discrete dynamical systems with limited degrees of freedom, a critical issue appears in their implementation on systems consisting of a large number of degrees of freedom or on infinite-dimensional structures. The reason being that the presence of delay in the control loop renders the characteristic polynomial of the transcendental type which produces infinite number of eigenvalues for every discrete controller\u27s gain and time delay. As a result, choosing a gain-delay combination that stabilizes the lower vibration modes can easily destabilize the higher modes. To address this problem, this dissertation introduces the concept of filter-augmented delayed-feedback control algorithms and applies it to mitigate vibrations of various structural systems both theoretically and experimentally. In specific, it explores the prospect of augmenting proper filters in the feedback loop to enhance the robustness of delayed-feedback controllers allowing them to simultaneously mitigate the response of different vibration modes using a single sensor and a single gain-delay actuator combination. The dissertation goes into delineating the influence of filter\u27s dynamics (order and cut-off frequency) on the stability maps and damping contours clearly demonstrating the possibility of effectively reducing multi-modal oscillations of infinite-dimensional structures when proper filters are augmented in the feedback loop. Additionally, this research illustrates that filters may actually enhance the robustness of the controller to parameter\u27s uncertainties at the expense of reducing the controller\u27s effective damping. To assess the performance of the proposed control algorithm, the dissertation presents three experimental case studies; two of which are on structures whose dynamics can be discretized into a system of linearly-uncoupled ordinary differential equations (ODEs); and the third on a structure whose dynamics can only be reduced into a set of linearly-coupled ODEs. The first case study utilizes a filter-augmented delayed-position feedback algorithm for flexural vibration mitigation and external disturbances rejection on a macro-cantilever Euler-Bernoulli beam. The second deals with implementing a filter-augmented delayed-velocity feedback algorithm for vibration mitigation and external disturbances rejection on a micro-cantilever sensor. The third implements a filter-augmented delayed-position feedback algorithm to suppress the coupled flexural-torsional oscillations of a cantilever beam with an asymmetric tip rigid body; a problem commonly seen in the vibrations of large wind turbine blades. This research also fills an important gap in the open literature presented in the lack of studies addressing the response of delay systems to external resonant excitations; a critical issue toward implementing delayed-feedback controllers to reduce oscillations resulting from persistent harmonic excitations. To that end, this dissertation presents a modified multiple scaling approach to investigate primary resonances of a weakly-nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear feedback with small control gains; this effort proposes an approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified procedure lumps the delay state into unknown linear damping and stiffness terms that are function of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the Method of Multiple Scales to match that obtained directly by solving the forced linear problem. Through several examples, this research examines the validity of the modified procedure by comparing its results to solutions obtained via a Harmonic Balance approach demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses