Coprime Factorization Research Articles

Continuous-time robust frequency regulation in isolated microgrids with decentralized fixed structure μ-synthesis and comparative analysis with PID and FOPID controllers

Isolated microgrids, which are crucial for supplying electricity to remote areas using local energy sources, have garnered increased attention due to the escalating integration of renewable energy sources in modern microgrids. This integration poses technical challenges, notably in mitigating frequency deviations caused by non-dispatchable renewables, which threaten overall system stability. Therefore, this paper introduces decentralized fixed structure robust μ-synthesis controllers for continuous-time applications, surpassing the limitations of conventional centralized controllers. Motivated by the increasing importance of microgrids, this work contributes to the vital area of frequency regulation. The research challenge involves developing a controller that not only addresses the identified technical issues but also surpasses the limitations of conventional centralized controllers. In contrast to their centralized counterparts, the proposed decentralized controllers prove more reliable, demonstrating enhanced disturbance rejection capabilities amidst substantial uncertainties, represented through normalized co-prime factorization. The proposed controllers are designed using the D-K iteration technique, incorporating performance weight filters on control actions to maintain low control sensitivity and ensure specific frequency band operation for each sub-system. Importantly, the design considers unstructured uncertainty up to 40%, addressing real-world uncertainties comprehensively. Rigorous robust stability and performance tests underscore the controller's superiority, demonstrating its robustness against elevated uncertainty levels. Robust stability is verified for all controllers, with the proposed controller showing robust stability against up to 171% of the modeled uncertainty. Notably, the controller boasts a fixed structure with lower order compared to other H-infinity controllers, enhancing its practical implementation. Comparative analyses against Coronavirus Herd Immunity Optimizer tuned Proportional-Integral-Derivative (CHIO-PID) controller and CHIO tuned Fractional-Order Proportional-Integral-Derivative (CHIO-FOPID) controller further validate the superior performance of the proposed solution, offering a significant step towards ensuring the stability and reliability of microgrid systems in the face of evolving energy landscapes.

Operator-Based Triboelectric Nanogenerator Power Management and Output Voltage Control.

In this paper, an operator-based voltage control method for TENGs is investigated, achieving output voltage tracking without compensators and uncertainty suppression using robust right coprime factorization. Initially, a comprehensive simulation-capable circuit model for TENGs is developed, integrating their open-circuit voltage and variable capacitance characteristics. This model is implemented to simulate the behavior of TENGs with a rectifier bridge and capacitive load. To address the high-voltage, low-current pulsating nature of TENG outputs, a storage capacitor switching model is designed to effectively transfer the pulsating energy. This switching model is directly connected to a buck converter and operates under a unified control strategy. A complete TENG power management system was established based on this model, incorporating an operator theory-based control strategy. This strategy ensures steady output voltage under varying load conditions without using compensators, thereby reducing disturbances. Simulation results validate the feasibility of the proposed TENG system and the efficacy of the control strategy, providing a robust framework for optimizing TENG energy harvesting and management systems with significant potential for practical applications.

Filtered Right Coprime Factorization and Its Application to Control a Pneumatic Cylinder

The main objective of this research is to expand right coprime factorization based on operator theory in nonlinear systems. A novel method for right coprime factorization is proposed by introducing an operator that can deform the system’s response into an arbitrary shape. This enables the design of control systems that are highly effective against noise. As an application, we use a pneumatic stage. The effectiveness of this method is verified through simulations and real-world experiments.

Performance-Based Hierarchical Fault-Tolerant Control for Closed-Loop Systems With Multiplicative Faults: A Data-Driven Design Method.

Fault-tolerant control (FTC) is vital for the safety and reliability of automatic systems. Most of the existing FTC methods are developed for open-loop systems subject to additive faults, regardless of the widely present control loops and multiplicative faults within systems. In this article, a performance-based FTC strategy is proposed for the closed-loop systems with multiplicative faults. Considering the high efforts in modeling complex systems, the proposed FTC strategy is realized in the data-driven context. Specifically, a nominal feedback-feedforward controller is first established for the fault-free systems. By selecting the system stability and reference tracking behavior as the key performance indices, two performance evaluators are constructed to detect and classify the occurred multiplicative faults based on the fault-induced effects on the system performance. Then, with the aid of the coprime factorization technique, the multiplicative faults, in the form of additive perturbations to the system coprime factors, are estimated utilizing the closed-loop process data. Furthermore, based on the fault knowledge, a hierarchical fault-tolerant tracking controller is developed according to the levels of system performance degradations, where the functional controller parameters are reconfigured with different priorities. Finally, case studies are provided to validate the effectiveness of the proposed method.

Right Coprime Factorization-Based Simultaneous Control of Input Hysteresis and Output Disturbance and Its Application to Soft Robotic Finger

In a nonlinear control system, hysteresis exists usually as common characteristics. In addition, external output disturbances like modelling error, machine friction and so on also occur frequently. Both of them are considered to cause instability and unsatisfactory performance. In this paper, a practical nonlinear control system design is proposed so as to achieve the simultaneous control of input hysteresis and output disturbance. The system is based on RCF (right coprime factorization theory). Additionally, the proposed design has been applied to a soft robotic finger system and the results of simulations and practical experiments are exhibited, which show the effectiveness of the proposed system.

Coprime Factor Model Reduction of Multidimensional Systems

In this work, the right coprime factor for balancing and truncating unstable systems is extended to parameter-varying multidimensional systems employing recent works on coprime factor model reduction of one-dimensional uncertain systems. Since the balanced truncation method cannot be applied directly to unstable systems, state feedback gains should be computed and incorporated in order to stabilize the given system and to be able to apply the balanced truncation technique via defining the so-called stable coprime factor, as coprime factorization overcomes the stability condition required for model reduction. Parameter-dependent state feedback and parameter-dependent Gramians are considered in this work yielding less conservative techniques. In addition, the Gramians are defined as block diagonal matrices, which are partitioned according to the structure of the multidimensional systems. The application to a simulation example demonstrates the applicability and validity of the proposed reduction approach leading to small error bounds between the full and the reduced models.

Robustness Analysis for Sundry Disturbed Open Loop Dynamics Using Robust Right Coprime Factorization

In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback loop. In such a control framework, the nominal plant is firstly right coprime factorized and a feed forward and a feedback controllers are designed based on Bezout identity to ensure the overall stability. Subsequently, considering the sundry disturbed open loop dynamics, a new condition formulated as a disturbed Bezout identity is put forward to achieve the closed loop stability of the system, even in the presence of disturbances existing in sundry open loops, where in the feedback loop a disturbed identity operator is defined. This approach guarantees the system robustness if a specific inequality condition is satisfied. And, it should be noted that the proposed approach is applicable to both linear and nonlinear systems with sundry disturbed open loop dynamics. Simulations demonstrate the effectiveness of our methodology.

Observer‐based residual‐driven dynamic compensation strategy for performance improvement of grid‐forming inverter

SummaryGrid‐forming (GFM) inverters offer stable frequency support for microgrid systems, even in the absence of synchronous generators. However, the GFM inverters have low inertia and vulnerability to system uncertainties and external disturbances. The conventional dual‐loop proportional integral (PI) control strategy, while widely used for its simplicity and robustness, suffers from poor dynamic performance. Motivated by this, this paper presents an observer‐based residual‐driven dynamic compensation (RDDC) strategy based on the coprime factorization technic and the Youla parameterization theory to achieve the primary control of the GFM inverter. The observer‐based RDDC strategy comprises four components: a PI controller for tracking control, a linear quadratic regulator (LQR) controller for dynamic adjustment, a residual generator based on the Kalman filter for state estimation and residual generation, and a residual compensation controller designed using model matching theory and solved through linear matrix inequality (LMI) methods for disturbance suppression. Simulation and experiment results consistently demonstrate that the observer‐based RDDC strategy ensures system robustness, enhances the dynamic and steady‐state performance of the GFM inverter system, and strengthens the ability of the GFM inverter to suppress disturbances.

Operator-based adaptive robust control for uncertain nonlinear systems by combining coprime factorization and fuzzy control method1

In this paper, the robust stability of nonlinear system with unknown perturbation is considered combining operator-based right coprime factorization and fuzzy control method from the input-output view of point. In detail, fuzzy logic system is firstly combined with operator-based right coprime factorization method to study the uncertain nonlinear system. By using the operator-based fuzzy controller, the unknown perturbation is formulated, and a sufficient condition of guaranteeing robust stability is given by systematic calculation, which reduces difficulties in designing controller and calculating inverse of Bezout identity. Implications of the results related to former results are briefly compared and discussed. Finally, a simulation example is shown to confirm effectiveness of the proposed design scheme of this paper.

Optimal Error Quantification and Robust Tracking under Unknown Upper Bounds on Uncertainties and Biased External Disturbance

This paper addresses a problem of optimal error quantification in the framework of robust control theory in the 𝓁1 setup. The upper bounds of biased external disturbance and the gains of coprime factor perturbations in a discrete-time linear time invariant SISO plant are assumed to be unknown. The computation of optimal data-consistent upper bounds under a known bias of external disturbance has been simplified to linear programming. This allows for the computation of optimal estimates in real-time and their application to achieve optimal robust steady-state tracking even when facing an unknown bias in the external disturbance. The presented results have been illustrated through computer simulations.

Fully actuated system approach for high-order linear systems with high-order input derivatives

This paper presents a frequency-domain method for transforming a class of high-order linear systems with high-order derivatives of input into the corresponding full-actuated system (FAS) models. Based on the coprime factorisation theory, it is proved that under the condition that the factorisation polynomial matrix is column reduced, a direct method can be given to transform the original system into an FAS model. Regard the non-square problem faced by the direct method, two different methods are given to present the explicit expression of the transformed state. The first one is given for the case the order of system is higher than that of input, in which the system is transformed into the corresponding observability canonical form. The other one considers the opposite case, in which the model reduction method is adopted. Three numerical examples are given to illustrate the proposed theories.

Adaptive optimal ℓ∞‐induced robust stabilization of minimum phase SISO plant under bounded disturbance and coprime factor perturbations

AbstractThis paper addresses the problem of optimal robust stabilization of a discrete‐time minimum‐phase plant in the framework of robust control theory in the setup and under poor a priori information. Coefficients of the transfer function of the plant nominal model with stable zeros are unknown and belong to a known bounded polyhedron in the space of coefficients. The gains of coprime factor perturbations of the plant and the upper bound of external disturbance are also unknown. The problem under consideration is to design adaptive controller that minimizes, with the prescribed accuracy, the worst‐case asymptotic upper bound of the output. Solution of the problem is based on set‐membership estimation of unknown parameters and treating the control criterion as the identification criterion. A hard nonconvex problem of on‐line computation of optimal estimates is reduced, under additional nonrestrictive assumption, to a linear‐fractional programming via a nonlinear transformation of estimated parameters. Despite the non‐identifiability of the unknown parameters, the proposed adaptive controller guarantees, with the prescribed accuracy, the same optimal asymptotic upper bound of the output of adaptive system as the optimal controller for the plant with known parameters. In addition to the optimality of adaptive control, the proposed solution provides on‐line verification/validation of current estimates and a priori assumptions.

A Generalized Anti-windup Approach for Internal Model Control

In this paper, we develop a generalized anti-windup synthesis procedure for saturating internal model control (IMC). Within this framework, the anti-windup design problem reduces to a search over a class of left coprime factorization of the linear internal model controller. An appropriate anti-windup is constructed by solving a set of linear matrix inequalities. As compared to other static anti-windup constructions in the literature, the IMC anti-windup is always feasible with guaranteed closed-loop stability and performance. Using a simulation example, we benchmark the behaviour of the constructed anti-windup with existing state-of-the-art methods.

Robust model reference tracking control for high‐order descriptor linear systems subject to parameter uncertainties

AbstractThis paper investigates the robust model reference tracking control problem for high‐order descriptor linear systems (HODLS) subject to norm‐bounded parameter uncertainties. The problem is divided into two subproblems: a robust proportional plus derivative state feedback stabilization (RPPDSFS) problem and a robust feed‐forward compensation (RFFC) problem. In the presence of uncertainties, by using the controller consisting of RPPDSFS part and RFFC part, all the signals of the closed‐loop system are bounded. The latter is equivalent to seeking three coefficient matrices such that a series of linear matrix equations are satisfied, and simultaneously the effect caused by uncertainties is minimized. Based on the right coprime factorization, the RFFC problem is further converted into a minimization problem with a quadratic performance index and some linear constraints. Thereafter, a linear matrix equation that determines the optimal solution is achieved. The restrictive condition that the coefficient matrix of the augmented reference system is non‐defective is relaxed, which reduces the conservativeness. A numerical example and the application of a three‐axis dynamic flight motion simulator manifest the effectiveness and the practicability of the proposed algorithm.

Robust control design for suspension system in LIGO

In this article, we present a relatively new state-space methodology, called synthesis, to design a robust controller for a subsystem of the Laser Interferometer Gravitational Wave Observatory. In particular, we examine the quadruple pendulum subsystem, which is simplified for the purpose of the study to a two wire simple pendulum system. Structured and unstructured uncertainties are considered in the modelling, though the structured ones are incorporated in the unstructured framework based on a design philosophy presented in the literature. Incorporating weighting functions that emphasize the spectral band of the uncertainties as well as the band in which disturbances are to be attenuated, we design a robust control law using the coprime factorization technique of the paradigm.

Optimal tracking performance of network time‐delay systems with multiple constraints

AbstractThis paper investigates the optimal tracking performance based on the codec, colored noise, and bandwidth constraints in the network time‐delay systems. Choosing the optimal two‐degree‐of‐freedom compensator and the dominant expression of the system optimal tracking performance is obtained after Youla parameterization by using the coprime factorizations method and spectral decomposition technology. The result shows that the tracking performance of the system is determined by the nonminimum phase zero and unstable pole of the controlled plant. Moreover, codec, colored noise, and bandwidth constraints can also significantly impact the performance. Finally, two simulation examples prove the accuracy of the theory.

H‐infinity loop shaping control of wireless power transfer system based on generalized state space averaging model

SummaryThe transmission distance and the output load of wireless power transfer (WPT) system are particularly prone to change, which results in higher requirements for the robustness and dynamic performance of the control. In order to reduce the influence of these characteristics, an H∞ loop shaping controller based on generalized state space averaging (GSSA) method is proposed in this paper. Based on the GSSA model, an appropriate compensation function is constructed for the WPT system to the target loop transfer function. The nominal coprime factorization of WPT system is carried out, and on this basis, H∞ controller is achieved by Riccati equation to minimize the infinite norm from disturbance to error output. The designed controller meets the requirements of robust. Furthermore, the obtained high‐order controller is discretized by a reasonable discretization method, and the details and limitations of its practical application are discussed based on the hardware computational capacity. The experimental results show that the proposed controller has better robustness and dynamic performance and flexibility than the mixed sensitivity H∞ controller and the Z‐N PI controller.

An Intelligent Robust Operator-Based Sliding Mode Control for Trajectory Tracking of Nonlinear Uncertain Systems

This paper investigates the problem of trajectory tracking control in the presence of bounded model uncertainty and external disturbance. To cope with this problem, we propose a novel intelligent operator-based sliding mode control scheme for stability guarantee and control performance improvement in the closed-loop system. Firstly, robust stability is guaranteed by using the operator-based robust right coprime factorization method. Secondly, in order to further achieve the asymptotic tracking and enhance the responsiveness to disturbance, a finite-time integral sliding mode control law is designed for fast convergence and non-zero steady-state error in accordance with Lyapunov stability analysis. Lastly, the controller’s parameters are automatically adjusted by the proved stabilizing particle swarm optimization with the linear time-varying inertia weight, which significantly saves tuning time with a remarkable performance guarantee. The effectiveness and efficiency of the proposed method are verified on a highly nonlinear ionic polymer metal composite application. The extensive numerical simulations are conducted and the results show that the proposed method is superior to the state-of-the-art methods in terms of tracking accuracy and high robustness against disturbances.

Ripple Voltage Suppression of Dual Active Bridge Converter in Electric Locomotive Using H∞ Loop Shaping Control Method

The next-generation electric locomotive drive system has the problem of double grid frequency ripple voltage in the DC side of a single-phase PWM rectifier. In this paper, the intermediate stage dual-active-bridge DC-DC converter is studied, and an H∞ loop shaping ripple voltage suppression control strategy is proposed to restrain the harmonic voltage and optimize the dynamic and robust performance of the closed-loop system. Firstly, the mathematical model of the DAB closed-loop control system with fluctuating input voltage is established. Then, based on the H∞ loop shaping control theory, the dynamic performance, the ability of ripple voltage suppression, and the robust performance of the system are analyzed in the frequency domain. According to the results, a reasonable compensator is selected to reconstruct the magnitude frequency characteristic curve of the nominal system. After reconstruction, the solution of the robust stability margin of the corrected object is transformed into the H∞ robust control problem under the coprime factor uncertainty. The suboptimal H∞ robust controller is solved to obtain the best robustness boundary of the constructed system. Finally, the H∞ loop shaping ripple voltage suppression controller is obtained by combining the H∞ controller with the compensator. The experimental environment is set up in MATLAB-Simulink software. Compared with the traditional voltage loop control method and the input voltage feedforward control method, the dynamic performance of the proposed control strategy is slightly improved, and while within the allowable robust boundaries of the system, it has better ripple voltage suppression performance. When the system magnitude frequency curve is outside the tolerant robust stability margin, the ripple voltage suppression performance fails, but the harmonic content of output voltage is still far less than the other two control methods. The experiment results also prove the correctness of the theoretical analysis and that the proposed control method is effective and proper.

Frequency-Domain Estimation of a Transfer Matrix of an Uncommon Quadrotor in Hover

In this article, system identification of an uncommon quadrotor in hover is discussed. Two counter-rotating big rotors on the longitudinal axis and two counter-rotating small tilt rotors on the lateral axis form this quadrotor configuration. First, the nonlinear dynamic model of this vehicle is derived from Newton–Euler formulation. Next, using the approximate linear hover model and the suitable rotor mixing matrix for axes decoupling, simplified attitude dynamics of this quadrotor are obtained. However, the effects of sensor delays, flexible modes of the airframe, and inexact decoupling are not visible in this diagonal model, which is mainly based on physical principles. Therefore, a system identification method is used to obtain a more accurate model, which is required for control design. Hence, during parametric identification of nominal model coprime factors, a small robust control criterion is targeted. Then, the frequency response function of the proposed quadrotor prototype in hover is obtained. Next, the linear parametric model of the vehicle is estimated by solving optimally conditioned least squares and subsequent Gauss–Newton problems. After that, using validation-based uncertainty quantification, the uncertain model set is constructed around the estimated coprime factors. The resulting model set and its robust control relevance are depicted.