Multivariable Nonlinear Systems Research Articles

Stability Analysis of Systems with Fuzzy PI Controllers Applied to Electric Drives

This paper analyzes the stability of fuzzy control systems with applications for electric drives. Ensuring the stability of these systems is a necessity in practice. The purpose of the study is the analysis of the dynamic characteristics of the speed control systems of electric drives based on fuzzy PI controllers in the context of performing stability analyses, both internal and input–output, finding solutions to stabilize these systems and provide guidance on fuzzy regulator design. The main methods of treatment applied are as follows: framing the control system in the theory of stability of multivariable non-linear systems, application of Lyapunov’s theory, performing an input–output stability analysis, and verification of the stability domain. The article presents the conditions for correcting the fuzzy controller to ensure internal and external stability, determines the limits of the stability sector, and gives indications for choosing the parameters of the controller. The considerations presented can be applied to various structures for regulating the speed of electric drives which use various PI fuzzy controllers.

Control design for a class of multivariable nonlinear system with uncertain control direction: A laser cladding case study

In this work, the design of a controller for a multiple-input multiple-output (MIMO) nonlinear system with uncertain control direction is proposed. A formal Lyapunov stability analysis is presented for the design of the MIMO controller by considering the system with and without parametric uncertainties. The design is applied to an additive manufacturing technology known as Laser Cladding (LC). The proposed MIMO controller is composed of a feedback linearization term (aimed at canceling the known nominal dynamics) and a sliding mode term with the boundary layer technique (aimed at compensating the system uncertainties). The proposed control scheme was tested via simulations by considering tracks of two different powder feedstock, Ti6Al4V and IN718. Simulation results confirm that the control objective is achieved under parametric uncertainties, including variations up to 40% of the convection coefficient. Moreover, the control law is easy to implement and the control signal has a continuous behavior precluding any high-frequency oscillation (chattering effect).

Particle Swarm Optimization-Based Fuzzy PID Controller for Stable Control of Active Magnetic Bearing System

In this paper, the application of particle swarm optimization (PSO) algorithm based fuzzy PID control for the stable control of an active magnetic bearing (AMB) system is studied. Active magnetic bearings are known to be highly nonlinear multivariable systems. An AMB system is used in motors, turbines and various other machineries in different industries to provide an active suspension to the rotor shafts. The heuristic PSO algorithm is applied to optimize the parameters of the PID controller offline. The fuzzy PID controller based on PSO algorithm is designed to adjust the control parameters of AMB system online. The comparison of controlled responses of closed-loop systems resulting from the use of conventional PID, Incomplete differential PID and particle swarm optimization-based fuzzy logic PID control strategies is discussed. The proposed controller performance is superior as compared to PID and incomplete differential PID controllers in improving the system response of an AMB under normal conditions as well as under external disturbances.

State feedback linearization using block companion similarity transformation

In this research work, a new method is proposed for linearizing a class of nonlinear multivariable system; where the number of inputs divides exactly the number of states. The idea of proposed method consists in representing the original nonlinear system into a state-dependent coefficient form and applying block similarity transformations that allow getting the linearized system in block companion form. Because the linearized system’s eigenstructure can determine system performance and robustness far more directly and explicitly than other indicators, the given class multivariable system is chosen. Examples are used to illustrate the application and show the effectiveness of the given approach.

RcdMathLib: An Open Source Software Library for Computing on Resource-Limited Devices.

We developped an open source library called RcdMathLib for solving multivariate linear and nonlinear systems. RcdMathLib supports on-the-fly computing on low-cost and resource-constrained devices, e.g., microcontrollers. The decentralized processing is a step towards ubiquitous computing enabling the implementation of Internet of Things (IoT) applications. RcdMathLib is modular- and layer-based, whereby different modules allow for algebraic operations such as vector and matrix operations or decompositions. RcdMathLib also comprises a utilities-module providing sorting and filtering algorithms as well as methods generating random variables. It enables solving linear and nonlinear equations based on efficient decomposition approaches such as the Singular Value Decomposition (SVD) algorithm. The open source library also provides optimization methods such as Gauss–Newton and Levenberg–Marquardt algorithms for solving problems of regression smoothing and curve fitting. Furthermore, a positioning module permits computing positions of IoT devices using algorithms for instance trilateration. This module also enables the optimization of the position by performing a method to reduce multipath errors on the mobile device. The library is implemented and tested on resource-limited IoT as well as on full-fledged operating systems. The open source software library is hosted on a GitLab repository.

The identification of fractional order systems by multiscale multivariate analysis

Abstract The recognition of complex systems plays a vital role in system dynamics. To distinguish chaotic systems and stochastic processes, the complex-entropy causality plane (CECP) was proposed and received considerable attention. However, few methods have been applied to the identification of fractional order chaotic systems. In this paper, we propose multiscale multivariate fractional dispersion entropy (MMFDE) and MMFDE plane to study the structural complexity of multivariate nonlinear systems. Unlike the ordinal pattern in CECP, the dispersion pattern is used in MMFDE plane to solve the computational burden in fractional order systems. Moreover, the fractional order α in MMFDE can be used to adjust the ability of MMFDE plane to distinguish between different systems. Through simulation data, we find that the MMFDE plane can classify not only chaotic systems and stochastic processes but also different fractional order chaotic systems even if adding noise to the signals. It is then applied to financial time series and epileptic EEG recordings. The results show that the MMFDE plane can identify developed and emerging markets. For epileptic EEG recordings, both the EEG subbands and the original data without filtering can be used to distinguish different states of healthy subjects and patients with epilepsy. The coarse-graining process is also discussed to demonstrate the consequences further.

Improved Metaheuristic Optimization Algorithm Applied to Hydrogen Fuel Cell and Photovoltaic Cell Parameter Extraction

As carriers of green energy, proton exchange membrane fuel cells (PEMFCs) and photovoltaic (PV) cells are complex and nonlinear multivariate systems. For simulation analysis, optimization control, efficacy prediction, and fault diagnosis, it is crucial to rapidly and accurately establish reliability modules and extract parameters from the system modules. This study employed three types of particle swarm optimization (PSO) algorithms to find the optimal parameters of two energy models by minimizing the sum squared errors (SSE) and roots mean squared errors (RMSE). The three algorithms are inertia weight PSO, constriction PSO, and momentum PSO. The obtained calculation results of these three algorithms were compared with those obtained using algorithms from other relevant studies. This study revealed that the use of momentum PSO enables rapid convergence (under 30 convergence times) and the most accurate modeling and yields the most stable parameter extraction (SSE of PEMFC is 2.0656, RMSE of PV cells is 8.839 · 10−4). In summary, momentum PSO is the algorithm that is most suitable for system parameter identification with multiple dimensions and complex modules.

A New Active Fault Tolerant Control System: Predictive Online Fault Estimation

This study presents a new approach for active fault-tolerant controller (FTC) design for constrained nonlinear multi-variable systems. The proposed approach utilize the nonlinear model predictive controller (NMPC) and fault estimation method which is on basis of extended kalman filters (EKFs). The deficiency of actuators and sensors and also the plant states measurement errors are estimated by the suggested approach. A supervisor unit using the fault information and fault modeling per sampling time, corrects the predictor model of the controller and compensates actuator and sensor faults in control system. Furthermore, by the presented feedback compensation, the robustness of the designed method against plant faults and uncertainties is ensured. The important advantages of the proposed method are: (1) The suggested FTC scheme based on NMPC leads to calculate more accurate control action than MPC in nonlinear processes, (2) it is comprehensive in fault accommodation point of view because it is able to compensate all types of faults in control systems simultaneously, (3) it has low computational cost because of using NMPC by analytical solution, (4) it can handle control and states constraints to prevent of actuator saturations and unsafe situations, (5) the simplicity and effectiveness of the designed FTC scheme for real applications is more significant. Simulation results on continuous stirred tank reactor process verifies the superiority and capability of the designed approach.

A New Proposal for the Closed-Loop Orientation Control of a Windfloat Turbine System

It is believed that offshore wind power will play an important role in fulfilling the environmental commitments made in the Paris Agreement signed in 2015. The use of floating solutions in offshore wind power is gaining prominence, since it allows the expansion of these technologies to a large number of regions in which this has not been technically feasible until now. One of the current issues that needs to be studied in detail as regards offshore wind structures are the oscillations produced by the waves and the wind that could cause important problems in the turbine such as the reduction of the efficiency and performance or the appearance of structural damages owing to undesired loads on the blades, among others. This signifies that the inclination of the structure must not exceed a certain small value in the operating phase. This paper, therefore, presents a simple dynamic modeling and a nonlinear multivariable control model-based system for a Windfloat turbine whose main function is to stabilize the system and to keep it vertical in order to improve its operation and, therefore, maintain the offshore wind turbine in the optimal conditions required to harness energy from the wind. The effectiveness of both the dynamic model and the proposed control algorithm has been demonstrated by developing various simulations in a MATLAB-Orcaflex environment, in which the Windfloat structure proved to have very small vertical deviations when wind impinges over the rotor, despite the undesirable effects caused by parametric uncertainties (e.g., the effect of the anchoring cables) in the system and unmodeled unmatched perturbations (such as the effects of the wind and the waves on the Windfloat turbine).

Sensor Systems for Greenhouse Microclimate Monitoring and Control: a Review

Sensors are the primary component of a monitoring and control system. Effective monitoring and control of the microclimatic environment in a greenhouse is the key necessity for protecting crops from adverse environments. Moreover,the greenhouse microclimate is influenced by various factors. In the large-scale greenhouse facilities, several sensors and actuators are needed to control the system. Manual monitoring and control of such a large and complex system is labor-intensive and impractical. Therefore, an automatic monitoring and control system in the greenhouse becomes indispensable. In addition, microclimatic parameters such as temperature, humidity, and solar irradiance in the greenhouse are non-linearly interlinked, thereby forming a non-linear multivariate system. Thus, an appropriately designed sensor system is needed for monitoring and controlling the greenhouse microclimate. Research articles on greenhouse microclimate monitoring and control published in the last 6 years were considered. The sensor devices and technologies applied to control particular environmental parameters in the greenhouse and their key achievements were systematically reviewed. In addition, different approaches to determine the optimum number of sensors and their placement inside the greenhouse were investigated. It was found that spatially installed sensor devices above the plant height reflect the actual information of the environment getting by plant. Furthermore, both hardware and software-based sensing techniques control the greenhouse microclimate optimally. The proper positioning of sensors and their protection from harsh environmental factors is also essential. It can be concluded that modern sensor devices and systems are driving the greenhouse monitoring and control system toward an intelligent, real-time, remotely accessible, and fully automatic system.

H∞ tracking control for nonlinear multivariable systems using wavelet-type TSK fuzzy brain emotional learning with particle swarm optimization

This paper studies the H∞ tracking control for uncertain nonlinear multivariable systems. We propose a control strategy, which combines the adaptive wavelet-type Takagi-Sugeno-Kang (TSK) fuzzy brain emotional learning controller (WTFBELC) and the H∞ robust tracking compensator. As for the adaptive WTFBELC, it is a main controller designed to mimic the ideal controller. The proposed WTFBELC is to obtain much better ability of handling nonlinearities and uncertainties, but the proposed H∞ robust tracking compensator is to compensate the residual error between the adaptive WTFBELC and the ideal controller. Furthermore, the optimal learning rates of the adaptive WTFBELC are searched quickly by using the particle swarm optimization (PSO) algorithm, and the parameter updated laws are derived based on the steepest descent gradient method. The robust tracking performance of this novel control scheme is guaranteed based on Lyapunov stability theory. The mass-spring-damper mechanical system and the three-link robot manipulator, are used to verify the effectiveness of the proposed adaptive PSO-WTFBELC H∞ control scheme.

Filtering adaptive tracking control for uncertain switched multivariable nonlinear systems

ABSTRACT This paper synthesizes a filtering adaptive tracking control method for uncertain switched multivariable nonlinear systems. The multivariable nonlinear systems under consideration have both matched and mismatched uncertainties, which satisfy the semiglobal Lipschitz condition. A piecewise constant adaptive law will generate adaptive parameters which represent the uncertainty approximations by solving the error dynamics between the real system and the state predictor with the neglection of unknowns, whereas a filtering control law is designed to handle the nonlinear uncertainties and deliver a good tracking performance. The matched uncertainties are cancelled directly by adopting their opposite, whereas a dynamic inversion of the system is required to eliminate the effect of the mismatched uncertainties on the output. The uniform performance bounds are derived for the states and control signals compared with the virtual reference system which defines the best performance that can be achieved by the closed-loop system.

Recursive filtering adaptive neural fault-tolerant control for uncertain multivariable nonlinear systems

This paper synthesizes a recursive filtering adaptive neural fault-tolerant controller for uncertain multivariable nonlinear systems. The proposed control scheme adopts an estimation/cancellation strategy to deal with uncertainties and/or disturbances. The nonlinear uncertainties are approximated by a Gaussian radial basis function (GRBF)-based neural network incorporated with a piecewise constant adaptive law, where the adaptive law will generate adaptive parameters by solving the error dynamics between the real system and the state predictor with the neglection of unknowns, and recursive least squares (RLS) is applied to distribute the total uncertainty estimates into matched and mismatched components. The cooperation of GRBF learning method and piecewise constant adaptive law relaxes the stringent constraint on the hardware CPU speed and achieves fast adaptation. The filtering control law delivers a satisfactory performance with guaranteed robustness. Two numerical examples are provided to illustrate the effectiveness of the proposed control architecture via comparisons.

A three-stage PSO-based methodology for tuning an optimal PD-controller for robotic arm manipulators

Six degrees-of-freedom (DoF) robot manipulators are dynamically coupled and highly nonlinear multi-variable systems. Calculating the optimal joint angles from the operational space (i.e. inverse kinematics) and designing the optimal joint controller parameters are two important research topics. This paper proposes a 3-stage Particle Swarm Optimization (PSO)-based methodology for solving the inverse kinematics and optimizing the controller parameters. In the first stage, a PSO algorithm solves the inverse kinematics problem by minimizing a multi-objective cost function in the operational space (i.e. the error in the end-effector’s pose) and therefore finds the optimal joint angles. In the second stage, polynomial functions generate the desired trajectory between the initial and final poses in the joint space. Finally, a second PSO algorithm tunes six proportional-derivative (PD) controllers, one for each joint, to track the desired trajectory by minimizing another multi-objective cost function in the joint space. Two case studies, based on six DoF Puma 560, validate the performance of the proposed methodology. Simulation results show that the proposed 3-stage methodology provides fast and accurate results as the PSO algorithms are effective in solving the inverse kinematics problem and tuning the optimal PD parameters.

Identification of the long-term variations of groundwater and their governing factors based on hydrochemical and isotopic data in a river basin

A clear identification of the hydrochemical processes in groundwater systems is critical for the management and protection of alternative water resources. The Dagu River Basin (DRB) has become one of the most concerning areas due to the typical geographical conditions and intensive human disturbance. In this study, the spatio-temporal variations of groundwater hydrochemical characteristics were discriminated based on the self-organizing maps (SOM) clustering results from 2001 to 2017. Simultaneously, their main governing factors were also identified according to the composite properties of hydrogen and oxygen isotopic evidence, hydrochemical facies evolution diagram (HFE-D), chloro-alkaline indices (CAI) values, and the enrichment of nitrogen. Analysis results indicated that the long-term evolution of hydrochemical signatures basically conformed to the trends of rising in the dry season and falling in the rainy season. Total dissolved solids (TDS) fluctuated from 187.90 mg/L to 2294.81 mg/L, with an average of 881.77 mg/L. Moreover, the nitrate concentrations in the river basin varied significantly from 0.70 mg/L to 605.32 mg/L. The unconfined aquifer beneath DRB was experiencing freshening gradually, while some other samples were still subjected to the interference of environmental pollution. Spatio-temporal variations in the respective clusters demonstrated that natural factors including precipitation infiltration, evaporation, and water–rock interactions played vital roles in the samples of Clusters I-III, V, and VIII, while the samples of Cluster VII were influenced by seawater intrusion; and those of Clusters IV and VI were mainly affected by nitrate and nitrite contamination. This study suggests that the SOM connected with stable isotope analysis, HFE-Diagram, CAI values, and correlation analysis could be successfully used to interpret nonlinear and high-dimensional multivariate systems and provide supplementary information about the dominant mechanisms controlling the regional groundwater evolution.

Exploring the potential of combining chemometric approaches to model non-linear multi-way data with quantitative purposes – A case study

Second-order based calibration methods have been widely investigated capitalizing on the inherent benefits of the data structure and the decomposition models, demonstrating that second-order advantage is a property that conspires to a high likelihood success in the resolution of systems of varying complexity. This work aims to demonstrate the applicability of a combined chemometric strategy to solve non-linear multivariate calibration systems in the presence of non-multilinear multi-way data. The determination of histamine by differential pulse voltammetry at different pH is presented as case study. The experimental system has the outstanding difficulty arisen from the large displacement along the potential axis by the pH, which was successfully overcome by implementation of the presented combined strategy. For data modeling, MCR-ALS, U-PLS/RBL and U-PCA/RBL-RBF were used. MCR-ALS allowed unraveling the non-linear behavior between the signal and the concentration, and extracting the underlying profiles of the constituent. Quantitative analysis was performed through the three models, and a comparative evaluation of the predictive performance was done. The best results were achieved with U-PCA/RBL-RBF (mean recovery = 101%) whereas, MCR-ALS yield the lowest mean recovery for all samples (70%)

Event-triggered-based adaptive super-twisting attitude tracking for RLV in reentry phase

An event-triggered control scheme is proposed for attitude tracking of Reusable Launch Vehicle (RLV) by combining adaptive and super-twisting techniques. Firstly, an adaptive event-triggered super-twisting control (AETSTC) strategy for multi-variable second-order nonlinear systems with unknown external disturbances is designed, which can adaptively adjust the control gains with respect to the unknown external disturbances and save computational cost or limited network resources while ensuring the desired control performance. Then, a formal proof of the stability of the closed-loop systems under the proposed method is derived based on Lyapunov stability theory, and the Zeno-free execution of triggering sequences is also guaranteed. Furthermore, the proposed AETSTC strategy is applied on attitude control of RLV without any knowledge of the bounds of external disturbances. Finally, simulation results on attitude tracking of RLV are provided to illustrate the efficiency and advantage of the developed algorithm.

“Class-Type” Identification-Based Internal Models in Multivariable Nonlinear Output Regulation

The paper deals with the problem of output regulation in a "non-equilibrium" context for a special class of multivariable nonlinear systems stabilizable by high-gain feedback. A post-processing internal model design suitable for the multivariable nature of the system, which might have more inputs than regulation errors, is proposed. Uncertainties in the system and exosystem are dealt with by assuming that the ideal steady state input belongs to a certain "class of signals" by which an appropriate model set for the internal model can be derived. The adaptation mechanism for the internal model is then cast as an identification problem and a least square solution is specifically developed. In line with recent developments in the field, the vision that emerges from the paper is that approximate, possibly asymptotic, regulation is the appropriate way of approaching the problem in a multivariable and uncertain context. New insights about the use of identification tools in the design of adaptive internal models are also presented.

Optimally designed Lyapunov–Krasovskii terminal costs for robust stable–feasible model predictive control of uncertain time-delay nonlinear dynamical systems

This article details a new Model Predictive Control algorithm ensuring robust stability and control feasibility for uncertain nonlinear multi-input multi-output dynamical systems considering uncertain time-delay effects. The proposed control algorithm is based on construction of a Lyapunov–Krasovskii functional as terminal cost. Incorporation of this terminal cost into the Model Predictive Control optimization problem and calculation of the associated admissible set result in robust feasibility and robust stability of closed-loop system in presence of uncertain time-delay effects and bounded disturbance signals. The Lyapunov–Krasovskii functional term is constructed with respect to predicted sliding functions over the prediction horizon and considers the effects of dynamical variations over the prediction horizon in generation of control inputs. As dynamical variations are investigated in a sample-to-sample basis, feasible sliding regions are updated at each sample as well. Finally, based on expression of sliding functions as a combination of dynamical variations and input-based terms, required control inputs are calculated in the admissible bound by the optimization algorithm. Construction of control scheme on this basis permits straightforward calculation of robust stability and feasibility conditions for a general class of uncertain nonlinear system in finite prediction horizon whereas in the previous works, often-restrictive conditions were considered for the investigated dynamical systems. Numerical illustrations indicate precision and efficiency of control algorithm and improved stability and convergence rate for multivariable nonlinear dynamical systems considering uncertain time-delay effects. Finally, hardware-in-the-loop implementation indicates applicability of the proposed scheme in real-time control applications particularly in case appropriate compromises between optimality and calculation speed are considered.

Filtering adaptive output feedback control for multivariable nonlinear systems with mismatched uncertainties and unmodeled dynamics

SummaryThis article synthesizes a filtering adaptive output feedback controller for multivariable nonlinear systems with mismatched uncertainties and unmodeled dynamics. The multivariable nonlinear systems under consideration have both matched and mismatched uncertainties, which satisfy the semiglobal Lipschitz condition. The unmodeled dynamics are bounded‐input bounded‐output stable. By adopting an estimation/cancellation strategy, a piecewise constant adaptive law drives the estimation error to zero at every time instant, which yields the adaptive parameters; a disturbance rejection control law is designed to compensate the nonlinear uncertainties within the bandwidth of low‐pass filters. The matched uncertainties are cancelled directly by adopting their opposite in the control signal, while a dynamic inversion of the system is required to eliminate the effect of the mismatched uncertainties on the output. A feedforward control law is designed to track a given command. Since the virtual reference system defines the best performance that can be achieved by the closed‐loop system, the uniform performance bounds are derived for the states and control signals via comparison. Both numerical and practical examples are provided to illustrate the effectiveness of the proposed filtering adaptive output feedback control architecture, comparisons with the model reference adaptive control demonstrates the superiority of the proposed control method.