Multivariable Nonlinear Systems Research Articles

Neural algorithm for optimization of multidimensional object controller parameters

Optimal control of multivariable systems is a complex dynamic process that minimizes the cost function to obtain the optimal control strategy. Unfortunately, for nonlinear systems, it is not possible to use the traditional linear quadratic regulator (LQR), which would be optimal over the entire range of parameter variation. The problem of nonlinear multivariable systems and their optimal control is very momentous. The solution presented in this paper is based on the application of Reinforcement Learning (RL) networks in controlling a five-degree-of-freedom overhead crane system. Additionally, unlike the classical approach, the algorithm is adapted to directly analyze tabular data of inputs and outputs of the controlled model instead of analyzing its state as feedback (model-free). Implementing the new control structure for the multivariable system improved control quality compared to the classical LQR controller with linearization at the operating point. In addition to quality, the resource indicators, which in the LQR controller are represented by the matrix R, have been significantly improved. The architecture of the neural control system is presented, ensuring that over the entire range of nonlinearity, the quality of control is preserved while reducing the cost of its resource intensity. Obtaining optimal control with reduced resources for its implementation induces a wide range of applications of such neural control in engineering systems. The effectiveness of the proposed control system has been demonstrated in simulation studies. The simulation results present the system’s excellent control performance and adaptability over the entire range of object nonlinearity. The neural algorithm resulted in significantly shorter adjustment time and better control quality with significantly less system resource consumption and increased system dynamics.

Structure modification based PID neural network decoupling control for nonlinear multivariable systems

In this research, a structure modification based PID neural network (PIDNN) decoupling strategy is proposed to solve the difficulty in controller caused by the strong coupling in nonlinear multivariable systems. Incomplete differential neurons are first introduced into the hidden layer of the PIDNN, which achieving a decreased amount of control overshoot and better control stability. Then, an improved integration strategy is incorporated in the hidden layer structure, resulting in expedited convergence speed in the PIDNN control. Furthermore, intelligent optimization algorithms are employed to improve the convergence rate of the proposed PIDNN. The stability of the proposed controller is analyzed, and an adaptive learning rate scheme is derived. Finally, to confirm the effectiveness of the controller, three examples and the analysis of experimental results are provided.

Nonlinear Multivariable System Identification: A Novel Method Integrating T-S Identification and Multidimensional Membership Functions

In this paper, a new multidimensional Takagi–Sugeno (T-S) identification technique is proposed for multivariable nonlinear systems. In this technique, multidimensional membership functions are designed using concepts from solid mechanics. The design of membership functions is carried out in multidimensional space, defining the principal axes from the eigenvectors of the inertia matrix, and it has the characteristic of dividing the space into regions with the same inertia. These regions are analyzed to define the center of gravity for each rule. Illustrative examples of multivariable nonlinear systems, such as a thermal mixing process and a binary distillation column, are selected to evaluate the effectiveness of the proposed method. The proposed method is compared with traditional T-S identification that uses one-dimensional membership functions and shows a reduction in the relative identification error and the algorithm execution time. Additionally, the proposed method prevents rules from being positioned outside the system’s range, thereby avoiding the generation of unnecessary rules.

Limit hypersurface state of the art Gaidai multivariate risk evaluation approach for offshore Jacket

Current study presents state of the art approach to risk and reliability assessment for multivariate nonlinear dynamic systems. Specifically, novel hypersurface Gaidai risks evaluation methodology has been presented for evaluating offshore structural risks. Advocated approach being particularly suitable for offshore engineering multidimensional dynamic systems, possessing large number of critical components, that have either been physically observed/measured, or numerically modeled over a representative timelapse. Advocated Gaidai hypersurface structural risks evaluation methodology is applicable for a wide range of engineering and industrial systems. This study demonstrates that given in situ environmental conditions, it is well possible to assess accurately risks of system’s failures, damages or hazards, caused by excessive structural dynamics. Multivariate dynamic systems, possessing nonlinear cross-correlations between critical system’s components often present design challenges, when utilizing classic structural risks evaluation approaches, as those are mostly only univariate or bivariate. Dynamic offshore Jacket hot spot stresses have been utilized as an example in this reliability investigation. Modeling system’s excessive dynamics is challenging because of the non-stationarity of the system and the intricacy of fluid-structural dynamic interactions, arising from in situ wave-induced loads, influencing Jacket’s structural dynamics. Nonlinearities greatly affect structural dynamics, e.g., 2nd, 3rd, higher order effects within fluid-structural interactions. Methodology presented in this work offers a straightforward, effective, yet precise means of assessing the risks of failure or hazard for multivariate, non-stationary, non-linear dynamic offshore systems. For bivariate case, verification note has been added.

Exponential-form predefined-time convergent controller and its applications to Van-der-Pol system and 2-DOF helicopter

This paper proposes a predefined-time convergent stabilization controller for a class of nonlinear systems. First, a controller is introduced for linear systems where the convergent time is selected in advance, independent of initial conditions, and can be assigned as a parameter of a control input. The corresponding result is then obtained for a multidimensional system consisting of [Formula: see text] nonlinear differential equations. The block control technique is employed to consequently stabilize each state component by a virtual control until the real control appears. Finally, the presented result is applied to a multivariable nonlinear system to design a predefined-time convergent robust controller. To show effectiveness of the proposed convergent controller, numerical simulations have been carried out for a Van-der-Pol system and a 2-degree-of-freedom (2-DOF) helicopter model.

Meshing principle and numerical study of classic conic worm drive

To explore the meshing mechanism of classic conic worm drive and provide a theoretical basis for creating other types of worm drive, a complete meshing theory on the proposed worm pair and its meshing performance are investigated. A new mathematical model containing a full set of geometric parameters on this drive is established using differential geometry. The formulae for calculating the reference point coordinates in different coordinate systems are attained, and then the blank size of the conical worm wheel is given. To build tooth surface boundaries, the meshing theory based on reference point is applied, which ensures the location of this point on the tooth surface. By solving multivariate nonlinear equation system in MATLAB environment, these conjugate regions and instantaneous contact lines are plotted within the axial section of the conic worm pair, at the same time, these values of sliding angle and induced principal curvature at meshing points are calculated. Finally, a numerical study is performed and the results verify that the conic worm pair has a relatively high tooth surface utilization ratio and favorable conditions for contact stress and lubrication, which is consistent with the practical application.

Real-Time HIL Simulation of Nonlinear Generalized Model Predictive-Based High-Order SMC for Permanent Magnet Synchronous Machine Drive

The dynamics of the permanent magnet synchronous motor (PMSM) are described by nonlinear equations, which present challenges. Variations in external factors such as unidentified disturbances (loads) and evolving motor properties add complexity to control efforts. To tackle these intricacies and limitations, a nonlinear control approach is essential. Recent attention has turned to employing predictive control techniques for nonlinear multivariable systems, offering an intriguing avenue for research. In this context, this study introduces a novel hybrid control approach that addresses nonlinearity, parametric fluctuations, and external disturbances. The method combines two essential components: first, the outer loop utilizes high-order sliding mode control (HSMC) to optimize torque and trajectory speed, mitigating chattering phenomena while preserving the PMSM’s convergence and robustness traits. The inner loop, known as the current control, employs the newly developed nonlinear robust generalized predictive control (RNGPC) technique. Importantly, this strategy circumvents the need for direct measurement and observation of external disturbances and parameter uncertainties. The proposed strategy follows a two-phase process. Initially, the reference quadratic current is designed using the electromagnetic torque computed via HSMC, subsequently determining the necessary current to achieve the desired torque. The second phase involves computing the controller law through the robust generalized nonlinear predictive control technique. The approach’s strength lies in its ability to maintain stability and convergence in the face of external disturbances and parameter fluctuations, without necessitating precise measurements or knowledge of the disturbances. To validate the proposed control approach, simulation and experimental tests have been conducted across various operational scenarios. The obtained results demonstrate the method’s robustness against external disturbances and parameter changes while ensuring rapid convergence and reliable performance.

Filtered generalized iterative parameter identification for equation‐error autoregressive models based on the filtering identification idea

SummaryBy using the collected batch data and the iterative search, and based on the filtering identification idea, this article investigates and proposes a filtered multi‐innovation generalized projection‐based iterative identification method, a filtered generalized gradient‐based iterative identification method, a filtered generalized least squares‐based iterative identification method, a filtered multi‐innovation generalized gradient‐based iterative identification method and a filtered multi‐innovation generalized least squares‐based iterative identification method for equation‐error autoregressive systems described by the equation‐error autoregressive models. These filtered generalized iterative identification methods can be extended to other linear and nonlinear scalar and multivariable stochastic systems with colored noises.

Parameter estimation of multivariable Wiener nonlinear systems by the improved particle swarm optimization and coupling identification

This paper investigates the parameter estimation of multivariable Wiener nonlinear systems. To solve the inconsistency problem of the parameter vector and the parameter matrix, the coupling identification concept is applied. Combined with particle swarm optimization (PSO) and an auxiliary model, the partially coupled improved particle swarm optimization (PC-IPSO) method is proposed. In this algorithm, the adaptive feedback inertia weight is improved to accelerate the convergence speed, and the retirement update mechanism is introduced to improve the optimization ability of the basic PSO algorithm. To verify the performance of PC-IPSO, we also derive a multivariable improved PSO (M-IPSO) method for comparison. The computational complexity analysis shows that the PC-IPSO algorithm requires less computational resources than the M-IPSO algorithm. Then, the convergence of the improved PSO method is analyzed. The simulation results indicate that the PC-IPSO method has a faster convergence speed and higher identification accuracy than the M-IPSO and several existing state-of-the-art methods for multivariable Wiener system identification.

Application of artificial intelligence for the optimization of advanced oxidation processes to improve the water quality polluted with pharmaceutical compounds

Sulfamethoxazole and metronidazole are emerging pollutants commonly found in surface water and wastewater. These compounds have a significant environmental impact, being necessary in the design of technologies for their removal. Recently, the advanced oxidation process has been proven successful in the elimination of this kind of compounds. In this sense, the present work discusses the application of UV/H2O2 and ozonation for the degradation of both molecules in single and binary systems. Experimental kinetic data from O3 and UV/H2O2 process were adequately described by a first and second kinetic model, respectively. From the ANOVA analysis, it was determined that the most statistically significant variables were the initial concentration of the drugs (0.03 mmol L−1) and the pH = 8 for UV/H2O2 system, and only the pH (optimal value of 6) was significant for degradation with O3. Results showed that both molecules were eliminated with high degradation efficiencies (88–94% for UV/H2O2 and 79–98% for O3) in short reaction times (around 30–90 min). The modeling was performed using a quadratic regression model through response surface methodology representing adequately 90 % of the experimental data. On the other hand, an artificial neural network was used to evaluate a non-linear multi-variable system, a 98% of fit between the model and experimental data was obtained. The identification of degradation byproducts was performed by high-performance liquid chromatography coupled to a time mass detector. After each process, at least four to five stable byproducts were found in the treated water, reducing the mineralization percentage to 20% for both molecules.

Possibility of Quenching of Limit Cycles in Multi Variable Nonlinear Systems with Special Attention to 3X3 Systems

The present work proposes novel methods of Quenching self-sustained oscillations in the event of the existence of limit cycles (LC) in 3x3 non-linear systems. It explores the possibility of Stabilising/Quenching the LC by way of signal stabilization using high frequency dither signals both deterministic and random when 3X3 systems exhibit such self-sustained nonlinear oscillations under autonomous state. The present work also explores the suppression limit cycles of 3X3 systems using state feedback by either arbitrary pole placement or optimal selection of pole placement. The complexity involved, in implicit non-memory type nonlinearity for memory type nonlinearities, it is extremely difficult to formulate the problem. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Furthermore, the method is made simpler assuming the whole 3X3 system exhibits the LC predominantly at a single frequency. It is equally a formidable task to make an attempt to suppress the limit cycles for 3X3 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, the automation industry, and other occasions of modern applications. The proposed methods are well illustrated through examples and substantiated by digital simulation (a program developed using MATLAB CODES) and the use of the SIMULINK Toolbox of MATLAB software.

Generalized fuzzy hyperbolic model based ship course system control in the presence of complex noise

To solve the ship course control problem with nonlinear terms, input saturation, and complex noise, this paper proposes a saturated ship course control method with complex noise based on generalized fuzzy hyperbolic model (GFHM). Due to the characteristics of fewer identification parameters, GFHM can simplify the complexity of traditional ship fuzzy models. GFHM is more suitable for multi-variable nonlinear systems such as ships of which variables are limited and difficult to measure. Furthermore, to deal with the input rudder angle saturation problem caused by the limited capability to compensate the shipboard equipment controller, an auxiliary system is proposed. And the complex noise in the navigation environment is described by the random process. Then, a new type of ship fuzzy course controller is designed based on the theoretical framework of random differential equations (RDEs) and it is proved that the ship course system under the proposed GFHM-based controller is noise-to-state stable in probability (NSS-P) and the state is an asymptotic gain in probability (AG-P). The simulation results show that the proposed algorithm in this paper can effectively control the ship’s course.

Closed-Loop Stability of a Non-Minimum Phase Quadruple Tank System Using a Nonlinear Model Predictive Controller with EKF

The dynamics of a quadruple tank system (QTS) represent an extensive class of multivariate nonlinear uncertain systems found in the industry. It has been established that changes in split fractions affect the transmission zero location, thereby altering the operating conditions between the minimum and non-minimum phase regions. The latter is difficult to control as more fluid flows into the two upper tanks than into the two bottom tanks, resulting in competing effects between the initial and final system responses. This attribute, alongside nonlinearity, uncertainties, constraints, and a multivariate nature, can degrade closed-loop system performance, leading to instability. In this study, we addressed the aforementioned challenges by designing controllers for the regulation of the water flow in the two bottom tanks of the QTS. For comparative analysis, three controller algorithms—a nonlinear model predictive controller (NMPC), NMPC augmented with an extended Kalman filter (i.e., NMPC-EKF) and linear model predictive controller (LMPC)—were considered in the analysis and design of the control mechanism for the quadruple water level system in a non-minimum phase condition via the Matrix Laboratory (MATLAB) simulation package environment. The simulated and real-time results in the closed loop were analyzed, and the controller performances were considered based on faster setpoint responses, less oscillation, settling time, overshoot, and smaller integral absolute error (IAE) and integral square error (ISE) under various operational conditions. The study showed that the NMPC, when augmented with an EKF, is effective for the control of a QTS in the non-minimum phase and could be designed for more complex, nonlinear, and multivariable dynamics systems, even in the presence of constraints.

Investigation of the Existence of Limit Cycles in Multi Variable Nonlinear Systems with Special Attention to 3X3 Systems

The proposed work addresses the dynamics of a general system and explores the existence of limit cycles (LC) in multi-variable Non-linear systems with special attention to 3x3 nonlinear systems. It presents a simple, systematic analytical procedure as well as a graphical technique that uses geometric tools and computer graphics for the prediction of limit cycling oscillations in three-dimensional systems having both explicit and implicit nonlinear functions. The developed graphical method uses the harmonic balance/harmonic linearization for simplicity of discussion which provides a clear and lucid understanding of the problem and considers all constraints, especially the simultaneous intersection of two straight lines & one circle for determination of limit cycling conditions. The method of analysis is made simpler by assuming the whole system exhibits the limit cycling oscillations predominantly at a single frequency. The discussions made either analytically/graphically are substantiated by digital simulation by a developed program as well as by the use of the SIMULINK Toolbox of MATLAB Software.

Non-Gaussian disturbance rejection control for multivariate stochastic systems using moment-generating function

In this paper, a non-Gaussian disturbance rejection control algorithm for a class of nonlinear multivariate stochastic systems is studied. Based on the moment-generating functions obtained from the deduced probability density functions of the output tracking errors, a new criterion representing the stochastic properties of the system is proposed, motivated by a minimum entropy design. A time-variant linear model can be established by the sampled moment-generating functions. Using this model, a control algorithm is developed that minimizes the newly developed criterion. Moreover, a stability analysis is performed for the closed-loop control system. Finally, simulation results of a numerical example demonstrate the effectiveness of the presented control algorithm. The contribution and novelty of this work can be summarized as follows: (1) a novel non-Gaussian disturbance rejection control scheme is proposed based on the minimum entropy principle, (2) the randomness of the multi-variable non-Gaussian stochastic nonlinear system is attenuated based on the new performance criterion, (3) a theoretical convergence analysis has been given for the proposed control system, and (4) a potential framework has been established for the design of a general stochastic system control.

Discrete sliding mode control method for particle swarm optimization-based brushless DC motor of electric vehicle

Brushless DC motor (BLDCM) is a multivariable nonlinear time-varying system, which is difficult to control. The discrete sliding mode control method for BLDCM of electric vehicle on the basis of particle swarm optimization (PSO) is studied to improve the application of BLDCM in electric vehicle. The mathematical model of BLDCM of electric vehicle is established using the state formula. Based on the mathematical model of BLDCM, through the analysis of electromagnetic torque control of BLDCM, it is clear that controlling the angle between rotor flux and stator flux can accurately control the electromagnetic torque of BLDCM. The adaptive discrete sliding mode controller (SMC) is set to control the electromagnetic torque of BLDCM of electric vehicle, and the PSO algorithm is adopted to obtain the optimal parameters of the adaptive discrete SMC to realize the discrete sliding mode control of BLDCM of electric vehicle. According to experimental results, the proposed method can achieve the accurate control of torque and speed of BLDCM of electric vehicle, and increase the application of BLDCM in electric vehicle.

Filtered auxiliary model recursive generalized extended parameter estimation methods for Box–Jenkins systems by means of the filtering identification idea

AbstractFor equation‐error autoregressive moving average systems, that is, Box–Jenkins systems, this paper presents a filtered auxiliary model generalized extended stochastic gradient identification method, a filtered auxiliary model multi‐innovation generalized extended stochastic gradient identification method, a filtered auxiliary model recursive generalized extended gradient identification method, a filtered auxiliary model multi‐innovation recursive generalized extended gradient identification method, a filtered auxiliary model recursive generalized extended least squares identification method, and a filtered auxiliary model multi‐innovation recursive generalized extended least squares identification method by using the filtering identification idea and the auxiliary model identification idea. The proposed filtered auxiliary model recursive generalized extended identification methods can be generalized to other linear and nonlinear multivariable stochastic systems with colored noises.

Design Method for Concrete Mix Proportion Based on Indices of Strength and Durability of Concrete Exposed to Chloride Attack

In the current design code, concrete mix proportions are not based on durability parameters that can quantitatively design the requirement of service life for chloride-exposed structures. Multifactor calculation models of durability index are developed, and the replacement of mineral admixtures in concrete is determined according to predetermined concrete strength and durability requirements. The chloride diffusion coefficient and aging factor of concrete are chosen as index parameters for the resistance of concrete to chloride salts. A multifactor model of the chloride diffusion coefficient of concrete is then established by regression analysis based on the experimental data, and explicit expressions for the water-to-binder ratio and replacement of admixture were determined by solving the multivariate nonlinear equation system. By calculating the average compressive strength and chloride diffusion coefficient from their specified values, the mix proportion of concrete is determined. The superiority of the proposed method is demonstrated by the comparison with the current code. Lastly, an orthogonal experiment was conducted to demonstrate the validity of the proposed method for designing concrete mix proportions by referencing engineering practices. The results show that major parameters of concrete mix proportion can be determined quantitatively from structural design parameters such as compressive strength, chloride diffusion coefficient, and aging factor, satisfying the requirements of ultimate bearing capacity and service life for concrete structures in chloride environments, as demonstrated in this paper.

A third‐order sliding mode controller for nonlinear multivariable systems

AbstractThis paper proposes a third‐order sliding mode controller for nonlinear multivariable systems with uncertain parameters and subject to external disturbances. The controller achieves fast convergence rate, high tracking accuracy, and a reduced level of chattering. The stability of the controller and its global ultimately uniform convergence is proved by the Lyapunov stability theory. Simulation results on a single inverted pendulum system are given to illustrate the effectiveness of the proposed control scheme by comparing it with methods such as a second‐order supertwisting controller, a third‐order supertwisting controller, and an integral terminal third‐order sliding mode controller.

A Review on Fractional-Order Modelling and Control of Robotic Manipulators

Robot manipulators are widely used in many fields and play a vital role in the assembly, maintenance, and servicing of future complex in-orbit infrastructures. They are also helpful in areas where it is undesirable for humans to go, for instance, during undersea exploration, in radioactive surroundings, and other hazardous places. Robotic manipulators are highly coupled and non-linear multivariable mechanical systems designed to perform one of these specific tasks. Further, the time-varying constraints and uncertainties of robotic manipulators will adversely affect the characteristics and response of these systems. Therefore, these systems require effective modelling and robust controllers to handle such complexities, which is challenging for control engineers. To solve this problem, many researchers have used the fractional-order concept in the modelling and control of robotic manipulators; yet it remains a challenge. This review paper presents comprehensive and significant research on state-of-the-art fractional-order modelling and control strategies for robotic manipulators. It also aims to provide a control engineering community for better understanding and up-to-date knowledge of fractional-order modelling, control trends, and future directions. The main table summarises around 95 works closely related to the mentioned issue. Key areas focused on include modelling, fractional-order modelling type, model order, fractional-order control, controller parameters, comparison controllers, tuning techniques, objective function, fractional-order definitions and approximation techniques, simulation tools and validation type. Trends for existing research have been broadly studied and depicted graphically. Further, future perspective and research gaps have also been discussed comprehensively.