Control Of Nonlinear Multivariable Systems Research Articles

Fuzzy Control of Multivariable Nonlinear Systems Using T–S Fuzzy Model and Principal Component Analysis Technique

In this work, a new nonlinear control method is proposed, which integrates the Takagi–Sugeno (T–S) fuzzy model with the Principal Component Analysis (PCA) technique. The approach uses PCA to reduce the system’s dimensionality, minimizing the number of fuzzy rules required in the T–S fuzzy model. This reduction not only simplifies the system variables but also decreases the computational complexity, resulting in a more efficient control with smooth transient responses and zero steady-state error. To validate the performance of this PCA-based approach for both system identification and control, an interconnected double-tank system was employed. The results demonstrate the method’s capacity to maintain control accuracy while reducing computational load, making it a promising solution for applications in industrial and engineering systems that require robust, efficient control mechanisms.

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.

A multi‐strategy fuzzy control method based on the Takagi‐Sugeno model

AbstractIn this work, a new multi‐strategy fuzzy control (MSFC) based on fuzzy fusion techniques for the control of multivariable nonlinear systems is proposed. This new nonlinear control method is a mixture of different control techniques by fuzzy interpolation using the fuzzy Takagi‐Sugeno model. It combines various control strategies to achieve smooth transient responses and zero steady state error. The following techniques are merged within the proposed MSFC structure: an optimal state feedback control that yields an optimal transient response, optimal control through an incremental state model that returns a zero steady state error, and a constant input control to maintain the system behavior within predefined boundaries whenever the MSFC method is applied.

Robust Control and Co-simulation of a Multivariable Nonlinear Liquid Level Control System

Robust Control and Co-simulation of a Multivariable Nonlinear Liquid Level Control System

Incremental state model in predictive control: A new fuzzy control proposal for nonlinear systems

In this work, a fuzzy predictive optimal control for multivariable nonlinear systems with pure time delays is presented. Therefore, dynamic local linear state models are used at each point of the state space obtained by fuzzy Takagi-Sugeno (T-S) modelling. The modelling error is considered as white noise, and the state is observed using the Kalman Filter (KF). This method does not take into account possible input constraints. Therefore, it applies to systems where there are no saturation problems. In this work, a new approach in the Model Predictive Control (MPC) method is proposed by calculating the control signal increment as a function of the error between a reference state vector and the prediction at N-steps of the state vector instead of using the traditional MPC approach which is based on calculating the error between a reference and the predicted output. The proposed method has a satisfactory tracking performance. The main feature of the proposed method is that it attains computational savings compared to other methods that have used incremental state models, which making it more appropriate for real-time applications.

Local input‐to‐state stability analysis on compact set for disturbance observer‐based uncertain multivariable nonlinear control systems

AbstractThis article proposes a local input‐to‐state stability (LISS) analysis as a practical stability analysis approach. In existing LISS analysis approaches, a region including a state trajectory is not specified, and the decrement of a Lyapunov function must be guaranteed in the entire domain of the system state. This reduces the advantage of LISS by limiting the state region, and limits the class of systems for which stability can be guaranteed. To expand the class of systems for which stability can be guaranteed, the LISS Lyapunov function on a compact set (LISS LF CS) is proposed to guarantee LISS. By using the LISS LF CS, the region containing the state trajectory was revealed and the LISS condition was relaxed. In addition, the convergence rate and steady state error bound were determined for systems with LISS LF CS. The input‐to‐state stability (ISS) and LISS were analyzed for a multivariable disturbance observer‐based control system using the ISS and LISS LFs, respectively. The advantages of the proposed LISS analysis are demonstrated by comparing the ISS and LISS conditions. The analytical results were verified by conducting numerical simulations.

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 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.

Stochastic data-driven model predictive control using gaussian processes

Nonlinear model predictive control (NMPC) is one of the few control methods that can handle multivariable nonlinear control systems with constraints. Gaussian processes (GPs) present a powerful tool to identify the required plant model and quantify the residual uncertainty of the plant-model mismatch. It is crucial to consider this uncertainty, since it may lead to worse control performance and constraint violations. In this paper we propose a new method to design a GP-based NMPC algorithm for finite horizon control problems. The method generates Monte Carlo samples of the GP offline for constraint tightening using back-offs. The tightened constraints then guarantee the satisfaction of chance constraints online. Advantages of our proposed approach over existing methods include fast online evaluation, consideration of closed-loop behaviour, and the possibility to alleviate conservativeness by considering both online learning and state dependency of the uncertainty. The algorithm is verified on a challenging semi-batch bioprocess case study.

Filtering adaptive neural network controller for multivariable nonlinear systems with mismatched uncertainties

AbstractThis paper synthesizes a filtering adaptive neural network controller for multivariable nonlinear systems with mismatched uncertainties. The multivariable nonlinear systems under consideration have both matched and mismatched uncertainties, which satisfy the semiglobal Lipschitz condition. 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. The combination of GRBF‐based neural network and piecewise constant adaptive law relaxes hardware limitations (CPU). A filtering control law is designed to handle the nonlinear uncertainties and deliver a good tracking performance with guaranteed robustness. The matched uncertainties are cancelled directly by adopting their opposite in the control signal, whereas a dynamic inversion of the system is required to eliminate the effect of the mismatched uncertainties on the output. 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. To validate the theoretical findings, comparisons between the model reference adaptive control method and the proposed filtering adaptive neural network control architecture with the implementation of different sampling time are carried out.

Adaptive Fuzzy Control for Multivariable Nonlinear Systems with Indefinite Control Gain Matrix and Unknown Control Direction

In this paper, we focus on the direct adaptive fuzzy control design for a class of uncertain MIMO nonlinear systems with indefinite control gain matrix and unknown control direction. The control design is based on the approximation of an unknown ideal control law that can meet the control objective by using fuzzy systems. The adjustable parameters of the used fuzzy systems are adjusted online using the error between the unknown ideal controller and the fuzzy controller. In this paper, unlike most existing works, the Nussbaum gain technique is not used to overcome the obstacle of the unknown control direction. In fact, with the help of a matrix decomposition technique, the unknown control direction is redefined as an unknown constant vector, which is estimated online by a suitable update law. The stability of the closed-loop system is studied using the Lyapunov direct approach. Numerical simulation results are provided to illustrate the effectiveness of the proposed control design approach.

A Loop Pairing Method for Non-Linear Multivariable Control Systems Under a Multi-Objective Optimization Approach

In the multivariable control literature there are few techniques that face the problem of selecting suitable loop pairings in non-linear multivariable systems. Most techniques analyze the linearized system at a specific operating point. This paper proposes a new methodology to optimally and simultaneously select the loop pairings and the tuning of the parameters of the decentralized control by applying a multi-objective optimization approach directly on the non-linear system. The main contribution of this work is that the proposed methodology enables a detailed multi-dimensional analysis of the performances and trade-offs in the available loop pairings to control a multivariable non-linear system. The methodology is applied in this paper to three examples that analyze how the different types of loop pairings conflict. In one of the examples, the proposed methodology was applied first in the linearized system and later in the non-linear system. The results were contradictory and show how the application of loop pairing techniques for linear systems can be inaccurate when they are applied on a non-linear system previously linearized at an operating point. The following examples show that the operating point of a non-linear system, the design objectives of each multi-objective problem, as well as the designer's preferences have important roles in the selection of an optimal loop pairing.

Variable-structure backstepping controller for multivariable nonlinear systems with actuator nonlinearities based on adaptive fuzzy system

In this paper, a novel robust adaptive fuzzy control is presented for a quite general class of multivariable nonlinear systems with actuators’ nonlinearities (saturation with dead zone) and uncertain dynamics. The backstepping concept in combination with the variable-structure control framework and Lyapunov approach is used to design this adaptive fuzzy control. The fuzzy systems are incorporated in the controller for approximating online the unknown system dynamics. In the controller design and stability analysis, the control gain matrices, which are not necessarily symmetric and definite, are decomposed via the so-called SDU matrix decomposition lemma into a product of three main useful matrices, namely a symmetric definite-positive matrix, a diagonal constant matrix with + 1 or − 1 in its main diagonal and a unity upper triangular matrix. It is shown that the proposed adaptive fuzzy control is able to ensure the uniform ultimate boundedness of all solutions of the closed-loop system, as well as the convergence of the underlying tracking errors. Finally, in a numerical simulation framework, the effectiveness of the presented controller is illustrated on two practical examples.

Multivariable Nonlinear Data-Driven Control With Application to Autonomous Vehicle Lateral Dynamics

Complex engineering systems are usually described by the interaction of several agents and characterized by highly nonlinear dynamics. Control of multivariable nonlinear systems is a widely explored topic, and many different studies have been presented in the scientific literature. However, most of the existing methods strongly rely upon an accurate model of the system, which is generally costly and/or hard to undertake in practice. In this work, we propose a multivariable extension of the data-driven inversion-based control (D2-IBC) method, where a model of the system is derived from data and considered relevant or not, based only on its weight on the final control performance. This method will prove its effectiveness on a challenging application: the stability control of a four-wheel steering autonomous vehicle.

Sliding mode control assisted by GPI observers for tracking tasks of a nonlinear multivariable Twin-Rotor aerodynamical system

This paper explores the techniques based on Generalized Proportional Integral Observers (GPI), under the approach of active disturbance rejection, applied to the control of multivariable nonlinear systems with the same number of inputs and outputs (square). As a central axis, a sliding mode control scheme assisted by GPI observers is proposed. These observers are responsible for estimating the disturbances associated with the dynamics of the sliding surfaces produced by non-linearities, not modeled elements, parameter uncertainty and external disturbances. This process allows the creation of sliding regimes under the point of view of decoupled control. As a case study, the position control of a small-scale 2-DOF helicopter affected by adverse operating conditions, such as unknown external disturbances and actuator faults, is considered. Finally, experimental results that validate the proposed control strategy and a comparison with a linear GPI observer-based active disturbance rejection control scheme are presented.

Newton-like phasor extremum seeking control with application to cooling data centers

We design a Newton-like phasor-based Extremum Seeking Control (ESC) aimed at multi-variable nonlinear control systems. We first augment the derivative estimator underlying the phasor approach to capture the Hessian of the plant index. Then we use these estimates to steer the system along a Newton-like direction. We assess the closed-loop stability of the proposed scheme and demonstrate its effectiveness in a data center cooling scenario.

A fast online multivariable identification method for greenhouse environment control problems

Abstract Growing and pruning radial basis function (GAP-RBF) is extended for identification and control of multivariable nonlinear systems in this work. The proposed MGAP-RBF algorithm utilizes a sliding data window in the growing criterion and limits the number of hidden neurons by introducing a soft constraint in the pruning strategy to reduce the effect of disturbance and to improve learning speed, respectively. The performance of the proposed method is tested through some benchmark problems, and the results show that the proposed method can gain faster speed than the original GAP-RBF method and Ran algorithm, and more importantly, it can obtain an overwhelming advantages especially for some large-scale data sets with some complex attributes. Finally, the proposed method is applied to online PID tuning on a greenhouse environment control process. Simulation results show the proposed MGAP-RBF algorithm has better performance than the traditional RBF method and the original GAP-RBF method, in particular, it is faster and provides a more compact network with reduced computational complexity than the original GAP-RBF method.

Adaptive population extremal optimization-based PID neural network for multivariable nonlinear control systems

Abstract The connection weights parameters play important roles in adjusting the performance of PID neural network (PIDNN) for complex control systems. However, how to obtain an optimal set of initial values of these connection weight parameters in a multivariable PIDNN called MPIDNN is still an open issue for system designers and engineers. This paper formulates this issue as a typical constrained optimization problem firstly by minimizing the cumulative sum of the product of exponential time and the system errors, and a real-time penalty function for overshoots of the system outputs, and then proposes an adaptive population extremal optimization-based MPIDNN method called PEO-MPIDNN for the optimal control issue of multivariable nonlinear control systems. The simulation results for two typical multivariable nonlinear control systems have demonstrated the superiority of the proposed PEO-MPIDNN to real-coded genetic algorithm (RCGA) and particle swarm optimization (PSO)-based MPIDNN, traditional MPIDNN with back propagation algorithm, and population extremal optimization-based multivariable PID control algorithm in terms of transient-state, steady-state, and robust control performance.

Linear extended state observer based sliding mode disturbance decoupling control for nonlinear multivariable systems with uncertainty

In this paper, the sliding mode dynamic disturbance decoupling tracking control method based on the linear extended state observer (LESO) is proposed for a class of square multivariable nonlinear uncertain system. The model plant contains the known linear dynamics, the unknown nonlinear dynamics and the internal and external disturbances, and the various input-output pairs are interacted. The system states are not available for measurement. An improved LESO is developed. The leading feature which is different from the typical ESO lies in that its extended state does not contain the known linear dynamics. The improved LESO can guarantee the error variables to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. So this ball radius can be arbitrarily as small as desired by tuning design parameters. And we give a simple method by which the gain parameters of LESO can be computed easily. This estimation to the total disturbance of the original system is introduced into the sliding mode control design to complete disturbance rejection and decoupling. Rigorous stability analysis shows that the system output can track the desired signal closely. Finally, a class of mass–spring–damper system is taken to make the numerical simulation analysis to illustrate the effectiveness of the proposed control method.

Controller design for multivariable nonlinear control systems based on gradient-based ant colony optimisation

In the present study, a swarm intelligent algorithm called gradient-based continuous ant colony optimisation (GCACO) was utilised for controller design in multi-variable nonlinear control systems. The time domain objectives subject to the frequency domain constraints were considered in the proposed procedure. In order to be able to incorporate the time domain requirements in an overall controller design technique, the appropriate linearisation techniques utilised here was the exponential input describing function (EIDF). Experimental results obtained showed that GCACO technique developed here for controller design of nonlinear multivariable feedback control systems was capable to produce optimal and practical controllers that meet the design requirement in an efficient manner and computationally acceptable amount of time.