Multi-input Multi-output Nonlinear Systems Research Articles

Four actor-critic structures and algorithms for nonlinear multi-input multi-output system

The action-dependent heuristic approximate dynamic (ADHDP) for nonlinear multi-input multi-output (MIMO) system needs different forms to adapt to variable practical objects. Due to some inappropriate network structure or training algorithm, unsuccessful designs or undesirable control effect is common in reality. Thus, at first, this paper addresses the chain rule problem of the compound derivative in training the nonlinear MIMO ADHDP. Then, this paper researches and proposes four actor-critic algorithms systematically according to four typical nonlinear systems. That is, the action-network extension, the sub-network, the cascaded action-network and the combined method. To illustrate the four methods, their detailed structures, derivation procedures and training algorithms are derived. The Lyapunov stability for the nonlinear MIMO ADHDP is proved as well. Through examples of an idling engine and aircraft controlling, the simulation results show the effectiveness of these methods. Besides, the property, advantages, disadvantages and the applicability of these methods are compared and highlighted. The four methods can be used to meet the design requirement of almost all the nonlinear MIMO ADHDP control systems. For incoming scholars in search of a nonlinear MIMO ADHDP to achieve the best control effect, the four actor-critic structures and algorithms can be a reference.

A Unified Approach to Design Robust Controllers for Nonlinear Uncertain Engineering Systems

This paper presents new theorems, which allow to design in a unified way robust proportional-derivative (PD)-type control laws without chattering for a broad class of uncertain nonlinear multi-input multi-output (MIMO) systems, subject to bounded disturbances and noises, of great theoretical and engineering relevance. These controllers are used to track a reference signal with bounded second derivative with the tracking error norm smaller than a prescribed value. The proposed control laws are simple to design and implement, above all for robotic systems, both in the case of a trajectory assigned in the joint space and in the workspace. The obtained theoretical results can have numerous applications. In this paper four significant applications are provided. The first one concerns the solution of a nonlinear equations system or the determination of an equilibrium point of a nonlinear system. The second case study deals with the inversion of a nonlinear vectorial function or the kinematic inversion of a robot. The third application concerns: (A) the tracking control of a robot with parametric uncertainties, with and without measurement noise on velocity, both in the joint space and the workspace; (B) the impedance control of a robot interacting with a human operator. The fourth case study addresses the tracking control of an uncertain nonlinear system that does not belong to the class of mechanical systems. Finally, an appendix is included, providing six easy examples, which show how the results proposed in the paper can eliminate and/or reduce serious disadvantages existing in the robust control literature for significant classes of linear and nonlinear uncertain systems.

Robust Sliding Mode-Based Learning Control for MIMO Nonlinear Nonminimum Phase System in General Form.

The tracking control of a multi-input multioutput nonlinear nonminimum phase system in general form is discussed. This system is assumed to be suffering from parameter uncertainties and unmodeled dynamics, and the priori information of them is unknown. By considering both the exact model and uncertain model, the sliding mode-based learning controller is proposed. By designing an appropriate sliding surface and a learning controller, the stability of the closed-loop system is guaranteed for both the exact model and uncertain model. To overcome the disadvantage caused by parameter uncertainties and unmodeled dynamics, a fuzzy logical system is adopted here. A numerical simulation result carried on vertical takeoff and landing aircraft is taken as an example to validate the effectiveness of the presented controller.

Adaptive model predictive control‐based attitude and trajectory tracking of a VTOL aircraft

A novel adaptive model-based predictive controller for attitude and trajectory tracking of a vertical take-off and landing (VTOL) aircraft in the simultaneous presence of model uncertainties and external disturbances is introduced in this study. An important challenge of designing the model-based controllers is developing an accurate model, especially in the presence of model uncertainties. In this study, first, the nominal model of a ducted-fan air vehicle, which is a multi-input multi-output nonlinear system with non-minimum phase behaviour, is given as the test case of this research. After that, two modified robust and adaptive model predictive controllers are proposed for tracking a predefined path in the presence of external gusts and model uncertainties. The design is performed by introducing two novel observers to estimate the model uncertainties and external disturbances, simultaneously. Different types of input constraints, as well as the time delay of the system inputs, can be considered effectively in the design steps of the proposed control strategy. The stability of the closed-loop system is proved under a few realistic assumptions. The simulation results demonstrate the great capability of the proposed method for attitude and trajectory tracking of the VTOL aircraft under model uncertainties and wind turbulence.

Active disturbance rejection control to MIMO nonlinear systems with stochastic uncertainties: approximate decoupling and output-feedback stabilisation

ABSTRACTIn this paper, we apply the active disturbance rejection control, an emerging control technology, to output-feedback stabilisation for a class of uncertain multi-input multi-output nonlinear systems with vast stochastic uncertainties. Two types of extended state observers (ESO) are designed to estimate both unmeasured states and stochastic total disturbance which includes unknown system dynamics, unknown stochastic inverse dynamics, external stochastic disturbance without requiring the statistical characteristics, uncertain nonlinear interactions between subsystems, and uncertainties caused by the deviation of control parameters from their nominal values. The estimations decouple approximately the system after cancelling stochastic total disturbance in the feedback loop. As a result, we are able to design an ESO-based stabilising output-feedback and prove the practical mean square stability for the closed-loop system with constant gain ESO and the asymptotic mean square stability with time-varying gain ESO, respectively. Some numerical simulations are presented to demonstrate the effectiveness of the proposed output-feedback control scheme.

Adaptive sensor fault-tolerant control for a class of multi-input multi-output nonlinear systems: Adaptive first-order filter-based dynamic surface control approach

This paper is concerned with the adaptive fault-tolerant control (FTC) problem for a class of multivariable nonlinear systems with external disturbances, modeling errors and time-varying sensor faults. The bias, drift, loss of accuracy and loss of effectiveness faults can be effectively accommodated by this scheme. The dynamic surface control (DSC) technique and adaptive first-order filters are brought together to design an adaptive FTC scheme which can reduce significantly the computational burden and improve further the control performance. The adaptation laws are constructed using novel low-pass filter based modification terms which enable under high learning or modification gains to achieve robust, fast and high-accuracy estimation without incurring undesired high-frequency oscillations. It is proved that all signals in the closed-loop system are uniformly ultimately bounded and the tracking-errors can be made arbitrary close to zero. Simulation results are provided to verify the effectiveness and superiority of the proposed FTC method.

Adaptive Fuzzy Tracking Control Based Barrier Functions of Uncertain Nonlinear MIMO Systems With Full-State Constraints and Applications to Chemical Process

An adaptive control approach based on the fuzzy systems for a class of uncertain nonlinear multi-input multi-output (MIMO) systems is presented in this paper. This class of systems is in the nested multiple coupling structure and their states are constrained in the corresponding compact sets. The properties of the system structure are inevitable to bring about a complicated design and a difficult task. The fuzzy logic systems are employed to approximate the unknown functions of systems, and the decoupling backstepping way is proposed to design the stability controller and adaptation laws. Barrier Lyapunov functions (BLFs) are constructed in the backstepping design to guarantee that the constraint bounds are not violated. Based on Lyapunov analysis in barrier form, we can prove the stability of the closed-loop system. Two simulation examples are viewed to verify the feasibility of the approach.

A Runge–Kutta neural network-based control method for nonlinear MIMO systems

In this paper, a novel Runge–Kutta neural network (RK-NN)-based control mechanism is introduced for multi-input multi-output ( MIMO) nonlinear systems. The overall architecture embodies an online Runge–Kutta model which computes a forward model of the system, an adaptive controller with tunable parameters and an adjustment mechanism realized by separate online Runge–Kutta neural networks to identify the dynamics of each tunable controller parameter. Runge–Kutta identification block has the competency to approximate the time-varying parameters of the model and unmeasurable states of the controlled system. Thus, the strengths of radial basis function (RBF) neural network structure and Runge–Kutta integration method are combined in this structure. Adaptive MIMO proportional–integral–derivative (PID) controller is deployed in the controller block. The control performance of the proposed adaptive control method has been evaluated via simulations performed on a nonlinear three-tank system and Van de Vusse benchmark system for different cases, and the obtained results reveal that the RK-NN-based control mechanism and Runge–Kutta model attain good control and modelling performances.

Fault Tolerant Nonrepetitive Trajectory Tracking for MIMO Output Constrained Nonlinear Systems Using Iterative Learning Control.

Most works on iterative learning control (ILC) assume identical reference trajectories for the system state over the iteration domain. This fundamental assumption may not always hold in practice, where the desired trajectories or control objectives may be iteration dependent. In this paper, we relax this fundamental assumption, by introducing a new way of modifying the reference trajectories. The concept of modifier functions has been introduced for the first time in the ILC literature. This proposed approach is also a unified framework that can handle other common types of initial conditions in ILC. Multi-input multi-output nonlinear systems are considered, which can be subject to the actuator faults. Time and iteration dependent constraint requirements on the system output can be effectively handled. Backstepping design and composite energy function approach are used in the analysis. We show that in the closed loop analysis, the proposed control scheme can guarantee uniform convergence on the full state tracking error over the iteration domain, beyond a small initial time interval in each iteration, while the constraint requirements on the system output are never violated. In the end two simulation examples are shown to illustrate the efficacy of the proposed ILC algorithm.

Realization of Nonlinear Time-Delay Input–Output Equations

In this letter the problem of transforming a nonlinear multi-input multi-output time-delay control system, described by a set of higher order functional differential equations relating system inputs and outputs, into a set of first order functional differential equations, is studied. The globally linearized system equations are described in terms of polynomials in a delay operator and derivative operator. The Euclidean left division of polynomials is used to compute the basis of a certain free submodule of differential 1-forms. The integrability of the submodule is proved to be necessary and sufficient for the solvability of the problem.

Terminal adaptive output feedback variable structure control

Based on the Lyapunov stability theorem, an adaptive output feedback variable structure control (OFVSC) scheme is proposed in this study for a class of multi-input multi-output non-linear systems with matched and mismatched perturbations to solve regulation problems. First, a switching function is designed, and the system to be controlled is decomposed into two subsystems through a linear state transformation. Then the proposed OFVSC scheme is presented. A perturbation estimation algorithm is utilised in designing the proposed control scheme in order to overcome the problem of unmeasurable states. Adaptive mechanism is also employed so that the upper bounds of perturbations as well as perturbation estimation errors are not required to be known in advance. Furthermore, the resultant control system is capable of driving all the states into zero within a finite time and guaranteeing global stability. Finally, a numerical example is given for demonstrating the feasibility of the proposed control scheme.

A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing

The significant task for control performance assessment (CPA) is to review and evaluate the performance of the control system. The control system in the semiconductor industry exhibits a complex dynamic behavior, which is hard to analyze. This paper investigates the interesting crossover properties of Hurst exponent estimations and proposes a novel method for feature extraction of the nonlinear multi-input multi-output (MIMO) systems. At first, coupled data from real industry are analyzed by multifractal detrended fluctuation analysis (MFDFA) and the resultant multifractal spectrum is obtained. Secondly, the crossover points with spline fit in the scale-law curve are located and then employed to segment the entire scale-law curve into several different scaling regions, in which a single Hurst exponent can be estimated. Thirdly, to further ascertain the origin of the multifractality of control signals, the generalized Hurst exponents of the original series are compared with shuffled data. At last, non-Gaussian statistical properties, multifractal properties and Hurst exponents of the process control variables are derived and compared with different sets of tuning parameters. The results have shown that CPA of the MIMO system can be better employed with the help of fractional order signal processing (FOSP).

Extended state observer for MIMO nonlinear systems with stochastic uncertainties

ABSTRACTIn this paper, both linear extended state observer (ESO) and nonlinear ESO with homogeneous weighted functions are proposed for a class of multi-input multi-output (MIMO) nonlinear systems composed of coupled subsystems with large stochastic uncertainties. The stochastic uncertainties in each subsystem including internal coupled unmodelled dynamics and external stochastic disturbance without known statistical characteristics are lumped together as the stochastic total disturbance (extended state) of each subsystem. The linear ESO and nonlinear ESO are designed separately for real-time estimation of not only the unmeasured state but also the stochastic total disturbance of each subsystem. The practical mean square convergence of these two classes of ESOs are developed. Some numerical simulations are presented to demonstrate the effectiveness of the ESOs with the advantages of smaller peaking values and more accurate estimation by the nonlinear ESO.

Fuzzy adaptive control of nonlinear MIMO systems with unknown dead zone outputs

This paper addresses a novel fuzzy adaptive control method for a class of uncertain nonlinear multi-input multi-output (MIMO) systems with unknown dead-zone outputs and immeasurable states. The immeasurable states under consideration are estimated by designing a fuzzy state observer. Based on the properties of the Nussbaum-type function, the difficulty of fuzzy adaptive control caused by the unknown dead zone outputs of MIMO nonlinear uncertain systems is overcome. The presented design algorithm not only guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, but also ensures that the outputs of the MIMO system converge to a small neighborhood of the desired outputs. The main contributions of this research lie in that the developed MIMO systems are more general, and an efficient design method of output-feedback controller is investigated for the studied MIMO systems, which is more applicable in practical environment. Simulation results illustrate the effectiveness of the proposed scheme.

Sliding mode controller design for MIMO nonlinear systems: A novel power rate reaching law approach for improved performance

This paper proposes a novel approach to the design of reaching law based on Sliding Mode Controller (SMC) for multi input multi output (MIMO) non-linear systems so as to overcome the drawbacks associated with conventional reaching law based SMC design strategies. The modification is proposed with an aim to completely eliminate chattering, to ensure control inputs within admissible limits and to guarantee fast response when SMC is used. Modification to conventional power rate reaching law is the point of interest here in order to ensure complete elimination of chattering. Two different modifications to power rate reaching law are presented which incorporate control constraints during controller design so that admissible control input limits are not exceeded. The first modified method ensures limited control effort as well as complete chattering free response, but does not improve the reaching characteristics. So a second adaptive modification to power rate reaching law is also presented here. This method ensures fast reaching to the sliding surface along with properties of complete elimination of chattering and bounded control inputs. However, as in power rate reaching law these modified methods retain the limitation of not possessing robustness properties. The method is applied to a three degree of freedom robotic arm which is typically a non-linear MIMO system. The ability of the presented method to satisfy attributes, viz., chattering free operation, bounded control inputs and fast response is compared with the performance of various reaching law methods available in the literature. The performance of the proposed method is validated through simulation studies on the robotic arm example.

Indirect adaptive fuzzy fault-tolerant tracking control for MIMO nonlinear systems with actuator and sensor failures

In this paper, an active fuzzy fault tolerant tracking control (AFFTTC) scheme is developed for a class of multi-input multi-output (MIMO) unknown nonlinear systems in the presence of unknown actuator faults, sensor failures and external disturbance. The developed control scheme deals with four kinds of faults for both sensors and actuators. The bias, drift, and loss of accuracy additive faults are considered along with the loss of effectiveness multiplicative fault. A fuzzy adaptive controller based on back-stepping design is developed to deal with actuator failures and unknown system dynamics. However, an additional robust control term is added to deal with sensor faults, approximation errors, and external disturbances. Lyapunov theory is used to prove the stability of the closed loop system. Numerical simulations on a quadrotor are presented to show the effectiveness of the proposed approach.

Oscillation damping of nonlinear control systems based on the phase trajectory length concept: An experimental case study on a cable-driven parallel robot

In this paper, via shortening the phase trajectory length concept, a systematic approach for oscillation damping of nonlinear systems is introduced and the proposed procedure is applied in order to control an under-actuated cable-driven parallel robot both in simulation and experiment. In this regard, at first, the phase trajectory length concept in the state space of a multi-input multi-output nonlinear system is defined and then its lower and upper bounds are computed. Then, the concept of oscillation number index is defined and sufficient conditions for designing an anti-oscillation control system are presented. Moreover, the effect of including the time as a weight in the computation of the phase trajectory length in order to enforce the system response speed beside the oscillation damping is investigated. Finally, based on the governing equations of the robot under study, utilizing the genetic algorithm, the computed torque controller gains are explored in order to minimize the oscillation number index to alleviate the oscillation of the system. In this regard, the performance of the aforementioned index for alleviating the oscillation of the robot is compared with the integral absolute error index, experimentally. Moreover, the Fourier analysis of the obtained results is presented.

Zero-error tracking control with pre-assignable convergence mode for nonlinear systems under nonvanishing uncertainties and unknown control direction

This work addresses the tracking control problem for a class of high-order multi-input multi-output (MIMO) nonlinear systems with unknown control direction and nonparametric uncertainties. By integrating a matrix (rather than scalar) rate transformation with Nussbaum gain, we develop an accelerated robust adaptive control that exhibits several attractive features: (1) it is able to achieve full-state zero-error tracking despite unknown control direction and non-vanishing uncertainties; (2) with the proposed control scheme, the whole tracking process seamlessly consists of the first phase of steering the tracking error into an adjustable small residual region with accelerated convergence rate and the second phase of further driving the error to zero; (3) before reaching the residual region, each component of the tracking error is forced to decay at an accelerated rate that can be pre-assigned; and (4) the resultant control action is continuous differentiable everywhere without involving excessively large initial driving effort.

A robust type-2 fuzzy sliding mode controller for disturbed MIMO nonlinear systems with unknown dynamics

ABSTRACTIn this paper, in order to achieve the best tracking control of a class of multi-input multi-output (MIMO) nonlinear systems with unknown dynamics and unknown disturbances, a new robust adaptive interval type-2 fuzzy sliding mode control law (AIT2-FSMCL) has been proposed. Based on developing interval type-2 fuzzy local models for some operating points of the controlled system, an interval type-2 fuzzy logic system (IT2-FLS) has been designed to better estimate the unknown nonlinear dynamics of the studied system. Then, to enhance the tracking control performance and ensure the system robustness in the presence of approximation errors, parameter variations, un-modelled dynamics and external disturbances, a new AIT2-fuzzy sliding mode system (AIT2-FSMS), has been introduced. In order to avoid the chattering phenomenon while keeping the system performance, the AIT2-FSMS uses three AIT2-fuzzy logic systems (AIT2-FLSs) to estimate the optimal gains of the AIT2-FSMCL. The adaptation laws have been derived using the Lyapunov stability approach. The mathematical proof shows that the closed-loop system with the proposed control approach is globally asymptotically stable. Finally, the proposed design method is applied to a two-link robot arm to validate the effectiveness of the proposed control approach.

Adaptive Critic Design for Pure-Feedback Discrete-Time MIMO Systems Preceded by Unknown Backlashlike Hysteresis

This paper concentrates on the adaptive critic design (ACD) issue for a class of uncertain multi-input multioutput (MIMO) nonlinear discrete-time systems preceded by unknown backlashlike hysteresis. The considered systems are in a block-triangular pure-feedback form, in which there exist nonaffine functions and couplings between states and inputs. This makes that the ACD-based optimal control becomes very difficult and complicated. To this end, the mean value theorem is employed to transform the original systems into input-output models. Based on the reinforcement learning algorithm, the optimal control strategy is established with an actor-critic structure. Not only the stability of the systems is ensured but also the performance index is minimized. In contrast to the previous results, the main contributions are: 1) it is the first time to build an ACD framework for such MIMO systems with unknown hysteresis and 2) an adaptive auxiliary signal is developed to compensate the influence of hysteresis. In the end, a numerical study is provided to demonstrate the effectiveness of the present method.