Underactuated Nonlinear Systems Research Articles

Variable-structure trajectory correction control of small-caliber ammunition based on cascade fuzzy active disturbance rejection

In this study, a low-cost two-dimensional (2D) trajectory correction mechanism is proposed to address the large trajectory deviation of small-caliber extended-range ammunitions. By controlling the projectile’s roll, a single-channel 2D trajectory correction function can be realized. Moreover, based on the ideal trajectory, a five-degree-of-freedom approximate rigid-body trajectory equation is proposed. A high-order system is transformed into two low-order cascading subsystems using fuzzy control and the self-disturbance rejection control theory, which reduces the parameter tuning complexity. The entire control process is divided into three stages which realizes the second-order variable structure control of non-affine nonlinear subsystems. A low-altitude atmospheric disturbance model is established, and as an example, numerical calculations are performed on the control process and projectile impact point dispersion of a 35 mm small-caliber rocket. The results verify the performance of the proposed trajectory correction under atmospheric disturbances. This study provides a control approach for coupled underactuated nonlinear systems and provides guidance for the design of a low-cost 2D trajectory correction controller for small-caliber ammunitions.

Simplified internal models in human control of complex objects.

Humans are skillful at manipulating objects that possess nonlinear underactuated dynamics, such as clothes or containers filled with liquids. Several studies suggested that humans implement a predictive model-based strategy to control such objects. However, these studies only considered unconstrained reaching without any object involved or, at most, linear mass-spring systems with relatively simple dynamics. It is not clear what internal model humans develop of more complex objects, and what level of granularity is represented. To answer these questions, this study examined a task where participants physically interacted with a nonlinear underactuated system mimicking a cup of sloshing coffee: a cup with a ball rolling inside. The cup and ball were simulated in a virtual environment and subjects interacted with the system via a haptic robotic interface. Participants were instructed to move the system and arrive at a target region with both cup and ball at rest, 'zeroing out' residual oscillations of the ball. This challenging task affords a solution known as 'input shaping', whereby a series of pulses moves the dynamic object to the target leaving no residual oscillations. Since the timing and amplitude of these pulses depend on the controller's internal model of the object, input shaping served as a tool to identify the subjects' internal representation of the cup-and-ball. Five simulations with different internal models were compared against the human data. Results showed that the features in the data were correctly predicted by a simple internal model that represented the cup-and-ball as a single rigid mass coupled to the hand impedance. These findings provide evidence that humans use simplified internal models along with mechanical impedance to manipulate complex objects.

Adaptive neural hierarchical sliding mode control for uncertain switched underactuated nonlinear systems against unmodeled dynamics and input delay

Adaptive neural hierarchical sliding mode control for uncertain switched underactuated nonlinear systems against unmodeled dynamics and input delay

Constraint-following control design for nonlinear underactuated belt conveyors with uncertainty

There are many mismatched uncertainties in the nonlinear underactuated belt conveyor systems which leads to challenges for designing the controllers. This paper addresses this issue based on the concept of constraint-following by establishing the dynamic model and treating the desired trajectory as a constraint. First, the nominal control portion is studied in the absence of the deviation of initial condition and uncertainties. Second, the uncertainties are decomposed into matched and mismatched parts according to the match condition. Then, the robust constraint-following control portion that is solely based on matched uncertainty is designed. By utilizing the Lyapunov method, the proposed control with two portions is able to ensure the uniform boundedness and uniform ultimate boundedness of the underactuated systems with uncertainties. Simulations are additionally carried out for the purpose of verification.

Adaptive direct RBFNN consensus control for a class of unknown nonlinear underactuated systems

The consensus tracking control of a class of nonlinear underactuated multi-agent systems with uncertainties is studied in this paper. A radial basis function neural network (RBFNN)-based direct adaptive control algorithm is designed. Unlike many previous articles in which a neural network is employed to identify the unknown nonlinear functions in the system model or controller, this method directly approximates the ideal control law by a neural network, making the control law simpler. Based on the neural network direct control algorithm, a controller with a single-parameter learning scheme is designed. This new control method reduces the computational burden by reducing the amount of computational data. Finally, the closed-loop system is proved to be ultimately uniformly bounded stable; the application examples of the controllers are given by simulation.

Multiswitching Surface Based Sliding Mode Controller for a Class of Underactuated Nonlinear Systems: With Application to Ball and Plate System

Multiswitching Surface Based Sliding Mode Controller for a Class of Underactuated Nonlinear Systems: With Application to Ball and Plate System

Robust Control Design of Under-Actuated Nonlinear Systems: Quadcopter Unmanned Aerial Vehicles with Integral Backstepping Integral Terminal Fractional-Order Sliding Mode

In this paper, a novel robust finite-time control scheme is specifically designed for a class of under-actuated nonlinear systems. The proposed scheme integrates a reaching phase-free integral backstepping method with an integral terminal fractional-order sliding mode to ensure finite-time stability at the desired equilibria. The core of the algorithm is built around proportional-integral-based nonlinear virtual control laws that are systematically designed in a backstepping manner. A fractional-order integral terminal sliding mode is introduced in the final step of the design, enhancing the robustness of the overall system. The robust nonlinear control algorithm developed in this study guarantees zero steady-state errors at each step while also providing robustness against matched uncertain disturbances. The stability of the control scheme at each step is rigorously proven using the Lyapunov candidate function to ensure theoretical soundness. To demonstrate the practicality and benefits of the proposed control strategy, simulation results are provided for two systems: a cart–pendulum system and quadcopter UAV. These simulations illustrate the effectiveness of the proposed control scheme in real-world scenarios. Additionally, the results are compared with those from the standard literature to highlight the superior performance and appealing nature of the proposed approach for underactuated nonlinear systems. This comparison underscores the advantages of the proposed method in terms of achieving robust and stable control in complex systems.

Adaptive neural predefined-time hierarchical sliding mode control of switched under-actuated nonlinear systems subject to bouc-wen hysteresis

For switched under-actuated systems subject to bouc-wen hysteresis, an adaptive neural predefined-time control scheme is presented first time in this paper. To improve the robustness and response rate of the system under consideration, a hierarchical sliding mode control technique is introduced. Based on a predefined-time stability criterion, the stabilization time of the system can be directly preset under the designed controller. Besides, the convergence time depends only on the preset value and is independent of all other parameters. The projection algorithm is introduced to avoid the singularity problem in the controller design process. Meanwhile, it is proved that all signals of the closed-loop system are bounded after a preset time through the Lyapunov stability theory. Finally, a simulation example and a comparison example are given to further illustrate the effectiveness and superiority of developed control approach.

Barrier Function-Based Integral Sliding Mode Controller Design for a Single-Link Rotary Flexible Joint Robot

Abstract This paper proposes and evaluates a novel control approach for trajectory tracking, stability enhancement, and vibration reduction of a flexible joint robot (FJR). The FJR is a 2-degree-of-freedom underactuated nonlinear system that is challenging to control due to vibration, underactuation, uncertainties, and external disturbances. The control objectives are high trajectory tracking performance together with suppressing vibration. The proposed control approach uses integral sliding mode control (ISMC) combined with a barrier function based on back-stepping. This ensures robust and smooth performance by eliminating the reaching phase where sliding mode control (SMC) is typically not robust. Robustness is thus guaranteed from the start. The FJR is modeled as a 4th-order system using Lagrangian mechanics and decomposed into two 2nd-order subsystems for control design. Sliding variables are defined for each subsystem. Using these variables, the proposed control achieves robust trajectory tracking, stability, and vibration reduction. Numerical simulations in MATLAB validate the superior performance of the proposed ISMC-barrier function control compared to conventional ISMC for the FJR. This novel control approach addresses the challenges of controlling this FJR.

Enhanced free reaching phase SMC for UMS

In this paper, a new design method for sliding mode control (SMC) is presented for nonlinear underactuated mechanical systems. The aim is to eliminate the reaching phase in SMC and avoid the chattering phenomenon while achieving fast and robust tracking for a class of underactuated mechanical systems (UMS). An alternative method is presented to express the sliding domain equations by incorporating tracking errors. The fundamental concept behind the suggested control scheme is to adjust the tracking errors, enabling the system response to commence on the sliding surface regardless of the initial conditions. This modification guarantees the elimination of the reaching phase, avoiding the chattering phenomenon, and ensuring that the tracking error converges to zero. The stability analysis of the proposed approach is conducted using the Lyapunov method. Through numerical simulations, the validity of the approach is confirmed by implementing the control scheme on a crane system. Subsequently, the performance of the proposed approach is compared with other control methods, emphasizing its effectiveness in handling uncertainties. This comparative analysis aims to emphasize the advantages and efficiency of the proposed control strategy over alternative methods, in particular, to address uncertainties and achieve the desired control objectives.

Optimisation of nonlinear controllers for a quadrotor using metaheuristic algorithm

Unmanned aerial vehicles (UAVs) facilitate complex activities and are widely used for aerial transport. Quadrotor UAVs (QUAV), the most popular UAV containing four motors, are characterised by higher control properties since they have fewer actuators than degrees of freedom, implying a nonlinear underactuated system. In addition, the coupling of dynamics, flaws while modelling and parameter uncertainty are the factors that hinder the design and implementation of a controller. Here, we present the modelling, optimisation, simulation, and implementation methodology for controllers, proportional-integralderivative (PID), and super-twisting-sliding mode control (ST-SMC). We carry out the parameterisation problem of controllers using the hunger game search (HGS) metaheuristic algorithm. This process was developed offline, and the values obtained were successfully implemented in simulation and experimental form. The testing platform comprises a motion capture system, Vicon® Bonita cameras, linked by ROS, that allows the known position and the attitude of a Parrot® QUAV bebop1. The whole six dynamics of the QUAV are included in the implementation, translational trajectories X-Y are trapezoidal, and the altitude trajectory is a ramp. The results enabled the comparison of the statistics calculation of each controller. Successful tracking trajectories were obtained even with disturbance when the ST-SMC algorithm was implemented with root mean square error (RMSE)=0.0176.

Vibration Analysis and Motion Control Method for an Under-Actuated Tower Crane

In this paper, we developed a solution for controlling a tower crane thought as a no-rigid system, and therefore able to have deformation and, during the motion, vibrations. Particularly, large tower cranes show high structural dynamics. Under external excitations, the payload tends to sway around its vertical position and this motion is coupled to the resulting dynamic vibration of the crane structure. These induced vibrations may cause instability and serious damage to the crane system. Furthermore, the energy stored in the flexible structure of a tower crane causes vibrations in the structure during the acceleration and deceleration of slewing movements. A crane operator perceives these vibrations as an unstable speed of the boom. Such behavior involves the control of the crane, particularly precise positioning and manual control of the crane movement at low pivoting speed. We define an Elastic model of the Slewing crane and analyze the bending and Torsional elasticity of the Tower, and the Jib Elasticity. With an approximated method, we calculate the natural wavelengths of the crane structure in the slewing direction. We consider the tower crane as a nonlinear under-actuated system. The motion equations are obtained considering both the normal vibration modes of the tower crane and the sway of the payload. An elastic model of the Slewing crane is achieved, modeling the crane jib as an Euler-Bernoulli beam. Even the payload dynamic is considered, developing an Anti-sway solution by the equation of the movement. We define an iterative calculation of the sway angles and obtain the corresponding velocity profiles, implementing two kinds of solution: an input-shaping control in open-loop, to be used with automatic positioning, and a “command smoothing” method in open-loop, used for reducing the sway of the payload with the operator control. These solutions lead to a reduction of the vibrations of the crane structure. As a consequence, the tower crane is not subject to the strong horizontal and vertical oscillations during the motion of the elastic structure.

Adaptive fuzzy sliding-mode control of under-actuated systems with unknown input gain function

This research addresses the design of an Adaptive Fuzzy Sliding-Mode Control (AFSMC) for a group of nonlinear under-actuated systems with unknown input gain functions. In the proposed approach, to provide a structure for controlling the under-actuated subsystem, the sliding manifold corresponding to the whole dynamic system is derived based on the backstepping technique. In addition, two separate fuzzy inference systems are utilized for approximating the system’s internal dynamics and the unknown input gain function. In order to derive the robust part of the controller, a novel method is proposed to define the upper bound of the uncertain term in the form of a state-dependent second-degree polynomial. It should be mentioned that the output vectors of the fuzzy systems and the coefficients of the robust part of the controller are obtained by the designed adaptation laws. The asymptotic stability analysis for the proposed AFSMC structure is provided by the second theorem of Lyapunov and Barbalat’s lemma. In the end, the AFSMC effectuality is confirmed by the depth control of a REMUS Autonomous Underwater Vehicle (AUV).

Adaptive Stabilization Control for a Class of Non-Strict Feedback Underactuated Nonlinear Systems by Backstepping

Adaptive Stabilization Control for a Class of Non-Strict Feedback Underactuated Nonlinear Systems by Backstepping

A Robust Hybrid MRAPID Controller Design for an Underactuated Nonlinear System With Parametric Uncertainty

This paper presents an optimal design and simulation of a novel enhanced model reference adaptive proportional–integral–derivative controller (MRAPIDC) for underactuated nonlinear systems with parametric uncertainty. The controller is applied to a cart‐inverted pendulum mechanical system, a fourth‐order, nonminimum phase system with pronounced nonlinearity and unstable dynamics. This system includes a single input and multiple outputs, specifically the position and velocity of the cart, as well as the angle and angular velocity of the inverted pendulum arm. The main objective of the proposed controller is to maintain the pendulum in an upright position against gravity while moving the cart to a desired displacement regardless of parametric uncertainty, input variation, external disturbances, and random noise. The hybrid controller design is developed by combining the MRAPIDC design with controller gains applied to each output and incorporating the system adaptation error in a specific configuration. Optimal performance is achieved through parameter tuning via the social spider optimization (SSO) algorithm, and MIT (Massachusetts Institute of Technology) rule criteria are employed to ensure system stability. Performance improvements are demonstrated through several error indices, showing enhanced transient and steady‐state responses compared to both the reference model adaptive controller and the conventional proportional–integral–derivative (PID) controller. The proposed controller can effectively achieve system stability with a short settling time of 2.55 s and establishes a steady swinging balance within 2.9 s.

Control system design for a class of non-cascade nonlinear under-actuated systems with an application to a rotary crane system

For cascade under-actuated systems, several effective control methods have been established. For non-cascade under-actuated systems with one under-actuation degree, several control methods, such as cascade transformation and energy-based control, have been studied. For non-cascade nonlinear under-actuated systems with two or more under-actuation degrees, approximate linearization is necessary to apply these methods, thus only stability at the equilibrium point has been guaranteed. In this paper, control system design and stability analysis are presented for a class of non-cascade nonlinear under-actuated systems with two or more under-actuated degrees. The stability condition obtained in the analysis is a conservative sufficient condition because norm estimation is used. Based on the sufficient condition, we propose a stabilization parameter search algorithm to numerically obtain the practical control parameters. The proposed algorithm is applied to a rotary crane system as one of the nonlinear systems with two under-actuated degrees. The effectiveness of the proposed control system design is verified by numerical simulations and experiments.

Constrained model predictive control for 3-D offshore boom cranes

Offshore boom cranes are complex nonlinear underactuated systems, whose control problems, when considering the 3-dimensional (3-D) model affected by ship-induced disturbances, are full of various challenges. In fact, it still remains an open problem to efficiently control the 3-D offshore boom cranes. Moreover, most existing works on offshore cranes concentrate upon designing the force/torque controller, whose performance degrades badly in the presence of friction. In this paper, considering the above issues, a novel model predictive control (MPC) method, which successfully considers the constraints of both input and output of the system, is proposed to achieve satisfactory control performance even under the effect of persistent ship roll and heave perturbations. Specifically, a discrete model is first obtained by some careful transformation and discretization on the unactuated dynamics equations, based on which a novel model predictive controller is constructed. To reduce the complexity of the system, the unactuated states and the accelerations of actuated states are considered as system states and control inputs respectively. After that, the requirements for the payload position accuracy and swing suppression as well as other system constraints, are taken into full account by converting them into input constraints to facilitate subsequent handling. At last, hardware experiments are implemented on a self-built testbed, with the obtained results clearly illustrating the effectiveness and robustness of the proposed method.

Hierarchical sliding mode-based adaptive fuzzy control for uncertain switched under-actuated nonlinear systems with input saturation and dead-zone

PurposeThis paper aims to study the issues of adaptive fuzzy control for a category of switched under-actuated systems with input nonlinearities and external disturbances.Design/methodology/approachA control scheme based on sliding mode surface with a hierarchical structure is introduced to enhance the responsiveness and robustness of the studied systems. An equivalent control and switching control rules are co-designed in a hierarchical sliding mode control (HSMC) framework to ensure that the system state reaches a given sliding surface and remains sliding on the surface, finally stabilizing at the equilibrium point. Besides, the input nonlinearities consist of non-symmetric saturation and dead-zone, which are estimated by an unknown bounded function and a known affine function.FindingsBased on fuzzy logic systems and the hierarchical sliding mode control method, an adaptive fuzzy control method for uncertain switched under-actuated systems is put forward.Originality/valueThe “cause and effect” problems often existing in conventional backstepping designs can be prevented. Furthermore, the presented adaptive laws can eliminate the influence of external disturbances and approximation errors. Besides, in contrast to arbitrary switching strategies, the authors consider a switching rule with average dwell time, which resolves control problems that cannot be resolved with arbitrary switching signals and reduces conservatism.

Barrier function-based nonsingular prescribed performance terminal sliding mode tracker for under-actuated nonlinear systems: Experimental study

In this paper, an adaptive concave barrier function scheme coupled with the non-singular terminal sliding mode control technique is proposed for finite-time tracking control of the under-actuated nonlinear system in the existence of model uncertainty, external disturbance and input saturation. Firstly, the dynamical equation of under-actuated nonlinear n-order system is expressed under model uncertainty, external disturbance and input saturation. Secondly, for the improvement of stability performance of the system in the existence of input saturation, a compensation system is designed to overcome the constraint on the control input. Afterward, the tracking errors between actual states of the system and differentiable reference signals are defined and the sliding surface based on the defined tracking errors is presented. Then, for gaining the better transient and steady-state performance of the closed-loop system, the prescribed performance control scheme is adopted. Based on this method, the transformed prescribed form of the previous determined sliding surface is obtained to ensure that the sliding surface can reach to a predefined region. Afterward, for assurance of the finite-time reachability of transformed sliding surface, the nonsingular terminal sliding surface is recommended. In addition, for the compensation of the model uncertainty and external disturbance existed in the system, the adaptive-based concave barrier function technique is used to estimate the unknown bounds of uncertainty and exterior disturbance. Finally, for demonstration of the proposed control method, the simulations and experimental implementation are done on the air levitation system.

High-order fully actuated system approaches: Model predictive control with applications to under-actuated systems

The main results of this paper are concentrated on the nonlinear model predictive control (MPC) tracking optimization based on high-order fully actuated (HOFA) system approaches. The proposed HOFA MPC strategy makes full use of full-actuation property to eliminate the nonlinear dynamics of the system, and then the nonlinear optimization problem is equivalently transformed into a series of easy-solve linear convex optimization problems. Different from general nonlinear MPC methods and the current optimal control of the HOFA system approach, an analytical controller with smooth and less energy is obtained by the moving horizon optimization. And it is proven that the proposed controller can stabilize the corresponding tracking error closed-loop system. Finally, not limited to FA systems, as examples, a nonlinear numerical under-actuated model in the mathematical sense and a benchmark nonlinear under-actuated mechanical system are transformed into corresponding equivalent HOFA systems, the simulation results are given to verify the effectiveness of the proposed strategy.