Underactuated Nonlinear Systems Research Articles

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

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

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.

Finite-Time Control of a Class of Nonlinear Underactuated Systems with Application to Underactuated Axisymmetric Spacecraft

In this paper, the finite-time control problem of a class of nonlinear underactuated systems is addressed. By using the auxiliary state generated by a time-varying oscillator-like differential equation and constructing a novel Lyapunov-like function, a time-varying dynamic feedback controller is designed for such nonlinear underactuated systems. It is proved that the state of the closed-loop system converges to zero at the finite time and the controls are bounded. As an application, the finite-time tracking problem for underactuated axisymmetric systems is solved. Numerical simulations verify the effectiveness of the proposed methods.

Adaptive Fuzzy Sliding Exact Tracking Control Based on High-Order Log-Type Time-Varying BLFs for High-Order Nonlinear Systems

This article investigates the adaptive fuzzy sliding exact tracking control problem of a class of high-order nonlinear systems with full-state constraints and mismatched external disturbances. To achieve exact tracking control, the upper bound of a composite polynomial is estimated, which is composed of approximation error, mismatched disturbance, and virtual control law and its upper bound estimated information is employed in the controller design. In this way, once the tracking error deviates from the origin, the corresponding adaptive fuzzy sliding exact tracking control mechanism will be activated to force it to slide along the origin, and then exact tracking control is achieved. Meanwhile, high-order log-type time-varying barrier Lyapunov functions are introduced to ensure that the full-state constraint is not violated. A 2-D high-order nonlinear system and an underactuated nonlinear benchmark system with weak coupling are considered to illustrate the characteristics of the control mechanism in simulation.

A two stage Active Disturbance Rejection Control design for under-actuated nonlinear systems

A two-stage Active Disturbance Rejection Control (ADRC) design procedure is presented for the output trajectory tracking control problem in Single Input Single Output (SISO) nonlinear systems. Aside from the given scalar system output, the plant is assumed to exhibit a second, physically measurable, internal scalar output variable. The system output is controlled in an ADRC manner from an implicit static, or dynamic, nonlinear feedback controller using, in general, a nonlinear differential expression of the internal output variable. A desired internal output trajectory is thus generated. The internal output variable, in turn, is controlled from the original input and forced to track the desired trajectory synthesized in the previous step by the auxiliary controller. Each ADRC controller design is based on a simplified model of the corresponding subsystem, usually relegating interaction variables and additive nonlinearities to be part of a local (total) disturbance input term. The control scheme is reduced to a set of linear ADRC controllers with, possibly, cancellation of nonlinear input gains which are assumed to be known. Two illustrative aerospace examples are presented where the proposed procedure is illustrated via digital computer simulations.

Computing of LQR Technique for Nonlinear System Using Local Approximation

The main idea behind the present research is to design a state-feedback controller for an underactuated nonlinear rotary inverted pendulum module by employing the linear quadratic regulator (LQR) technique using local approximation. The LQR is an excellent method for developing a controller for nonlinear systems. It provides optimal feedback to make the closed-loop system robust and stable, rejecting external disturbances. Model-based optimal controller for a nonlinear system such as a rotatory inverted pendulum has not been designed and implemented using Newton-Euler, Lagrange method, and local approximation. Therefore, implementing LQR to an underactuated nonlinear system was vital to design a stable controller. A mathematical model has been developed for the controller design by utilizing the Newton-Euler, Lagrange method. The nonlinear model has been linearized around an equilibrium point. Linear and nonlinear models have been compared to find the range in which linear and nonlinear models’ behaviour is similar. MATLAB LQR function and system dynamics have been used to estimate the controller parameters. For the performance evaluation of the designed controller, Simulink has been used. Linear and nonlinear models have been simulated along with the designed controller. Simulations have been performed for the designed controller over the linear and nonlinear system under different conditions through varying system variables. The results show that the system is stable and robust enough to act against external disturbances. The controller maintains the rotary inverted pendulum in an upright position and rejects disruptions like falling under gravitational force or any external disturbance by adjusting the rotation of the horizontal link in both linear and nonlinear environments in a specific range. The controller has been practically designed and implemented. It is vivid from the results that the controller is robust enough to reject the disturbances in milliseconds and keeps the pendulum arm deflection angle to zero degrees.

Model-Free Controller Design for Nonlinear Underactuated Systems With Uncertainties and Disturbances by Using Extended State Observer Based Chattering-Free Sliding Mode Control

Most of the control strategies require a mathematical model or reasonable knowledge that is difficult to obtain for complex systems. Model-free control is a good alternative to avoid the difficulties and complex modeling procedures, especially if the knowledge about the system is insufficient. This paper presents a new control scheme completely independent of the system model. The proposed scheme combines sliding mode control (SMC) with intelligent proportional integral derivative (iPID) control based on a local model and extended state observer (ESO). Although the iPID control makes the proposed method model-free, it cannot guarantee that the tracking errors converge to zero asymptotically except the system is in a steady-state regime. Therefore, the SMC is added to the control scheme to ensure the convergence by minimizing the estimation errors of the observer. The proposed iPIDSMC controller is tested in the presence of different parameter variations and external disturbances on an inverted pendulum - cart (IPC), which is a highly unstable underactuated system with nonlinear coupled dynamics. The proposed controller is compared with the PID, iPID and Hierarchical Sliding Mode Control (HSMC) for a clearer evaluation. Simulation results showed that the proposed controller is extremely insensitive to parameter variations, matched and mismatched disturbances and the control signal of the proposed method is chattering-free, even though it is based on a discontinuous control action.

Combined algorithms for analytical inverse kinematics solving and control of the Q-PRR aerial manipulator*

This article presents the design and modeling of a new aerial manipulator system, called Q-PRR, composed of three joints with a fixed base in the center of mass of the multirotor considered as a whole system. This structure has a prismatic joint as a first joint which allows to keep the center of gravity of the Q-PRR as close as possible to the center of gravity of the multirotor. This will also allow to reduce the influence of arm motion on the multirotor roll thus to ensure the stability of the system on trajectory tracking with dynamic changes in the multirotor’s center of gravity. Furthermore, the configuration of the manipulator arm for the desired position of the end-effector given by the inverse kinematics model is kept without any change in the position and attitude of the multirotor. This article develops both forward and inverse kinematics models for a nonlinear underactuated system using the Denavit–Hartenberg notation. When a new algorithm is presented for the inverse kinematics based on Levenberg–Marquardt algorithm. Then, the dynamic model in the joint spaces is developed with the Lagrangian formalism. The Q-PRR is controlled using a model-free control with a comparison of two states, a free fly and disturbance forces applied to the whole system with manipulator arm movement.

A Constructive Procedure for Orbital Stabilization of a Class of Underactuated Mechanical Systems

In this note, we present a constructive procedure to solve the orbital stabilization problem of a class of nonlinear underactuated mechanical systems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> degrees of freedom and underactuation degree one using the Immersion and Invariance technique. We define sufficient conditions to solve explicitly the partial differential equations arising in the Immersion and Invariance methodology, so that mechanical systems with gyroscopic terms can be considered. At the end, we use two practical systems as examples to illustrate the design procedure, as well as validating performance via simulations and experiments.