Underactuated Nonlinear Systems Research Articles

Immersion and Invariance-Based Adaptive Controller for Quadrotor Systems

This paper develops a novel immersion and invariance (I&I)-based adaptive controller for nonlinear under-actuated quadrotor systems subject to uncertain inertial parameters. The objective is to drive the quadrotor to accurately track a reference trajectory. The controller synthesis is based on a hierarchical framework. First, the solution to the trajectory tracking problem is transformed to stabilizing its corresponding error system asymptotically. Second, to avoid singularity in the command attitude extraction, a saturated backstepping control strategy with smooth hyperbolic tangent functions is developed to synthesize a command force in the position loop, where an I&I adaptive methodology is introduced to update the mass estimation. The I&I adaption guarantees the asymptotic convergence of the mass estimation to its actual value. Third, in the attitude loop, the projection algorithm is used to update the inertial tensor estimations, which enables the estimations to be maintained within a prescribed range. Then, a rotation matrix-based applied torque with an initial condition constraint is synthesized. It is proven that, with appropriate control parameters, the proposed controller guarantees the asymptotic stability of the error system and then the trajectory tracking of the quadrotor is achieved. Finally, numerical simulations are given to verify and highlight the performance of the proposed controller.

Stabilization of underactuated nonlinear systems to non‐feasible curves

AbstractThis paper focuses on the development of stability conditions for systems of nonlinear non‐autonomous ordinary differential equations and their applications to control problems. We present a novel approach for the study of asymptotic stability properties for nonlinear non‐autonomous systems based on considering a parameterized family of sets. The proposed approach allows to state asymptotic stability conditions for a family of sets representing the level sets of a time‐varying Lyapunov function and to estimate the rate of convergence of solutions to a prescribed neighbourhood of the given curve. The obtained stability results are applied to the trajectory tracking problem for a class of nonholonomic systems.

An optimal adaptive hybrid controller for a fourth-order under-actuated nonlinear inverted pendulum system

In this paper, a combination of approximate feedback linearization and sliding mode control approaches is applied to stabilize a class of fourth-order nonlinear systems. In order to improve the performance of the proposed controller, design parameters of the sliding surface are adjusted using the adaptation laws, based on the gradient descent technique. Eventually, a new version of particle swarm optimization is implemented to optimize the control gains with respect to the integral of time-weighted absolute errors as the objective function, and a constraint on the maximum control effort value. The introduced idea is put into practice for the under-actuated inverted pendulum system, and the procedure is described with details step by step. The time responses of cart position, pendulum angle, and control effort of the proposed method is then compared with those of several recently published references so as to evaluate the effectiveness of this scenario.

Stabilization of Uncertain Feedforward Nonlinear Systems With Application to Underactuated Systems

This note presents saturated controllers to stabilize uncertain feedforward nonlinear systems whose nominal dynamics contains uncertain “gains” and is subject to linear perturbations. We assign a class of saturated controllers within which multiplicative coefficients appear before both states and saturation functions, and the convergence analysis is conducted in the following way: to verify the reduction of saturated terms, we calculate the time derivative of a boundary surface in a small domain when the states reach the boundary surface; to prove the asymptotic stability of the corresponding reduced system, we use an $M$ -matrix-based comparison principle. In this way, the suggested algorithm does not depend on the small-gain theory, and is suitable for dealing with strongly nonlinear underactuated systems including the cart-pendulum system, the overhead crane system, and the uncertain vertical takeoff and landing vehicle.

A holistic optimization approach for inverted cart-pendulum control tuning

The inverted cart-pendulum (ICP) is a nonlinear underactuated system, which dynamics are representative of many applications. Therefore, the development of ICP control laws is important since these laws are suitable to other systems. Indeed, many nonlinear control strategies have emerged from the control of the ICP. For these reasons, the ICP remains a canonical and fundamental benchmark problem in control theory and robotics that is of interest to the scientific community. Till now, the trial-and-error method is still widely applied for ICP controller tuning as well as the sequential tuning referring to tune the swing-up controller and thereafter, the stabilization controller. Therefore, the aim of this paper is to automate and facilitate the ICP control in one step. Thus, this paper proposes to holistically optimize ICP controllers. The holistic optimization is performed by a simplified Ant Colony Optimization method with a constrained Nelder–Mead algorithm (ACO-NM). Holistic optimization refers to a simultaneous tuning of the swing-up, stabilization and switching mode parameters. A new cost function is designed to minimize swing-up time, achieve high stabilization performance and consider system constraints. The holistic approach optimizes four controller structures, which include controllers that have never been tuned by a specific method besides by the trial-and-error method. Simulation results on a ICP nonlinear model show that ACO-NM in the holistic approach is effective compared to other algorithms. In addition, contrary to the majority of work on the subject, all the optimized controllers are validated experimentally. The simulation and experimental results obtained confirm that the holistic approach is an efficient optimization tool and specifically responds to the need of optimization technique for the potential-well controller structure and for the Q [diagonal of the matrix and the full matrix] in the linear–quadratic regulator (LQR) technique. Moreover, ICP experimental response analysis demonstrates that using the full Q provides greater experimental stabilization performance than using its diagonal terms in the LQR technique.

Switching Logic-Based Nonlinear Feedback Control of Offshore Ship-Mounted Tower Cranes: A Disturbance Observer-Based Approach

We propose, in this paper, an observer-based nonlinear feedback controller for four degrees of freedom (4-DOF) offshore ship-mounted tower cranes, which includes a robust-type term to address the observation uncertainty, and a switching logic tuning mechanism to update the involved unknown parameter. As a typical nonlinear underactuated mechatronic system, unlike the land-fixed tower cranes, an offshore ship-mounted tower crane also suffers from persistent disturbances caused by sea waves or currents, which leads to many difficulties and challenges in the controller design of such systems. Existing offshore crane control methods require either linearizations or approximations when performing analysis; moreover, most of them are only applicable for simplified 2-DOF crane models, and the ranges of system state errors cannot be guaranteed during the overall process. Motivated by these facts, to achieve simultaneous accurate jib/trolley positioning and fast payload swings suppression against complex unknown external disturbances, in particular, a disturbance observer is first designed in this paper, based on which a novel nonlinear switching logic-based control scheme for offshore cranes with jib rotation and horizontal transportation is proposed. As far as we know, the proposed method yields the first observer-based feedback closed-loop control without simplification operations to the original 4-DOF offshore tower crane dynamics and achieves the first asymptotic stability of the closed-loop system’s equilibrium point for offshore cranes. To support the theoretical derivation, the corresponding stability analysis of the closed-loop system’s equilibrium point is implemented by Lyapunov techniques. Numerous simulation and hardware experimental results are presented to demonstrate the superior performance of the proposed method. Note to Practitioners —This paper is motivated by the issue of controlling a four- degree-of-freedom (4-DOF) offshore ship-mounted tower crane system under the effects of unknown external disturbances. In practical applications, the persistent disturbances caused by sea waves or currents can always happen and make the offshore crane operation very challenging. Moreover, it brings more difficulties in the controller design for such a system due to its nonlinear underactuated property. Existing control approaches for offshore cranes are developed based upon linearized or oversimplified crane models or require exact model knowledge, and they may not work well in the presence of harsh sea conditions. Toward this end, this paper suggests a new control approach for 4-DOF offshore tower cranes suffering from sea wave effects to achieve a satisfactory performance. The asymptotic stability of the closed-loop system’s equilibrium point is derived, implying that the accurate jib/trolley positioning and fast payload swings suppression against complex unknown external disturbances are achieved simultaneously. Preliminary physical experiments implemented on a self-built offshore tower crane hardware test bed verify the effectiveness and the robustness of the proposed observer-based control method. In the future research, we will apply the suggested control approach to industrial offshore crane systems to improve their working efficiency.

Sliding mode control design for stabilization of underactuated mechanical systems

Stabilization of underactuated mechanical systems is one of the fundamental benchmark problem in the field of control theory. In this article, a hierarchical sliding mode control based on state-dependent switching gain is proposed for stabilization of underactuated mechanical systems. This controller is based on the so called first-level and second-level sliding surfaces. The asymptotic stability of these surfaces is proved by the Lyapunov stability theory. The proposed control technique is applied to two nonlinear underactuated mechanical systems and its feasibility is verified by numerical simulation. The proposed controller efficiently tackles the bounded external disturbance and shows robust performance. The convergence rate of the proposed controller is much faster as compared with the conventional decoupled sliding mode control and the integral sliding mode control.

Design of motion trajectory and tracking control for underactuated cart‐pendulum system

SummaryA cart‐pendulum system is a nonlinear underactuated mechanical system with two degrees of freedom. This paper addresses the motion trajectory design and tracking control problems for this underactuated system. First, a friction‐like control law is designed for the system. Then, the characteristics of the closed‐loop control system are analyzed. Second, a new method of constructing an optimal trajectory for the system is developed. Then, a tracking control law is designed to quickly track the constructed trajectory. It guarantees that the motion control of the cart‐pendulum system is achieved along a reference trajectory. Finally, a numerical example is presented to demonstrate the effectiveness of the theoretical results. This study constructs an optimal trajectory for the cart‐pendulum system in its whole motion space and solves the motion control objective by tracking the constructed trajectory. It has many advantages compared with other motion control methods, eg, the optimal motion control objective of the system is achieved by a single control law; and the motion process and transient characteristics (eg, the settling time) of the control system can be accurately predicted.

A novel adaptive robust control approach for underactuated mobile robot

Underactuated mobile robot (UMR) is a typical nonlinear underactuated system with nonholonomic and holonomic constraints. Based on the model of UMR, we propose a novel adaptive robust control to control the UMR and compensate the uncertainties from the view of constraint-following. The uncertainties, which are (possibly fast) time-varying and bounded, include modeling error, initial condition deviation, friction force and other external disturbances. However, the bounds are unknown. To estimate the bounds of the uncertainties, we design an adaptive law which is of leakage type. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by Lyapunov method. Furthermore, the effectiveness of the control is shown via numerical simulation of a case.

Adaptive nonlinear control of an autonomous underwater vehicle

Autonomous underwater vehicles (AUVs) are highly nonlinear underactuated systems with uncertain dynamics and a challenging control problem. The main focus of this paper is to present a control law that shows desirable performance in the presence of modeling uncertainties. In this study, uncertainties are considered to be bounded and the AUV mathematical model is obtained in the presence of such uncertainties. Forces and torques applied to the AUV are also designed using a nonlinear dynamic controller. Appropriate adaptive rules are also presented to overcome system uncertainties and external disturbances. The adaptive nonlinear dynamic controller is designed based on upper bounds of system uncertainties and its stability is proven using the Lyapunov theory. In this article, the performance of the proposed control algorithm for tracking reference trajectories in an obstacle-rich environment is investigated. Therefore, the control algorithm is combined with potential fields for obstacle avoidance. Obtained results show the efficiency of the proposed controller.

Optimal self-tuning decoupled sliding mode control for a class of nonlinear systems

A decoupled sliding mode controller (DSMC) is well-known as a simple way to achieve asymptotic stability of the nonlinear under-actuated systems. This method has many advanced features such as good performance and robustness against parameter variations. However, because of changing the system states, the controller with the constant parameters would not be optimum in any state of the system, and designing DSMC requires an adaptation for the controller parameters. Hence, in this paper, a robust self-tuning decoupled sliding mode controller (RSDSMC) is presented and optimised with five conflicting objective functions and twelve design variables using a multi-objective genetic algorithm. The proposed controller is applied to a highly nonlinear inverted pendulum and cart system via the computer simulation. The final results depict the appropriate performance of this new controller and demonstrate its superiority in comparison with those reported in literature.

Optimal self-tuning decoupled sliding mode control for a class of nonlinear systems

A decoupled sliding mode controller (DSMC) is well-known as a simple way to achieve asymptotic stability of the nonlinear under-actuated systems. This method has many advanced features such as good performance and robustness against parameter variations. However, because of changing the system states, the controller with the constant parameters would not be optimum in any state of the system, and designing DSMC requires an adaptation for the controller parameters. Hence, in this paper, a robust self-tuning decoupled sliding mode controller (RSDSMC) is presented and optimised with five conflicting objective functions and twelve design variables using a multi-objective genetic algorithm. The proposed controller is applied to a highly nonlinear inverted pendulum and cart system via the computer simulation. The final results depict the appropriate performance of this new controller and demonstrate its superiority in comparison with those reported in literature.

A novel chattering-free pi sliding mode control for a class of nonlinear underactuated systems

For the tracking and stabilisation control problem of nonlinear underactuated systems, a novel chattering-free sliding mode control approach is proposed in this paper. According to Lyapunov theorem of stability, sliding mode can be around the sliding surface in a finite time by the control law. Moreover, the chattering phenomenon caused by the discontinuous control law can be eliminated by the proposed approach. Simulation results for a bipedal walking robot system demonstrate the feasibility and efficiency of the introduced design. Furthermore, it is noteworthy that the developed approach can be widely applied to kinds of underactuated nonlinear control problems.

A novel chattering-free pi sliding mode control for a class of nonlinear underactuated systems

A novel chattering-free pi sliding mode control for a class of nonlinear underactuated systems

Time-Scale Expansion-Based Approximated Optimal Control for Underactuated Systems Using Projection Neural Networks

In this paper, a time-scale expansion-based scheme is proposed for approximately solving the optimal control problem of continuous-time underactuated nonlinear systems subject to input constraints and system dynamics. By time-scale Taylor approximation of the original performance index, the optimal control problem is relaxed into an approximated optimal control problem. Based on the system dynamics, the problem is further reformulated as a quadratic programming problem, which is solved by a projection neural network. Theoretical analysis on the closed-loop system synthesized by the controlled system and the projection neural network is conducted, which reveals that, under certain conditions, the closed-loop system possesses exponential stability and the original performance index converges to zero as time tends to infinity. In addition, two illustrative examples, which are based on a flexible joint manipulator and an underactuacted ship, are provided to validate the theoretical results and demonstrate the efficacy and superiority of the proposed control scheme.

A novel trajectory planning-based adaptive control method for 3-D overhead cranes

ABSTRACT A three-dimensional (3-D) overhead crane is a complicated nonlinear underactuated mechanical system, for which high-speed positioning and anti-sway control are the kernel objective. Existing trajectory-based methods for 3-D overhead cranes focus on combining efficient and smooth trajectories with anti-sway tracking controllers without regard for payload motion; moreover, the exact value of plant parameters is required for accurate compensation during the control process. Motivated by these facts, we present a two-step design tracking strategy which consists of a trajectory planning stage and an adaptive tracking control design stage for 3-D overhead cranes. As shown by Lyapunov techniques and Barbalat's Lemma, the proposed controller guarantees asymptotic swing elimination and trolley positioning result in the presence of system uncertainties including unknown parameters and external disturbances. Simulation results also showed the applicability of the proposed method with good robustness against parameter uncertainties and external disturbances.

Adaptive IDA‐PBC for underactuated mechanical systems with constant disturbances

SummaryThis work investigates the control of nonlinear underactuated mechanical systems with matched and unmatched constant disturbances. To this end, a new control strategy is proposed, which builds upon the interconnection‐and‐damping‐assignment passivity‐based control, augmenting it with an additional term for the purpose of disturbance compensation. In particular, the disturbances are estimated adaptively and then accounted for in the control law employing a new matching condition of algebraic nature. Stability conditions are discussed, and for comparison purposes, an alternative controller based on partial feedback linearization is presented. The effectiveness of the proposed approach is demonstrated with numerical simulations for three motivating examples: the inertia wheel pendulum, the disk‐on‐disk system, and the pendulum‐on‐cart system.

Optimal interval type-2 fuzzy fractional order super twisting algorithm: A second order sliding mode controller for fully-actuated and under-actuated nonlinear systems

In this paper, a novel interval type-2 fuzzy fractional order super twisting algorithm (IT2FFOSTA) which is essentially a second order sliding mode controller is presented. The proposed IT2FFOSTA enhances fractional order super twisting algorithm (FOSTA) by taking advantage of an interval type-2 fuzzy fractional order sliding surface (IT2FFOSS) for some classes of fully-actuated and under-actuated nonlinear systems in presence of uncertainty. The FOSTA significantly reduces the amount of chattering and the IT2FFOSS results in decreasing the tracking error, control effort, and chattering level. In order to control under-actuated systems, a hierarchical sliding surface is employed. The multi-tracker optimization algorithm is utilized to adjust the controller’s parameters; this leads to an optimal performance for the IT2FFOSTA. To examine the performance of the IT2FFOSTA, some simulation and experimental tests on three examples of different classes of fully-actuated and under-actuated systems, including ball and plate, inverted pendulum, and ball and beam systems are carried out. The simulation and experimental results demonstrate the superiority of the IT2FFOSTA in reducing the amount of chattering, tracking error, and control effort compared to those of the other control methods.

Continuous Sliding Mode Control Strategy for a Class of Nonlinear Underactuated Systems

Sliding mode control (SMC), which is known to present strong robustness against various disturbances, has been extensively employed on underactuated systems, whose control problem has been a research focus in recent years. However, though received great attention, the study on SMC for underactuated systems still remains quite open since it is very challenging to construct a proper manifold which can simultaneously stabilize both actuated and unactuated system states. Motivated to promote the corresponding research, we propose an improved second-order SMC (IS-SMC) method for a class of nonlinear underactuated systems in this paper, which guarantees that the closed-loop system's equilibrium point is asymptotically stable even in the presence of unknown nondiminishing disturbances. Different from traditional SMC methods presenting with chattering problem, the control inputs of the proposed method are essentially continuous, which brings much convenience for practical applications. Moreover, for the disturbances which are second-order differentiable, an enhanced IS-SMC (EIS-SMC) method is derived to guarantee the smoothness for both the control inputs and their derivatives. Finally, the proposed approach is applied to a practical underactuated system: The overhead crane system, with sufficient simulation results provided to validate the efficiency of the proposed method.

An Anti-Swing Closed-Loop Control Strategy for Overhead Cranes

The payload swing of an overhead crane needs to be controlled properly to improve efficiency and avoid accidents. However, the swing angle is usually very difficult to control to zero degrees or for it to even remain within an acceptable range because the overhead crane is a complex nonlinear underactuated system, especially when the actual working environment is accompanied by strong disturbances and great uncertainty. To resolve this, a real-time anti-swing closed-loop control strategy is proposed that considers external disturbances. The swing angle is measured in time and it functions with the load displacement as feedback inputs of the closed-loop system. The nonlinear model of the crane is simplified by a linear system with virtual disturbances, which are estimated by the equivalent input disturbance (EID) method. Both simulation and experimental results for a 2-D overhead crane system are investigated to illustrate the validity of the proposed method.