Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

In this paper, bifurcation theory is employed to classify different dynamical behaviors arising in an underactuated mechanical system subject to bounded controls. The methodology is applied to an inertia wheel pendulum consisting of a simple pendulum with a rotating disk at the end. Restricting the magnitude of the control action places an important obstacle to the design of a continuous controller capable of swinging-up and stabilize the pendulum at the inverted position: the arm only can reach that position by means of oscillations of increasing amplitude. The controller is derived from a simple nonlinear state-feedback law, followed by a saturating device that limits the maximum amplitude of the control action applied to the system. This bound gives birth to a rich dynamical behavior, including pitchfork and Hopf bifurcations of equilibria, saddle-node bifurcations of periodic orbits, homoclinic and heteroclinic bifurcations. The global dynamics is analyzed in terms of certain control gains and a two-parameter bifurcation diagram is derived. It is shown that the dynamics on this bifurcation diagram is organized in a pair of codimension-two rotationally symmetric bifurcation points. Finally, it is found out that when the control gains lie on a certain region in the parameter space simultaneous stabilization of the upright position together with a large basin of attraction is obtained. Simulation results show that almost global stabilization of the system can be achieved.     

A feedback linearization design for the control of vehicle’s lateral dynamics

In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s dynamical behavior.     

Resilient anti-disturbance dynamic surface control of uncertain strict-feedback systems subject to partial loss of actuator effectiveness

This paper is concerned with the tracking control of a class of uncertain strict-feedback systems subject to partial loss of actuator effectiveness, in addition to uncertain model dynamics and unknown disturbances. A resilient anti-disturbance dynamic surface control method is proposed to achieve stable tracking regardless of partial actuator faults. First, data-driven adaptive extended state observers are designed based on memory-based identifiers, such that the uncertain model dynamics, external disturbances and the unknown input gains due to actuator faults can be estimated. Next, a resilient anti-disturbance dynamic surface controller is developed based on recovered information from the data-driven adaptive extended state observers. After that, it is proven that the cascade system formed by the observer and controller is input-to-state stable. Finally, comparative studies are performed to validate the efficacy of the resilient anti-disturbance dynamic surface control method for nonlinear strict-feedback systems subject to partial loss of actuator effectiveness.

     

DYNAMICAL APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS II. GLOBAL ASYMPTOTIC BEHAVIOR OF TIME DISCRETIZATIONS ∗

SUMMARY

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how “numerical” basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.     

Velocity-based tuning of degree of homogeneity for finite-time stabilization and fault tolerant control of an ROV in the presence of thruster saturation and rate limits

This paper introduces a homogeneous controller along a fixed-time state and fault observer for finite-time stabilization and fault accommodation of a remotely-operated vehicle in the presence of actuator saturation and rate limits. For this, a novel tuning algorithm is improvised for manipulating the degree of homogeneity in homogeneous controllers to effectively acquire different properties from the overall control system. The tuning of degree of homogeneity is based on vehicle’s velocity. The proposed algorithm results in a switching-type controller, which undergoes three different stages during the operation, to eliminate the sensitivity of conventional finite-time and fixed-time controllers to large initial errors in the presence of thruster constraints. In addition, a new fixed-time fault and state observer is designed for the realization of output feedback control and fault tolerance by combining a fixed-time state observer with a fault estimation unit. In contrast to conventional extended-state observers, this observer considers the dynamics of the thruster system in its formulation so that better performance can be provided for the control system upon thruster failures. Control allocation is utilized to accommodate thruster failures and faults and to take account of thruster saturation and rate limits. Stability analyses are carried out for the overall control system and the proposed observer. It is shown that the closed-loop control system would be globally finite-time stable. The state estimation subsystem is fixed-time stable and the fault estimation unit is input-to-state stable. Simulations are carried out and comparisons are made with several finite-time and fixed-time controllers to outline the advantages of the proposed homogeneous controller and the benefits of the overall fault-tolerant control system.

     

Trajectory tracking of a class of under-actuated thrust-propelled vehicle with uncertainties and unknown disturbances

This paper deals with the problem of designing a controller for a thrust-propelled vehicle which steers the vehicle to track a 3D spatial path, while effective compensation for both time-varying disturbances and uncertainties is achieved as well. Taking advantage of extraction algorithm, we separate the design for the translational and rotational dynamics. A back-stepping-based controller and a sliding mode controller are, respectively, designed for the translational and rotational dynamics in succession. The stability of the control framework is established through Lyapunov analysis. A numerical simulation is also included in the paper to render the effectiveness of the proposed control scheme.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

Global motion analysis of energy-based control for 3-link planar robot with a single actuator at the first joint

This paper studies nonlinear control of a 3-link planar robot moving in the vertical plane with only the first joint being actuated while the two other revolute joints are passive (called the APP robot below). A nonlinear energy-based controller is proposed, whose objective is to drive the APP robot into an invariant set where the first link is in the upright position and the total mechanical energy converges to its value at the upright equilibrium point (all three links are in the upright position). By presenting and using a new property of the motion of the APP robot, without any condition on its mechanical parameters, this paper proves that if the control gains are larger than specific lower bounds, then only a measure-zero set of initial conditions converges to three strictly unstable equilibrium points instead of converging to the invariant set. This paper presents numerical results for a physical 3-link planar robot to validate the obtained theoretical results and to demonstrate a switch–and–stabilize maneuver in which the energy-based controller is switched to a linear state feedback controller that stabilizes the APP robot at its upright equilibrium point.     

Attitude control of a rigid spacecraft with one variable-speed control moment gyro

Nonlinear controllability and attitude stabilization are studied for the underactuated nonholonomic dynamics of a rigid spacecraft with one variable-speed control moment gyro(VSCMG), which supplies only two internal torques.Nonlinear controllability theory is used to show that the dynamics are locally controllable from the equilibrium point and thus can be asymptotically stabilized to the equilibrium point via time-invariant piecewise continuous feedback laws or time-periodic continuous feedback laws. Specifically,when the total angular momentum of the spacecraft-VSCMG system is zero, any orientation can be a controllable equilibrium attitude. In this case, the attitude stabilization problem is addressed by designing a kinematic stabilizing law, which is implemented through a nonlinear proportional and derivative controller, using the generalized dynamic inverse(GDI)method. The steady-state instability inherent in the GDI controller is elegantly avoided by appropriately choosing control gains. In order to obtain the command gimbal rate and wheel acceleration from control torques, a simple steering logic is constructed to accommodate the requirements of attitude stabilization and singularity avoidance of the VSCMG. Illustrative numerical examples verify the efcacy of the proposed control strategy.     

Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

Controller design for fractional-order interconnected systems with unmodeled dynamics

This paper focuses on the output feedback tracking control for fractional-order interconnected systems with unmodeled dynamics. The reduced order high gain K-filters are designed to construct the estimation of the unavailable system state. Unmodeled dynamics is extended to the general fractional-order dynamical systems for the first time which is characterized by introducing a dynamical signal r(t). An adaptive output feedback controller is established using the fractional-order Lyapunov methods and proposed by novel dynamic surface control strategy. Then, it is confirmed that the considered system is semi-globally bounded stable and the errors between outputs and the desired trajectories can concentrate to a small neighborhood of the origin. Finally, a simulation example is introduced to demonstrate the correctness of the supplied controller.

     

Lyapunov-Based Controller for the Inverted Pendulum Cart System

A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.Contributed by Prof. F. Pfeiffer.     

bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

The pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce bSTAB, an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language Matlab featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of bSTAB for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The bSTAB projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.

     

Synchronizations of rotating pendulums via self-learning terminal sliding-mode control subject to input nonlinearity

In this study, under the presentations of system uncertainties, external disturbances, and input nonlinearity in control, a self-learning terminal sliding-mode control scheme is adopted to control the synchronization and anti-synchronization between two chaotic rotating pendulums with different periods of harmonic parametric excitation. Without known bounds of the unmatched nonlinear dynamics, system uncertainties, and external disturbances, the proposed controller, which is associated with time-varying feedback gains, can achieve the control goals. Meanwhile, the feedback gains are not determined beforehand but are self-learning according to the learning rules. Some sufficient conditions for stable synchronizations are performed in the sense of the Lyapunov stability theorem. Besides, numerical simulations are provided to demonstrate the effectiveness of the present scheme.     

Stabilization of the Furuta Pendulum Based on a Lyapunov Function

We propose a Lyapunov-function-based control for the stabilization of the under-actuated Furuta pendulum. Firstly, by a suitable partial feedback linearization that allows to linearize only the actuated coordinate of the system, we proceed to find a candidate Lyapunov function. Based on this candidate function, we derive a stabilizing controller, in such away that the closed-loop system is locally and asymptotically stable around the unstable vertical equilibrium rest, with a computable domain of attraction.     

索-梁耦合结构Hopf分岔的反控制

本文研究了索-梁耦合结构的Hopf分岔的反控制,动态窗口滤波反馈控制器在反控制领域有着很广泛的应用。本文通过使用这种控制器,可以使得受控系统在指定的平衡点处产生Hopf分岔。最后,根据庞加莱截面和级数展开法,证明了上述方法的有效性及可行性。     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Mechanism analysis and optimum control of negative airgap eccentricity effect for in-wheel switched reluctance motor driving system

In this paper, the generation mechanism of the negative airgap eccentricity effect for the in-wheel switched reluctance motor (SRM) driving system is analyzed. An independent current chopping control strategy is proposed to achieve optimum control between the response characteristic of the in-wheel motor driving system and the dynamic performance of electric vehicle (EV). Firstly, the electromagnetic characteristic of the studied SRM under airgap eccentricity is studied based on electromagnetic coupling model and circuit driving equation, and the radial electromagnetic force under different airgap eccentricity is verified by adopting the built experiment device. Then, combined with the excitation characteristics of the radial electromagnetic force, the negative dynamic effect of the in-wheel motor driving system is analyzed in the time–frequency domain. Finally, an independent current chopping control strategy for the in-wheel SRM driving system based on vehicle vibration feedback is proposed. The controller parameters including the turn-off angle and chopping current threshold are optimized by data interpolation. Results show that the proposed control strategy can achieve the optimum control between the response characteristics of the in-wheel motor driving system and the vehicle dynamic performance, especially to suppress the vehicle sprung mass acceleration and tire bounce while starting EV.

     

Displaced orbits generated by solar sails for the hyperbolic and degenerated cases

Displaced non-Keplerian orbits above planetary bodies can be achieved by orientating the solar sail normal to the sun line. The dynamical systems techniques are employed to analyze the nonlinear dynamics of a displaced orbit and different topologies of equilibria are yielded from the basic configurations of Hill’s region, which have a saddlenode bifurcation point at the degenerated case. The solar sail near hyperbolic or degenerated equilibrium is quite unstable. Therefore, a controller preserving Hamiltonian structure is presented to stabilize the solar sail near hyperbolic or degenerated equilibrium, and to generate the stable Lissajous orbits that stay stable inside the stabilizing region of the controller. The main contribution of this paper is that the controller preserving Hamiltonian structure not only changes the instability of the equilibrium, but also makes the modified elliptic equilibrium become unique for the controlled system. The allocation law of the controller on the sail’s attitude and lightness number is obtained, which verifies that the controller is realizable.