Semi-global Asymptotic Stability Research Articles

Rotary inverted pendulum control using neuro-adaptive robust generalized dynamic inversion

The objective of this research is to design a robust control system for trajectory tracking of the under-actuated Rotary Inverted Pendulum (RIP). The focus is on achieving semi-global asymptotic stability in the absence of knowledge about the RIP geometric and inertia parameters. The control design commences by formulating differential servo constraint dynamics (VCD) that encapsulates the control objectives for the RIP. The baseline generalized dynamic inversion (GDI) control law is derived by inverting the VCD for the control voltage input using the Moore–Penrose generalized inverse. To enhance robustness against modeling uncertainties and exogenous disturbances, a sliding mode control (SMC) element is integrated within the GDI control system, resulting in the robust GDI (RGDI) control system. Furthermore, an adaptive estimator that is based on Radial Basis Function Neural Networks (RBF-NN) is adopted to eliminate the dependency of the RGDI control system on the RIP mathematical model. The weighting matrices of the RBF-NN are updated with the aid of a Lyapunov control function. The proposed control system offers robust solution and guarantees semi-global asymptotic stability to the unstable equilibria of the RIP dynamics in the absence of a mathematical model. The proposed control system’s performance is evaluated through computer simulations and experimental tests on a real RIP system. Comparative analyses with traditional SMC and linear quadratic control methodologies are conducted to show the effectiveness of the proposed approach.

Reinforcement learning swarm of self-organizing unmanned surface vehicles with unavailable dynamics

In this paper, self-organizing a swarm of unmanned surface vehicles (USVs) is creatively solved by optimal reinforcement learning control approach without acquiring USV dynamics. Self-organizing swarm mechanism for flexible decision-making is established by combing control theory with repulsive potentials, thereby lifting orientation constraints on USV swarm coordination. Distributed finite-time extended state observers are devised to sufficiently accommodate multiple unknowns covering internal dynamics and external uncertainties/disturbances. Swarming coordination of individual USV is optimally synthesized by reinforcement learning-based distributed control architecture. As a consequence, semi-global asymptotic stability of the entire self-organizing swarm can be rigorously ensured by Lyapunov analysis. Virtual-reality experiments show that reinforcement learning swarm is skilled in coordinating a herd of USVs tracking a trajectory.

Semi-global robust stabilisation of disturbed nonlinear systems with state-dependent uncertainties: a DOBC approach

The robust stabilisation for a general class of disturbed nonlinear systems subject to state-dependent uncertainties is addressed in this paper. Different from most existing results, the lumped uncertainty, including external disturbances and model uncertainties, is relaxed to be bounded by some nonlinear functions rather than constants in this paper. Based upon the classical design procedure of disturbance observer-based control (DOBC), a new continuous saturation-based dynamic controller is proposed to estimate the lumped uncertainty. By the delicate design of several compact sets, the qualitative robust stability of the closed-loop system is guaranteed in the sense that all the signals of the closed-loop system can converge to an arbitrarily small compact set in a finite time. Furthermore, when the external disturbances are constants, the semi-global asymptotic stability can also be guaranteed by the proposed controller. Finally, the relationship between system uncertainties and observer gain is shown by a numerical simulation example.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

Adaptive RISE Feedback Control for Robotized Machining With PKMs: Design and Real-Time Experiments

The development of high-precision tasks, such as machining, needs a positioning device for the cutting tool with the smallest possible error. Multiple design factors need to be considered to ensure a mechatronic device successfully performs such tasks. One of these factors may be attributed to the control scheme, which is responsible for controlling the position of the machine. In view of the importance of designing a good control scheme for a robotic system, in this article, we propose a new extension of the robust integral sign of the error (RISE) for the positioning device a parallel kinematic machine (PKM). This extension consists in including a nominal feedforward term based on the inverse dynamic model of the robot and replacing the RISE fixed feedback gains with adaptive ones. The RISE part of the proposed controller ensures semiglobal asymptotic stability. Moreover, it can accommodate sufficiently smooth bounded disturbances. The feedforward part cancels the nonlinearities of the system, improving the tracking performance of the controller. The adaptive feedback gains produce corrective actions when an increase in the tracking errors is due to the contact forces that occur during the machining process. A Lyapunov-based stability analysis is conducted to prove the semiglobal asymptotic stability of the proposed control solution. To show its effectiveness, real-time experiments are performed for two case studies; the first one is on a free motion trajectory and the second on machining experiments under three different forward speeds on SPIDER4, a redundantly actuated PKM.

Rotary Inverted Pendulum Control using Robust Generalized Dynamic Inversion with Adaptive Neural Estimation

A novel robust generalized dynamic inversion control system with adaptive neural estimation (RGDI-ANE) is designed for trajectory tracking of the under actuated rotary inverted pendulum (RIP). The RGDI control design begins by prescribing a virtual differential constraint dynamics (VCD) that imitates the desired tracking control objectives. The baseline RGDI control law is derived by inverting the VCD using dynamically scaled Generalized Inversion, and then augmenting a sliding mode control (SMC) element to provide robustness against performance deterioration due to dynamic scaling of the Moore-Penrose generalized inverse. Furthermore, an adaptive estimator that is based on radial basis function neural networks (RBF-NN) is adopted to limit the dependency of RGDI control on the RIP mathematical model. The weighting matrices of the RBF-NN are updated using a Lyapunov control function. The closed loop RGDI-ANE control system is shown to guarantee semi-global asymptotic stability. Computer simulations along with experimental tests have been conducted on Quanser's RIP system to validate the performance of RGDI-ANE control, and the results are compared with the results obtained by using the SMC and the linear quadratic control methodologies.

Nonlinear PI control for semi-global asymptotic stabilization and robustness of SIQR model with inflow perturbations

For mitigation of the spread of human infectious diseases, this paper proposes nonlinear PI control laws of vaccination, isolation, and contact regulation. Using the SIQR model, the control laws are designed to achieve not only internal stability on an arbitrarily large domain in the state space, but also input-to-state stability with respect to inflow perturbations of immigrants and newborns. The control laws get rid of the precise knowledge of the nominal rates required by feedback control. This paper pursues incorporation of integral action in control to render a coarse estimate a real equilibrium. It is demonstrated how to accomplish it in the presence of severe range limitations of control inputs.

A Semiglobal Approach for Stabilizing Nonlinear Systems With State Delays by Memoryless Linear Feedback

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic state</i> feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">static linear</i> feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article.

Neural Network-Based Dual-Cylinder Synchronous Control of a Multi-Link Erection Mechanism

A dual-cylinder erection mechanism, in which two telescopic cylinders physically connect to a load, is a nonlinear system with model uncertainties and coupled dynamics. In this paper, a novel synchronous control algorithm with thrust-allocation law is proposed for eliminating the excessive internal forces caused by the unbalanced rotation and lateral moments during the erection process. With regulated internal forces, the “pull and drag” issue is attenuated and better synchronization performance is attained. For improved tracking accuracy, the inter-stage collision dynamics of the telescopic cylinder are considered for model compensation to enhance stage-changing and in-position performance. A radial basis function (RBF) neural network is utilized to estimate the model uncertainties and external disturbances, which alleviates reliance upon the accuracy of a system model for controller implementation. As a result, theoretical analysis revealed that the semi-global asymptotic stability and synchronized motion performance with decreased internal forces can be achieved via the presented synchronous controller with thrust-allocation strategy. Contrasting simulations were implemented on a multi-link erection mechanism and the results confirmed the superiority and effectiveness of the proposed synchronous control algorithm.

Intelligent Dynamic Inversion Controller Design for Ball and Beam System

The Ball and beam system (BBS) is an attractive laboratory experimental tool because of its inherent nonlinear and open-loop unstable properties. Designing an effective ball and beam system controller is a real challenge for researchers and engineers. In this paper, the control design technique is investigated by using Intelligent Dynamic Inversion (IDI) method for this nonlinear and unstable system. The proposed control law is an enhanced version of conventional Dynamic Inversion control incorporating an intelligent control element in it. The Moore-Penrose Generalized Inverse (MPGI) is used to invert the prescribed constraint dynamics to realize the baseline control law. A sliding mode-based intelligent control element is further augmented with the baseline control to enhance the robustness against uncertainties, nonlinearities, and external disturbances. The semi-global asymptotic stability of IDI control is guaranteed in the sense of Lyapunov. Numerical simulations and laboratory experiments are carried out on this ball and beam physical system to analyze the effectiveness of the controller. In addition to that, comparative analysis of RGDI control with classical Linear Quadratic Regulator and Fractional Order Controller are also presented on the experimental test bench.

Nonlinear PID global regulators with selftuned PD gains for robot manipulators: theory and experimentation

It is known that the Proportional Integral and Derivative (PID) regulator with constant and symmetric positive definite gain matrices (Kp, Kv and Ki), in closed loop with robotic manipulators, guarantees semiglobal asymptotic stability of the equilibrium point. In this paper, a fresh approach for control of robots manipulators based on a class of PID regulators with variable gain matrices Kp and Kv, which guarantee global asymptotic stability, is proposed. This PID control scheme, proposed with PD variable gains and fixed integral gains, improves the performance of the transient response and the handling of constraints, such as the limits of the actuator torques. Based on the authors experience, our proposal is the first scheme of a nonlinear PID regulator for robot manipulators whose PD gains are variable and its stability proof is global asymptotic in the sense of Lyapunov. The PD variable gains are directly dependent on the joint position errors and are fine tuned by fuzzy logic methodology. Good results were obtained on an experimental direct-drive robot arm of two degrees of freedom, when it was tested in order to verify the effectiveness of the new controller.

New results and examples in semiglobal asymptotic stabilization of nonaffine systems by sampled-data output feedback

For a class of SISO non-affine systems, we address the problem of how semiglobal asymptotic stabilization (SGAS) can be achieved by n-dimensional sampled-data output feedback. The new result of this paper gives a partial answer to the open question in the recent work Lin (2020), where (2n−1)-dimensional output feedback controllers were designed based on the dynamic extension method.

Semiglobally Asymptotically Stable Nonlinear Observer for Camera Aided Navigation

This brief presents a nonlinear observer that performs range estimation as well as gyro-bias estimation by using velocity, angular rate, and bearing angle measurements from landmarks at unknown locations. The observer is proved to have semiglobal asymptotic stability, and its performance is verified in simulations and on experimental data. The observer is demonstrated on an unmanned aerial vehicle (UAV) with a sensor setup consisting of camera, inertial measurement unit (IMU), and velocity measured by a global navigation satellite system (GNSS). This sensor suite is sufficient to replace the magnetometer and the altimeter.

Stabilization of discrete-time upper triangular nonlinear cascade systems using cross term constructed Lyapunov functional

In this study, semi-global practical asymptotic stability of a class of nonlinear cascade systems with upper triangular configuration has been investigated. In particular, using general results presented on stabilization of the discrete-time systems, a semi-global practical asymptotic stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear cascade systems have been presented. The controller-design framework is based on the approximate discrete-time model of the system and the corresponding cross term constructed Lyapunov function. To illustrate the effectiveness of the proposed scheme, it has been applied to some examples and also been compared with other counterpart results in this context.

Semiglobal asymptotic stabilization of nonlinear systems with triangular zero dynamics by linear feedback

This paper studies the problem of semiglobal asymptotic stabilization (SGAS) by state feedback for a class of nonlinear systems beyond the normal form. The zero dynamics is of a triangular structure and assumed to satisfy minimum-phase and ISS conditions. By taking advantage of the triangular structure, we develop a semiglobal backstepping approach for the construction of a set of parameterized Lyapunov functions, associated sublevel sets and nested high-gain linear state feedback laws, step-by-step. To prevent the shrinking of domains of attraction and overcome the complexity of the semiglobal design based on the rational Lyapunov functions (Teel and Praly, 1995), we introduce the notion of sublevel set and a set of parameterized Lyapunov functions that are well-defined in the entire space, and show how they can be constructed recursively, which are instrumental in solving the SGAS problem by partial linear state feedback.

Model reference adaptive tracking control for hydraulic servo systems with nonlinear neural-networks

It is well known that hydraulic systems typically suffer from heavy disturbances including parametric uncertainties and unknown disturbances. In order to attain high performance tracking control, this paper proposes a composite design of nonlinear neural-networks (NN) and continuous robust integral of the sign of the error (RISE) feedback controller. The control development incorporates a NN feedforward component to have a compensation for unknown state-dependent disturbances and to further improve the accuracy of feedforward compensation, meanwhile input parameter is updated online. To achieve asymptotic stability, a novel RISE term with NN-based feedforward component is developed for the first time to enable the incorporation of model reference adaptive control structure where acceleration signal is not employed. The proposed controller guarantees controlled hydraulic system a semi-global asymptotic stability. For the experimental results, the prescribed transient performance is tested under rectangular trajectory and the steady state performance is tested under sinusoidal trajectory.

RISE-based adaptive control for EICoSI exoskeleton to assist knee joint mobility

Exoskeleton devices are used to assist joint motion of subjects suffering from mobility deficiencies. Controlling an exoskeleton subjects to high nonlinearities, which are mainly due to the mechanical coupling, external disturbances, parameter uncertainties, and modeling errors. Keeping in view the requirement of a relatively accurate movement tracking while reducing the disturbances effects, there is a need for a robust controller. In this paper, an adaptive RISE (Robust Integral of Sign Error) controller is developed and implemented on the EICoSI (Exoskeleton Intelligently Communicating and Sensitive to Intention) knee exoskeleton. RISE has an advantage over standard controllers that it achieves semi-global asymptotic tracking even in the presence of unstructured disturbances. But to achieve this tracking, high control gains are required. Thus, to limit such high gains, RISE control strategy is combined with an adaptive controller, which has the advantage of improving the tracking performance while reducing the eventual overshoots. The stability of the coupled human/ exoskeleton system is analyzed based on Lyapunov theory and the system has shown semi-global asymptotic stability. The adaptive RISE controller gives better SNR (signal to noise ratio) by 11% and 2% as compared with adaptive and RISE controller respectively. In terms of tracking error, the adaptive RISE controller shows 9% more RMSE than adaptive controller but 41% less when compared with RISE controller Three experimental scenarios are analyzed to validate the proposed controller, namely (i) external disturbances, that could come from the ground during walking; (ii) induced payload, that could come from the resistive/assistive torque from the muscles and (iii) payload with external disturbances. The system is found to be robust and efficient in tracking the reference trajectories while maintaining limited error and high signal to noise ratio.

Adaptive RISE Control of Hydraulic Systems With Multilayer Neural-Networks

This paper focuses on developing an advanced nonlinear controller for hydraulic system to achieve asymptotic tracking with various disturbances. To accomplish this study, a multilayer neural-networks (NNs) estimator is first developed to improve the compensation accuracy of model-based feedforward control terms, which can greatly reduce the uncompensated disturbance, then a continuous robust integral of the sign of the error (RISE) control approach is effectively integrated with the multilayer NNs estimator to deal with the residual mismatched disturbance, in which the RISE feedback gain is adapted online to further decrease the high-gain feedback. At last, by considering the inwardness of matched disturbance in hydraulic systems, it is estimated by another multilayer NNs fully with the residual functional reconstruction inaccuracies handled by a novel adaptive term. As a result, theoretical analysis reveals that the proposed controller guarantees a semiglobal asymptotic stability. Extensively comparative experimental results verify the priority of the proposed control strategy, and a 0.2% dynamic tracking accuracy is achieved.

Robust object manipulation for tactile-based blind grasping

Tactile-based blind grasping addresses realistic robotic grasping in which the hand only has access to proprioceptive and tactile sensors. The robotic hand has no prior knowledge of the object/grasp properties, such as object weight, inertia, and shape. There exists no manipulation controller that rigorously guarantees object manipulation in such a setting. Here, a robust control law is proposed for object manipulation in tactile-based blind grasping. The analysis ensures semi-global asymptotic and exponential stability in the presence of model uncertainties and external disturbances that are neglected in related work. Simulation and hardware results validate the effectiveness of the proposed approach.

Sampled-data control of underwater gliders: digital redesign approach

This paper discusses a sampled-data control problem for a nonlinear underwater glider based on the digital redesign (DR) technique. An analog controller is pre-designed based on the fully feedback-linearised model. The condition for the DR problem is proposed in terms of linear matrix inequality, using the delta-operator discrete-time models. We show that the DRed sampled-data controller guarantees the semiglobal practical asymptotic stability of the nonlinear steady gliding motion in the sampling period. The numerical simulations demonstrate the advantage of the developed methodology in terms of the larger sampling period and the better preservation of the original analog control performance than the conventional emulation technique.