Search Results (21)

This work proposes an Adaptive Neural Control for the asymptotic stabilization of a chain of integrators at the origin. The proposed approach addresses the stabilization of the integrator chain by means of a control law whose applied signal is structurally bounded to $(-1,1)$ by the hyperbolic tangent architecture, i.e., $u\left(t\right)=tanh\left(z\right)$, where z represents a weighted linear combination of the system states and a bias term. Furthermore, an adaptation law for the weights is proposed, based on the classical backpropagation algorithm for neural networks. The stability analysis is conducted using singular perturbation theory, demonstrating that, under a sufficiently high learning rate, the closed-loop system exhibits a Standard Singular Perturbation Form. This formulation allows for the analysis of the system across two distinct time scales: the adaptation dynamics (fast subsystem) and the state dynamics (slow subsystem). Based on this formulation, explicit conditions on the learning rate and the initial conditions are derived to guarantee local asymptotic stability using Tikhonov’s theorem. These conditions characterize the region of attraction and ensure that the adaptive neural controller stabilizes the system. Numerical simulations were carried out to evaluate the controller’s performance under three different scenarios: ideal conditions, initialization outside the region of attraction, and a low learning rate. These scenarios illustrate the closed-loop system behavior and validate the theoretical conditions required for asymptotic stability. Furthermore, comparative numerical simulations were conducted on an Inverted Pendulum on a Cart system to benchmark the proposed Adaptive Neural Control against Linear Quadratic Regulator, Sliding Mode Control, and Nested Saturation Function controllers. Based on the Integral of Time-weighted Squared Error performance index, the Adaptive Neural Control demonstrated a significant reduction in control effort, achieving performance improvements of up to 95.02% compared to the aforementioned strategies. Full article

This paper investigates uncertainty-resilient stabilization of an inverted pendulum on a cart (IPOC) using an interval type-2 Takagi–Sugeno (IT2 T–S) fuzzy model and an LQR-based control framework. The IPOC dynamics are represented as a weighted combination of local linear subsystems, where interval firing strengths derived from upper and lower membership functions capture modeling uncertainties. An LQR state-feedback controller is designed for each subsystem, and the final control input is obtained by blending the local controllers according to the normalized firing strengths. To analyze stability, an LMI-based verification condition is established as a sufficient condition for the subsystem LQR controllers. Simulation results show that this condition is satisfied only in a limited operating region, while the closed-loop system can still remain stable even when the condition is violated, demonstrating the reduced conservatism and flexibility of the proposed approach. Furthermore, comparisons with the conventional PDC structure confirm that the proposed method provides greater design flexibility and enables a trade-off between robustness and transient-state performance. Full article

This study proposes a discrete multi-agent Q-learning framework for the online tuning of PID controllers in continuous dynamic systems with limited observability. The approach treats the adjustment of each PID gain ($k_p$, $k_i$, $k_d$) as an independent learning process, in which each agent operates within a discrete state space corresponding to its own gain and selects actions from a tripartite space (decrease, maintain, or increase its gain). The agents act simultaneously under fixed decision intervals, favoring their convergence by preserving quasi-stationary conditions of the perceived environment, while a shared cumulative global reward, composed of system parameters, time and control action penalties, and stability incentives, guides coordinated exploration toward control objectives. Implemented in Python, the framework was validated in two nonlinear control problems: a water-tank and inverted pendulum (cart-pole) systems. The agents achieved their initial convergence after approximately 300 and 500 episodes, respectively, with overall success rates of $49.6\%$ and $46.2\%$ in 5000 training episodes. The learning process exhibited sustained convergence toward effective PID configurations capable of stabilizing both systems without explicit dynamic models. These findings confirm the feasibility of the proposed low-complexity discrete reinforcement learning approach for online adaptive PID tuning, achieving interpretable and reproducible control policies and providing a new basis for future hybrid schemes that unite classical control theory and reinforcement learning agents. Full article

In this study, a nonlinear dynamic model is developed to examine the stability and vibration behavior of a self-balancing electric Segway operating over irregular stochastic terrains. The Segway is treated as a three-degrees-of-freedom cart–inverted pendulum system, incorporating elastic and damping effects at the wheel–ground interface. Road irregularities are generated in accordance with international standard using high-order filtered noise, allowing for representation of surface classes from smooth to highly degraded. The governing equations, formulated via Lagrange’s method, are transformed into a Lorenz-like state-space form for nonlinear analysis. Numerical simulations employ the fourth-order Runge–Kutta scheme to compute translational and angular responses under varying speeds and terrain conditions. Frequency-domain analysis using Fast Fourier Transform (FFT) identifies resonant excitation bands linked to road spectral content, while Kernel Density Estimation (KDE) maps the probability distribution of displacement states to distinguish stable from variable regimes. The Lyapunov stability assessment and bifurcation analysis reveal critical velocity thresholds and parameter regions marking transitions from stable operation to chaotic motion. The study quantifies the influence of the gravity–damping ratio, mass–damping coupling, control torque ratio, and vertical excitation on dynamic stability. The results provide a methodology for designing stability control systems that ensure safe and comfortable Segway operation across diverse terrains. Full article

This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural network optimization. Nevertheless, their integration with fractional-order error models within adaptive control paradigms remains unexplored and represents a promising avenue for research. The proposed control scheme extends the classical MRAC architecture by embedding Caputo fractional derivatives into the adaptive law governing parameter updates, thereby improving both convergence dynamics and control flexibility. To ensure optimal performance across multiple criteria, the controller parameters are systematically tuned using a multi-objective Particle Swarm Optimization (PSO) algorithm. Two fractional-order error models (FOEMs) incorporating fractional gradients (FOEM2-FG, FOEM3-FG) are investigated, with their stability formally analyzed via Lyapunov-based methods under conditions of sufficient excitation. Validation is conducted through both simulation and real-time experimentation on a physical pendulum-cart setup. The results demonstrate that the proposed fractional-order MRAC (FOMRAC) outperforms conventional MRAC, proportional-integral-derivative (PID), and fractional-order PID (FOPID) controllers. Specifically, FOMRAC-FG achieved superior tracking performance, attaining the lowest Integral of Squared Error (ISE) of $2.32\times {10}^{-5}$ and the lowest Integral of Squared Input (ISI) of 6.40 in simulation studies. In real-time experiments, FOMRAC-FG maintained the lowest ISE ($5.11\times {10}^{-6}$). Under real-time experiments with disturbances, it still achieved the lowest ISE ($1.06\times {10}^{-5}$), highlighting its practical effectiveness. Full article

This work will present an algorithmic approach for robust control focusing on hydraulic–mechanical systems. The approach is applied to a hydraulic actuator driving a cart with an inverted pendulum. The algorithmic approach aims to satisfy two robust control requirements for single input single output (SISO) linear systems with nonlinear uncertain structure. The first control requirement is robust stabilization, and the second is robust asymptotic command following for arbitrary reference signals. The approach is analyzed in two stages. In the first stage, the stability regions of the controller parameters are identified. In the second stage, a Particle Swarm Optimization Algorithm (PSO) is applied to find suboptimal solutions for the controller parameters in these regions, with respect to a suitable performance cost function. The application of the approach to a hydraulic actuator, driving a cart with an inverted pendulum, satisfies the goal of achieving precise control of the pendulum angle, despite the system’s inherent physical uncertainties. Full article

Stabilization and tracking problems for cart inverted pendulums under disturbances and uncertainties have posed significant challenges for control engineers. While various controllers have been designed for an inverted pendulum, they often overlook the calibration error of the pendulum angle in practical implementations, which degrades the control performance. Incorrect calibration of the pendulum angle in upright equilibrium position generates an offset of cart position errors. To solve this problem, an augmented model comprising integral cart position errors was first constructed. Afterwards, a sliding mode control was designed for this system based on a linear quadratic controller, to facilitate implementation. Additionally, a stepper motor was employed in the inverted pendulum to enhance the control performance and widen applicability in industrial settings. The effectiveness and performance of the proposed controller were validated by means of experimental studies, focusing on stabilization control and tracking control of a cart inverted pendulum actuated by a stepper motor. Full article

This paper addresses the robust control problem for under-actuated mechanical systems subject to uncertainties. The key challenge lies in achieving precise control with insufficient degrees of freedom while maintaining robustness against system uncertainties. We propose a novel control framework that characterizes bounded, time-varying uncertainties through fuzzy set theory, leading to a fuzzy dynamical system formulation. The main contributions are threefold: (1) the development of a deterministic robust controller that eschews traditional IF-THEN rules while guaranteeing system stability through a Lyapunov–Minimax analysis; (2) the formulation of a performance optimization scheme that minimizes both fuzzy system average performance and control costs, with proven existence and uniqueness of the analytical solution; and (3) the establishment of stability conditions using the Lyapunov theory for time-varying systems with bounded uncertainties. The theoretical framework is validated through both numerical simulations and experimental implementation on a linear motor-driven inverted pendulum system. The experimental results demonstrate significant performance improvements over conventional approaches: the optimal robust controller achieves 34.89% and 29.20% reductions in cart position and pendulum angle errors, respectively, from the initial conditions. A comparative analysis with traditional PD control shows a reduction in steady-state errors from 0.00318 m to 0.00057 m for the cart position and from 0.01117 rad to 0.00055 rad for the pendulum angle, validating the effectiveness of the proposed methodology. Full article

With advancements in bipedal locomotion for humanoid robots, a critical challenge lies in generating gaits that are bounded to ensure stable operation in complex environments. Traditional Model Predictive Control (MPC) methods based on Linear Inverted Pendulum (LIP) or Cart–Table (C-T) methods are straightforward and linear but inadequate for robots with flexible joints and linkages. To overcome this limitation, we propose a Flexible MPC (FMPC) framework that incorporates joint dynamics modeling and emphasizes bounded gait control to enable humanoid robots to achieve stable motion in various conditions. The FMPC is based on an enhanced flexible C-T model as the motion model, featuring an elastic layer and an auxiliary second center of mass (CoM) to simulate joint systems. The flexible C-T model’s inversion derivation allows it to be effectively transformed into the predictive equation for the FMPC, therefore enriching its flexible dynamic behavior representation. We further use the Zero Moment Point (ZMP) velocity as a control variable and integrate multiple constraints that emphasize CoM constraint, embed explicit bounded constraint, and integrate ZMP constraint, therefore enabling the control of model flexibility and enhancement of stability. Since all the above constraints are shown to be linear in the control variables, a quadratic programming (QP) problem is established that guarantees that the CoM trajectory is bounded. Lastly, simulations validate the effectiveness of the proposed method, emphasizing its capacity to generate bounded CoM/ZMP trajectories across diverse conditions, underscoring its potential to enhance gait control. In addition, the validation of the simulation of real robot motion on the robots CASBOT and Openloong, in turn, demonstrates the effectiveness and robustness of our approach. Full article

This paper addresses the robust stabilization problem of a cart–pole system. The controlled dynamics of this interconnected system are deduced by following the analytic framework of Lagrangian mechanics, and the residual terms are formulated as a bias depending on the angle and angular velocity. A geometric definition of Proportional–Integral–Derivative (PID) control algorithm is proposed, and a Lyapunov function is explicitly constructed through two stages of variable change. Local exponential stability of the stable equilibrium is proved, and a criterion for parameter tuning is provided by ensuring an exponential decrease in the Lyapunov function. Enlarging the control parameters to infinity allows for the extension of attraction region almost to the half circle. The effectiveness of geometric PID controller and the local exponential stability of the resulting close system are verified by simulating a numerical example. Full article

This paper aims to develop a novel hierarchical recursive nonsingular terminal sliding mode controller (HRNTSMC), which is designed to stabilize the inverted pendulum (IP). In contrast to existing hierarchical sliding mode controllers (HSMC), the HRNTSMC significantly reduces the chattering problem in control input and improves the convergence speed of errors. In the HRNTSMC design, the IP system is first decoupled into pendulum and cart subsystems. Subsequently, a recursive nonsingular terminal sliding mode controller (RNTSMC) surface is devised for each subsystem to enhance the error convergence rate and attenuate chattering effects. Following this design, the HRNTSMC surface is constructed by the linear combination of the RNTSMC surfaces. Ultimately, the control law of the HRNTSMC is synthesized using the Lyapunov theorem to ensure that the system states converge to zero within a finite time. By invoking disturbances estimation, a linear extended state observer (LESO) is developed for the IP system. To validate the effectiveness, simulation results, including comparison with a conventional hierarchical sliding mode control (CHSMC) and a hierarchical nonsingular terminal sliding mode control (HNTSMC) are presented. These results clearly showcase the excellent performance of this approach, which is characterized by its strong robustness, fast convergence, high tracking accuracy, and reduced chattering in control input. Full article

The cart–inverted pendulum system (CIPS) is a typical example of underactuated mechanical systems. For the CIPS with friction and disturbances, a gain-scheduled model predictive control method is proposed to achieve the upright stabilization objective of the single inverted pendulum (SIP) while controlling the cart to reach a desired new position. To this end, first, a dynamic equation of the CIPS with friction and disturbances is formulated based on the Newton–Euler equation. On the basis of the dynamic equation of the CIPS, its motion characteristics and control process are analyzed. Next, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models at seven different pendulum angles. Then, seven model predictive controllers (MPCs) are designed based on the above-linearized models, respectively. Introducing the idea of the gain-schedule, a gain-scheduled MPC (GSMPC) is designed to switch one of these seven MPCs to realize the control objective of the CIPS, according to the actual pendulum angle of the SIP during the control process. Finally, multi-group simulations that consider the friction and disturbances of the CIPS are implemented to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method. Full article

This paper investigates secure control of Networked Inverted Pendulum Visual Servo Systems (NIPVSSs) based on Active Disturbance Rejection Control (ADRC). Firstly, considering the network- and image-induced delays in conjuction with computational errors caused by image processing and image attacks, the model of NIPVSSs is established. Secondly, the limitations of the traditional Single-Input-Single-Output (SISO) ADRC used in NIPVSSs with disturbance are revealed. The limitations are that the ESO used in the traditional SISO ADRC brings large steady-state error, and the NLSEF used in the traditional SISO ADRC can achieve stable control of pendulum angle, but cannot achieve stable control of cart position. Thirdly, a new Single-Input-Multi-Output (SIMO) ADRC method is proposed for NIPVSSs with disturbance. In the new SIMO ADRC method, the new ESO is designed by introducing additional first and second derivatives of error to reduce the steady-state error. In addition, the new NLSEF is developed by taking both the calculated cart position and pendulum angle as inputs to achieve dual stable control of pendulum angle and cart position. Finally, combined with the designed ADRC parameter-tuning strategy, the results from simulation and real-world experiments confirm the effectiveness and feasibility of the proposed method. Full article

The double-inverted pendulum (DIP) constitutes a classical problem in mechanics, whereas the control methods for stabilizing around the equilibrium positions represent the classic standards of control system theory and various control methods in robotics. For instance, it functions as a typical model for the calculation and stability of walking robots. The present study depicts the controlling of a double-inverted pendulum (DIP) on a cart using a fuzzy logic controller (FLC). A linear-quadratic controller (LQR) was used as a benchmark to assess the effectiveness of our method, and the results showed that the proposed FLC can perform significantly better than the LQR under a variety of initial system conditions. This performance is considered very important when the reduction of the peak system output is concerned. The proposed controller equilibration and velocity tracking performance were explored through simulation, and the results obtained point to the validity of the control method. Full article

In this paper, a pole-independent, single-input, multi-output explicit linear MPC controller is proposed to stabilize the fourth-order cart–inverted-pendulum system around the desired equilibrium points. To circumvent an obvious stability problem, a generalized prediction model is proposed that yields an MPC controller with four tuning parameters. The first two parameters, namely the horizon time and the relative cart–pendulum weight factor, are automatically adjusted to ensure a priori prescribed system gain margin and fast pendulum response while the remaining two parameters, namely the pendulum and cart velocity weight factors, are maintained as free tuning parameters. The comparison of the proposed method with some optimal control methods in the absence of disturbance input shows an obvious advantage in the average peak efficiency in favor of the proposed SIMO MPC controller at the price of slightly reduced speed efficiency. Additionally, none of the compared controllers can achieve a system gain margin greater than 1.63, while the proposed one can go beyond that limit at the price of additional degradation in the speed efficiency. Full article