Inverted Pendulum Cart Research Articles

Fuzzy swinging-up with sliding mode control for third order cart-inverted pendulum system

Cart Inverted Pendulum (CIP) system is a benchmark problem in nonlinear automatic control. In this paper, two third-order differential equations were derived to create a combining model for the cart-pendulum with its DC motor dynamics. Motor voltage was considered the system input in the presented model. The friction between the cart and rail was included in the system equations through a nonlinear friction model. Fuzzy Swinging-up controller was designed to swing the pendulum to the upright position, once reaching the upward position; Sliding Mode Controller (SMC) is activated, to balance the system. In order to verify the performance of the proposed SMC, a Linear Quadratic Regulator Controller (LQRC) was suggested and compared with the proposed SMC. Simulation and experimental results have shown a significant improvement of the proposed SMC over LQRC where, the pendulum angle oscillations were decreased by 80% in the real implementation.

Stabilization of non-minimum phase switched nonlinear systems with the concept of multi-diffeomorphism

In this paper we propose a control approach for the stabilization of a class of switched nonlinear systems where each mode may be non-minimum phase. The proposed approach is based on the exact input–output linearization and the Lyapunov stability theory. The main contribution in this work is to elaborate a strategy of switching that recourse to the concept of multi-diffeomorphism makes it possible to guarantee an improvement of the transient state compared to a feedback linearization based on one diffeomorphism. Specifically, we show the sufficient condition for the exponential stability and the exponential upper bound of the trajectory of the switched system. The theoretical results are applied to a non-minimum phase inverted cart–pendulum in order to illustrate the effectiveness of the proposed approach.

Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

Hierarchical Sliding Mode Control of an Inverted pendulum

This paper presented a hierarchical sliding mode controller for an inverted pendulum system. The structure characteristic of the inverted pendulum system could be treated as two subsystems. According to the decomposition of cart-inverted pendulum system we select sliding mode surfaces for both subsystems. The sliding mode surfaces of the two subsystems were defined at first. The sliding mode surface of one subsystem was selected as the first layer sliding mode surface. The second layer sliding mode surface was constructed by the first layer sliding mode surface and the sliding mode surface of the other subsystem. The hierarchical sliding mode control law was deduced in terms of Lyapunov stability theorem. Finally, simulation results showed the validity of this control method and its robustness for external disturbances.

Adaptive Optimal Control of Nonlinear Inverted Pendulum System Using Policy Iteration Technique

This paper presents the adaptive optimal control of nonlinear inverted pendulum-cart dynamic system using policy iteration technique based adaptive critic scheme. Inverted pendulum is a highly nonlinear unstable system which is used as a benchmark for implementing the control methods. The optimal control design using Hamilton-Jacobi-Bellman (HJB) equation requires the complete knowledge of the system dynamics with an offline solution. Policy iteration technique which is based on actor-critic structure consists of two-step iteration: policy evaluation and policy improvement. The online policy iteration technique gives online continuous-time adaptive optimal control solution without using the complete knowledge of the system's internal dynamics. In this paper the state regulation control problem of cart position control and stabilization of inverted pendulum in vertically upright position is considered. The simulation results justify the effectiveness of the control scheme.

Stabilization of an Inverted Pendulum Cart with a Balancing Mechanism by Consistent Trajectories in Acceleration Behavior

New personal vehicles are required in current society because of necessity of low-carbon system. At present, most personal vehicles require a driver to have a certain level of physical ability. Low-carbon personal vehicles should, however, be made available to people with lower levels of physical ability, such as the elderly. The present paper therefore proposes an inverted pendulum cart that incorporates a balancing mechanism with actuating a seated driver. In the case of the proposed inverted pendulum cart, since the driver does not have to be able to maintain balance in order to cause the vehicle to accelerate, the proposed cart can be used by almost anyone. In addition, a control method using both input generated by desired acceleration trajectories and dynamics-canceling input is proposed. The effectiveness of the proposed control method is confirmed by simulation.

2A2-R01 スライド式重心制御機構を有する倒立振子型車両の開発 : 小型車両の設計製作および姿勢制御実験(車輪型/クローラ型移動ロボット(2))

A lot of researches on inverted pendulum cart are done in recent years. And it is attracted as a personal vehicle which realizes energy saving for many practical applications, such as the Segway and the P.U.M.A. The personal mobility has the advantage of energy efficiency for transportation because the cart is small and light. However, since most conventional personal vehicles require a certain level of physical ability from the driver, it is not suitable for elderly and disabled people to drive. Therefore, as an inverted pendulum cart capable of driving to anyone, I develop an inverted pendulum cart with a sliding mechanism for posture control, and aim for realization of acceleration and deceleration while keeping perpendicular cart angle. In this research, proposed inverted pendulum cart with a sliding mechanism for posture control was designed and manufactured, and the posture control performance was verified through experiment.

Decoupled Third-order Fuzzy Sliding Model Control For Cart-Inverted Pendulum System

Cart Inverted Pendulum (CIP) system is a benchmark problem in nonlinear automatic control. It is commonly used to verify the robustness of any proposed nonlinear controller. CIP is mostly represented by two second order differential equations to avoid complexity due to the DC motor dynamics. This representation is not practical for the real CIP dynamics and might lead to instability. Therefore in this paper, two third-order differential equations were derived to combine the pendulum system and DC motor dynamics to have a more realistic mathematical model. Friction between the cart and rail was included in the system equations through a nonlinear friction model. To stabilize the obtained nonlinear electromechanical CIP model, a third-order Fuzzy Sliding Mode Controller (FSMC) was designed. The chattering of the control signal was eliminated using general bill shape membership functions for the Fuzzy controller. Simulation results proved the robustness of the proposed FSMC over Linear Quadratic Regulator Controller (LQRC). For instant, the overshoot in the cart position response was reduced by 300%.

Output feedback stabilization of the inverted pendulum system: a Lyapunov approach

In this work, we present an output feedback stabilization method for the Inverted Pendulum Cart (IPC) system around its unstable equilibrium point. The pendulum is initialized in the upper-half plane, and the position of the cart and the pendulum angular positions are always available. Our strategy was accomplished introducing a suitable coordinate change to obtain a nonlinear version of the original system, which is affine in the unmeasured velocities state. This fact allows us to adapt an observer based controller devoted to render the closed-loop system to the origin. The proposed observer based controller was designed using the direct Lyapunov method. This allows estimating the corresponding attraction domain for the whole system, which can be as large or as small as desired. While the corresponding closed-loop stability analysis was made using the LaSalle Invariance Theorem. Convincing numerical simulations were included to show the performance of the closed-loop system.

Robust proportional–integral–derivative compensation of an inverted cart–pendulum system: an experimental study

This study designs a two-loop proportional-integral-derivative (PID) controller for an inverted cart-pendulum system via pole placement technique, where the (dominant) closed-loop poles to be placed at the desired locations are obtained from an Linear quadratic regulator (LQR) design. It is seen that in addition to yielding better responses (because of additional integral action) than this LQR (equivalent to two-loop PD controller) design, the proposed PID controller is robust enough. The performance and robustness of the PID compensation are verified through simulations as well as experiments.

Sliding mode-like learning control for SISO complex systems with T-S fuzzy models

In this paper, a sliding mode-like learning control scheme is developed for a class of single input single output (SISO) complex systems. First, the Takagi-Sugeno (T-S) fuzzy modelling technique is employed to model the uncertain complex dynamical systems. Second, a sliding mode-like learning control is designed to drive the sliding variable to converge to the sliding surface, and the system states can then asymptotically converge to zero on the sliding surface. The advantages of this scheme are that: The sufficient conditions for the sliding mode-like learning control to stabilise the global fuzzy model are discussed in detail. A simulation example for the control of an inverted pendulum cart is presented to demonstrate the effectiveness of the proposed control scheme.

A Flatness‐Based Controller for the Stabilization of the Inverted Pendulum

We propose a flatness based approach for controlling the inverted pendulum cart system, under the assumption that the pendulum state is always located on a vicinity of its unstable equilibrium point. This is achieved by representing the original system, as a chain of integrators with an additive nonlinear state dependent perturbation. After discarding the small nonlinear perturbation, we may directly use design tools provided by the the flatness approach. The effectiveness and robustness of the obtained control law, which turned out to have a large domain of attraction, were numerically assessed in the context of stabilization and, also, for a reference oscillatory trajectory tracking task.

Design and Implementation of Robust Hybrid Control of Vision Based Underactuated Mechanical Nonminimum Phase Systems

This paper presents a vision based Cart-Inverted Pendulum (CIP) system under a hybrid feedback configuration: the continuous cart’s position measured by encoder and the delayed & sampled inverted pendulum’s upper coordinates, obtained from a visual sensor. The challenge here is to stabilize the CIP from a big inclined initial angle by using a low cost CCD camera. Under this scheme, we propose a hybrid control which consists in a Jumpingup (Bang-Bang) control and a two causal stabilization loops control: the first one (inner loop) realizes a linearization and the stabilization control of the pendulum based on an innovative Piecewise Continuous Reduced Order Luenberger Observer coupled with a linearization module, the second one (the outer loop) realizes a Lyapunov based control for the unstable internal system with lower dynamics than that of the pendulum. This hybrid control method is capable of balancing the CIP system within small cart’s displacement. Performances issues of the proposed method are illustrated by the experimental figures and videos.

Stabilization of a Cart-Inverted Pendulum with Interconnection and Damping Assignment Passivity-Based Control Focusing on the Kinetic Energy Shaping

Interconnection and damping assignment passivity-based control (IDA-PBC) is a nonlinear control method which stabilize a system shaping its total energy and utilizing passivity. In previous studies, IDA-PBC is shown to be a powerful method to stabilize underactuated mechanical systems. However, the transient performances tend to be slow. In this study, to improve the slow responses and realize fast transient ones, the IDA-PBC is applied for a cart-inverted pendulum and free parameters in a closed-loop inertia matrix which have not been used in the previous studies are utilized. This means modifying the kinetic energy more actively. On the other hand, the additional use of the free parameters makes the selection of them more complicated because several assumptions must be satisfied to derive an IDA-PBC controller. To deal with this problem, we also propose a systematic method to select them graphically from a simple two dimensional region. Finally, we show that the transient performances of the IDA-PBC can be as fast as the LQR with the active modification of the kinetic energy. The IDA-PBC with suitable parameters also theoretically estimates as large domain of attraction as the upper half plane preserving the fast transient performances.

Stabilization of Inverted Pendulum Cart with Balancing Mechanism by Integrity Trajectories in Acceleration Behavior

Stabilization of Inverted Pendulum Cart with Balancing Mechanism by Integrity Trajectories in Acceleration Behavior

4C16 Stabilization of a Cart-Inverted Pendulum with Interconnection and Damping Assignment Passivity-Based Control Focusing on the Kinetic Energy Shaping

4C16 Stabilization of a Cart-Inverted Pendulum with Interconnection and Damping Assignment Passivity-Based Control Focusing on the Kinetic Energy Shaping

A hybrid controller design for bi‐axial inverted pendulum system

AbstractThis study presents the use of Tustin's friction model and a disturbance observer (DOB) to improve the steady‐state error (SSE) of a bi‐axial inverted pendulum–cart system (IPCS). Furthermore, a hybrid controller contains a feedback linearization control for pendulum angle in the region of 3–12° to enlarge the angle of operation and an H∞ control using loop shaping design procedure (LSDP) for cart position and pendulum angle in the region of 0–3° to stabilize the IPCS, respectively. Experimental results reveal that the pendulum maximum angle of operation is improved from 7 to 12°; the SSE of the angle of the pendulum is reduced from 0.85 to 0.1°, and the SSE of the position of the cart is reduced from 10 to 1.4 mm. Experimental results are illustrated and films are provided at the web site http://hinfinity.myweb.hinet.net to show the effectiveness and robustness of the hybrid controller with Tustin's friction model and DOB compensation. Copyright © 2008 John Wiley & Sons, Ltd.

Fuzzy hierarchical swing-up and sliding position controller for the inverted pendulum–cart system

In this paper, a fuzzy hierarchical swing-up and sliding position controller is proposed for the swing-up and position controls of an inverted pendulum–cart system. The proposed fuzzy hierarchical swing-up and sliding position controller includes a fuzzy switching controller, a fuzzy swing-up controller, and a twin-fuzzy-sliding-position controller. The fuzzy switching controller is designed to smoothly switch between the swing-up and position controls. With the inverted pendulum–cart system decomposed into a pendulum subsystem and a cart subsystem, fuzzy sliding position controllers are presented to robustly balance the pendulum and cart at the desired positions. The sliding mode and stability of the fuzzy sliding position control systems are guaranteed. Moreover, an adaptive mechanism is provided to on-line adjust the parameters in the fuzzy sliding position controllers. Simulation results are included to illustrate the effectiveness of the proposed fuzzy hierarchical swing-up and sliding position controller.

A simple model matching for the stabilization of an inverted pendulum cart system

AbstractWe present a simple model matching controller for the stabilization of the inverted pendulum cart system, assuming that the pendulum is initialized above the horizontal plane. The control strategy consists of forcing the closed‐loop system to behave as another nonlinear target system with some stability properties. To this end, we solve two matching conditions that allow us to shape a suitable target system. Having satisfied both matching conditions, the stabilizing controller is directly obtained from the proposed target system. The obtained close‐loop system is locally asymptotically exponentially stable, with a very large domain of attraction. Copyright © 2007 John Wiley & Sons, Ltd.

CAN을 통신매체로 하는 역진자 시스템의 제어

In a networked control system (NCS), time delays which are larger than one sampling period can change the control period. As a result, it may cause system instability. This paper presents a control method for an NCS using the controller area network (CAN), where time delays arise in the control loop. Specifically, a simple yet efficient method is proposed to improve control performance in the presence of time delays. The proposed method, which can be regarded as a gain scheduling method, selects a suitable LQ control gain among several gains to deal with the problems due to the change of control period. It is found that the gain can be scheduled in terms of the relation between the gain and the sampling period, which is represented by first-order algebraic equations. The proposed method is evaluated with an inverted cart pendulum system where the actuator and sensors are connected through the CAN. Experiment results are presented to show the efficiency of the proposed method.