Design of Fuzzy Controller for Double Inverted Pendulum

In this design, through the analysis of the physical model of the linear double inverted pendulum, the motion equation of the system is obtained by using the classical Newtonian mechanical formula, and the obtained equation is linearized at the stable point, and the state space model of the system is obtained. In order to realize the stable control of the trolley position and the secondary linkage in the linear inverted pendulum system, the linear quadratic optimal controller is used in this design to obtain an excellent output response. In order to design the optimal state feedback control matrix K, the shortcomings of weight matrices Q and R in general LQR control design that depend on experience selection and trial and error are overcome. This design attempts to apply PSO algorithm to LQR controller parameter optimization. PSO algorithm is applied to optimize Q parameters and R parameters of LQR weighting matrix of linear quadratic optimal controller, and K matrix is obtained. The matrix is applied to a fuzzy PID controller designed by the fusion algorithm to overcome the disadvantage that a single PSO cannot adapt itself. Finally, a comparative simulation study is carried out in the Simulink environment of Matlab software toolbox, which verifies the effectiveness of the fuzzy PID controller based on particle swarm optimization algorithm, laying a theoretical foundation for the practical application of the inverted pendulum system, and has certain application value.

In this study, interval type2 fuzzy logic (IT2FL) and PID controller is designed for swing-up position control of double inverted pendulum (DIP) system. The double inverted pendulum system consists of two rigid bars connected by a revolute joint. Mass of the revolute joint is included in the dynamic model.Rigid bars in the system are assumed to experience planar motion. The pendulum system is connected to the base by means of a revolute joint. Torque provided through a motor mounted to the base is used for position control of the system. PID (Proportional-Derivative-Integral) and interval type2 fuzzy logic controllers are developed by using the same performance criteria for position control of double inverted pendulum system. IT2FL controller is similar with type1 fuzzy logic controller. IT2FL system provides soft decision boundaries, whereas a type-1 fuzzy logic system provides a hard decision boundary. Membership function in interval type2 fuzzy logic set as an area called Footprint of Uncertainty (FOU) which limited by two type1 membership function those are upper membership function (UMF) and lower membership function (LMF).System behaviour is obtained by computer simulation using developed controllers respectively. Computer simulation results are compared in order to evaluate applicability of developed controllers. MATLAB/Simulink software is used in computer simulations.

Deals with the modeling of a flexible inverted pendulum. The mathematical model of the controlled object used is a simplified one, which is approximated by rigid links and springs. The possibility of driving a simplified model of the flexible inverted pendulum using a double inverted pendulum with an elastic joint is studied. The experimental results show that the simplified model is sufficient to describe the flexible inverted pendulum, but with some modeling and identification errors. These uncertainty will excite the vibration. An H/sub infinity / control to stabilize the flexible inverted pendulum using the simplified model is presented. >

In this paper a fuzzy optimal control for stabilizing an upright position a double inverted pendulum (DIP) is developed and compared. Modeling is based on Euler-Lagrange equations. This results in a complicated nonlinear fast reaction, unstable multivariable system. Firstly, the mathematical models of double pendulum system are presented. The weight variable fuzzy input is gained by combining the fuzzy control theory with the optimal control theory. Simulation results show that the controller, which the upper pendulum is considered as main control variable, has high accuracy, quick convergence speed and higher precision.

Mathematical models and parameters with high accuracy are required for the swing-up control of a double inverted pendulum. This study proposes a mechanical structure of a double inverted pendulum having excellent matching characteristics with a mathematical model. The implementation of a parameter estimation method based on nonlinear optimization, which is essential for swing-up control, is also proposed. The proposed structure of a double inverted pendulum eliminates backlash using the direct drive of a BLDC motor, minimizes the unmodeled frictional force through bearing cleaning and oiling, minimizes unnecessary load torque using the double support of a timing pulley shaft and power transmission through a coupling, and constrains the pendulum rotation to one degree-of-freedom by adopting a dual-bearing joint. In addition, both 3D printing and CNC milling are appropriately used to satisfy the requirements for each mechanical part. The swing motion of the manufactured double inverted pendulum is measured using the lab-built light-weight rapid control prototyping 02 (LW-RCP02). The proposed parameter estimation method based on nonlinear optimization is implemented to match the response trajectory of the actual double inverted pendulum with that of a Simulink model implemented as an S-function. Finally, the effectiveness of the proposed double inverted pendulum design and parameter estimation method is confirmed by showing that the mathematical model that, based on the parameters estimated using the proposed method, accurately describes the response characteristics of the double inverted pendulum.

The linear fusion function is constructed on the basis of LQR control in this paper. Therefore it is possible to decrease the dimensions of fuzzy controller input variables and reduce the rules for fuzzy control. The fuzzy controller based on fusion function is applied to the approximate linear model of double inverted pendulum, and the experimental results show that the system's response rate is well improved and the robustness is also enhanced.

A new type of adaptive neural network fuzzy controller based on the stability for the double inverted pendulum control problem is introduced. The method uses a fusion function to reduce the dimension of the system, reducing the number of input variables to solve the fuzzy rule explosion problem. In order to optimize and amend the front-part and later-part parameter of Takagi-Sugeno fuzzy model, a mixed algorithm of backward propagation (BP) and least square method (LSE) algorithm are used. Using the collected original input and output data to establish adaptive neural network fuzzy inference system (ANFIS), and to control the double inverted pendulum system. Simulation results show that the controller is better than the other controller.

In this paper, The mathematical model is established using analytical dynamics method for planar double inverted pendulum, based on optimal control theory, parameter adaptive fuzzy controller is designed by use of variable fusion technology, thereby it can reduce the dimension of the input variables of the fuzzy controller, greatly decrease the number of the fuzzy rules, and the impact of the quantization factor on the control effectiveness is studied. The designed controller for system is applied in simulation experiments, the results show that the fuzzy controller can guarantee good control precision, fast response, control planar inverted pendulum stability.

In this paper, adaptive network based fuzzy inference system (ANFIS) was used in control applications of different type nonlinear systems as interval type-2 fuzzy logic controller (IT-2FL). Two adaptive network based fuzzy inference systems were chosen to design type-2 fuzzy logic controllers for each control applications. Membership functions in interval type-2 fuzzy logic controllers were set as an area called footprint of uncertainty (FOU), which is limited by two membership functions of adaptive network based fuzzy inference systems; they were upper membership function (UMF) and lower membership function (LMF). The double inverted pendulum, a single flexible link and a flexible link carrying pendulum systems were used to test the performances of designed interval type-2 fuzzy logic controllers. System behaviours were defined by Lagrange formulation and MATLAB/SimMechanics computer simulations. The performances of the proposed controllers were evaluated and discussed on the basis of the simulation results. An experiment set up of flexible link carrying pendulum system was built and used to verify the performance of IT-2FL controller. Key words: Adaptive network based fuzzy inference system, interval type-2, inverted pendulum, a single flexible link, control, simulation.

Realizing balance control of single inverted pendulum without mathematic model by reinforcement learning has gained great success. However, further applying it to the multilevel inverted pendulum faces problems such as curse of dimensionality and difficulty in convergence. In this paper, we propose a hierarchical reinforcement learning algorithm to control parallel double inverted pendulum. Firstly, control of single inverted pendulum is learned by Q-learning algorithm. Then the learned control states of single inverted pendulum are used to direct control process of parallel double inverted pendulum so that the double one is controlled.

Double inverted pendulum system is nonlinear and unstable. Fuzzy control uses some expert's experience knowledge and learns approximate reasoning algorithm. For it does not depend on the mathematical model of controlled object, it has been widely used for years. In practical engineering applications, most systems are nonlinear time-varying parameter systems. As the fuzzy control theory lacks of on-line self-learning and adaptive ability, it can not control the controlled object effectively. In order to compensate for these defects, it introduced adaptive, self-organizing, self-learning functions of neural network algorithm. We called it adaptive neural network fuzzy inference system (ANFIS). ANFIS not only takes advantage of the fuzzy control theory of abstract ability, the nonlinear processing ability, but also makes use of the autonomous learning ability of neural network, the arbitrary function approximation ability. The controller was applied to double inverted pendulum system and the simulation results showed that this method can effectively control the double inverted pendulum system.

Inverted pendulum systems, such as double or single, rotational or translational inverted pendulums are non-linear and unstable, which have been the most dominant approaches for control systems. The double inverted pendulum is one kind of a non-linear, unstable system, multivariable, and strong coupling with a wide range of control methods. To model these types of systems, many techniques have been proposed so that motivating researchers to come up with new innovative solutions. The Linear Quadratic Regulator (LQR) controller has been a common controller used in this field. Meanwhile, the Artificial Bee Colony (ABC) technique has become an alternative solution for employing Bee Swarm Intelligence algorithms. The research solutions of the artificial bee colony algorithm in the literature can be beneficial, however, the utilization of discovered sources of food is ineffective. Thus, in this paper, we aim to provide a double inverted pendulum system for stabilization by selecting linear quadratic regulator parameters using a bio-inspired optimization methodology of artificial bee colony and weight matrices Q and R. The results show that when the artificial bee colony algorithm is applied to a linear quadratic regulator controller, it gains the capacity to autonomously tune itself in an online process. To further demonstrate the efficiency and viability of the suggested methodology, simulations have been performed and compared to conventional linear quadratic regulator controllers. The obtained results demonstrate that employing artificial intelligence (AI) together with the proposed controller outperforms the conventional linear quadratic regulator controllers by more than 50 % in transient response and improved time response and stability performance

Design and analysis of controllers for a double inverted pendulum

According to the Lagrange equation, the mathematical model for the double inverted pendulum is first presented. For the fuzzy controller, the dimension of input varieties of fuzzy controller is depressed by designing a fusion function using optimization control theory, and it can reduce the rules of fuzzy sharply, `rule explosion' problem is solved. The result shows that the LQR-Fuzzy logic controller not only has quite good stability, but also be able to make the cart of the pendulum moving to the designed in advance place along the orbit.

The double inverted pendulum is a nonlinear system with unstable balance and rapid response. When two pendulums are allocated vertically, and there is no movement, the system is stable. In this article, we propose a control mechanism based on reinforcement learning. The system can improve its performance and balance by implementing the Q-Learning techniques besides the PID control scheme. The Q-Learning algorithm balances the pendulum to its inverted vertical position in the first step. In the second step, the Q-Learning algorithms are combined with classical PID to balance the double inverted pendulum and perform swing-up control. Eventually, simulation findings showed the efficiency of the proposed controller.