In this paper, first a fuzzy controller is proposed for systems comprising two inverted pendulums placed in the same plane, w here one pendulum moves along the X-axis and the other moves along the Y-axis; rules of a Fuzzy-Mamdani controller is improved by considering the behavior of an operator and a the developed expert system and by trial and error in the observation of the system’s behav ior. Simulation results obtained in MATLAB are proposed and then two other intelligent methods are presented. In the second proposed method, a TSK controller is designed with Anfis training approach and its simulation results are presented. Finally, a non-fuzzy controller (linearization feedback) is used as the supervisor fuzzy controller and the simulation results are presented. At last, performance of the three controllers is compared and the results are demonstrated.
In the design of modern control system, optimal control theory is playing an increasingly vital role, whose main objective is to determine control signals that will cause a process (plant) to satisfy some physical constraints and at the same time maximize or minimize a chosen performance criterion. The designed approach used in this paper is the Linear Quadratic Regulator (LQR) controller for linear invertedpendulum system and doubleinvertedpendulum system. Linear Quadratic Gaussian (LQG) controller is a combination of LQR designed with that of a Kalman Filter. In this paper LQG controller is used only for linear invertedpendulum system.
T HE doubleinvertedpendulum (DIP) system is an extension of the single invertedpendulum (with one additional pendulum added to the single invertedpendulum), mounted on a moving cart. The DIP sys- tem is a standard model of multivariable, nonlinear, unstable system, which can be used for pedagogy as well as for the introduction of intermediate and advanced control concepts. There are two types of control synthesis for an invertedpendulum, swing- up and stabilization. One of the most popular control methods for swinging up the invertedpendulum is based on the energy method (see  and the refer- ences therein). The stabilization problem of an invertedpendulum is a classical control example for testing of linear and nonlinear controllers (see, e.g., , , ). Several control design approaches have been applied for the stabilization of the doubleinvertedpendulum including the linear quadratic regulator , the state- dependent Riccati equation, optimal neural network , and model predictive control . To our knowl- edge, these studies only use numerical simulations to
Secondly, a control strategy with a state variables feedback was implemented by the design of a controller using the root locus (LQR) method, in order to locating the system poles in closed loop and controlling the angular position of both pendulums. Additionally, the control system must have an integral compensator, which eliminates the positional error of the system. Furthermore, using the expanded expression arrays of the state variables , which can be observed in equation (12), the symmetric matrices "Q" and "R" were proposed with the aim of estimate the feedback constants "K" of the system. Figure-2 presents the results of the controlled system using a step input, where it can be visualized that the system outputs are asymptotic to zero.
Abstract: A self-erecting single invertedpendulum (SESIP) is one of typical nonlinear systems. The control scheme running the SESIP consists of two main control loops. Namely, these control loops are swing-up controller and stabilization controller. A swing-up controller of an invertedpendulum system must actuate the pendulum from the stable position. While a stabilization controller must stand the pendulum in the unstable position. To deal with this system, a lot of control techniques have been used on the basis of linearized or nonlinear model. In real-time implementation, a real invertedpendulum system has state constraints and limited amplitude of input. These problems make it difficult to design a swing-up and a stabilization controller. In this paper, first, the mathematical models of cart and single invertedpendulum system are presented. Then, the Position-Velocity controller is designed to swing- up the pendulum considering physical behavior. For stabilizing the invertedpendulum, a Takagi- Sugeno fuzzy controller with Adaptive Neuro-Fuzzy Inference System (ANFIS) architecture is used to guarantee stability at unstable equilibrium position. Experimental results are given to show the effectiveness of these controllers.
widely used as a teaching aid and in research experiments around the world. As a teaching aid, the invertedpendulum used because it’s an imaginable unstable nonlinear dynamics problem commonly discussed in control engineering (Hauser et al., 2005), and various control algorithms, ranging from conventional through to intelligent control algorithms, has been applied and evaluated (Jung and Kim, 2008). There are a number of associated control problems that can be derived from the invertedpendulum models such as rocket control, the dynamic balance of skiing, bicycle/motorcycle dynamics (Hauser et al., 2005). Recently, a mobile invertedpendulum model with two wheels has been applied to various robotic problems such as designing walking or legged humanoid robot, robotic wheelchairs and personal transport systems (Kim et al., 2006).
The disturbance is introduced to the system to test for the performances and robustness of the proposed controllers. The white noise of 0.01 power parameter value is added to the process output (feedback) after stabilised at 40 seconds. Figure 16 and 17 shows the simulation results of the lower pendulum angle, upper pendulum angle and arm angle. It can be seen from Figure 16 and 17 that the proposed hybrid Fuzzy-LQR controller is able to control both upper and lower pendulums to remain at stabilization position with some oscillation. This oscillation is due to the introduced disturbance. On the other hand, from the same Figure we can see that as soon as the disturbance is
of the obstacles by using the PID and PD controllers are that they alone cannot effectively control all of the pendulum state variables (modes) since they are of lower order than the pendulum itself. Hence, they are usually replaced by a full-order controller . A linear state feedback controller based on the linearized invertedpendulum model can instead be used, and may also be extended with a disturbance observer (Kalman filter) to improve the disturbance rejection performance.
There are two set of pole in InvertedPendulum system Fast & Slow. Angle Dynamics is determined by fast set of poles and position Dynamics is determined by other set of poles. Performances of different controllers like Sliding Mode, PD, Fuzzy; Neural Network is shown in . Comparison for different Energy based Controllers to stabilization of InvertedPendulum is mentioned in . An Energy based gradient method is described in . A feedback control law is derived in . A method for Controlled Lagrangians described in . A combined Controller is described in . For global stabilization of Invertedpendulum a hybrid Controller is designed in . A non-linear controller is designed in  considering non-linearity for stabilization of InvertedPendulum. A simple design to stabilize the system is described in .
Baili Zhang (2011), three methods used to design the controller of the invertedpendulum, where for PID controller, it depends on the parameter of the Proportion, P, Integral, I and Derivative, D. For this method, it used transfer function that can be achieved by the state space equation and developed by using double closed loop control. For the state feedback controller, there were used of state space equation for the plant model and it will directly feedback to the input signal where the value of gain proportion link, K can be obtained. Lastly, the Linear Quadratic Regulator, LQR, the linear state that is to be formed can be used in state space equation. The used of matrix equation shows that weighting matrix, R and Q are for balancing state variable and also input variable, and the value of P can be obtained by using Riccati equation. Therefore, value of K can be formed using this formula : K = (𝑅 −1 )(𝐵 𝑇 )
Jadlovska and Sarnovsky  have revealed an approach for the control of the rotary single invertedpendulum system. State feedback control techniques are used as pole placement and the LQR optimal control. Mathew et al.  performed a study on swing up and stabilization control of a RIP system. Two control schemes are performed for stabilization as LQR and SMC. Chen and Huang  have proposed an adaptive controller for RIP with time-varying uncertainties to bring the pendulum close to the upright position regardless of the various uncertainties and disturbances. Its underactuated dynamics was first decoupled by Olfati’s transformation into a cascade form. Oltean  has proposed solution for swing up and stabilization of RIP using PD and fuzzy PD controllers. The models are performed in MATLAB/ Simulink environment. Ding and Li  have proposed a cascade fuzzy controller based on Mamdani for the outer loop and Sugeno for the inner loop. The simulation graphs are performed with Simulink. Dang et al.  have designed a robust Takagi-Sugeno (T-S) fuzzy descriptor approach for a stabilizing controller for the RIP with real-time implementation. Chandran et al.  derived the nonlinear dynamics of the RIP. Artificial neural network (ANN) is applied to identify the model.
In this paper, two controllers such as SMC and PID are successfully designed. Based on the results and the analysis, a conclusion has been made that both of the control method, modern controller (SMC) and conventional controller (PID) are capable of controlling the nonlinear invertedpendulum system angular and linear position. All the successfully designed controllers were compared. The responses of each controller were plotted in one window and are summarized in Table III and Table IV. Simulation results and bar charts in Fig. 7 and Fig. 8 show that SMC controller has better performance compared to PID controller in controlling the nonlinear invertedpendulum system. Further improvement need to be done for both of the controllers. PID controller should be improved so that the maximum overshoot and maximum undershoot for the linear and angular positions do not have very high range as required by the design criteria. On WKH RWKHU VLGH /45 FRQWUROOHU FDQ EH LPSURYHG VR WKDW LW¶V settling time for angular position might be reduced as faster as PID controller.
T HE classical Proportional Integral Derivative (PID) controller has remained the most popular industrial controller over the last six decades, despite the enormous hosts of development over the same period . Various PID tuning methods have been developed by a number of re- searchers in the last 40 years. Developments in evolutionary algorithms and particle swarm optimization have led to the application of these methods for PID tuning , . Other PID tuning approaches include the direct search algorithms and online optimization based approaches , . Although these methods have resulted in the automatic tuning of the PID controllers, they require significant computational loading, and are not suitable for real time applications.
Modeling and robust control design of nonlinear system is investigated in this thesis. This thesis also presented an overview of working with H∞ (Robust control method) of designing controllers. Although the application of H∞ requires understanding of the linear algebra and intricate mathematics therefore the aim of this thesis to give a clear picture of the working procedure and how to apply it to practical problems in hand.
Accurate model of systems is required to design a controllers especially if the controllers are model based controllers . Many researchers have worked on modelling and control of the TWIP mobile robot. Euler Lagrange methods are implemented in [1, 2, 4, 5], Newton-Euler equations of motion method is used in [6, 7]. In [8, 9] the modelling of TWIP is carried out using Kane’s method of modelling. A Takagi–Sugeno (TS) fuzzy model was used in . All models developed using the various methods were able to emulate the dynamics characteristics of the TWIP to some extent, the Kane’s method gives best model presentation.
The trajectory 1 was entered via Command menu and step and setup was also selected. The step size was at 2000, dwell time = 4000ms and repetitions = 2. The execute was selected with Normal Data sampling and execute Trajectory 1 only was checked and ran the trajectory. Step move of 2000 counts a dwell of 4 seconds and return step move were noticed respectively. It was noted that base motion of the cart was reversed into the initial part of the step response non minimum phase behaviour of the controlled system. The base moved near its command position and subsequently returned. It was also noted that the pendulum rod moved initially in a direction ‘’pointing forward’’ the new set point as the base accelerated in that direction. It latter moved in the direction as the base decelerated and maintained near zero (vertical) orientation during regulation at the new position. All these actions were automatically executed by closed loop control.
Velocity tracking is one of the important objectives of vehicle, machines and mobile robots. A two wheeled invertedpendulum (TWIP) is a class of mobile robot that is open loop unstable with high nonlinearities which makes it difficult to control its velocity because of its nature of pitch falling if left unattended. In this work, three soft computing techniques were proposed to track a desired velocity of the TWIP. Fuzzy Logic Control (FLC), Neural Network Inverse Model control (NN) and an Adaptive Neuro-Fuzzy Inference System (ANFIS) were designed and simulated on the TWIP model. All the three controllers have shown practically good performance in tracking the desired speed and keeping the robot in upright position and ANFIS has shown slightly better performance than FLC, while NN consumes more energy.
The invertedpendulum is a classical problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms’ (PID controllers, SFB etc). Balancing the cart & pendulum is related to rocket or missile guidance where the centre of gravity is located behind the centre of drag causing aerodynamic instabity. where the centre of gravity is located behind the centre of drag causing aerodynamic instabity. As a more realistic example of the design and simulation of controlled systems we now focus on the development and analysis of an invertedpendulum on a motor driven cart. A sketch of this system is shown in fig(2).
The LQR method is a powerful technique for designing controllers for complex systems that have stringent performance requirements and it seeks to find the optimal controller that minimizes a given cost function . The cost function is parameterized by two matrices, Q and R, that weight the state vector and the system input respectively. LQR method is based on the state-space model and it tries to obtain the optimal control input by solving the algebraic Riccatti equation. In this paper, the state feedback controller is designed using the linear quadratic regulator and the linear model of the system. Briefly, the LQR/LTR theory says that, given an n th order stabilizable system
This paper deals with design of State feedback controller for Rotary InvertedPendulum based on Kharitnov polynomial approach . Rotary InvertedPendulum (RIP) is a nonlinear and unstable system. The key idea here is to stabilize the position of the pendulum at inverted position for the rotary base. This paper deals with mathematical modeling , design of state feedback controller and stability analysis for the system specified.