Many studies are found in the area of control issues related to invertedpendulum systems. Some of the studies dealing with invertedpendulum control are summarized herein. Yan  developed a tracking control law for underactuated RIP by applying nonlinear back stepping, differential flatness, and small gain theorem. Mirsaeid and Zarei  presented a mechatronic system case study on adaptive modeling and control of an invertedpendulum. Hassanzadeh et al.  presented an optimum Input-Output Feedback Linearization (IOFL) cascade controller. Genetic Algorithm (GA) was applied for the inner loop with PD controller forming the outer loop for balancing the pendulum in an inverted position. The control criterion was to minimize the Integral Absolute Error (IAE) of the system angles. The optimal controller parameters are found by minimizing the objective function related to IAE using Binary Genetic Algorithm (BGA). Ozbek and Efe  focused on the swing up and stabilization control of a rotaryinvertedpendulum (RIP) system with linear quadratic regulator (LQR). Sliding Mode Control (SMC) is based on hard boundary switching law and fuzzy logic control (FLC). Akhtaruzzaman et al.  have described different controller designs for rotarypendulum. Experimental and MATLAB based simulation results are given. Hassanzadeh et al.  also studied control by using evolutionary approaches. GA, Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) are used for designing the rotational invertedpendulum. Quyen et al.  presented the dynamic model of RIP. ANN controller is used for controlling the system.
Adaptive system is defined by (Narendra and Annaswamy,1989)  as a system which is provided with a means of continuously monitoring its own performance in relation to a given figure of merit or optimal condition and a means of modifying its own parameters by a closed loop action to approach a optimum condition. MRAS that uses Model Reference Adaptive Control (MRAC) is an adaptive system that makes overt use of such models for identification or control purposes. MRAC as adaptive controller is chosen to control the RIP system based on the performance wise and other characteristics. Tracing back chronologically from 1950s [5,7] until now, the automatic control of physical processes has been an experimental technique deriving more from art than from scientific bases. When implementing a high-performance control system, the poor characteristic plant dynamic characteristics starts to arise. Besides that large and unpredictable variations occur. As a result, a new class of control systems called adaptive control systems has evolved which provides potential solutions. In the late 1950s,manysolutionshavebeenproposedinordertomakeacontrolsystem“adaptive” and among of them is a special class of adaptive systems called Model Reference Adaptive System.
The invertedpendulum system is a standard problem in the area of control systems. They are often useful to demonstrate concepts in linear control such as the stabilization of unstable systems by varying the pendulum position and using the gains. Since the system is inherently nonlinear, it has also been useful in illustrating some of the ideas in nonlinear control . This work consists of two parts of experimental procedures. Part 1 deals with finding or calculating the vectors of gains K using pole placement method and performing the closed loop simulations for the non-linear using simulink model with a full state feedback controller of gains calculated while Part 2 is implementing the controllers by using the gain calculated on the laboratory physical invertedpendulum system and compares their stability.
Where A with n x n matrix, B with n x r matrix, C with m x n matrix and D with m x r matrix where this matrices need for control design of the system, while u is input applied to the system and x is the state variable of the dynamic system where this variables making up the smallest set of variable determine the state of the dynamic system [15,17]. The state space method is based on system equation in term of n differential equation. The vector-matrix notation is simplification of the mathematical model of the system. The state space approach also enable the engineer to design control system with respect to given performance index and also can include the initial condition in the design. Figure 2.7 show the block diagram for discrete time control system for state space approach.
Generally, first task involved to accomplish this project is to do a literature review regarding the invertedpendulum system. Next task is to analyze the mathematical model of the system. From the analysis, a nonlinear final state space equation is obtained. However, the LQR controllerdesign is based on a linear state equation. Therefore, the nonlinear equation is linearized about the origin.
Abstract - The research on two-wheel invertedpendulum or commonly call balancing robot has gained momentum over the last decade at research, industrial and hobby level around the world. This paper deals with the modeling of 2- wheels InvertedPendulum and the design of Proportional Integral Sliding Mode Control (PISMC) for the system. The mathematical model of 2-wheels invertedpendulum system which is highly nonlinear is derived. The final model is then represented in state-space. A robust controller based on Sliding Mode Control is proposed to perform the robust stabilization and disturbance rejection of the system. A computer simulation study is carried out to access the performance of the proposed control law.
 S. Kawamoto, "Nonlinear Control and Rigorous Analysis Based On Fuzzy System for Interved Pendulum," The Fifth IEEE International Conference on Fuzzy Systems, vol. 2,, pp. 1429-1432, Sep. 1996.  H. K. Lam, F. H. F. Leung, and P. K. S. Tam, “Stability
The disturbance rejection ability of the controller strategy is explained in this section. After the pendulum swing up and stabilization phase, disturbance is introduced into the pendulum at 15 th second as shown in Fig. 8. The zoomed view of pendulum velocity response is shown in Fig. 9 to highlight the magnitude of deviation in angle. The magnitude of pendulum velocity deviates to a maximum of 60 deg when the disturbance signal is introduced, but the controller is able to reduce the oscillation in less than 2 seconds which makes the pendulum to maintain its upright position to track the given signal.
The simulation result can be seen in Figure 4.12. The result shows that the settling time for controller based on LQR control strategy is about one and half second i.e. the controller can stabilise the pendulum within two and half second and has overshoot about 5 %. When applying a prescribed velocity input, the two DoF invertedpendulum system is able to follow it while keep the pendulum stable although there exists a lagging for about one second as can be seen in Figure 4.13. Compared with the pole placement method, the LQR result gives time to stabilise the system about one second slower but LQR use smaller gains of K matrix. It means that the system uses lesser effort or energy for stabilising the system. This is the advantage to use LQR control strategy the poles are placed in such way through the cost function to get optimal gains for not only in stabilising the system but also in controlling effort. Therefore, based on simulation results, it can be said that the LQR control strategy can be useful to determine controller gain values for the two DoF invertedpendulum system with optimal performance.
Generally, all systems are initially checked with conventional controllers including P, PI, and PID  since it is easy to develop and implement. Various methods are available for tuning these controllers. If the response is not satisfactory advanced, controllers are considered. When the system is non-linear and with significant delay, conventional controllers cannot give a satisfactory result . LQR controller is a suitable alternative in such case. It can deal with non-linear systems efficiently. Pole placement methods like Ackerman’s formula are very popular in designing the state feedback gain K and hence to place the poles in desired locations -. But in these methods, we need to specify the desired poles to seek the SVFB gain. Also these methods are only appli- cable for single input systems. However, it is very inconvenient to specify all the closed loop poles and we would like to have a technique that works for many numbers of inputs. Due to these constrains, we make use of the theory of optimal control for the design of a better controller. Optimal controllers are designed in sense of using the least required control effort to maintain equilibrium . Optimal control principle is inspired from naturally occurring systems which are optimal.
There are basically divided to classical controller and advanced controller in proposed controllers to stabilize the RIP system. Proportional-Integral-Derivative (PID) controller is one of the most widely used controller in field of control engineering. As the RIP system is an underactuated and non-linear system, PID controller is common to be designed in the RIP system, as it improves overshoot percentage and steady state error of the system with an easy approach. PID controller can be used although mathematical model of the system is not known. When the mathematical model is not known, Ziegler-Nichols rules can be applied. Ziegler Nichols tuning rules give an educated guess for the parameter values and provide a starting point for fine tuning. Thus, from year 2009 to 2012, 2DOF PID or Double-PID was designed as a controller in the RIP system [6,8,11,13,15]. As a stabilization controller in the RIP system, the controller stabilized the invertedpendulum and as well as the rotary arm.
The standard hybrid Fuzzy-LQR controller is constructed by choosing the inputs to be an error (e) and derivative of error (Δe) as shown and the output is the control signal (u). As indicated in Meshkov & Sokolov , among the three categories of hybrid Fuzzy-LQR controller structure, double input type is the most robust structure for unstable pole systems. As can be seen from Figure 8, the handled hybrid Fuzzy-LQR controller structure has two input and two output scaling factors. The input SFs Ke (for error (e)) and Kd (for the change of error (Δe)). While the FLC output (U) is mapped onto the respective actual output (u) domain by output scaling factors β and α.
Among the all Teqniques the LQR design is the simplest Teqnique to stabilize the InvertedPendulum System. This is similar to two-loop PD controllerDesign for stabilizes the system. Stabilization of the system is followed by linearization, finding state feedback gain for LQR and Swing by an energy based Controller. Position states are more penalized in compare to Velocity states.
After having specified the initial weighting factors, one important task is then to simulate to check if the results correspond with the specified d ign goals given in the introduction If not, an adjustment of the weighting factors to get a controller more in line with t e specified design goals must be performed. However, difficulty in finding the right weighting factors limits the application of the LQR based controller synthesis [?]. By an iterative study when changing Q values and running the
T HE double invertedpendulum (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 double invertedpendulum 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
This paper is to find a way to regulate the IRP to work with non-zero input for the arm position. In order to do that, State Space Control method is used to determine a control law. To handle with the error causing by the nonzero position of the arm, one more state is added to the IRP state vector, increasing the considered states into five. This will make the controller more complicated.
This paper proposes an intelligent control approach towards InvertedPendulum in mechanical engineering. InvertedPendulum is a well known topic in process control and offering a diverse range of research in the area of the mechanical and control engineering. Fuzzy controller is an intelligent controller based on the model of fuzzy logic i.e. it does not require accurate mathematical modelling of the system nor complex computations and it can handle complex and non linear systems without linearization. Our objective is to implement a Fuzzy based controller and demonstrate its application to InvertedPendulum. Model design and simulation are done in MATLAB SIMULINK ® software.
In supervisory control, at least, two levels as high-level (supervisor) and low-level controllers are used together to control a system under different conditions . As stated before, a controller with linearization feedback is used as the supervisor in the supervisory control system. Equations of the supervisor controller are as follows.
Being an under-actuated mechanical system and inherently open loop unstable with highly non-linear dynamics, the invertedpendulum system is a perfect test-bed for the design of a wide range of classical and contemporary control techniques. Its applications range widely from robotics to space rocket guidance systems. Originally, these systems were used to illustrate ideas in linear control theory such as the control of linear unstable systems. Their inherent non-linear nature helped them to maintain their usefulness along the years and they are now used to illustrate several ideas emerging in the field of modern non-linear control.
The model used in this paper is nonlinear in nature and are linearized using standard linearizing methods. Using this model, a set of transfer functions are derived which show the dynamics of the system. The same model is used to design various controller such as conventional PID controller, Fuzzy Logic Controller and H∞ Controller and comparative study of the performance is done in the face of model uncertainties, disturbances, rise time, settling time, maximum overshoot and finally we conclude that robust control using fuzzy Logic Controller is best controller compared to conventional and fuzzy controller.