control iteration, making their real-time implementation on systems with fast dynamics challenging. MPCs with linear prediction models and constraints have been well studied in the context of cart and pole invertedpendulum, but their appli- cation to TWIPs has been limited. Dini  implements a finite horizon MPC, but only for the control of the yaw and pitch states. Hirose  and Yue  also control the system’s forward velocity, incorporating both pitch and velocity constraints. Both still rely on an externally generated velocity trajectory in order avoid infeasibility when performing point-to-point manoeuvres. Ohhira  comes close to implementing a full MPC position controller, in which a stabilising inner LQR is used to provide a closed loop system that is then augmented by an outer optimal predictive controller, in a similar manner to dual-mode MPC. However, the inner loop is calculated with a 0.01 s sample time, whilst the outer optimiser is computed with a 0.08 s sample time, meaning that constraint satisfaction can only be guaranteed when the inner and outer loop sampling instants coincide. This intermittent enforcement of the hard input constraints could allow a large disturbance with an aggressive controller to demand a sufficiently large wheel torque as to induce wheel slip and loss of control.
Control system design for wheeledinverted pendulums has been intensively studied in the literature. Many of early studies adopt linear controllers – based on a linear approximation of a nonlinear model around an equilibrium point. A limitation of linear controllers is that they are not necessarily effective in cases where the system state is far from the equilibrium point, such as in our transformation problem. More recent studies have proposed various nonlinear control methods for wheeledinverted pendulums –. They can be classified by how to deal with the underactuation, which occurs mainly because both the velocity (or position) of the vehicle and the pitch angle of the body need to be controlled by a single actuator.
Grasser et al.  built a scaled-down prototype of a DSP-controlled two-wheeled vehicle based on the invertedpendulum, and Pathak et al.  studied the dynamic equations of the wheeledinvertedpendulum by partial feedback linearization. Though the control system used to guarantee stability of the system in [1, 2], developing a highly accurate model of WIP system is complex. Ren et al.  proposed a self-tuning PID controller which the controller parameters are tuned automatically to overcome the disturbances and parameter variations. However, the system stability can not be guaranteed. A sliding-modecontrol method is proposed to be capable of handling both parameter uncertainties and external disturbances . The algorithm of sliding-modecontrol required the system dynamics; however, it is difficult to be obtained in real-time application. Chiu  used an adaptive output recurrent cerebellar model articulation controller. The proposed scheme was implemented in a PC-based experimental system to verify its effectiveness but the design procedure is overly complex.
A two-wheeledinvertedpendulum (TWIP) mobile robot is a three-degree-of-freedom under- actuated mechanical system with highly nonlinear dynamics. It is quiet challenging to control such system due to its unstable and under-actuated nature. Numerous works on modeling and control of TWIP mobile robot have been presented in literature. Kim et al  investigated the exact dynamics of the TWIP mobile robot, and a Linear Quadratic Regulator (LQR) controller was developed for balancing the robot. Fiacchini et al  proposed linear and nonlinear controllers for stabilizing a personal pendulum vehicle. To compensate for the measurable disturbances, the work in  compared the performance of ModelPredictive Controller and LQR. Multipoint pole placement control for velocity tracking of the TWIP is shown in . In Jones and Stol , the performance of the two wheeled mobile robot in low-traction environment was investigated by designing a LQR controller based on linearized model of the robot which includes wheel slip effects. Pathak et al  proposed velocity and position controllers for the TWIP robot via partial feedback linearization. Dai et al  proposed sliding mode controllers for self-balancing and yaw motion and designed independently. While Kim et al  investigated a nonlinear motion control using the State-Dependent Riccati Equation (SDRE) control framework. Kharola et al  discussed a fuzzy logic control strategy for control and stabilization of TWIP.
It is important to be noted that for the case of the tracking per- formance, where the controller effort should make y(t) ≈ r(t) as t → ∞, hence the DC gain of the transfer function should be approximately 1. Therefore, it is necessary to scale the reference input using pregain N ¯ . Simulation experiments were systematically conducted by varying the configuration of Q and R. Simulation results have been discussed in detail. The simulation results have shown that the larger Q and R the more you penalize the state and the control effort. Choosing a large value for R means that the controller stabilises the system with less (weighted) energy, it is called expensive control strategy. This control strategy is used when the control signal is constrained . On the contrary, choosing a small value for R means that the controller penalises the control signal (cheap control strategy), causing a large control signal. Large Q implies less concern about the changes in the states.
A new kind of quadruped-imitating walking robot is designed, which is composed of a body bracket, leg brackets and walking legs. The walking leg of the robot is comprised of a first swiveling arm, a second swiveling arm and two striding leg rods. Each rod of the walking leg is connected by a rotary joint, and is directly controlled by the steering gear. The walking motion is realized by two striding leg rods alternately contacting the ground. Three assumptions are put forward according to the kinematic characteristics of the quadruped-imitating walking robot, and then the centroid equation of the robot is established. On this basis, this paper simplifies the striding process of the quadruped-imitating walking robot into an invertedpendulummodel with a constant fulcrum and variable pendulum length. According to the invertedpendulummodel, the stability of the robot is not only related to its centroid position, but also related to its centroid velocity. Takes two typical movement cases for example, such as walking on flat ground and climbing the vertical obstacle, the centroid position, velocity curves of the invertedpendulummodel are obtained by MATLAB simulations. The results show that the quadruped-imitating walking robot is stable when walking on flat ground. In the process of climbing the vertical obstacle, the robot also can maintain certain stability through real-time control adjusted by the steering gears.
In order to perform any research work on any system, the first and foremost thing is to know about the system dynamics. The dynamics of the double invertedpendulum can be explained using a series of differential equations called the equations of motion ruling over the double invertedpendulum response to the applied force. The double invertedpendulum is shown in the figure(4) below:
SMC is a method in modern control theory that uses state- space approach to analyze such a system. Using state-space methods it is relatively simple to work with a multi-output system .The typical structure of a sliding mode controller (SMC) is composed of a nominal part and additional terms to deal with model uncertainty. The way SMC deals with uncertainty is to drive the plants state trajectory onto a sliding surface and maintain the error trajectory on this surface for all subsequent times. The advantage of SMC is that the controlled system becomes insensitive to system disturbances. The sliding surface is defined such that the state tracking error converges to zero with input reference. With the perspective to achieve zero steady state error, Cao and Xu (2001) and Sam et al. (2002) have proposed the proportional integral sliding modecontrol (PISMC) in their studies . The proportional factor in this controller gives more freedom in selecting some parameters matrices that will make the output response faster and the stability condition to be more easily satisfied. The proportional integral sliding surface equation can be represented as (14).
• Objects and functions do not map to each other: the architecture of functionally decomposed system is significantly different from the architecture of an object-oriented system; from UML point of view, it is impossible to map directly from use case model, sequence diagram, system sequence diagram…etc. to domain model or object model and vice versa. Also using use cases within the context of object-oriented development - so called “differing localization strategies” - will result in the introduction of significant errors .
The effect of quantization factor and scale factor on the control effect of fuzzy controller is further analyzed. Because the fuzzy controller is obtained by the linear state feedback controller, in order to maintain the approx- imate linear relationship between the control variables and the state variables, the value of the quantization fac- tor and scale factor can be changed under the following conditions:
It is seen from the above end products and analysis that all the planned models work to stabilize the InvertedPendulum. The implemented PID’s control design has the slowest response due to Integral term, while the PD’s controlling proposal improves this transient response. Furthermore, outcomes also reveal that, experimented SMC scheme depicts the best results, when compared to PID’s and PD’s. This methodology has less settling time as well as the best response, when it reaches to the unstable point. However, due to applied discontinuous control signal, fast and finite amplitude oscillations (known as chattering) are generated, which travel to the plant in a controlled manner. In addition, the future scope belongs to the cutback of produced chattering phenomenon, exclusive of upsetting the basic nature of the controller itself.
ABSTRACT: For at least fifty years, the invertedpendulum has been the most popular benchmark, among others, for teaching and researches in control theory and robotics. This paper presents the key motivations for the use of that system and explains, in details, the main reflections on how the invertedpendulum benchmark gives an effective and efficient application. Several real experiences, virtual models and web-based remote control laboratories will be presented with emphasis on the practical design implementation of this system. A bibliographical survey of different design control approaches and trendy robotic problems will be presented through applications to the invertedpendulum system. In total, 150 references in the open literature, dating back to 1960, are compiled to provide an overall picture of historical, current and challenging develop- ments based on the stabilization principle of the invertedpendulum
Invertedpendulum system is a nonlinear unstable system, an ideal experiment platform for teaching control theories and conducting various control experiments. Many abstract control concepts, such as the stability and the controllability of a control system, can all be shown visually through the invertedpendulum system. In addition to educational purposes, an invertedpendulum is also a research area for many researchers of modern control theories. Through the continuous research on new ways of controlling invertedpendulum, researchers have developed new control methods, and apply them to the high tech areas such as aeronautical engineering and robotics.
Next problem in this project is faced by hardware part, where the selection of the appropriate tools and equipment to ensure the system is always running smoothly. The device selection is involved in this project including microcontroller circuit and also the device to allow the pendulum to know differ angles depending on the situation. In terms of equipment, the selection of designing a cart, the materials used for the pendulum and also the selection motor to drives a cart from dropping the pendulum. Lastly, the problem is faced from installation software into the hardware part which is to ensure the pendulum always maintain in upward position by using controlling the speed of wheel. Then to strength the result for performance this system, data from simulation will be compared where all the equipment and tools will combine together to get the InvertedPendulum (IP) systems with movement a cart in full functioning.
A robust MPC algorithm is developed in Chapter 3 for continuous-time uncertain non- linear systems. This R-MPC (Robust and re-solvable MPC) algorithm generates online a nominal feedforward control policy based on a modified FHC. An additional offline-designed feedback policy is added in R-MPC to generate an invariant tube that ensures the actual states remain in the proximity of the nominal feedforward trajectory. The tube provides an explicit characterization of the R-MPC robustness, which can accommodate uncertainties and disturbances in the actual dynamics up to the level considered in the feedback policy design. An important contribution of the R-MPC algorithm is the modified FHC, which puts a relaxation on the initial nominal state for re-solves. The initial nominal state is allowed to be within the invariant tube rather than fixed to the actual state, as in the base- line MPC; this relaxation guarantees continued FHC feasibility during re-solves and leads to robust asymptotic stability of the R-MPC algorithm. The R-MPC algorithm places no requirements on the re-solve rate, which is useful in online applications with computational limitations. Additionally, specific design methods are provided for a class of continuous-time systems with uncertain nonlinear terms that have norm-bounded derivatives. The R-MPC algorithm development was joint work performed with Beh¸ cet A¸cıkme¸se .
Fig. 11 and Fig. 12 show two sets of results comparing the application of fuzzy control and conventional control (PID controller) techniques to the invertedpendulum problem simulation. For the same system parameters here, the PID controller proportional gain, Kp, derivative gain, Kd and integral gain, Ki are found to be 9, 14, and 0.06 respectively. The first two graph show that the fuzzy logic controller gives a smaller overshoot and shorter settling time. In the second set, the mass of the cart is changed without modifying the controllers. Fig. 14 shows that the conventional controller totally failed to balance the pendulum as it was designed for the nominal value of cart mass. On the other hand, the fuzzy logic controller exhibited small performance degradation due to this parameter change as shown in Fig. 13. This proves that fuzzy logic is not based on the mathematical model of the invertedpendulum and more robust to mass variations.
Abstract : Here modeling and simulation study of basically two control strategies of an invertedpendulum system are presented. The invertedpendulum represents a challenging control problem, which continually moves toward an uncontrolled state. Certain modern techniques are now available because of the development of artificial intelligence and soft computing methods. Fuzzy control based learning makes the system more reliable and taking the advantage of precise control at different levels of existence as fuzzy is applicable to non value zero to full value as one, rest of the interim values are also considered . ANFIS is one of an example of fused neuro fuzzy system in which the fuzzy system is incorporated in such a framework which is adaptive in nature . The fuzzy controllers are used to sort out the problem of learning and when the ANFIS is used the adaptability of the system get improves which is based on that learning feature for non-linear system of invertedpendulummodel . D. K. Somwanshi ,Department of EIC & Electrical Engineering ,Jagannath gupta institute of engineering & technology sitapura, jaipur,