This paper describes the method for **stabilizing** and **trajectory** **tracking** of Self Erecting Single **Inverted** **Pendulum** (SESIP) using Linear Quadratic Regulator (**LQR**). A **robust** **LQR** is proposed in this paper not only to stabilize the **pendulum** in upright position but also to make the cart system to track the given reference signal even in the presence of disturbance. The control scheme of **pendulum** system consists of two controllers such as swing up **controller** and **stabilizing** **controller**. The main focus of this work is on the **design** of **stabilizing** **controller** which can accommodate the disturbance present in the system in the form of wind force. An optimal **LQR** **controller** with well tuned weighting matrices is implemented to stabilize the **pendulum** in the vertical position. The steady state and dynamic characteristics of the proposed **controller** are investigated by conducting experiments on benchmark linear **inverted** **pendulum** system. Experimental results prove that the proposed **LQR** **controller** can guarantee the **inverted** **pendulum** a faster and smoother **stabilizing** process with less oscillation and better robustness than a Full State Feedback (FSF) **controller** by pole placement approach.

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ABSTRACT: Double Rotary **Inverted** **Pendulum** (DRIP) is a member of the mechanical under-actuated system which is unstable and nonlinear. The DRIP has been widely used for testing diﬀerent control algorithms in both simulation and experiments. The DRIP control objectives include Stabilization control, Swing-up control and **trajectory** **tracking** control. In this research, we present the **design** of an intelligent **controller** called “hybrid Fuzzy-**LQR** **controller**” for the DRIP system. Fuzzy logic **controller** (FLC) is combined with a Linear Quadratic Regulator (**LQR**). The **LQR** is included to improve the rules. The proposed **controller** was compared with the Hybrid PID-**LQR** **controller**. Simulation results indicate that the proposed hybrid Fuzzy-**LQR** controllers demonstrate a better performance compared with the hybrid PID-**LQR** **controller** especially in the presense of disturbances.

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This paper present real time control of an **inverted** **pendulum**. The system is inherently unstable and multivariable. It is mostly used in laboratories to study, verify and validate new control ideas. The dynamic model of the system was derived based on Lagrange approach and it was linearized. Linear Quadratic Regulator (**LQR**) **controller** was designed to stabilize the system in an upright position. The robustness of the control algorithm was tested based on disturbance rejection. Simulation and experimental results showed a good performance was achieved and the **controller** is **robust** to external disturbances.

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This thesis begins with the explanation of CIPS together with the hardware setup used for research, its state space dynamics and transfer function models after linearizing it. Since, **Inverted** **Pendulum** is inherently unstable i.e. if it is left without a **stabilizing** **controller** it will not be able to remain in an upright position when disturbed. So, a systematic iterative method for the state feedback **design** by choosing weighting matrices key to Linear Quadratic Regulator (**LQR**) **design** is presented assuming all the states to be available at the output. After that, Kalman Filter, which is an optimal Observer has been designed to estimate all the four states considering process and measurement noises in the system.

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Based on the pole placement **design** technique, a full state feedback **controller** using separation factor is proposed to stabilize a real single **inverted** **pendulum** [11]. The strategy is to start with selection two dominant poles that achieve a certain desired performance, using a separation factor between the selected dominant poles and the other poles to eliminate their effect on the system performance, and finally Ackermann’s formula can be used to calculate the feedback gain matrix to place the system poles at the desired locations. Simulation and experimental results demonstrate the effectiveness of the proposed **controller**, which offers an excellent **stabilizing** and also ability to overcome the external resistance acting on the **pendulum** system.

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Jadlovska and Sarnovsky [8] have revealed an approach for the control of the rotary single **inverted** **pendulum** system. State feedback control techniques are used as pole placement and the **LQR** optimal control. Mathew et al. [9] 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 [10] 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 [11] 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 [12] 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. [13] 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. [14] derived the nonlinear dynamics of the RIP. Artificial neural network (ANN) is applied to identify the model.

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Based on the review stated above, velocity **tracking** has been one of the major objectives of controlling the TWIP. Many works in the past years have been published in the area. Model based controllers like **LQR** [32, 34], and Pole placement **controller** [27], which are designed based on the linearized model of the robot making the model uncertainties to affect the controllers gain. Nonlinear controllers like partial feedback linearization [11, 19, 33] and SMC [1, 12, 22] performs well in rejecting modelling inaccuracies, parameter variations and disturbances, but SMC has the problem of chattering. Intelligent controllers were used by researchers to track a desired speed of the TWIP, in [6] a direct adaptive model reference control scheme was used, fuzzy logic which is a non-model based **controller**, was used in [15, 20]. Adaptive neural SMC method for **trajectory** **tracking** was shown in [35]. In this paper, three

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It is a well-established benchmark problem which provides many challenging problems on the **design** of control **design**. The system is nonlinear, unstable, no minimum phase and under actuated. Because of their nonlinear nature pendulums have maintained their usefulness and they are now used to illustrate many of the ideas emerging in the field of nonlinear control [1]. The challenges of control made the **inverted** **pendulum** systems become a classic tool in control laboratories.

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the system for angles up-to maximum degrees, it is also seen that it does not perform well in terms of speed. In order to solve this problem, we have used an supervisory **controller**. A non- Fuzzy supervisor (linearization feedback) is designed for the Fuzzy **controller** such that, if the **inverted** pendulum’s angle exceeds a limit (23 degrees); supervisor **controller** takes action and reduces the angle immediately. In normal conditions (angles below 23 degrees) the fuzzy **controller** has to control the system. Thus not only the system becomes stable for a larger initial angle but also a simpler fuzzy **controller** (with less rules and inputs) is used, less calculation is required and the speed becomes higher.

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widely used as a teaching aid and in research experiments around the world. As a teaching aid, the **inverted** **pendulum** 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 **inverted** **pendulum** models such as rocket control, the dynamic balance of skiing, bicycle/motorcycle dynamics (Hauser et al., 2005). Recently, a mobile **inverted** **pendulum** 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).

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Figure 2.2 shows a block diagram of a Full State Feedback (FSF) **controller** in the RIP system. The RIP system is was designed in state space model. Poles of the closed loop system may be placed at any desired locations by means of state feedback through an appropriate state feedback gain matrix K. There are a few approach to tune FSF **controller**, one of them is by pole placement method. According to M. Akhtaruzzaman [5], he designed FSF **controller** by placing stable poles of the RIP system, then used Ackermann’s formula and Integral of Time-weighted Absolute Error (ITAE) table, state feedback control gain matrix, K which is a 4 × 1 matrix was obtained. FSF **controller** was considered relatively ease of **design** and effective procedure to obtain the gain matrix K. There are some drawbacks of designing FSF **controller**. It requires successful measurement of all state variables or a state observer in the system, where it needs control system **design** in state space model. It also requires experienced researcher to determine the desired closed loop poles of the system, especially when the system has a higher order system than second or third order.

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3.1 **Inverted** **Pendulum** on cart (Modeled by K.Ogata, 1978) 30 3.2 **Inverted** **Pendulum** plant in MATLAB’s Simulink 35 4.1 Basic structure of state feedback **controller** 37 4.2 Block diagram of a plant with a state feedback **controller** 37 4.3 State feedback block diagram in the Simulink 41 4.4 Gain of the state feedback **controller** in Simulink 42 4.5 State feedback controls with steady state (SS) gain 43 4.6 State feedback with integral block diagram at Simulink 44 5.1 The block diagram representing the interface process 45

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simulated **inverted** **pendulum** system and these K values were implemented in the physical laboratory **inverted** **pendulum**. The non – minimum phase behaviour of both simulated and laboratory **inverted** **pendulum** systems were noticed. Although, the results obtained show no direct correlation between time - taken to swing the **pendulum** to its upright position for the two systems. These variations may be due to high K values which make the system approach the upright position with too high velocity. The physical **inverted** **pendulum** shows high robustness for any external disturbance as the cart is homing back and front during the measurement as stability attained.

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Synopsis: Purpose of this paper is to **design** and build a two wheeled upright robot. The robot was embedded with sensors, in order to the robot interact with surrounding the distance sensor in combination with temperature sensor was implemented on it. The robot also has a bowl on top for carrying load purpose. The **controller** that used to balance the robot is PID **controller** and linear quadratic Gaussian (LQG) **controller**. For testing the sensor fusion between the accelerometer and gyro, the Kalman filter and complementary filter was used. As result, Kalman filter was chosen because Kalman filter sufficient for both linear movement and noise of the accelerometer as well as estimate and compensate for the gyros drift and bias. In this project, the microcontroller has been used is Arduino Mega35. Unfortunately, the LQG **controller** cannot be implementing in hardware this is because Arduino is not fast enough to do calculation for the **controller** in the required loop time. As result, PID **controller** was used to balance the robot and it was successful.

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C ONTROLLING a Cart-**Inverted** **Pendulum** System (CIPS) is a challenging problem which is widely used as benchmark for testing control algorithms such as PID controllers [1], neural networks [2], [3], fuzzy control [4], genetic algorithms [5]. The CIPS features as higher order, nonlinear, strong coupling and multivariate system, which has been studied by many researchers. It is used to model the field of robotics and aerospace field, and so has important significance both in the field of the theoretical study research and practice. It has good practical applications right from mis- sile launchers to segways, human walking, earthquake resistant building **design** etc. The CIPS dynamics resembles the missile or rocket launcher dynamics as its center of gravity is located behind the centre of drag causing aerodynamic instability. The CIPS has two equilibrium points [6], one of them is stable while the other is unstable. The stable equilibrium corresponds to a state in which the **pendulum** is pointing downwards. The control challange is to maintain the **pendulum** at the unstable equilibrium point where the **pendulum** upwards, with minimum control energy. In recent times optimal control provides the best possible solution to process control problems for a given set of performance objectives. Detail review on optimal control has been presented in [7]. Survey on optimal control approaches and their applications have been conducted, for instance, an optimal control approach for inventory systems [8] and energy optimisation [9].

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being an inherently unstable system, is a common control problem, and so being one of the most significant classical problems, the control of **inverted** **pendulum** has been a research interest in the field of control engineering. Due to its importance this is a choice of dynamic system to analyze its dynamic model. The aim of this case study is to stabilize the **Inverted** **Pendulum** (IP) such that the position of the cart on the track is controlled rapidly and accurately so that the **pendulum** is always in erected in its **inverted** position during such movements even the system is affected with disturbance input such as wind force or due to others cause. Realistically, this mechanical system is representative of a class of altitude control problems whose aim is to maintain the desired vertically oriented position at all times [6-9].

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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** [3]. A linear state feedback **controller** based on the linearized **inverted** **pendulum** model can instead be used, and may also be extended with a disturbance observer (Kalman filter) to improve the disturbance rejection performance.

The PID **controller** has been used to control about 90% of industrial processes worldwide [3]. The main problem of that simple **controller** is the correct choice of the PID gains and the fact that by using fixed gains, the **controller** may not provide the required control performance, when there are variations in the plant parameters and operating conditions. Therefore, a tuning process must be performed to insure that the **controller** can deal with the variations in the plant [4]. To tune the PID **controller**, there are numbers of strategies, the most famous, which is frequently used in industrial applications, is the Ziegler-Nichols method [3] , genetic algorithm GA, etc. Moreover, PSO was another method for tuning procedure. PSO first introduced by Kennedy and Eberhart is one of the modern heuristic algorithms, it has been motivated by the behavior of organisms, such as fish schooling and bird flocking [5]. Generally, PSO is characterized as a simple concept, easy to implement, and computationally efficient. Unlike the other heuristic techniques, PSO has a flexible and well-balanced mechanism to enhance the global and local exploration abilities [6]. In this paper, a novel PSO-based approach to optimally **design** a PID **controller** for a mobile robot **trajectory** **tracking** is proposed. This paper has been organized as follows: in section 2 both kinematics and dynamic models of mobile robot are described. In section 3, the particle swarm optimization method is reviewed. Section 4, describes how PSO is used to **design** t`he PID **controller** optimally for mobile robot to control the velocity and

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Although, some investigations have been developed on spherical robots recently, motion control of these robots is still one of the major problem in robotic researches. Since the spherical shell is assumed to roll over the ground surface without any slipping, its motion is subjected to two nonholonomic constraints. On the other hand, according to Brocket’s theorem, the stabilization of the equilibrium points of the nonholonomic systems through time invariant state-feedback is not possible [9]. By the way, for the control of the spherical robot, Zhao et all have derived the dynamical model of the spherical robot merely for straight line motions and a PID **controller** has been proposed for the robot’s motion on straight line trajectories [10]. The control of the spherical robot by uses of a **pendulum** as a control actuator has been developed by feedback linearization method in straight line trajectories [11]. Furthermore, **trajectory** **tracking** control of the spherical robot on straight paths has been investigated using SMC method [12], adaptive hierarchical sliding mode approach [13]; and also using combined adaptive neuro-fuzzy and SMC method [14].

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