I declare that this thesis entitled “**Design** of **linear** **quadratic** **regulator** **controller** with **adjustable** **gain** **function** for **rotary** **inverted** **pendulum** **system**” is the result of my own research except as cited in the references. The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.

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Many studies are found in the area of control issues related to **inverted** **pendulum** systems. Some of the studies dealing with **inverted** **pendulum** control are summarized herein. Yan [1] developed a tracking control law for underactuated RIP by applying nonlinear back stepping, differential flatness, and small **gain** theorem. Mirsaeid and Zarei [2] presented a mechatronic **system** case study on adaptive modeling and control of an **inverted** **pendulum**. Hassanzadeh et al. [3] 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 [4] focused on the swing up and stabilization control of a **rotary** **inverted** **pendulum** (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. [5] have described different **controller** designs for **rotary** **pendulum**. Experimental and MATLAB based simulation results are given. Hassanzadeh et al. [6] also studied control by using evolutionary approaches. GA, Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) are used for designing the rotational **inverted** **pendulum**. Quyen et al. [7] presented the dynamic model of RIP. ANN **controller** is used for controlling the **system**.

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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** [12]. 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**

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In the research being carried out the **system** is taken as **rotary** **inverted** **pendulum**. It is a highly nonlinear and unstable **system**. Linearization is done with the help of state space equations. From the linearized model, the transfer **function** of the **system** with **pendulum** angle as output and input voltage to the motor is taken. Then from the characteristic polynomial that is the Kharitnov polynomial is formed with state feedback **controller** transfer **function**. By giving different ranges for state feedback **gain** matrix, the range which stabilizes the **system** is found ie all closed loop poles lies on the left side of the s plane in that range.

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This project is a contribution to developed **Linear** **Quadratic** **regulator** (LQR) speed control for DC motor using MC68HC11. The performance measure is a **quadratic** **function** composed of state vector and control input. Based on LQR control law (**linear** time invariant **system**) will be obtained via solving the algebraic Riccati equation. The **system** is applied for the speed control of servo motor. To measure the performance have to be minimized contains output error signal and differential control energy. The LQR **controller** doesn’t need to feedback full state just the **controller** received signal error only. Matrix Q can be determined from the roots of the characteristic equation. When the poles for the closed loop **system** are assigned, the existences of LQR **controller** are derived. At the DC motor control **system**, the feedback information on the motor can provide error detector signal. The control loops and to improve the reliability fault condition that may damage the motor based on comparator. The scopes of this project are to choose the optimal value feedback **gain** on the order grab the stable **system** and to describe how a MC68HC11 can be used to implement a speed **Linear** **Quadratic** **Regulator** feedback control in the unstable **system**. The project used G340 servo driver, the driver is a monolithic DC servo motor **controller** proved all active functions necessary for a complete closed loop **system**. To developed this **system** used MATLAB software, C programming and WP11 software window.

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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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For high-level **controller**, different control techniques, such as model predictive control [6], [7], robust control [8]–[10] and sliding mode control [11], [12], have been proposed in the technical literature. However, **Linear** **Quadratic** Regulators (LQR) is one of the most commonly used control method for reference yaw moment generation. To enhance tracking performance for a wide range of longitudinal velocities, the solution of the Jacobi-Riccati equation of the LQR optimisation was exploited by Refs. [13]–[15] to formulate variable feedback and feedforward gains as functions of vehicle longitudinal velocity. However, closed-loop stability of the resulting control systems were not systematically investigated for time varying longitudinal velocities, and represent a concern for such methods. Furthermore, LQR algorithms suffer from a limited **gain** margin against parameter perturbation and external disturbances [16], [17]. The robustness of LQR methods depend on the selection of the weights in the cost **function** to be minimized which also affect the closed-loop response [16]. Usually, such weights are the result of time-consuming trial and error numerical procedures aiming to find a satisfactory trade-off between robustness and performance [18]. Alternatively, to increase **system** robustness against disturbances and uncertainties without increasing the complexity of the **design**, LQR methods have been augmented with Variable Structure Control (VSC) actions. For example, a robust optimal sliding-mode yaw rate **controller** was proposed in [15] to address the tracking problem under uncertain conditions. A sliding mode **controller** with time-varying sliding surfaces to solve the optimal control problem for both **linear** and nonlinear systems was presented in Ref. [19]. Furthermore, a LQR/VSC method based on the Planes Cluster Approaching Mode (PCAM) to guarantee global asymptotic stability in the presence of parameter perturbation and unmodelled dynamics was developed in [20]. However, despite their theoretical effectiveness to suppress bounded disturbances, discontinuous control terms, typically embedded in sliding mode control solutions, induce chattering on the control action. In automotive applications, chattering may results either in stress and wear of mechanical or electrical parts, with a consequence damage of the **system** in short time, or vibrations during implementation [21]. In addition, when the discontinuous control action is smoothed to mitigate chattering, often it is not possible to prove the asymptotic convergence to zero of the tracking error but only its boundedness [21].

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Baili Zhang (2011), three methods used to **design** the **controller** of the **inverted** **pendulum**, 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 )

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LQG is a type of compensator. LQG, it is a combination of **Linear** **Quadratic** **Regulator** (LQR) designed with that of Kalman Filter. Here LQR is a **linear** **regulator** that minimized a **Quadratic** Objective **Function**, which includes transient, terminal, and control penalties. Kalman Filter is an optimal observer for multi output plant in the presence of process and measurement noise, modelled as white noise. The block diagram of LQG **regulator** is shown in figure (7).The software is based upon the equation (26) in order to calculate the state estimates.

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There are four ways of doing code in MATLAB. One can directly enter code in a terminal window. This amounts to using MATLAB as a kind of calculator, and it is good for simple, low-level work. The second method is to create a script M-file. Here, one makes a file with the same code one would enter in a terminal window. When the file is run, the script is carried out. The third method is the **function** M-file. This method actually creates a **function**, with inputs and outputs. The fourth method will not be discussed here. It is a way to incorporate C code or FORTRAN code into MATLAB and this method uses .mex files.

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Servo motors have received great attention in recent decades. Compare to other motor types, servo motors can deliver more torque and power for the same size. Besides that, it also provides safe and reliable operation and requires less maintenance which make the used of servo motor received a great attention. Specifically, with latest development in microprocessor and power electronics, very accurate and highly efficient servo motor control systems have been developed. The need for a feedback element was required in order to make control **system** stable.However, the feedback not only increases the cost but also makes the control of position and speed of servo motor becomes complex Further, the control accuracy is directly proportional the precision of servo motor control. However, the precision of servo motor control has been difficult to achieve due to the ignorance of the plant disturbance and inaccuracies in feedback measurement cause by sensor noise. Therefore, **Linear** **Quadratic** Gaussian (LQG)**controller** has been proposed to overcome the sensor less position and speed control due to this problem.The main proposed alternative of this project were to control and estimate the motor position and speed besides to overcome the weakness of other digital **controller**.

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based on **system** behavior so that they can control the intended **system**. Lot A. Zadeh proposed that \as the complexity of the **system** increases, the possibility of describing the **system** with deterministic terms diminishes" [18]. In the following, the fuzzy logic and fuzzy **controller** will be expressed briey. One of the most important characteristics of fuzzy logic is the use of linguistic variables, instead of numerical variables. Linguistic variables are dened as variables in a natural language such as small/large, almost/certainly, etc. Fuzzy sets are not similar to classic sets to which a member belongs or not. Fuzzy sets allow partial membership in a set, which means that an element belongs to more than one set [19]. Each fuzzy set is characterized by its MF whose member is allocated a number between zero and one. Based on these MFs, logical operations were redened among sets.

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The **inverted** **pendulum** is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. It is unstable without control. The process is non **linear** and unstable with one input signal and several output signals. It is hence obvious that feedback of the state of the **pendulum** is needed to stabilize the **pendulum**. The aim of the study is to stabilize the **pendulum** such that the position of the carriage on the track is controlled quickly and accu- rately. The problem involves an arm, able to move horizontally in angular motion, and a **pendulum**, hinged to the arm at the bottom of its length such that the **pendulum** can move in the same plane as the arm. The conventional PID **controller** can be used for virtually any process condition. This makes elimination the offset of the proportional mode possible and still provides fast response. In this paper, we have modelled the **system** and studied conventional **controller** and LQR **controller**. It is observed that the LQR method works better compared to conventional **controller**.

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the state feedback **controller** was loaded in the file menu of the editor and saved. It should be noted here that this process provides self-erecting functionality and implement a high performance control as the **inverted** **pendulum** once erected. The editor file was exit without any changes to the algorithm. The **pendulum** was displaced using ruler to approximate 20 0 and withdrawn quickly, if not (i.e. the **controller** is not active) the **pendulum** rod and carriage will be repositioned and the process will be repeated again. The real-time algorithm kicked the **pendulum** at properly phased movement of its oscillation cycle and it increased the amplitude of its swung. The **pendulum** approached the **inverted** position and algorithm switched to a control law that captured the **pendulum** in an **inverted** position, if not the process as to be repeated.

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In the right hand side, the hardware component consists of the mechanical **system** of **inverted** **pendulum**, together with DAQ card and I/O port for interfacing purpose. The hardware responses through the signal received from the software. At the same time, it connects to the **controller** for corrective task.

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In paper [14], stabilization of RIP **system** is achieved by mapping **linear** optimal control law to the fuzzy inference **system** (FIS). A Mamdani FIS is designed which stabilizes the **pendulum** in the **linear** zone, emulating LQR control around the equilibrium point. The **linear** state feedback law is mapped to the rules of the fuzzy inference engine. The **system** consists of two fuzzy inference subsystems, one taking as inputs the angular position and speed of the **pendulum** arm, and other one taking as inputs the angular position and speed of the **pendulum**. The two output signals from both subsystems are then added to give a single control signal [14]. In the experiment, the observed performance of the **system** is smooth, and it is experimentally shown that the closed loop balancing **system** based on the fuzzy **controller** exhibits greater robustness to unmodeled dynamics and uncertain parameters than the LQR **controller** that it emulates [14]. Fuzzy logic **controller** requires complex linguistic expression. The linguistic expression which are the basic of the rule-base of the fuzzy logic **controller** must generated based on experts’ whom have done many relevant experiments.

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Graph based modelling have been used or developed in three paradigms: bond graphs, **linear** graphs and block diagrams. Bond graph modelling is based on junctions that transform elements through bonds (Rossenberg and Karnopp, 1983). The bonds represent power flow between modelling elements. Bond graphs are domain independent but they are not convenient for 3D mechanics and continuous hybrid systems. **Linear** graph modelling is similar to bond graphs, these **linear** graphs represent the energy flow through the **system**, expressed by through and across variables (also called terminal variables) (Durfee et.al., 1991). **Linear** graphs are domain independent and, unlike bond graphs, they can be easily extended to model 3D mechanics and hybrid **system**. The third modelling technique is based on block diagrams, as in Simulink. Block diagrams modelling are specified by connecting inputs and outputs of primitive models such as integrators, multipliers, or adders.

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for swing up and stabilization control of an **inverted** **pendulum**. In addition, Chakraborty et al (2013) investigated the optimization of PID **controller** using Genetic Algorithm. A dynamic modelling and optimal control of wheel **inverted** **pendulum** were proposed in Shamsudin et al (2013) using an optimally tuned partial-state PID. Swing up and stabilization of double **inverted** **pendulum** using LQR and LQR based fuzzy was proposed in Bhangal (2013), and their performances were analyzed and compared. An intelligent control algorithm has also been proposed in A-hadithi (2012) and implemented for swing up and stabilization of a double **inverted** **pendulum** **system**. The combinations of intelligent and conventional control have shown good performances and robustness of the control algorithms. An Adaptive Neural Network for motion control of a wheeled, **inverted** **pendulum** has also been presented in Yang et al (2014). In Chalupa and Bobal (2008), Model Predictive **Controller** has been proposed. A novel PSO based Sliding Mode Control for stabilization of an **inverted** **pendulum** was presented by Singh et al (2014). In Brisilla and Sankaranarayanan (2015), a stabilization of an **inverted** **pendulum** using a nonlinear control algorithm has been presented.

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174 The selection of PID **controller** parameters (KP, KI, KD) is important as incorrect selection of these parameters can make controlled process input unstable. The control parameters are adjusted to optimum values for the desired response. This is called Tuning of the control loop. Fuzzy **controller**:

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