The thesis presents a number of control approaches such as LQR and LQG for CIPS. These design methods have been successful in meeting the stabilization goals of the CIPS, simultaneously satisfying the physical constraints in track limit and control voltage. Due to the non-linear cart friction behavior there is a deviation from the ideal behavior that leads to undesired oscillations mainly in state feedback based control methods. The Linear Quadratic Regulator (LQR) weight selection for the cart-invertedpendulum has been thoroughly presented. The choice of LQR is well known that unlike ordinary state feedback the LQR solution obtained after LQR weight selection automatically takes care of physical constraints. LQG compensator design also considers the white noises such as process noise and measurement noise. While designing LQG compensator a Kalman Filter was used as an optimal estimator. Lastly, Loop Transfer Recovery (LTR) analysis has been performed for suitably selecting the tuning parameter for observer design. By LTR, a set of possible tuning parameters representing the state and process noise covariances can be selected depending on the trade-off between noise suppression and system robustness.
The thesis presents a number of control approaches such as LQR, Two-Loop PID Controller, Sub-optimal LQR, and ISM. These design methods have been successful in meeting the stabilization goal of the CIPS, simultaneously satisfying the physical constraints in track limit and control voltage. The LQR, Two-Loop-PID and ISM are successful in ensuring good robustness on the input side of the CIPS. The ISM and Two-Loop-PID give good tolerability towards multichannel gain variation on the output side. Due to the non-linear cart friction behavior there is a deviation from the ideal behavior that leads to undesired stick slip oscillations mainly in state feedback based control methods. The Linear Quadratic Regulator (LQR) weight selection for the cart-invertedpendulum has been systematically presented together with robustness analysis. The choice of LQR is well known that unlike ordinary state feedback the LQR solution obtained after LQR weight selection automatically takes care of physical constraints. The LQR poles guarantee minimum robustness of ± 6 dB gain margin and 60 o phase margin.
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:
This thesis includes system and hardware description of InvertedPendulumSystem, Dynamics of the system, State space model, Derivation of Transfer Functions. In Past a lot of research work has already been done in InvertedPendulum for development of Control Strategy. Here in this thesis we have done a very small work to design Control Strategy and also validate them with real-time experiments.
Rotary InvertedPendulumSystem (RIPS) has become one of the most challenging task even it is an interesting system in this era. RIPS is a classical system that includes in this engineering system field with characteristics which is an unstable system, multi- variables and non-linear system. The stucture of this invertedpendulumsystem is made up of a vertical pendulum rod, a horizontal arm and an encoder which will drives the pendulum to rotate. This project intends to obtain a mathematical modeling for the RIPS and to develop the controller for the system. The mathematical model can be obtained by deriving the model of invertedpendulumsystem by using Euler’s equation to perform state space equation. Therefore, the Pole Placement controller is proposed in this project to regulate the RIPS to drive it in upright position. The mathematical derivations proved that the designed controller is required to stabilize the invertedpendulumsystem. Simulation study is performed to prove the effectiveness of the designed controller and the result shows that the controller is capable to maintain the pendulum in the stable inverted position at the desired value of parameters.
This paper describes the method for stabilizing and trajectory tracking of Self Erecting Single InvertedPendulum (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 pendulumsystem 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 invertedpendulumsystem. Experimental results prove that the proposed LQR controller can guarantee the invertedpendulum a faster and smoother stabilizing process with less oscillation and better robustness than a Full State Feedback (FSF) controller by pole placement approach.
In this paper modeling of an invertedpendulum is done using Euler – Lagrange energy equation for stabilization of the pendulum . The controller gain is evaluated through state feedback and Linear Quadratic optimal regulator controller techniques and also the results for both the controller are compared. The SFB controller is designed by Pole- Placement technique. An advantage of Quadratic Control method over the pole- placement techniques is that the former provides a systematic way of computing the state feedback control gain matrix. LQR controller is designed by the selection on choosing. The proposed system extends classical invertedpendulum by incorporating two moving masses. The motion of two masses that slide along the horizontal plane is controllable.
and thus designing a controller on the linearized dynamical model itself would likely result in a sub-optimaldesign. To achieve optimaldesign the entire system is thus divided into two regions in which the reference angle θ=0 is taken from the upright position: the normal pendulum region and the invertedpendulum region, where the normal pendulum region is between -45 ◦ ≤ θ ≤ 45 ◦ and the invertedpendulum region is between 45 ◦ < θ < -45 ◦ as shown in Figure 4. This division is similar to the control zoning approaches used in many industrial PLCs in level and temperature controls. For future references the invertedpendulum region will also be termed the linear region. Figure 5 shows the flowchart describing the implementation of the invertedpendulum control through a PLC. It is seen that the pendulum regions defined in Figure 4 is specified into the PLC system.
The PID controller is the most common form of feedback. It was an essential element of early governors and it became the standard tool when process control emerged in the 1940s. The PID controller calculation involves three separate parameters; the proportional, the integral and derivative values. The proportional value determines the reaction to the current error, the integral value determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve or the power supply of a heating element. . Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability  .
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.
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 invertedpendulumsystem 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.
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.
In this paper, modelling design and analysis of a triple invertedpendulum have been done using Matlab/Script toolbox. Since a triple invertedpendulum is highly nonlinear, strongly unstable without using feedback control system. In this paper an optimal control method means a linear quadratic regulator and pole placement controllers are used to stabilize the triple invertedpendulum upside. The impulse response simulation of the open loop system shows us that the pendulum is unstable. The comparison of the closed loop impulse response simulation of the pendulum with LQR and pole placement controllers results that both controllers have stabilized the system but the pendulum with LQR controllers have a high overshoot with long settling time than the pendulum with pole placement controller. Finally the comparison results prove that the pendulum with pole placement controller improve the stability of the system.
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
Te invertedpendulum offers a very good example for control engineers to verify a modern control theory. This can be explained by the facts that invertedpendulum is marginally stable, in control sense, has distinctive time variant mathematical model. The double invertedpendulum is a highly nonlinear and open-loop unstable system. The invertedpendulumsystem usually used to test the effect of the control policy, and it is also an ideal experimental instrument in the study of control theory [1, 2]. To stabilize a double invertedpendulum is not only a challenging problem but also a useful way to show the power of the control method (PID controller, neural network, FLC, genetics algorithm, etc.).
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 invertedpendulumsystem 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 invertedpendulumsystem with optimal performance.
Invertedpendulumsystem has a great role in real application of engineering fields. Thus, companies in industry and researches have studied the invertedpendulum. In this study, the SRV02 rotary invertedpendulum is studied with controllers which have different damping ratios. Experiments have been performed in Mechatronics Laboratory, Turkish Aeronautical Association University Mechatronics Engineering Department. Having dynamic models of the rotary invertedpendulum, four controllers are designed. δ = 0.8 and δ = 0.85 of damping ratios show better responses than the other controllers. The experimental results for 5 seconds with a 10-degree step input are obtained for all designed controllers. The 4 th controller which has δ = 0.85 of damping ratio with w n = 4 rad/s is the best controller. The pendulum is held at upright position.
The invertedpendulumsystem 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 invertedpendulumsystem and compares their stability.
This paper present real time control of an invertedpendulum. 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.
A system cannot be separated from their errors caused by interference. This error can cause changed behavior of the system. This changed can lead to unstable condition of the system. To maintain the stability of the system, utilizing controller devices in inevitable. The function of the controller is to reduce error signal. The smaller the error, the better the performance of the control system is applied. Proportional, integral and derivative is popular controller among industry practitioners due to fast response of the controller [5, 6].