Stability Analysis of an Inverted Pendulum on a Cart with the Presence of Restoring and Frictional Forces Disturbance using Observer Based and Full State Feedback H2 Controllers

In this paper, the stability control of the inverted pendulum on a cart with a disturbance forces has been done using observer based and full state feedback H2 controllers. The Lagrangian equation has been used to model the system equation of motions and linearized the system to the unstable upward position. Comparison of the system stability has been simulated by comparing the proposed controllers using Matlab/Scripts and a promising results has been analyzed successfully.


Introduction
An inverted pendulum on a cart is a pendulum system that has its center of mass above its pivot point. It is nonlinear and unstable without an additional feedback control system. The inverted pendulum stability is a major problem in dynamics and control strategy and is used as a controller testing mechanism. Most of the applications used in this system is a 1 degree of freedom by arranging the pole to a fixed axis of rotation. In order to improve the stability of an inverted pendulum in its inverted position, a feedback control theory based controller can be used, which regulates the angle of the pendulum and moving the pendulum mass when the system starts to fall over, to keep it in upraise zero-degree angle from the vertical position. One of the inverted pendulum application is rocket or missile guidance.

System Discription
The inverted pendulum on a cart system model is shown in Figure 1 below.
These equations are nonlinear, but since the goal of a control system would be to keep the pendulum upright the equations can be linearized around 0 Then after linearization, Equation (9) and Equation (10) becomes Rearranging Equation (11) and Equation (12) The parameters of the system are shown in Table 1 below The state space representation numerically becomes  

The Proposed Controllers Design 3.1 Observer-Based Controller Design
The deal with the general claim where only a subset of the states, or linear combinations of them, are obtained from measurements and are available to our controller. Such a guide lines is referred to as the output feedback problem.
The output of the system is  

y Cx Du 
The block diagram of the inverted pendulum on a cart with the observer-based controller is shown in Figure 2 bellow.

Full State Feedback
Assuming that d(t) is the white noise vector with bunch intensity The system block diagram with full state feedback H2 controller is shown in Figure 4 below.

.1 Comparison of the Inverted Pendulum on a Cart with Observer Based and Full State Feedback H 2 Controllers for an Impulse Input Force Signal
The comparison response simulation of the inverted pendulum on a cart with observer based and full state feedback H 2 controllers for an impulse input of 1 N force signal is shown in Figure 5 below. The simulation result shows that the inverted pendulum on a cart with observer based controller has a high oscillation and overshoot with a long settling time as compared to the inverted pendulum on a cart with full state feedback H 2 controller. The impulse response of the system with restoring and frictional forces does not affect the stability of the system but it makes the system to oscillate for a short time with 0.04 to -0.13 degree to a final zero-degree position of the pendulum.

Comparison of the Inverted Pendulum on a Cart with Observer Based and Full State Feedback H 2 Controllers for a Step Input Force Signal
The comparison response simulation of the inverted pendulum on a cart with observer based and full state feedback H 2 controllers for a step input of 1 N force signal is shown in Figure 6 below. Figure 6 Step response The simulation result shows that the inverted pendulum on a cart with full state feedback controller has a small oscillation and overshoot with a small settling time as compared to the inverted pendulum on a cart with observer based controller. The step response of the system with restoring and frictional forces disturbance does not affect the stability of the system but it makes the system to oscillate for a short time with 0.012 to -0.004 degree to a final zero-degree position of the pendulum.

Conclusion
In this paper, the comparison of the inverted pendulum on a cart stability has been done using observer based and full state feedback H2 controllers. Comparison of the system stability has been simulated by comparing the proposed controllers using Matlab/Scripts for an impulse and step input force signals. The comparison result of the impulse response shows that the inverted 9 pendulum on a cart with observer based controller has a high oscillation and overshoot with a long settling time as compared to the inverted pendulum on a cart with full state feedback H 2 controller while the comparison result of the step response shows that the inverted pendulum on a cart with full state feedback controller has a small oscillation and overshoot with a small settling time as compared to the inverted pendulum on a cart with observer based controller.