Stability Control of a Rotational Inverted Pendulum using Augmentations with Weighting Functions Based Robust Control System Stability Control of a Rotational Inverted Pendulum using Augmentations with Weighting Functions Based Robust Control System.

: This paper mainly analyzes the design and control of the rotational inverted pendulum, and presents a state space expression. Since the system is highly unstable a feedback control system is used. Augmentations with weighting functions based mixed sensetivity and H2 optimal control methods are used to make the system stable for uprise position. The rotational inverted pendulum have been simulated and compared with the proposed controllers and a promising results have been analyzed sussesfuly.


1.
Introduction Inverted pendulum system is a typical multivariable nonlinear strong coupling unstable system. Inorder to control this system, the theory of controlling the system stability, controllability, robustness and tracking the system. Its control method are widly used in military, industry, robot and in the field of general industrial process control, such as the balance in the process of the robot control and satellite attitude control in flight, Inverted pendulum system which is more ideal experimental apparatus control theory is often used to test the effect of the control strategy. This article aims at single stage of nonlinear rotational inverted pendulum control problem, the design has realized the single inverted pendulum with robust control based theory.

2.
Mathematical Modelling of the Rotational Pendulum Figure 1 shows the structural design of the rotational pendulum.

For the DC Motor
Assume The stator current is constant therefore the magnetic flux is constant The motor torque is proportional to the armature current and the flux The voltage vb is proportional to the angular speed of the motor Applying KVL to the motor circuit neglecting the coil inductance For the pendulum The rotational equation of the pendulum is Torque of the motor based on the pendulum   P T t Torque of the pendulum Torque of the motor based on the pendulum is Note that the sign of K3 depends on whether the pendulum is in the inverted or non-inverted position.
The torque of the pendulum is (11) and Equation (12) into Equation (10) and rearranging yields Linearization of these equations about the vertical position (i.e., P The Values of the parameters of the system is shown in Table 1 below

Proposed Controllers Design 3.1 Augmentations of the Model with Weighting Functions
In this section, we will focus on the weighted control structure shown in Figure 2, where W1(s), W2(s), and W3(s) are weighting functions or weighting filters. We assume that G (s), W1(s), and W3(s) G (s) are all proper; i.e., they are bounded when s →∞. It can be seen that the weighting function W3(s) is not required to be proper. One may wonder why we need to use three weighting functions in Figure 7.13. First, we note that the weighting functions are, respectively, for the three signals, namely, the error, the input, and the output. In the two-port state space structure, the output vector y1 = [y1a, y1b, y1c] T is not used directly to construct the control signal vector u2. We should understand that y1 is actually for the control system performance measurement. So, it is not strange to include the filtered "input signal" u (t) in the "output signal" y1 because one may need to measure the control energy to assess whether the designed controller is good or not. Clearly, Figure 2 represents a more general picture of optimal and robust control systems. We can design an H 2 optimal and mixed sensitivity controllers by using the idea of the augmented state space model.

Open Loop Response of the Rotational Pendulum
The open loop response for an impulse input of the rotational pendulum is shown in Figure 3 below. The open loop response for a step input of the rotational pendulum is shown in Figure 4 below.

Figure 4 Open loop step response of the rotational pendulum
The simulation results of the open loop system shows that the rotational pendulum is unstable so the need of feedback control system is essential.

Comparison of the Rotational Pendulum with Mixed Sensitivity and H2 Optimal Controllers for an Impulse Input Voltage
The simulation result of the rotational pendulum with mixed sensitivity and H2 optimal controllers for an impulse input voltage signal is shown in Figure 5 below.

Figure 5 Impulse response of the rotational pendulum
The simulation result shows that the rotational pendulum with H2 optimal controller improve the settling time and the overshoot and the angular position returns to zero means the rotational pendulum is in upward stable position.

Comparison of the Rotational Pendulum with Mixed Sensitivity and H2 Optimal Controllers for a Step Input Voltage
The simulation result of the rotational pendulum with mixed sensitivity and H2 optimal controllers for a step input voltage signal is shown in Figure 6 below. Figure 6 Step response of the rotational pendulum The simulation result shows that the rotational pendulum with H2 optimal controller improve the settling time and the overshoot and the angular position returns to zero means the rotational pendulum is in upward stable position.

5.
Conclusion In this paper, modeling, simulation and comparison of the rotational inverted pendulum have been done using Matlab/Script Toolbox. Augmentations with weighting functions based mixed sensitivity and H2 optimal controllers have been used to control the system instability. The open loop response of the system for a step and impulse voltage input shows that the system is unstable. Comparison of the rotational inverted pendulum with mixed sensitivity and H2 optimal controllers have been done for a step and impulse voltage input and the simulation results prove that the rotational inverted pendulum with H2 optimal controller improves the settling time and overshoot and the angular position returns to its position successfully.