Modeling and Simulation of a Horizontally Moving Suspended Mass Pendulum Base using H  Optimal Loop Shaping Controller with First and Second Order Desired Loop Shaping Functions

: In this paper, a horizontally moving suspended mass pendulum base is designed and controlled using robust control theory. H  optimal loop shaping with first and second order desired loop shaping function controllers are used to improve the performance of the system using Matlab/Simulink Toolbox. Comparison of the H  optimal loop shaping with first and second order desired loop shaping function controllers for the proposed system have been done to track the desired angular position of the pendulum using step and sine wave input signals and a promising result has been obtained succesfully.


1.
Introduction A pendulum with suspended mass is a system that has a mass suspended in its base and a mass suspended from a pivot so that it can swing freely. When the suspended mass in the base of the pendulum forced to move horizontally by applying a force, the pendulum become displaced sideways from its resting, equilibrium position, then the pendulum will be subjected to a restoring force due to gravity that will accelerate it back toward the equilibrium position. The restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.

2.
Mathematical Modeling of the System A system consists of two point masses, m 1 and m2, connected with a weightless rigid rod of length l ( Figure 1). The motion occurs in a gravity field and is considered to be in a plane, i.e. is considered in the coordinates x, y, t. The location of point of a mass m1 (suspension) is not fixed, and can move along the axis x. The mathematical model of the system will be as follow.

Figure 1 Pendulum with suspended mass
The four functions of time of the system are, x1(t), y1(t), x2(t), y2(t), i.e. the Cartesian coordinates of the first and second points.
The suspension cannot move vertically Y1=0 While the second is described by equation We choose the generalized coordinates as where  is the angle between the vertical is and the axis of rod.
In so far as the considered motion is potential, it is necessary to use the Lagrangian equations 1 2 The state space representation of the system becomes The parameters of the system are shown in Table  1 below.
A MIMO stable min-phase shaping pre-filter W, the shaped plant G s = GW, the controller for the shaped plant K s = WK, as well as the frequency range {ω min , ω max } over which the loop shaping is achieved. The block diagram of the pendulum on the free suspension with H  Optimal Loop Shaping Controller is shown in Figure 2 below. In this paper, the plant has been desired loop shaped with a first order and second order system.
For the first order, the desired loop shaping function is 1

Result and Discussion
Here in this section, the investigation of the open loop response and the closed loop response with the proposed controller have been done. Finally the comparison of the system with the proposed controllers for a first order and a second order desired loop shaping design have been done.

Open Loop Response of the Pendulum
The open loop response of the system for a 0.1 Newton suspended mass force simulation is shown in Figure 3 below.

Comparison of the Step Response of Pendulum with Suspended Mass using H  Optimal Loop Shaping Controller with First and Second Order Desired Loop Shaping Function Controllers
The simulation result of the step response of pendulum with suspended mass using H  optimal loop shaping controller with first and second order desired loop shaping function is shown in Figure 4 below.

Figure 4.
Step response The data of the rise time, percentage overshoot, settling time and peak value is shown in Table 2. As Table 2 shows that the pendulum with suspended mass using H  optimal loop shaping controller with first order desired loop shaping function controller improves the performance of the system by minimizing the rise time, percentage overshoot and settling time.

Comparison of the Sine Wave Response of Pendulum with Suspended Mass using H  Optimal Loop Shaping Controller with First and Second Order Desired Loop Shaping Function Controllers
The simulation result of the sine wave response of pendulum with suspended mass using H  optimal loop shaping controller with first and second order desired loop shaping function is shown in Figure 5 below. As Figure 5 shows that the pendulum with suspended mass using H  optimal loop shaping controller with first order desired loop shaping function controller improves the performance of tracking the set point input to the system.

Conclusion
In this paper, the design and simulation of a horizontally moving suspended mass pendulum base is done using H  optimal loop shaping with first and second order desired loop shaping function controllers. Comparison of the proposed system with H  optimal loop shaping with first and second order desired loop shaping function controllers have been done to track the desired angular position of the pendulum using step and sine wave input signals. The step input signal response shows that the pendulum with suspended mass using H  optimal loop shaping controller with first order desired loop shaping function controller improves the performance of the system by minimizing the rise time, percentage overshoot and settling time while the sine wave input signal response shows that the pendulum with suspended mass using H  optimal loop shaping controller with first order desired loop shaping function controller improves the performance of tracking the set point input to the system. Finally the simulation comparison results prove that the system with H  optimal loop shaping controller with first order desired loop shaping function controller improved the system performance better.