A Better Stability Control of Inverted Pendulum System Using FMINCON Based FOPID Controller Over Fractional Order Based MRAC Controller

Introduction

A case study has been approached to implement model reference adaptive control on inverted pendulum system which is one of the most challenging task of control engineering, using normal MIT rule and fractional order MIT rule for analyzing performance evaluation on stability of angular movement of inverted pendulum. The MIT rule was designed in Massachusetts Institute of technology. Various schemes have been proposed by several experts to analyze the performance of inverted pendulum using different control techniques. Traditional PID controller on inverted pendulum cart system has been introduced by Mohammad Ali. GA based PID controller for inverted pendulum has been approached by Reza. MRAC using Lyapunov theory and fuzzy model reference control for inverted pendulum has been proposed by Adrian-Duka. Indirect fractional order pole assignment based adaptive control has been proposed by Ladaci. Fractional order adaptive controller for stabilized systems has been also proposed by Ladaci. Fractional order PID controller using GA has been approached by Pankaj Rai. Fractional order multivariable composite model reference adaptive control has been designed by Cheng. So, it has been studied that basically adaptive gain has been approached with integer order model as well as fractional order model under MRAC controller by researchers and only global optimization technique has been approached to tune fractional order PID controller parameters which has been used to make stable inverted pendulum. Adaptive controller has been chosen as it is more effective than fixed gain PID controller to handle difficult situations.

In this work, design of model reference adaptive control for inverted pendulum has been suggested by normal MIT rule based on adaptation gain but this rule does not provide stability and adaptive controller designed using MIT rule is very sensitive to the amplitudes of the signals. So, the adaptive gain is usually kept small to make stable the angular movement and MIT rule has been modified as fractional order MIT rule with extra degree of freedom varying with adaptation gain to test how the performance of inverted pendulum tracks better the reference model minimizing error between them and to analyze performance nature of pendulum using fractional order rule over integer order MIT rule. Next, optimal FOPID controller has been designed using FMINCON local optimization algorithm to achieve superior performance over fractional order MIT rule of MRAC controller minimizing objective function which has been defined as a performance measure that has to be minimized. Cost includes total power consumption, integrated error and derivation from a reference value of a signal. The objective or cost function as a functional equation, which has been figured a set of points in a time series to single scalar value. FOPID controller unlike integer order PID controller has two extra degrees of freedom which help to achieve desired nature of system with better adjustment of tuning.

Benchmark Process

The inverted pendulum is a pendulum that has its center of mass above its pivot point and has been taken as benchmark for studying control strategies. Inverted pendulum has been taken as nonlinear dynamic system including a stable resting point when pendulum is at pending position and upright position. In this work, inverted pendulum has been assumed which is shown in Figure 1.

Figure 1.

Rotational inverted pendulum system that shows a free body diagram

Inverted pendulum mainly carries a weight which is unstable in one direction and does never swing like classical pendulum system. In the above figure, considered = length of pendulum, =frictional constant, = mass of pendulum, = acceleration, = half length at centre and T = tension.

The equation of motion for pendulum is shown in Equation (1).

(1)

Taking the Laplace transform

(2)

The parameters are given in below:

• a.

  (Inertia) = .2453 N/S

• b.

  (Frictional Constant) = 0

• c.

  (Mass) = 900 g

• d.

  (Acceleration) = 9.81

• e.

  (Length of Pendulum) = .102 m

• f.

  (Half length at center) = 0.945 cm

The transfer function of the actual plant is shown in Equation (3).

(3)