Output Feedback Control of a Ball and Beam System Based on Jacobian Linearization under Sensor Noise

In this paper, we consider a control problem of a ball and beam system with sensor noise on feedback sensor. If sensor noise exists in sensor's signal, it can make feedback signals deformed and then it can lead to control performance degradation or even system failure. Therefore, we need to design a robust controller to deal with the possible sensor noise in the feedback information. We develop an output feedback controller with a gain-scaling factor in order to minimize the effect of AC sensor noise in output feedback information. Our proposed controller is applied to a ball and beam system and verified by analysis and simulation. As a result, our controller reduces the effect of sensor noise to arbitrarily small by increasing a gain-scaling factor.


Introduction
We consider a control problem of a ball and beam system with sensor noise on feedback channel.Control systems operate via measurement data and it is usually assumed that the measured signals are clean.If sensor noise exists in sensor's signal, it can make feedback signals deformed and then it can result in performance degradation or even system failure.Therefore, we need to design a robust controller to accommodate the possible sensor noise in the feedback information [2], [5], [6].In this paper, we assume that an output sensor of a ball and beam system is coupled with AC noise signal.We apply Jacobian linearization to a ball and beam system and propose an output feedback controller compensating sensor noise of feedback sensor for a ball and beam system.Our controller is equipped with a gain-scaling factor to minimize the effect of sensor noise in output feedback information.We give a theoretical analysis of the controlled system using Lyapunov stability theorems and Laplace transform [2], [4].We illustrate the improved control performance via simulation for a ball and beam system.

A ball and beam system model based on Jacobian linearization and sensor noise
The ball and beam system can be modeled as [7] sin where r is the position of ball, θ is the angle of motor, bb K is the model gain of the ball and beam system, srv K is a steady-state gain, µ is a time constant and u is an input motor voltage.
We define the states of the system as follows , , , , Then, we obtain the following state equation The Jacobian linearlized system is as follows where We suppose that sensor noise enters at the output sensor of a ball and beam system and the measured output becomes as follows ( ) where ( ) s t is sensor noise.Sensor noise is often modeled as a sinusoidal function [1].Therefore, we introduce the following condition on ( ) s t .Assumption 1.There exist constants 0

Design and analysis of proposed controller
In the presence of sensor noise, the proposed output feedback controller equipped with a gain-scaling factor is given by where [ , , , ] T e e e e e = where ˆ, 1, , 4 By subtracting (8) from (4) and with the controller (9), the augmented closed-loop dynamics is where ( ) ( ) are Hurwitz.The Lyapunov equation for We set ( ) ( ) V e e P e ε = for (10).Then, along the trajectory of (10), we have

V e e E e e E P E L s t e E P E x x
E e E e P Ls t P E e E x x

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We have 13), and then we can obtain If ε is increased, the ultimate bounds of 2 , 3 e and 4 e except for 1 e can be made arbitrarily small.
Because the Lyapunov stability theorem is a sufficient condition, we analyze the ultimate bound of 1 e via Laplace transform.Applying Laplace transform for the system (10), 1 ( ) E s is as follows where ( ) ( ) If ε is increased, the size of We set ( ) ( ) for (11).Then, along the trajectory of (11), we have where We set a Lyapunov function for entire system (10)-( 11) to ( , ) ( ) ( ), 0 along the trajectories of ( 10) and (11), we obtain the following from ( 14) and (21).
{ } { } ( ) where is satisfied, M becomes positive definite matrix.Thus, the ultimate bound of ( , , , ) x x x x is as follows ( )

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Substituting the ultimate bound of 1 2 3 4 ( , , , ) e e e e into (23), we can obtain If ε is increased in (24), the ultimate bounds of 2 x , 3 x and 4 x is decreased except for 1 x .Thus, we analyze the ultimate bound of 1 x via Laplace transform.Applying Laplace transform for the system (11), 1 ( ) X s is as follows where η is a Hurwitz polynomial, the terms with initial value for (25) become 0 by the final-value theorem and the remaining terms( 1 ( ) r X s ) is as follows We assume that ( ) s η has separate roots for analysis and substitute ( ), Because the ultimate bounds of 1 2 3 4 ( , , , ) x x x x are reduced by increasing ε as shown in ( 15) and (19), we set maximum value of ( ) bi e t as , 1, , 4 The exponential terms ( ) x in steady state is as follows If ε is increased in (30), the ultimate bounds of 1 x , 2 x , 3 x and 4 x can be reduced.

Simulation
In simulation for a ball and beam system, we set the initial conditions as 1 (0) 10 x cm = , 2 (0) 0 x = , x is reduced when ε is increased.Thus, our proposed controller can reduce the effect of sensor noise by increasing ε .

Conclusions
In this paper, we assume that AC sensor noise is included in the output sensor of a ball and beam system.We design an output feedback controller with a gain-scaling factor to minimize the effect of sensor noise.We analyze the ultimate bounds of e and x by the Lyapunov stability theorem and Laplace transform for controlled system.As a result, we show that the effect of sensor noise is arbitrarily reduced by increasing ε via analysis and simulation.