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Linear Quadratic Gaussian (LQG) Control Design for Position and Trajectory Tracking of the Ball and Plate System.

1 Introduction

Balancing systems is one of the very popular; and challenging test platforms in the field of control engineering. Example of these systems include double and multiple inverted pendulums, the ball and beam system and the traditional cart-pole system (Mohajerin et al., 2010). The ball and plate system (BPS), consists of a ball that can roll easily on a horizontal flat plate. The ball is required to track a desired path by adjusting the angle of tilt of the plate with respect to two mutually perpendicular directions (Roy et al., 2016b). The motion of the ball in the BPS has four degrees of freedom (DOF), but it is only controlled by just two actuating inputs. This implies under-actuation (Das & Roy, 2017), and it is an open loop unstable system, in which the ball's position becomes unbounded whenever the plate is tilted around either its x-axis or y-axis (Farooq et al., 2013), which poses a challenge for modelling and control (Roy et al., 2016b). The BPS is generalization of the ball and beam benchmark system (Moarref et al., 2008).

However, the BPS is a more complex system than the traditional ball and beam system due to its coupling of multivariable. This under-actuated system has two actuators, and it gets stabilized by the two control inputs (Ghiasi & Jafari, 2012). The BPS finds application in areas like humanoid robot, satellite control, unmanned aerial vehicle (UAV) and rocket system (Mukherjee et al., 2002) in the field of path planning, trajectory tracking and friction compensation (Oriolo & Vendittelli, 2005).

A servo system controller which consists of two servo motors, and a motor controller card is used for the tilt of the angle of the plate. However, an intelligent vision system from a CCD camera is used for measuring the ball on the plate (Dong et al., 2009a; Dong et al., 2011).

The control of motion of the BPS, is to control the position of the ball on the plate for both stabilization, and also path trajectory tracking. The plate's slope can be controlled in two perpendicular directions, so that the angle of tilt of the plate along the x-y axis will allow the movement of the ball on the plate (Dong et al., 2009b; Dong et al., 2011).

However, various control techniques have been used in recent years for the ball and plate system. A controller design for two electro-mechanical ball and plate systems based on classical and modern theories was presented by Knuplez et al., (2003). Ker et al., (2007) proposed back-stepping control design based on Lyapunov stability theory. The system was constructed using two magnetic suspension actuators for two DOF controls. Various conditions of dynamic operation which include oscillatory stabilization and circular trajectory tracking were tested to prove the system's performance and capability.

Lin et al., (2014) designed a controller, which also confirmed the BPS stability by using a loop shaping technique that is based on Normalized Right Coprime Factors (NRCF) perturbation technique, in which the familiar lead-lag series compensation technique were originally designed to find the suitable pre- and post- compensators as the weighting functions to guarantee the BPS time domain performance requirements. Debono & Bugeja, (2015) proposed and examined the application of sliding mode control (SMC) to the control problem of the ball and plate. The linear full-state feedback controller was compared with the SMC, and the performance between the SMC and linear full-state feedback controller was tested experimentally on a designed and constructed physical test bed that was meant for the purpose of the research.

However, a unique motion controller, based on the evolved lookup tables has been developed by Beckerleg & Hogg, (2016) to move a ball on a set-point on a typical BPS, in order to overcome the problem of under-actuation, instability and nonlinearity, which is attributed to the BPS. Roy et al., (2016a) proposed the design of a cascaded SMC for the ball's position control in a BPS. The efficiency of the proposed controller was also tested through simulation analysis, in which the ball was allowed to follow a circular and square path trajectories. However, the chattering effect was found to be within the acceptable limit.

One of the most effective control schemes for the BPS is the design of the double feedback loop structure, i.e. a loop within a loop. The inner loop works as dc motor servo position controller, while the outer loop control's the position of the ball (Liu & Liang, 2010).

In this paper, the ball and plate is considered as a double feedback loop structure for position and trajectory control. The inner feedback loop will be designed based on linear algebraic method, by solving a set of Diophantine equations, while the outer loop will be designed using linear quadratic Gaussian (LQG) controller, which is one of the robust controllers.

The rest of the paper is organized as follows: Section 2 introduces the mathematical modelling of the BPS, while section 3 discusses the design of the controllers. Also, section 4 displays the simulation results of the trajectory tracking and finally, the conclusion is presented in section 5.

2 Mathematical Model of the Ball and Plate System

Figure 1 shows a typical laboratory model of the BPS by Humusoft

Figure 2 Shows the schematic model diagram of the BPS.

The plate rotates about the x-axis and y-axis in two perpendicular directions. The kinematic differential equations of the BPS are derived using the Euler-Lagrange equation, that is given as follows (Kassem et al., 2015; Morales et al., 2017; Yildiz & Goren-Sumer, 2017):

[d/dt][[partial derivative]T/[partial derivative][[??].sub.i]]-[[partial derivative]T/[partial derivative][q.sub.i]]+[[partial derivative]V/[partial derivative][q.sub.i]] = [Q.sub.i] (1)

Whrere [q.sub.i] represents the [i.sup.th] -direction coordinate; T represents the kinetic energy of the system; V and Q represents the potential energy and the composite force respectively.

The BPS can be simplified into a system of particle that is made up of two rigid bodies; the geometry of the plate has a limits in translation along x, y and z-axis. It also has a geometry limit in rotation about its z-axis. The plate has two DOF in rotation about the x and y-axis. The ball's geometry has a limit in translation along the z-axis. It also has two DOF along the x and y-axis respectively. The BPS system model has four degrees of freedom (DOF), in which the generalized coordinates are (Hongrui et al., 2008; Hussein et al., 2017):

[q.sub.1] = x, [q.sub.2] = y, [q.sub.3] = [alpha], [q.sub.4] = [beta].

The total kinetic energy of the BPS is given as:

T = [T.sub.ball] + [T.sub.plate] (2)

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

The potential energies of the BPS along the x and y-axis is given as:

[V.sub.x] = [m.sub.b]gx sin [alpha] (5)

[V.sub.y] = [m.sub.b]gy sin [beta] (6)

And the mathematical equation of the BPS is given as:

([m.sub.b] + [[J.sub.b]/[R.sub.b.sup.2]])[??] - [m.sub.b]x[([??]).sup.2] - [m.sub.b]y[??][??] + [m.sub.b]g sin [alpha] = 0 (7)

([m.sub.b] + [[J.sub.b]/[R.sub.b.sup.2]])[??] - [m.sub.b]y[([??]).sup.2] - [m.sub.b]x[??][??] + [m.sub.b]g sin [alpha] = 0 (8)

([m.sub.b][x.sup.2] + [J.sub.b] + [J.sub.Px])[??] + 2[m.sub.b]x[??][??] + [m.sub.b]xy[??] + [m.sub.b] ([??]y + x[??])[??] + [m.sub.b]gxcos[alpha] = [[tau].sub.x] (9)

([m.sub.b][y.sup.2] + [J.sub.b] + [J.sub.Py])[??] + 2[m.sub.b]y[??][??] + [m.sub.b]xy[??] + [m.sub.b] ([??]y + x[??])[??] + [m.sub.b]gycos[beta] = [[tau].sub.y] (10)

[m.sub.b] (kg) represents the mass of the ball, [J.sub.b] (kg[m.sup.2]) is the rotational moment of inertia of the ball, [J.sub.[P.sub.x] (kg[m.sup.2]) and [R.sub.b] (m) are the rotational moment of inertia of the plate and the radius of the ball; x(m) and y(m) gives the position of the ball along the x and y-axis; [??](m/s) and [??](m/[s.sup.2]) are the velocity and acceleration along the x-axis; [??](m/s) and [??](m/[s.sup.2]) gives the velocity and acceleration along the y-axis; a(rad) and [??](rad/sec) gives the plate deflection angle, and angular velocity along x-axis; [beta](rad) and [??](rad/sec) gives the plate deflection angle and angular velocity along y-axis; and [[tau].sub.x](Nm) and [[tau].sub.y](Nm) gives the torques on the plate in the x and y-axis.

Equations (7) and (8) describes the ball's movement on the plate; they also shows how the effect of the ball's acceleration relies on the plate's angular deflection and its angular velocity. However, equations (9) and (10) describes how the dynamics of the plate's deflection rely on the external forces driving it, and the ball's position (Duan et al., 2009). Considering the state variable assignment of the BPS (Fan et al., 2004):

X = [([x.sub.1],[x.sub.2],[x.sub.3],[x.sub.4],[x.sub.5],[x.sub.6],[x.sub.7],[x.sub.8]).sup.T] = [(x,[??],a,[??],y,[??],[beta],[??]).sup.T] (11)

And the state space equation of the BPS is as follows (Fan et al., 2004):

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

B = [[m.sub.b]/([m.sub.b] + [J.sub.b]/[R.sub.b.sup.2])] (14)

In the steady state, the plate should be at a position that is horizontal, where both the inclination angles of the x and y-axis are equal to zero. If the inclination angle of the plate does not have much change, i.e. [+ or -][5.sup.0], then, the sine function can be substituted by its argument (Fabregas et al., 2017). The mathematical model of the BPS can be simplified and decomposed into x and y-axis as:

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

3 Controller Design

3.1 Determination of the Actuator Parameters

The actuator with a permanent DC motor is considered in the inner loop design. Also, the relationship between [[theta].sub.L] and [e.sub.a] is given as (Golnaraghi & Kuo, 2010):

[mathematical expression not reproducible] (17)

The parameters of the DC motors are derived based on the requirements of the load torque, the speed of the motor, and the moment of inertia. This is given in Table 1.

Equation (17) is given as (Umar, 2017):

[[[theta].sub.L]/[e.sub.a]] = [0.105/[0.47005[s.sup.2] + 421.113s]] (18)

= [0.2234/s(s + 895.89)] [approximately equal to] [2.49x[10.sup.-4]/s] (19)

3.2 Determination of the Actuator Parameters

The two-port parameter configuration was used in the design of the inner loop. To find the value of [[omega].sub.o], a step response is required, which could settle in 0.4 seconds. From the Humusoft ball and plate system manual, through simulation, [[omega].sub.o] = 20 rad/secs was found to give the steady state response. The ITAE optimal overall transfer function with zero position error of the system, which is [G.sub.0] (s) is (Chen, 1995):

[G.sub.0] (s) = [[[omega].sub.0.sup.2]/[s.sup.2] + 1.4[[omega].sub.0]s + [[omega].sub.0.sup.2]] (20)

An additional gain of 494 was provided to limit the step response using a preamplifier. [G.sub.0](s) is implemented as shown in Figure 3 using the two-port configuration.

Where L(s), M(s) and A(s) are the polynomials that defines the compensator, p which is the input disturbance. Solving the Diophantine equation, the compensator and the DC motor actuator has the following values (Umar, 2017):

A(s) = 28 + s (21)

M(s) = 3252 + s (22)

L(s) = 3252 (23)

3.3 Linear Quadratic Gaussian (LQG) Control

Linear quadratic Gaussian (LQG) control is also considered a robust control method because in the LGQ, the state output equations are considered explicitly. However, the quantitative information on the noise is taken into consideration in the controller design (Dingyu et al., 2007). Given the plant state space equation which is given as (Umar, 2017):

[??](t) = Ax(t)+Bu(t) + [GAMMA][xi](t) (24)

y(t) = Cx(t) + [theta](t) (25)

[xi](t) is the random noises in the state equation and [theta](t) is the output measurements. Assuming that [xi](t) and [theta](t) to be the zero mean Gaussian random process which has a covariance matrix that is given as (Dingyu et al., 2007):

E[[xi](t)[[xi].sup.T](t)] = [XI] [greater than or equal to] 0, e[[theta](t)[[theta].sup.T] (t)] = [THETA] > 0 (26)

E[x], which is the mean value of x, E[[xx.sup.T]], is the covariance matrix of the zero mean Gaussian signal x. However, [xi](t) and [theta](t) are the random signals which are also assumed to be mutually independent (Dingyu et al., 2007. This is written as:

E[[xi](t)[[xi].sup.T] (t)] = 0 (27)

Also, using a Kalman filter, the states can be optimally approximated, rather than using an observer. An optimal state estimation [??](t), in which the covariance e[[(x - [??])(x - [??]).sup.T]] is minimized, and also, the signal is estimated, [??](t) is also used as a replacement of the actual state variables in a way that the problem can be adjusted to suit the LQ optimal control problem. The Kalman filter gain matrix is written as (Dingyu et al., 2007):

[K.sub.f] = [P.sub.f][C.sup.T][[THETA].sup.-1] (28)

In which [P.sub.f] satisfies the algebraic Riccatti equation (ARE), which is also written as (Dingyu et al., 2007):

[P.sub.f][A.sup.T] + A[P.sub.f] - [P.sub.f][C.sup.T][[THETA].sup.-1]C[P.sub.f] + [GAMMA][XI][[GAMMA].sup.T] = 0 (29)

[P.sub.f] which is also a symmetrical semi-positive-definite matrix [P.sub.f] = [P.sub.f.sup.T] [greater than or equal to] 0. When the optimal filter signal [??](t), is obtained, the LQG compensator can be represented in a block diagram as given in Figure 4.

The optimal control [u.sup.*](t) is given as:

[u.sup.*](t) = -[K.sub.c][??](t) (30)

Also, the optimal state feedback matrix [K.sub.c] is written as:

[K.sub.c] = [R.sup.-1][B.sup.T][P.sub.c] (31)

The symmetrical semi-positive-definite matrix should satisfy the algebraic Riccatti equation (ARE), which is also given in equation (32)

[A.sup.T][P.sub.c] + [P.sub.c]A - [P.sub.c][BR.sup.-1][B.sup.T][P.sub.c] + [M.sup.T]QM = 0 (32)

In the LQG optimal control problem, it was observed that the optimal estimation and the optimal control problem could be solved separately; this is the well-known separation principle. In the design of the LQG controller, the state estimator is designed first, then the estimated state is used as if the states were accurately quantifiable for the LQR state feedback controller design.

The optimization criterion is given as (Dingyu et al., 2007):

[mathematical expression not reproducible] (33)

Normally, [N.sub.c] can be chosen as a zero matrix, Q and R are the weighted matrices; Q is semi-positive-definite, and R is positive-definite (Jafari et al., 2012). The state feedback matrix is:- [K.sub.c], and Kalman filter gain matrix is [K.sub.f] which were both obtained from the well-known separation principle. The dynamic equation of the Kalman filter is written as (Dingyu et al., 2007):

[??] = A[??] + Bu + [K.sub.f] (y - C[??] - Du) (34)

The observer based LQG controller can be written as (Dingyu et al., 2007):

[mathematical expression not reproducible] (35)

4 Results of the Simulation

The following simulation results were obtained from MATLAB 2017a software.

4.1 Inner Loop Design

The step response of the actuator is shown in Figure 5. Table 2 shows the properties of the actuator.

From Table 2, the settling time of the actuator is 0.2989 seconds, which shows that the plate will settle before 0.4 seconds that was set for it. However, the actuator:- U(s) due to a step input should not exceed the rated DC motor voltage of the actuator, which is 75V. Figure 6 shows the step response of the actuator with open parameters (rated power and voltage) of the DC motor.

From Figure 6, the plate stabilized at 0.3546 seconds. Also, the peak voltage is 74.56V, which is closer to the rated voltage of 75V. From this, it shows that a proper inner loop design of the DC motor actuator has the properties, that are given in Table 3.

4.2 Outer Loop Design

The parameters of the LQG controller are:

R = 0.1

[XI] = 7e - 03

[THETA] = 1e - 04

The step response of the LQG controller is given in Figure 7.

From Figure 7, the properties of the LQG controller is given in Table 4.

Table IV shows that the LQG controller stabilized the ball at 1.5075 seconds, with an overshoot of 33.4996%. This shows a good indication of tracking the ball on the desired path on the plate.

A circular trajectory of radius 0.4m, and a sinusoidal reference input signal of x = 0.4(1 - cos[omega]t) and y = 0.4(sin[omega]t) were taken into consideration, and were used to demonstrate the trajectory tracking performance of the ball. The angular frequency of the sinusoidal reference signal used was 1.33 rad/sec. This is shown in Figure 8.

The ball was allowed to track a circular trajectory of 0.46 rad/sec at a complete revolution of 13.6 seconds. When the speed was increased to 0.8 rad/sec, at a complete revolution of 8 seconds, the trajectory tracking error increased.

However, it was observed that using the LQG controller, the steady state tracking error of the ball is 0.0112m, which shows that the ball was able to track the circular reference signal with a trajectory tracking error of 0.0112m.

5 Conclusion

The position and trajectory tracking control of the BPS using a double feedback loop structure i.e. a loop within a loop has been proposed. The inner loop was designed using linear algebraic method via solving a set of Diophantine equation, while the outer loop was designed using LQG controller. The results of the simulation shows that the controllers has strong robustness, adaptability and high control performance for the BPS. However, future research work will consider incorporating artificial intelligent techniques with the controllers for optimal path trajectory tracking of the ball on the plate.

Acknowledgment

This research work is supported by the Control-and-Computer Research Group of the Department of Computer Engineering, Faculty of Engineering, Ahmadu Bello University, Zaria Kaduna State Nigeria. Also, the authors will like to thank the anonymous reviewer and the chief Editor, Dr. Abel Usoro, for his insightful comments and suggestions, which have been greatly beneficial for improving the quality of this work.

References

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Umar, A. (2017). Development of a Position and Trajectory Tracking Control of Ball and Plate System using a Double Feedback Loop Structure. (Masters of Science Thesis), Department of Electrical and Computer Engineering, Faculty of Engineering, Ahmadu Bello University, Zaria, Published.

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Abubakar Umar (1*), Muhammed B. Mu'azu (1), Aliyu D. Usman (2), Umar Musa (3), Ore-ofe Ajayi (1), and Abdulmumin M. Yusuf (1)

(1) Department of Computer Engineering, Ahmadu Bello University, Zaria, Kaduna-State Nigeria

(2) Department of Communication Engineering, Ahmadu Bello University, Zaria, Kaduna-State Nigeria

(3) Department of Electrical Engineering, Ahmadu Bello University, Zaria, Kaduna-State Nigeria

Email: abubakaru061010@gmail.com, muazumb1@yahoo.com, aliyuusman@gmail.com, umarnadada@yahoo.co.uk ajayi.oreofe17@gmail.com, abdulmumin.myusuf@gmail.com

(*) Corresponding Author
Table 1: BPS System Parameters (Humusoft, 2012b)

S/N   Description       Symbol        Unit       Value

 1    Mass of the         m            kg       0.11
          ball
 2   Radius of the        R             m       0.02
          ball
 3     Dimension         lxb        [m.sup.2]   0.16
      of the plate
        (square)
 4        Mass       [J.sub.Px,y]  kg[m.sup.2]  0.5
       moment of
     inertia of the
         plate
 5        Mass        [J.sub.b]    kg[m.sup.2]  1.76e-5
       moment of
     inertia of the
          ball
 6      Maximum          [nu]          m/s      0.04
      velocity of
        the ball

Table 2: Properties of the Actuator

  System Response    Value

Settling Time (sec)  0.2989
   Overshoot (%)     4.5989

Table 3: Properties of the Actuator with Open Parameters

Actuator System Response   Value

  Settling Time (sec)      0.3546
    Peak Voltage (V)      74.5631

Table 4: Properties of the LQG Controller

LQG Controller System Response   Value

     Settling Time (sec)         1.5075
        Overshoot (%)           33.4996
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Author:Umar, Abubakar; Mu'azu, Muhammed B.; Usman, Aliyu D.; Musa, Umar; Ajayi, Ore-ofe; Yusuf, Abdulmumin
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Date:Feb 1, 2019
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