36-150-1-optimal robust controller design for the ball and plate system

7/23/2019 36-150-1-Optimal Robust Controller Design for the Ball and Plate System

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Optimal Robust Controller Design for the Ball andPlate System

Amir Rikhtehgar Ghiasi#1, Hooshang Jafari#2 # Department of Electrical and Computer Engineering, University of Tabriz

University of Tabriz, Tabriz, Iran1agi asi @t abri zu. ac. i r

2h. j af ar i 90@ms. t abr i zu. ac. i r

Abstract— The problem of controlling an open-loop unstable

system presents many unique and interesting challenges and ball

and plate system is specific example for these kinds of systems. Ball

and plate system is inherently nonlinear and under-actuated

system. This paper proposes an optimal robust controller design via

H-infinity approach. Simulation results show that the proposed

controller has the strong robustness and satisfactory and eliminate

the effect of linearization problems, unknown external disturbances

and time-varying uncertain friction while the ball rolling on the

plate surface, thus the trajectory tracking precision is improved.

Keywords— Ball and plate system; nonlinear system; robust

control; optimal control; trajectory tracking; H-infinity;

uncertainty; under-actuate;

I. I NTRODUCTION

The ball and plate system is extension of the well-known ball and beam system. The latter has two degree of freedomwhere a ball can roll on a rigid beam while the former is thefour degree of freedom system consisting of a ball that canroll freely on a plate. However, it is more complicated than ball and beam due coupling of multi-variables. This under-

actuated system has only two actuators and should bestabilized by just two control inputs. The system is a benchmark to test various nonlinear control schemes and it isinfluenced by nonlinearities such as motion resistance between the ball and plate, backlash in transmission set and soon. The experimental platform includes a flat plate, a ball,motors and their driving system, a CCD camera and so on.The plate rotates around its x and y axis in two perpendiculardirections. Inclinations of the plate are changed by a servocontrol system mainly including photo encoders, two stepmotors and two step motor drivers. Position of the ball ismeasured by a machine vision system.

Difficulty of creating a ball and plate mechanism with

respect to a ball and beam mechanism is the main drawback ofsuch systems but the enormous potential of performingmanifold control strategies on ball and plate systems makethem desirable.

Various control methods have been introduced for ball and plate system [1]-[9]. For instance, a controller design for twodimensional Electro-mechanical ball and plate system basedon the classical and modern control theory [1]. A supervisoryfuzzy controller was proposed for studying motion control ofthe system included the set-point problem and the tracking problem along desired trajectory, which was composed of two

layers [2]. A nonlinear velocity observer for output regulationof ball and plate system proposed in [3] where ball velocitiesare estimated by state observer. In [4] position of the ballregulated with double feedback loops. The sufficientconditions for the controllability of affine nonlinear controlsystems on Poisson manifolds are discussed in [5]. The activedisturbance rejection control is applied to the trajectorytracking design of the ball and plate system in [6] and the PIDneural network controller based on genetic algorithm [7] isanother work in this topic. Also, recently some works

proposed based on sliding mode control [8]-[9].This paper proposes an optimal robust controller design for

linear and nonlinear model of ball and plate system withuncertainties. H-infinity approach is considered. The problemsof linearization, unknown external disturbances and time-varying uncertain friction make large estimation and outputerror in system response, so it is necessary to design acontroller witch eliminate all external and internal

perturbations.The rest of this paper is organized as follows. Section 2

present ball and plate system model. Section 3 introduces the

H-infinity optimal controller design and finally theeffectiveness of the proposed method is demonstrated throughsimulation results in section 4.

II. SYSTEM DESCRIPTION

The mathematical model for ball and plate system is shownin Fig. 1. The plate rotates around its x and y axis in two

perpendicular directions. Kinematics differential equations ofthe ball and plate system are obtained using Lagrange method.

Fig. 1 The scheme of the model simplification for ball and plate system

System variables are selected as following: x(m) is thedisplacement of the ball along the x-axis, y(m) is thedisplacement of the ball along the y-axis, α (rad) is the angle

between x-axis of the plate and horizontal plane, β (rad) is the

angle between y-axis of the plate and horizontal plane, τx (N·m) is the torque exerted on the plate in x-axis, τy (N·m) isthe torque exerted on the plate in the y-axis. Ball and plate



system can be simplified into a particle system made by tworigid bodies. The plate has three geometry limits in thetranslation along the x-axis, y-axis and z-axis. It also has ageometry limit in rotation about z-axis. The plate has twodegree of freedom (DOF) in the rotation about x-axis and y-axis. The two DOF are depicted in α and β. They are limited

in a certain range. The ball has a geometry limit in thetranslation along the z-axis, and it has two DOF in translations

along x-axis and y-axis. Generalized coordinates are chosenas, q 1 = x, q 2 = y, q 3 = α, q4 = β. Generalized forces or torque

Q of the generalized coordinates are obtained with virtualwork principle. Generalized forces Q Corresponding togeneralized coordinates are:

Qx = – mg sin α + f x, Qy = – mg sin β + f y

Qα = τx – mgx cos α, Qβ = τy – mgy cos β

Where f x and f y are motion resistance acted on the ballalong the x-axis and y-axis respectively. Motion resistanceincludes friction between the ball and plate [10], collision power between the ball and plate and etc.

Suppose the ball remains in contact with the plate and therolling occurs without slipping at any time. Dynamicalequations of the n-DOF system are obtained using Euler-Lagrange’s equation [11].ddt ∂ℒ∂q

∂ℒ∂q = Qk (1)

Where ℒ is the deference of the kinetic and potential energy,Qk is generalized force associated with q k and k = 1,..., n.Kinetic energy of the ball T ball is

Tball = 12 [m +

Ir (x 2 + y 2) + Ib α 2+ β 2 + mxα + yβ 2

] (2) (2)

Kinetic energy of the ball and plate system is

Tplate = 12 [Ipxα 2 + Ipy β 2 + m + Ir (x 2 + y 2) + Ib α 2 +

β 2 + mxα + yβ 2] (3)

Potential energies of system along the x-axis and y-axis are

Vx = mgx sin α, Vy = mgy sin β

Applying the Euler-Lagrange’s equation the mathematicalmodel for ball and plate system is shown as:m +

Ir x mxα 2 myα β + mgsin α = 0 (4)

m + Ir y mxβ 2 myα β + mg sinβ = 0 (5)

Ipx + Ib + mx2 α + 2mxx α + mx yβ + mxy β + mxyβ +mgxcos α = τx (6)Ipy + Ib + my2 β + 2myy β + my xα + myx α + mxyα +mgycos β = τy (7)

Where m is the mass of the ball, g is gravity acceleration, I b is ball inertia, I px is plate inertia to x-axis and I py is plateinertia to its y-axis state variables are selected as

x1 = x, x2 = x , x3 = α, x4 = α x5 = y, x6 = y , x7 = β, x8 = β

State equations of the ball and plate system are:

x 1 = x2

x 2 = C(x1x42 + x1x4x8 g sin x3)

x 3 = x4

x 4 = ux

x

5= x

6 x 6 = C(x5 x82 + x4x5x8 g sin x7)

x 7 = x8

x 8 = uy

Where C = mr mr +I and Ib =

25 mr2 for a spherical ball so it

easily can be shown C = 57. The value of ux and uy considered

to be zero. In the ball and plate system, it is supposed that the ball remains in contact with the plate and the rolling occurswithout slipping, which imposes a constraint on the rotationacceleration of the plate. Because of the low velocity and

acceleration of the plate rotation, the mutual interactions of both coordinates can be negligible. So the model of the balland plate system can be approximately decomposed asfollows:

x 1x 2x 3x 4

= x2C(x1x42 g sin x3)

x40

+ 0

00

1

τx (8-a)

x 5x 6x 7x

8

= x6C(x5 x82 g sin x7)

x80

+ 0

00

1

τy (8-b)

Namely, the ball–plate system can be regarded as twoindividual sub-systems and both coordinates can be controlledindependently. Linear state space model along x-axis is:

x 1x 2x 3x 4

= 0 1 0 00 0 7 000

00

00

10

x1x2x3x4

+ 0

00

1

τx (9-a)

y=[1 0 0 0]x (9-b)

x = Ax + Bu, y = Cx + Du (10)

The state space model is similar along y-axis. According tocontinue system control theory, system is state completelycontrollable if and only if vector B, AB,..., An-1B are linearindependent, that is the rank of matrix [B AB ... An-1B] equalto n, the dimension of state system. Moreover, system (10) iscompletely observable if and only if the rank of matrix [C CA... CAn-1] equal to n, too. Therefore the Rank of controllabilitymatrix and observability matrix of system (9) are both 4 whichare computed in MATLAB, which equal to the dimension ofsystem state. Thus some kind of controller can be designed tomake the system (9) stable.

III. OPTIMAL ROBUST CONTROLLER DESIGN



A. Robust Control and Uncertainty

Robust control is a branch of control theory that explicitlydeals with uncertainty in its approach to controller design. Thekey idea of robust control is to separate the known part andthe uncertain part from the knowledge about the uncertainsystem under investigation. This is illustrated in Fig. 2, whereM(s) denotes the known part of the uncertain system and Δ(s)

denotes the uncertain part [12]. Usually, we have some limitedknowledge about Δ(s) such as the upper bound information.

Fig. 2 general structure of uncertain system

A general description of robust control system structure isshown in Fig. 3, where P(s) is the augmented plant model andF(s) is the controller model. The transfer function from theinput u1(t) to the output y1(t) is denoted by Ty1u1(s). It should

be emphasized at this point that the block diagram shown inFig. 3, is fairly general. The signal vector u1(t) can include both reference and disturbance signals. P(s) can include boththe plant model and the disturbance generation model.Moreover, uncertainties can also be included in P(s).

Fig. 3 standard feedback control

The unstructured uncertainties can be classified into theadditive and multiplicative uncertainties. The feedback systemstructure with uncertainties is shown in Fig. 4. In general, theuncertain model can be represented by

Gp(s) = ΔA(s) + G(s) [I + ΔM(s)] (11)

If ΔA(s) ≡ 0, one has G p(s) = G(s) [I + ΔM(s)], and theuncertainty is referred to as the multiplicative uncertainty.

When ΔM(s)≡ 0, the uncertainty is referred to as the additiveuncertainty with the model G p(s) = G(s) + ΔA(s). Based onsmall gain theorem [12] with no loss of generality, if weassume that the uncertainty norm bound shown in Fig. 2, isunity, then we can concentrate on Fig. 3, as if there were no

uncertainty in P(s). But now our control design task amountsto designing F(s) such that ||Ty1u1(s)||∞ < 1. We shouldunderstand that it is always possible to scale Δ(s) such that the

scaled uncertainty bound is less than 1.

Fig. 4 Feedback control with uncertainties

B. H ∞ Controller Design

The system structure for H∞ control described in Fig. 3considered. Based on the above arguments, considering on theconfiguration shown in Fig. 3, where an augmented plantmodel can be constructed as the controller can be represented

by

p(s) = A B

1B

2C1 D11 D12C2 D21 D22 (12)

With the augmented state space description as follows:

x = A x + [B1 B2] u1u2 (13-a)

y1y2 = �C1

C2 x + �D11 D12D21 D22 u1

u2 (13-b)

Straightforward manipulations give the following closed-loop transfer function:

Ty1u1(s) =P11(s) +P12(s) [I−F(s) P22(s)]-1F(s) P21(s) (14)

The above expression is also known as the linear fractionaltransformation (LFT) of the interconnected system. Theobjective of robust control is to find a stabilizing controlleru2(s) =F(s) y2(s) such that ||Ty1u1(s)|| < 1.

Based on (14), the standard H∞ robust control obtained by

||Ty1u1(s)||∞<1.The design objective is to find a robust controller Fc(s)

guaranteeing the closed-loop system with an H∞-norm bounded by a given positive number γ [13].

Fc(s) = Af ZL

F 0 (15)

Where

Af = A + γ−2B1B1 TX + B2 F+ZLC2

F = B2 TX, L = YC2 T, Z = (I γ−2YX)−1

And X and Y are, respectively, the solutions of thefollowing two AREs:

A X + XT AT + X(γ−2 B1B1T B2B2T)XT + C1C1T = 0

AY + YT AT + YT(γ−2C1TC1 C2TC2)Y + B1TB1 = 0

The conditions for the existence of an H∞ controller are asfollows:

• D11 is small enough such that D11 < ;• The solution X of the controller ARE is positive-definite;• The solution Y of the observer ARE is positive-definite;• λ max (XY) < 2, which indicates that the eigenvalues of

the product of the two Riccati equation solution matrices are

all less than 2.

The returned tree variable FC(s) is the designed H∞ controller in state space form.

In optimal H∞ controller design, the optimal criterion isdefined as:

maxγ ‖Ty1 u1‖ < 1 (16)

http://en.wikipedia.org/wiki/Control_theory

http://en.wikipedia.org/wiki/Control_theory



IV. SIMULATION RESULTS

With the discussion above, the proposed optimal H∞ controller can be realized. The simulation object for tracking problem is to make ball follow a specific path. The desiredtrajectory for tracking is a circle. Simulation results showreasonable output for both linear and nonlinear models. Theradius of desired circle is 400 mm. Simulation results for

linear model by applying robust controller are shown in Fig. 5and Fig. 6. For linear model system error shown in Fig. 6 isless than 0.5 mm and ball settle down in circle path after 4second.

Fig. 5 Circular trajectory tracking for linear system

Fig. 6 Circular trajectory tracking error for linear system

Simulation results for nonlinear system are shown in Fig. 7and Fig. 8. Results demonstrate the effectiveness of controlleron nonlinear system.

Fig. 7 Circular trajectory tracking for nonlinear system

Fig. 8 Circular trajectory tracking error for nonlinear system

For nonlinear model tracking error shown in Fig. 8 is lessthan 1mm at the steady state.

V. CONCLUSTION

This paper proposes a trajectory tracking optimal robustcontroller for the ball and plate system with uncertainty.Simulation is conducted in MATLAB, and the resultsdemonstrate that the proposed controller can make the ball

precisely track the given trajectory despite of the existence ofuncertainty. The controller can be implemented on the balland plate system. Despite being based on a linear model, thecontrollers performed extremely well with the nonlinearsystem. Also, the problems of linearization, unknown externaldisturbances and time-varying uncertain friction that makelarge estimation and output error in system responseeliminated with designed controller.

R EFERENCES

[1] Andrej Knuplez, Amor Chowdhury and Rajko Svecko “Modelingand Control Design for Ball and Plate System” ICIT, 2003,

Maribor, Slovenia[2] Ming Bai, Huiqiu Lu, Jintao Su and Yantao Tian “Motion Control

of Ball and Plate System Using Supervisory Fuzzy Controller” 6thWorld Congress on Intelegent Control and Automation, June, 2006

[3] Hongrui Wang, Yantao Tian, Le Ding, Qing Gu and Feng Guo“Output Regulation of the Ball and Plate System With a NonlinearVelocity Observer” 7th World Congress on Intelegent Control andAutomation, june, 2008

[4] Wang Hongrui, Tian Yantao, Fu Siyan, Sui Zhen” NonlinearControl for Output Regulation of Ball and Plate System”Proceedings of the 27th Chinese Control Conference, July 16-18,2008, Kunming, Yunnan, China

[5] Siyan Fu, Zhen Sui, Hongrui Wang and Yantao Tian“Controllability of the Nonlinear Under actuated Ball and PlateSystem” Proceedings of the 2009 IEEE International Conference onMechatronics and Automation, August 9 - 12, Changchun, China

[6] Huida Duan, Yantao Tian and Guangbin Wang” TrajectoryTracking Control of Ball and Plate System Based on Auto-Disturbance Rejection Controller” Proceedings of the 7th AsianControl Conference, Hong Kong, China, August 27-29, 2009

[7] Xiucheng Dong, Zhang Zhang, Chao Chen “Applying GeneticAlgorithm to On-Line Updated PID Neural Network Controllers forBall and Plate System” 4th International Conference on InnovativeComputing Information and Control, 2009

[8] Dejun Liu, Yantao Tian and Huida Duan “Ball and Plate ControlSystem Based on Sliding Mode Control with Uncertain ItemsObserve Compensation”

[9] Hong Wei Liu, Yanyang Liang “Trajectory Tracking Sliding ModeControl of Ball and Plate system” 2012 2nd International Asiaconference on Informatics in Control, Automation and Robotics.

[10] Zhang X F, Tian Y T, Wang H R. “Friction estimation of ball and plate system based on extended Kalman filter” 26th ChineseControl Conference. Beijing: Beihang University Press, 2007

[11] Mark W. spong, M. Vidyasagar “Robot Dynamics and Control” byJohn Wiley & Sons, Inc. 1989, pp: 239-243

[12] Dingyu Xue, Yang Quan Chen and Derek P.Atherton “LinearFeedback Control: Analysis ND Design with MATLAB” 2007, pp:235-282

[13] Kemin Zhou, John C. Doyle Keith Glover “Robust and OptimalControl” pp: 405-440

36-150-1-optimal robust controller design for the ball and plate system

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