Ball and Beam Control SystemTran Thien DungFaculty of Electronics, Thai Nguyen University of Technology2015

OverviewIntroductionWhat is ball and beam systemMathematical ModelLinear ModelLQR ControllerConclusion

IntroductionThe ball and beam is one of the most popular andimportant bench systems for studying controlalgorithms.

Its a laboratory equipment with interested in its dynamics: high nonlinearity, open loop unstable.The system is simple to understand.Many classical and modern control methods have been used to stabilize the ball and beam system.

What is the Ball and BeamThe system is simple: a steel ball rolling on the top of the long beam. The beam is mounted on the shaft of an electric motor can be tilted by applying an voltage signal to motor.

What is the Ball and BeamThe other configuration: one end of the beam is coupled to the motor through a lever arm and gears,the other end is fixed.

What is the Ball and BeamPosition sensor: the beam consists of a steel rod in parallel with awire-wound resistor which is biased and the position along the track is obtained by measuring the voltage at the steel rod.

What is the Ball and BeamAngle sensor: the angle of beam is obtained by measuring the voltage at the output of a rotary potentiometer which is attached to output gear of sevo-motor

What is the Ball and BeamBall and beam system in Laboratory of Automatic Control Division

What is the Ball and BeamIt has a very important property: open loop unstable the ball does not stay in one place on the beam but move with an acceleration that is proportial to the tilt of the beam.

Controller has to automatically regulate the position of the ball on the beam by changing the angle of the beam. This is a difficult control task.

Mathematical ModelSchematic of Ball and Beam System

Mathematical Model

LLengh of the beam (m)rPosition of ball on the beam (m).Torque applied on the beam (Nm)The angle of beam (rad)Torque of motor (Nm)

Mathematical ModelLagrangian Method

Kinetic energy:

+) kinetic of the beam.

+) kinetic of the ball.

+) kinetic of rotor.

Mathematical ModelPotential energy:

Lagrange function:Dynamical equations:

Mathematical ModelWe have:And:

Mathematical ModelEquations of DC servo motor:

and

Mathematical ModelSet: , we have state space equations of ball and beam system:

Parameters of System

mBBall mass21,64/ (g)mbBeam mass81,27(g)RBall radius0,9cm)LBeam length43 (cm)dArm length2,5(cm)KbVelocity constant of motor0,1174KtTorque constant of motor0,1174

Parameters of System

KgGear ratio5(120/24)RaResistor of armature10,4()JbBeam moment of inertia0.005(kg.m2)JBBall moment of inertia7,0114.10-7(kg.m2)JmRotor moment of inertia5,5136.10-5(kg.m2)KuAmplifier gain10(V/round)

Linear ModelIn order to design controllers such as: LQR, PID,LQG,for Ball and Beam system, we must have thelinear approximation of the dynamic equations.

Using Jacobian linearization method to give the linear dynamic equation around operating point, dynamic equations were rewritten:

Linear Model

In that:

Linear ModelOperating point is selected to be:

With: we have:

Linear Model

Or:

Linear ModelSummary, after some calculations, we have linearitymodel of ball and beam system:

With matrices A, B as following:

Linear Model

Linear ModelUsing above parameters, we have matrices:

Linear ModelInner structure of linear four states model:

ControllerController has to automatically regulate the positionof the ball on the beam. Based on dynamic model, there are various controllers:

It could be modeled as a fourth order multivariable system and a state feedback controller could be developed.A simpler method, it was modeled as two second order system, and two control loops were designed

ControllerWe can design many classical or modern feedback controller for ball and beam system. Examples:

PID controllers.LQR, LQGPID-LQRFuzzy, Sliding mode,

ControllerState Feedback Controller:The idea of feedback in state space control is similar to that in the classical control system. The control signal in full state feedback control is defined as follow:

where K is proportional control gains.

ControllerState Feedback Controller:

LQR ControllerLQR Method:The LQR stands for Linear Quadratic Regulator, which seeks a gain matrix the minimises performanceindex J:

Where: Q is the state weighting matrix, R is the control weighting matrix.

LQR ControllerThe optimal gain matrix is calculated:

with P is determined by solving the algebraic Riccati Equation:

LQR ControllerMatrices are:

Control gains calculated using above Q, R is presented as follow:

LQR ControllerSimulation:

LQR ControllerResults from Simulation:

LQR ControllerExperiment:To implement, we can use PC, or analog circuit. In thiscase, an data acquisition card NI USB 6008 and PCis used to implement LQR controller.

Experimental schematic:

LQR Controller

LQR ControllerInterface between the physical system and NI-USB-6008

LQR ControllerExperimental Results:Ball Position - r

LQR ControllerTilt angle of the beam

LQR ControllerVelocity of ball -

LQR ControllerAngle velocity of beam -

ConclusionState feedback controller using LQR method can stabilize ball and beam system a open loop unstable. System is robust with noises, and initial conditions.Controller can be implemented using analog circuits, or PC.

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