control of a double inverted pendulum on a

8/3/2019 Control of a Double Inverted Pendulum on a

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Control of a Double

Inverted Pendulum on acart



Mechanical Setup Modifications

• The first link of the existing single invertedpendulum setup had to be replaced by anotherarm to accommodate the second link.

• A hollow shaft miniature encoder was used.

• The problem faced was how to support the

second link.



Mathematical Model

Euler-Lagrange equation:

Where L is the Lagrangian function which isdefined as the difference between the totalkinetic and potential energies of the systemand R represents the system’s energydissipation due to viscous friction.



Mathematical Model

• König’s theorem

• Lagrange equations in a compact matrix form:

Where represent the inertia matrix, centripetal-Coriolis matrix and the gravitational vector respectively. q = (q 1, q 2 , q 3 )

T =(x , θ 1, θ 2 )

T and H = (1, 0, 0) T.

• These equations were used to implement the open-loop model of thesystem on SIMULINK.



Linearization

• The equations of motion were linearized in the neighbourhood of the upperunstable equilibrium point, in order to design the linear Stabilizing controller.

• The following are the assumptions taken in order to linearize the Lagrangeequations:

• The linearized model in valid state-space matrix form is given by:



Linear Stabilizing Controller • The design procedure of the linear controller is the same as that used for

the SISO system although our system is a SIMO system, since duringsteady-state all the states will converge to zero except for the cartdisplacement which is taken as the output.

• There are various techniques which are used to design state-feedback

linear regulator controllers, like the linear quadratic regulator or the pole-placement technique.

• State-feedback regulator controller:



Simulation Results



Inner current control loop

• The inner-loop takes care, that the required force to be applied to the cart isdeveloped instantaneously.

• The main blocks constituting the inner current control loop:

• The inner current controller was designed by using a pole-placement trackercontroller, designed in the digital domain.



Electronic and Power Electronic Circuitry

The electronic and power electronic circuitry consists of the following fourcircuit boards:

• Current sensor and anti-aliasing filter circuit for the motor current feedback.

• Regulator circuitry in order to power the electronic circuitry.

• PM DC motor drive which is composed of an IPM IGBT H-bridge and anIPM-Driver Board.

• The Interface board used to interface the hardware with the DS1104 board.



Experimental Results



THANK YOU

control of a double inverted pendulum on a

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