TORSIONAL POSITION CONTROL SYSTEM

Figure 6.7 illustrates the block diagram of torsional system, which consists of a

DC motor, an instrumented bearing block, and a torsional load with two masses attached

to the shaft of the DC motor. DC motor shaft is free to rotate inside the bearing block,

and the shaft position and torsion module position are measured by encoders. The control

objective is to control the position of the torsional load with minimum vibration using full

state feedback controller design via LQR. Since the shaft of the torsion and DC motor are

coupled, by controlling the voltage applied to the motor the position of both motor shaft

angle and torsion shaft angle can be controlled. So it is a type of Single Input Multi

Output (SIMO) system, which has DC motor voltage as input and motor shaft angle and

torsion shaft angle as outputs. Moreover, such a system emulates torsional compliance

and joint flexibility that are common characteristics in mechanical systems namely high-

gear-ratio harmonic drives or lightweight transmission shafts.

Figure 6.7 Block diagram of torsional system

6.7.1 Mathematical Modeling

The mathematical model of the flexible torsional system is obtained from the

first principles. Applying Newton’s second law of motion,

J1( d2

d t2 θ1( t))+B1( ddt

θ1(t))+K s (θ1 ( t )−θ2 ( t ) )=τ (6.75)

The mass balance equation at the torsion load can be represented as,

J2( d2

dt 2 θ2(t))+B2( ddt

θ2(t))+K s (θ2 (t )−θ1(t))=0 (6.76)

Four system variables namely motor shaft angle (θ1), motor shaft velocity (θ1),

torsion shaft angle (θ2) and torsion shaft velocity (θ2 ¿ are taken as state variables, and the

motor voltage (V m) is considered a input variable. Hence the state and input variables are,

x1=θ1 ( t ) , x2=θ2 ( t ) , x3=ddt

θ1

( t ) , x4=ddt

θ2 (t ) (6.77)

By substituting the state variables in above equations,

J1( ddt

x3

(t))+B1 x3+K s ( x1−x2 )=τ1 (6.78)

J2( ddt

x4

(t ))+B2 x4+K s ( x2−x1 )=0 (6.79)

Rearranging the Equations (6.78) and (6.79),

ddt

x3( t)=−B1 x3

J1

−K s x1

J 1

+K s x2

J 1

+τ1

J 1

(6.80)

ddt

x4(t)=−B2 x4

J2

−K s x2

J 2

+K s x1

J 2

(6.81)

The state space representation of 1-DOF torsion system is

[ x1

x2

x3

x4]=[

0 0 1 00 0 0 1

−K s

J1

K s

J1

−B1

J1

0

K s

J2

−K s

J2

0−B2

J 2

][ x1

x2

x3

x4]+[ 0

01J 1

0]u (6.82)

y=[1 0 0 00 1 0 0] [ x1

x2

x3

x 4] (6.83)

6.8 RESULTS AND DISCUSSION

Figure 6.8 Snapshot of experimental set up of torsional system

Table 6.3 Torsional system parameters

Symbol Description Value

J1 Equivalent moment of inertia at motor shaft 0.0022 kg.m2B1 Equivalent Viscous damping at motor shaft 0.0150

Mb Disc weight mass 0.0022 kg

Dw Disc weight diameter 0.0380 m

Ks Flexible coupling stiffness 1 N.m/rad

Lb Load support bar length 0.044 m

J2 Equivalent moment of inertia at torsion load shaft 5.45×10-4

kg.m2B2 Equivalent Viscous damping at torsion load shaft 0.015 N.m.s/rad

The experimental set up, as shown in Figure 6.8, consists of a DC servo unit,

torsion module, power amplifier and a PC. The proposed control algorithm is realized in

the PC using the real time algorithm, QUARC, which is similar to C like language. The

sampling interval is chosen to be 0.001s. By substituting the parameter values of torsional

system from Table 6.3 into Equation (6.78) and (6.79), the following mathematical

model is obtained in state-space form.

[ X1

X2

X3

X4]=¿

Y=[1 0 0 00 1 0 0] [X1

X2

X3

X4] (6.84)

Using similarity transformation the above state space model is converted into

the controllable canonical form as given below.

[ X1

X2

X3

X4]=[0 1 0 0

0 0 1 00 0 0 10 −24760 2472 −33.81

] [X1

X2

X3

X4]+[ 0

00

206500]U

Y=[1 0 0 00 1 0 0] [X1

X2

X3

X4] (6.85)

An optimal state feedback regulator via LQR is designed to control the motion

of the torsion system with reduced vibrations. The objective is to control the position of

the torsion load shaft by controlling the DC motor shaft. For one sample value of settling

time and damping ratio, the coefficients of Q and R matrices are explained below.

The controller should result in a response which has a settling time of 0.2s and

an overshoot of less than 5%. The value of damping ratio from the settling time is

calculated using the following expression.

t s=4

ζ ωn (6.86)

The damping ratio of the system for the given specification is found to be 5.

The control input which is voltage applied to the DC motor is restricted to ± 10V . Since it

is a single input system, the value of R can be fixed to any scalar which will meet the

constraint on the control input. Then, fixing the value of R makes the selection of Q

matrix straight forward, and the diagonal elements of Q matrix are found to be

Q=[60.0344 0 0 00 8.7682 0 000

00

0.42410

00.001

]Then, the weighting matrices are used to find the following transformation

matrix P.

P=[36.0206 5.7033 0.0901 0.00045.7033 3.1884 0.0517 0.00020.09010.0004

0.05170.0002

0.00820.00003

0.000030.0003

]Using the Lyapunov optimization method, the corresponding state feedback

gain is found to be

K= [0.7748 0.3450 0.0616 0.001 ]

Angular position response of both the torsion load and motor shaft are shown

in Figure 6.9 and 6.10. The time domain parameters of torsion load shaft angular

response is given in Table 6.4. It is worth to note that in the real time results the settling

time and the overshoot of the torsion load shaft is found to be 0.18 and 4% which is very

close to the design specifications. The motor shaft velocity and torsion load shaft velocity

are shown in Figures 6.11 and 6.12. Figure 6.13 illustrates the control signal (Vm) applied

to the DC motor, and it is worth to mention that the control input is maintained below the

saturation value which is 10V in the present case.