Pendulum-Exp_7_

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7AAD0021Control of Mechanical SystemsINVERTED PENDULUM, BRIEF

Typically, the process industries are concerned with obtaining quicker response withnear critical damping from slow systems and many mechanical systems have the samerequirements, differing only in time scale. Generally these are systems with real,negative plant poles. Another common class of mechanical systems have lightlydamped or undamped complex plant poles or unstable poles. In these cases the task ofthe control system designer is to stabilize and damp the system response. Often speedof response is sacrificed to gain otherwise acceptable behavior. Modern aircraft tend tohave a pair of lightly damped complex poles and a pair of real poles with one being closeto the origin in either the left or right (unstable) half plane. The Bytronic apparatus has alightly damped pair of poles when the pendulum hangs down and two real poles with one

of them unstable when the pendulum is upright.

In order to design a compensator and predict performance, the transfer function of each

block in the block diagram of Figure 1 must be identified:

1. Identify Gp(s), the transfer function of the pendulum in both the upright and hangingconfiguration. Adjust the mass so that it is flush with the end of the rod. The length of apendulum is the distance from the pivot centerline to the center of mass. The center of


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mass can be found by unscrewing it from the carriage and finding where it balances.The mathematical model is easily derived for the upright pendulum and the model of thehanging pendulum is obtained by changing the sign of the acceleration due to gravity.Applying Newtons law in the vertical and horizontal directions produces two equations.Using one equation to eliminate the tension T results in one nonlinear, second-order

differential equation relating input motion x(t) and plumb-bob angle (t). Show that onegets

2 2

2 2cos sin 0

d d xL g

dt dt

+ =

.Solve for the second derivative of (t) and isolate the nonlinearity in function f. Thenexpand f in a Taylor series and truncate after the first-order terms to get an approximatelinear model in terms of perturbation variables and X. Show that the plant transferfunction is

2

2

/( )

/

g X s LG s

L L s g L

= =

&&

&&

Include all of the steps and assumptions required to get to this result. The poles of thisplant transfer function are clearly real with one positive and one negative, since g and Lobviously have positive real values. So the upright pendulum is unstable when near thevertical position.


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If the sign of the acceleration due to gravity is reversed, the poles of the planttransfer function become imaginary and the model is that of a pure oscillator with an

undamped natural frequency of /n g L= . With a storage oscilloscope, measurethe period of free vibration when the pendulum is in the hanging orientation andcalculate L. Does this agree with the length of the pendulum as measured directly?

Real mechanical systems are never completely free of energy dissipation. If weassume viscous damping is the dominant energy loss mechanism, we can use thelogarithmic decrement method to estimate the viscous damping.The logarithmic decrement, , can be computed from two amplitudes, x1 and xn+1,separated by n complete cycles and, for lightly damped systems, the damping factor, , can be estimated from the log decrement. Use an oscilloscope to look at(t)during free vibration.

1

1

1ln

2n

x

n x

+

=

The damping should be small but not zero. Thus we add to the denominator of bothplant transfer functions a damping term

2 2 2 22

n n ns s s

Strictly speaking, what was measured was d, but you can check to see that thecorrection is indeed small enough to be neglected

21

d n = .

Now the plant model is complete for both configurations (Inv Pend/Gantry Crane)

( )2 22 2

/( ) /

2

n

p

n n

g sG s rad inch

s s

=

+ + where g must be expressed in units of inches/sec2 if Gp is to have units ofradians/inch. The next task is to determine the characteristics of the internal servoloop.


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2. Identify Gs(s), the transfer function of the servo and carriage subsystem. Unscrewthe pendulum and set it aside. Connect a function generator to the input of the servocontrol loop (terminal H) and drive it with a 1 Hz square wave at an amplitude thatmoves the carriage through a range of about two inches. Observe the carriageposition on the PicoScope. Dont drive the carriage continuously back and forthunnecessarily. This apparatus isnt very robustAdjust the servo gain and the velocity feedback to achieve critically dampedresponse. Record the knob settings so you can go back to them if they get moved.

Are the knob settings amplitude dependent? Switch the function generator tosinewave output and obtain enough of a Bode plot to determine the transfer functionof the servo and carriage subsystem from terminal H to terminal J1. Since you knowthe subsystem is critically damped. It is enough to establish the slope and value ofthe low frequency asymptote and the bandwidth, assuming the transfer functioniszero order in s in the numerator and second order in s in the denominator.Unfortunately, the output wave form is not sinusoidal, indicating that the servosubsystem is nonlinear. Indeed it is, due to hysteresis.The bandwidth, the frequency at which the amplitude is one half the low frequencyamplitude, is on the order of 1 to 6 Hz. Since the system is critically damped,=0.7,the bandwidth and n are equal. Dont mix Hz and radians/second. The closed-loop

transfer function of the internal servo and carriage loop is

2

2( )

2

ns s

n n

G ss s

=

+ + .3. Determine Kr, the conversion factor from tick marks on the setpoint slider tovoltage. Use the PicoScope to Measure this Voltage.


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4. Determine Kx, the conversion factor from carriage-position feedback voltage toinches. Do this by driving the carriage by connecting a wire from the setpoint slider(terminal A) to the servo amplifier input (terminal H) and measuring position with ascale or tape measure.

5. Determine Kp, the conversion from pendulum angle to volts. Do this by measuringV (terminal K) and the angle of the pendulum.6. Determine Ka, the constant used to calculate the center of gravity position Vy fromcarriage position Vx and pendulum angle V.

y x aV V K V = + Since Vx cannot be scaled, Vy must be scaled so that it also has the same scale (sothat when the carriage is stationary, but the center of mass moves one inch, Vy hasthe same value as when the carriage moves one inch but the pendulum angle

remains zero). The value calculated for Ka will be used to set the scalingpotentiometer. It is important that this value be correct. Verify your results by seeingthat Vy (terminal L) does not change when the carriage is moved, but the center ofmass position remains stationary.Simulation & Modelling1. Model the system using simulink.2. Confirm the open-loop characteristics3. Determine the Transfer function of the compensator.For the Tasks below, Use MATLAB.Compensator Design and Verification

Complete the following work for the system in both the upright and hangingconfigurations.1. Obtain the root locus for the system (both) without compensator. For each case:(a) can the system be stabilized by setting Kc? (b) what kind of stable performancecan be achieved (overshoot, settling time, etc.)?2. Design a compensator for each configuration using the two parallel RC plugsprovided. Note that the compensator is an OpAmp wired in an invertingconfiguration with a gain of Zf/Zi. The console denotes the open loop OpAmp gainas A because the internal wiring of the console inverts the gain to make thecompensator gain Zf/Zi. What type of compensator did you choose and why? Whatis the compensator transfer function?

3. Obtain the root locus plot for both systems with the compensators included. Foreach case, what range of Kc results in stable operation? Plot step responses forseveral values of Kc. Include the following trials: (a) all marginally stable values, (b)a value to produce quick settling, (c) a value to produce low overshoot, (d) a good


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compromise (use your judgement, but describe what you are trying to accomplish)and (e) a value that ensures stability in the face of parameter variation (a robustdesign).4. For both configurations, wire up the control system using your compensatordesigns and test the response using the values of Kc chosen above. How does the

system perform in comparison to your theoretical expectations? Any surprises?Explain any differences you see.

5. Simulate the effects of non-linearities, and confirm the results on the rig,comment on the accuracy of the simulation.

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