# Control System Help (not an EE)

## Homework Statement

(See Attachment for image)

Part 1)
This system is a model of an actuator with internal springiness, also called a series-elastic actuator.The system naturally decomposes into two subsystems: i) the mass of the actuator m1 with viscous friction b1 due to sliding on the floor, and ii) the mass of the load m2 with viscous friction b2. We assume that the magnitude of the force generated by the spring is proportional to (x1 - x2 )k . Each of these subsystems has its own inputs and outputs, e.g., f and fleft are the inputs to the actuator m1. fright is the input to the load system m2. f is the force input from the motor. f_left and fright are forces generated by the spring deflection. The position of the first mass is x1 and the position of the second mass is x2 , and x2 $$\geq$$ x1 . Write down the differential equations that characterize the equation of motion of each the two subsystems. You also need to replace f_left and f_right by the correct force that is generated by the spring.

Part 2)
Transform the two subsystems into the frequency domain, and write down the transfer functions for each subsystem.

Part 3)
Draw a control diagram of the entire system, i.e., with f as input and x1 and x2 as output. Hint: you will need a feedback loop to make this a
proper control diagram.

Part 4)
Write down the transfer function of the entire system, i.e., the system that receives f as input and has and x1 and x2 as output. There is no need to simplify the transfer function, i.e., you could use H1, and H2 for the two different transfer functions as abbreviation

Part 5)
Write down a PD controller to provide the feedback control for the actuator. The PD controller is supposed to provide the force f such x2, $$\dot{x}$$2 are as close as possible to a desired xdesired, $$\dot{x}$$desired .

N/A

## The Attempt at a Solution

For system 1 (mass 1), I have:

m1$$\ddot{x}$$1 - k(x2-x1) - b1$$\dot{x}$$1 = f

converting to frequency domain I have:

X1(s)*(m1*s^2 + k - b1*s) = k*X2(s) + F(s)

(I am assuming that the right hand side of the equation above is considered the input Correct me if I am wrong)

Therefore the Transfer function = H1 = output / input = X1(s)/(k*X2(s) + F(s)) = 1/(m1*s^2 + k - b1*s)

Similarly with subsystem 2 (mass 2), I come up with:

H2 = X2(s)/(k*X1(s)) = 1/(m2*s^2 + k - b2*s)

Question 1

What will this Control Diagram look like? Will the system be:
H(s) = H2(s) * H1(s),
or
H(s) = H1(s) * (I + H2(s)*H1(s))^-1

Question 2

Can I create the PD controller by taking the inverse Laplace transform of the total system transfer function (H(s)) and then setting f = Kp(x2desired - x2) + kd(x1desired - x1)?

Thanks for your help in advance!! (My TA has not responded to my e-mail and I cannot make it to his office hours)[/b/

#### Attachments

• springMass.jpg
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I think your system has two degrees of freedom. You should have two differential equations, one for acceleration of x_{1} and one for acceleration of x_{2}.

Also, It seems from the problem statement that you have one input f, and two outputs x_{1} and x_{2}. Which means that you should have two transfer functions.

So I think:

1. You derive two second order differential equations.
2. You use the Laplace transform and get linear equations.
3. After getting linear equations, you should be able to get one equation having only X_{1}(s) and F(s), and the other having only X_{2}(s) and F(s).
4. Now do some algebra, and get expressions for X_{1}(s)/F(s) and X_{2}(s)/F(s).
5. Now for formality, you say that your first output is Y_{1}(s) = X_{1}(s) and second output is Y_{2}(s) = X_{2}(s). Just substitute in your expressions for X_{1}(s) and X_{2}(s) to get your transfer functions.

After that, I am a little lost because I don't understand how you can draw a control diagram without a controller or a control objective. Do you want to drive the outputs to a specified reference?

My 2 cents.