Example: Modeling a Bus Suspension System using Transfer Functions
Physical setup
Design requirements
Equations of motion
Transfer function equation
Entering equations into MATLAB
Open-loop response
Control Block Diagram
State Space Method

Physical setup

Designing an automatic suspension system for a bus turns out to be an interesting control problem. When the suspension system is designed, a 1/4 bus model (one of the four wheels) is used to simplify the problem to a one dimensional spring-damper system. A diagram of this system is shown below:

Where:

* body mass (m1) = 2500 kg,
* suspension mass (m2) = 320 kg,
* spring constant of suspension system(k1) = 80,000 N/m,
* spring constant of wheel and tire(k2) = 500,000 N/m,
* damping constant of suspension system(b1) = 350 Ns/m.
* damping constant of wheel and tire(b2) = 15,020 Ns/m.
* control force (u) = force from the controller we are going to design.

Design requirements:

A good bus suspension system should have satisfactory road holding ability, while still providing comfort when riding over bumps and holes in the road. When the bus is experiencing any road disturbance (i.e. pot holes, cracks, and uneven pavement),the bus body should not have large oscillations, and the oscillations should dissipate quickly. Since the distance X1-W is very difficult to measure, and the deformation of the tire (X2-W) is negligible, we will use the distance X1-X2 instead of X1-W as the output in our problem. Keep in mind that this is an estimation.

The road disturbance (W) in this problem will be simulated by a step input. This step could represent the bus coming out of a pothole. We want to design a feedback controller so that the output (X1-X2) has an overshoot less than 5% and a settling time shorter than 5 seconds. For example, when the bus runs onto a 10 cm high step, the bus body will oscillate within a range of +/- 5 mm and return to a smooth ride within 5 seconds.

Equations of motion:

From the picture above and Newton's law, we can obtain the dynamic equations as the following:

Transfer Function Equation:

Assume that all of the initial condition are zeroes, so these equations represent the situation when the bus wheel goes up a bump. The dynamic equations above can be expressed in a form of transfer functions by taking Laplace Transform of the above equations. The derivation from above equations of the Transfer Functions G1(s) and G2(s) of output,X1-X2, and two inputs,U and W, is as follows.

Find the inverse of matrix A and then multiple with inputs U(s)and W(s) on the right hand side as the following:

When we want to consider the control input U(s) only, we set W(s) = 0. Thus we get the transfer function G1(s) as the following:

When we want to consider the disturbance input W(s) only, we set U(s) = 0. Thus we get the transfer function G2(s) as the following:

Entering equations into MATLAB

We can put the above Transfer Function equations into MATLAB by defining the numerator and denominator of Transfer Functions in the form, nump/denp for actuated force input and num1/den1 for disturbance input, of the standard transfer function G1(s) and G2(s):

G1(s) = nump/denp

G2(s) = num1/den1

Now, let's create a new m-file and enter the following code:

```
m1=2500;
m2=320;
k1=80000;
k2=500000;
b1 = 350;
b2 = 15020;

nump=[(m1+m2) b2 k2];
denp=[(m1*m2) (m1*(b1+b2))+(m2*b1) (m1*(k1+k2))+(m2*k1)+(b1*b2) (b1*k2)+(b2*k1) k1*k2];
P=tf(nump,denp);

num1=[-(m1*b2) -(m1*k2) 0 0];
den1=[(m1*m2) (m1*(b1+b2))+(m2*b1) (m1*(k1+k2))+(m2*k1)+(b1*b2) (b1*k2)+(b2*k1) k1*k2];
G2=tf(0.1*num1,den1);
```

Open-loop response

We can use MATLAB to display how the original open-loop system performs (without any feedback control). Add the following commands into the m-file and run it in the MATLAB command window to see the response of unit step actuated force input and unit step disturbance input. Note that the step command will generate the unit step inputs for each input.

`step(P)`

From this graph of the open-loop response for a unit step actuated force, we can see that the system is under-damped. People sitting in the bus will feel very small amount of oscillation and the steady-state error is about 0.013 mm. Moreover, the bus takes an unacceptably long time to reach the steady state (the settling time is very large). The solution to this problem is to add a feedback controller to the system.

`step(G2)`

To see some details, you can change the axis:

```axis([0 10 -.1 .1])
```

From this graph of open-loop response for 0.1 m step disturbance, we can see that when the bus passes a 10 cm high bump on the road, the bus body will oscillate for an unacceptably long time(100 seconds) with larger amplitude, 13 cm, than the initial impact. People sitting in the bus will not be comfortable with such an oscillation. The big overshoot (from the impact itself) and the slow settling time will cause damage to the suspension system. The solution to this problem is to add a feedback controller into the system to improve the performance. The schematic of the closed-loop system is the following:

From the above transfer functions and schematic, we can draw the bus-system block diagram as the following:

From the schematic above we see that:

Plant = nump/denp

F * Plant=num1/den1

so that

F=num1/(den1*Plant)

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