1. INTRODUCTION TO CONTROL SYSTEMS
POSITION CONTROL OF A SPRING-MASS SYSTEM
Goal: Positioning the mass m at the desired position x. To achieve this goal, a control force Fc is used.
To perform a classical control task, the transfer function of the considered system should be known. The transfer function of an engineering system can be obtained using its governing equation of motion. The Lagrange method is a practical way of deriving the equation of motion. For this purpose, we first write the kinetic and potential energies of the system together with the virtual work expression.
Lagrange’s Equation:

2. The equation of motion of the spring-mass system:
Application of the Laplace Transform to the both side of the equation of motion gives the TRANSFER FUNCTION of the system. The TRANSFER FUNCTION relates the Input of the system (Fc ) to the Output (x) assuming all initial conditions as zero.
The Laplace transform of the output for a step input with magnitude 20 N.

3. The response of the spring-mass system to a control force Fc in the form of a step function with magnitude 20 N can be obtained by using the step command in Matlab. 20
Fc(t)
t (sec)
step command calculates the response of the system to a unit step input. So, the magnitude of the step input should be specified in the numerator of the output.

4. clc;clear
m=50;c=50;k=2000;
num=20;
den=[m,c,k];
step(num,den)
Magnitude of the step input
Spring constant

5. The response of the system can be obtained by using Matlab Simulink.

6. Considering that the the spring constant k in the system is constant, the only factor, which determines the magnitude of the response is the amplitude of the control force Fc.
The stedy-state response of the system xss will be different when a disturbance acts on the system. If the disturbance is shown as Ft
Disturbance
The form of equation of motion with a disturbing (uncontrolled) force

7. We will add an actuator , which converts the electric signal to a force /moment and a sensor, which converts a physical output to an electric signal (voltage)
K2=10 Volt/m
The relation between the inputs (V1 and Ft) and the output V2 can be written as:
Main input
Disturbance

8. If the desired output of the mass is xss=0
If the desired output of the mass is xss=0.01 m (1 cm), the voltage output of the displacement sensor should be V2ss=xss*K2=0.01*10=0.1 V. (The steady state value of the sensor output). The input voltage, which produces the desired sensor output can be calculated using the relation given below
The s term denotes the time derivative and in the steady-state case , all changes (derivatives) are zero.
Input voltage in order to obtain mass displacement as 1 cm.

9. Now, we calculate the response of the system to a step input with magnitude 0.2 V. In this calculation, we take the magnitude of disturbing input Ft as zero.

10. What will the system response be if the disturbing force is different from zero? Let’s assume the step disturbance Ft = 10 N.
Laplace transform of the disturbance

11. Δt should be choosen properly !
Disturbance
Input
Voltage output
Displacement output
Ft=0
Ft=10 N
Δt should be choosen properly !
V2(t)
Time (s)

12. For Ft=0, V2ss=0. 1 V and for Ft=10 N, V2ss=0. 05
For Ft=0, V2ss=0.1 V and for Ft=10 N, V2ss=0.05. This result show that the output of the open-loop system is highly influenced from the disturbance. This undesired effect may be removed by converting the open loop system to a closed loop control system.
Disturbance
Controller
Reference signal
(Desired output)
Sensor E
Error
Actuator
System
The expression for the output of the system can be writen as
In a Closed Loop Control system, the output of the system is sent back and compared with the Reference signal.

13. Where Hk is the transfer function of the controller used in the closed loop control system. The transfer function of the controller determines the type of the operations which will be applied to the error signal.

14. V2 R2 - + V1 R1
GAIN CIRCUIT (PROPORTIONAL) R2 - + V1 R1 V2 R
V2'

15. INTEGRAL CIRCUIT C - + V1 V2 R

16. DERIVATIVE CIRCUIT R - + V1 V2 C

17. - + V1 Vc R V2 V3 R V2 V1 - + Vc

18. PID CONTROLLER R2 - + R1 R4 C4 C3 R3 R V1 V2

19. P Control (PROPORTIONAL CONTROL)
In this type of control, the error signal is amplified in order to eliminate the negative effect of the disturbance on the system output.
Choose Hk as 60
clc;clear
m=50;c=50;k=2000;
k1=100;
k2=10;
hk=60;
num=[k1*k2*hk];
den=[m,c,k+k1*k2*hk];
step(num*0.1,den)
Without disturbance

20. With disturbance Ft=10 N
clc;clear
m=50;c=50;k=2000;
k1=100;k2=10;
hk=60;
num=[k1*k2*hk];
den=[m,c,k+k1*k2*hk];
step(num*0.1-k2*10,den)

21. P-I Control (PROPORTIONAL-INTEGRAL CONTROL)

23. P-I-D Control (PROPORTIONAL-INTEGRAL –DERIVATIVE CONTROL)