Project - Position control of mass spring damper system

AIM;

Obtain the transfer function of a mass spring damper system and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal.

given data;

Take m = 3.6; %kg k = 400; %N/m c = 100; %Ns/m

solution;

transfer function of a mass spring damper system and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal. the matlab model of the pid control and transfer function pulse wavefrome generator.

This example shows a controlled mass-spring-damper. A controller adjusts the force on the mass to have its position track a command signal. The initial velocity for the mass is 10 meters per second. The controller adjusts the force applied by the Force Source to track the step changes to the input signal.

PID controller;

PID control stands for proportional–integral–derivative control. PID control is a feedback mechanism used in a control system. This type of control is also termed as three-term control, and is implemented by a PID Controller. By calculating and controlling three parameters – the proportional, integral and derivative of how much a process variable deviates from the desired set point value – we can achieve different control actions for specific work.

PID controllers are considered to be the best controller in the control system family. Nicholas Minorsky published the theoretical analysis paper on PID controller. For PID control the actuating signal consists of proportional error signal added with derivative and integral of the error signal. Therefore, the actuating signal for PID control is:

The Laplace transform of the actuating signal incorporating PID control is

There are some control actions which can be achieved by using any of the two parameters of the PID controller. Two parameters can work while keeping the third one to zero. So PID controller becomes sometimes PI (proportion-integral), PD (proportional-derivative) or even P or I. The derivative term D is responsible for noise measurement while the integral term is meant for reaching the targeted value of the system. In early days PID controller was used as a mechanical device. These were pneumatic controllers as they were compressed by air. Mechanical controllers include spring, lever or mass. Many complex electronic systems are provided with a PID control loop. In modern days PID controllers are used in PLC (programmable logic controllers) in the industry. The proportional, derivative and integral parameters can be expressed as – K_p, K_d and K_i. All these three parameters have an effect on the closed loop control system. It affects rise time, settling time and overshoot and also the steady state error.

PID control combines the advantages of proportional, derivative and integral control actions. Let us discuss these control actions in brief.

A PID controller has some limitations also apart from being one of the best controllers in control action system. PID control is applicable to many control actions but it does not perform well in case of optimal control. Main disadvantage is the feedback path. PID is not provided with any model of the process. Other drawbacks are that PID is linear system and derivative part is noise sensitive. Small amount of noise can cause great change in the output.

m = 3.6; % kg
k = 400; % N/m
c = 100; % Ns/m
% Pole placement design
fn = 1.0; 
wn = 2*pi*fn; 
% A & B matrix value
A = [0 1; -k/m -c/m]; 
B = [0; 1/m]; % Design values
zeta = 0.5; % for an under-damped behavior
% Desied poles value
p1 = -zeta*wn+wn*sqrt(zeta^2-1);
p2 = -zeta*wn-wn*sqrt(zeta^2-1);
p= [p1;p2];
K = place(A,B,p); 
Kp = K(1);
Kd = K(2);
% End

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