Project - Speed Control of a Direct Current (DC) motor

Q1)-

ans-as per the digram we build the simulink as screenshot(133).

• we are using first clock and the workspace and rename to time and connect them.
• we take step block having the step time 0,and initial time also zero and the final value we set in firstly take zero.
• step block are connect to the sum block .
• the positive sign are connect to the x_1ref and -ve sign are connect to the initial condition integrator.
• then we are using five gain block and rename to kp,kd and ki and 1/m and k/m .
• we are using two workspace block name out.u and out.x1.
• we are using three integrator block for diplacement, velocity and acceleration.
• finaly connect all the block .
• change the model setting and save it and run it.

Key Highlights:

Mathematical modeling from the first principles:-

First principle models:

putting knowledge to work And they can be used to connect different unit operations by mass and heat balance in a process flowsheet. First principle modelling has been used extensively to help design new plants.

-we are using the mathematical calculation like this script we are using this model expression

• In this model we are using
• m = 100.0; % mass of the spring system
  k = 3944.0; % stiffness constant of the spring-mass
  % Equivalent wn
  wn = 6.2801; % sqrt(k/m)rad/s
  fn = 1.0; % wn/2/pi in Hz
  % Design parameters
  zeta = 0.5; % for under damped condition
  % Desire poles
  p1 = -zeta*wn+wn*sqrt(zeta^2-1);
  p2 = -zeta*wn-wn*sqrt(zeta^2-1);
  P = [p1;p2];
  % A & B matrices
  A = [0 1;-k/m 0];
  B = [0;1/m];
  % pole placement design
  K = place(A,B,P);
  % kp and kd Gain are obtained from earlier Design using the PLACE
  % command
  Kp = 3.86;
  Kd = 628.32;
  Ki = 0.0;
  % Dynamics
  % x1_dot = x2
  % x2_dot = (-k/m)*x1 + (1/m)*u
  % x2_dot = (-k/m)*x1 + (1/m)*[kp*x1ref-x1) - kd*x2 + ki*int(x1ref-x1)*dt]

Modeling an Electrical Engineering system

Electrical circuits involving resistors, capacitors and inductors are considered. The behaviour of such systems is governed by Ohm’s law and Kirchhoff’s laws.

• Resistor: Consider a resistance of ‘R’ Ω carrying current ‘I’ Amps as shown in Fig (a), then the voltage drop across it is v = R I
• Inductor: Consider an inductor ― L’ H carrying current ’i ’ Amps as shown in Fig (a), then the voltage drop across it can be written as v = L di/dt
• Capacitor: Consider a capacitor ’C’ F carrying current ’i ’ Amps as shown in Fig (a), then the voltage drop across it can be written as v = (1/C)∫ i dt
• Steps for modeling of electrical system
  Apply Kirchhoff's voltage law or Kirchhoff's current law to form the differential equations describing electrical circuits comprising of resistors, capacitors, and inductors.
• Form Transfer Functions from the describing differential equations.

Simulink based modeling

• we are using first clock and the workspace and rename to time and connect them.
• we take step block having the step time 0,and initial time also zero and the final value we set in firstly take zero.
• step block are connect to the sum block .
• the positive sign are connect to the x_1ref and -ve sign are connect to the initial condition integrator.
• then we are using five gain block and rename to kp,kd and ki and 1/m and k/m .
• we are using two workspace block name out.u and out.x1.
• we are using three integrator block for diplacement, velocity and acceleration.
• finaly connect all the block .
• change the model setting and save it and run it.
• for subplot and plot we are using plot command like,
• subplot(211);plot(out.time,out.x1,'b-');hold;grid;
• xlabel('Time');title('position - x1'); for title as assume the screenshot (131,132).
• subplot(212);plot(out.time,out.u,'b-');hold;grid;
• xlabel('Time');title('control - u');

Control law design and test

A control law is a mathematical formula used by the controller to determine the output u that is sent to the plant. In a feedback control scenario, the output u can lead to robustness to uncertainty, and can be used to design system dynamics.

control design objective

• setting time < 2s
• overshoot < 5%
• steady state error < 1%

Stability of a dynamic system

• dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small neighborhood. Various criteria have been developed to prove stability or instability of an orbit.
• Dynamics