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MATLAB AND CONTROLS

PRESENTED BY:-

AGILESWARI K. RAMASAMY

DR. FARRUKH HAFIZ NAGI

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Controls & MATLAB

INRTRODUCTION Control system consists of subsystems

and processes assembled for the purpose of controlling the outputs of the processes.

DC MOTOR Physical Modeling of a DC Motor Physical Modeling of a DC Motor

in STATE SPACE Designing the full-state feedback controller Bode Plot PID CONTROLLER LTIVIEW

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A common actuator in control systems is the DC motor.

It directly provides rotary motion and, coupled with wheels or drums and cables, can provide transitional motion.

Physical Modeling of a DC Motor

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The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back emf, e, is related to the rotational velocity by the following equations:

e

t

Ke

iK Kt (armature constant) = Ke (motor constant).

Physical Modeling of a DC Motor

rotational speed is the output voltage is the input

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)t(KV)t(Ri)t(dt

diL

)t(Ki)t(b)t(J

Newton's law combined with Kirchoff's law:

)s(Ks)s(V)s(I)RLs(

)s(KI)s()bJs(s

The modeling equations in Laplace Transforms

e

t

Ke

iK

Physical Modeling of a DC Motor

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Open-loop transfer function

The value of the constants:- moment of inertia of the rotor (J) = 0.01 kg.m^2/s^2

damping ratio of the mechanical system (b) = 0.1 Nms electromotive force constant (K=Ke=Kt) = 0.01 Nm/Amp electric resistance (R) = 1 ohm electric inductance (L) = 0.5 H

rotational speed is the output voltage is the input

Physical Modeling of a DC Motor

2K)RLs)(bJs(

K

)s(V

)s(

)s(V

)s(s

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Motor speed design criteria based on step

input

Settling time less than 2 seconds Overshoot less than 5% Steady-state error less than 1%

Physical Modeling of a DC Motor

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MATLAB representation of Open loop transfer function is as follows:

Create a new m-file Enter the following commands in m-file:

J=0.01 %Defining constants b=0.1 K=0.01 R=1 L=0.5 num=K %Defining the numerator den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)] %Defining

denominator step(num,den,0:0.1:3) %Obtaining the step response title('Step Response for the Open Loop System')

Physical Modeling of a DC Motor

))Kb*R()]b*L()J*R[(ss)L*J((

K

)s(V

)s(22

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Alternative MATLAB representation J=0.01; b=0.1; K=0.01; R=1; L=0.5; num=K den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)] T=tf(num,den) %Defining transfer

function step(T,0:0.1:3) %Obtaining step response title('Step Response for the Open Loop

System') %Title of the figure

Physical Modeling of a DC Motor

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Physical Modeling of a DC Motor

When 1 volt is applied to the system, the motor can only achieve a maximum speed of 0.1 rad/sec, ten times smaller than our desired speed.

It takes the motor 3 seconds to reach its steady-state speed; this does not satisfy our 2 seconds settling time criterion.

Overshoot less than 5%Settling time less than 2 secondsSteady-state error less than 1%

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Physical Modeling of a DC Motorin STATE SPACE

DC Motor Differential Equation is represented in state-space by choosing rotational speed and

electric current as the state variables

voltage as an input rotational speed as output

)t(KV)t(Ri)t(dt

diL

)t(Ki)t(b)t(J

State space representation

DC Motor Differential Equation

DuCxy

BuAx

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Physical Modeling of a DC Motorin STATE SPACE

MATLAB representation of Open loop transfer function using the state-space equations.

Create a new m-file Enter the following commands in m-file:

J=0.01; b=0.1; K=0.01; R=1;L=0.5; A=[-b/J K/J ; -K/L -R/L] %Defining the matrix A,B,C,D B=[0 ; 1/L] C=[1 0] D=0 step(A, B, C, D) % Obtaining the step

response

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Designing the full-state feedback controller

The schematic of full state feedback system is

CxY

BuX)BKA(X

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Designing the full-state feedback controller

The characteristic polynomial for this closed-loop system is the determinant of (sI-(A-BK)).

The matrices A and B*K are both 2x2 matrices, there should be 2 poles for the system.

The two poles will be placed at -5 + i and -5-i (note that this corresponds to a zeta = 0.98 which gives 0.1% overshoot and a sigma = 5 which leads to a 1 sec settling time).

MATLAB will find the controller matrix,K using these two poles.

Settling time less than 2 seconds Overshoot less than 5% Steady-state error less than 1%

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J=0.01; b=0.1; K=0.01; R=1;L=0.5; A=[-b/J K/J ; -K/L R/L]; B=[0 ; 1/L]; C=[1 0]; D=0; p1 = -5 + i % Pole 1 p2 = -5 - i % Pole 2 K = place(A,B,[p1 p2]) % Obtaining the value for the

controller t=0:0.01:3; %Defining the time step step(A-B*K,B,C,D,1,t) %Obtaining the step response

MATLAB representation of full state feedback system .

Designing the full-state feedback controller

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Step response with K controller

Designing the full-state feedback controller

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BODE PLOT

The main idea of frequency-based design is to use the Bode plot of the open-loop transfer function to estimate the closed-loop response.

J=0.01; b=0.1; K=0.01; R=1;L=0.5; num=K; den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)]; bode(num,den)%Obtaining the bode

plot

MATLAB representation to draw bode plot.

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A controller will be designed to satisfy the design

requirements.

The closed loop DC motor with the controller

Overshoot less than 5%Settling time less than 2 secondsSteady-state error less than 1%

PID CONTROLLER

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PID CONTROLLER

• Transfer function of PID controller:

Kp=porportional gain

KD=derivative gain

Ki=intergral gain

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PID CONTROLLER

J=0.01; b=0.1; K=0.01; R=1;L=0.5; num=K; den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)]; Kp=100 %Defining the Proportional

constant numa=Kp*num %Obtaining the new numerator dena=den %Obtaining the denominator [numac,denac]=cloop(numa,dena) %Obtaining the num and den

of the %closed loop system t=0:0.01:5; step(numac,denac,t) title('Step response with Proportion Control')

MATLAB representation for proportional controller.

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PID CONTROLLER

J=0.01; b=0.1; K=0.01; R=1;L=0.5; num=K; den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)]; Kp=100 %Defining the Proportional constant numa=Kp*num %Obtaining the new numerator dena=den %Obtaining the denominator T=tf(numa,dena) %Obtaining the new OL transfer

function G=feedback(T,1) %Obtaining the new CL transfer

function t=0:0.01:5; step(G,t) title('Step response with Proportion Control')

Alternative MATLAB representation for proportional controller

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PID CONTROLLER

Step response with Proportional Control

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PID CONTROLLER Derivative control – reduce overshoot Integral control – reduce steady state

J=0.01; b=0.1; K=0.01;

R=1; L=0.5; num=K;

den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)];

Kp=100; %Defining proportional contstant

Ki=1; Kd=1; %Defining integral and derivative constant

numc=[Kd, Kp, Ki]; %Obtaining num and den of the controller

denc=[1 0];

numa=conv(num,numc) %Obtaining the num and den of the whole % system

dena=conv(den,denc)

[numac,denac]=cloop(numa,dena) %Obtaining the num and den of the closed %loop system

step(numac,denac)

title('PID Control with small Ki and Kd')

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PID CONTROLLER Derivative control – reduce overshoot Integral control – reduce steady state

J=0.01; b=0.1; K=0.01;

R=1; L=0.5; num=K;

den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)];

Kp=100; %Defining proportional constant

Ki=1; Kd=1; %Defining integral and derivative constant

numc=[Kd, Kp, Ki]; %Obtaining num and den of the controller

denc=[1 0];

C=Tf(numc,denc) %Transfer function of the controller

P=tf(num,den) %Transfer function of the Plant

G=series(C,P) %Transfer function of the plant and controller

T=feedback(G,1) %Closed Loop transfer function

step(T)

title('PID Control with small Ki and Kd')

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PID CONTROLLER

Step response of the system with small Ki and Kd for the PID controller

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PID CONTROLLER

Increase Ki to reduce settling time Ki=200

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PID CONTROLLER

Increase Kd to reduce overshoot Kd=10 Ki=200 Kp=100

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LTIVIEW

LTIVIEW (GUI Tools)- convenient way to obtain time and frequency response plots of LTI transfer function.

Define the transfer function in the command window

Type LTIVIEW Import the function Right click on the plot and then select the

type of responses.

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THE ENDTHE END