Control Example Using Matlab.ppt

Control Example using Matlab Control Example using Matlab Cruise Control Modeling a Cruise Control System • The inertia of the wheels is neglected • Aerodynamic Drag is neglected – is proportional to[.]

Control Example using Matlab

Cruise Control

Modeling a Cruise Control System

neglected

• Aerodynamic Drag is neglected

– is proportional to the square

of the car’s speed

Physical setup and system equations

• The problem is reduced to the simple mass and damper system

–is proportional to the car's speed

Modeling a Cruise Control System

• Using Newton's law, the dynamic equation for this system is:

u b

m x  x 

where u is the force from the engine.

• There are several different ways to describe a system of

linear differential equations.

• To calculate the Transfer Function, we shall use the Stare Space representation then transform it to TF using ss2tf

State-space equations

• The state-space representation is given by the

equations:

where

is an n by 1 vector representing the state

(commonly position and velocity variables in mechanical systems),

u is a scalar representing the input

(commonly a force or torque in mechanical systems), and

y is a scalar representing the output

DuC

y

Bu

Adt

d

x

x x

State Equation for our CC model

The state vector is [ x, ] and y is x x x

The equation

um

1m

m x  x 

or

 0 1  x   0 u y

u m

1 0

x m

b 0

1 0

x

x x

– Rise time < 5 sec

Overshoot < 10%

Steady state error < 2%

Matlab representation and open-loop response

Get the Transfer Function

• A step input can be described as a change in the input from zero to a finite value at time t = 0

• By default, the step command performs a unit step (i.e the input goes from zero to one at time t = 0)

• The basic command to use the step function is one

of the following

– (depending if you have a set of state-space equations or a transfer function form):

step(A,B,C,D) step(num,den)

P controller

• The steady state error is more than 10%,

and the rise time is still too slow

• Adjust the proportional gain to make the

response better

 but you will not be able to make

the steady state value go to 10m/sec

without getting rise times that are too

fast

• You must always keep in mind that you are designing a real system,

and for a cruise control system to respond 0-10m/sec in less than half

of a second is unrealistic

• To illustrate this idea, adjust the k value to equal 10,000, and you

should get the following velocity response:

P controller

• The solution to this problem is to add some

integral control to eliminate the steady state

error

• Adjust k until you get a reasonable rise time

– For example, with k = 600, the response should now look like:

PI controller

k = 600;

ki = 1; % small to get feeling for integral response

num1 = [k ki]; % PI has two gains

PI controller

• Now you can adjust

the ki value to reduce

the steady state error.

As you can see, this step

response meets all of the

design criteria, and

therefore no more iteration

is needed

It is also noteworthy that

no derivative control was

needed in this example

Modeling a Cruise Control System

•.ךוכיחה חוכ תוחפ עינמה חוכל הווש היצרניאה חוכ

Physical setup and system equations

Let’s now add Drag and assume the friction to be non velocity

dependent (the static friction, proportional to normal force).

) ( )

(

)

t Bv t

Cu dt

t

dv

) ( )

(

)

t u

t dt

(

0  t u 

:רשאכו

ב תכרעמ תינב Simulink

0

- ל 1

- ב שמתשהל יאדכ ןכל ,

Saturation

Very Short Simulink Tutorial

• In the Matlab command window write simulink

• The window that has opened is the Simulink Library Browser

– It is used to choose various Simulink modules to use in your simulation

• From this window, choose the File menu , and then New

(Model).

– Now we have a blank window, in which we will build our model

– This blank window and the library browser window, will be the

windows we’ll work with

• We choose components from the library browser, and then drag them to our work window

– We’ll use only the Simulink library (also called toolbox) for now

Simulink Tutorial

As we can see, the Simulink library is divided into several

categories:

1 Continuous – Provides functions for continuous

time, such as integration, derivative, etc.

2 Discrete – Provides functions for discrete time.

3 Funcitons & Tables – Just what the name says.

4 Math – Simple math functions.

5 Nonlinear – Several non-linear functions, such as

switches, limiters, etc.

6 Signals & Systems – Components that work

with signals Pay attention to the mux/demux

7 Sinks – Components that handle the outputs of the

system (e.g display it on the screen).

8 Sources – Components that generate source signals

for the system.

Simulink Tutorial: First simulation

• Drag the Constant component

from the Simulink library

(Sources) to the work window,

and then drag the Scope module

(Sinks) in the same manner

• Now, click the little triangle

(output port) to the right of the

holding the mouse button down,

drag the mouse to the left side of

the scope (input port) and then

release it You should see a

pointed arrow being drawn.

• Double click the Constant module to open its dialog window

• Now you can change this module’s parameters

• Change the constant value to 5

Simulink Tutorial: First simulation

• Double-click the scope to view its window

• You can choose Simulation

change the time limits for your simulation

• Choose Start from the Simulation menu

(or press Ctrl+T, or click the play button

on the toolbar) to start the simulation

Simulink Tutorial: First simulation

• Choose Start from the Simulation menu

(or press Ctrl+T, or click the play button

on the toolbar) to start the simulation

• Now this is a rather silly simulation All

it does is output the constant value 5 to the

graph (the x-axis represents the simulation

time)

• Right-click on the scope’s graph window

and choose Autoscale to get the following

result:

Simulink Tutorial

• Now let’s try something a little

bit more complicated

• First, build the following

system (you can find the Clock

module in the Sinks category

• The Trigonometric Function

and Sum modules reside in the

Simulink Tutorial

• Now, let’s see if the

derivative is really a cosine.

• Build the following system

(the Derivative module is

located in the Continuous

category):

We can see that this is indeed a cosine, but something is wrong

at the beginning

Simulink Tutorial

• This is because at time 0,

the derivative has no

prior information for

calculation

– there’s no initial value for

the derivative,

– so at the first time step ,

the derivative assumes

that its input has a

constant value (and so the

derivative is 0 ).

An other example

• Let’s say that we have a differential equation that we want

to model The equation is:

0

A

We’ll notice 2 simple facts:

1 If we have A, then we have A' (multiplication).

2 If we have A', then we have A (integration).

We can get out of this loop by using the initial condition We know

that A0 = 0.5, so now we can calculate A' and then recalculate the new A, and so on

Simulink Tutorial

• double-click the

Integrator and choose

Initial Condition Source:

External

• Note that pressing ctrl

when clicking the mouse

button on a line, allows

you to split it into 2 lines :

• Set the simulation stop time to

3.5 seconds

–the solution goes to infinity

• and see the results in the scope:

ב תכרעמ תינב

Simulink Automatic Cruise Control

0

- ל 1

- ב שמתשהל יאדכ ןכל ,

Saturation

Automatic Cruise Control

• בכרה לש היצלומיסה תא הנבנ )

plant

(

Automatic Cruise Control

Automatic Cruise Control

• הלחתהה יאנת רובע

היוצר תוריהמו

,

תכרעמה תוגהנתה תא ןחב

– רקב םע P

רקב םע , I

רקב םעו PI

.(דרפנב הרקמ לכ)

– הרעה

:תויהל הרומא היוצר הבוגת :

.הריהמ ) לש ילמיסקמ ךרע - t

ךרעה לעמ "ידמ" הלעי אל (

.דימתמ בצמב שרדנה (ןתינש לככ הנטק) "ספא" איה דימתה בצמה תאיגש -

5.0)0( 



8 0



r





Automatic Cruise Control

P controller – No wind

Automatic Cruise Control

P controller – with wind

Automatic Cruise Control

I controller – No wind

Automatic Cruise Control

PI controller – No wind

Automatic Cruise Control

PI controller – with wind

Automatic Cruise Control

Automatic Cruise Control