Using MATLAB and Simulink For

7/30/2019 Using MATLAB and Simulink For

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2/22

Outline

Overview (and review) of MATLAB and Simulink

Using MATLAB and Simulink for dynamical system

analysis and simulation, and control design

Nonlinear vs. linear simulation and analysis

Application to Pan-Tilt platform


3/22

Last Time

Equation of motion for dynamical systems:

1-link:

General:

As a control system, we may regard is the input, lis

the output. Today, we will see how to use MATLAB

and Simulink to simulate the response of the system

for a given input trajectory.

2( ) ( ) sinm m m c

I N I B NB N mg + + + = +! ! ! ! !

"" " !

( ) ( ) ( , ) ( )M B C G + + + ="" " " "


4/22

MATLAB

A powerful package with built-in math functions, array

and matrix manipulation capabilities, plotting and lots

of add-on toolboxes (e.g., control, image processing,symbolic manipulation, block diagram programming,

i.e., Simulink, etc.)


5/22

MATLAB

Vectors: theta=[theta_1;theta_2];

Matrices: M=[M11 M12; M21 M22];

Polynomials: p=[a3 a2 a1 a0];

Transfer functions: G=tf(num,den);

Linear simulation:

step response: step(G);

impulse response: impulse(G);

general response: y=lsim(G,u,t);


6/22

MATLAB (Cont.)

Solving ODE

[t,x]=ode23(func,[0 tf],xinit); f=func(t,x)

Plotting: plot(t,x1,t,x2);xlabel(time (sec));ylabel(theta(deg)); title(theta(t)); legend(\theta_1,\theta_2);

Printing (to printer or file)

print -f -d Using m-files in MATLAB

use any editor (or MATLAB built-in editor, just type in edit)

function f=func(t,x) ...

Getting help in MATLAB: help or just help

on-line tutorial: http://www.engin.umich.edu/group/ctm/

( , )x f t x="


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8/22

Simulation

Considerinput as the motor torques (2x1) and output as

the joint angles (link) (2x1). Simulation involves find the

output response for a given input trajectory.

You will use a high fidelity simulation to validate your design

(including nonlinearity, friction, saturation, etc.). For your

control design, you will need to use a linearized model.

pan-tilt

dynamics


9/22

Linearization

Equation of motion is nonlinear (M, C, Gare nonlinear

functions of). To facilitate control system design,

we first linearize about an operating point( , ) ( ,0)d ="


10/22

Linearization: 1-D example

Taylor series expansion about

and keep the linear term. Consider the 1-D example

from last class:

#++= 2)(sin5.0)(cossinsinddddd

may be cancelled ortreated as a disturbance

Linearized system:

cos sin

( )

v g d g d

d

I F mgl mgl

+ = +

=

"" "

$%&%'

sgn sinc v gI F F mgl + + ="" " "

( , ) ( ,0)d ="


11/22

Linearization: General Mechanical Systems

(

cancelled or

treated asdisturbance

( ) ( )( ) ( )d d d d G

M D G

+ + =

"" "

1 1

1 2

2 2

1 2

G G

G

G G

=

For pan-tilt platform, input is motor torque (2x1), outputis (2x1).

linearized

pan-tilt

dynamics


12/22

Description of LTI Systems

Input/Output

(differentialequation)

Frequency

Domain State Space

What does LTI mean?

Input output

state


13/22

Description of LTI Systems

Input/Output(differentialequation)

Frequency

Domain State Space

( ) ( )( )d d dM D G + + ="" "

2 1

( )

( ) ( ( ) ( )) ( )d d

G s

s s M Ds G s

= + +

$%%%%%&%%%%%'[ ]

(

1 1 1

0 0

0 0A B

DC

Ix x

M G M D M

y I x

= +

= +

"

$%%%&%%%' $&'

$&'

1

2

dxxx

= =

"


14/22

MATLAB Description of LTI

Each LTI is treated as an objectwith a variety of

possible description:

transfer function: tf(num,den) (numerator and

denominator polynomials)

pole/zero/gain: zpk(z,p,k) (zeros, poles, gain)state space: ss(A,B,C,D) (state space parameters)

Take a look ofpantilt_init.mon the webpage


15/22

Open Loop Linear System Response

Impulse response:

y=impulse(G)

step response:

y=step(G)

general response:

y=lsim(G,u,t)

Bode plot:

y=bode(G)

poles/zeros/dampings

pole(G),zero(G), damp(G)

pole/zero plot:

pzmap(G)

gain/phase margin

(robustness):

margin(G)


16/22

Incorporation of Control

Interconnection of LTI systems:

G

H

KFr

-

+

Gcl = feedback(G*K,H)*F


17/22

Simulink

Instead of command line entries, it may be easier to use

a block diagram programming tool:

LTI block

Take a look ofpantiltlinear.mdl for linearizedpan-tilt under PID control


18/22

Effect of Sampling

Most control systems these days are digital in nature so

sampling is inherent (through A/D for sensor, which

contains a sampler, and D/A for actuators, whichcontains a zero-order-hold).

To analyze the effect of sampling, we can find the

equivalent discrete time system:

Gd = c2d(G,ts);%ts=sampling period (sec)

Gd is also an LTI object and the commands for LTI maybe applied.


19/22

Adding Sampling to Simulink Diagram

To add sampling to your continuous time simulation,

just add a zero-order-hold block (in the discrete time

system library) to the input, then set the sampling time.


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Nonlinear System Simulation

G(s) may be replaced by a nonlinear block


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What You Need to Do

!Turn your qualitative spec into more quantitative specin terms of speed and precision, ability to reject

disturbance, etc.

! Develop a Simulink diagram for your design iteration. Use the Simulink diagram on-line as a starting

template.

Add motor and gear parameters to the pan-tiltskeleton (generate composite m, I, pfor each body).

Use your design parameters for simulation.

Desired input should be based on your spec.

You need to tweak the controllers also (e.g., gravity

compensation, removing the mass matrix coupling,

tune gains for each axis, avoid saturation, etc.)


22/22

Today at 5pm, Next Tuesday, 2/4 (5pm),

Next Wednesday 2/5 (5pm)

Work on your project proposal (include preliminary

design using MATLAB/Simulink)

Next Wednesday, 2/5 (9am)

Components of control systems: amplifier, encoder,

motor

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Using MATLAB and Simulink For