ME 431 System Dynamics

Dept of Mechanical Engineering

Lecture 1: Overview and Intro

• Introduction to the control system design process

• Control system example• open loop vs. closed loop

• Introduction to modeling

• Solving differential equations• Free response• Forced response

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Control System Design Process

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TranslatePlant

Design (andConstruction)

Model Analyze Controller Design

customerinput / gov’tregulations eng specs physical

systemdiagrams

math behavior controlsystem

purpose of models• analysis• design• verification

types of models• physical vs. empirical• mathematical• graphical

types of analysis• time domain• frequency domain• simulation• hardware in the loop (HIL)

types of control• supervisory logic control• on/off control• P, PI, PD, PID• advanced techniques

Control System Example

• Cruise Control Example

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ControlAlgorithm Engine Car

desiredspeed

throttleangle

(voltage)

forceactualspeed

Open-loop Control [feedforward]

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+-

disadvantages• sensitive to errors in model• sensitive to disturbances• needs periodic recalibration

advantages• simple to design• inexpensive• doesn’t affect stability• fast response wind force,

gravity force

ControlAlgorithm Engine Car

throttleangle

(voltage)

forceactualspeed

desiredspeed

Closed-loop Control [feedback]

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disadvantages• extra complexity• extra cost• can affect stability• can be slow to respond

advantages• robust to errors in model• robust to disturbances

+-

wind force,gravity force

ControlAlgorithm Engine Car

throttleangle

(voltage)

forceactualspeed

Speedometer

+-

measuredspeed

R E

D

U YCONTROLLER ACTUATOR PLANT

SENSOR

desiredspeed

Introduction to Modeling

• A model is an abstraction of the physical world

• Used for analysis and design, possibly before physical system exists

• Can be obtained from first principles or experimentally

• Purpose determines level of abstraction, form

• Complex enough, but no more

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Model Derivation

• From first principles• Use physical laws to derive models• Provides understanding• Can use empirical data to determine

parameters, validate model ME

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Model Derivation

• From empirical data• Feed a known input and observe output, fit model to

data

• Good for complicated systems (IC engine, battery) • Good for black-box systems (driver model)• Does not provide intuition, can’t be widely applied

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SYSTEM

Complexity Depends on Purpose

• Design/analysis model: simpler• Simple enough to generate closed-form solution• Less accurate, but provides intuition

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a a a b a

diR i L K e

dt

T b J

Complexity Depends on Purpose

• Simulation model: more accurate

Static vs. Dynamic SystemsStatic Systems

• Output is determined only by the current input, reacts instantaneously

• Relationship does not change (it is static!)

• Relationship is represented by an algebraic equation

Dynamic Systems• Output takes time to react• Relationship changes with

time, depends on past inputs and initial conditions (it is dynamic!)

• Relationship is represented by a differential equation M

E 4

31 L

ectu

re 1

12

SYSTEMinput output

Static vs. Dynamic SystemsMotor from a Dynamic ViewpointMotor from a Static Viewpoint

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0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

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350

400

450

500

Mot

or S

peed

Time

0 0.5 1 1.5 2 2.5 30

1

2

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4

5

6

Mot

or T

orqu

e

Time

speed

torq

ue

T

Tstall

ea1

ea2

wwno-load

Solving Differential Equations

• Homogenous differential equations

• Righthand side of equation equals 0• Represents free response of system• Solution consists of exponentials

where exponents are roots of the characteristic eq.

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0mx bx kx

1 21 2( ) t tx t a e a e

Solving Differential Equations

• Homogenous differential equations

• For the above, the characteristic equation is

• Roots can be found from the quadratic formula ME

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0mx bx kx

2 0m b k

2

1,2

4

2 2

b b km

m m

Solving Differential Equations

• Recalling that• If the roots are completely real, then the solution

is exponential• If all negative, stable• If any positive, unstable

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1 21 2( ) t tx t a e a e

time

disp

lace

men

t, x

Solving Differential Equations

• If the roots are complex, then can rewrite in sines and cosines using Euler’s identity:

• Therefore, ME

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cos sinj te t j t

( ) ( )1 2

d dj t j ta e a e

( cos sin )td de A t B t

1 2d djt jtt ta e e a e e

Solving Differential Equations

• Above follows when have complex roots of char. eq.

real part = rate of decay (growth)imag part = freq of oscillation

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d j

( ) ( cos sin )td dx t e A t B t

Solving Differential Equations

• Forced differential equations

• Solution consists of two parts

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( )mx bx kx F t

( ) ( ) ( )h px t x t x t xh is the homogenous solution

- same form as before, natural response of systemxp is the particular solution

- generally same form as F(t), due to the input

Example

has a solution of the form

where

the homogenous portion dies out (transient) the particular portion remains (steady state)

8 25 2x x x t

4( ) ( cos3 sin 3 )tx t e A t B t at b

determined from characteristic equationhx

21,2where +8 +25=0 has roots 4 3 j

and has same form as ( )px F t

xh(t) xp(t)

Example

• Consider other types of forcing functions:

8 25 5x x x 4( ) ( cos3 sin 3 )tx t e A t B t a

28 25 3 tx x x e 4 2( ) ( cos3 sin 3 )t tx t e A t B t ae

8 25 5sin 2x x x t 4( ) ( cos3 sin 3 ) sin(2 )tx t e A t B t a t

cos 2 sin 2C t D t

Example

• Find the solution x(t) for 3 0, (0) 5x x x

Example

• Find the solution x(t) for23 , (0) 5tx x e x

Example (continued)