EE 1166 Controls Module-

Today : a Module Overview

• State Space Representation• Equilibrium Points• Linearization Of non-linearsystems

Module Overnice-

Study of Systems (physical orwhich have free inputs

.

Virtual )Examples :-

. Building - HVAC• cars - cruise control• Airplanes - Gps flight

Path tracking

Questions we should be able to

answer at the end of this Module :

consider a dynamical system :⑧ how do we Model * this system

mathematically ?• Can we use data to learnparameters of the Model of the system ?

• How does the System evolve

through time ?

• How do we Make the system do

what we want ?•what is the most efficient wayto get it to do what we want?

• Is the system inclined to'

come to resist ? If so,

in which

configuration ?• Can we change the propertiesAt the system by choice of

control ?

-

preliminaries-

Ict) :< d- Xlt)de

Ict ) :> DZXCEI-

④E) 2

' ÷:÷÷÷.

State Space Representation-

Interested in how Systemsevolve through time

Use ODES (ordinary differential

equations) to model Our systems

Example : pendulum-

stink )

"¥I GH( M

l

l

Ml Ect ) = - Kl CH - mg sina.cat Tinta

"t' :=t¥i:L """ Este's.mx. -¥xu*+¥NE ) fully represents the

state of the systembecause we can express the

time evolution of the Systemas a function OF thosevariables

,constants

,and inputs

.

Xf't ) is called the

state vector of this system,

X. ( t) and Xzdfl are the

State variables

Xct ) C- IR' UCH : -Tink) EIRt

For this example,

1/22 is State space

112 is control space

Itt) = f- (NH,UCH )--

T-

- Tinct) for

pendulumf : #***U → YT a¥÷:n

.T 972:c:

State control Not alwaysspace space same as X

Pend 1122 x HR → 1122

Definition : l.

- Equilibrium point :

(Kee, Uea ) c- XXUSit . f-(Xe , ,Ueq)=0

Find *ear , TeaSit .

liar fisherman. I:o)

⇒ Xea,= O

⇒ eq= Hsin a.)

For

T.in#=o:.*kea--qCnqojpnc- IN }o-ei-nn.Y.ae !

go.eu-

- Yeun"unstable

equilibrium" b "

esgtgswei.vn "

El : RLC circuit

run-mallVinet,

R L •

cdu.ae#*ieHdt-=icCt)=ieCtyLdiqH=V*LH=-VeCt)-VrltltVinltl=-Vcu) - Riecttvinctl

xuii-f.ie":D :-

-Kit:DVe - Xzlt )Mt) = (h . x. a ,

- Rrexzltlttvin

x.* =f%!fN#ft vine"

Linear System

Find (Xea,Uea ) s -t . Htt = o

Keg,= o

-Ikea ,

- Rzxqz t ¥HUeq=oI

↳ Xea,= Uea

9-

System is at an equilibrium

point if X ,= Vc = Vin

Example : car-

n

""""

Peat

↳ Peony

'

Ii:"" ⇐i÷÷÷:

Non-linear system

Equilibrium points :

" '÷ :c:*. }

Meg :-c foo )-

Linearsystenssxlts- Axel * BUGS

• Explicit solutions at

XCES ( for;np:{integrable

")• Straightforward analysis atstability

• Easy to design controllers (stabilizing ,• Can Serve as local

Optimal,etc .)

approximations to non- linear systems

""