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
""