ECE 460
Lecture 1
Prerequisite
Laplace transforms
u (t ) L
1
s
t.u (t ) L
1
s2
12
1
.t .u (t ) L
3
2
s
1
e a.t .u (t ) L
s+a
b
( s + a) 2 + b 2
(s + a)
cos[b.t ].e a.t u (t ) L
(s + a) 2 + b 2
sin[b.t ].e a.t u (t ) L
1
Prerequisite
Electronic cir
ECE 460
Lecture 19
1
We want the closed loop system, with proper gain (K)
adjustment, to meet all four of these performance
requirements:
Ts = 1s
%OS = 20%
Stable
epos() = 102
2
Root Locus of
Antenna Position Control System
0.85
100
1.71
3
t
La
Ra
:
E
E
:>
>
d
E
E
E
s
H (s) =
&
K
D
m ( s)
ea ( s)
K
Y
W
1  t 1 
(a) x(t ) =
0
(b)
K
?
D
L
W
Z
t [0,2]
L
?
[10Pts]
otherwise
1
s 2 + 4 s + 20
[10Pts]
Z
d
Z
Z
K
D
& &
W
Z
,
Z
s
s
Wd^
&
t
,
K
y
&
D
K
ECE 460 Sample Quiz #2
1. Consider the closedloop control system below when the openloop system
K
G (s) =
:
( s + 1)( s + 7)
X(s)
G(s)
+
Y(s)
(a) If K=30, plot the precise location of the poles of the system and characterize the
stepresponse of the sy
ECE 460
Sample Quiz #3
1. Consider the following closedloop control system:
X(s)
Let G ( s ) =
+
Y(s)
K.G(s)
1
.
( s + 2)( s + 5) 2
A. Do a basic sketch of the rough locus of the poles of this closed loop system as a
function of K.
[10 Pts]
B. Refine th
ECE 460
Lecture 22
R(s)
+

K.G(s)
C(s)
We used the Bode plot of the openloop G(s) to
adjust K so that the closedloop system meets
certain performance requirements
%OS, epos(), Stability
Estimation of gain and phase margin is the key!
One more param
ECE 460 Sample Quiz #4
1.
Consider the transfer function G(s) =
25( s 2)
s( s 5)( s 10)
Plot the asymptotic magnitude and phase Bode plot for the function. In each of the plots, determine and
mark all the salient points along the abscissa and ordinate.
Qu
ECE 460
Lecture 21
R(s)
+

K.G(s)
C(s)
We want this system to perform:
%OS, epos(), Ts, Stability
But G(s) is unknown !
We do have an experimentally obtained frequency
response of G(s)
Bode plot !
How do we use the Bode plot to adjust K?
1
Typical
ECE 460
Lecture 20
1
The transfer function of the various system
components is not always known !
In others words we may dealing with a closed loop
control system:
R(s)
K.G(s)
+
C(s)

For which we do not know the plant G(s) !
2
How to proceed ?
Measu
ECE 460
Lecture 15
Lecture 13  basics of root locus
Lecture 14  first refinement of the root locus
Breakaway/in point(s)
This lecture is about additional refinements and
special cases
Complex poles/zeros
Angle of departure and arrival
Calibratio
ECE 460
Lecture 18
Compensating a deficient open loop system
(plant)
%OS
epos()
Lag compensation using root locus
1
Consider the following closedloop system
Suppose that we wish that this system perform as
follows:
1. %OS 20% (K 52)
2. epos() 0.01
ECE 460
Lecture 2
Control systems are all around us
Homes
Satellite
Heating/Cooling
Cars
Throttle/Steering/Braking
Cruise control
Airplane
Speed/Pitch/Yaw
Finance
Inflation
Liquidity
Biological
Pest
Disease
Environment
Pollution
1
Contro
ECE 460
Lecture 3
1
Example
( J1s 2 + D1s + K )
K
=
K
( J 2 s 2 + D2 s + K )
K is usually assumed to be zero, and the second bearing is
not always present
2
Freebody diagrams
3
Rotational systems often involve gears to
transmit force
Motion
Conservati
ECE 460
Lecture 5
Characterizing the behavior of a closed loop
control system
Step response
x(t)=u(t) X(s)=1/s
H ( s) =
Y (s)
X (s)
y(t) Y(s)
Performance measures
Overshoot (%OS)
Settling time (Ts)
Steady state error (e()
Stability (BIBO)
1
Location o
ECE 460
Lecture 7
Step response is dependent on pole locations
1
Stepresponse also determined by n and
2
2
n
n
b
=2
=
2
2
2
s + as + b s + 2 n s + n ( s + n ) 2 + n (1 2 )
n = 3
= 1.5
n = 3
= 0.33
n = 3
=0
n = 3
=1
2
Focus on underdamped system
ECE 460
Lecture 4
Electromechanical system (DC motor) transfer
function
K = K t /( Ra J m )
Kt
Dm + ( ) K b
Ra
=
Jm
m ( s)
K
=
Ea (s) s(s + )
1
N1 2
Jm = Ja + JL ( )
N2
N1 2
Dm = Da + DL ( )
N2
Ra
Tm + K bm = ea
Kt
TorqueSpeed Test
(Dynamometer)
(Dynam
ECE 460
Lecture 8
Step response is dependent on pole locations
1
Pole location, n, , %OS, and Ts
%OS = e
(
%OS = e
(
.
1
2
tan( )
)
)
100
100
4
Ts =
.n
4
Ts =
2
What happens when the system is higher order
Antenna position control system (additiona
ECE 460
Lecture 9
System stability
X(s)
+

G(s)
Y(s)
A system is stable if x(t) is bounded (energy or
power signal) implies that y(t) is also bounded
1
Systems with feedback have a particular
propensity for being unstable
2
The antenna position syste
ECE 460
Lecture 10
System stability
X(s)
+

G(s)
Y(s)
A system is stable if x(t) is bounded (energy or
power signal) implies that y(t) is also bounded
1
How do we ensure that a closed system is stable?
Poles of the system should be in the lefthalfp
ECE 460
Lecture 13
System performance
1
Adjust K in order to achieve
Desired %OS
Desired Ts
Desired e()
Stability
?
?
2
Lets look at %OS
Closedloop poles should lie at a (no more than)
desired angle w.r.t. to the ve realaxis
= cos (
1
ln(%OS / 10
ECE 460
Lecture 14
Root Locus Continued
Last lecture
Basic rules
This lecture
Rule refinement
Multiple examples
It is impossible to verify this for every s. Instead, we
seek a locus of the poles of the closedloop system T(s)
as a function of K
Ru
ECE 460
Lecture 11
Recall the antenna position control system
1
A case study of unityfeedback closedloop control
system
A system whose behavior can be dramatically
altered by just changing the value of K (gain)
gain
Stepresponse (%OS, Ts )
Stabili
ECE 460
Lecture 16
Further review of gain adjustment (calibration) using
root locus
%OS
Ts ?
Stability ?
1
Stability Using RootLocus
Consider the following closedloop system
Suppose that we wish to calculate the values of K,
using the rootlocus,