ully distributed formation-containment control protocol for networked MASs with timevarying formation reference. Two detailed case studies are considered in Section 4.4 to show the effectiveness of the proposed methodology. One of them deals with the formationcontainment of a team of networked satellites, and the other one shows experimental validation using nonholonomic mobile robots. Section 4.5 concludes the chapter mentioning the future research directions

1. 1
DC−DC Buck Converter

2. 2
Objective – to efficiently reduce DC voltage
DC−DC Buck
Converter
+
Vin
−
+
Vout
−
Iout
Iin
Lossless objective: Pin = Pout, which means that VinIin = VoutIout and
The DC equivalent of an AC transformer
out
in
in
out
I
I
V
V


3. 3
Here is an example of an inefficient DC−DC
converter
2
1
2
R
R
R
V
V in
out



+
Vin
−
+
Vout
−
R1
R2
in
out
V
V
R
R
R



2
1
2

If Vin = 39V, and Vout = 13V, efficiency η is only 0.33
The load
Unacceptable except in very low power applications

4. 4
Another method – lossless conversion of
39Vdc to average 13Vdc
If the duty cycle D of the switch is 0.33, then the average
voltage to the expensive car stereo is 39 ● 0.33 = 13Vdc. This
is lossless conversion, but is it acceptable?
Rstereo
+
39Vdc
–
Switch state, Stereo voltage
Closed, 39Vdc
Open, 0Vdc
Switch open
Stereo
voltage
39
0
Switch closed
DT
T
Taken from “Course Overview” PPT

5. 5
Convert 39Vdc to 13Vdc, cont.
Try adding a large C in parallel with the load to
control ripple. But if the C has 13Vdc, then
when the switch closes, the source current
spikes to a huge value and burns out the
switch.
Rstereo
+
39Vdc
–
C
Try adding an L to prevent the huge
current spike. But now, if the L has
current when the switch attempts to
open, the inductor’s current momentum
and resulting Ldi/dt burns out the switch.
By adding a “free wheeling” diode, the
switch can open and the inductor current
can continue to flow. With high-
frequency switching, the load voltage
ripple can be reduced to a small value.
Rstereo
+
39Vdc
–
C
L
Rstereo
+
39Vdc
–
C
L
A DC-DC Buck Converter
lossless
Taken from “Course Overview” PPT

6. 6
C’s and L’s operating in periodic steady-state
Examine the current passing through a capacitor that is operating
in periodic steady state. The governing equation is
dt
t
dv
C
t
i
)
(
)
(  which leads to 



t
o
t
o
t
o dt
t
i
C
t
v
t
v )
(
1
)
(
)
(
Since the capacitor is in periodic steady state, then the voltage at
time to is the same as the voltage one period T later, so
),
(
)
( o
o t
v
T
t
v 

The conclusion is that






T
o
t
o
t
o
o dt
t
i
C
t
v
T
t
v )
(
1
0
)
(
)
(
or
0
)
( 

T
o
t
o
t
dt
t
i
the average current through a capacitor operating in periodic
steady state is zero
which means that

7. 7
Now, an inductor
Examine the voltage across an inductor that is operating in
periodic steady state. The governing equation is
dt
t
di
L
t
v
)
(
)
(  which leads to 



t
o
t
o
t
o dt
t
v
L
t
i
t
i )
(
1
)
(
)
(
Since the inductor is in periodic steady state, then the voltage at
time to is the same as the voltage one period T later, so
),
(
)
( o
o t
i
T
t
i 

The conclusion is that






T
o
t
o
t
o
o dt
t
v
L
t
i
T
t
i )
(
1
0
)
(
)
(
or
0
)
( 

T
o
t
o
t
dt
t
v
the average voltage across an inductor operating in periodic
steady state is zero
which means that
Taken from “Waveforms and Definitions” PPT

8. 8
Capacitors and Inductors
In capacitors:
dt
t
dv
C
t
i
)
(
)
( 
Capacitors tend to keep the voltage constant (voltage “inertia”). An ideal
capacitor with infinite capacitance acts as a constant voltage source.
Thus, a capacitor cannot be connected in parallel with a voltage source
or a switch (otherwise KVL would be violated, i.e. there will be a
short-circuit)
The voltage cannot change instantaneously
In inductors:
Inductors tend to keep the current constant (current “inertia”). An ideal
inductor with infinite inductance acts as a constant current source.
Thus, an inductor cannot be connected in series with a current source
or a switch (otherwise KCL would be violated)
The current cannot change instantaneously
dt
t
di
L
t
v
)
(
)
( 

9. 9
Vin
+
Vout
–
iL
L
C iC
Iout
iin
Buck converter
+ vL –
Vin
+
Vout
–
L
C
Iout
iin
+ 0 V –
Iout
0 A
• Assume large C so that
Vout has very low ripple
• Since Vout has very low
ripple, then assume Iout
has very low ripple

10. 10
The input/output equation for DC-DC converters
usually comes by examining inductor voltages
Vin
+
Vout
–
L
C
Iout
iin
+ (Vin – Vout) –
iL
(iL – Iout)
Reverse biased, thus the
diode is open
,
dt
di
L
v L
L 
L
V
V
dt
di out
in
L 

,
dt
di
L
V
V L
out
in 

,
out
in
L V
V
v 

for DT seconds
Note – if the switch stays closed, then Vout = Vin
Switch closed for
DT seconds

11. 11
Vin
+
Vout
–
L
C
Iout
– Vout +
iL
(iL – Iout)
Switch open for (1 − D)T seconds
iL continues to flow, thus the diode is closed. This
is the assumption of “continuous conduction” in the
inductor which is the normal operating condition.
,
dt
di
L
v L
L 
L
V
dt
di out
L 

,
dt
di
L
V L
out 

,
out
L V
v 

for (1−D)T seconds

12. 12
Since the average voltage across L is zero
      0
1 






 out
out
in
Lavg V
D
V
V
D
V
out
out
out
in V
D
V
V
D
DV 




in
out DV
V 
From power balance, out
out
in
in I
V
I
V 
D
I
I in
out 
, so
The input/output equation becomes
Note – even though iin is not constant
(i.e., iin has harmonics), the input power
is still simply Vin • Iin because Vin has no
harmonics

13. 13
Examine the inductor current
Switch closed,
Switch open,
L
V
V
dt
di
V
V
v out
in
L
out
in
L



 ,
L
V
dt
di
V
v out
L
out
L



 ,
sec
/
A
L
V
V out
in 
DT (1 − D)T
T
Imax
Imin
Iavg = Iout
From geometry, Iavg = Iout is halfway
between Imax and Imin
sec
/
A
L
Vout

ΔI
iL
Periodic – finishes
a period where it
started