Page 209 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 173

To guarantee Lyapunov stability the inequality _ V 2 # 0 has to be valid.
Again, c 2 . 0 is a design parameter. This can be achieved by setting the
term inside of the brackets to zero. Combining Eqs. (5.167) and (5.168)
leads to a duty cycle:

L f C f x 1 x 2 R f 2
d 5 1 1 ðc 2 1Þz 1 2 ðc 1 1 c 2 Þz 2
1
E L f C f L f C f
_
x 2 x 1 I d _ I d
1 2 2 1 Þ (5.169)
2
R L C 2 R C 2 R L C 2 C f
f L f f
The duty cycle is also the final control law for the virtual disturbance
Backstepping procedure. This leads to:
_ V 2 5 _ V 1 1 z 2 _z 2
2
1
ð
52 c 1 z 1 z 1 z 2 1 z 2 2 c 2 z 2 2 z 1 Þ (5.170)
2
2
52 c 1 z 2 c 2 z , 0; z 1 ; z 2 6¼ 0:
1 2
Therefore, ensuring Eq. (5.169) guarantees that the derivative of the
Lyapunov function V 2 is less than zero. According to the Lyapunov direct
method, this means that the system is asymptotically stable around the
point where the voltage equals the reference voltage. It is noteworthy that
the tuning of the design parameters c 1 and c 2 influences how much each
state overshoots. An example can be given by decreasing the design
parameter c 1 , which has as a consequence a slower convergence of the
voltage and less overshoot in the inductor current, since the weight of
error z 1 is also decreased.
A very interesting conclusion can be taken when rewriting (5.170):
_
2
2
2c 1 z 2c 2 z 5V 2 #2 2θV 2 ; (5.171)
1 2 θ 5 minðc 1 ; c 2 Þ
Therefore (5.171) is described by an exponential function, which
means that V converges exponentially fast to zero, and as z 1;2 are the
error variables consequently the error converges exponentially fast to
zero. This has consequences for (5.158), which represents the deviation of
the bus voltage from V ref , where the steady-state error disappears. This
implies that the DC bus is tightly controlled. Extending this thought to
(5.164), which represents the current, implies that the disturbance current
I d , which is the reference, is perfectly tracked. Taking now into account
(5.163), (5.166), and (5.170), where V is positive definite and _ V is