Download Una propuesta de control por modo deslizante para convertidores

REVISTA INGENIERÍA UC. Vol. 11, No 2, 26-38, 2004
Una propuesta de control por modo deslizante para convertidores
electrónicos de potencia
Rubén Rojas, Alfredo R. Castellano, Ramón O. Cáceres, Oscar Camacho
Postgrado en Automatización e Instrumentación. Av. Tulio Febres Cordero. Facultad de Ingeniería.
Escuela de Ingeniería Eléctrica, ULA. Mérida – Venezuela
Email: [email protected]
Resumen
Este artículo describe la síntesis de un controlador por modo deslizante basado en un modelo lineal
reducido de segundo orden usando una superficie integro-diferencial de error, con salida aproximadamente
continua. A diferencia de estrategias similares, los parámetros de sintonización guardan estrecha relación con la
dinámica del sistema en términos de especificaciones convencionales de respuesta transitoria. El controlador
propuesto requiere solamente la realimentación de la variable de salida del sistema y podría ser empleado
satisfactoriamente en control de sistemas eléctricos no lineales de una entrada y una salida, tal como el caso de los
convertidores electrónicos de potencia a PMW (modulación por ancho de pulsos).
Palabras clave: Modo deslizante, control no lineal, convertidor de potencia, PWM.
A sliding mode control proposal for electronic power converters
Abstract
This article describes the synthesis of a sliding mode controller SMCr based on a reduced second order
linear model using an integral-differential surface of the tracking-error. Different from similar strategies, the
tuning parameters keep a close relationship with the system dynamics in terms of conventional specifications of
transient response. The proposed controller only needs the output feedback of system and could be satisfactorily
used in control of single input-single output nonlinear electric systems such as electronic power converters with
pulse-width modulation.
Keywords: Sliding mode, nonlinear control, power converter, Pulse-width modulation
1. INTRODUCTION
Numerous electrical systems exhibit a behavior
whose dynamics can be represented by a single inputsingle output (SISO), second order model. However,
such approach discards process inherent nonlinearities that sometimes would cause degradation of
control system, such as when the conventional PID
strategy is used. This becomes more evident when it
is working in extreme conditions or in zones where the
parametric uncertainty of the model becomes more
accented.
dynamics of the system in the sliding mode have stimulated development of multiple procedures for the
synthesis of controllers in a wide spectrum of applications. Such as are the cases of processes with multiinput / multi-output configuration [1], with strong non
linearities [2], with variable dead time [3] or non minimum phase [4]. In addition, excellent results have
been reported in control of electric motors [5] and
electronic power converters [6, 7] where discontinuous action in Variable Structure Control (VSC) is
compliant with the nature of their elements.
In this framework, robustness proper of the sliding mode control strategy (SMC) should provide
better performance than conventional control strategies. Properties such as order reduction and invariant
The main drawback in VSC is the chattering
generally associated with a high control activity
that sometimes could not be tolerated by the system.
26 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
Rojas, Castellano, Cáceres y Camacho
It could excite high frequency unmodeled dynamics
or decrease their efficiency [5, 8]. This last aspect
highlights the convenience of synthesizing the SMCr
under an approximately continuous control law [9].
Based on the SMC robustness, this article
shows the synthesis of a controller based on an integral-differential surface of the error and a continuous
approximation of nonlinear part of the controller. The
design is pointed to obtain a simple structure that only
needs the output feedback making it attractive for
control of nonlinear electric systems such as power
converters with schemes of pulse width modulation
(PWM).
In a sliding mode control strategy, the system
dynamics is forced by the controller to stay confined
in a subset of the state space denominated sliding surface, σ(t). The system is directed toward and reaches
the sliding surface at a finite time due to the control
action. Finally, once the system dynamics reach the
user-chosen sliding surface, it will behave according
to that surface, which is of a lower order than that of
the system and independent of the model parametric
uncertainties [10].
Let M be the second order model of system
(lower order approximation of the real system Q),
with an output variable θ1(t) that tracks the reference
input θr(t) with a tracking error e1(t) under control
law U(t) of the SMCr . The dynamics of the controlled system M can be described by the following state
space equation:
.
θ 1 (t ) = θ 2 (t )
.
(2)
y (t ) = e1 (t )
here, f(t) contains the external disturbances m(t), the
reference signal qr(t) and its derivatives. Whenever
the external disturbance acts in the same state space
of the control U, a control Uf will exist such that:
bU
f
(t ) = f (t )
(3)
and the sliding mode existence will be ensured if a
known upper bound, the supreme F(t), for f(t) exists
such that for any instant t:
2. SYSTEM MODELING
θ 2 (t ) = −a1θ1 (t ) − a2θ 2 (t ) + m(t ) + bU (t )
y (t ) = θ1 (t )
.
e1 ( t ) = e 2 ( t )
.
e 2 (t ) = − a1e1 (t ) − a 2 e2 (t ) + f (t ) + bU (t )
(1)
where a1, a2, b are the parameters of the second order
model approximation of the system and m(t)
represents the external disturbances. This equation
can be rewritten in terms of the tracking error as
follows:
f (t ) < F (t )
(4)
3. CONTROLLER SYNTHESIS
The desired performance of the system is
usually described in terms of the transient response
specifications. This performance must be satisfied in
presence of disturbances and set point changes. The
transient response specifications are thoroughly used
in conjunction with PID control to regulate systems
with dominant second order dynamics. This approach
is used in this article to evaluate the system response.
In general, the most used set of specifications for the
transient response can be summarized as follows:
S1.
S2.
S3.
S4.
Steady State Error, ess.
Percent Overshoot, Mp.
Peak Time, Tp.
Settling Time, Ts.
The design problem consists first on choosing
the sliding surface so that the system exhibits the
desired dynamics defined by the given specifications;
and second to guarantee the conditions so that the
system reaches that surface at a finite time. There are
many options to choose s(t); the Sliding Surface
selected in this work is an integral-differential
equation acting on the tracking-error [11], which is
represented by the following expression:
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
27
Una propuesta de control por modo deslizante
⎞
⎛d
+ λ⎟
⎠
⎝ dt
σ (t ) = ⎜
∫
t
(5)
e(t )dt
de (t )
+ 2λ1λ0 e(t ) + λ0
dt
(9)
∀ t ≥ t0
0
The advantage of this surface is that it contains
an integral term, which ensures the cancellation of the
steady state error. Then s(t) is a function of the error
between the reference, qr, and the output, q1, values as
was described in Eq. (2), such that:
σ (t ) =
σ& (t ) = e& 2 (t ) + 2 λ 1 λ 0 e 2 (t ) + λ 0 2 e1 (t ) = 0
2
t
∫ e (t )dt = 0
(6)
0
λ0 , λ1 > 0
represents the system dynamics in the sliding mode.
The satisfaction of expression s(t) = 0 means that in
steady state the tracking-error of the system should
decrease to zero with a dynamics that depends on the
selection of parameters l0 and l1.
For all SISO systems [10], the condition for
existence of a sliding mode is satisfied if:
σ (t ) σ& (t ) < 0
(7)
Geometrically, this inequality means that the
time derivatives of the state error vector always point
toward the sliding surface when system is in reaching
mode, and therefore, the system dynamics will
approach to the surface dynamics in a finite time. To
satisfy the given specifications S1–S4 and Eq. (6), an
augmented equivalent control law [1] was used:
U ( t ) = U eq ( t ) + U N ( t )
(
)
As can be appreciated in Eq. (10), Ueq(t)
evidences a proportional-derivative nature, attenuated
by the static gain of the model. Finally, the chattering
problem could be solved satisfactorily if the control
UN(t) is designed according to Zinober [12]:
U N (t ) = K d
σ (t )
The continuous part Ueq(t) could be determined
supposing that in the instant t0, the state trajectory of
the system intercepts the surface and enters the sliding
mode. The existence of sliding mode implies that
Eq. (6) is satisfied, so:
(11)
; Kd ,δ > 0
σ (t ) + δ
where Kd is a tuning parameter to assure the reaching
of the sliding surface and d is set to obtain
suppression of chattering. An estimate of the Kd
parameter could be obtained from Eq. (3) and Eq. (4).
The overall control law structure is shown in
Figure 1.
d/dt
+
2λ0 λ1
e
∫ dt
+
σ (t)
Σ
+
Kd × ( )
abs( ) + δ
UN
+
λ02
Σ
U
+
(8)
Ueq(t) is denominated the Equivalent Control,
and represents the continuous part of the controller
that maintains the output of the system restricted to
the sliding surface. UN(t) is the nonlinear part of the
controller that ensure to reach the sliding surface and
therefore, it should satisfy the inequality given
in Eq. (7).
28 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
Then the equivalent control Ueq(t) is obtained
from Eq. (9), after substituting the error derivatives by
the state equations (2), when the system is not
disturbed (f (t) = 0):
1
2
Ueq ( t ) = ⎡ λ0 − a1 e1 ( t ) + ( 2 λ0 λ1 − a2 ) e2(t ) ⎤ (10)
⎣
⎦
b
d/dt
2λ0 λ1 – a2
+
Σ
1/ b
+
Ueq
λ02 – a1
Figure 1. Structure of the proposed SMCr.
4. CONTROLLER TUNING
In sliding mode, system dynamics depends on
the sliding surface σ(t), and therefore the desired
transient response will depend on (10) and (11)
parameters selection. Both parameters could be
determined solving Eq. (6). Depending on values
range in (11), it is possible to obtain three different
solutions for the set of specifications:
Rojas, Castellano, Cáceres y Camacho
If 0 < λ1 < 1; λ0 > 0
4.1
In this case, the dynamics of the surface will
present a behavior strongly underdamped and steadystate error will approach to zero after an oscillation
around the equilibrium point (see Figure 2). The
response of the system can be evaluated through the
following equations:
e ss = 0
Ts ≈
Mp = 13.53%
(18)
Tp =
exp
α
⎛ 2λ β ⎞
−⎜ 1 ⎟
⎝ α ⎠
sin β
2
(19)
λ0
5.4
Ts =
(20)
λ0
(12)
100
Tp = 2
(17)
1
(13)
λ0 = 15
λ1 = 1
0.8
β
λ0α
Tracking error e(t)
Mp =
e ss = 0
(14)
4
(15)
λ0 λ1
0.6
0.4
Mp = 13. 53 %
Tp = 0. 13 s
0.2
Ts = 0. 36 s
0
Mp
-0.2
α = 1 − λ12
⎛α ⎞
; β = tan −1 ⎜ ⎟
⎝ λ1 ⎠
(16)
-0.4
Ts
Tp
0
0.1
0.2
0.3
0.4
0.5
0.6
Time t , (s)
Figure 3. Dynamics of the sliding surface when λ1 = 1.
Notice that the percent overshoot Mp only
depends on λ1.
1
Tracking error e(t)
In this case, the system response can be
evaluated through the following equations:
λ0 = 15
λ1 = 0.7
0.8
0.6
Mp = 21. 03 %
0.4
Tp = 0. 15 s
0
Mp
-0.2
Ts
Tp
-0.4
0
0.1
0.2
0.3
0.4
0.5
e ss = 0
(21)
−⎜ −1⎟ Ln ⎜
⎟
50 ⎡
⎢( λ1 − α ) exp ⎝ α ⎠ ⎝ λ1 −α ⎠
Mp =
α ⎢
⎣
⎛ λ ⎞ ⎛ λ +α ⎞ ⎤
− ⎜ 1 +1⎟ Ln ⎜ 1
⎟
α
λ −α
− ( λ1 + α ) exp ⎝ ⎠ ⎝ 1 ⎠ ⎥
⎥⎦
⎛ λ1
Ts = 0. 33 s
0.2
If λ1> 1; λ0 > 0
4.3
0.6
Time t , (s)
Figure 2. Dynamics of the sliding surface when 0 < λ1 < 1.
⎛ λ1 +α ⎞
(22)
Tp =
⎛ λ +α ⎞
Ln ⎜ 1
⎟
λ0α
⎝ λ1 − α ⎠
(23)
Ts ≈
⎛ 50 ( λ1 − α ) ⎞
Ln ⎜
⎟
λ0 ( λ1 − α ) ⎝ α em % ⎠
(24)
4.2 If λ1= 1; λ0 > 0
In this case, the system response will present a
moderate underdamped behavior (see Figure 3).
Because Mp only depends on λ 1 , it is constant
(13.53%), and the response of the system can be
evaluated through the following equations:
⎞
1
1
α = λ12 − 1
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
(25)
29
Una propuesta de control por modo deslizante
where em% is the specified error band inside which Ts
is specified. The response, Figure 4, exhibits a
reduced overshoot but a larger settling time.
response in terms of peak time or settling time. This
tuning procedure is straightforward and it is as
simpler as a PID controller tuning.
Due to overdamping, the settling time will be
severely influenced by the specified error band.
Damping ratio associated with λ1 parameter is shown
in Figure 5. Note that overshoot only could be
reduced if λ1 >> 1.
Another interesting aspect is the strong
relationship between the controller parameters and the
system natural response. In Eq. (1), the system
approximated by the second order model should
contain the dominant poles of Q, so that an alternative
representation of M could be obtained in terms of
natural frequency ωn and damping radio ξ from the
original system at any operation point, such that:
1
λ0 = 15
λ1 = 1. 5
Tracking error e(t)
0.8
a1 = ω n
0.6
Mp = 7. 56 %
Tp = 0. 12 s
Ts = 0. 37 s
0.4
0.2
-0.2
-0.4
a 2 = 2ξω n
Mp
0
Ts
Tp
0
0.1
0.2
0.3
0.4
0.5
0.6
Time t , (s)
Figure 4. Dynamics of the sliding surface when λ1> 1.
Percent Overshoot Mp, (%)
14
12
10
8
Mp = 7. 56%
6
4
2
0
1
1.5
2
2.5
3
3.5
4
4.5
2
5
λ1 parameter
Figure 5. Overshoot -λ1 relationship.
(26)
(27)
From Eq. (10) it is evident that l0 is related with
ωn and λ1 with ξ. As usual, the controlled system must
respond faster than the open-loop system and
therefore, it is desirable that λ0 > ω n.
According to Eqs. (13), (18) and (22), the
percent overshoot only depends on λ1, which provides
a straight way to achieve the desired percent
overshoot in system response. For instance, choosing
λ1 > ξ is convenient if ξ < 0.707. Then, next step is to
adjust λ0 to provide the desired response in terms of
settling time or peak time, which are totally
determined according to the λ1 range.
A reasonable value for Kd could be determined
assuming that in the meantime of an external
disturbance or when step change is introduced, the
sliding surface value s(t) >> d. Therefore, the
nonlinear part of the controller UN approximates
Kdsign(s(t)) and its magnitude will be Kd. In order to
guarantee the reaching condition in any circumstance,
from Eq. (3) and Eq. (4), it is enough that:
Kd >
1
F (t )
b
(28)
As shown in the above results for the three
cases, the relationship between the controller
parameters and the transient response of the closeloop system is evident. This facilitates the controller
tuning considerably. For instance, if a desired
overshoot should be lower than 8 % , λ1 >1.5 would
be a suitable election (see Figure 5).
For systems with saturated single input, the
superior estimate F(t), in Eq. (4), is given by the
control input bounds:
Once defined the percent overshoot Mp, the
parameter λ0 is selected to satisfy the desired time
because is impossible to allow, without loss of
stability, a disturbance in the system that requires a
30 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
f ( t ) < b (U max − U min )
(29)
Rojas, Castellano, Cáceres y Camacho
5. EXAMPLES
In order to show the proposed controller
performance and its parameter tuning simplicity, a
couple of examples of SISO systems, increasing in
complexity, are presented in this section, beginning
with the original system state variables description
and their inherent characteristic dynamics. When the
order of the original system is higher (greater than
two) a second order model approximation around the
operation point is determined for tuning the controller.
Finally, the proposed control strategy is evaluated by
simulation using the original state variable description
in MATLAB and the controller with the parameters
set at the previous established tuning; in each case the
resulting response is compared an contrasted with the
predicted response given by the controller equations.
5.1 A boost converter
In general, if energy conversion is performed in
continuous conduction mode, the boost converter
can be described in the average state space according
to:
diL ( t ) Vs 1
=
− v0 ( t ) (1 − u ( t ) )
dt
L L
dv0 ( t )
1
1
=−
v0 ( t ) + iL ( t ) (1 − u ( t ) )
dt
RC
C
d (s)
=−
R
2⎞
⎛
⎜ s − (1 − D ) ⎟
L
⎝
⎠
2
⎛ 2
1− D)
(
1
s+
⎜s +
⎜
RC
LC
⎝
Vs
RC (1 − D )
2
Switch PWM
L
D’T
2
3
Vs
1
+
DT
where iL(t) is the inductor average current, v0(t) is the
output voltage, R is the load in ohms, vs is the
external dc source voltage and u(t) is the control
input. This control input is assumed as saturated
continuous function in the open interval (0, 1).
In order to derive an approximate model of the
converter, the electronic switch is replaced by the
PWM switch [13], resulting in the circuit shown in
Figure 6.
Substituting the PWM switch by its
fundamental frequency model, around a stationary
point of operation D, the dynamics of system in Eq.
(30) is described approximately as follows:
+
R
C
v0
–
Figure 6. Equivalent circuit of the boost converter.
The behavior of the open-loop system
corresponding to reference step changes and load
disturbances is shown in Figure 7.
21.5
25
21
20.5
(30)
⎞ (31)
⎟
⎟
⎠
where d(s) represents the incremental duty cycle.
Equation (31) characterizes a non-minimum phase
system with respect to the output voltage, and
therefore, it is difficult to control.
Output voltage v0 , (V)
Finally, the parameter d (always positive) is
adjusted to suppress the chattering.
v0 ( s )
Output voltage v0 , (V)
control magnitude larger than the range of values of
U(t) for an indefinite period of time.
20
0.5
0.6
0.7
0.8
Time t , (s)
0.9
1
20
15
10
0
(a)
100 to 50Ω
in t = 0. 5 s
0.5
50 to 100Ω
in t = 1 s
1
Time t , (s)
1.5
(b)
Figure 7. Open loop response of the converter against a:
(a) Reference step change response (20 - 21V)
(b) Step change in the load resistor (±50%R).
To obtain the tuning parameters of the
controller, the system in Eq. (31) is described in terms
of the natural response of the approximate model M:
ωn =
(1 − D ) ξ
LC
=
LC
2 RC (1 − D )
(32)
where D is the steady state duty cycle. A parameter
summary is shown in Table 1, according
to the following circuit elements: inductor, 170 mH
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
31
Una propuesta de control por modo deslizante
capacitor, 1000 µF; resistor, 100 Ω; and power supply
voltage, 10 Vdc. The duty cycle D is computed for an
output voltage of 20 V, (D = 0.5).
Table 1. Model parameters M(s).
Parameter
Value
b
5.882x104
a1
1.471x103
a2
10
ωn , natural frequency
38.35
ξ , damping ratio
0.130
Lower limits for l0 and l1 are immediately
obtained from the natural response:
λ0 > ω n = 38.35
λ1 > ξ = 0.130
(33)
Considering the closed-loop gain restriction,
that imposes an unstable zero dynamics of the system,
is advisable to choose moderate gains, for instance,
λ0 = 1.05 ωn or 1.1 ωn and then increase λ1 from a
value equal to ξ until the desired response is achieved.
The gain Kd is obtained from the control input bounds
(0 < u < 1). Hence, to reach the sliding surface is
enough to choose Kd = 1.
In systems with high natural frequency, λ0 will
be larger than λ1 and therefore, the instantaneous
value of the surface s(t) could reach an unusual
magnitude. Because d y s(t) have the same order of
magnitude, for practical implementation purposes, the
sliding surface s(t) can be normalized with respect to
the natural frequency without degrading the UN
behavior.
5.2 A Buck-boost converter
The buck-boost DC-AC converter is an inverter
of commuted mode that works in for quadrants. It is
characterized by its capacity to generate sinusoidal
voltages that can be greater or lower than the DC
supply voltage, depending on the duty cycle. Its main
advantage is that it does not need a second power
conversion stage [8].
The voltage control on this inverter topology is
32 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
a troublesome task due to the inherent nonlinear
behavior of the system and the sensitivity to abrupt
load changes that eventually could drive it to
instability, particularly when classical PID control
methods are used [13]. Control is even more difficult
because of the non minimum phase characteristic of
the system.
The design and experimentation of a boost type
similar converter show very satisfactory experimental
results following a sliding mode control strategy with
a frequency modulation scheme [8]. To reach the
control goal, nevertheless, two controllers (one for
each branch of the inverter) and four state variables
feedback (two voltages and two currents for each
branch), were needed. In the proposed control
strategy, the control input is a continuous signal to be
used beside a PWM auxiliary circuit with a fix
commutation frequency.
The DC-AC conversion is obtained using two
DC-DC
bidirectional
buck-boost
converters,
modulating the output voltage in a sinusoidal way in
counter-phase as it is shown by the basic electrical
scheme of Figure 8. Each converter generates a
unipolar sinusoidal voltage with a DC component.
This unipolar component of the voltage DC is
suppressed by the differential connection of the load.
S1
Vs
+
S3
L1
u
PWM
S2
L2
C1
–
C2
v0 +
S4
R
Figure 8. The Buck-boost DC-AC converter.
The average state variable representation of the
converter is shown in Figure 9 according with the
following relationships [14]:
diL1 1
⎛ Vs − vC1 ⎞
= vC1 + ⎜
⎟u
dt
L
⎝ L ⎠
dvC1 1 ⎛ vC 2 − vC1
⎞ i
= ⎜
− i L1 ⎟ + L1 u
dt
C⎝
R
⎠ C
(34)
Rojas, Castellano, Cáceres y Camacho
diL 2 1
⎛ Vs − vC 2 ⎞
= Vs + ⎜
⎟u
dt
L
L ⎠
⎝
(35)
dvC 2 1 ⎛ vC1 − vC 2 ⎞ iL 2
= ⎜
⎟− u
dt
C⎝
R
⎠ C
v0 = vC 2 − vC1
+ Va
where iL1 and iL2 are the average currents in L1 and
L2 inductors, respectively, vC1 are vC2 the average
voltages through C1 and C2, respectively. Vs is the DC
external voltage supply, R is the load connected to the
converter, v0 is the average voltage on the load, and u
is the average control input.
–
X
u
0<u<1
X
•
1
L1
Σ
+
Σ
–
+
1
C1
vC1
+
X
( )
∫
•
1
L2
Σ
iL2
iL2
∫
v0
+
X
X
+
•
1
C2
Σ
–
vC2
vC2
∫
After replacing each pair of commuters S1-S4,
S2-S3 by the PWM switch to the equivalent diagram
of the inverter is obtained as it is shown in Figure 10.
During the time interval, DT, energy is stored on the
L1, inductor, simultaneously the storage energy L2 is
transferred to the RC net. Similarly, during the
following interval, D’T, energy is transferred from the
L1 inductor to the RC net while energy is stored on L2
inductor.
a
2
1
D’T
3
Vs
–
C1
L1
v0
b
+
vC1
–
D’T
1
2
DT
R
+
+
+
Vs
3
Ib
1
Vs
1− D
(36)
1
Vs
D
(37)
Vb = −
2D−1 Vs
2
D(1− D) R
Ib = −
Figure 9. Average state variable model of the converter.
DT
–
+
DVb
Ia
Ia =
–
DVa
3
Va =
vC1
Σ
1
Vs
2
DIb
The following equations are obtained from the
steady state model:
iL1
∫
–
+
R
iL1
•
–
X
–
+
+
–
1
R
DIa
+ Vb
+
Figure 11. Buck-boost converter steady state model.
+
+
X
1
V0
Σ
+
–
–
–
2
3
+
vC2
–
+
Vs
C2
L2
(38)
2D −1 Vs
D2 (1− D) R
(39)
V0
2D − 1
=
Vs D (1 − D )
(40)
Equation (40) defines the nonlinear static gain
of the converter shown in Figure 12. It can be
observed that the duty cycle is constrained to the open
interval (0,1) and the output voltage is zero when
D = 0.5.
4
Voltage Static Gain, v0 / Vs
Vs
Assuming converter operates on the DC
conduction mode during all the commutation period T
and this commutation occurs on a sufficiently high
frequency, the average model of the PWM switch is
used to obtain the steady state model (see, Figure 11).
2
0
-2
-4
0
0.2
0.4
0.6
0.8
1
Duty Cycle, D
Figure 10. Buck-boost converter equivalent diagram.
Figure 12. Static gain characteristic.
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
33
Una propuesta de control por modo deslizante
Converter fundamental frequency model is
obtained replacing the PWM switch by its equivalent
around steady state (Figure 13). The converter control
input is obtained by adding the incremental variation
d and the steady state duty cycle D.
+ va –
2
– v0
D ia
+
1
D ib
R
–
+ D va
d Ia
1
–
+ d Va
ia
3
+ vb –
d Ib
+
+
vC1
vC2
–
–
2
–
D vb +
d Vb
–
+
ib
L
3
L
(
z1 = −L D2 Ia + (1 − D) Ib
2
)
P2 = RLC(1−2D+2D2 )
z 2 = LC ( (1 − D ) Va − DVb )
p3 = 2 L2 C
z 3 = − L2 C ( I a + I b )
p 4 = R ( LC )
2
Figure 14 show the root locus of Q(s) around an
operation point defined in V0 = 200 V (D=0.707) with
L = 1 mH, C = 10 µF, R = 60 ohm, Vs = 100 Vdc.
From Eq. (44) it can be observed that the system
presents right half-plane zeros that causes a nonminimum phase response respect to the output
voltage.
Figure 13. Buck-boost converter
fundamental frequency model.
1.5
x j 104
Analyzing the above model in the frequency
domain the following equations are obtained:
v0 ( s ) = v a ( s ) + vb ( s )
(41)
⎛ (1 − D ) Va − I a L s ⎞
va ( s ) = d ( s ) ⎜
⎟
⎜ (1 − D )2 + LC s 2 ⎟
⎝
⎠
Imaginary Axis
1
0.5
0
-0.5
-1
(42)
-1.5
-0.5
0
0.5
1
1.5
2
⎛
⎞
Ls
−v0 ( s ) ⎜
⎟
2
⎜ R (1 − D ) + LC s 2 ⎟
⎝
⎠
(43)
⎛
⎞
Ls
⎟
−v0 ( s ) ⎜
⎜ R ( D 2 + LC s 2 ) ⎟
⎝
⎠
The non linear characteristic of the static gain,
Eq. (40), causes the open loop output voltage
distortion during a sinusoidal reference tracking as it
is shown in Figure 15.
300
Vref: 250 Vpk
R: 60 ohm
Equations from (41) to (43) represent the open
loop converter dynamics around the operation point
D. The transfer function Q(s) = vo(s)/d(s), derived
from Figure 13 model [12], is given by:
z3 s 3 + z 2 s 2 + z1 s + z 0
p 4 s 4 + p3 s 3 + p 2 s 2 + p1 s + p0
(44)
Output Voltage v0, (V)
200
where:
3
x 10 4
Figure 14. Q(s) root locus.
⎛ DV + I L s ⎞
vb ( s ) = − d ( s ) ⎜ 2 b b 2 ⎟
⎝ D + LC s ⎠
Q (s) = k
2.5
Real Axis
100
0
-100
-200
-300
0
10
20
30
40
50
Simulation time t , (ms)
k=R
p 0 = R (1 − D ) D 2
Figure 15. Static gain effect on the average output voltage.
z0 = D (1 − D) ( DVa − (1− D)Vb )
p1 = L (1 − 2 D + 2 D 2 )
Figures, 16 and 17, show the converter response against abrupt changes in load. Due to energy
2
34 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
Rojas, Castellano, Cáceres y Camacho
storage on the inductor and filters, the converter
experiments severe variations in the loop gain when
abrupt decrements in load occur. On the other hand, it
is less sensitive to overloads.
400
Output Voltage v0, (V)
Vref: 200 Vpk
200
-200
0
10
20
30
40
50
Simulation time t , (ms)
b=k
z0 2
ωn
p0
(47)
Vref: 200 Vpk
200
Table 2 Converter and system Q(s) parameters.
Converter Design Parameters
Figure 16. Step load change converter response
(from 60 to 6000 ohm).
Output Voltage v0, (V)
(46)
ξ
where Re(p) and Im(p) are the dominant pole, p, real
and imaginary part magnitudes, respectively, k, z0, p0
are the original system Q(s) coefficients. The
converter design parameters, Q(s) coefficients and the
parameters given by the root locus, are summarized in
Table 2.
0
-400
Re ( p )
ωn =
D
L1, L2
C1, C2
R
Vs
0.5
1 mH
10 µF
60 W
100 Vdc
System Q(s) coefficients for Vo = 200Vdc
(D=0.707)
100
k
0
z3 x10-11
z2x10-6
z1 x 10-3
z0
-6. 67
2.00
-5.29
58.58
p1x10
p0
60
-100
p4x10
-200
-15
p3x10
6.00
0
10
20
30
40
3.51
-4
5.86
2.57
Table 3 Model M(s) parameters.
Figure 17. Step load change converter response
(from 60 to 42 ohm).
M(s) model parameters obtain by root locus
5.2.1 Controller tuning
The control aim is a 60 Hz sinusoidal reference
tracking to generate output voltages from 0 to 200
volts approximately, using a supply source of 100
Vdc.
As it was described in the controller synthesis
procedure, first of all the system is approximated by a
second order model that contains the dominant poles
of Q(s). The model parameters static gain, b, natural
frequency, ωn, and damping coefficient, ξ , can be
estimated using linear systems simplification
techniques such as the root locus as follows:
⎞
⎟
⎟⎟
⎠
2.00
p2x10
-7
50
Simulation Time t , (ms)
2
⎛
⎛ Im ( p ) ⎞
⎜
ξ = 1 + ⎜⎜
⎟
⎜⎜
Re ( p ) ⎟⎠
⎝
⎝
-11
−1
(45)
Re (p)
895
Im (p)
2 898
ξ
0. 295
ωn
3 033
K
1. 256x1010
a1
9. 199x106
a2
1. 789x103
Figure 18 shows the frequency response of the
original system Q(s) and the second order
approximation M(s).
Using the second order model parameters, the
tuning parameters range for l0 and l1 are defined as:
λ0 > ω n = 3.033
(48)
λ1 > ξ = 0.295
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
35
Una propuesta de control por modo deslizante
even better than the predicted by the tuning equations.
The ITAE performance index, Figure 8 (b), becomes
constant confirming the annulment of steady state
error.
Q(s)
-50
100
M(s)
101
102
103
104
21.5
105
106
0
M(s)
Mp = 12.93 %
Tp = 0.092 s
Ts = 0.186 s
ess = 0
20
Q(s)
30
Ts
20.5
-360
ess = 0
ess
21
0.5
-720
0.6
0.7
0.8
Time t , (s)
0.9
20
10
0.5
1
0.6
(a)
10 2
10 3
10 4
10 5
10 6
Frequency (rad/s)
Figure 18. Open loop systems frequency responses.
Considering the non-minimum phase
characteristic (that constrains the system direct loop
gain) and the calculated natural frequency, the
procedure to follow is to choose initial moderated
gains as l0 = 1.1 ω n (or 1.2 ω n ) . Then, increase
progressively the l1 parameter from a base value,
equal to ξ , up to reach another value that guaranties a
satisfactory reference tracking. The gain of the
controller nonlinear part, Kd, is obtained from the
control input signal bounds (in this case: 0 < u < 1).
The sufficiency condition to reach the sliding surface
is satisfied if:
As in the case of the buck converter, the
controller must include a DC component Ubias = 0.5 to
cancel the output voltage.
6. RESULTS
6.1 Boost converter
Figure 19, shows the system step response
when a reference step change is introduce at t = 0.5 s.
It shows a small overshoot and a short settling time
36 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
1
A good system response behavior to load
disturbances is shown in Fig. 20 (a), when the output
voltage is set to 20 V. The load disturbances are
simulated by step changes of ±50% of the nominal
load. Figure 20 (b) shows that the steady state error is
annulled in similar times.
25
0.3
ess= 0
20
50 to 100Ω
in t = 1 s
15
0
0.5
1
Time t , (s)
0.2
ess= 0
0.1
100 to 50Ω
in t = 0. 5 s
ess= 0
10
0
1.5
0
(a)
0.5
1
Time t , (s)
1.5
(b)
Figure 20. Attenuation of load disturbances, step changes
± 50% nominal load. ITAE performance index.
1
0.8
0.6
0.4
0.2
0
Using the average description according to Eq.
(30), the system was simulated using MATLAB with
step time equal to 1x10–4 s. The controller synthesis
was performed with the following parameters: l0 =
42.18 (1.1 ωn); l1 = 0.417 (3.2x); Kd = 1; d = 1.5; the
error surface was normalized with respect to a factor
of 10 ωn.
0.9
Figure 19. System step response, from 20 up to 21 V, in
t = 0.5 s. ITAE performance index.
(49)
Kd ≥ 1
0.7
0.8
Time t , (s)
(b)
ITAE
10 1
Inductor current iL , (A)
10 0
Output voltage v0, (V)
Phase φ , (degrees)
Frequency (rad/s)
-3
40 x 10
Mp ; Tp
ITAE
0
Output voltage v0, (V)
50
Control u (normalized)
Gain |G(jω)|, (dB)
100
0
0.5
1
Time t , (s)
(a)
1.5
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Time t , (s)
1.5
(b)
Figure 21. Load step change response, control input
magnitude (normalized), average inductor current.
Figure 21 shows the control signal and the
average inductor current when the load change was
simulated. The results are very similar to those
obtained using a SMC adaptive strategy (Escobar et
al., 1997) evaluated on experimental prototype of the
boost converter.
Rojas, Castellano, Cáceres y Camacho
6.2 Buck-boost converter
250
vref
200
0. 109 × 10–6
d/dt
+
0. 321 × 10–3
+
e
σN
Σ
+
+
Ubias
+
0. 5
Σ
+
d/dt
0. 154 ×
Output Voltage v0, (V)
0
-50
-100
-150
-250
0
10
20
30
40
50
Simulation Time t , (ms)
40
Vref: 200 Vpk
30
20
10
0
-10
-20
-40
u
Σ
Ueq
+
50
-30
+
0. 093 × 10–6
100
-200
UN
1. 0 × ( )
abs ( ) + 0. 35
∫
1. 210
v0
150
Tracking Error e (t), (V)
Using the average model, Figure 9, the system
was simulated using MATLAB with a sampling
period of 5 ms, following 60 Hz sinusoidal references
of amplitude 150, 200 and 250 V. The controller
realization is shown in Figure 22 with the tuning
parameters set as follows: l0 = 3.336 (1.1ωn); l1 =
2
0.443 (1.5 ξ ); Kd = 1; d = 0.35; sN = s/ ωn (sliding
2
surface normalized respect to ωn ). Figure 23, 24 and
25 shows the system tracking response for a 200 V
amplitude, 60 Hz sinusoidal reference signal against
abrupt load and DC voltage changes at t =21 ms,
when the system is more sensitive to disturbances (on
the top of the sinusoidal wave approximately).
0
10
20
30
40
50
Simulation time t , (ms)
Figure 24. System tracking response for a 60 Hz
sinusoidal, reference signal of amplitude 200 V, when
abrupt load was applied. (a) Step load change from 60 to 42
ohm. (b) Instantaneous tracking error.
10–3
Figure 22. Controller realization for the
buck-boost converter.
300
vref
v0
Output Voltage v0, (V)
200
100
Output Voltage v0, (V)
250
0
200
vref
150
v0
100
50
0
-50
-100
-150
-200
-250
-100
0
10
20
30
40
50
Simulation time t , (ms)
-200
0
10
20
30
40
50
40
Simulation time t , (ms)
Vref: 200 Vpk
30
40
Vref: 200 Vpk
20
Tracking Error e (t), (V)
30
Tracking Error e (t), (V)
20
10
0
-10
-20
-30
-40
0
-10
-20
-30
-40
-50
-60
10
0
10
20
30
40
50
0
10
20
30
40
50
Simulation time t , (ms)
Simulation time t , (ms)
Figure 23. System tracking response for a 60 Hz sinusoidal,
reference signal of amplitude 200 V, when abrupt load was
applied. (a) Step load change from 60 to 6000 ohm.
(b) Instantaneous tracking error.
Figure 25. System tracking response for a 60 Hz sinusoidal,
reference signal of amplitude 200 V, when abrupt DC
voltage changes were applied. (a) Step DC voltage change
from 100 to 90 V. (b) Instantaneous tracking error.
Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004
37
Una propuesta de control por modo deslizante
7. CONCLUSIONS
The designed SMC controller, based on an
integral-differential surface of the tracking-error,
performed very well when reference and load step
changes were introduced, with zero steady state error
and without chattering.
Its main attribute is the close relationship
between the controller tuning parameters and the
desired closed-loop transient response. The possibility
to obtain the desired percent overshoot unilaterally
(adjusting l1) allows achieving demanding agreements
between percent overshoot and settling time.
Since the control law has been synthesized
from a second order model of the system, a simple
structure controller structure is obtained, and it
requires only an output feedback loop.
Just one parameter (d) in the controller
structure is subject to the designer selection. Once a
satisfactory Kd/d relationship has been determined,
the sliding surface s(t) can be normalized with respect
to the natural frequency, without degrading the UN
behavior because it only depends on the magnitude
and sign of the sliding surface  s (t) with respect to the
numeric value of d. This aspect is very important for
practical implementation of the SMCr in analog
schemes.
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