Constant Switching Frequency Strategy

Practically, the switching power converters cannot operate at an infinite switching frequency and a hysteresis band is usually used by the SMC-based controllers.

However, the switching frequency of the hysteresis-type SMC-based controller is affected by line and load variations and thus, several methods as introduced in Chapter 1 have been proposed to modify the SMC-based controllers to control switching power converters at a constant switching frequency.

In order to achieve the same purpose, the bang-bang sliding motions inside the hysteresis region will first be discussed. Then, a brief review of existing methods for SMC-based controllers will be given. Finally, the simplest method of adding a periodic ramp signal to control bang-bang sliding motions at a constant frequency will be discussed.

A) Bang-Bang Sliding Motions Inside the Hysteresis Region

In the sliding mode theory, the input is assumed to be capable of switching at an infinite frequency such that the controlled system can be theoretically in the sliding mode. However, for the switching power converters, the switching devices cannot be operated with an infinite switching frequency and thus a hysteresis band is usually

introduced to avoid this unachievable requirement. While a hysteresis band is introduced around the s=0, the control law in (3.11) will become

⎩⎨

where ε is the width of the hysteresis band. In the steady-state operation, the bang-bang sliding motions inside the hysteresis region are illustrated in Fig. 3.14, in which f(z,u⁻) and f(z,u⁺) denote the vectors of state variable velocity with u⁻=1 and u⁺=0 respectively. The time taken for f(z,u⁻) to move from point P to point Q and the time taken for f(z,u⁻) to move from point Q to point R are denoted as Δt1 and Δt2, which can be calculated by

( )

^⋅

( )

⁻ approximately obtained as

L

From (3.38), fsw is mainly affected by the variations of input voltage and output voltage. Besides, if the resistive load is not exactly known (i.e., λ cannot be exactly set as 1/RC), it will also affect fsw. With the hysteresis band being used, the control scheme of the hysteresis-type bang-bang sliding mode controller is shown in Fig.

3.15.

grad(s)

s(z)>0 s(z)<0

t Q

f(z,u^-) f(z,u⁺)

P R

t1

Δ Δt₂

( )z =ε s

( )^{z =0}

s

( )z =−ε s

Fig. 3.14 Bang-bang sliding motions inside the hysteresis region.

R

v

_L

v

_C

v

_o

E

L

C i

_L

i

_C

i

_o

D

S

+ + + _

Hysteresis-Type Bang-Bang Sliding Mode Controller

α α γ β

Vref

s

_g

s

Fig. 3.15 Hysteresis-type bang-bang sliding mode controller

B) Existing methods for fixed frequency sliding mode controller

As discussed in previous paragraph, fsw is highly dependent on the input voltage, output voltage and the resistive load. To control the switching power converters at a constant fsw, several methods have been briefly introduced in Chapter 1. In [20], an extra state defined as the integral of output voltage error is used to formulate the specific sliding function, which results in a PID-type PWM controller. Then, the equivalent control ueq can be derived from the invariance conditions in the sliding mode. However, R is still involved in the representation of ueq such that ueq cannot be perfectly realized. Compared with [20], the sliding function in [24] is designed as the function of inductor current, output voltage and the integral of output voltage error.

With this sliding function, the system is initially in the sliding mode, i.e., s(t=0)=0. As a result, ueq can be theoretically obtained for all time. Significantly, an adaptive law is proposed to eliminate the effect of the un-modeled parasitic resistance in the circuit.

In [21], the idea of adopting a variable hysteresis band adjusted by adaptive feedforward and feedback control is proposed. The adaptive feedforward loop is used to reduce the deviation of fsw resulting from the input voltage variation. As for the adaptive feedback loop, it is used to adjust the coefficients of the sliding function such that the deviation of fsw resulting from the load variation can also be reduced. Actually, the measuring of input voltage is needed for the adaptive feedforward control scheme to directly change the width of hysteresis band ε, and the measuring of the output load current io is also needed for the adaptive feedback control scheme to compute the resistive load R. The simplest method is that proposed in [26], which is implemented by adding a periodic ramp signal and a PI-type compensator into the hysteresis-type sliding mode controller. This method possesses the advantages that fsw is constant under all operation conditions and it can be directly controlled by varying the frequency of the periodic ramp signal. Therefore, it will be used in this dissertation

and some design considerations will be given in the next.

C) Hysteresis-Type Bang-Bang Sliding Mode Controller at fixed-frequency

First, define a periodic ramp signal sramp with the amplitude, frequency and period being denoted as εramp, framp and Tramp respectively. Then, add sramp into the original control scheme as the block diagram shown in Fig. 3.16, in which sc is the summation of s and sramp, sg is the gate driver signal and εs is the hysteresis band of s. The ideal steady-state waveforms of s, sramp and sc are given Fig. 3.17. From Fig. 3.17, it can be found that these signals should satisfy the following equations:

ε ε

Note that (3.39) and (3.40) are used to guarantee that S will switch off during each cycle and it can immediately switch on at the beginning of each cycle. To satisfy (3.40), εramp is typically chosen to be larger than ε.

While a converter operates at a constant switching frequency, Δton/Tramp is equivalent to the duty cycle d and (3.39) can be expressed as

( )

^{ε ε}

ε_s+ 2d−1 _ramp = (3.41)

Besides, (3.40) can be rewritten

ε ε

ε − <−

− _{s ramp} (3.42)

Obviously, when fsw is synchronizing to framp, the bang-bang sliding motions inside the hysteresis region will be governed by (3.41) and (3.42). From (3.41), if the averaging value of s is zero, εs will be smaller than ε when d>0.5, and εs will be larger than ε when d<0.5. The waveforms with d>0.5 have been shown in Fig. 3.17 and the waveforms with d<0.5 are given in Fig. 3.18 as a comparison.

R

v

L

v

C

v

o

E

L

C i

L

i

C

i

o

D S

+ + + _

α α γ β

Vref

+ +

s

s

c

Hysteresis-Type Bang-Bang Sliding Mode Controller at Constant Frequency

s

ramp

s

g

Fig. 3.16 Hysteresis-type bang-bang sliding mode controller at switching constant frequency.

sc

sramp

s

εramp

＋

εramp

−

( ) s

sz =−ε

( ) s

sz =ε

( )z =0 s

ε

− +ε

t ton

Δ Δtoff

Fig. 3.17 Waveforms of the signals inside the hysteresis region (d>0.5).

sc

sramp

s

εramp

＋

εramp

−

( ) s

sz =−ε

( ) s

sz =ε

( )z =⁰ s

ε

− ε +

t ton

Δ Δtoff Δton

Fig. 3.18 Waveforms of the signals inside the hysteresis region (d<0.5).

Ideally, sramp should not affect the averaging value of s. But, it may deteriorate s such that the averaging value of s will be a nonzero value ε0. In this situation, the waveforms are non-ideal as given in Fig. 3.19, where ε0=0.5(εs1−εs2), and εs1 and εs2

are not necessary to be positive. The equations of (3.41) and (3.42) will become

( )

ε ε

ε_s1+ 2d−1 _ramp = (3.43)

and

ε ε

ε − <−

− _{s2 ramp} (3.44)

The waveforms with positive ε0 have been shown in Fig. 3.19 and the waveforms with negative ε0 are also given in Fig. 3.20 as a comparison. Compared with Fig. 3.18, d is smaller when ε0 is positive, and d is larger when ε0 is negative. Thus, a nonzero ε0

may lead to steady-state error in the output voltage. In order to reduce the steady-state error, a PI-type compensator will be further incorporated into the control scheme and the block diagram is shown in Fig. 3.21.

In next section, numerical simulations based on the control scheme in Fig. 3.16 will

first be given to show the effectiveness of adding an additional periodic ramp signal for achieving constant switching frequency. Then, a PI-type compensator will further be incorporated into the controller and numerical simulations will demonstrate the reduction of the steady-state error in the output voltage.

sc

sramp

s

εramp

＋

εramp

−

( ) s2

sz =−ε

( ) s1

sz =ε

( )z =ε0

s

ε

− +ε

t ton

Δ Δtoff

( )z =0 s

Fig. 3.19 Waveforms of the signals inside the hysteresis region (ε₀>0).

εramp

＋

εramp

−

( ) s2

sz =−ε

( ) s1

sz =ε

( )z =ε0

s

−ε +ε

t ton

Δ Δtoff

( )z =0 s

sc

sramp

s

Fig. 3.20 Waveforms of the signals inside the hysteresis region (ε₀<0).

R

v

_L

v

_C

v

_o

E

L

C i

_L

i

_C

i

_o

D

S

+ + + _

α α γ β

Vref

PI Compensator

+ +

s

s

_c

Hysteresis-Type Fixed-Frequency Bang-Bang Sliding Mode Controller

s

_ramp

s

g

Fig. 3.21 Hysteresis-type fixed-frequency bang-bang sliding mode controller with a PI-type compensator.

在文檔中 Bang-Bang順滑控制在切換式電源轉換器之設計 (頁 39-47)