CN104734505A - Voltage-current double closed-loop terminal sliding mode control method of Buck converter - Google Patents

Abstract

The invention relates to a voltage-current double closed-loop terminal sliding mode control method of a Buck converter. The voltage-current double closed-loop terminal sliding mode control method of the Buck converter aims to solve the problems that the traditional linear sliding mode control method is slow in response speed and low in stability precision, and includes the following steps: 1 mathematics model building of the Buck converter; 2 load observer design according to output voltage VC of the Buck converter and the loaded current iL; 3 voltage terminal sliding mode controller design: a voltage controller outputs an inductive current demand signal iL* according to a demand input direct-current voltage tracking error ev; 4 current linear sliding mode controller design: a current controller outputs a control signal mu of a controllable switch tube V of the Buck converter according to the inductive current demand signal iL*. The voltage-current double closed-loop terminal sliding mode control method is applied to the electricity field.

Description

Buck converter voltage-current double closed-loop TSM control method

Technical field

The present invention relates to converter voltage-current double closed-loop TSM control method.

Background technology

(1) conventional linear sliding-mode control has Asymptotic Behavior For Some, affects response speed and the stable state accuracy of Buck converter output voltage.

Different sliding-mode control determines the difference change of capacitance voltage and inductive current in Buck converter, then directly affects the quality of its output voltage.At present, Buck converter is mainly to apply traditional linear sliding mode method, but its convergence is asymptotic and there is steady-state error, makes the continuous convergence of system mode but forever cannot arrive given trace, therefore directly affecting response speed and the precision of Buck converter output voltage.

(2) Mathematical Modeling Methods many employings State-space Averaging Principle of current Buck converter, modeling accuracy is not high and affect its voltage output quality.

Buck converter comprises the energy-storage travelling wave tube such as electric capacity and inductance, the switching devices such as MOSFET, IGBT on circuit is formed, and belongs to typical nonlinear time_varying system.Modeling method directly affects design and the control performance of controller, but the model of current Buck converter many employings approximate linearization, namely suppose that disturbance is very little, in a switch periods, state variable weighting is averaging, cancellation time-variant nonlinear item and the continuous time model obtained, therefore only can ensure the local stability of system.

Summary of the invention

The present invention will solve conventional linear sliding-mode control response speed and the low problem of stable state accuracy, and provides Buck converter voltage-current double closed-loop TSM control method.

Buck converter voltage-current double closed-loop TSM control method, it realizes according to the following steps:

One, the Mathematical Models of Buck converter;

Two, according to the output voltage V of Buck converter _cwith the current i by load _l, carry out Load Torque Observer design;

Three, the design of voltage termination sliding mode controller: voltage controller is according to given input direct voltage tracking error e _voutputting inductance given value of current signal i _l ^*;

Four, the design of the linear sliding mode controller of electric current: inductive current Setting signal i followed the tracks of by current controller _l ^*, export the control signal u of Buck converter controlled tr tube V.

Invention effect:

Kirchhoffs law is utilized to set up the unified Differential Model of Buck converter under switch turns on and off two kinds of situations, and according to the different control overflow of capacitance voltage and inductive current, under load resistance unknown situation, utilize the circuit characteristic of TSM control method Flexible Control Buck converter, to strengthen the robustness of system to unknown load disturbance, improve the quality of output voltage.

1) the TSM control method adopted has finite time convergence control characteristic, and fast response time higher than traditional linear sliding mode control method control precision.2) realize the Flexible Control to capacitance voltage (output voltage) and inductive current simultaneously, and then make Buck converter voltage and electric current all be stabilized in desired value.3) use based on the circuit differential equation under kirchhoffs law, reduce the conservative of Controller gain variations.

Accompanying drawing explanation

Fig. 1 is Buck converter circuit figure in embodiment two;

Fig. 2 is Buck converter System with Sliding Mode Controller figure in embodiment one;

Fig. 3 (a) is DC input voitage E adopts the control signal u of the controlled tr tube V under two close cycles TSM control method and conventional linear sliding-mode control when there is disturbance handoff procedure figure in emulation experiment;

Fig. 3 (b) is DC input voitage E load resistance in emulation experiment when there is disturbance measured value figure;

Fig. 3 (c) is inductive current convergence oscillogram DC input voitage E adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 3 (d) is output voltage convergence oscillogram DC input voitage E adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 4 (a) is sliding-mode surface reference voltage adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 4 (b) is reference voltage adopts the control signal u of the controlled tr tube V under two kinds of methods when there is disturbance handoff procedure figure in emulation experiment;

Fig. 4 (c) is inductive current convergence oscillogram reference voltage adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 4 (d) is output voltage convergence oscillogram reference voltage adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 5 (a) is sliding-mode surface load voltage adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 5 (b) is load voltage adopts the control signal u of the controlled tr tube V under two kinds of methods when there is disturbance handoff procedure figure in emulation experiment;

Fig. 5 (c) is inductive current convergence oscillogram load adopts two kinds of methods when there is disturbance in emulation experiment under;

Fig. 5 (d) is output voltage convergence oscillogram load adopts two kinds of methods when there is disturbance in emulation experiment under;

Embodiment

Embodiment one: present embodiment 1, Buck converter voltage-current double closed-loop TSM control method, it is characterized in that it realizes according to the following steps:

Buck converter voltage-current double closed-loop TSM control method, it realizes according to the following steps:

One, the Mathematical Models of Buck converter;

Two, according to the output voltage V of Buck converter _cwith the current i by load _l, carry out Load Torque Observer design;

Three, the design of voltage termination sliding mode controller: voltage controller is according to given input direct voltage tracking error e _voutputting inductance given value of current signal i _l ^*;

Four, the design of the linear sliding mode controller of electric current: inductive current Setting signal i followed the tracks of by current controller _l ^*, export the control signal u of Buck converter controlled tr tube V.

Embodiment two: present embodiment and embodiment one unlike: step one is specially:

As shown in Figure 1, wherein E is the direct voltage source of input to the circuit theory diagrams of Buck converter, and V is controlled tr tube, and its operating state u represents, V _cfor output voltage, VD is afterflow diode, and L is filter inductance, and C is filter capacitor, and R is load resistance, and Vc is output voltage, i _lfor inductive current.

First analyze Buck converter at the circuit characteristic of controlled tr tube V in " open-minded " and " shutoff " two kinds of situations, the mode of operation of its correspondence represents with u=1 and u=0 respectively:

(1) when controlled tr tube V conducting, i.e. u=1, afterflow diode VD bears reverse biased and ends, input DC power E connects with inductance L, be now the accumulation of energy stage, based on kirchhoffs law, the differential equation obtained when Buck converter is opened is

$\left\{\begin{array}{c}\frac{d{i}_L}{\mathrm{dt}}=\frac{1}{L}(E-V_C)\\ \frac{d{V}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(1\right)$

Wherein, described VD is afterflow diode, and L is filter inductance, and C is filter capacitor, and R is load resistance, and Vc is output voltage, i _lfor inductive current;

(2) and as u=0, controlled tr tube V ends, afterflow diode VD bears forward bias and conducting, and forms closed loop circuit with inductance L and load resistance R, is now freewheeling period.The differential equation that can obtain similarly when Buck converter turns off is

$\left\{\begin{array}{c}\frac{d{i}_L}{\mathrm{dt}}=-\frac{V_C}{L}\\ \frac{d{V}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(2\right)$

Combinatorial formula (1)-(2), then the uniform mathematical model of Buck converter under controlled tr tube V opens u=1 and turns off u=0 two kinds of mode of operations is

$\left\{\begin{array}{c}\frac{d{i}_L}{\mathrm{dt}}=\frac{1}{L}(\mathrm{uE}-V_C)\\ \frac{d{V}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(3\right).$

Other step and parameter identical with embodiment one.

Embodiment three: present embodiment and embodiment one or two unlike: the design of step 3 Load Torque Observer is specially:

Voltage of carrying-current double closed-loop Buck converter TSM control scheme as shown in Figure 2.Outer shroud is Voltage loop, and design (calculated) load observer overcomes the disturbing influence of unknown load resistance, and according to input direct voltage tracking error e _voutputting inductance given value of current signal i _l ^*, inner ring is electric current loop, and inductive current Setting signal i followed the tracks of by current controller _l ^*, export control signal u=1 or the u=0 of Buck converter controlled tr tube V.Lower mask body provides the design process of current controller and voltage controller.

Here the situation of actual control system load R the unknown is considered.The load supposing Buck converter is purely resistive load, and load rating value during system stable operation is R ₀, Load Torque Observer design is as follows

$\hat{R}=\left\{\begin{array}{ccc}\frac{V_C}{i_R},& \mathrm{if}& \frac{1}{2}{R}_0<\frac{V_C}{i_R}\\ \frac{1}{2}{R}_0,& \mathrm{if}& \frac{V_C}{i_R}<\frac{1}{2}{R}_0\end{array}\right.---\left(4\right)$

Wherein, i _rfor flowing through the electric current of load resistance R, for the measured value of load resistance, visible, the Load Torque Observer that through type (4) designs, can make output load current be limited in rated range, prevent overcurrent situations from occurring, and then the power device connect after protection Buck converter output terminal.

By the measured value of load resistance substitution formula (3), then Buck converter Mathematical Modeling is rewritten as

$\left\{\begin{array}{c}\frac{d{i}_L}{\mathrm{dt}}=\frac{1}{L}(\mathrm{uE}-V_C)\\ \frac{d{V}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{\hat{R}})\end{array}\right.---\left(5\right).$

Other step and parameter identical with embodiment one or two.

Embodiment four: one of present embodiment and embodiment one to three unlike: the design of step 3 voltage termination sliding mode controller is specially:

Definition direct voltage tracking error wherein, for output dc voltage reference value, design terminal sliding-mode surface is

$s_v=C{\stackrel{\&CenterDot;}{e}}_v+\mathrm{C\&beta;sig}{n}^{q/p}{e}_v---\left(6\right)$

In formula, design parameter β >0, p and q is odd number, and 1<p/q<2;

System arrives and maintains terminal sliding mode face s _vtime, i.e. s _v=0, then the dynamic behaviour of direct voltage tracking error system is expressed as

$C{\stackrel{\&CenterDot;}{e}}_v+\mathrm{C\&beta;sig}{n}^{q/p}{e}_v=0---\left(7\right)$

Known in phase plane, (0,0) is terminal attractors, namely has s _v=0, substitute into formula (5), can obtain

$\mathrm{C\&beta;sig}{n}^{q/p}(V_C^{*}-V_C)+C\frac{d{V}_C^{*}}{\mathrm{dt}}+\frac{V_C}{\hat{R}}-i_L=0---\left(8\right)$

Calculate voltage controller outputting inductance given value of current signal i _l ^*for

$i_L^{*}=\mathrm{C\&beta;sig}{n}^{q/p}(V_C^{*}-V_C)+C\frac{d{V}_C^{*}}{\mathrm{dt}}+\frac{V_C}{\hat{R}}---\left(9\right)$

According to the finite convergence characteristic of TSM control, suppose that direct voltage follows the tracks of initial value e by mistake _v(0) ≠ 0, then by obtain system mode converge to the finite time t of zero _s

$t_s=\frac{p}{\mathrm{\&beta;}(p-q)}{\left|e_v\left(0\right)\right|}^{(p-q)/p}---\left(10\right).$

Other step and parameter identical with one of embodiment one to three.

Embodiment five: one of present embodiment and embodiment one to four unlike: the design of the linear sliding mode controller of step 2 (three) electric current is specially:

Definition current track error can derive current track error system by formula (5) is

$e_{\mathrm{iL}}=\frac{d{i}_L^{*}}{\mathrm{dt}}-\frac{d{i}_L}{\mathrm{dt}}=\frac{d{i}_L^{*}}{\mathrm{dt}}+\frac{V_C}{L}-\frac{E}{L}u---\left(11\right)$

Here linear sliding mode face is directly chosen

s _iL＝e _iL(12)

The object of design is: due to when inner ring current track error is in sliding-mode surface s _iL=0, e _iLafter converging to zero, from formula (9), outer shroud direct voltage tracking error system also will enter sliding formwork state s simultaneously _v=0;

For ensureing linear sliding mode face s _iLsliding-mode surface s is arrived in finite time _iL=0, then demand fulfillment sliding formwork reaching condition according to s _iLpositive and negative situation, be divided into following two kinds of situation discussion:

1) s is worked as _iLduring >0, then demand fulfillment namely by formula (11), then need the control signal u=1 of visible now controlled tr tube V;

2) s is worked as _iLduring <0, then demand fulfillment namely the control signal u=0 of visible now controlled tr tube V;

Comprehensive above two kinds of situations, can derive and ensure that the constraints of Buck converter steady operation is

$0<L\frac{d{i}_L^{*}}{\mathrm{dt}}+V_C<E---\left(13\right)$

The control signal u that then can obtain controlled tr tube V is

$u=\frac{1}{2}(1+\mathrm{sign}{s}_i)---\left(14\right)$

The switching frequency of real system breaker in middle pipe can not be infinitely fast, therefore adopts boundary layer method to revise formula (14), namely have

$u=\frac{1}{2}(1+\mathrm{sign}{s}_i)\text{=}\left\{\begin{array}{cc}1& \left|s_i\right|>\mathrm{\&Delta;}\\ 0& \left|s_i\right|<\mathrm{\&Delta;}\end{array}\right.---\left(15\right)$

In formula, Δ is the width in boundary layer.

Other step and parameter identical with one of embodiment one to four.

Emulation experiment:

For prove put forward the validity of Buck converter two close cycles TSM control method, by carrying out simulation comparison with conventional linear sliding-mode control.The circuit parameter of Buck converter is: inductance L=27mH, electric capacity C=120 μ F, load resistance R=20 Ω, DC input voitage E=20V, nominal load resistance R ₀=20 Ω, output voltage is

Linear sliding mode controller adopts conventional method, namely with inductive current tracking error for variable directly constructs linear sliding-mode surface, wherein

$s_l=i_L^{*}-i_L=\frac{V_c}{\hat{R}}-i_L---\left(16\right)$

The control signal u of corresponding controlled tr tube V is

$u=\frac{1}{2}(1+\mathrm{sign}{s}_l)\text{=}\left\{\begin{array}{cc}1& \left|s_l\right|>\mathrm{\&Delta;}\\ 0& \left|s_l\right|<\mathrm{\&Delta;}\end{array}\right.---\left(17\right)$

Wherein, width Delta=0.01 in boundary layer.Visible, conventional linear sliding-mode control (16)-(17) are the output realizing rated voltage by controlling inductive current, but the non-linear energy-storage travelling wave tube in Buck converter circuit also comprises electric capacity, the change of its voltage also must affect the performance of output voltage, reduces the control performance of Buck converter.

Compare down, the TSM control dominance of strategies of the Buck converter of voltage carried here-current double closed-loop structure is embodied in: 1) realize the Flexible Control to capacitance voltage (output voltage) and inductive current simultaneously, and then make Buck converter voltage and electric current all be stabilized in desired value; 2) the TSM control method adopted has finite time convergence control characteristic, and fast response time higher than traditional linear sliding mode control method control precision.Wherein, design parameter is: β=900, q=3, p=5, width Delta=0.01 in boundary layer.

Consider respectively below at input direct voltage E, output reference voltage V _c* these the three kinds of situations that there is disturbance with load resistance R carry out simulation comparison.

Situation 1: DC input voitage E exists disturbance, supposes DC input voitage saltus step between 20V and 40V from t=0.05s, and every 0.025s saltus step once

Fig. 3 (a) is respectively the handoff procedure figure of the control signal u of the controlled tr tube V under employing two close cycles TSM control method and conventional linear sliding-mode control, visible owing to adopting boundary layer method to make switching tube there is necessary switching delay, cause load resistance measured value (as Fig. 3 (b)) be delayed acquisition too; Fig. 3 (c)-(d) is inductive current in two kinds of situations and output voltage convergence waveform, all stationary value can be converged to as seen, and be not subject to the disturbing influence of DC input voitage E, but the response speed under two close cycles TSM control method is adopted obviously to want fast.

Situation 2: output reference voltage V _c* there is disturbance, suppose V _c* saltus step between 10V and 15V from during t=0.02s, and every 0.02s saltus step is once.

As can be seen from Fig. 4 (a), fast than conventional linear sliding formwork of the convergence rate of terminal sliding mode, and output reference voltage V _c* disturbance can make system depart from sliding-mode surface; Fig. 4 (b) is respectively the handoff procedure figure of the control signal u of the controlled tr tube V under employing two kinds of methods; Fig. 3 (c)-(d) is inductive current in two kinds of situations and output voltage convergence waveform, all can converge to stationary value as seen, but under two close cycles TSM control method, inductive current conversion amplitude is large, and fast convergence rate.

Situation 3: during load disturbance, supposes to add load disturbance from t=0.05s, makes load resistance saltus step between 20 Ω and 10 Ω, and every 0.025s change once

Fig. 5 is the simulation result contrast of load R two kinds of methods when there is disturbance, and the analysis result deposited in disturbance cases with Fig. 4 output reference voltage is similar.

Claims (5)

1.Buck converter voltage-current double closed-loop TSM control method, is characterized in that it realizes according to the following steps:

One, the Mathematical Models of Buck converter;

Two, according to the output voltage V of Buck converter _cwith the current i by load _l, carry out Load Torque Observer design;

Three, the design of voltage termination sliding mode controller: voltage controller is according to given input direct voltage tracking error e _voutputting inductance given value of current signal i _l ^*;

Four, the design of the linear sliding mode controller of electric current: inductive current Setting signal i followed the tracks of by current controller _l ^*, export the control signal u of Buck converter controlled tr tube V.

2. Buck converter voltage according to claim 1-current double closed-loop TSM control method, is characterized in that step one is specially:

(1) when controlled tr tube V conducting, i.e. mode of operation u=1, afterflow diode VD bears reverse biased and ends, input DC power E connects with filter inductance L, for the accumulation of energy stage, based on kirchhoffs law, the differential equation obtained when Buck converter is opened is

$\left\{\begin{array}{c}\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{1}{L}(E-V_C)\\ \frac{{\mathrm{dV}}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(1\right)$

Wherein, described VD is afterflow diode, and L is filter inductance, and C is filter capacitor, and R is load resistance, and Vc is output voltage, i _lfor inductive current;

(2) and as mode of operation u=0, controlled tr tube V ends, afterflow diode VD bears forward bias and conducting, and forms closed loop circuit with filter inductance L and load resistance R, and be freewheeling period, the differential equation obtained when Buck converter turns off is

$\left\{\begin{array}{c}\frac{{\mathrm{di}}_L}{\mathrm{dt}}=-\frac{V_C}{L}\\ \frac{{\mathrm{dV}}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(2\right)$

Combinatorial formula (1)-(2), then the uniform mathematical model of Buck converter under controlled tr tube V opens u=1 and turns off u=0 two kinds of mode of operations is

$\left\{\begin{array}{c}\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{1}{L}(\mathrm{uE}-V_C)\\ \frac{{\mathrm{dV}}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{R})\end{array}\right.---\left(3\right).$

3. Buck converter voltage according to claim 2-current double closed-loop TSM control method, is characterized in that the design of step 2 Load Torque Observer is specially:

The load supposing Buck converter is purely resistive load, and load resistance rated value during system stable operation is R ₀, Load Torque Observer design is as follows

$\hat{R}=\left\{\begin{array}{ccc}\frac{V_C}{i_R},& \mathrm{if}& \frac{1}{2}{R}_0<\frac{V_C}{i_R}\\ \frac{1}{2}{R}_0,& \mathrm{if}& \frac{V_C}{i_R}<\frac{1}{2}{R}_0\end{array}\right.---\left(4\right)$

Wherein, i _rfor flowing through the electric current of load resistance R, for the measured value of load resistance;

By the measured value of load resistance substitution formula (3), then Buck converter Mathematical Modeling is rewritten as

$\left\{\begin{array}{c}\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{1}{L}(\mathrm{uE}-V_C)\\ \frac{{\mathrm{dV}}_C}{\mathrm{dt}}=\frac{1}{C}(i_L-\frac{V_C}{\hat{R}})\end{array}\right.---\left(5\right).$

4. Buck converter voltage according to claim 3-current double closed-loop TSM control method, is characterized in that the design of step 3 voltage termination sliding mode controller is specially:

Definition direct voltage tracking error wherein, for output dc voltage reference value, design terminal sliding-mode surface is

$s_v=C{\stackrel{.}{e}}_v+\mathrm{C\&beta;}{\mathrm{sign}}^{q/p}{e}_v---\left(6\right)$

In formula, design parameter β >0, p and q is odd number, and 1<p/q<2;

System arrives and maintains terminal sliding mode face s _vtime, i.e. s _v=0, then the dynamic behaviour of direct voltage tracking error system is expressed as

$s_v=\mathrm{C\&beta;}{\mathrm{sign}}^{q/p}{e}_v+C{\stackrel{.}{e}}_v=\mathrm{C\&beta;}{\mathrm{sign}}^{q/p}({V_C}^{*}-V_C)+C(\frac{{{\mathrm{dV}}_C}^{*}}{\mathrm{dt}}-\frac{{\mathrm{dV}}_C}{\mathrm{dt}})=0---\left(7\right)$

? in phase plane, (0,0) is terminal attractors, namely has s _v=0, substitute into formula (5), can obtain

$\mathrm{C\&beta;}{\mathrm{sign}}^{q/p}(V_C^{*}-V_C)+C\frac{d{V}_C^{*}}{\mathrm{dt}}+\frac{V_C}{\hat{R}}-i_L=0---\left(8\right)$

Calculate voltage controller outputting inductance given value of current signal for

$i_L^{*}=\mathrm{C\&beta;}{\mathrm{sign}}^{q/p}(V_C^{*}-V_C)+C\frac{{\mathrm{dV}}_C^{*}}{\mathrm{dt}}+\frac{V_C}{\hat{R}}---\left(9\right)$

According to the finite convergence characteristic of TSM control, suppose that direct voltage follows the tracks of initial value e by mistake _v(0) ≠ 0, then by obtain system mode e _v, converge to the finite time t of zero _s

$t_s=\frac{p}{\mathrm{\&beta;}(p-q)}{\left|e_v\left(0\right)\right|}^{(p-q)/p}---\left(10\right).$

5. Buck converter voltage according to claim 4-current double closed-loop TSM control method, is characterized in that the design of the linear sliding mode controller of step 4 electric current is specially:

Definition current track error deriving current track error system by formula (5) is

$e_{\mathrm{iL}}=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}-\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}+\frac{V_C}{L}-\frac{E}{L}u---\left(11\right)$

Choose linear sliding mode face

s _iL＝e _iL(12)

Due to when inner ring current track error is in sliding-mode surface s _iL=0, e _iLafter converging to zero, enter sliding formwork state s by formula (9) outer shroud direct voltage tracking error system simultaneously _v=0;

For ensureing linear sliding mode face s _iLsliding-mode surface s is arrived in finite time _iL=0, then demand fulfillment sliding formwork reaching condition $s_{\mathrm{iL}}{\stackrel{.}{s}}_{\mathrm{iL}}<0;$

1) s is worked as _iLduring >0, then demand fulfillment namely by formula (11), then need ${\stackrel{.}{s}}_{\mathrm{iL}}=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}-\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}+\frac{V_C}{L}-\frac{E}{L}u<0,$ The control signal u=1 of controlled tr tube V;

2) s is worked as _iLduring <0, then demand fulfillment namely ${\stackrel{.}{s}}_{\mathrm{iL}}={\stackrel{.}{e}}_i=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}-\frac{{\mathrm{di}}_L}{\mathrm{dt}}=\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}+\frac{V_C}{L}-\frac{E}{L}u>0,$ The control signal u=0 of controlled tr tube V;

Derive and ensure that the constraints of Buck converter steady operation is

$0<L\frac{{\mathrm{di}}_L^{*}}{\mathrm{dt}}+V_C<E---\left(13\right)$

The control signal u obtaining controlled tr tube V is

$u=\frac{1}{2}(1+{\mathrm{signs}}_i)---\left(14\right)$

Adopt boundary layer method to revise formula (14), namely have

$u=\frac{1}{2}(1+{\mathrm{signs}}_i)=\left\{\begin{array}{cc}1& \left|s_i\right|>\mathrm{\&Delta;}\\ 0& \left|s_i\right|<\mathrm{\&Delta;}\end{array}\right.---\left(15\right)$

In formula, Δ is the width in boundary layer.