# Stable Switching Rule Design for Buck Converter via Affine Switched Systems

Authors: Jian Li; Jingjing Wang; Quankang Wang; Yingzhou Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper presents design and analysis of the switching rule based on affine switched systems, and simulation and experimental verifications are carried out for the Buck converter. Firstly, the switched system is modeled according to the dynamic characteristics of the Buck converter, and the affine switched systems are introduced. Secondly, a switching rule based on affine switched systems is introduced which takes the maximum Lyapunov function between the two sets of Lyapunov functions. Finally, the experiment verifies the closed loop control based on the Buck converter after the simulation. The real-time switching signal of the model is the real-time PWM waveform in the experiment. The experimental results show that the switching rule makes the system asymptotically stable.


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## Body

## 1. Introduction

In recent years, DC-DC converters have become more complex and have been widely used in many fields, especially in the fields of new energy power generation, electric vehicle, communication equipment, medical equipment, industrial instrument, and military instrument [1–5]. It is of great significance in security monitoring, energy saving, and environmental protection. Electric energy is applied in all aspects of life, which needs to be transformed into appropriate forms of electricity to serve people’s life perfectly. With the development of DC-DC converters, various control technologies are developing rapidly.DC-DC converter is a typical nonlinear time-varying system. Its output ripple voltage is large. The circuit is sensitive to parameter changes and easy to be affected by the environment. It is known that the anti-interference ability of DC-DC converter is poor. The improvement and optimization of topology can only improve the performance of the system to a small extent, but using advanced control theory to control the converter can make up for the deficiency of topology and make the whole system stable quickly.Compared with the traditional linear control strategy, various nonlinear control strategies are widely used in the field of control, such as fuzzy control [6], adaptive control [7], sliding mode control [8], predictive control [9], and so on. Many methods based on the Lyapunov function are applied in the control system [10–12]. However, few studies have focused on the maximum Lyapunov function [13, 14]. It is significant that the maximum Lyapunov function control rule is used in the experimental verification of the affine switched system. Relevant experimental verification is still in a blank state.In this paper, we use the Buck converter to prove the feasibility of the switching rule based on affine switched systems and complete the experimental verification. This experiment proves that the two states of the system are asymptotically stable. Firstly, the dynamic characteristics of the model are analyzed, and the model is built. Then, the method of Lyapunov function comparison is adopted to ensure the system asymptotically stable. Finally, the closed loop experiment is verified. The main contribution is to complete the experimental closed loop verification of the Buck converter by the switching rule.The remainder of the paper is described as follows. Section2 establishes the mathematical model. Section 3 describes the control scheme of the switched system. The simulation results and the experimental verification are presented in Section 4. Section 5 concludes the paper.


## 2. Preliminary

### 2.1. Circuit and Control Platform of Buck Converter

Figure1 shows the closed loop system based on the Buck converter. It includes two parts: circuit and control platform of the Buck converter. The sensing devices obtain current from the inductor branch and voltage from both ends of the capacitor. The switching rule is written in the controller, and the real-time inductive current and capacitor voltage of the circuit are read from the sensing devices. The controller generates real-time variable switching signals according to the switching rule, and controls the driver to generate actions for T.Figure 1
Closed loop system based on the Buck converter.


### 2.2. Modeling of Buck Converter

The mathematical model is described in this section for the Buck converter. The output voltage of the Buck converter is less than or equal to the input voltage. When the switch is switched on, the power supply supplies the load, and the inductor stores energy and inductor current increases. When the switch is off, the diode continues to flow, the inductor current decreases, and the inductor keeps the load running by releasing energy.According to Kirchhoff’s law of circuit and equivalent circuits of the Buck converter in Figure2, the following equation is obtained:(1)i˙Ltv˙Ct=0−1L1C−1RCiLtvCt+skL0Vit,where iLt is the value of inductive current, vCt is the value of capacitor voltage, and Vit is the supply voltage.Figure 2
Equivalent circuits of the Buck converter.
(a)
(b)The switched system is described as follows:(2)x˙t=Aσtxt+Bσtut,yt=Cxt,where xt is the state vector, yt is the output vector, Aσt is the system matrix, Bσt is the input matrix, ut is the input vector, C is the output matrix, and σt is the switching signal (see Table 1). The linear-switched state-space model of the Buck converter in the continuous conduction mode is given by:(3)xt=iLtvCt,ut=Vit,Aσt=0−1L1C−1RC,Bσt=skL0,C=1001.Table 1
Possible open/close switch positions for (T) and (D).
σ
t
1
2
T
1
0
D
0
1The Buck converter has two modes of working as shown in Figure2. Table 1 shows the switching signal σt=1, where sk=1 indicates open T and close D. Its working mode is shown in Figure 2(a). The second mode is the switching signal σt=2, where sk=0 indicates open D and close T in Figure 2(b).The switched system finally reaches the equilibrium pointx¯ by choosing the appropriate switching rule. Thus, the error system et=xt−x¯ is studied.The system is composed of two affine subsystems. The state-space expression for affine switched systems [15, 16] is shown as follows:(4)e˙t=Aσtet+kσt,kσt=Aσtx¯+Bσtut,yet=Cet,where kσt is the affine item. As et=0, affine switched system achieves asymptotic stability. The objective of this paper is to make the system (3) asymptotically stable under the action of the control signal and to adopt closed loop experiment to verify the feasibility of the theory.


## 2.1. Circuit and Control Platform of Buck Converter

Figure1 shows the closed loop system based on the Buck converter. It includes two parts: circuit and control platform of the Buck converter. The sensing devices obtain current from the inductor branch and voltage from both ends of the capacitor. The switching rule is written in the controller, and the real-time inductive current and capacitor voltage of the circuit are read from the sensing devices. The controller generates real-time variable switching signals according to the switching rule, and controls the driver to generate actions for T.Figure 1
Closed loop system based on the Buck converter.


## 2.2. Modeling of Buck Converter

The mathematical model is described in this section for the Buck converter. The output voltage of the Buck converter is less than or equal to the input voltage. When the switch is switched on, the power supply supplies the load, and the inductor stores energy and inductor current increases. When the switch is off, the diode continues to flow, the inductor current decreases, and the inductor keeps the load running by releasing energy.According to Kirchhoff’s law of circuit and equivalent circuits of the Buck converter in Figure2, the following equation is obtained:(1)i˙Ltv˙Ct=0−1L1C−1RCiLtvCt+skL0Vit,where iLt is the value of inductive current, vCt is the value of capacitor voltage, and Vit is the supply voltage.Figure 2
Equivalent circuits of the Buck converter.
(a)
(b)The switched system is described as follows:(2)x˙t=Aσtxt+Bσtut,yt=Cxt,where xt is the state vector, yt is the output vector, Aσt is the system matrix, Bσt is the input matrix, ut is the input vector, C is the output matrix, and σt is the switching signal (see Table 1). The linear-switched state-space model of the Buck converter in the continuous conduction mode is given by:(3)xt=iLtvCt,ut=Vit,Aσt=0−1L1C−1RC,Bσt=skL0,C=1001.Table 1
Possible open/close switch positions for (T) and (D).
σ
t
1
2
T
1
0
D
0
1The Buck converter has two modes of working as shown in Figure2. Table 1 shows the switching signal σt=1, where sk=1 indicates open T and close D. Its working mode is shown in Figure 2(a). The second mode is the switching signal σt=2, where sk=0 indicates open D and close T in Figure 2(b).The switched system finally reaches the equilibrium pointx¯ by choosing the appropriate switching rule. Thus, the error system et=xt−x¯ is studied.The system is composed of two affine subsystems. The state-space expression for affine switched systems [15, 16] is shown as follows:(4)e˙t=Aσtet+kσt,kσt=Aσtx¯+Bσtut,yet=Cet,where kσt is the affine item. As et=0, affine switched system achieves asymptotic stability. The objective of this paper is to make the system (3) asymptotically stable under the action of the control signal and to adopt closed loop experiment to verify the feasibility of the theory.


## 3. Control Scheme

Inspired by [17], the switching signal is obtained by the maximum value of multiple Lyapunov functions for the affine switched systems. The switching rule is described as(5)σet=argmaxvσet,where σet is two cases, so the switching signal can be calculated by comparing the largest Lyapunov function. The Lyapunov function is shown below:(6)vσet=etTPσet+2etTSσ,where Pσ∈R2×2 is a positive definite matrix and Sσ∈R2×1 is a matrix.DenoteVet=maxvσet and V˙et=e˙tTPσet+etTPσe˙t+2e˙tTSσ. In order to make the system (3) asymptotically stable, we have V˙et<0. Now, applying the S-procedure, we get(7)V˙et+2aθtVet−etTP¯et<0,taking into account that aθ=a1θ1+a2θ2>0 and θ1+θ2=1, where θ1≥0, θ2≥0 and 0≤a1≤α1, 0≤a2≤α2, α1, and α2 are the absolute values of the real part of the eigenvalues of the imaginary axes closest to A1 and A2, respectively. In addition, Pθ=θ1P1+θ2P2 and P¯=Pθθ¯. Aθ,Kθ,andSθ can follow the same principle from Pθ. In order to calculate easily, we convert (8) as(8)et1TAθTPθ+PθAθ+2aθPθ−P¯∗KθTPθ+SθTAθ+2SθTaθ2KθTSθet1<0.Therefore, (6) can be realized ifAθTPθ+PθAθ+2aθPθ−P¯∗KθTPθ+SθTAθ+2SθTaθ2KθTSθ<0, where Pθ and Sθ are obtained by linear matrix inequality (6) through simulation.The switching signal of the Buck converter can be obtained according to the following switching rule. Ifv1et is greater than v2et, σet is equal to 1, as shown in Figure 2(a). Otherwise, σet is equal to 2, as shown in Figure 2(b). The real-time PWM waveform is generated to control the Buck converter.


## 4. Simulation and Experimental Verification

In this section, the feasibility of the switching rule is verified by simulation and experiment. With Buck circuit as the model and DSP as the digital signal processor, we complete the closed loop control.


### 4.1. Simulation Results and Analysis

The switching rule achieves the effect that the state of Buck converter is asymptotically stable, and the switching frequency changes in real time according to the collected voltage and current. Inductance is4.5mh, when a certain margin is satisfied in the experimental design. Capacitance is 100μf that related to inductance and frequency, and the power resistance is 12ohm (see Table 2). The initial frequency is low and changes in real time. The frequency gradually increases when the state reaches a gradual stability.Table 2
Simulation parameters.
Inductance
4.5 mh
Resistance
12 ohm
Capacitance
100μf
Power supply
30 VIn order to verify the feasibility of the control scheme, we describe and analyze two cases. The objective of the first case is to achieve an effect where the output voltage is12V and the output power is 12W, when the input voltage is 30V. Pθ and Sθ are obtained by simulation according to the control scheme. They are described as follows:(9)P1=0.03900.00040.00040.0011,S1=−0.0138−0.0088,P2=0.0644−0.0025−0.00250.0018,S2=0.03280.0049.The objective of the second case is to achieve an effect where the output voltage is15.6V and the output power is 20.28W, when the input voltage is 30V. Pθ and Sθ are shown as follows:(10)P1=0.0173−0.0005−0.00050.0005,S1=−0.0091−0.0011,P2=0.0163−0.0004−0.00040.0004,S2=0.00840.0013.The simulation results of the first case show that the inductive current tends to be stable at about2.6ms in Figure 3(a), and the capacitor voltage tends to be stable at 2.9ms in Figure 3(b). The speed is fast, and the adjustment time is much less than 1 s. Finally, the inductive current is stable around 1A and the waveform is triangular wave, while the capacitor voltage is stable at 12V and the waveform is a straight line. The simulation results of the second case are shown in Figures 4(a) and 4(b). The inductive current reaches a stable value of 1.3A at 3.3ms, while the capacitor voltage is 15.6V at 4.5ms.Figure 3
Two states of the Buck converter. (a) The inductive current stabilizes at 1 A in 2.6 ms. (b) The capacitor voltage stabilizes at 12 V in 2.9 ms.
(a)
(b)Figure 4
Two states of the Buck converter. (a) The inductive current stabilizes at 1.3 A in 3.3 ms. (b) The capacitor voltage stabilizes at 15.6 V in 4.5 ms.
(a)
(b)


### 4.2. Experimental Verification

The structure of the system is shown in Figure5, including four parts: control unit, Buck converter, driver, and sampling module. Control unit is realized by DSP28335. Buck converter, driver, and sampling module are introduced in the circuit model. In this experiment, DSP is used as the digital signal processor, TLP250 is used as the driver, and hall element is used as the sampling module. The relevant parameters of the Buck converter are shown in Table 3. The complete experimental testbed setup is shown in Figure 5.Figure 5
Experimental verification platform.Table 3
Experimental parameters.
Inductance
4.5 mh
Capacitance
100μf
Diode
SB3100
Controller
DSP
Drive chip
TLP250
Mos
IRF540 N
Sampling chip1
LV25-P
Sampling chip2
LA55-P/SP50


#### 4.2.1. Control Unit

DSP28335 has played an important role in the fields of industrial control because of its high processing speed and abundant peripheral interfaces. The voltage and current of the Buck converter are collected through AD sampling pins. Closed loop real-time control can be realized programmatically by comparingv1 with v2. The control logic for programming is shown in Figure 6. The closed loop experimental verification combines the conversion between analog and digital quantities.Figure 6
DSP control logic diagram.


#### 4.2.2. Circuit Model

MOSFET can work to hundreds of KHZ, and its driver is simple; so, we choose IRF540N to switch the Buck converter. The driver adopts TLP250 to drive MOSFET. Schottky diode SB3100 is selected to meet the design requirements that the maximum current is3A with a withstand voltage of 100V.DSP is used to collect inductor current and capacitor voltage. Hall voltage sensing LV25-P is used to collect the voltage of the capacitor. The input resistance is 3 kohm, and the output resistance is120ohm. The capacitor voltage can be reduced by 10 times, and the AD pin in DSP can be collected. The inductive current adopts LA55-P/SP50, and the output resistance is 1kohm. The current can be reproduced into voltage and collected to another pin of DSP.


#### 4.2.3. Experimental Results and Analysis

The experimental results are proposed for the switching rule to ensure the stability of the system. We select the power resistance of12ohms and select two sets of images for discussion and analysis (see Figures 7 and 8). The average inductive current is 0.923A in Figure 7(a) and the capacitor voltage is equal to the load voltage with an average value of 11.6V in Figure 7(b), while the target value of current is 1A and the target value of voltage is 12V. The other set of images are shown in Figures 8(a) and 8(b). The average inductive current is 1.2A and the capacitor voltage is 15.1V, while the target values are 1.3A and 15.6V, respectively. The feasibility of the switching rule is verified by experiments.Figure 7
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 0.923 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 11.6 V.
(a)
(b)Figure 8
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 1.2 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 15.1 V.
(a)
(b)


## 4.1. Simulation Results and Analysis

The switching rule achieves the effect that the state of Buck converter is asymptotically stable, and the switching frequency changes in real time according to the collected voltage and current. Inductance is4.5mh, when a certain margin is satisfied in the experimental design. Capacitance is 100μf that related to inductance and frequency, and the power resistance is 12ohm (see Table 2). The initial frequency is low and changes in real time. The frequency gradually increases when the state reaches a gradual stability.Table 2
Simulation parameters.
Inductance
4.5 mh
Resistance
12 ohm
Capacitance
100μf
Power supply
30 VIn order to verify the feasibility of the control scheme, we describe and analyze two cases. The objective of the first case is to achieve an effect where the output voltage is12V and the output power is 12W, when the input voltage is 30V. Pθ and Sθ are obtained by simulation according to the control scheme. They are described as follows:(9)P1=0.03900.00040.00040.0011,S1=−0.0138−0.0088,P2=0.0644−0.0025−0.00250.0018,S2=0.03280.0049.The objective of the second case is to achieve an effect where the output voltage is15.6V and the output power is 20.28W, when the input voltage is 30V. Pθ and Sθ are shown as follows:(10)P1=0.0173−0.0005−0.00050.0005,S1=−0.0091−0.0011,P2=0.0163−0.0004−0.00040.0004,S2=0.00840.0013.The simulation results of the first case show that the inductive current tends to be stable at about2.6ms in Figure 3(a), and the capacitor voltage tends to be stable at 2.9ms in Figure 3(b). The speed is fast, and the adjustment time is much less than 1 s. Finally, the inductive current is stable around 1A and the waveform is triangular wave, while the capacitor voltage is stable at 12V and the waveform is a straight line. The simulation results of the second case are shown in Figures 4(a) and 4(b). The inductive current reaches a stable value of 1.3A at 3.3ms, while the capacitor voltage is 15.6V at 4.5ms.Figure 3
Two states of the Buck converter. (a) The inductive current stabilizes at 1 A in 2.6 ms. (b) The capacitor voltage stabilizes at 12 V in 2.9 ms.
(a)
(b)Figure 4
Two states of the Buck converter. (a) The inductive current stabilizes at 1.3 A in 3.3 ms. (b) The capacitor voltage stabilizes at 15.6 V in 4.5 ms.
(a)
(b)


## 4.2. Experimental Verification

The structure of the system is shown in Figure5, including four parts: control unit, Buck converter, driver, and sampling module. Control unit is realized by DSP28335. Buck converter, driver, and sampling module are introduced in the circuit model. In this experiment, DSP is used as the digital signal processor, TLP250 is used as the driver, and hall element is used as the sampling module. The relevant parameters of the Buck converter are shown in Table 3. The complete experimental testbed setup is shown in Figure 5.Figure 5
Experimental verification platform.Table 3
Experimental parameters.
Inductance
4.5 mh
Capacitance
100μf
Diode
SB3100
Controller
DSP
Drive chip
TLP250
Mos
IRF540 N
Sampling chip1
LV25-P
Sampling chip2
LA55-P/SP50


### 4.2.1. Control Unit

DSP28335 has played an important role in the fields of industrial control because of its high processing speed and abundant peripheral interfaces. The voltage and current of the Buck converter are collected through AD sampling pins. Closed loop real-time control can be realized programmatically by comparingv1 with v2. The control logic for programming is shown in Figure 6. The closed loop experimental verification combines the conversion between analog and digital quantities.Figure 6
DSP control logic diagram.


### 4.2.2. Circuit Model

MOSFET can work to hundreds of KHZ, and its driver is simple; so, we choose IRF540N to switch the Buck converter. The driver adopts TLP250 to drive MOSFET. Schottky diode SB3100 is selected to meet the design requirements that the maximum current is3A with a withstand voltage of 100V.DSP is used to collect inductor current and capacitor voltage. Hall voltage sensing LV25-P is used to collect the voltage of the capacitor. The input resistance is 3 kohm, and the output resistance is120ohm. The capacitor voltage can be reduced by 10 times, and the AD pin in DSP can be collected. The inductive current adopts LA55-P/SP50, and the output resistance is 1kohm. The current can be reproduced into voltage and collected to another pin of DSP.


### 4.2.3. Experimental Results and Analysis

The experimental results are proposed for the switching rule to ensure the stability of the system. We select the power resistance of12ohms and select two sets of images for discussion and analysis (see Figures 7 and 8). The average inductive current is 0.923A in Figure 7(a) and the capacitor voltage is equal to the load voltage with an average value of 11.6V in Figure 7(b), while the target value of current is 1A and the target value of voltage is 12V. The other set of images are shown in Figures 8(a) and 8(b). The average inductive current is 1.2A and the capacitor voltage is 15.1V, while the target values are 1.3A and 15.6V, respectively. The feasibility of the switching rule is verified by experiments.Figure 7
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 0.923 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 11.6 V.
(a)
(b)Figure 8
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 1.2 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 15.1 V.
(a)
(b)


## 4.2.1. Control Unit

DSP28335 has played an important role in the fields of industrial control because of its high processing speed and abundant peripheral interfaces. The voltage and current of the Buck converter are collected through AD sampling pins. Closed loop real-time control can be realized programmatically by comparingv1 with v2. The control logic for programming is shown in Figure 6. The closed loop experimental verification combines the conversion between analog and digital quantities.Figure 6
DSP control logic diagram.


## 4.2.2. Circuit Model

MOSFET can work to hundreds of KHZ, and its driver is simple; so, we choose IRF540N to switch the Buck converter. The driver adopts TLP250 to drive MOSFET. Schottky diode SB3100 is selected to meet the design requirements that the maximum current is3A with a withstand voltage of 100V.DSP is used to collect inductor current and capacitor voltage. Hall voltage sensing LV25-P is used to collect the voltage of the capacitor. The input resistance is 3 kohm, and the output resistance is120ohm. The capacitor voltage can be reduced by 10 times, and the AD pin in DSP can be collected. The inductive current adopts LA55-P/SP50, and the output resistance is 1kohm. The current can be reproduced into voltage and collected to another pin of DSP.


## 4.2.3. Experimental Results and Analysis

The experimental results are proposed for the switching rule to ensure the stability of the system. We select the power resistance of12ohms and select two sets of images for discussion and analysis (see Figures 7 and 8). The average inductive current is 0.923A in Figure 7(a) and the capacitor voltage is equal to the load voltage with an average value of 11.6V in Figure 7(b), while the target value of current is 1A and the target value of voltage is 12V. The other set of images are shown in Figures 8(a) and 8(b). The average inductive current is 1.2A and the capacitor voltage is 15.1V, while the target values are 1.3A and 15.6V, respectively. The feasibility of the switching rule is verified by experiments.Figure 7
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 0.923 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 11.6 V.
(a)
(b)Figure 8
Experimental results of the Buck converter. (a) The inductive current is converted to an equal value of voltage at 1.2 V. (b) The capacitor voltage is equal to the load voltage and stabilizes at 15.1 V.
(a)
(b)


## 5. Conclusion

This paper describes the switching rule of taking the maximum Lyapunov function based on affine switched systems and takes Buck converter as an example to verify the closed loop experiment. Firstly, the mathematical model of the affine switched systems is established. Secondly, the switching rule is introduced. Finally, the control scheme is verified by simulation and experimental verification. The correctness of the switching rule is proved, and the stable closed loop control is realized.



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*Source: 4640789-2019-10-23.xml*