# Nonlinear Voltage Regulation Algorithm for DC-DC Boost Converter with Finite-Time Convergence

Authors: Chun Duan; Di Wu
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

For the DC-DC Boost converter system, this paper employs the finite-time control technique to design a new nonlinear fast voltage regulation control algorithm. Compared with the existing algorithm, the main advantage of the proposed algorithm lies in the fact that it can offer a fast convergent rate, i.e., finite-time convergence. Based on the average state space model of Boost converter system and finite-time control theory, rigorous stability analysis showed that the output voltage converges to the reference voltage in a finite time. Simulation results demonstrate the efficiency of the proposed method. Compared with PI control algorithm, it is shown that the proposed algorithm has a faster regulation performance and stronger robust performance on load-variation.


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## Body

## 1. Introduction

As a kind of important power electronic devices, the main function of DC-DC converters is used to achieve energy conversion, which has been widely applied in many industrial occasions, such as switching power supply, direct current motor drives, and communication equipment. Boost type direct current-direct current (DC-DC) converter is a typical power converter which has many industrial applications such as direct current (DC) motor drives, computer systems, and communication equipment [1]. With the development of distributed power generations, it is required that the DC-DC converter has the high-quality, reliable, efficient power supplies, and other features. However, from the control viewpoint, how to improve the control system performance for DC-DC converter is challenging since DC-DC converters are usually time-varying systems due to their switching operation.In the past decades, many researchers from automatic control and power electronic have investigated the control problem for this kind of devices. It is well-known that DC-DC power converters are typical switch systems and the Boost type DC-DC converter is non-minimum-phase system, which is a main obstacle for controller’s design.So far, many researchers have employed different nonlinear control methods to design control algorithms for Boost converter. In [2, 3], based on sliding mode control (SMC) method, some SMC algorithms were given and the analog control circuits were set up. In [4], combining SMC and feedback linearization technique, the corresponding SMC algorithm was also designed. Based on LMI technique and saturation control technique, the work [5] proposed a saturated nonlinear control algorithm. In [6], time-delayed compensation technique was employed to design a time-delayed control algorithm.Actually, for a control system, the steady-state and dynamical performances (e.g., convergent rate) are two key indexes. Note that the most of existing voltage regulation algorithms for Boost converter system only guarantee that the convergence is at best exponential with infinite settling time. Clearly, in practice, it is more desirable if the output voltage can converges to the reference voltage in a finite time. Motivated by this, finite-time control theory has been introduced and developed in the literate [7–13], which guarantees that the system states converge to equilibrium in a finite time. Besides faster convergence rates, the closed-loop systems under finite-time control usually have some other nice features such as higher accuracies and better disturbance rejection properties [7, 14]. Because of the advantages of finite-time control, the work [15] employed the terminal sliding mode technique to design the finite-time voltage regulation algorithm for Buck converter. The works [16, 17] considered the finite-time control law for the DC-DC buck converter system. As for the Boost converter, the work [18] designed a class of finite-time voltage control algorithms, where the input voltage and load resistance are assumed to be known.This paper will also employ the finite-time control technique to solve the voltage regulation problem for DC-DC Boost converter systems. Different from the existing work [18], this paper considers the design of fast finite-time control algorithm. The contribution/novelty of this paper is that a new nonlinear control algorithm is designed, i.e., the finite-time control algorithm. The main advantage of this algorithm is that the fast convergent rate of the closed-loop system can be guaranteed when the state is near the equilibrium. First, since the DC-DC Boost converter system has a nonlinear structure, a coordinate transform based on the total energy storage function is used to the average state space mode. Then the voltage control problem is equivalent to control the total storage energy. Based on the finite-time control theory, a second-order finite control algorithm is given. Finally, a rigorous proof is given to prove the global finite-time stability of the closed-loop system under the proposed controller. At the end, simulation results are provided to show the potentials of the proposed techniques.


## 2. Preliminaries and Problem Formulation

### 2.1. System Model

Figure1 shows a typical boost type DC-DC converter. Vin is a DC input voltage source, S is a controlled switch, D is a diode, Vo is sensed output voltage, iL denotes the inductance current, and L,C,R are the inductance, capacitance, and load resistance, respectively. If the switching frequency for S is sufficiently high, the dynamic of DC-DC converters can be described by an average state space model [3]. Based on the average state space model [3], the dynamic equation for the Boost converter is(1)i˙L=-VoL+VinL+VoLμ,V˙o=iLC-1RCVo-iLCμ,where μ is the duty ratio function (called control input) and μ∈[0,1]. The Boost type DC-DC converters are used in applications where the required output voltage is larger than the input voltage. Let Vref be the desired DC output reference voltage; then the reference current iLref can be described by(2)iLref=Vref2VinR.Figure 1
DC-DC Boost converter.Letz1=1/2LiL2+1/2CVo2 be the total energy storage function for Boost circuit. Then it can be followed from system (1) that the state z1 satisfies(3)z˙1=z2=-Vo2R+ViniL,z˙2=-2RCVoiL+2R2CVo2-VinLVo+Vin2L+2VoiLRC+VinVoLμ.≔uy=z1.Based on this model, the reference values for system states z1,z2 are(4)z1ref=12LiLref2+12CVref2,z2ref=0.Then, the control objective of this paper is to design a nonlinear control algorithm such that the system’s output y=z1 can track the reference signal z1ref ina finite time. Actually, if there is a time T such that z1(t)≡z1ref,∀t≥T, then (5)12iL2+12Vo2=12iLref2+12Vref2.Meanwhile, note that iL=Vo2/VinR when the system state is kept steady. As a result, z1(t)→z1ref is equivalent to Vo→Vref and iL→iLref.Define the tracking error as(6)e1=z1-z1ref,e2=z2-z2ref;then the error dynamic equation is given as(7)e˙1=e2,e˙2=u.


### 2.2. Some Useful Definitions and Lemmas

As that in [16, 17], in order to prove that the closed-loop system is finite-time stable, some essential definitions and lemmas are given.Definition 1 (see [7, 16, 17]).
Consider the nonlinear system(8)x˙=fx,f0=0,x∈Rn,where f(·):Rn→Rn is a continuous vector function. The origin is finite-time stable equilibrium if it is Lyapunov stable and finite-time convergence. The finite-time convergence means that there is a function T(x0) such that limt→T(x0)⁡x(t,x0)=0 and x(t,x0)≡0,∀t≥T(x0).The homogeneous theory method will be used in this paper to construct the finite-time controller. The definition of homogeneity is given below.Definition 2 (see [16, 17, 19]).
Consider system (8) and define the dilation (r1,…,rm)∈Rm with ri>0, i=1,…,m. Let (9)fx=f1x,…,fmxTbe a continuous vector field. f(x) is said to be homogeneous of degree k∈R with respect to dilation (r1,…,rm) if, for any given ε>0,i=1,…,m, (10)fiεr1x1,…,εrmxm=εk+rifix,∀x∈Rm,where k>-min⁡{ri,i=1,…,m}. We can say that the system x˙=f(x) is homogeneous if f(x) is homogeneous.In addition, the following definition is given to make the controller design convenient.Definition 3.
Denotesigα(x)=sign(x)xα, where α≥0, x∈R, and sign(·) is the standard sign function.Lemma 4 (see [20]).
For the following system,(11)x˙=fx+f^x,f0=0,f^0=0,x∈Rn,where f(x) is a continuous homogeneous vector space and is homogeneous of degree k<0 with respect to the dilation (r1,…,rn). If x=0 is the asymptotically stable equilibrium point of the system x˙=f(x), and ∀x≠0,(12)f^iεr1x1,…,εrnxnεri+k=0,i=1,2,…,n,then x=0 is a locally finite-time stable equilibrium of system (11). In addition, if system (11) is not only globally asymptotically stable, but also locally finite-time stable, then it is globally finite-time stable.


## 2.1. System Model

Figure1 shows a typical boost type DC-DC converter. Vin is a DC input voltage source, S is a controlled switch, D is a diode, Vo is sensed output voltage, iL denotes the inductance current, and L,C,R are the inductance, capacitance, and load resistance, respectively. If the switching frequency for S is sufficiently high, the dynamic of DC-DC converters can be described by an average state space model [3]. Based on the average state space model [3], the dynamic equation for the Boost converter is(1)i˙L=-VoL+VinL+VoLμ,V˙o=iLC-1RCVo-iLCμ,where μ is the duty ratio function (called control input) and μ∈[0,1]. The Boost type DC-DC converters are used in applications where the required output voltage is larger than the input voltage. Let Vref be the desired DC output reference voltage; then the reference current iLref can be described by(2)iLref=Vref2VinR.Figure 1
DC-DC Boost converter.Letz1=1/2LiL2+1/2CVo2 be the total energy storage function for Boost circuit. Then it can be followed from system (1) that the state z1 satisfies(3)z˙1=z2=-Vo2R+ViniL,z˙2=-2RCVoiL+2R2CVo2-VinLVo+Vin2L+2VoiLRC+VinVoLμ.≔uy=z1.Based on this model, the reference values for system states z1,z2 are(4)z1ref=12LiLref2+12CVref2,z2ref=0.Then, the control objective of this paper is to design a nonlinear control algorithm such that the system’s output y=z1 can track the reference signal z1ref ina finite time. Actually, if there is a time T such that z1(t)≡z1ref,∀t≥T, then (5)12iL2+12Vo2=12iLref2+12Vref2.Meanwhile, note that iL=Vo2/VinR when the system state is kept steady. As a result, z1(t)→z1ref is equivalent to Vo→Vref and iL→iLref.Define the tracking error as(6)e1=z1-z1ref,e2=z2-z2ref;then the error dynamic equation is given as(7)e˙1=e2,e˙2=u.


## 2.2. Some Useful Definitions and Lemmas

As that in [16, 17], in order to prove that the closed-loop system is finite-time stable, some essential definitions and lemmas are given.Definition 1 (see [7, 16, 17]).
Consider the nonlinear system(8)x˙=fx,f0=0,x∈Rn,where f(·):Rn→Rn is a continuous vector function. The origin is finite-time stable equilibrium if it is Lyapunov stable and finite-time convergence. The finite-time convergence means that there is a function T(x0) such that limt→T(x0)⁡x(t,x0)=0 and x(t,x0)≡0,∀t≥T(x0).The homogeneous theory method will be used in this paper to construct the finite-time controller. The definition of homogeneity is given below.Definition 2 (see [16, 17, 19]).
Consider system (8) and define the dilation (r1,…,rm)∈Rm with ri>0, i=1,…,m. Let (9)fx=f1x,…,fmxTbe a continuous vector field. f(x) is said to be homogeneous of degree k∈R with respect to dilation (r1,…,rm) if, for any given ε>0,i=1,…,m, (10)fiεr1x1,…,εrmxm=εk+rifix,∀x∈Rm,where k>-min⁡{ri,i=1,…,m}. We can say that the system x˙=f(x) is homogeneous if f(x) is homogeneous.In addition, the following definition is given to make the controller design convenient.Definition 3.
Denotesigα(x)=sign(x)xα, where α≥0, x∈R, and sign(·) is the standard sign function.Lemma 4 (see [20]).
For the following system,(11)x˙=fx+f^x,f0=0,f^0=0,x∈Rn,where f(x) is a continuous homogeneous vector space and is homogeneous of degree k<0 with respect to the dilation (r1,…,rn). If x=0 is the asymptotically stable equilibrium point of the system x˙=f(x), and ∀x≠0,(12)f^iεr1x1,…,εrnxnεri+k=0,i=1,2,…,n,then x=0 is a locally finite-time stable equilibrium of system (11). In addition, if system (11) is not only globally asymptotically stable, but also locally finite-time stable, then it is globally finite-time stable.


## 3. Main Results

In this section, we present the main results.Theorem 5.
For system (7), if the controller is designed as(13)u=-k1sigα1e1+e1-k2sigα2e2+e2,where k1>0,k2>0,0<α1<1,α2=2α1/(1-α1), then the states of system (7) will converge to zero in a finite time; i.e., (e1(t),e2(t))→0 in a finite time.Proof.
The proof can be divided into two steps, i.e., global asymptotic stability and local finite-time stability. The closed-loop system is(14)e˙1=e2,e˙2=-k1sigα1e1+e1-k2sigα2e2+e2.
Step 1 (proof of global asymptotic stability). Choose Lyapunov function as follows:(15)V=k1∫0e1sigα1ρdρ+12e22+12k1e12=k1α11+α1e11+α1+12e22+12k1e12,Clearly, the Lyapunov function V is positive definite and radially unbounded. Since (16)d∫0e1sigα1ρdρdt=sigα1e1e2,then(17)V˙=k1sigα1e1·e2+k1e1e2+e2-k1sigα1e1-k1e1-k2sigα2e2-k2e2=-k2sigα2e2e2-k2e22.Noticing that(18)e2sigα2e2=e2α2+1,then(19)V˙=-k2sigα2e2e2-k2e22≤0.Define the set Ψ={e1,e2∣V˙}≡0. It follows from (19) that V˙≡0 means that e2≡0, and further e˙2≡0. By (14), it can be obtained that (e2,e˙2)≡0 implies that e1≡0. Thus, based on LaSalle invariant principle [21], it can be concluded that (e1(t),e2(t))→0 as t→0. That is to say, system (14) is globally asymptotically stable.
Step 2 (proof of local finite-time stability). Rewrite system (14) as follows:(20)e˙1=e2,e˙2=-k1sigα1e1-k2sigα2e2+ge1,e2,where g(e1,e2)=-k1e1-k2e2.
Choose Lyapunov function as(21)W=k1∫0e1sigα1ρdρ+12e22.According to (20), the derivative of (21) is(22)W˙=-k2e2sigα2e2=-k2e2α2+1≤0.Similar to the proof in Step 1, it can be first proved that system (20) is asymptotically stable.
In addition, note that0<α1<1,α2=2α1/(1+α1); according to Definition 2, it can be verified that system (20) is homogeneous of degree m=(α1-1)/2<0 with respect to the dilation (r1,r2)=(1,(α1+1)/2).
For any(e1,e2)≠(0,0), according to the definition of function g(·), we obtain(23)limε→0⁡gεr1e1,εr2e2εr2+m=0.Thus, according to Lemma 4, it can be concluded that system (20) is globally finite-time stable.
By the proposed results in Steps 1 and 2, it can be found that system (7) under the controller (13) is globally finite-time stable. The proof is completed.Based on this result, we now can design a finite-time voltage regulation algorithm.Theorem 6.
For the Boost converter system (1), if the duty ratio function μ is designed as(24)μ=12VoiL/RC+VinVo/L·2RCVoiL-2R2CVo2+VinLVo-Vin2L-k1sigα1e1-k1e1-k2sigα2e2-k2e2,e1=12LiL2+12CVo2-12LVref2VinR2-12CVref2,e2=-Vo2R+ViniL,where k1>0,k2>0,0<α1<1,α2=2α1/(1+α1), then the output voltage Vo will reach the reference voltage Vref in a finite time.Remark 7.
Note that although the sign function is employed in the controller (24), the combination of the sign function and the term e1α1 and e2α2, i.e., sign(e1)e1α1 and sign(e2)e2α2), is continuous. In addition, it should be pointed out that there is no singularity problem in the proposed controller (24) since 0<α<1.


## 4. Numerical Simulations

All the simulation data is based on the PSIM (Power Simulation which is developed for power electronics and motor drive) software.


### 4.1. Simulation Parameters

The parameters of Boost converter are chosen as follows: input voltageVin=15V, inductance L=30mH, capacitance C=50μF, load resistance R=30Ω, and reference voltage Vref=40V.To have a comparison, two kinds of control algorithms are used. One is the proposed finite-time control (FC) algorithm (24) in this paper; the other one is PI control algorithm.For the FC algorithm, the control parameters are chosen as follows:α1=1/5,α2=1/3,k1=2×104,k2=1×103.For the PI control algorithm, the proportional gain is chosen asKp=0.004 and the integral gain Ki=0.007.Respectively, consider the system dynamical response performances under the conditions of boot process, reference voltage change, and load variation.


### 4.2. Dynamic Responses under Different Reference Voltages

The reference voltage is changed from40V to 30V at 0.3 seconds, and the other parameters are kept unchanged. Under the two kinds of control algorithms, the response curves of output voltage are shown in Figure 2. By comparisons, the finite-time control algorithm can offer a faster convergent speed than that of PI control.Figure 2
The response curves for output voltage under two control algorithms: finite-time control (FC) and PI control.


### 4.3. Dynamic Responses in the Presence of Load Variations

The load resistance is changed as follows:(25)R=30,0≤t≤0.2s;20,0.2s<t≤0.4s;30,t>0.4s,and the other parameters are kept unchanged. Under the finite-time control algorithm and PI control algorithm, the response curves are given in Figure 3. Clearly, the finite-time control algorithm also demonstrates faster regulation speed and stronger disturbance rejection ability.Figure 3
The response curves for output voltage under two control algorithms in the presence of load variations: finite-time control (FC) and PI control.


## 4.1. Simulation Parameters

The parameters of Boost converter are chosen as follows: input voltageVin=15V, inductance L=30mH, capacitance C=50μF, load resistance R=30Ω, and reference voltage Vref=40V.To have a comparison, two kinds of control algorithms are used. One is the proposed finite-time control (FC) algorithm (24) in this paper; the other one is PI control algorithm.For the FC algorithm, the control parameters are chosen as follows:α1=1/5,α2=1/3,k1=2×104,k2=1×103.For the PI control algorithm, the proportional gain is chosen asKp=0.004 and the integral gain Ki=0.007.Respectively, consider the system dynamical response performances under the conditions of boot process, reference voltage change, and load variation.


## 4.2. Dynamic Responses under Different Reference Voltages

The reference voltage is changed from40V to 30V at 0.3 seconds, and the other parameters are kept unchanged. Under the two kinds of control algorithms, the response curves of output voltage are shown in Figure 2. By comparisons, the finite-time control algorithm can offer a faster convergent speed than that of PI control.Figure 2
The response curves for output voltage under two control algorithms: finite-time control (FC) and PI control.


## 4.3. Dynamic Responses in the Presence of Load Variations

The load resistance is changed as follows:(25)R=30,0≤t≤0.2s;20,0.2s<t≤0.4s;30,t>0.4s,and the other parameters are kept unchanged. Under the finite-time control algorithm and PI control algorithm, the response curves are given in Figure 3. Clearly, the finite-time control algorithm also demonstrates faster regulation speed and stronger disturbance rejection ability.Figure 3
The response curves for output voltage under two control algorithms in the presence of load variations: finite-time control (FC) and PI control.


## 5. Conclusion

In this paper, a new finite-time control algorithm has been designed for boost converters. It has been proven that the output voltage can track the reference voltage in a finite time. The effectiveness of the proposed method has been verified through simulations.



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