# PWM-Based Finite-Time Tracking of Switched Buck Power Converters

Authors: Hui Zhang; Zhaojing Wu
Journal: Complexity
 (2020)

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## Abstract

In this paper, the problem of finite-time tracking is investigated for switched buck power converters based on the pulse width modulation (PWM) technique. For the continuous model, an equivalent continuous controller is solved by the backstepping technique, such that all signals are finite-time stable. PWM-based finite-time tracking with the equivalent control input is proposed for the switched buck converter, such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable. Simulation results are given to demonstrate the effectiveness of the proposed schemes.


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## Body

## 1. Introduction

Electrical systems described by the multiple circuit topologies associated with the regulating switch are switched DC-DC power converters [1], which have widespread applications in power systems, photovoltaic systems, communication equipment, computers, and industrial electronics. The three basic topologies of switched DC-DC power converters are buck, boost, and buck-boost. The research of these power converters has drawn a great deal of attention for the development in both power electronics and control theory [2–4]. A good deal of results on point stabilization and trajectory tracking has been obtained under the switching method [5] and PWM technique [6]. Some switching methods have been investigated for stabilization of power converters, i.e., optimal switching instants based on a numerical optimization approach [7], state-dependent switching with the aid of variable-structure control [8], and switching law by combining sliding mode control with an equivalent control input [9–12]. The references mentioned above regard the switching signal as the control input directly based on switched systems theory.The output voltage regulation of PWM-based DC-DC power converters has already been an extremely active research. The output voltage tracking was discussed by selecting passivity-based control [13–15], energy shaping control [16, 17], and state feedback indirect control with nonminimum phase [18, 19]. In order to estimate the uncertain load resistances and enhance output performance, the adaptive backstepping controllers were proposed with better robustness and adaptability [20–23]. Most of the results focused on asymptotical convergence rate for DC-DC power converters. Compared with asymptotical stability, finite-time stability has a faster convergence performance in terms of time optimization. Finite-time control has aroused a great deal of interest in recent years. For DC-DC buck converters, finite-time tracking control via integral terminal sliding modes was presented in [24]. The output voltage regulation control is investigated to guarantee the finite-time convergence rate based on finite-time convergent observer [25, 26] or via adaptive saturated finite-time control algorithm [27]. However, all these references above only considered point stabilization for continuous average models with the continuous control as a control input.Little work has been done on PWM control to achieve the trajectory tracking problem of switched power converters so far. In this paper, we consider PWM-based finite-time tracking of switched buck power converters, where the system input is a digital control. The main work consists of the following aspects.(1)
Finite-time stability of a nonlinear system with continuous control is given as a priori information, and it is firstly proved that the closed-loop digital system is practically finite-time stable in the condition of the continuous control replaced by physical PWM control.(2)
For the corresponding continuous system, an equivalent continuous controller is designed using the backstepping method, such that the tracking error tends to zero in a finite time, and the closed-loop error system is finite-time stable in terms of time optimization compared with [22, 23].(3)
For the switched buck converter, PWM-based finite-time tracking with the equivalent control input is proposed such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable.The paper is organized as follows. Problem formulation is given in Section2. PWM and its implementation are presented in Section 3. PWM-based finite-time tracking is researched in Section 4. Simulation results for buck converters are presented in Section 5. The paper is concluded in Section 6.Notations: For a vectorx, x denotes its usual Euclidean norm and xT denotes its transpose; X denotes the Frobenius norm of a matrix X defined by X=TrXXT1/2, where Tr⋅ denotes the trace of a square matrix; ℝ+ denotes the set of all nonnegative real numbers; ℝn denotes the real n-dimensional space; ℝn×r denotes the real n×r matrix space; Ci denotes the set of all functions with continuous i-th partial derivative; AB denotes the distance between A and B; SABCD denotes the area of a rectangle ABCD.


## 2. Problem Formulation

Consider the switch-regulated buck converter circuit as shown in Figure1. E is the voltage parameter of the supply voltage, R is the resistance of the resistor, C is the capacitance of the capacitor, L is the inductance of the inductor, VD is the diode, and VT is a PWM gate drive controlled switch. The control variable σ is introduced to denote the switch state, σ=1 when VT=ON, and σ=0 when VT=OFF. That is, such a control input takes values in the discrete set 0,1. It is assumed that there is no noise in the circuit; Figure 2 is the equivalent circuit of Figure 1.Figure 1
The buck converter circuit.Figure 2
Equivalent circuit of Figure1.In order to establish the circuit model associated with the regulating switch position, we need to make an explanation for those physical symbols. The inductive currentq˙L respects the derivative of the circulating electric charge qL, qC is the electrical charge stored in the capacitor, and qC/C is the capacitor voltage. The switch-regulated buck converter circuit is presented as(1)qCC=−Rq˙C−q˙L,Lq¨L=−Rq˙L−q˙C+σE.Substituting the second equation into the first, it can be rewritten as(2)q˙C=q˙L−qCRC,q¨L=−qCLC+ELσ.Letx1=qC/C,x2=q˙L; the dynamic equation is derived as(3)x˙1=−1RCx1+1Cx2,x˙2=−1Lx1+1LEσ,where x1 and x2 represent the capacitor voltage and the inductor current, respectively, and the variable σ is the digital control which represents time series produced by servo amplifiers and only take values from binary set 0,1. Let y=x1, that is, the output is the capacitor voltage, which is also the resistor voltage.Given a smooth reference signalyrt∈C2ℝ, where yrt,y˙rt,y¨rt are bounded with known constants. The aim of this paper is to design a switching signal σ for the switched buck converter (3), such that the capacitor voltage y=x1 can be driven to track a given reference signal yrt in a finite time; simultaneously, all signals in the resulting system are required to be practically finite-time stable.To study finite-time tracking, we introduce some basic concepts and lemmas that will serve as the basis for the development of our digital switching control.Definition 1.
(see [29, 28]). Consider a nonlinear system(4)x˙=fx,x0=x0,x∈D⊆ℝn,where f:D⟶ℝn is continuous, f0=0, and D is an open neighborhood of the origin. The origin of system (4) is finite-time stable (FS), if it is Lyapunov stable and finite-time convergent in a neighborhood D0⊆D of the origin. By “finite-time convergence,” we mean that there is a settling time T∗∈0,∞, such that for ∀x0∈D0/0, xt∈D0/0 for t∈0,T∗, limt⟶T∗xt=0, that is,(5)xt=0,∀t>T∗.
WhenD0=D=ℝn, the origin is a globally finite-time stable equilibrium.Definition 2.
(see [30]). The origin of system (4) is said to be practical finite-time stable (PFS), if for all initial conditions x0, there exist a constant εT>0 and a settling time T∗εT,x0<∞, such that(6)xt≤εT,∀t>T∗.Remark 1.
For PFS, ifεT depends on T and T⟶0, εT⟶0, then the effect of PFS is the same as FS. As T gets smaller, the bound εT of PFS here becomes smaller and even tends to zero compared with boundedness of [30].Lemma 1.
(see [31]). For any real number xi, i=1,…,n, and 0<p≤1, the following inequality holds:(7)x1+⋯+xnp≤x1p+⋯+xnp.Lemma 2.
(see [28]). For system (4), suppose there exist constants a>0, 0<β<1, and a positive definite function Vx∈C1:D¯⟶ℝ (D¯⊂D⊂ℝn is a neighborhood of the origin), such that(8)V˙x≤−aVβx,∀x∈D¯,then, the origin of system (4) is finite-time stable, and the settling time T∗ satisfies(9)T∗≤1a1−βVx01−β.IfD¯=ℝn and Vx is radially unbounded, the origin of system (4) is globally finite-time stable.


## 3. PWM and Its Implementation

For a nonlinear system(10)x˙=fx+Bu,x0=x0,where x∈ℝn, f∈ℝn×1 is a locally Lipschitz continuous function, B∈ℝn is a constant vector, and u∈ℝ is the digital control produced by servo amplifiers; it can only take values from binary set 0,1.In order to design such digital control, we first give a priori hypothesis for system (10).Assumption 1.
There exists a continuous static control(11)u=μx∈0,1,such that the origin of the closed-loop system (10) with continuous static control (11) is finite-time stable.
To achieve digitization, regardingμt:=μxt as a duty ratio, from the viewpoint of mathematics, a PWM control is a strategy defined as [32](12)u=σ¯Tt=1,tk≤t<tk+μtkT,0,tk+μtkT≤t<tk+T,tk+1=tk+T,t0=0,k=0,1,2,…,where tk is a sampling instant and T represents the sampling period, where T is parameter-tuning. However, it is very difficult to implement the PWM control (12). The one is how to get exact μtk at tk with variable-step resolution, the other one comes from the fact that PWM control should be a physical signal produced by a servo circuit instead of numerical signal directly.Remark 2.
By [1], systems (10) and (11) are the average models of the PWM controlled system (10) with (12). As T⟶0, the duty ratio μt as the equivalent control replaces the PWM control σ¯Tt in (12). The duty ratio μt may be designed in the form of a state feedback function, and thus it is easy to achieve some performance index of the closed-loop PWM switched system. In practice, the PWM control is generated as a discrete gate pulse signal [33], comparing a desired analog control signal with a fixed-frequency ramp.
One approximate implementation of PWM control (12) is presented as(13)u=σTt=1,μt>u0t,0,μt<u0t,where μt is the function as (11) and u0t is a triangular wave with magnitude A=1 and same period T as (12) (see Figure 3).
If continuous control (11) is changed to PWM implementation (13), the stability analysis of system (10) will be given as follows.
For PWM implementation (13), we will discuss the relationship between σTt and μt on the interval 0,t.(1)
Ifμt=a (a∈0,1 is a constant), we have (see Figure 4)(14)∫0T/2μsds=SOFGE=aT2=OFOAT2=OHOCT2=ODOESOACE=SOABD=∫0T/2σTsds.Case (1):∃n∈ℕ+, t=nT; from (14),(15)∫0tσTsds=∫0nTσTsds=∫0nTμsds=∫0tμsds.Case (2):∃n∈ℕ+, t∈nT,n+1T; from (15) and the values of μt and σTt,(16)limT⟶0∫0tσTsds=limT⟶0∫0tμsds−limT⟶0∫nTtμsds+limT⟶0∫nTtσTsds=∫0tμsds.From two cases above, ifμt=a, one gets that(17)limT⟶0∫0tσTsds=∫0tμsds,∀t≥0.(2)
Ifμt is a continuous function, there exists simple function series ∑i=1naiIEit, such that(18)μt=limn⟶∞∑i=1naiIEit,where E1,…,En are n disjoint closed sets of 0,∞. Let the length of Ei be less than that of T; then, T⟶0 leads to n⟶∞.(19)∫0tμsds=limn⟶∞∑i=1n∫0taiIEisds=limn⟶∞∑i=1nlimT⟶0∫0tσTsIEisds=limT⟶0∫0tσTsds,∀t≥0.
The solution of system (10) with (11) is denoted as xμ, and the solution of system (10) and PWM implementation (13) is denoted as xσT; by formula (19), the following relationship holds(20)limT⟶0xσTt=x0+∫0tfxsds+BlimT⟶0∫0tσTsds=x0+∫0tfxsds+B∫0tμsds=xμt,∀t≥0.which implies that(21)xσTt=xμt+DT,limT⟶0DT=0,∀t≥0.
According to Assumption1, the origin of system (10) with (11) is finite-time stable. By Definition 1, there exists a settling time T∗∈0,∞, such that(22)xμt=0,∀t>T∗.
From (22) and 21, we have that(23)xσTt≤DT,limT⟶0DT=0,∀t≥0.
As a consequence, there existseT, such that DT≤eT, and(24)xσTt≤eT,limT⟶0eT=0,∀t≥0.
According to Definition2, the origin of closed-loop digital system (10) and (13) is practically finite-time stable.
From the above analysis, we can get the result.Figure 3
Producing pulse using comparator.Figure 4
The pulse generated comparingμ=a with a triangular wave.Theorem 1.
For system (10) under Assumption 1, if continuous static control μt (11) is replaced by PWM control σTt (13), then the origin of closed-loop digital system (10) with (13) is practically finite-time stable.Remark 3.
According to Theorem1, system (10) with (13) is practically finite-time stable. From (24), as T⟶0, eT⟶0. This implies that the sampling period T can be tuned smaller, and system (10) with (13) is closer to be finite-time stable. Moreover, all signals can be made arbitrarily small for small enough T.


## 4. PWM-Based Finite-Time Tracking

For the switched buck converter (3), in order to achieve the tracking target, we give the ideas for design as follows:(1)
Design the equivalent continuous control by the backstepping method(2)
The digital control is produced using the equivalent continuous control based on the PWM technique


### 4.1. Design of a Finite-Time State Feedback Controller

System (3) with equivalent continuous control μ is presented as(25)x˙1=−1RCx1+1Cx2,x˙2=−1Lx1+1LEμ.For the continuous system (25), the backstepping method is adopted to design a finite-time state feedback controller μ, such that the capacitor voltage x1 tracks the given reference voltage yrt in a finite time.Introduce the track error(26)z1=x1−yr.Step 1.
From (25) and (26), the derivative of z1 with respect to time turns out to be(27)z˙1=−1RCx1+1Cx2−y˙r.
Choosing the Lyapunov functionV1=1/2z12, the derivative of V1 along the system (27) is given by(28)V˙1=z1−1RCx1+1Cx2−y˙r.
As1/Cx2 is just a variable and not an effective control input, (27) cannot be enforced for all t≥0. Nevertheless, it shows that the desired value for the variable 1/Cx2 is(29)α1=−c1z12β−1+1RCx1+y˙r,where c1>0 and β∈1/2,1 are design parameters. Substituting (29) into (28), we have(30)V˙1=−c1z12β+z11Cx2−α1.Step 2.
Indeed, design the error(31)z2=1Cx2−α1.
From (25) and (31), deriving z2 with respect to time yields(32)z˙2=−1LCx1+1LCEμ−α˙1.
For the second Lyapunov-like functionV=V2=V1+1/2z22, the derivative of V along the system (32) satisfies(33)V˙=−c1z12β+z1z2+z1z˙2=−c1z12β−c2z22β+z2c2z22β−1+z1−1LCx1+1LCEμ−α˙1,where c2>0 is a design parameter. The finite-time state feedback controller is designed as(34)μ=LCE−z1−c2z22β−1+1LCx1+α˙1.
Substituting (34) into (33) results in(35)V˙=−c1z12β−c2z22β.
In view of the above control analysis, we give the stability result of the continuous system in the following.Theorem 2.
For system (3) with continuous control μ (34), all the signals of the closed-loop error systems (27), (32), and (34) are finite-time stable, and the tracking error converges to the origin in a finite time. Moreover, the equivalent continuous control (34) is bounded, and by choosing design parameters appropriately, μ can be obtained such that(36)μt∈0,1,t≥0.Proof.
By definingz=z1T,z2TT and Vz=1/2z12+1/2z22, Vz is a positive definite function. From (35) and Lemma 1,(37)V˙=−c1z12β−c2z22β≤−2c1z122β−2c2z222β≤−c12z12+12z22β≤−cVβ,where c=min2c1,2c2 and β∈1/2,1. According to Lemma 2, the closed-loop error system is finite-time stable. Besides, based on Definition 1, we can obtain that(38)zt=0,∀t≥T∗withT∗≤1c1−βV1−βz0,zt=0,∀t≥T∗.
By the definition ofz,(39)x1−yr=0,∀t>T∗,that is, the tracking error converges to the origin in a finite time. Furthermore, from (38), this implies that z is bounded. For the boundedness of yr,y˙r, and y¨r, it can be obtained that α1 and states x1,x2 are bounded. After that, νt:=LC−z1−c2z22β−1+1/LCx1+α˙1 is bounded, that is, there exists a positive constant E>0, such that νt≤E. Then, μ=νt/E∈−1,1. Let the beginning values of inductor current x1 and capacitor voltage x2 of circuits be zero, and choose design parameters c1,c2 appropriately; we can get μt∈0,1,t≥0.


### 4.2. Design of PWM-Based Finite-Time Tracking Controller

For the switched buck converter (3), there exists a equivalent continuous finite-time state feedback control (34), that is,(40)μ=LCE−z1−c2z22β−1+1LCx1+α˙1∈0,1,t≥0,by choosing positive parameters E,c1,c2 appropriately. Based on PWM implementation (13) and continuous signal (40), the binary PWM control is designed as(41)σ=σTt=1,μt>u0t,0,μt<u0t,where σ is the digital switch in the buck converter (3) and u0t is the triangular wave with period T.In order to indicate the proposed PWM-based finite-time tracking control approach more clearly, the control block diagram is shown in Figure5, where the switching signal σt is generated by the PWM technique comparing equivalent continuous control μt with fixed-frequency triangle wave by comparator.Figure 5
The control framework of buck converters under PWM-based finite-time tracking control.Theorem 3.
For the switched buck converter (3), if continuous control μ (40) is replaced by binary PWM control σ (41), then the closed-loop error systems (27) and (32) with (41) are practically finite-time stable. Furthermore, the tracking error can be made arbitrarily small for small enough T.Proof.
According to Theorem2, the closed-loop error systems (27) and (32) with (40) are finite-time stable. Based on Theorem 1, the closed-loop error systems (27) and (32) with (41) are practically finite-time stable.
Furthermore,z1=x1−yr is a component of states z of the closed-loop error system; based on Remark 3, when T is small enough, the tracking error x1−yr can be made arbitrarily small.Remark 4.
Finite-time point regulation problem was discussed for continuous average models in [25–27] with the continuous control as a control input. Compared with [25–27], PWM control based on the equivalent control input (i.e., the duty ratio function) is considered for finite-time trajectory tracking of switched power converters in this paper, and here the digital control is the system input.


## 4.1. Design of a Finite-Time State Feedback Controller

System (3) with equivalent continuous control μ is presented as(25)x˙1=−1RCx1+1Cx2,x˙2=−1Lx1+1LEμ.For the continuous system (25), the backstepping method is adopted to design a finite-time state feedback controller μ, such that the capacitor voltage x1 tracks the given reference voltage yrt in a finite time.Introduce the track error(26)z1=x1−yr.Step 1.
From (25) and (26), the derivative of z1 with respect to time turns out to be(27)z˙1=−1RCx1+1Cx2−y˙r.
Choosing the Lyapunov functionV1=1/2z12, the derivative of V1 along the system (27) is given by(28)V˙1=z1−1RCx1+1Cx2−y˙r.
As1/Cx2 is just a variable and not an effective control input, (27) cannot be enforced for all t≥0. Nevertheless, it shows that the desired value for the variable 1/Cx2 is(29)α1=−c1z12β−1+1RCx1+y˙r,where c1>0 and β∈1/2,1 are design parameters. Substituting (29) into (28), we have(30)V˙1=−c1z12β+z11Cx2−α1.Step 2.
Indeed, design the error(31)z2=1Cx2−α1.
From (25) and (31), deriving z2 with respect to time yields(32)z˙2=−1LCx1+1LCEμ−α˙1.
For the second Lyapunov-like functionV=V2=V1+1/2z22, the derivative of V along the system (32) satisfies(33)V˙=−c1z12β+z1z2+z1z˙2=−c1z12β−c2z22β+z2c2z22β−1+z1−1LCx1+1LCEμ−α˙1,where c2>0 is a design parameter. The finite-time state feedback controller is designed as(34)μ=LCE−z1−c2z22β−1+1LCx1+α˙1.
Substituting (34) into (33) results in(35)V˙=−c1z12β−c2z22β.
In view of the above control analysis, we give the stability result of the continuous system in the following.Theorem 2.
For system (3) with continuous control μ (34), all the signals of the closed-loop error systems (27), (32), and (34) are finite-time stable, and the tracking error converges to the origin in a finite time. Moreover, the equivalent continuous control (34) is bounded, and by choosing design parameters appropriately, μ can be obtained such that(36)μt∈0,1,t≥0.Proof.
By definingz=z1T,z2TT and Vz=1/2z12+1/2z22, Vz is a positive definite function. From (35) and Lemma 1,(37)V˙=−c1z12β−c2z22β≤−2c1z122β−2c2z222β≤−c12z12+12z22β≤−cVβ,where c=min2c1,2c2 and β∈1/2,1. According to Lemma 2, the closed-loop error system is finite-time stable. Besides, based on Definition 1, we can obtain that(38)zt=0,∀t≥T∗withT∗≤1c1−βV1−βz0,zt=0,∀t≥T∗.
By the definition ofz,(39)x1−yr=0,∀t>T∗,that is, the tracking error converges to the origin in a finite time. Furthermore, from (38), this implies that z is bounded. For the boundedness of yr,y˙r, and y¨r, it can be obtained that α1 and states x1,x2 are bounded. After that, νt:=LC−z1−c2z22β−1+1/LCx1+α˙1 is bounded, that is, there exists a positive constant E>0, such that νt≤E. Then, μ=νt/E∈−1,1. Let the beginning values of inductor current x1 and capacitor voltage x2 of circuits be zero, and choose design parameters c1,c2 appropriately; we can get μt∈0,1,t≥0.


## 4.2. Design of PWM-Based Finite-Time Tracking Controller

For the switched buck converter (3), there exists a equivalent continuous finite-time state feedback control (34), that is,(40)μ=LCE−z1−c2z22β−1+1LCx1+α˙1∈0,1,t≥0,by choosing positive parameters E,c1,c2 appropriately. Based on PWM implementation (13) and continuous signal (40), the binary PWM control is designed as(41)σ=σTt=1,μt>u0t,0,μt<u0t,where σ is the digital switch in the buck converter (3) and u0t is the triangular wave with period T.In order to indicate the proposed PWM-based finite-time tracking control approach more clearly, the control block diagram is shown in Figure5, where the switching signal σt is generated by the PWM technique comparing equivalent continuous control μt with fixed-frequency triangle wave by comparator.Figure 5
The control framework of buck converters under PWM-based finite-time tracking control.Theorem 3.
For the switched buck converter (3), if continuous control μ (40) is replaced by binary PWM control σ (41), then the closed-loop error systems (27) and (32) with (41) are practically finite-time stable. Furthermore, the tracking error can be made arbitrarily small for small enough T.Proof.
According to Theorem2, the closed-loop error systems (27) and (32) with (40) are finite-time stable. Based on Theorem 1, the closed-loop error systems (27) and (32) with (41) are practically finite-time stable.
Furthermore,z1=x1−yr is a component of states z of the closed-loop error system; based on Remark 3, when T is small enough, the tracking error x1−yr can be made arbitrarily small.Remark 4.
Finite-time point regulation problem was discussed for continuous average models in [25–27] with the continuous control as a control input. Compared with [25–27], PWM control based on the equivalent control input (i.e., the duty ratio function) is considered for finite-time trajectory tracking of switched power converters in this paper, and here the digital control is the system input.


## 5. Simulation Results

Example 1.
To investigate the effectiveness of the proposed control method, we give the simulation results as follows.
The corresponding closed-loop systems are considered with two different cases:(i)
The continuous model (25) based on finite-time state feedback controller (40)(ii)
The switched buck converter (3) based on PWM-based finite-time tracking (41)
In the simulation, suppose that the output voltage reference for tracking isyrt=2+sint; the circuit parameters are E=8V, L=0.2H, C=0.001F, and R=5Ω; the initial values are x10=qC0/C=0andx20=q˙L0=0; and the design parameters are c1=80, c2=80, β=99/101, and T=0.001.
To make comparisons of simulation results, the two system cases are discussed keeping other parameters intact, respectively. Figures6 and 7 show the responses of the corresponding closed-loop system with the different cases. Figure 6 implies that the tracking error tends to the origin in a finite time. From Figure 7, it can be learned that the tracking error converges to a small neighborhood of the origin in a finite time. Compared with Figure 6, Figure 7 illustrates the validity of PWM-based finite-time tracking control of the closed-loop system.Figure 6
The response of closed-loop continuous model (25) with equivalent continuous control (40).Figure 7
The response of closed-loop switched buck converter (3) with PWM control (41).


## 6. Conclusions

The problem of finite-time tracking for switched buck converters is considered in this paper. PWM-based tracking control with the equivalent control input is proposed for the switched buck converter, such that the tracking error converges to an arbitrarily small neighborhood of the origin in finite time, and the origin of the closed-loop system is practically finite-time stable. Simulation results are given to demonstrate the effectiveness of the proposed schemes.It is an interesting and challenging direction under current research. The finite-time control method can be further generalized to solve other converters, such as boost converter, buck-boost converter, and Ćuk converters; another future research is regarding stochastic PWM control of systems based on random theory [34, 35].



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