In recent years, the combination of fractional calculus (FC) and sliding-mode control (SMC) has been gaining more and more interests due to fusion characteristics of SMC and FC. This paper presents the fractional-order terminal sliding-mode control (FTSMC) which has a new fractional-order sliding surface and assures the finite time convergence of the output voltage error to the equilibrium point during the load changes. TSMC is a special case of FTSMC. Through mathematical analysis, the system can reach the sliding-mode surface in finite time. The theoretical considerations have been verified by numerical simulations. And a Buck DC/DC converter application is presented and compared to illustrate the effectiveness of the proposed method. It is shown that the novel fractional terminal sliding-mode control exhibits considerable improvement in terms of a faster output voltage response during load changes.
National Natural Science Foundation of China5150713451279161Shaanxi Provincial Education Department15JK1537Doctoral Scientific Research Foundation of Xi’an Polytechnic UniversityBS150251. Introduction
The concept of fractional calculus as an extension of ordinary calculus can stretch back to 1695. In the letter to L’Hospital, Leibniz proposed the possibility of generalizing the operation of half-order derivative [1]. Though fractional calculus has a long history, only in recent years have the applications of fractional calculus to physics and engineering become an important aspect of modern technology. In comparison with the classical elementary calculus, the main advantage of fractional calculus is that it can provide an elegant description for the memory and hereditary properties of various real objects. The list of the applied fields of fractional calculus has been ever growing and includes the electrode-electrolyte polarization, viscoelastic fluids, chaotic systems, and power converters [2–6]. Most striking of all, the fractional differentials and integrals are applied to the control theory, when the controller or the controlled system is described by fractional calculus [7–11]. In 1996, Oustaloup et al. developed the first fractional-order controller which is the so-called CRONE [12]. Then some fractional-order control strategies are proposed one after another, such as fractional-order PID controllers [13], fractional-order sliding-mode controllers [14–16], fractional-order optimal controllers [17], and fractional-order adaptive controllers [18, 19].
DC/DC power converters, which work in switch mode, are applied in a wide variety of applications, including DC motor drives, active filters, computers, power supplies, and medical instrumentation. The Buck converter is one of the simplest but most useful power converters that can convert a DC input to a DC output at a lower voltage. There are some other basic DC/DC converters, namely, the Boost, Buck-Boost, Cuk, Zeta, and Sepic. Each of these converters consists of the passive power switch, the active power switch, and the storage elements. The main objective of most closed-loop feedback controlled DC/DC converters is to ensure that the converter operates with fast dynamical response, small steady-state output error, and low overshoot, while maintaining high efficiency and low noise emission in terms of rejection of input voltage changes, parameter uncertainties, and load variations [20]. The designing of high performance control strategy is always a challenge, because DC/DC converters are inherently time-varying variable-structure nonlinear systems. Nonlinear control strategies [21–23] are better candidates in DC/DC converters than other linear ones. Among the nonlinear control strategies, the sliding-mode control (SMC) has been well known due to its fast dynamic response, robustness to disturbances, guaranteed stability, parameter variations, and simplicity in implementation. Moreover, compared with other nonlinear control strategies, the SMC method is easy to implement and has a high degree of flexibility in the designing process.
The sliding-mode control is an effective robust control strategy with the feature of switching the control law to force the state trajectories of the system from the initial states onto some predefined sliding surface which exhibit desired dynamics. When in the sliding mode, the closed-loop response becomes totally insensitive to both internal parameter uncertainties and external disturbances. Compared with the SMC, the terminal sliding-mode control (TSMC) has some superior properties, such as the state of the system which converges in finite time and the higher steady-state tracking accuracy. Near the equilibrium point the rate of the convergence is being speeded up, so the TSMC is more suitable for high precision control. In recent years, the combination of fractional-order control (FOC) and sliding-mode control (SMC) has been gaining more and more interests from the systems control community. An introductory work on the application in fractional-order sliding mode is reported in [24]; the double integrator is introduced as the plant under control. A fuzzy fractional-order sliding-mode controller is adopted to control nonlinear systems in [25]. In [26], fractional-order sliding-mode control strategies for power electronic Buck converters are presented, where pulse-width modulation (PWM) sets the basis for the regulation of switched mode converters. In [27], authors collect different methods to apply FOC in SMC through the use of fractional-order surfaces which are fractional-order PID or fractional-order PI and propose a direct Boolean control strategy in order to avoid using PWM.
In this paper, we focus on the introduction of the FOC in TSMC for switching systems. The fractional-order terminal sliding-mode control (FTSMC) method has been adopted for controlling the Buck converter. The idea of such method is used to design a novel nonlinear sliding surface function which is a fusion of characteristics of TSMC and FOC. Based on the Lyapunov stability theory, a fractional-order sliding-mode control law is derived to assure finite time convergence of the output voltage error to the equilibrium point. Through mathematical analysis, we obtain the finite time tsFTSMC of FTSMC. It is shown that TSMC is a special case of FTSMC. The theoretical considerations have been verified by numerical simulations and a Buck converter application is given to show the superiority of the proposed strategy. The fractional terminal sliding-mode control exhibits considerable improvement in terms of a faster output voltage response during load changes. FTSMC can overcome the influence which disturbances bring about to control system and guarantee that DC/DC converter keeps good dynamic and steady performances. The output can follow the given well, and the disturbances almost do not affect the output.
The rest of this paper is organized as follows. In Section 2, some basic concept and definitions of the fractional calculus are introduced. Then the basic principle of the Buck converter in CCM is introduced. In Section 3, the integer-order TSMC is briefly reviewed. In Section 4, FTSMC is described for the Buck converter. In Section 5, numerical simulation results are presented to verify the theoretical considerations and an application to Buck converter is given to show the superiority of the proposed strategy. Finally, some concluding remarks of this paper are drawn in Section 6.
2. Preliminaries
In this section, firstly, some basic concept and definitions of fractional calculus are introduced. Afterwards, a brief introduction to the basic principle of the Buck converter in CCM is presented.
2.1. The Basis of Fractional Calculus
In fractional calculus, the operator Dtαa is the differintegral operator which takes both integrals and derivatives in one single expression. It denotes generalization of integrals and derivatives to arbitrary order and can be defined as follows:(1)Dtαa=dαdtα,Reα>01,Reα=0∫atdτ-α,Reα<0,where α∈R can be an arbitrary real number, which denotes the order of the operation, and a and t are the lower and upper limits.
Definition 1.
In the fractional calculus, the Gamma function proposed by Euler is defined by the integral(2)Γz=∫0∞e-ttz-1dt,which satisfies Rez>0, and converges in the right half of the complex plane.
Several reputed definitions for fractional derivatives are put forward including Riemann-Liouville definition, Grunwald-Letnikov definition, Caputo definition, Weyl definition, and Marchaud definition. Among them, being the best-known one, Riemann-Liouville definition is precisely studied.
Definition 2 (see [<xref ref-type="bibr" rid="B7">7</xref>]).
Riemann-Liouville definition of the αth-order fractional derivative operator Dtα0 is given by(3)Dtα0ft=1Γ1-αddt∫0tt-τ-αfτdτ,where 0<α<1.
Definition 3 (see [<xref ref-type="bibr" rid="B7">7</xref>]).
Riemann-Liouville definition of the αth-order fractional derivative operator Dt-α0 is defined by(4)Dt-α0ft=1Γα∫0tt-τα-1fτdτ.According to this definition, the derivative of a time function with f(t)=tβ, β>-1, is evaluated as(5)Dtα0tβ=Γα+1Γα-β+1tβ-α.
Definition 4 (see [<xref ref-type="bibr" rid="B7">7</xref>]).
If the fractional derivative Dtα0f(t)(k-1<α<k) and a function f(t) are integrable, then(6)Dt-αaDt-α0ft=ft-∑j=1kDtα-j0ftt=at-aα-jΓμ-j+1.
2.2. Brief Introduction of the Buck Converter in CCM
The Buck converter, sometimes called a step-down power stage, is a popular nonisolated power stage topology. A simplified schematic of the Buck converter is shown in Figure 1(a).
(a) Buck converter; (b) the switch S is turned on; (c) the switch S is turned off.
The basic operation of the Buck converter can be regarded as piecewise switched dynamical systems, involving toggling among a set of linear or nonlinear circuit topologies. In continuous conduction mode (CCM), there are two switching modes. (1) When the switch (labeled as S in Figure 1(b)) is turned on, the inductor current passes through the switch, and the diode is reverse-biased with the inductor current iL ramping up. (2) When the switch (labeled as S in Figure 1(c)) is turned off, the inductor maintains current flow in the same direction so that the diode is forward-biased. And this causes the inductor current to ramp down. The process repeats cyclically. For the operation described above, we have two state equations as follows.
When S is turned on,(7)diLtdtdvotdt=0-1L1C-1CRiLtvot+1L0vint.When S is turned off,(8)diLtdtdvotdt=0-1L-1C-1CRiLtvot+00vint.Combining (7) and (8) gives(9)diLtdtdvotdt=0-1L1C-1CRiLtvot+1L0uvint,where u is the control input. The switch is in the ON state for u=1 and in the OFF state for u=0.
Define the output voltage error x1 as follows:(10)x1=vo-Vref,where Vref is the reference value of the output voltage. By taking the time derivative of (10), x2 which is the rate of change of voltage error can be expressed as(11)x2=x˙1=v˙o.Then the dynamics of x2 can be given as(12)x˙2=1LCuVin-Vref-x1-x2RC.So, it can be concluded that(13)x˙1x˙2=01-1LC-1RCx1x2+0uVin-Vref1LC.
3. The Integer-Order Terminal Sliding-Mode Control for the Buck Converter
The conventional TSM concept can be described as(14)s=αx1β+x2,where α is a designed positive constant, β=q/p, and p and q are both positive odd integers which satisfy the following condition: 0<q/p<1. When the system is in the terminal sliding mode (s=0), we can get its dynamics from (14):(15)x˙1=-αx1β.Rewriting (15) yields(16)dt=-1αx1βdx.Given any initial state x1(0)≠0 and x1(ts)=0 which is the terminal attractor of the system. Taking integral of both sides of (16), the finite time is ts obtained by(17)s=αx1β+Dtμ-1t0x2.
This means that state x1 can converge to zero in a finite time. In other words, the term “terminal” referred to equilibrium zero and can be reached in finite time and it is stable:(18)s˙=αβx1β-1x˙1+x˙2=αβx1β-1x2+1LCuVin-Vref-x1-x2RC=0.
The equivalent control can be expressed as(19)ueq=x2RC-αβx1β-1x2LCVin+Vref+x1Vin.
4. The Fractional-Order Terminal Sliding-Mode Control Method for the Buck ConverterTheorem 5.
In order to introduce the so-called fractional-order terminal sliding-mode model, a fractional-order sliding surface function can be defined as(20)s=αx1β+Dtμ-1t0x2,where μ∈(0,1], α is a designed positive constant, β=q/p, and p and q are both positive odd integers which satisfy the following condition: 0<q/p<1. For Buck converter with FTSMC, if the control is designed as(21)u=x2RC-D1-μαβx1β-1x2LCVin+Vref+x1Vin-Ksigns-Ks,where 0<β<1, k>0, 0<μ<1, then states x1 and x2 will converge to zero in finite time. Furthermore, the system will be in the terminal sliding mode.
Proof.
Select the Lyapunov function as(22)V=12s2.Taking the derivative of (22),(23)V˙=ss˙=sαβx1β-1x˙1+Dμ-1x˙2.Substituting (12) into (23) results in(24)V˙=sαβx1β-1x2+sDμ-11LCuVin-Vref-x1-x2RC.Substituting (21) into (24) results in(25)V˙=s-Ksigns-Ks=-Ks+s2.Because the controller parameter K is positive, we can find V>0 and V˙>0. In other words, the controlled system satisfies the reaching condition.
When the system reaches the sliding surface (s=0), it is in the terminal sliding mode. Its dynamics can be determined by the following equation:(26)Dtsμ-10x˙1=-αx01β.Taking the concept of fractional integral and derivative operators into account, one obtains(27)Dts1-μ0Dtsμ-10x˙1=-αD0ts1-μx1β.From (6), we can get(28)x˙1t-Dtμ-20x˙1tt=0t-trμ-2Γμ-1=-αDts1-μ0x1βt,where Dtμ-20x˙1tt=0(t-trμ-2/Γμ-1)=0. So, it can be concluded that(29)x˙1t=-αDts1-μ0x1βt.According to Definition 3, we have(30)D1-μx1βt=Γβ+1Γβ+μx1β+μ-1t.Substituting (30) into (29) results in(31)x˙1t=-αΓβ+1Γβ+μx1β+μ-1t.Equation (31) can be rewritten as(32)dx1t=-αΓβ+1Γβ+μx1β+μ-1tdt.Taking integral of both sides of (32),(33)∫0tsx11-β-μtdx1t=-αΓβ+1Γβ+μ∫0tsdt.The finite time ts is obtained by(34)tsFTSMC=Γβ+μx02-μ-βαΓβ+12-μ-β.Therefore, it can be concluded that system trajectories can reach the equilibrium point in finite time. When μ=1, it is obvious that tsTSMC is equivalent to tsFTSMC. It means that the finite time taken to attain to the equilibrium point of the FTSMC system is the same as the one of the TSMC system as given in (17).
5. Simulation Results
The block diagram of the DC/DC Buck converter with the proposed FTSMC method is depicted in Figure 2. In order to show the performance of the proposed methods, the DC/DC Buck converter system has been tested by simulations. Simulations are carried out using Matlab/Simulink. Parameters of Buck converter are given in Abbreviations.
Basic structure of an FTMSC Buck converter.
In other literatures, the influences of parameters α and β have been studied. This paper focuses on the influence of the order parameter μ on the control effect. Table 1 shows the parameters of the controller.
SMC parameters.
Description
α
β
μ
SMC
100
—
—
TSMC
100
0.6
—
FTSMC1
100
0.6
0.9
FTSMC2
100
0.6
0.8
FTSMC3
100
0.6
0.7
FTSMC4
100
0.6
0.6
As shown in Figure 3, when α=100 and β=0.6, the value range of μ is between 0 and 1. μ-Ts curve is not a monotonic function. When the value of μ is about 0.8, Ts get the minimum value. The parameter μ increases the degree of freedom of the original terminal sliding-mode controller. The sliding-mode surface is selected by the original 2 parameters (α,β) and becomes the three parameters (α,β,μ).
Ts responses due to the changes in μ.
Figure 4 shows the simulated start-up and transient responses of the output voltage obtained by FTSMC strategies with different μ values. It is interesting to note that the output voltage responses become faster with decreasing the value of μ. Therefore, the value of μ is chosen to make compromise between start-up and transient responses of the converter. With the decrease of μ, the output voltage response is gradually reduced. When μ=0.7, the overshoot of the system appears. The reason is that, with the decrease of μ, the integral weight is gradually increasing; that is, the cumulative effect on the time scale increases and then affects the system overshoot, which also verifies the influence of μ on Ts as shown in Figure 3. When μ=0.6, the overshoot of the system exceeds 20%.
Output voltage responses due to the changes in μ.
In order to compare the control effect of TSMS, FTSMC, and SMC, select α=100, β=0.6, and μ=0.8. As shown in Figure 5, the response time of the system with FTSMC is less than others. At t=0.15s, the load resistance is changed from 2Ω to 8Ω. Because the load is lighter, the output current will be reduced. Then the output voltage has a short step-up. It can be seen that the output voltage returns faster to 5V in FTSMC.
Output voltage responses due to the step changes in R from 2Ω to 8Ω at t=0.15s.
Figure 6 shows the output voltage Vo responses due to the step changes in R from 8Ω to 2Ω. We can find the same results as above. The control effect of FTSMC is better.
Output voltage responses due to the step changes in R from 8Ω to 2Ω at t=0.15s.
It is obvious that the FTSMC method acts faster than others in correcting the output voltage when a step change takes place in Vin. On the other hand, as shown in Figure 7, it is clear that SMC and TSMC result in a slight steady-state error. By contrast, the FTSMC method has the lesser steady-state error.
Output voltage responses due to the step changes in Vin from 10V to 9V at t=0.15s.
6. Conclusions
This paper presents the fractional-order terminal sliding-mode control (FTSMC) which has a new fractional-order surface and assures the finite time convergence of the output voltage error to the equilibrium point during the load changes. It introduces a novel tuning parameter, the order μ, which allows the adjustment of the system output accordingly. Through mathematical analysis, we obtain the finite time of fractional-order terminal sliding-mode control. It is shown that when the order μ of FTSMC is one, the finite time taken to attain to the equilibrium point of the FTSMC system is the same as the one of the TSMC system. In other words, it means that TMSC is a special case of FTMSC. In addition, simulation results for a Buck converter are given to show the good performance of the FTSMC in comparison with the use of the traditional TSMC and SMC. The novel fractional terminal sliding-mode control exhibits considerable improvement in terms of a faster output voltage response during load changes.
AbbreviationsSpecifications of Buck ConverterVin:
Input voltage (10 V)
C:
Capacitance (1000 μF)
L:
Inductance (1 mH)
Rmin:
Minimum load resistance (2 Ω)
Rmax:
Maximum load resistance (8 Ω)
Vo:
Desired output voltage (5 V).
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 51507134 and 51279161), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 15JK1537), and Doctoral Scientific Research Foundation of Xi’an Polytechnic University (Grant no. BS15025).
OldhamK. B.SpanierJ.XuB. B.ChenD. Y.ZhangH.WangF.The modeling of the fractional-order shafting system for a water jet mixed-flow pump during the startup processYangN.-N.LiuC.-X.WuC.-J.Modeling and dynamics analysis of the fractional-order Buck—boost converter in continuous conduction modeYangN. N.WuC. J.JiaR.LiuC.Modeling and characteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusWuC.SiG.ZhangY.YangN.The fractional-order state-space averaging modeling of the Buck–Boost DC/DC converter in discontinuous conduction mode and the performance analysisYangN.-N.LiuC.-X.WuC.-J.A hyperchaotic system stabilization via inverse optimal control and experimental researchPodlubnyI.ChenD. Y.ZhangR. F.LiuX. Z.MaX.Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networksHegaziA. S.AhmedE.MatoukA. E.On chaos control and synchronization of the commensurate fractional order Liu systemWuC.-J.ZhangY.-B.YangN.-N.The synchronization of a fractional order hyperchaotic system based on passive controlChenD. Y.ZhangR. F.SprottJ. C.MaX.Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode controlOustaloupA.MoreauX.NouillantM.The CRONE suspensionPodlubnyI.Fractional-order systems and PIλDμ-controllersWangX.ZhangX.MaC.Modified projective synchronization of fractional-order chaotic systems via active sliding mode controlYangN. N.LiuC. X.A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode controlChenD. Y.ZhangR. F.MaX. Y.LiuS.Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control schemeTricaudC.ChenY. Q.An approximate method for numerically solving fractional order optimal control problems of general formLadaciS.CharefA.On fractional adaptive controlWuC. J.SiG. Q.ZhangY. B.YangN.Adaptive inverse optimal control of a novel fractional-order four-wing hyperchaotic system with uncertain parameter and circuitry implementationTanS.-C.LaiY.-M.TseC. K.JafarianM. J.NazarzadehJ.Time-optimal sliding-mode control for multi-quadrant buck convertersRamosR.BielD.FossasE.GriñoR.Sliding mode controlled multiphase buck converter with interleaving and current equalizationLeungK. K.-S.ChungH. S.-H.A comparative study of boundary control with first- and second-order switching surfaces for buck converters operating in DCMVinagreB. M.CalderonA. J.On fractional sliding mode controlProceedings of the 7th Portuguese Conference on Automatic ControlSeptember 2006Lisbon, PortugalDelavariH.GhaderiR.RanjbarA.MomaniS.Fuzzy fractional order sliding mode controller for nonlinear systemsCalderónA. J.VinagreB. M.FeliuV.Fractional order control strategies for power electronic buck convertersHosseinniaS. H.TejadoI.VinagreB. M.SierociukD.Boolean-based fractional order SMC for switching systems: application to a DC-DC buck converter