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Sliding mode controller with modified sliding function for DC-DC Buck Converter
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Research article
Sliding mode controller with modified sliding function for DC-DC Buck Converter B.B. Naik a,n, A.J. Mehta b a Sarvajanik College of Engineering & Technology, Instrumentation & Control Engineering Department, Dr. R.K. Desai Marg, Near Mission Wockhardt Hospitals, Athwalines, Surat 395001, India b Electrical Engineering, Institute of Infrastructure Technology Research And Management (IITRAM), Ahmedabad 380026, India
art ic l e i nf o
a b s t r a c t
Article history: Received 28 February 2015 Received in revised form 17 April 2017 Accepted 18 May 2017
This article presents design of Sliding Mode Controller with proportional integral type sliding function for DC-DC Buck Converter for the controlled power supply. The converter with conventional sliding mode controller results in a steady state error in load voltage. The proposed modified sliding function improves the steady state and dynamic performance of the Convertor and facilitates better choices of controller tuning parameters. The conditions for existence of sliding modes for proposed control scheme are derived. The stability of the closed loop system with proposed sliding mode control is proved and improvement in steady state performance is exemplified. The idea of adaptive tuning for the proposed controller to compensate load variations is outlined. The comparative study of conventional and proposed control strategy is presented. The efficacy of the proposed strategy is endowed by the simulation and experimental results. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: DC-DC converters Buck convertor Sliding mode control
1. Introduction The buck converter is the most popular in industry due to its high efficiency and simplicity which emanates from its linear and minimum phase type by nature. Also known as step down converter, a buck converter finds its applications for majority of electronics devices including power supplies for computers. Buck converters and other Power Electronic Converters(PEC) functioning in closed loop control mode is the need of many industrial as well as domestic applications like instrumentation systems, ac-dc drives, telecommunications and renewable energy systems. High quality controlled power supply with a good control system is required for better dynamic and steady state performance. Although controllers with Pulse Width Modulation(PWM) techniques are already in action since long, the Sliding Mode Control (SMC) enters with a new dimension in the field of control systems for PEC. The modeling and control techniques for PEC are available in the wide variety of literature [1,2]. Cuk [3] proposed a general unified approach to modeling switching power converters. The control and modeling of various PECs can be also be found in [4] along with various control methodologies including linear and non n
Corresponding author. E-mail addresses: [email protected] (B.B. Naik), [email protected] (A.J. Mehta).
linear control techniques. The important task in many of the PECs is control of the output/load voltage. In [5], a large-signal nonlinear control technique(One Cycle Control) of switching power converters is developed to dynamically control the duty ratio of a switch such that in each cycle the average value of the controlled variable is proportional to the control reference. The technique is good at rejecting source power perturbations. However, the load disturbances are remained to study. The voltage regulation of Pulse Width Modulation(PWM) based DC-DC switching converter is discussed in [6] with the controller designed in frequency domain. It uses the averaged small signal model of the converter. The SMC tightened its grip in the field of Power Electronics after Utkin [1] discussed about wide class of PECs which may be controlled with the SMC technique. Since then SMC is applied to the various PEC and Electromechanical Systems [1,7]. The fundamentals of SMC can be found in [8]. The SMC has been the choice of many researchers to control the output voltage of PECs. Caceres and Barbi [9] presented design, analysis and experimentation of DC-AC boost converter with SMC. The technique is useful for design of Uninterruptible Power Supply(UPS) and Inverters. The SMC for DCDC PEC is available in [10] with graphical and analytical explanations. Maity [11] proposed Fixed Frequency Hysteresis Controller (FFHC) that uses both SMC and FFHC with hysteresis band. In [12] and [13] they have described SMC based technique for control of PECs. Hasan [14] suggested the Adaptive Terminal SMC for DC-DC Buck Converter having non-linear sliding surface with finite time reaching law. But the Region Of Existence(ROE)of sliding modes
http://dx.doi.org/10.1016/j.isatra.2017.05.009 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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was not defined. However, the load variation is examined for its effects on load voltage. Tan [15]suggested the implementation of SMC for DC-DC Buck Converters with Analog Components. Although control circuit are the heart of PEC functioning in closed loop mode, its design is not straightforward. In [16] they described practical issues related to the design and implementation of these control circuits, with a focus on the presentation of the state-ofthe-art control solutions, including circuit technology, design techniques, and implementation issues of analog and digital electronic circuits. The conventional SMC for many PEC applications results in poor steady state performance. More precisely, the main obstacle in application of SMC for power converters is steady state error due to practical limitations and varying switching frequency [17]. The limitations of the conventional SMC are addressed in [18,17]. The tuning of parameters play very vital role for the steady state and dynamic performance of the converter under load variation. And also the power converters may become unstable due to improper selection of tuning parameters or due to the distortion of control signal caused by the saturation of the amplified feedback signal [17]. Moreover, by using conventional sliding function, the choice in selection of tuning parameter is restricted because the one and only tuning parameter has to perform multiple tasks: (i) govern speed of response of states; (ii)compensate load variations. As the conventional SMC has only one tuning parameter [17] so in case of load variations the parameter must be varied according to load variations to prevent variations in the Region Of Existence (ROE)where Sliding Mode exists. In this paper, authors have extended the theoretical results given in [19,20] to attain both the tasks as above simultaneously by tuning the parameters. The proposed SMC with the two tuning parameters proved to be better in the sense that one of them can be adjusted according to the load variations and the other is adjusted according to the required speed of response of the system states. The proposed modified sliding function improves the steady state performance at the cost of increased order of the sliding surface. The analysis to evaluate steady state performance in case of conventional and proposed SMC is carried out with mathematical arguments. It is proved that the steady state error in the load voltage does occur if conventional SMC is used and it can be eliminated with proposed SMC. The performance of DC-DC Buck Converter with modified sliding function based SMC is evaluated through analysis, simulation and experimentation. The analysis is carried out to prove stability in the ROE of Sliding Mode and to guarantee Reaching Mode. The discussion for tuning of controller parameters is presented. The paper is organized as follows: In Section 2, the mathematical model of DC DC Buck Converter is briefly explained. The main contribution i.e. modified sliding function is proposed followed by ROE and stability in Section 3. The simulation and experimental results are discussed in Section 4 and V respectively.
Fig. 1. DC-DC Buck Converter with modified sliding function based SMC.
compared to the load resistor to avoid loading effect. If Vref is a reference voltage, Vμ = βVref , is the scaled down version of reference voltage. L is an inductor, C is a capacitor, D is the freewheeling diode, Vo is the output or load voltage, Vi is the input voltage and rL is the load resistance. The SW is a n channel MOSFET switch turned ON or OFF with the output of SM controller which is in the form of pulses. Noting that u ¼1 means SW is closed and u¼ 0 means SW is open, the state space model of the electrical system can be derived by defining the states as follows:
x1 = Vμ − βVo,
(1)
x2 = x1̇ .
(2)
From Eqn. (2),
x2 = − β
dVo β = − iC . dt C
Where, iC is the capacitor current which can be measured as shown in the Fig. 1. Let iL and ir be the inductor current and load current respectively. The equation for x2 can be rewritten as,
x2 = −
β⎡ ⎣ iL − ir ⎤⎦. C
(3)
Let the voltage drop across inductor vL given by,
vL = (uVi − V0) = L
⟹iL =
∫
diL . dt
uVi − Vo dt. L
2. Mathematical modeling
So, Eq. (3) can be rewritten as,
A Sliding Mode Voltage Controlled (SMVC) DC-DC Buck Converter model is derived using state space approach [17]. A mathematical model of DC-DC Buck Converter as shown in Fig. 1 in open loop can be derived using basic circuit analysis laws. The converter is with the pure resistive load whose output voltage is to be controlled with SMC. The converter is assumed to be operated R in continuous current conduction mode [21]. Let β = R +2R is the
x2 = x1̇ =
1
2
voltage divider ratio. The values or R1 and R2 are very high
β ⎛ Vo − ⎜ C ⎝ rL
∫
uVi − Vo ⎞ dt ⎟. L ⎠
(4)
Hence,
x2̇ = −
Vμ βV 1 1 x1 − x2 − i u + . LC rLC LC LC
(5)
From Eqs. (2) and (5), the state space model of buck converter system is obtained as,
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
B.B. Naik, A.J. Mehta / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
⎡ ⎡ 0⎤ ⎡ 0 ⎤ 1 ⎤ ⎡ x1̇ ⎤ ⎢ 0 ⎥⎡ x1⎤ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥=⎢ 1 1 ⎥⎢ ⎥ + ⎢ βVi ⎥u + ⎢ Vμ ⎥. − ⎣ x2 ⎦ − ⎣ x2̇ ⎦ ⎢ − ⎢⎣ LC ⎥⎦ ⎣ LC ⎦ rLC ⎥⎦ ⎣ LC
(6)
3
about the use of modified sliding function so that restriction in selection of α is eliminated and better responses with load variation compensation is achieved through additional tuning parameter. The proposed sliding function in Section 3 not only eliminates the restriction in the choice of α but also gives better performance.
2.1. Sliding mode control Let the sliding function S which establishes linear relationship among states be defined as,
S = αx1 + x2 = Jx.
(7)
Where, J = [α , 1], state vector x = [x1, x2]T , α is a scalar, α > 0, and it controls the first order dynamics of Eq. (7). By reducing the value of α one can slow down the reaching mode dynamics. It is known that Sliding mode control occurs if the reaching condition
SS ̇ < 0,
(8)
is satisfied [1,7].The sliding mode control law can be defined as per the following rule:
⎧ 1 = ON , S > 0 u=⎨ ⎩ 0 = OFF , S < 0
(9)
The above control law Eq. (9), will trigger the switching across the sliding manifold S. The switching device has operating frequency limitations and hence for practical implementation the following control law is preferred:
⎧ 1 = ON , S > ε u=⎨ ⎩ 0 = OFF , S < − ε
(10)
where ε is a small positive number. Control law in Eq. (10) can reduce the severity of chattering by limiting chattering frequency and hence it is feasible for implementation. The existence of sliding modes requires the following two conditions to be satisfied [1]. From Eqs. (6), (7), (8) and (9),
⎛ Vμ − βVi 1 ⎞ 1 < 0, B1 = ⎜ α − x1 + ⎟ x2 − rLC ⎠ LC LC ⎝
(11)
3. Modified sliding function Let a Proportional Integral type function of sliding function be defined as,
E=S+γ
(12)
⎧ B = Jx ̇ for 0 < S < ε 1 Where, ⎨ . ⎩ B2 = Jx ̇ for − ε < S < 0 The Region Of Existence(ROE) of sliding modes on the phase plane is the region where condition SS ̇ < 0 is satisfied. From Eqs. (11) and (12), the ROE is determined by the three lines B1 = 0, B2 = 0 and S¼ 0 on the phase plane. It can be seen that there are two possibilities, α >
1 rLC
and α <
1 . rLC
It may be observed that the
slopes of lines B1 = 0 and B2 = 0 changes due to variation in rL. This imposes a limitation on dynamic behaviour of the system as α determines the speed of response. The choice of α is important due to the first order dynamics are given by,
x1(t ) = x1(t0)exp−α(t − t0) .
(14)
(15)
Similarly taking Laplace Transformation of Eq. (14) and from above Equation,
E(s ) E(s ) S(s ) (s + α )(s + γ ) = = x1(s ) S(s ) x1(s ) s
(16)
Obviously the above equation has PID (Proportional Integral Derivative) controller type of characteristics. It is well known that increasing the order of the controller can lead to better performance. However, in this case smooth operation is not possible due to the presence of switch and hence commutations around E ¼ 0 occur as per the following rule.
⎧ 1 = ON , E > ε u=⎨ . ⎩ 0 = OFF , E < − ε
(17)
Ideally the Eqn. (17) can also be stated as,
1 [1 + sgn(E )]. 2
(18)
The control law in Eq. (17) is used for actual system to obtain simulation and experimental results. Switching across E ¼ 0 yields the following important result. Setting E ¼ 0 in Eq. (14),
S=−γ
∫0
t
Sdt.
(19)
This is equivalent to,
S ̇ = − γS.
(20)
Which leads to minimization of S. The existence of sliding modes is possible on the phase plane if the following two conditions are satisfied.
lim E ̇ < 0
E → 0+
(21)
(13)
Where x1(t0) is the initial state at time t0. The choice of α is important for the speed of response of states and allowing the maximum possible ROE on the phase plane. To meet the requirements, α = r 1C is chosen [17]. This restricts the selection of α as it L
Sdt.
S(s ) =s+α x1(s )
⎪
⎪
t
Where γ > 0 is a scalar. Here, the integral term is used to achieve better steady stage performance of the load voltage under closed loop control. This fact can be visualized from the following equations. Taking Laplace Transformation of Eq. (7),
u= ⎛ Vμ 1 ⎞ 1 > 0. B2 = ⎜ α − x1 + ⎟ x2 − rLC ⎠ LC LC ⎝
∫0
governs the first order dynamics of Eq. (13). If it may be possible to introduce one more tuning parameter that can compensate the load variation, the goal can be achieved. The next section discusses
lim E ̇ > 0
E → 0−
(22)
The following theorem proves the existence of Reaching Mode (RM) [7]and to understand the stability of the system with proposed control strategy. Theorem 1. The switching across Modified Sliding Function E leads the phase trajectory to reach to E, S = 0 manifold in finite time and
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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then to minimization of E, S i.e. Sliding Modes occur if the condition EE ̇ < 0 is satisfied. Proof. If
γ is integral gain and E > 0, then, Ė < 0,
1) Stability and Equivalent Control Law: The sufficient condition for existence of sliding modes is,
⇒S ̇ + γ S < 0 ⇒S ̇ < − γ S ⇒SS ̇ < − γS2 ⇒
EE ̇ < 0.
(25)
The Equivalent Control law, ueq can be derived as follows:
1d 2 (S ) < − γS2 2 dt
E ̇ = 0 ⇒ S ̇ + γS = 0
⇒S2(t ) < 2S2(t0) exp [−2γ (t − t0)].
From above equation and Eqs. (5), (6) and (7),
Where, t0 is the initial time and γ > 0. The above condition is important as it states that the S2 must be decreasing any time if the sliding function E assumes positive value. Similarly, S2 must be increasing if E assumes negative value. Once the phase trajectory hits the sliding manifold in the ROE, where EE ̇ < 0 is satisfied, the Sliding Mode, SM [7] occurs and stability prevails. The following subsection will be helpful to study the situation for which the reaching condition, EE ̇ < 0, is satisfied and to find the ROE. The ROE is the region on the phase plane for which the reaching condition is valid. 3.1. The ROE and stability
αx1̇ + x2̇ + γαx1 + γx2 < 0. Now from above equation and Eq. (6), with u¼ 1,
⎛ −βVi + Vμ ⎞ ⎛ 1 ⎞ ⎛ 1⎞ ⎟ + γαx1 + γx2 < 0. αx2 + ⎜ − ⎟ x1 + ⎜ − ⎟ x2 + ⎜ ⎝ LC ⎠ LC ⎝ rLC ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ βV ⎞ ⎛ Vμ ⎞ ⎛ 1 1⎞ ⎟ x1 + ⎜ − i ⎟ + ⎜ ⎟ < 0 ⇒A1 = ⎜ α − + γ ⎟ x2 + ⎜ αγ − ⎝ ⎝ LC ⎠ ⎝ LC ⎠ rLC LC ⎠ ⎝ ⎠ (23) Similarly, from Eq. (19) and considering u¼ 0 for E < 0 we may write,
⎛ Vμ ⎞ ⎛ ⎞ ⎛ 1 1⎞ ⎟ x1 + ⎜ ⎟ > 0. + γ ⎟ x2 + ⎜ αγ − A2 = ⎜ α − ⎝ ⎠ rLC LC ⎝ ⎠ ⎝ LC ⎠
ueq =
⎛ ⎛ LC ⎡ Vμ 1⎞ 1 ⎞ ⎤ ⎢ + ⎜ αγ − ⎟ x1 + ⎜ α + γ − ⎟ x2⎥. βVi ⎢⎣ LC ⎝ LC ⎠ rLC ⎠ ⎥⎦ ⎝
(24)
It can be seen from Eqs. (23) and (24) that the ROE is determined with lines A1 = 0, A2 = 0 and S ¼ 0 on the phase plane [19]. It may be noticed that the dynamics of system states and the ROE on the phase plane can be determined with two independent parameters α and γ respectively. By selecting γ = r 1C L
the ROE on phase plane is insensitive to load variations. However, it should be noted that the higher values of α can lead to sustained oscillations or peak overshoots/undershoots in the load voltage [17]. Hence the values of α and or γ must be low enough to prevent oscillations or overshoots in the load voltage. Selecting α as high as possible,faster dynamics can be achieved. The parameter γ can be adaptively tuned as per the changes in the load resistance. Practically this is possible with the following relationship:
(26)
The equivalent control law alone is not enough and practically we must add some switching action to it. Now let us define the control law u as,
u=
⎤ ⎛ ⎛ LC ⎡ Vμ 1⎞ 1 ⎞ ⎢ + ⎜ αγ − ⎟ x1 + ⎜ α + γ − ⎟ x2 + Ksgn(E )⎥. ⎥⎦ βVi ⎢⎣ LC ⎝ LC ⎠ rLC ⎠ ⎝
(27)
Where K is the switching control gain. Rewriting Eq. (5) as,
x2̇ = −
To evaluate ROE on the phase plane, consider S ̇ + γS < 0 can be simplified as,
γ=
3.2. Further analysis on stability, switching frequency and steady state performance
Vμ βV 1 1 + d(t ), x1 − x2 − i u + LC rLC LC LC
(28)
where, d(t) represents the parametric uncertainty, we can now derive the condition for the robustness for parametric uncertainties of the system model. Analytically, it is known that the following condition must be satisfied for the sliding mode control to exist.
̇ 0 EE< ⇒E(S ̇ + γS )< 0
(29)
⇒E(αx1 + x2̇ + γαx1 + γx2) < 0
(30)
Now, if u in action, Eqs. (5), (23), (24) and from Eq. (28) above Eq. (29) can be shown as
E[ − Ksgn(E ) + d(t )] < 0.
(31)
Hence, for the sliding modes to exist, the following condition must be satisfied.
K > d(t )
(32)
2) Switching Frequency: Practically there are physical limitations of the switching element hence there exists upper limit in the switching frequency of the converter. Let the dead-zone in the switching element be represented by a small positive number ε. Then the switching across E can be viewed as per the following Fig. 2.
i 1 = r rLC VoC
Where, ir is the load current. Hence, the setup for adaptive control requires the measurement of load current. Note that due to variations in the load resistance the ROE on the phase plane does not vary. The more practical way to fix the ROE on the phase plane, and which is independent on load variation, is to select α, γ very high compared to 1 in Eqs. (23), (24). rLC
Fig. 2. Switching with sliding function with dead zone.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Now, for the time period TON , we get,
TON =
δ = αx1 + x1̇ .
2ε − Ė
(33)
2ε +. Ė
(34)
and,
TOFF =
+ − Here, E ̇ and E ̇ denotes E ̇ > 0 and E ̇ < 0 respectively. Considering that the systems states are very near to zero during the sliding modes, we can write the following set of equations during TON and TOFF. Considering this fact and Eqs. (6), (17), (21) and (22) we get,
⎛ βV ⎞ ⎛ Vμ ⎞ Ė = ⎜ − i ⎟ + ⎜ ⎟ ⎝ LC ⎠ ⎝ LC ⎠
(35)
and,
(36)
From Eqs. (31), (32), (33) and (34) we can derive the expressions for TON and TOFF as follows:
TON
2εLC = ( − βVi + Vμ)
(44)
Taking the Laplace Transformation and rearranging the terms one can write the expression of x1(s ) which is the x1 in so called s domain, and s is the Laplace Operator as per the following equation.
x1(s ) =
x (0) δ + 1 . s(s + α ) s+α
lim x1(t ) = lims (x1(s )) = s→0
2εLC = Vμ
TON
(38)
δ=S+γ
∫0
t
Sdt,
1 + TOFF
(39)
(Vμ − βVi )Vμ 2εLC[2Vμ − βVi ]
(40)
1
Defining w 2 = LC and considering the fact that Vμ = βVref , Eq. (39)can be rewritten as,
fs = Vref
(47)
∫0
t
(αx1 + x1̇ )dt.
(48)
Rearranging the terms and finding out the expression for x1(s ) it can be written,
x1(s ) =
From Eqs. (35), (36) and (37) the switching frequency can be given by,
fs =
(46)
(37)
Now the switching frequency fs is given by,
fs =
δ α
It is very clear from Eq. (46) that the state x1(t ) is not forced to zero, instead, the higher values of α can help to minimize the state which is the goal. But the state practically can't reach to zero and hence some steady state error is expected as can be observed with simulation results. Now, repeating the same procedure as above and assuming the value of E to be δ and from Eq. (7) one can rewrite Eq. (14) as,
δ = αx1 + x1̇ + γ TOFF
(45)
where, x1(0) is the initial condition of the state. It is now obvious that the final value of the state x1(t ) can be obtained by the final value theorem as, t →∞
−
⎛ Vμ ⎞ + E ̇ = ⎜ ⎟. ⎝ LC ⎠
5
δ + x1(0)s s 2 + (α + γ )s + γα
.
(49)
Hence, in case of Eq. (49),
lim x1(t ) = lims (x1(s )) = 0.
t →∞
s→0
(50)
It can now be clear that the switching across the proposed sliding function E ¼0 can lead to zero steady state error compared to the case of conventional sliding function S ¼0 which gives small but non zero steady state error. This fact can be visualized by observing Fig. 3.
(Vref − Vi )βw 2 2ε(2Vref − Vi )
(41)
Note that the switching frequency is inversely proportional to the size of the dead zone in the switching element ε and it depends on other systems parameters as mentioned in Eq. (40). 3) Steady State Performance: Exact analysis of steady state error is complicated due to the presence of switching element. However, in this section it is tried to evaluate mathematically the performance of proposed control law in terms of steady state performance. It is reasonable to assume V as the uniform neighbourhood of Sr which is a set of all the numbers in S those are very small in magnitude such that,
Sr = ⋃ Br (p), p∈S
Br (p) = {x ∈ S|d(x, p) < r} .
(42)
(43)
For p ¼0 and r is the centre and radius of an open ball, from the above equations one can immediately notice that for any point δ ∈ Sr from (1), (2) and (7),
4. Simulation results To show the efficacy of the proposed algorithm, a MATLAB simulation is carried out with the system parameters as given in Table 1. It can be observed from Fig. 3 that for the tuning parameter settings, α ¼ 600 and γ ¼ 3.3, the load voltage tracks well the reference voltage 12.5 V (dashed line). However, the load disturbance occurs at time 2.5 s i.e. the load resistance suddenly drops from 100 Ω to 32 Ω, the proposed SMC tracks the reference very accurately (Fig. 3(b)). With the conventional SMC, load voltage fails to track the reference and does exhibit small amount of steady state errors before and after the load disturbance (Fig. 3(a)). For the reference voltage of 19.5 V, the step response of load voltage is studied. The load resistance is suddenly varied from 100 Ω to 32 Ω. The parameter for Fig. 4 are α ¼ 6000 and γ ¼ 0.0020. The Fig. 4(a) shows the load voltage, reference and controller output. The better view is presented in Fig. 4(b) to study the effects of the load disturbance. Here, we can observe spike in load voltage due to small value of parameter γ chosen just to demonstrate its effect.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Fig. 3. Simulation results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Load disturbance from 100Ω, to 32Ω at 2.5 s and reference voltage ¼ 12.5 V. Table 1 System Parameters.
Fig. 5. Load Voltage for α ¼ 1000, γ ¼ 100.
The Fig. 5 shows the load voltage with tuning parameter set
α ¼ 1000 and γ ¼ 100 and for the reference of 19.5 V.
5. Experimental results To validate the proposed strategy, the implementation is carried out on an experimental set up. The set up includes STM32F407VG Digital Signal Controller (DSC) from ST Microelectronics [22]. It is a 32 bit DSC with ARM CORTEX M4 Architecture
operated with 168 MHz. The Analog to Digital Converter (ADC) clock is set to 84 MHz and the states are measured every 10 microseconds for the feedback control. The buck converter with the parameters as mentioned in Table 1 are used for the experimental set up. The current sensor CTSR 0.6P from LEM is used for sensing the current. The Fig. 6 shows the load voltage response under conventional SMC (a) and proposed SMC (b). It can be observed that the load voltage in Fig. 6(a) is 13.3 V for a given reference of 12.5 V while with the proposed SMC the load voltage is quite near around 12.47 V. The parameters are set as α = 600, and
Fig. 4. Load voltage and controller output for α ¼ 6000, γ ¼ 0.0020. Load disturbance from 100 Ω, to 32 Ω introduced at 0.2 s and reference voltage ¼ 19.5 V.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Fig. 6. Experimental results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Reference voltage ¼ 12.5 V.Load is fixed to 100 Ω.
Fig. 7. Experimental results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Load is 32 Ω. Reference voltage ¼ 12.5 V.
Fig. 8. Load voltage with conventional SMC when load varied from 100 to 32 Ω,α ¼ 6000 in (a) and when load is fixed to 100 Ω,α ¼ 6000, γ ¼ 0.0020 in (b). Reference voltage is 12.5 V.
α = 600, γ = 3.3 for conventional and proposed SMC respectively. The Fig. 7 shows that the experimental result for load voltage for the same tuning parameters set in Fig. 6 but with the load disturbance already introduced from 100 Ω to 32 Ω. However, it can be noticed that with the proposed SMC, load voltage remains unchanged while the same drops just below the reference 12.5 V in
presence of load disturbance. The proposed control strategy as mentioned earlier, results in better steady state performance by reducing steady state error. It can be observed from Fig. 8a and b that with the conventional SMC, the adverse effects on the load voltage due to load disturbances are quite apparent compared to the case of proposed SMC where there is almost no change in load
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Fig. 9. Load voltage with proposed SMC. when load is fixed to 100 Ω,α ¼ 6000, γ ¼ 0.0020. Reference voltage is 12.5 V.
Fig. 10. Load voltage, pulses when Load is 100 Ω,α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
Fig. 11. Load voltage,pulses, with fixed load 100 Ω, α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
voltage due to the load disturbances. Just to compare with Fig. 8b results, the Fig. 9 shows the load voltage under conventional SMC with fixed load of 100 Ω. The Figs. 10 to 13 Showing the load voltage responses with mentioned conditions and tuning parameters seemed quite satisfactory for the reference voltage of 19.5 V. It can be noticed from Fig. 13 that the higher value of the tuning parameter α, which is set 10000, results in a big overshoot. The DC DC Buck Converter used for experiment is shown in Fig. 14. More
Fig. 12. Load voltage,pulses, when Load changed from 100 Ω to 32 Ω, α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
Fig. 13. Load voltage, pulses when Load is 100 Ω,α ¼ 10000, γ ¼ 0.4. Reference is 19.5 V.
Fig. 14. The DC DC Buck Converter circuit.
understanding of the practical procedures can be acquired with Fig. 15. The green lines are ground lines of DSC. We have shown the connections of the current sensor powered by DSC board itself. The connections of the photocoupler/MOSFET Driver TLP250 from TOSHIBA are shown. The General Purpose Input Output (GPIO) port C pin 1 and 2 are used for sensing the analog values of the states for feedback control. The GPIO port E pin 9 is selected as controller output.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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References
Fig. 15. The DC DC Buck Converter with closed loop control. Note that the current sensor's output is scaled down(not shown in the figure)to be sensed by DSC GPIO pins.
6. Conclusion This paper proposed the sliding surface which is a PI type function of conventional sliding function for SMC. The SMC with the proposed sliding surface for a dc-dc buck converter eliminates the steady state error in the load voltage which does occur in case of conventional SMC. The proposed strategy is verified by simulation and experimentation. The steady state and dynamic behaviour of the system with the proposed control strategy was studied. The influence of tuning parameters on system performance are examined. The proposed strategy facilitates flexible choices of tuning parameters such that the ROE can be insensitive to load variations. The analysis for Region of Existence and the stability is carried out for the proposed control algorithm. The simulation as well experimental results shows that the proposed control strategy is quite satisfactory in presence of abrupt load variations. It is observed that the proposed SMC improves steady state error in load voltage.
[1] Utkin V, Guldner J, Shi J. Sliding mode control in electromechanical systems. London: Taylor & Francis; 1999. [2] Ang S, Oliva A. Power switching converters.Boca Raton: CRC Press, Taylor and Francis Group; 2005. [3] Middlebrook R, Cuk S. A general unified approach to modelling switching converter power stages, pp. 73–86, IEEE Power Electronics Specialists Conference, 1976. [4] Sira-Remirez H, Silva-Ortigoza R. Control design techniques in power electronics devices.London: Springer-Verlag; 2006. [5] Smedley KM, Cuk S. One-cycle contol of switching converters. IEEE Trans Power Electron 1995;10(6):625–33. [6] Martinez-Salamero L, Cid-Pastor A, Aroudi AE, Giral R, Calvente J, Ruiz-Magaz G. Sliding-mode control of dc-dc switching converters, pp. 1910–1916, Preprints of the 18th IFAC World Congress, Milano, Italy; 2011. [7] Edwards C, Spurgeon S. Sliding Mode Control Theory and Applications.London: Taylor & Francis; 1998. [8] Mehta A, Bandyopadhyay B. Frequency-shaped and observer-based discretetime sliding mode control. Springer; 2015. [9] Caceres R, Barbi I. A boost dc-ac converter analysis, design and experimentation. IEEE Trans Power Electron 1999;14(1):134–41. [10] Spiazzi G, Mattavelli P, Rossetto L. Sliding Mode Control of DC-DC Converters, University of Padova, Italy. [11] Maity S, Suraj Y. Analysis and modeling of an ffhc-controlled dc-dc buck converter suitable for wide range of operating conditions. IEEE Trans Power Electron 2012;27(12):4914–24. [12] Wang Q, Li T, Feng J. Discrete time synergetic control for dc-dc converter, pp. 1086–1091, PIERS Proceedings, Xi'an, China; 2010. [13] Perry AG, Fen G, Liu Y-F, Sen P. A new sliding mode like control method for buck converter, pp. 3688–3693, In: Proceedings of the 35th Annual IEEE Power Electronics Specialists Conference, Aachen, Germany; 2004. [14] Komurcugil H. Adaptive terminal sliding-mode control strategy for dc-dc buck converters, ISA Transactions, Elsevier, vol. 51, pp. 673–681, 2012. [15] Tan SC, Lai YM, Tse CK. On the practical design of sliding mode voltage controlled buck converter. IEEE Trans Power Electron 2005;20(2):425–36. [16] Castilla M. Control Circuits in Power Electronics: Practical issues in design and implementation. IET; 2016. [17] Tan SC, Lai YM, Tse CK. Sliding mode control of switching power converters. Boca Raton: CRC Press; 2012. [18] Tan SC, Lai YM, Tse CK. Indirect sliding mode control of power converters via double integral sliding surface. IEEE Trans Power Electron 2008;23(2):600–11. [19] Naik B, Mehta A. Sliding mode controller with modified sliding function for buck type converters, In: Proceedings of the 22nd IEEE International Symposium on Industrial Electronics,Taipei; 2013. [20] Naik B, Mehta A. Adaptive sliding mode controller with modified sliding function for dc dc boost converters, IEEE International Conference on Power Electronics Drives and Energy Systems,Mumbai; 2014. [21] Mohan N, Undeland T, Robbins W. Power electronics.Hoboken, NJ: John Wiley & Sons Inc; 2003. [22] STMicroelectronics, Reference manual:STM32F4 advance ARM based 32 bit MCUs; September 2011.
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Research article
Sliding mode controller with modified sliding function for DC-DC Buck Converter B.B. Naik a,n, A.J. Mehta b a Sarvajanik College of Engineering & Technology, Instrumentation & Control Engineering Department, Dr. R.K. Desai Marg, Near Mission Wockhardt Hospitals, Athwalines, Surat 395001, India b Electrical Engineering, Institute of Infrastructure Technology Research And Management (IITRAM), Ahmedabad 380026, India
art ic l e i nf o
a b s t r a c t
Article history: Received 28 February 2015 Received in revised form 17 April 2017 Accepted 18 May 2017
This article presents design of Sliding Mode Controller with proportional integral type sliding function for DC-DC Buck Converter for the controlled power supply. The converter with conventional sliding mode controller results in a steady state error in load voltage. The proposed modified sliding function improves the steady state and dynamic performance of the Convertor and facilitates better choices of controller tuning parameters. The conditions for existence of sliding modes for proposed control scheme are derived. The stability of the closed loop system with proposed sliding mode control is proved and improvement in steady state performance is exemplified. The idea of adaptive tuning for the proposed controller to compensate load variations is outlined. The comparative study of conventional and proposed control strategy is presented. The efficacy of the proposed strategy is endowed by the simulation and experimental results. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: DC-DC converters Buck convertor Sliding mode control
1. Introduction The buck converter is the most popular in industry due to its high efficiency and simplicity which emanates from its linear and minimum phase type by nature. Also known as step down converter, a buck converter finds its applications for majority of electronics devices including power supplies for computers. Buck converters and other Power Electronic Converters(PEC) functioning in closed loop control mode is the need of many industrial as well as domestic applications like instrumentation systems, ac-dc drives, telecommunications and renewable energy systems. High quality controlled power supply with a good control system is required for better dynamic and steady state performance. Although controllers with Pulse Width Modulation(PWM) techniques are already in action since long, the Sliding Mode Control (SMC) enters with a new dimension in the field of control systems for PEC. The modeling and control techniques for PEC are available in the wide variety of literature [1,2]. Cuk [3] proposed a general unified approach to modeling switching power converters. The control and modeling of various PECs can be also be found in [4] along with various control methodologies including linear and non n
Corresponding author. E-mail addresses: [email protected] (B.B. Naik), [email protected] (A.J. Mehta).
linear control techniques. The important task in many of the PECs is control of the output/load voltage. In [5], a large-signal nonlinear control technique(One Cycle Control) of switching power converters is developed to dynamically control the duty ratio of a switch such that in each cycle the average value of the controlled variable is proportional to the control reference. The technique is good at rejecting source power perturbations. However, the load disturbances are remained to study. The voltage regulation of Pulse Width Modulation(PWM) based DC-DC switching converter is discussed in [6] with the controller designed in frequency domain. It uses the averaged small signal model of the converter. The SMC tightened its grip in the field of Power Electronics after Utkin [1] discussed about wide class of PECs which may be controlled with the SMC technique. Since then SMC is applied to the various PEC and Electromechanical Systems [1,7]. The fundamentals of SMC can be found in [8]. The SMC has been the choice of many researchers to control the output voltage of PECs. Caceres and Barbi [9] presented design, analysis and experimentation of DC-AC boost converter with SMC. The technique is useful for design of Uninterruptible Power Supply(UPS) and Inverters. The SMC for DCDC PEC is available in [10] with graphical and analytical explanations. Maity [11] proposed Fixed Frequency Hysteresis Controller (FFHC) that uses both SMC and FFHC with hysteresis band. In [12] and [13] they have described SMC based technique for control of PECs. Hasan [14] suggested the Adaptive Terminal SMC for DC-DC Buck Converter having non-linear sliding surface with finite time reaching law. But the Region Of Existence(ROE)of sliding modes
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was not defined. However, the load variation is examined for its effects on load voltage. Tan [15]suggested the implementation of SMC for DC-DC Buck Converters with Analog Components. Although control circuit are the heart of PEC functioning in closed loop mode, its design is not straightforward. In [16] they described practical issues related to the design and implementation of these control circuits, with a focus on the presentation of the state-ofthe-art control solutions, including circuit technology, design techniques, and implementation issues of analog and digital electronic circuits. The conventional SMC for many PEC applications results in poor steady state performance. More precisely, the main obstacle in application of SMC for power converters is steady state error due to practical limitations and varying switching frequency [17]. The limitations of the conventional SMC are addressed in [18,17]. The tuning of parameters play very vital role for the steady state and dynamic performance of the converter under load variation. And also the power converters may become unstable due to improper selection of tuning parameters or due to the distortion of control signal caused by the saturation of the amplified feedback signal [17]. Moreover, by using conventional sliding function, the choice in selection of tuning parameter is restricted because the one and only tuning parameter has to perform multiple tasks: (i) govern speed of response of states; (ii)compensate load variations. As the conventional SMC has only one tuning parameter [17] so in case of load variations the parameter must be varied according to load variations to prevent variations in the Region Of Existence (ROE)where Sliding Mode exists. In this paper, authors have extended the theoretical results given in [19,20] to attain both the tasks as above simultaneously by tuning the parameters. The proposed SMC with the two tuning parameters proved to be better in the sense that one of them can be adjusted according to the load variations and the other is adjusted according to the required speed of response of the system states. The proposed modified sliding function improves the steady state performance at the cost of increased order of the sliding surface. The analysis to evaluate steady state performance in case of conventional and proposed SMC is carried out with mathematical arguments. It is proved that the steady state error in the load voltage does occur if conventional SMC is used and it can be eliminated with proposed SMC. The performance of DC-DC Buck Converter with modified sliding function based SMC is evaluated through analysis, simulation and experimentation. The analysis is carried out to prove stability in the ROE of Sliding Mode and to guarantee Reaching Mode. The discussion for tuning of controller parameters is presented. The paper is organized as follows: In Section 2, the mathematical model of DC DC Buck Converter is briefly explained. The main contribution i.e. modified sliding function is proposed followed by ROE and stability in Section 3. The simulation and experimental results are discussed in Section 4 and V respectively.
Fig. 1. DC-DC Buck Converter with modified sliding function based SMC.
compared to the load resistor to avoid loading effect. If Vref is a reference voltage, Vμ = βVref , is the scaled down version of reference voltage. L is an inductor, C is a capacitor, D is the freewheeling diode, Vo is the output or load voltage, Vi is the input voltage and rL is the load resistance. The SW is a n channel MOSFET switch turned ON or OFF with the output of SM controller which is in the form of pulses. Noting that u ¼1 means SW is closed and u¼ 0 means SW is open, the state space model of the electrical system can be derived by defining the states as follows:
x1 = Vμ − βVo,
(1)
x2 = x1̇ .
(2)
From Eqn. (2),
x2 = − β
dVo β = − iC . dt C
Where, iC is the capacitor current which can be measured as shown in the Fig. 1. Let iL and ir be the inductor current and load current respectively. The equation for x2 can be rewritten as,
x2 = −
β⎡ ⎣ iL − ir ⎤⎦. C
(3)
Let the voltage drop across inductor vL given by,
vL = (uVi − V0) = L
⟹iL =
∫
diL . dt
uVi − Vo dt. L
2. Mathematical modeling
So, Eq. (3) can be rewritten as,
A Sliding Mode Voltage Controlled (SMVC) DC-DC Buck Converter model is derived using state space approach [17]. A mathematical model of DC-DC Buck Converter as shown in Fig. 1 in open loop can be derived using basic circuit analysis laws. The converter is with the pure resistive load whose output voltage is to be controlled with SMC. The converter is assumed to be operated R in continuous current conduction mode [21]. Let β = R +2R is the
x2 = x1̇ =
1
2
voltage divider ratio. The values or R1 and R2 are very high
β ⎛ Vo − ⎜ C ⎝ rL
∫
uVi − Vo ⎞ dt ⎟. L ⎠
(4)
Hence,
x2̇ = −
Vμ βV 1 1 x1 − x2 − i u + . LC rLC LC LC
(5)
From Eqs. (2) and (5), the state space model of buck converter system is obtained as,
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⎡ ⎡ 0⎤ ⎡ 0 ⎤ 1 ⎤ ⎡ x1̇ ⎤ ⎢ 0 ⎥⎡ x1⎤ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥=⎢ 1 1 ⎥⎢ ⎥ + ⎢ βVi ⎥u + ⎢ Vμ ⎥. − ⎣ x2 ⎦ − ⎣ x2̇ ⎦ ⎢ − ⎢⎣ LC ⎥⎦ ⎣ LC ⎦ rLC ⎥⎦ ⎣ LC
(6)
3
about the use of modified sliding function so that restriction in selection of α is eliminated and better responses with load variation compensation is achieved through additional tuning parameter. The proposed sliding function in Section 3 not only eliminates the restriction in the choice of α but also gives better performance.
2.1. Sliding mode control Let the sliding function S which establishes linear relationship among states be defined as,
S = αx1 + x2 = Jx.
(7)
Where, J = [α , 1], state vector x = [x1, x2]T , α is a scalar, α > 0, and it controls the first order dynamics of Eq. (7). By reducing the value of α one can slow down the reaching mode dynamics. It is known that Sliding mode control occurs if the reaching condition
SS ̇ < 0,
(8)
is satisfied [1,7].The sliding mode control law can be defined as per the following rule:
⎧ 1 = ON , S > 0 u=⎨ ⎩ 0 = OFF , S < 0
(9)
The above control law Eq. (9), will trigger the switching across the sliding manifold S. The switching device has operating frequency limitations and hence for practical implementation the following control law is preferred:
⎧ 1 = ON , S > ε u=⎨ ⎩ 0 = OFF , S < − ε
(10)
where ε is a small positive number. Control law in Eq. (10) can reduce the severity of chattering by limiting chattering frequency and hence it is feasible for implementation. The existence of sliding modes requires the following two conditions to be satisfied [1]. From Eqs. (6), (7), (8) and (9),
⎛ Vμ − βVi 1 ⎞ 1 < 0, B1 = ⎜ α − x1 + ⎟ x2 − rLC ⎠ LC LC ⎝
(11)
3. Modified sliding function Let a Proportional Integral type function of sliding function be defined as,
E=S+γ
(12)
⎧ B = Jx ̇ for 0 < S < ε 1 Where, ⎨ . ⎩ B2 = Jx ̇ for − ε < S < 0 The Region Of Existence(ROE) of sliding modes on the phase plane is the region where condition SS ̇ < 0 is satisfied. From Eqs. (11) and (12), the ROE is determined by the three lines B1 = 0, B2 = 0 and S¼ 0 on the phase plane. It can be seen that there are two possibilities, α >
1 rLC
and α <
1 . rLC
It may be observed that the
slopes of lines B1 = 0 and B2 = 0 changes due to variation in rL. This imposes a limitation on dynamic behaviour of the system as α determines the speed of response. The choice of α is important due to the first order dynamics are given by,
x1(t ) = x1(t0)exp−α(t − t0) .
(14)
(15)
Similarly taking Laplace Transformation of Eq. (14) and from above Equation,
E(s ) E(s ) S(s ) (s + α )(s + γ ) = = x1(s ) S(s ) x1(s ) s
(16)
Obviously the above equation has PID (Proportional Integral Derivative) controller type of characteristics. It is well known that increasing the order of the controller can lead to better performance. However, in this case smooth operation is not possible due to the presence of switch and hence commutations around E ¼ 0 occur as per the following rule.
⎧ 1 = ON , E > ε u=⎨ . ⎩ 0 = OFF , E < − ε
(17)
Ideally the Eqn. (17) can also be stated as,
1 [1 + sgn(E )]. 2
(18)
The control law in Eq. (17) is used for actual system to obtain simulation and experimental results. Switching across E ¼ 0 yields the following important result. Setting E ¼ 0 in Eq. (14),
S=−γ
∫0
t
Sdt.
(19)
This is equivalent to,
S ̇ = − γS.
(20)
Which leads to minimization of S. The existence of sliding modes is possible on the phase plane if the following two conditions are satisfied.
lim E ̇ < 0
E → 0+
(21)
(13)
Where x1(t0) is the initial state at time t0. The choice of α is important for the speed of response of states and allowing the maximum possible ROE on the phase plane. To meet the requirements, α = r 1C is chosen [17]. This restricts the selection of α as it L
Sdt.
S(s ) =s+α x1(s )
⎪
⎪
t
Where γ > 0 is a scalar. Here, the integral term is used to achieve better steady stage performance of the load voltage under closed loop control. This fact can be visualized from the following equations. Taking Laplace Transformation of Eq. (7),
u= ⎛ Vμ 1 ⎞ 1 > 0. B2 = ⎜ α − x1 + ⎟ x2 − rLC ⎠ LC LC ⎝
∫0
governs the first order dynamics of Eq. (13). If it may be possible to introduce one more tuning parameter that can compensate the load variation, the goal can be achieved. The next section discusses
lim E ̇ > 0
E → 0−
(22)
The following theorem proves the existence of Reaching Mode (RM) [7]and to understand the stability of the system with proposed control strategy. Theorem 1. The switching across Modified Sliding Function E leads the phase trajectory to reach to E, S = 0 manifold in finite time and
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then to minimization of E, S i.e. Sliding Modes occur if the condition EE ̇ < 0 is satisfied. Proof. If
γ is integral gain and E > 0, then, Ė < 0,
1) Stability and Equivalent Control Law: The sufficient condition for existence of sliding modes is,
⇒S ̇ + γ S < 0 ⇒S ̇ < − γ S ⇒SS ̇ < − γS2 ⇒
EE ̇ < 0.
(25)
The Equivalent Control law, ueq can be derived as follows:
1d 2 (S ) < − γS2 2 dt
E ̇ = 0 ⇒ S ̇ + γS = 0
⇒S2(t ) < 2S2(t0) exp [−2γ (t − t0)].
From above equation and Eqs. (5), (6) and (7),
Where, t0 is the initial time and γ > 0. The above condition is important as it states that the S2 must be decreasing any time if the sliding function E assumes positive value. Similarly, S2 must be increasing if E assumes negative value. Once the phase trajectory hits the sliding manifold in the ROE, where EE ̇ < 0 is satisfied, the Sliding Mode, SM [7] occurs and stability prevails. The following subsection will be helpful to study the situation for which the reaching condition, EE ̇ < 0, is satisfied and to find the ROE. The ROE is the region on the phase plane for which the reaching condition is valid. 3.1. The ROE and stability
αx1̇ + x2̇ + γαx1 + γx2 < 0. Now from above equation and Eq. (6), with u¼ 1,
⎛ −βVi + Vμ ⎞ ⎛ 1 ⎞ ⎛ 1⎞ ⎟ + γαx1 + γx2 < 0. αx2 + ⎜ − ⎟ x1 + ⎜ − ⎟ x2 + ⎜ ⎝ LC ⎠ LC ⎝ rLC ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ βV ⎞ ⎛ Vμ ⎞ ⎛ 1 1⎞ ⎟ x1 + ⎜ − i ⎟ + ⎜ ⎟ < 0 ⇒A1 = ⎜ α − + γ ⎟ x2 + ⎜ αγ − ⎝ ⎝ LC ⎠ ⎝ LC ⎠ rLC LC ⎠ ⎝ ⎠ (23) Similarly, from Eq. (19) and considering u¼ 0 for E < 0 we may write,
⎛ Vμ ⎞ ⎛ ⎞ ⎛ 1 1⎞ ⎟ x1 + ⎜ ⎟ > 0. + γ ⎟ x2 + ⎜ αγ − A2 = ⎜ α − ⎝ ⎠ rLC LC ⎝ ⎠ ⎝ LC ⎠
ueq =
⎛ ⎛ LC ⎡ Vμ 1⎞ 1 ⎞ ⎤ ⎢ + ⎜ αγ − ⎟ x1 + ⎜ α + γ − ⎟ x2⎥. βVi ⎢⎣ LC ⎝ LC ⎠ rLC ⎠ ⎥⎦ ⎝
(24)
It can be seen from Eqs. (23) and (24) that the ROE is determined with lines A1 = 0, A2 = 0 and S ¼ 0 on the phase plane [19]. It may be noticed that the dynamics of system states and the ROE on the phase plane can be determined with two independent parameters α and γ respectively. By selecting γ = r 1C L
the ROE on phase plane is insensitive to load variations. However, it should be noted that the higher values of α can lead to sustained oscillations or peak overshoots/undershoots in the load voltage [17]. Hence the values of α and or γ must be low enough to prevent oscillations or overshoots in the load voltage. Selecting α as high as possible,faster dynamics can be achieved. The parameter γ can be adaptively tuned as per the changes in the load resistance. Practically this is possible with the following relationship:
(26)
The equivalent control law alone is not enough and practically we must add some switching action to it. Now let us define the control law u as,
u=
⎤ ⎛ ⎛ LC ⎡ Vμ 1⎞ 1 ⎞ ⎢ + ⎜ αγ − ⎟ x1 + ⎜ α + γ − ⎟ x2 + Ksgn(E )⎥. ⎥⎦ βVi ⎢⎣ LC ⎝ LC ⎠ rLC ⎠ ⎝
(27)
Where K is the switching control gain. Rewriting Eq. (5) as,
x2̇ = −
To evaluate ROE on the phase plane, consider S ̇ + γS < 0 can be simplified as,
γ=
3.2. Further analysis on stability, switching frequency and steady state performance
Vμ βV 1 1 + d(t ), x1 − x2 − i u + LC rLC LC LC
(28)
where, d(t) represents the parametric uncertainty, we can now derive the condition for the robustness for parametric uncertainties of the system model. Analytically, it is known that the following condition must be satisfied for the sliding mode control to exist.
̇ 0 EE< ⇒E(S ̇ + γS )< 0
(29)
⇒E(αx1 + x2̇ + γαx1 + γx2) < 0
(30)
Now, if u in action, Eqs. (5), (23), (24) and from Eq. (28) above Eq. (29) can be shown as
E[ − Ksgn(E ) + d(t )] < 0.
(31)
Hence, for the sliding modes to exist, the following condition must be satisfied.
K > d(t )
(32)
2) Switching Frequency: Practically there are physical limitations of the switching element hence there exists upper limit in the switching frequency of the converter. Let the dead-zone in the switching element be represented by a small positive number ε. Then the switching across E can be viewed as per the following Fig. 2.
i 1 = r rLC VoC
Where, ir is the load current. Hence, the setup for adaptive control requires the measurement of load current. Note that due to variations in the load resistance the ROE on the phase plane does not vary. The more practical way to fix the ROE on the phase plane, and which is independent on load variation, is to select α, γ very high compared to 1 in Eqs. (23), (24). rLC
Fig. 2. Switching with sliding function with dead zone.
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Now, for the time period TON , we get,
TON =
δ = αx1 + x1̇ .
2ε − Ė
(33)
2ε +. Ė
(34)
and,
TOFF =
+ − Here, E ̇ and E ̇ denotes E ̇ > 0 and E ̇ < 0 respectively. Considering that the systems states are very near to zero during the sliding modes, we can write the following set of equations during TON and TOFF. Considering this fact and Eqs. (6), (17), (21) and (22) we get,
⎛ βV ⎞ ⎛ Vμ ⎞ Ė = ⎜ − i ⎟ + ⎜ ⎟ ⎝ LC ⎠ ⎝ LC ⎠
(35)
and,
(36)
From Eqs. (31), (32), (33) and (34) we can derive the expressions for TON and TOFF as follows:
TON
2εLC = ( − βVi + Vμ)
(44)
Taking the Laplace Transformation and rearranging the terms one can write the expression of x1(s ) which is the x1 in so called s domain, and s is the Laplace Operator as per the following equation.
x1(s ) =
x (0) δ + 1 . s(s + α ) s+α
lim x1(t ) = lims (x1(s )) = s→0
2εLC = Vμ
TON
(38)
δ=S+γ
∫0
t
Sdt,
1 + TOFF
(39)
(Vμ − βVi )Vμ 2εLC[2Vμ − βVi ]
(40)
1
Defining w 2 = LC and considering the fact that Vμ = βVref , Eq. (39)can be rewritten as,
fs = Vref
(47)
∫0
t
(αx1 + x1̇ )dt.
(48)
Rearranging the terms and finding out the expression for x1(s ) it can be written,
x1(s ) =
From Eqs. (35), (36) and (37) the switching frequency can be given by,
fs =
(46)
(37)
Now the switching frequency fs is given by,
fs =
δ α
It is very clear from Eq. (46) that the state x1(t ) is not forced to zero, instead, the higher values of α can help to minimize the state which is the goal. But the state practically can't reach to zero and hence some steady state error is expected as can be observed with simulation results. Now, repeating the same procedure as above and assuming the value of E to be δ and from Eq. (7) one can rewrite Eq. (14) as,
δ = αx1 + x1̇ + γ TOFF
(45)
where, x1(0) is the initial condition of the state. It is now obvious that the final value of the state x1(t ) can be obtained by the final value theorem as, t →∞
−
⎛ Vμ ⎞ + E ̇ = ⎜ ⎟. ⎝ LC ⎠
5
δ + x1(0)s s 2 + (α + γ )s + γα
.
(49)
Hence, in case of Eq. (49),
lim x1(t ) = lims (x1(s )) = 0.
t →∞
s→0
(50)
It can now be clear that the switching across the proposed sliding function E ¼0 can lead to zero steady state error compared to the case of conventional sliding function S ¼0 which gives small but non zero steady state error. This fact can be visualized by observing Fig. 3.
(Vref − Vi )βw 2 2ε(2Vref − Vi )
(41)
Note that the switching frequency is inversely proportional to the size of the dead zone in the switching element ε and it depends on other systems parameters as mentioned in Eq. (40). 3) Steady State Performance: Exact analysis of steady state error is complicated due to the presence of switching element. However, in this section it is tried to evaluate mathematically the performance of proposed control law in terms of steady state performance. It is reasonable to assume V as the uniform neighbourhood of Sr which is a set of all the numbers in S those are very small in magnitude such that,
Sr = ⋃ Br (p), p∈S
Br (p) = {x ∈ S|d(x, p) < r} .
(42)
(43)
For p ¼0 and r is the centre and radius of an open ball, from the above equations one can immediately notice that for any point δ ∈ Sr from (1), (2) and (7),
4. Simulation results To show the efficacy of the proposed algorithm, a MATLAB simulation is carried out with the system parameters as given in Table 1. It can be observed from Fig. 3 that for the tuning parameter settings, α ¼ 600 and γ ¼ 3.3, the load voltage tracks well the reference voltage 12.5 V (dashed line). However, the load disturbance occurs at time 2.5 s i.e. the load resistance suddenly drops from 100 Ω to 32 Ω, the proposed SMC tracks the reference very accurately (Fig. 3(b)). With the conventional SMC, load voltage fails to track the reference and does exhibit small amount of steady state errors before and after the load disturbance (Fig. 3(a)). For the reference voltage of 19.5 V, the step response of load voltage is studied. The load resistance is suddenly varied from 100 Ω to 32 Ω. The parameter for Fig. 4 are α ¼ 6000 and γ ¼ 0.0020. The Fig. 4(a) shows the load voltage, reference and controller output. The better view is presented in Fig. 4(b) to study the effects of the load disturbance. Here, we can observe spike in load voltage due to small value of parameter γ chosen just to demonstrate its effect.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Fig. 3. Simulation results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Load disturbance from 100Ω, to 32Ω at 2.5 s and reference voltage ¼ 12.5 V. Table 1 System Parameters.
Fig. 5. Load Voltage for α ¼ 1000, γ ¼ 100.
The Fig. 5 shows the load voltage with tuning parameter set
α ¼ 1000 and γ ¼ 100 and for the reference of 19.5 V.
5. Experimental results To validate the proposed strategy, the implementation is carried out on an experimental set up. The set up includes STM32F407VG Digital Signal Controller (DSC) from ST Microelectronics [22]. It is a 32 bit DSC with ARM CORTEX M4 Architecture
operated with 168 MHz. The Analog to Digital Converter (ADC) clock is set to 84 MHz and the states are measured every 10 microseconds for the feedback control. The buck converter with the parameters as mentioned in Table 1 are used for the experimental set up. The current sensor CTSR 0.6P from LEM is used for sensing the current. The Fig. 6 shows the load voltage response under conventional SMC (a) and proposed SMC (b). It can be observed that the load voltage in Fig. 6(a) is 13.3 V for a given reference of 12.5 V while with the proposed SMC the load voltage is quite near around 12.47 V. The parameters are set as α = 600, and
Fig. 4. Load voltage and controller output for α ¼ 6000, γ ¼ 0.0020. Load disturbance from 100 Ω, to 32 Ω introduced at 0.2 s and reference voltage ¼ 19.5 V.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Fig. 6. Experimental results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Reference voltage ¼ 12.5 V.Load is fixed to 100 Ω.
Fig. 7. Experimental results for α ¼ 600 in (a) and α ¼ 600, γ ¼ 3.3 in (b). Load is 32 Ω. Reference voltage ¼ 12.5 V.
Fig. 8. Load voltage with conventional SMC when load varied from 100 to 32 Ω,α ¼ 6000 in (a) and when load is fixed to 100 Ω,α ¼ 6000, γ ¼ 0.0020 in (b). Reference voltage is 12.5 V.
α = 600, γ = 3.3 for conventional and proposed SMC respectively. The Fig. 7 shows that the experimental result for load voltage for the same tuning parameters set in Fig. 6 but with the load disturbance already introduced from 100 Ω to 32 Ω. However, it can be noticed that with the proposed SMC, load voltage remains unchanged while the same drops just below the reference 12.5 V in
presence of load disturbance. The proposed control strategy as mentioned earlier, results in better steady state performance by reducing steady state error. It can be observed from Fig. 8a and b that with the conventional SMC, the adverse effects on the load voltage due to load disturbances are quite apparent compared to the case of proposed SMC where there is almost no change in load
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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Fig. 9. Load voltage with proposed SMC. when load is fixed to 100 Ω,α ¼ 6000, γ ¼ 0.0020. Reference voltage is 12.5 V.
Fig. 10. Load voltage, pulses when Load is 100 Ω,α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
Fig. 11. Load voltage,pulses, with fixed load 100 Ω, α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
voltage due to the load disturbances. Just to compare with Fig. 8b results, the Fig. 9 shows the load voltage under conventional SMC with fixed load of 100 Ω. The Figs. 10 to 13 Showing the load voltage responses with mentioned conditions and tuning parameters seemed quite satisfactory for the reference voltage of 19.5 V. It can be noticed from Fig. 13 that the higher value of the tuning parameter α, which is set 10000, results in a big overshoot. The DC DC Buck Converter used for experiment is shown in Fig. 14. More
Fig. 12. Load voltage,pulses, when Load changed from 100 Ω to 32 Ω, α ¼ 6000, γ ¼ 0.0020. Reference is 19.5 V.
Fig. 13. Load voltage, pulses when Load is 100 Ω,α ¼ 10000, γ ¼ 0.4. Reference is 19.5 V.
Fig. 14. The DC DC Buck Converter circuit.
understanding of the practical procedures can be acquired with Fig. 15. The green lines are ground lines of DSC. We have shown the connections of the current sensor powered by DSC board itself. The connections of the photocoupler/MOSFET Driver TLP250 from TOSHIBA are shown. The General Purpose Input Output (GPIO) port C pin 1 and 2 are used for sensing the analog values of the states for feedback control. The GPIO port E pin 9 is selected as controller output.
Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i
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References
Fig. 15. The DC DC Buck Converter with closed loop control. Note that the current sensor's output is scaled down(not shown in the figure)to be sensed by DSC GPIO pins.
6. Conclusion This paper proposed the sliding surface which is a PI type function of conventional sliding function for SMC. The SMC with the proposed sliding surface for a dc-dc buck converter eliminates the steady state error in the load voltage which does occur in case of conventional SMC. The proposed strategy is verified by simulation and experimentation. The steady state and dynamic behaviour of the system with the proposed control strategy was studied. The influence of tuning parameters on system performance are examined. The proposed strategy facilitates flexible choices of tuning parameters such that the ROE can be insensitive to load variations. The analysis for Region of Existence and the stability is carried out for the proposed control algorithm. The simulation as well experimental results shows that the proposed control strategy is quite satisfactory in presence of abrupt load variations. It is observed that the proposed SMC improves steady state error in load voltage.
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Please cite this article as: Naik BB, Mehta AJ. Sliding mode controller with modified sliding function for DC-DC Buck Converter. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.05.009i