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Robust observer-based DC-DC converter control
Accepted Manuscript Original article Robust observer-based DC-DC converter control Montadher S. Shaker, Asaad A. Kraidi PII: DOI: Reference:
S1018-3639(17)30068-5 http://dx.doi.org/10.1016/j.jksues.2017.08.002 JKSUES 258
To appear in:
Journal of King Saud University - Engineering Sciences
Received Date: Accepted Date:
3 March 2017 16 August 2017
Please cite this article as: Shaker, M.S., Kraidi, A.A., Robust observer-based DC-DC converter control, Journal of King Saud University - Engineering Sciences (2017), doi: http://dx.doi.org/10.1016/j.jksues.2017.08.002
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Robust observer-based DC-DC converter control Montadher S. Shaker* and Asaad A. Kraidi* *
Department of Electrical Engineering, University of Technology, Baghdad, Iraq E-mail: [email protected] , [email protected] [email protected]
1
Corresponding author: Dr. Montadher Sami Shaker 30211@uotechno logy.edu.iq montadher_979@ yahoo.com
Robust observer-based DC-DC converter control Montadher S. Shaker* and Asaad A. Kraidi* *
Department of Electrical Engineering, University of Technology, Baghdad, Iraq E-mail: [email protected] , [email protected] [email protected]
Abstract: This paper develops a novel robust control strategy for DC-DC buck converter subjected to varying load and parameter uncertainty. The proposal exploits the robustness of the sliding mode control (SMC) incorporated with proportional-proportional-integral-observer (PPIO) to assure tight reference voltage tracking under a wide range of load change scenarios. Within this framework, an integral sliding mode surface based SMC (ISMC) is designed to guarantee closedloop robustness against the matched and mismatched disturbance components of the load uncertainty. Subsequently, a novel control structure comprises ISMC and PPIO is presented to overcome the design constraints and to mitigate the undesired transient response accompany the response of the closed-loop system based ISMC. Stability analysis has clearly demonstrated using linear matrix inequality (LMI) and Lyapunov approach. To illustrate the effectiveness of the proposal, a comparison between the closed-loop system responses of SMC, ISMC, and the combined ISMC and PPIO are presented in the simulation results. Keywords: Sliding mode control, proportional-proportional-integral-observer (PPIO), DC-DC converter, mismatched uncertainty, robust control. 2 Introduction: The DC-DC converters are an important part in modern technologies that require reliable power supplies such as computer systems and cellular phones or they could be used as power optimizers in renewable energy such as photovoltaic and wind turbine systems (Forsyth and Mollov, 1998). From circuit topology standing point, there are wide varieties of converter circuits ranging from simple to complex configuration of buck, boost, or buck/boost converters (Forsyth and Mollov, 1998). From control design standing point, tackling the effects of load variation and parameter uncertainty of DC-DC converter circuits have drawn the attention to develop control strategies that ensure reliable converters. Additionally, the time response of the converters must satisfy desired transient characteristics. However, owing to the nonlinear characteristics of the DC-DC converters and their highly uncertain model parameters, several challenges regarding their control have stimulated several researchers to exploit robust control schemes in order to cope with the required control objectives (El Fadil et al., 2009, Olalla et al., 2011, Nachidi et al., 2013, Wang et al., 2015). In this context, the authors in (El Fadil et al., 2009) proposed a backstepping-based control design algorithm that accounts for the effect of model parameter uncertainty. The work in (Olalla et al., 2011) exploits the potential of LMI-based multi-objective robust state feedback controller to account for model uncertainty, nonlinearity, and exogenous input. An -based robust output feedback fuzzy control for DC-DC converter is presented in (Nachidi et al., 2013) to account for the converter nonlinearity using Takagi-Sugeno fuzzy models. On the other hand, recent publications have shown interest in utilizing the relative design simplicity and the robustness of SMC to deal with DC-DC converter control problems (Gonzalez Montoya et al., 2016, Kanimozhi and Shunmugalatha, 2013, Liqun et al., 2015, Lopez-Santos et al., 2013, Oucheriah and Liping, 2013, Salimi et al., 2015, Shen et al., 2015, Yue et al., 2014, Wang et al., 2015). Specifically, the work presented in (Wang et al., 2015) combines the SMC and a disturbance observer such that the converter circuit exhibits robustness against mismatched uncertainties that have specific time behaviour. To guarantee robust reference voltage tracking of a converter circuit subjected to time varying external input, the authors in (Oucheriah and Liping, 2013) developed
an adaptive SMC scheme to provide asymptotic stability of the boost converter system. A sliding mode based boost converter control is presented in (Yue et al., 2014) to stabilize the DC power systems over the entire operating range in the presence of significant variations in load power and input voltage. An output voltage control of buck/boost converter based on a proportional-integral type hyper-plane SMC is presented in (Salimi et al., 2015) to provide robustness against parameters uncertainties, load disturbance and variations of the converter input voltage. In the aforementioned literature, the authors presented controllers that meet specific performance requirements and/or cope with a variety of DC-DC robustness issues. For instance, the work in (Wang et al., 2015) does not present systematic control system design procedure. Moreover, the proposal assumes pre knowledge of the upper bound of the disturbance signal. Additionally, the closed-loop robustness is constrained to step varying load change. On the other hand, the complexity of the proposal in (Nachidi et al., 2013) is attributed to the use of multiple model control. In (Kanimozhi and Shunmugalatha, 2013, Liqun et al., 2015, Lopez-Santos et al., 2013, Oucheriah and Liping, 2013, Salimi et al., 2015, Shen et al., 2015, Siew-Chong et al., 2006, Yue et al., 2014) the proposed sliding mode controllers are insensitive to the matched uncertainty and sensitive to the mismatched uncertainty. Hence, developing a control strategy that characterized by design simplicity and robustness against matched and time varying mismatched uncertainty is of significant contribution to the literature of DC-DC converter control. This work aims to design a robust control strategy to ensure acceptable performance and robustness of DC-DC converter circuit that affected by the variable load and/or parameter uncertainty. A novel control structure that comprises the ISMC and PPIO has been proposed to ensure robustness against matched and mismatched uncertainty. The proposal removes any offset caused by the mismatched uncertainty without the need for pre knowledge of the upper bound of the uncertainty. Moreover, unlike the work presented in (Wang et al., 2015), a systematic observer design is presented to ensure closed-loop robustness for a wide range of load change scenarios. 3 The DC-DC buck converter model This section establishes the state space model of a typical DC-DC buck converter. The buck converter circuit comprises a transistor switch, inductor, capacitor, and variable load resistance can be found in (Wang et al., 2015)(see Fig.1a). The circuit reveals two models arising from the particular position, ON (see Fig.1b) and OFF(see Fig.1c), of the switch. Based on the direct application of Kirchhoff’s voltage and current laws for the ON and OFF positions, the two models are given in Eq. (1) and Eq. (2) respectively.
L
Q
u DC
D
C R
Figure 1a DC-DC buck converter
L
DC
C R
Figure 1b the ideal switch is ON
L
DC
C R
Figure 1c the ideal switch is OFF
ν = i L − out R
(1)
ν out = iL − R
(2)
E = LiL + ν out C νout
LiL = −ν out C νout
Where in the input voltage, is the inductance, is the inductor current, is the output voltage, is the capacitance, and is the resistance. The average model that encompasses Eqs (1&2) is depicted in Eq. (3). LiL = µ E −ν out ν C νout = i L − out R
(3)
The average model presented in Eq.(3) assumes ideal circuit components and fixed load resistance. However, in order to consider the effects of load uncertainty, the model (3) has rewritten in term of nominal load ( Ro ) as follows: ν out ν out ν out = iL − + − R Ro R o
LiL = µ E −ν out C νout
(4)
Letting ( x 1 = ν out −ν r ) and ( x 2 =
i L ν out − ) yield, C R oC
x1 x2 x2 = − − + u + d2 LC Ro C
x1 = x2 + d1
(5)
where d1 =
ν out R oC
−
ν out RC
,u=
µ E −ν r LC
, and
d2 = −
1 d1 . R oC
where is the reference voltage, and the components ( d1 ) and ( d 2 ) represent the mismatched ( d1 ) and matched ( d 2 ) uncertainty. Hence, the objective is to design a robust controller for the system (5) despite the effects caused by uncertain load resistance.
4 The proposed control structure: This section explains the design steps of the proposed robust control structure that integrates the ISMC and PPIO within a feedback loop. The proposal follows the observer based robust control design methodology (Nazir et al., 2017, Sami and Patton, 2012) to tolerate a wide range of disturbance scenarios (i.e. load and parameter uncertainty). The closed-loop control system structure is shown in Figure 2.
$
"# !
Figure 2: the proposed control structure where is the DC input, is the control signal, is the reference DC voltage, and is the estimation of the mismatched uncertainty . Section 3.1 presents the ISMC design methodology for the buck converter model (5). On the other hand, subsection 3.2 will present a new control structure that combines the ISMC and PPIO in order to relax ISMC design constraints and performance limitation.
4.1
Sliding mode controller design
The ability of SMC to compensate matched disturbances has clearly investigated in the literature. Basically, the SMC design procedure encompasses two steps (Edwards and Spurgeon, 1998); (1) sliding surface design to which the states are confined during the sliding phase. (2) The control signal that makes the sliding surface attracts the states during the reaching phase. Specifically, the control law has two feedback control terms; (i) a linear control term, and (ii) a discontinuous term. While the linear control term guarantees the reachability condition, the discontinuous control term ensures the sliding condition.
Remark1: The proposed SMC needs the following assumption: Assumption1: The load uncertainty tackled by the proposed SMC must satisfy: 1-
β > δ + (d 2 + k 1d1 ) ,where () is the upper bound of uncertainty, ( δ ) is a small positive constant and k 1 is a design variable.
2-
The term d1 satisfies the limit lim d1 = 0 (i.e. has constant steady state value). t →∞
The objective is to guarantee acceptable system performance whilst the system confines into the sliding regime. Hence, to ensure acceptable sliding motion despite the effects of the matched and mismatched uncertainties (i.e. d1 and d 2 ) the following sliding surface has been proposed.
S = x 2 + k 1 x 1 + k 2 ∫ x 1dt
(6)
where () is the sliding variable, and and are design variables. Before proceeding with the second design step, analysis of system motion equation is given first. While the dynamics of system (5) confined into the sliding surface, the sliding variable must satisfies S = S = 0 . Hence, taking the time derivative of Eq. (6) yields:
S = x2 + k 1 x1 + k 2 x 1
(7)
Combining Eq.(5) and Eq.(7) gives:
1 1 S = u + ( )x 2 + (d 2 + k 1d 1 ) + k 2 )x 1 + (k 1 − LC RoC
(8)
To maintain sliding motion (i.e. S = 0 ), one could simply realize that the required control signal ( u ) should has the form: u = −(
1 1 )x 2 − (d 2 + k 1d1 ) + k 2 )x 1 − (k 1 − LC RoC
(9)
Clearly, the unknown disturbance terms makes the control signal (9) unrealistic. However, making use of assumption 1, a realistic form of the signal ( u ) is given in Eq.(10). u = −(
1 1 )x 2 − β sgn(S ) + k 2 )x 1 − (k 1 − LC RoC
(10)
Proper design of control (10) makes the sliding surface attracts the state of the system (5) into the sliding manifold within finite time ( t = t o ). Additionally, the states will remain within the sliding manifold for all t > t o . Now, consider a candidate Lyapunov function of the following form:
1 ∆= S2 2
(12)
the time derivative of Eq. (12) gives:
∆ = SS
(13)
For the control law (10), Eq.(14) gives simplified form of Eq. (8):
S = −β sgn(S ) + (d 2 + k 1d1 )
(14)
Combining Eq.(13) and Eq.(14) yields: ∆ = − β sgn(S )S + (d 2 + k 1d 1 )S
(15)
Eq.(15) can be rewritten into an inequality form as follows:
∆ ≤ S (−β + (d 2 + k1d1 ) )
(16)
Using assumption 1 gives:
∆ ≤ −δ S
(17)
Inequality (17) implies that, for initial condition outside the sliding manifold, i.e. S (t = 0) ≠ 0 , the solution to (17) becomes zero in finite time. Clearly, the consequence of (17) and (12) is:
∆ = − 2 δ ∆1/2 ,
(18)
which has the following solution:
∆ = ∆1/2 (0) − 2 δ t
(19)
Therefore, at time t = t o =
∆1/2(0)
2δ
, the states of system (5) reaches the surface (6).
Hence, using control law (10), the states of system (5) will converge toward the sliding surface (6) at time t o
< 0 ). provided that the control gain satisfies β > δ + (d 2 + k 1d1 ) (i.e. SS
While the state of (5) remains within the sliding vicinity, the dynamics of system (5) become:
x1 + k 1x1 + k 2 x 1 = d1
(20)
In fact, Eq.(20) becomes a homogeneous differential equation if the term d1 satisfies the limit of assumption1. This implies that; while sliding, the state in (5) moves toward the equilibrium point asymptotically.
Remark 2: • From Eq. (10), it should be noted that the model parameter uncertainty (i.e. the changes of L ,C ) represent matched uncertainty and hence can be tolerated via the control law (10). • While the control law (10) is capable of eliminating the steady state offset caused by ( d 1 ), conventional SMC lacks the capability to remove this offset. For comparison, consider the following conventional SMC:
Strad = x 2 + k1x 1 u=
(21)
1 1 )x 2 − g sgn(S trad ) x 1 − (k 1 − LC RoC
(22)
where Strad is the conventional sliding surface, ( g ) is the upper bound of the unknown input. If ( g ) is properly designed, the sliding motion will possess the dynamics (23): x1 = − k 1x + d 1
(23)
Clearly, it is impossible to eliminate the offset using conventional SMC.
Remark 3: The constraints listed in assumption 1 limit the applicability of the control law (10) because: • The upper bound of the unknown input (i.e. β > δ + (d 2 + k 1d1 ) ) increases the switching gain an thereby the negative effects of chattering. • The robustness of the closed-loop system in section 3.1 is restricted against disturbances that possess constant time behaviour (i.e. lim d1 = 0 ). t →∞
4.2
The design of LMI-based PPIO
This section presents the PPIO design steps and the vital role of PPIO in relaxing the constraints of section 3.1 (see remark 3). The converter system (5) can be rewritten in the following general form:
x = Ax + Bu + E d d1 y = Cx
(24)
T
T
where E d ∈ R n *m d = [1 −1/ R oC ] and x ∈ R n = [ x 1
x 2 ] . The PPIO is given as follows:
xˆ = Axˆ + Bu + E d dˆ1 + L ( y − yˆ ) yˆ = Cxˆ ˆ d 1 (t ) = P [ K o 1Cex + K o 2Ce x ]
(25)
where Ko1 and Ko 2 are the proportional and integral gains respectively, and P is a symmetric positive definite matrix. After subtracting the observer in (25) from the system (24) the state estimation error ( ex ) will be defined as: ex = ( A − LC ) e x + E d ed e y = C ex
(26)
Using eq.(25) the disturbance estimation error dynamics will become:
ed = d1 − dˆ1 = d1 − PK o 1CAe x + PK o 1CLCe x − PK o 2Ce x − PK o 1CE d ed
(27)
by combining Eqs. 26&27, the augmented estimation error dynamics can be constructed as defined in eq.(28):
ea (t ) = A ea + Nz
(28)
where A − LC A = K CA K o 1CLC − PK o 2C − P + P o1
e 0 Ed , ea = x , z = d1 , N = e − PK o1CE d I d
Now the objective is to compute the gains L , K o 1 , and K o 2 that attenuate the effects of the input z , in eq. (28), on the estimation error via minimizing the L2 norm
( z ) below desired level γ . 2
Remark4: decoupling the effects is of ( z ) is beyond the scope of this paper. However, based on the available information of E d , the following theorem ensures L2 norm minimization of ( z ) on ea .
Theorem1: The augmented estimation error in (28) is stable and the L2 performance is guaranteed with an attenuation level ,, Provided that the signals d1
( )
are bounded, rank (CEd ) = md , and the pair (A,C) is
observable, if there exists a symmetric positive definite matrices P1 , P −1 and G matrices H , K o 1 , K o 2 , and a scalar µo satisfying the following LMI constraint:
Minimize γ such that
Ψ11 Ψ12 * Ψ 22 * * * * * * * *
0 Ψ 23
0 Ψ 24
Ψ15 0
−γ I *
0 −G−1
0 0
* *
* *
−2µo P1 *
0 0 0 <0 0 µo I −G
(29)
where T
L = P1−1H , γ = γ , Ψ11 = P1A + ( P1A ) − HC − ( HC ) + w 1 , Ψ12 = P1E d − A T C T K o1T −C T K o 2T , Ψ15 = ( HC ) , T
T
T
Ψ22 = −K o1CE d − ( K o1CE d ) + w 2 , Ψ 23 = P −1 , Ψ 24 = K o 1C
Proof: See (Shaker, 2015). It is worth remarking that to tune the ℒ performance against the exogenous input .̃ , 3 0 4 has been nominated in the LMI formulation. the weighting matrix 0 = 2 0 3 The estimated disturbance ( ) plays a vital role in enhancing the overall closed loop performance. Specifically, the sliding surface (6) is modified to the following form:
S = x 2 + k1 x 1 + k 2 ∫ x 1dt + dˆ1
(30)
To maintain sliding motion (i.e. S = 0 ), the new control signal ( u ) has the form:
u = −(
1 1 )x 2 − (d 2 + k 1d 1 ) + (dˆ2 + k 1dˆ1 ) − dˆ1 − β new sgn(S ) + k 2 )x 1 − (k 1 − LC Ro C
(31)
Following the procedure of section 3.1, the closed-loop system stability can easily derive and hence omitted here.
Remark 5: the control signal in Eq.(31), the estimation dˆ1 compensates the effect of load uncertainty. Consequently, the design parameter β new can be selected much smaller than β , thereby minimising the chattering effect.
5 Simulation Results: In this section, the proposed SMC strategy has verified based on Matlab/Simulink software. Table 1 gives details of the considered converter parameter. The uncertain resistive load varies in the range (60-140 Ω) has subjected to the buck converter. Table 1: the parameters of the buck converter. Input voltage ( ) Desired voltage ( ) Capacitance ( ) Inductance () Load resistance ( )
20 V 10 V 1000 µ F
4.7 6 60Ω-140Ω
In the first simulation scenario, the results compare the output response of conventional SMC and SMC with integral action based buck converter control. The first design step of controller (10) is determining the switching gain ( β ) and the sliding surface parameters ( k 1 , k 2 ). The holding of assumption 1 ensures sliding motion governed by the dynamics of Eq. (20). Consequently, the performance parameters (the damping ratio ( ζ ) and natural undamped frequency ( ω n )) of
this system are adopted such that the maximum settling time ( T s ) satisfies T s ≤ 1sec and maximum peak time ( T p ) satisfies T p ≤ 0.5 sec . Hence,
T s ≤ 1sec ⇒
1
ζω n
T p ≤ 0.5sec ⇒
≤ 1sec ⇒
π ωn 1 − ζ
2
2 ≤ 1sec ⇒ 2 ≤ k 1 k1
≤ 0.5sec ⇒
π 2
k 2 (1 − ζ )
≤ 0.25sec ⇒
4π ≤ k2 , (1 − ζ 2 )
which led to k 1 = 25 , k 2 = 250 . On the other hand, the design parameter β = 500 is selected to satisfy
< 0 ). reachability condition (i.e. β > δ + (d 2 + k 1d 1 ) ⇒ SS In order to investigate controller robustness, the converter has subjected to nominal (100Ω), step-varying, and fast-varying load. Figure 3 shows tracking performance of the converter (5) when subjected to nominal load (100Ω). In this scenario, the controllers (10) and (22) can force the converter dynamics to start the sliding motion at time t o that depends on the variable δ (and consequently β ). 10.1 SMC I SMC
Output Voltage
10.08
10.06
10.04
10.02
10
9.98 0
2
4
6
8
10
Tim e (sec)
Figure 3: The tracking performance of the buck converter when subjected to nominal load. Figure 4 shows the output response of the buck converter when feeding a step-varying load. As demonstrated in remark 2, the traditional SMC is incapable of compensating the unmatched component d 1 . On the other hand, while the condition lim d1 = 0 holds for this scenario, the ISMC can compensate the effects of the unmatched t →∞
component d 1 . However, the closed-loop system based ISMC exhibit undesired transient response that disturbs the smoothness of the output voltage. 11.5 SMC ISMC
Output Voltage
11
10.5
10
9.5
9 0
2
4
6
8
10
Tim e (sec)
Figure 4: The tracking performance of the buck converter when subjected to step-varying load.
Figure 5 demonstrates how the controllers (10 & 22) are unable to maintain reference tracking when the unmatched disturbance does not satisfy the limit lim d1 = 0 . t →∞
10.25 10.2
SMC ISMC
Output Voltage
10.15 10.1 10.05 10 9.95 9.9 9.8 5 9.8 0
2
4
6
8
10
Tim e (sec)
Figure 5: The tracking performance of the buck converter when subjected to fast-varying load change. As mentioned in remark 2, the proposed ISMC and the SMC possess the ability to tolerate the converter parameter uncertainty. Figure 6 obviously demonstrates the closed-loop robustness against the following parameter change scenario: x1 x2 x2 = −α −α + u + d2 LC Ro C
x1 = x2 + d1
(32)
where (α ) represents the multiplicative uncertainty factor of the system (5). 10.05 SMC I SMC
10.04 10.03
Output voltage (v)
10.02 10.01 10 9.99 9.98
α =1
α =0.7
α =1.3
9.97 9.96 9.95
0
2
4
6
8
10
Tim e (sec)
Figure 6: The controller robustness against model parameter uncertainty The second simulation scenario highlights on the advantages of integrating the closed-loop system based-ISMC with the PPIO. Using the Matlab LMI toolbox, the gains of the PPIO are obtained as follows:
0.046
γ = 0.059, K o1 = [ −0.185 −3.248] , K o 2 = [ 0.003 −6.916] , L = −1.616
−0.001 . 9.954
Remark 6: •
The ISMC parameters ( k 1 , k 2 ) are maintained as given above (i.e. k 1 = 25 , k 2 = 250 ). However, in the combined ISMC and PPIO scheme, it becomes possible to reduce the switching gain to β new = 380 (see remarks 3 and 5).
•
While the PPIO guarantees precise estimation of load change when d1 ≠ 0 , the extended state observer (ESO) proposed in (Wang et al., 2015) is restricted to the case of d1 = 0 . Figure 7 shows the accuracy
of PPIO, compared with the ESO, for estimating d1 component.
(a) Time varying load change
(b) Step varying load change Figure 7 (a and b): PPIO and ESO based load change estimation. Figure 8 (a and b) shows the reference voltage tracking accuracy of buck converter-based SMC, ISMC, and combined ISMC and PPIO. Specifically, while tolerating the effects of time varying load change (i.e. d1 ≠ 0 ) is not possible via SMC and ISMC, the combined ISMC and PPIO has showed robust tracking performance against load change despite the fact that d1 ≠ 0 . 11.5
Output voltage
11
SMC I SMC I SMC+PPIO
10.5
10
9.5
9 0
2
4
6
8
10
Tim e(sec)
(a) Reference voltage tracking for step-varying load
10.25 10.2
SMC I SMC I SMC+PPIO
10.15
Output voltage
10.1 10.05 10 9.95 9.9 9.8 5 9.8 0
2
4
6
8
10
Tim e (sec)
(a) Reference voltage tracking for fast-varying load Figure 8 (a and b): Reference voltage tracking performance. With reference to Figure 7, it is clear that utilizing PPIO for load change estimation and compensation minimises the undesired transient response of ISMC. Finally, the duty ratio µ (t) is shown in figure 9(a and b). The combined ISMC and PPIO maintains the duty ratio unchanged despite the effects of load uncertainty.
0.5 8 0.56
SMC I SMC I SMC+PPIO
Duty ratio µ (t)
0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0
2
4
6
8
10
Tim e sec
(a) Duty ratio for step-varying load
0.52 0.5 15
SMC I SMC I SMC+PPIO
Duty ratio µ (t)
0.51 0.505 0.5 0.495 0.49 0.48 5 0.48 0
2
4
6
8
10
Tim e sec
(a) Duty ratio for fast-varying load Figure 7 (a and b): Duty ratio of the Buck converter. Tables 2&3 summarize the performance of SMC, ISMC, and ISMC+PPIO controllers for step and time varying load changes. Table 2: SMC, ISMC, and ISMC+PPIO controller performance for step load change.
SMC ISMC ISMC+PPIO
Min 9.087 9.355 9.913
Max 10.72 11.21 10.1
Mean 9.983 10 10
STD 0.3208 0.0767 0.0095
Table 3: SMC, ISMC, and ISMC+PPIO controller performance for time varying load change.
SMC ISMC ISMC+PPIO
Min 9.849 9.902 9.967
Max 10.15 10.1 10.11
Mean 10 9.998 10
STD 0.0728 0.0256 0.0066
6 Conclusions: This paper proposes a robust controller for buck converter feeding an uncertain load. The proposal extends the conventional sliding surface design methodology by integral action accounts for unmatched disturbance. However, the ISMC always brings undesired transient performance. Therefore, the proposal integrates the ISMC by PPIO to relax the design constraints, tolerate the undesired transient response, and estimate and compensate the effects of load change on tracking performance. Different simulation scenarios have examined the tracking performance of the proposed controller. The results have shown that the proposed ISMC+PPIO controller is capable of compensating the effects of load uncertainty regardless the time behaviour of the unmatched component. References EDWARDS, C. & SPURGEON, S. 1998. Sliding mode control: Theory and applications, Taylor & Francis. EL FADIL, H., GIRI, F., EL MAGUERI, O. & CHAOUI, F. Z. 2009. Control of DC–DC power converters in the presence of coil magnetic saturation. Control Engineering Practice, 17 (7), 849-862. FORSYTH, A. J. & MOLLOV, S. V. 1998. Modelling and control of DC-DC converters. Power Engineering J., 12 (5), 229-236. GONZALEZ MONTOYA, D., RAMOS-PAJA, C. A. & GIRAL, R. 2016. Improved Design of Sliding-Mode Controllers Based on the Requirements of MPPT Techniques. IEEE Trans. on Power Electronics, 31 (1), 235-247. KANIMOZHI, K. & SHUNMUGALATHA, A. 2013 Published. Pulse Width Modulation based sliding mode controller for boost converter. Int. Conference on Power, Energy and Control (ICPEC), 6-8 Feb. 2013. 341345. LIQUN, S., LU, D. D. C. & CHENGWEI, L. 2015. Adaptive sliding mode control method for DC-DC converters. IET Power Electronics, 8 (9), 1723-1732. LOPEZ-SANTOS, O., MARTINEZ-SALAMERO, L., GARCIA, G., VALDERRAMA-BLAVI, H. & MERCURI, D. O. 2013. Efficiency analysis of a sliding-mode controlled quadratic boost converter. Power Electronics, IET, 6 (2), 364-373. NACHIDI, M., EL HAJJAJI, A. & BOSCHE, J. 2013. An enhanced control approach for dc–dc converters. Int. J. of Electrical Power & Energy Systems, 45 (1), 404-412. NAZIR, M., KHAN, A. Q., MUSTAFA, G. & ABID, M. 2017. Robust fault detection for wind turbines using reference model-based approach. Journal of King Saud University - Engineering Sciences, 29 (3), 244-252. OLALLA, C., QUEINNEC, I., LEYVA, R. & EL AROUDI, A. 2011. Robust optimal control of bilinear DC– DC converters. Control Engineering Practice, 19 (7), 688-699. OUCHERIAH, S. & LIPING, G. 2013. PWM-Based Adaptive Sliding-Mode Control for Boost DC-DC Converters. IEEE Trans. on Industrial Electronics, 60 (8), 3291-3294. SALIMI, M., SOLTANI, J., ZAKIPOUR, A. & ABJADI, N. R. 2015. Hyper-plane sliding mode control of the DC-DC buck/boost converter in continuous and discontinuous conduction modes of operation. Power Electronics, IET, 8 (8), 1473-1482. SAMI, M. & PATTON, R. J. 2012. Wind turbine power maximisation based on adaptive sensor fault tolerant sliding mode control. 20th Mediterranean Conference on Control & Automation. Barcelona. SHAKER, M. S. 2015. A Robust Adaptive Observer-Based Time Varying Fault Estimation. Amirkabir Int. J. of Modeling, Identification, Simulation & Control, 47 (2), 11-19.
SHEN, L., LU, D. D.-C. & LI, C. 2015. Adaptive sliding mode control method for DC–DC converters. IET Power Electronics, 8 (9), 1723-1732. SIEW-CHONG, T., LAI, Y. M. & TSE, C. K. 2006. A unified approach to the design of PWM-based slidingmode voltage controllers for basic DC-DC converters in continuous conduction mode. IEEE Trans. on Circuits and Systems I, 53 (8), 1816-1827. WANG, J., LI, S., YANG, J., WU, B. & LI, Q. 2015. Extended state observer-based sliding mode control for PWM-based DC–DC buck power converter systems with mismatched disturbances. IET Control Theory & Applications, 9 (4), 579-586. YUE, Z., WEI, Q. & DAIHYUN, H. 2014. A Sliding-Mode Duty-Ratio Controller for DC/DC Buck Converters With Constant Power Loads. IEEE Trans. on Industry Applications, 50 (2), 1448-1458.
S1018-3639(17)30068-5 http://dx.doi.org/10.1016/j.jksues.2017.08.002 JKSUES 258
To appear in:
Journal of King Saud University - Engineering Sciences
Received Date: Accepted Date:
3 March 2017 16 August 2017
Please cite this article as: Shaker, M.S., Kraidi, A.A., Robust observer-based DC-DC converter control, Journal of King Saud University - Engineering Sciences (2017), doi: http://dx.doi.org/10.1016/j.jksues.2017.08.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust observer-based DC-DC converter control Montadher S. Shaker* and Asaad A. Kraidi* *
Department of Electrical Engineering, University of Technology, Baghdad, Iraq E-mail: [email protected] , [email protected] [email protected]
1
Corresponding author: Dr. Montadher Sami Shaker 30211@uotechno logy.edu.iq montadher_979@ yahoo.com
Robust observer-based DC-DC converter control Montadher S. Shaker* and Asaad A. Kraidi* *
Department of Electrical Engineering, University of Technology, Baghdad, Iraq E-mail: [email protected] , [email protected] [email protected]
Abstract: This paper develops a novel robust control strategy for DC-DC buck converter subjected to varying load and parameter uncertainty. The proposal exploits the robustness of the sliding mode control (SMC) incorporated with proportional-proportional-integral-observer (PPIO) to assure tight reference voltage tracking under a wide range of load change scenarios. Within this framework, an integral sliding mode surface based SMC (ISMC) is designed to guarantee closedloop robustness against the matched and mismatched disturbance components of the load uncertainty. Subsequently, a novel control structure comprises ISMC and PPIO is presented to overcome the design constraints and to mitigate the undesired transient response accompany the response of the closed-loop system based ISMC. Stability analysis has clearly demonstrated using linear matrix inequality (LMI) and Lyapunov approach. To illustrate the effectiveness of the proposal, a comparison between the closed-loop system responses of SMC, ISMC, and the combined ISMC and PPIO are presented in the simulation results. Keywords: Sliding mode control, proportional-proportional-integral-observer (PPIO), DC-DC converter, mismatched uncertainty, robust control. 2 Introduction: The DC-DC converters are an important part in modern technologies that require reliable power supplies such as computer systems and cellular phones or they could be used as power optimizers in renewable energy such as photovoltaic and wind turbine systems (Forsyth and Mollov, 1998). From circuit topology standing point, there are wide varieties of converter circuits ranging from simple to complex configuration of buck, boost, or buck/boost converters (Forsyth and Mollov, 1998). From control design standing point, tackling the effects of load variation and parameter uncertainty of DC-DC converter circuits have drawn the attention to develop control strategies that ensure reliable converters. Additionally, the time response of the converters must satisfy desired transient characteristics. However, owing to the nonlinear characteristics of the DC-DC converters and their highly uncertain model parameters, several challenges regarding their control have stimulated several researchers to exploit robust control schemes in order to cope with the required control objectives (El Fadil et al., 2009, Olalla et al., 2011, Nachidi et al., 2013, Wang et al., 2015). In this context, the authors in (El Fadil et al., 2009) proposed a backstepping-based control design algorithm that accounts for the effect of model parameter uncertainty. The work in (Olalla et al., 2011) exploits the potential of LMI-based multi-objective robust state feedback controller to account for model uncertainty, nonlinearity, and exogenous input. An -based robust output feedback fuzzy control for DC-DC converter is presented in (Nachidi et al., 2013) to account for the converter nonlinearity using Takagi-Sugeno fuzzy models. On the other hand, recent publications have shown interest in utilizing the relative design simplicity and the robustness of SMC to deal with DC-DC converter control problems (Gonzalez Montoya et al., 2016, Kanimozhi and Shunmugalatha, 2013, Liqun et al., 2015, Lopez-Santos et al., 2013, Oucheriah and Liping, 2013, Salimi et al., 2015, Shen et al., 2015, Yue et al., 2014, Wang et al., 2015). Specifically, the work presented in (Wang et al., 2015) combines the SMC and a disturbance observer such that the converter circuit exhibits robustness against mismatched uncertainties that have specific time behaviour. To guarantee robust reference voltage tracking of a converter circuit subjected to time varying external input, the authors in (Oucheriah and Liping, 2013) developed
an adaptive SMC scheme to provide asymptotic stability of the boost converter system. A sliding mode based boost converter control is presented in (Yue et al., 2014) to stabilize the DC power systems over the entire operating range in the presence of significant variations in load power and input voltage. An output voltage control of buck/boost converter based on a proportional-integral type hyper-plane SMC is presented in (Salimi et al., 2015) to provide robustness against parameters uncertainties, load disturbance and variations of the converter input voltage. In the aforementioned literature, the authors presented controllers that meet specific performance requirements and/or cope with a variety of DC-DC robustness issues. For instance, the work in (Wang et al., 2015) does not present systematic control system design procedure. Moreover, the proposal assumes pre knowledge of the upper bound of the disturbance signal. Additionally, the closed-loop robustness is constrained to step varying load change. On the other hand, the complexity of the proposal in (Nachidi et al., 2013) is attributed to the use of multiple model control. In (Kanimozhi and Shunmugalatha, 2013, Liqun et al., 2015, Lopez-Santos et al., 2013, Oucheriah and Liping, 2013, Salimi et al., 2015, Shen et al., 2015, Siew-Chong et al., 2006, Yue et al., 2014) the proposed sliding mode controllers are insensitive to the matched uncertainty and sensitive to the mismatched uncertainty. Hence, developing a control strategy that characterized by design simplicity and robustness against matched and time varying mismatched uncertainty is of significant contribution to the literature of DC-DC converter control. This work aims to design a robust control strategy to ensure acceptable performance and robustness of DC-DC converter circuit that affected by the variable load and/or parameter uncertainty. A novel control structure that comprises the ISMC and PPIO has been proposed to ensure robustness against matched and mismatched uncertainty. The proposal removes any offset caused by the mismatched uncertainty without the need for pre knowledge of the upper bound of the uncertainty. Moreover, unlike the work presented in (Wang et al., 2015), a systematic observer design is presented to ensure closed-loop robustness for a wide range of load change scenarios. 3 The DC-DC buck converter model This section establishes the state space model of a typical DC-DC buck converter. The buck converter circuit comprises a transistor switch, inductor, capacitor, and variable load resistance can be found in (Wang et al., 2015)(see Fig.1a). The circuit reveals two models arising from the particular position, ON (see Fig.1b) and OFF(see Fig.1c), of the switch. Based on the direct application of Kirchhoff’s voltage and current laws for the ON and OFF positions, the two models are given in Eq. (1) and Eq. (2) respectively.
L
Q
u DC
D
C R
Figure 1a DC-DC buck converter
L
DC
C R
Figure 1b the ideal switch is ON
L
DC
C R
Figure 1c the ideal switch is OFF
ν = i L − out R
(1)
ν out = iL − R
(2)
E = LiL + ν out C νout
LiL = −ν out C νout
Where in the input voltage, is the inductance, is the inductor current, is the output voltage, is the capacitance, and is the resistance. The average model that encompasses Eqs (1&2) is depicted in Eq. (3). LiL = µ E −ν out ν C νout = i L − out R
(3)
The average model presented in Eq.(3) assumes ideal circuit components and fixed load resistance. However, in order to consider the effects of load uncertainty, the model (3) has rewritten in term of nominal load ( Ro ) as follows: ν out ν out ν out = iL − + − R Ro R o
LiL = µ E −ν out C νout
(4)
Letting ( x 1 = ν out −ν r ) and ( x 2 =
i L ν out − ) yield, C R oC
x1 x2 x2 = − − + u + d2 LC Ro C
x1 = x2 + d1
(5)
where d1 =
ν out R oC
−
ν out RC
,u=
µ E −ν r LC
, and
d2 = −
1 d1 . R oC
where is the reference voltage, and the components ( d1 ) and ( d 2 ) represent the mismatched ( d1 ) and matched ( d 2 ) uncertainty. Hence, the objective is to design a robust controller for the system (5) despite the effects caused by uncertain load resistance.
4 The proposed control structure: This section explains the design steps of the proposed robust control structure that integrates the ISMC and PPIO within a feedback loop. The proposal follows the observer based robust control design methodology (Nazir et al., 2017, Sami and Patton, 2012) to tolerate a wide range of disturbance scenarios (i.e. load and parameter uncertainty). The closed-loop control system structure is shown in Figure 2.
$
"# !
Figure 2: the proposed control structure where is the DC input, is the control signal, is the reference DC voltage, and is the estimation of the mismatched uncertainty . Section 3.1 presents the ISMC design methodology for the buck converter model (5). On the other hand, subsection 3.2 will present a new control structure that combines the ISMC and PPIO in order to relax ISMC design constraints and performance limitation.
4.1
Sliding mode controller design
The ability of SMC to compensate matched disturbances has clearly investigated in the literature. Basically, the SMC design procedure encompasses two steps (Edwards and Spurgeon, 1998); (1) sliding surface design to which the states are confined during the sliding phase. (2) The control signal that makes the sliding surface attracts the states during the reaching phase. Specifically, the control law has two feedback control terms; (i) a linear control term, and (ii) a discontinuous term. While the linear control term guarantees the reachability condition, the discontinuous control term ensures the sliding condition.
Remark1: The proposed SMC needs the following assumption: Assumption1: The load uncertainty tackled by the proposed SMC must satisfy: 1-
β > δ + (d 2 + k 1d1 ) ,where () is the upper bound of uncertainty, ( δ ) is a small positive constant and k 1 is a design variable.
2-
The term d1 satisfies the limit lim d1 = 0 (i.e. has constant steady state value). t →∞
The objective is to guarantee acceptable system performance whilst the system confines into the sliding regime. Hence, to ensure acceptable sliding motion despite the effects of the matched and mismatched uncertainties (i.e. d1 and d 2 ) the following sliding surface has been proposed.
S = x 2 + k 1 x 1 + k 2 ∫ x 1dt
(6)
where () is the sliding variable, and and are design variables. Before proceeding with the second design step, analysis of system motion equation is given first. While the dynamics of system (5) confined into the sliding surface, the sliding variable must satisfies S = S = 0 . Hence, taking the time derivative of Eq. (6) yields:
S = x2 + k 1 x1 + k 2 x 1
(7)
Combining Eq.(5) and Eq.(7) gives:
1 1 S = u + ( )x 2 + (d 2 + k 1d 1 ) + k 2 )x 1 + (k 1 − LC RoC
(8)
To maintain sliding motion (i.e. S = 0 ), one could simply realize that the required control signal ( u ) should has the form: u = −(
1 1 )x 2 − (d 2 + k 1d1 ) + k 2 )x 1 − (k 1 − LC RoC
(9)
Clearly, the unknown disturbance terms makes the control signal (9) unrealistic. However, making use of assumption 1, a realistic form of the signal ( u ) is given in Eq.(10). u = −(
1 1 )x 2 − β sgn(S ) + k 2 )x 1 − (k 1 − LC RoC
(10)
Proper design of control (10) makes the sliding surface attracts the state of the system (5) into the sliding manifold within finite time ( t = t o ). Additionally, the states will remain within the sliding manifold for all t > t o . Now, consider a candidate Lyapunov function of the following form:
1 ∆= S2 2
(12)
the time derivative of Eq. (12) gives:
∆ = SS
(13)
For the control law (10), Eq.(14) gives simplified form of Eq. (8):
S = −β sgn(S ) + (d 2 + k 1d1 )
(14)
Combining Eq.(13) and Eq.(14) yields: ∆ = − β sgn(S )S + (d 2 + k 1d 1 )S
(15)
Eq.(15) can be rewritten into an inequality form as follows:
∆ ≤ S (−β + (d 2 + k1d1 ) )
(16)
Using assumption 1 gives:
∆ ≤ −δ S
(17)
Inequality (17) implies that, for initial condition outside the sliding manifold, i.e. S (t = 0) ≠ 0 , the solution to (17) becomes zero in finite time. Clearly, the consequence of (17) and (12) is:
∆ = − 2 δ ∆1/2 ,
(18)
which has the following solution:
∆ = ∆1/2 (0) − 2 δ t
(19)
Therefore, at time t = t o =
∆1/2(0)
2δ
, the states of system (5) reaches the surface (6).
Hence, using control law (10), the states of system (5) will converge toward the sliding surface (6) at time t o
< 0 ). provided that the control gain satisfies β > δ + (d 2 + k 1d1 ) (i.e. SS
While the state of (5) remains within the sliding vicinity, the dynamics of system (5) become:
x1 + k 1x1 + k 2 x 1 = d1
(20)
In fact, Eq.(20) becomes a homogeneous differential equation if the term d1 satisfies the limit of assumption1. This implies that; while sliding, the state in (5) moves toward the equilibrium point asymptotically.
Remark 2: • From Eq. (10), it should be noted that the model parameter uncertainty (i.e. the changes of L ,C ) represent matched uncertainty and hence can be tolerated via the control law (10). • While the control law (10) is capable of eliminating the steady state offset caused by ( d 1 ), conventional SMC lacks the capability to remove this offset. For comparison, consider the following conventional SMC:
Strad = x 2 + k1x 1 u=
(21)
1 1 )x 2 − g sgn(S trad ) x 1 − (k 1 − LC RoC
(22)
where Strad is the conventional sliding surface, ( g ) is the upper bound of the unknown input. If ( g ) is properly designed, the sliding motion will possess the dynamics (23): x1 = − k 1x + d 1
(23)
Clearly, it is impossible to eliminate the offset using conventional SMC.
Remark 3: The constraints listed in assumption 1 limit the applicability of the control law (10) because: • The upper bound of the unknown input (i.e. β > δ + (d 2 + k 1d1 ) ) increases the switching gain an thereby the negative effects of chattering. • The robustness of the closed-loop system in section 3.1 is restricted against disturbances that possess constant time behaviour (i.e. lim d1 = 0 ). t →∞
4.2
The design of LMI-based PPIO
This section presents the PPIO design steps and the vital role of PPIO in relaxing the constraints of section 3.1 (see remark 3). The converter system (5) can be rewritten in the following general form:
x = Ax + Bu + E d d1 y = Cx
(24)
T
T
where E d ∈ R n *m d = [1 −1/ R oC ] and x ∈ R n = [ x 1
x 2 ] . The PPIO is given as follows:
xˆ = Axˆ + Bu + E d dˆ1 + L ( y − yˆ ) yˆ = Cxˆ ˆ d 1 (t ) = P [ K o 1Cex + K o 2Ce x ]
(25)
where Ko1 and Ko 2 are the proportional and integral gains respectively, and P is a symmetric positive definite matrix. After subtracting the observer in (25) from the system (24) the state estimation error ( ex ) will be defined as: ex = ( A − LC ) e x + E d ed e y = C ex
(26)
Using eq.(25) the disturbance estimation error dynamics will become:
ed = d1 − dˆ1 = d1 − PK o 1CAe x + PK o 1CLCe x − PK o 2Ce x − PK o 1CE d ed
(27)
by combining Eqs. 26&27, the augmented estimation error dynamics can be constructed as defined in eq.(28):
ea (t ) = A ea + Nz
(28)
where A − LC A = K CA K o 1CLC − PK o 2C − P + P o1
e 0 Ed , ea = x , z = d1 , N = e − PK o1CE d I d
Now the objective is to compute the gains L , K o 1 , and K o 2 that attenuate the effects of the input z , in eq. (28), on the estimation error via minimizing the L2 norm
( z ) below desired level γ . 2
Remark4: decoupling the effects is of ( z ) is beyond the scope of this paper. However, based on the available information of E d , the following theorem ensures L2 norm minimization of ( z ) on ea .
Theorem1: The augmented estimation error in (28) is stable and the L2 performance is guaranteed with an attenuation level ,, Provided that the signals d1
( )
are bounded, rank (CEd ) = md , and the pair (A,C) is
observable, if there exists a symmetric positive definite matrices P1 , P −1 and G matrices H , K o 1 , K o 2 , and a scalar µo satisfying the following LMI constraint:
Minimize γ such that
Ψ11 Ψ12 * Ψ 22 * * * * * * * *
0 Ψ 23
0 Ψ 24
Ψ15 0
−γ I *
0 −G−1
0 0
* *
* *
−2µo P1 *
0 0 0 <0 0 µo I −G
(29)
where T
L = P1−1H , γ = γ , Ψ11 = P1A + ( P1A ) − HC − ( HC ) + w 1 , Ψ12 = P1E d − A T C T K o1T −C T K o 2T , Ψ15 = ( HC ) , T
T
T
Ψ22 = −K o1CE d − ( K o1CE d ) + w 2 , Ψ 23 = P −1 , Ψ 24 = K o 1C
Proof: See (Shaker, 2015). It is worth remarking that to tune the ℒ performance against the exogenous input .̃ , 3 0 4 has been nominated in the LMI formulation. the weighting matrix 0 = 2 0 3 The estimated disturbance ( ) plays a vital role in enhancing the overall closed loop performance. Specifically, the sliding surface (6) is modified to the following form:
S = x 2 + k1 x 1 + k 2 ∫ x 1dt + dˆ1
(30)
To maintain sliding motion (i.e. S = 0 ), the new control signal ( u ) has the form:
u = −(
1 1 )x 2 − (d 2 + k 1d 1 ) + (dˆ2 + k 1dˆ1 ) − dˆ1 − β new sgn(S ) + k 2 )x 1 − (k 1 − LC Ro C
(31)
Following the procedure of section 3.1, the closed-loop system stability can easily derive and hence omitted here.
Remark 5: the control signal in Eq.(31), the estimation dˆ1 compensates the effect of load uncertainty. Consequently, the design parameter β new can be selected much smaller than β , thereby minimising the chattering effect.
5 Simulation Results: In this section, the proposed SMC strategy has verified based on Matlab/Simulink software. Table 1 gives details of the considered converter parameter. The uncertain resistive load varies in the range (60-140 Ω) has subjected to the buck converter. Table 1: the parameters of the buck converter. Input voltage ( ) Desired voltage ( ) Capacitance ( ) Inductance () Load resistance ( )
20 V 10 V 1000 µ F
4.7 6 60Ω-140Ω
In the first simulation scenario, the results compare the output response of conventional SMC and SMC with integral action based buck converter control. The first design step of controller (10) is determining the switching gain ( β ) and the sliding surface parameters ( k 1 , k 2 ). The holding of assumption 1 ensures sliding motion governed by the dynamics of Eq. (20). Consequently, the performance parameters (the damping ratio ( ζ ) and natural undamped frequency ( ω n )) of
this system are adopted such that the maximum settling time ( T s ) satisfies T s ≤ 1sec and maximum peak time ( T p ) satisfies T p ≤ 0.5 sec . Hence,
T s ≤ 1sec ⇒
1
ζω n
T p ≤ 0.5sec ⇒
≤ 1sec ⇒
π ωn 1 − ζ
2
2 ≤ 1sec ⇒ 2 ≤ k 1 k1
≤ 0.5sec ⇒
π 2
k 2 (1 − ζ )
≤ 0.25sec ⇒
4π ≤ k2 , (1 − ζ 2 )
which led to k 1 = 25 , k 2 = 250 . On the other hand, the design parameter β = 500 is selected to satisfy
< 0 ). reachability condition (i.e. β > δ + (d 2 + k 1d 1 ) ⇒ SS In order to investigate controller robustness, the converter has subjected to nominal (100Ω), step-varying, and fast-varying load. Figure 3 shows tracking performance of the converter (5) when subjected to nominal load (100Ω). In this scenario, the controllers (10) and (22) can force the converter dynamics to start the sliding motion at time t o that depends on the variable δ (and consequently β ). 10.1 SMC I SMC
Output Voltage
10.08
10.06
10.04
10.02
10
9.98 0
2
4
6
8
10
Tim e (sec)
Figure 3: The tracking performance of the buck converter when subjected to nominal load. Figure 4 shows the output response of the buck converter when feeding a step-varying load. As demonstrated in remark 2, the traditional SMC is incapable of compensating the unmatched component d 1 . On the other hand, while the condition lim d1 = 0 holds for this scenario, the ISMC can compensate the effects of the unmatched t →∞
component d 1 . However, the closed-loop system based ISMC exhibit undesired transient response that disturbs the smoothness of the output voltage. 11.5 SMC ISMC
Output Voltage
11
10.5
10
9.5
9 0
2
4
6
8
10
Tim e (sec)
Figure 4: The tracking performance of the buck converter when subjected to step-varying load.
Figure 5 demonstrates how the controllers (10 & 22) are unable to maintain reference tracking when the unmatched disturbance does not satisfy the limit lim d1 = 0 . t →∞
10.25 10.2
SMC ISMC
Output Voltage
10.15 10.1 10.05 10 9.95 9.9 9.8 5 9.8 0
2
4
6
8
10
Tim e (sec)
Figure 5: The tracking performance of the buck converter when subjected to fast-varying load change. As mentioned in remark 2, the proposed ISMC and the SMC possess the ability to tolerate the converter parameter uncertainty. Figure 6 obviously demonstrates the closed-loop robustness against the following parameter change scenario: x1 x2 x2 = −α −α + u + d2 LC Ro C
x1 = x2 + d1
(32)
where (α ) represents the multiplicative uncertainty factor of the system (5). 10.05 SMC I SMC
10.04 10.03
Output voltage (v)
10.02 10.01 10 9.99 9.98
α =1
α =0.7
α =1.3
9.97 9.96 9.95
0
2
4
6
8
10
Tim e (sec)
Figure 6: The controller robustness against model parameter uncertainty The second simulation scenario highlights on the advantages of integrating the closed-loop system based-ISMC with the PPIO. Using the Matlab LMI toolbox, the gains of the PPIO are obtained as follows:
0.046
γ = 0.059, K o1 = [ −0.185 −3.248] , K o 2 = [ 0.003 −6.916] , L = −1.616
−0.001 . 9.954
Remark 6: •
The ISMC parameters ( k 1 , k 2 ) are maintained as given above (i.e. k 1 = 25 , k 2 = 250 ). However, in the combined ISMC and PPIO scheme, it becomes possible to reduce the switching gain to β new = 380 (see remarks 3 and 5).
•
While the PPIO guarantees precise estimation of load change when d1 ≠ 0 , the extended state observer (ESO) proposed in (Wang et al., 2015) is restricted to the case of d1 = 0 . Figure 7 shows the accuracy
of PPIO, compared with the ESO, for estimating d1 component.
(a) Time varying load change
(b) Step varying load change Figure 7 (a and b): PPIO and ESO based load change estimation. Figure 8 (a and b) shows the reference voltage tracking accuracy of buck converter-based SMC, ISMC, and combined ISMC and PPIO. Specifically, while tolerating the effects of time varying load change (i.e. d1 ≠ 0 ) is not possible via SMC and ISMC, the combined ISMC and PPIO has showed robust tracking performance against load change despite the fact that d1 ≠ 0 . 11.5
Output voltage
11
SMC I SMC I SMC+PPIO
10.5
10
9.5
9 0
2
4
6
8
10
Tim e(sec)
(a) Reference voltage tracking for step-varying load
10.25 10.2
SMC I SMC I SMC+PPIO
10.15
Output voltage
10.1 10.05 10 9.95 9.9 9.8 5 9.8 0
2
4
6
8
10
Tim e (sec)
(a) Reference voltage tracking for fast-varying load Figure 8 (a and b): Reference voltage tracking performance. With reference to Figure 7, it is clear that utilizing PPIO for load change estimation and compensation minimises the undesired transient response of ISMC. Finally, the duty ratio µ (t) is shown in figure 9(a and b). The combined ISMC and PPIO maintains the duty ratio unchanged despite the effects of load uncertainty.
0.5 8 0.56
SMC I SMC I SMC+PPIO
Duty ratio µ (t)
0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0
2
4
6
8
10
Tim e sec
(a) Duty ratio for step-varying load
0.52 0.5 15
SMC I SMC I SMC+PPIO
Duty ratio µ (t)
0.51 0.505 0.5 0.495 0.49 0.48 5 0.48 0
2
4
6
8
10
Tim e sec
(a) Duty ratio for fast-varying load Figure 7 (a and b): Duty ratio of the Buck converter. Tables 2&3 summarize the performance of SMC, ISMC, and ISMC+PPIO controllers for step and time varying load changes. Table 2: SMC, ISMC, and ISMC+PPIO controller performance for step load change.
SMC ISMC ISMC+PPIO
Min 9.087 9.355 9.913
Max 10.72 11.21 10.1
Mean 9.983 10 10
STD 0.3208 0.0767 0.0095
Table 3: SMC, ISMC, and ISMC+PPIO controller performance for time varying load change.
SMC ISMC ISMC+PPIO
Min 9.849 9.902 9.967
Max 10.15 10.1 10.11
Mean 10 9.998 10
STD 0.0728 0.0256 0.0066
6 Conclusions: This paper proposes a robust controller for buck converter feeding an uncertain load. The proposal extends the conventional sliding surface design methodology by integral action accounts for unmatched disturbance. However, the ISMC always brings undesired transient performance. Therefore, the proposal integrates the ISMC by PPIO to relax the design constraints, tolerate the undesired transient response, and estimate and compensate the effects of load change on tracking performance. Different simulation scenarios have examined the tracking performance of the proposed controller. The results have shown that the proposed ISMC+PPIO controller is capable of compensating the effects of load uncertainty regardless the time behaviour of the unmatched component. References EDWARDS, C. & SPURGEON, S. 1998. Sliding mode control: Theory and applications, Taylor & Francis. EL FADIL, H., GIRI, F., EL MAGUERI, O. & CHAOUI, F. Z. 2009. Control of DC–DC power converters in the presence of coil magnetic saturation. Control Engineering Practice, 17 (7), 849-862. FORSYTH, A. J. & MOLLOV, S. V. 1998. Modelling and control of DC-DC converters. Power Engineering J., 12 (5), 229-236. GONZALEZ MONTOYA, D., RAMOS-PAJA, C. A. & GIRAL, R. 2016. Improved Design of Sliding-Mode Controllers Based on the Requirements of MPPT Techniques. IEEE Trans. on Power Electronics, 31 (1), 235-247. KANIMOZHI, K. & SHUNMUGALATHA, A. 2013 Published. Pulse Width Modulation based sliding mode controller for boost converter. Int. Conference on Power, Energy and Control (ICPEC), 6-8 Feb. 2013. 341345. LIQUN, S., LU, D. D. C. & CHENGWEI, L. 2015. Adaptive sliding mode control method for DC-DC converters. IET Power Electronics, 8 (9), 1723-1732. LOPEZ-SANTOS, O., MARTINEZ-SALAMERO, L., GARCIA, G., VALDERRAMA-BLAVI, H. & MERCURI, D. O. 2013. Efficiency analysis of a sliding-mode controlled quadratic boost converter. Power Electronics, IET, 6 (2), 364-373. NACHIDI, M., EL HAJJAJI, A. & BOSCHE, J. 2013. An enhanced control approach for dc–dc converters. Int. J. of Electrical Power & Energy Systems, 45 (1), 404-412. NAZIR, M., KHAN, A. Q., MUSTAFA, G. & ABID, M. 2017. Robust fault detection for wind turbines using reference model-based approach. Journal of King Saud University - Engineering Sciences, 29 (3), 244-252. OLALLA, C., QUEINNEC, I., LEYVA, R. & EL AROUDI, A. 2011. Robust optimal control of bilinear DC– DC converters. Control Engineering Practice, 19 (7), 688-699. OUCHERIAH, S. & LIPING, G. 2013. PWM-Based Adaptive Sliding-Mode Control for Boost DC-DC Converters. IEEE Trans. on Industrial Electronics, 60 (8), 3291-3294. SALIMI, M., SOLTANI, J., ZAKIPOUR, A. & ABJADI, N. R. 2015. Hyper-plane sliding mode control of the DC-DC buck/boost converter in continuous and discontinuous conduction modes of operation. Power Electronics, IET, 8 (8), 1473-1482. SAMI, M. & PATTON, R. J. 2012. Wind turbine power maximisation based on adaptive sensor fault tolerant sliding mode control. 20th Mediterranean Conference on Control & Automation. Barcelona. SHAKER, M. S. 2015. A Robust Adaptive Observer-Based Time Varying Fault Estimation. Amirkabir Int. J. of Modeling, Identification, Simulation & Control, 47 (2), 11-19.
SHEN, L., LU, D. D.-C. & LI, C. 2015. Adaptive sliding mode control method for DC–DC converters. IET Power Electronics, 8 (9), 1723-1732. SIEW-CHONG, T., LAI, Y. M. & TSE, C. K. 2006. A unified approach to the design of PWM-based slidingmode voltage controllers for basic DC-DC converters in continuous conduction mode. IEEE Trans. on Circuits and Systems I, 53 (8), 1816-1827. WANG, J., LI, S., YANG, J., WU, B. & LI, Q. 2015. Extended state observer-based sliding mode control for PWM-based DC–DC buck power converter systems with mismatched disturbances. IET Control Theory & Applications, 9 (4), 579-586. YUE, Z., WEI, Q. & DAIHYUN, H. 2014. A Sliding-Mode Duty-Ratio Controller for DC/DC Buck Converters With Constant Power Loads. IEEE Trans. on Industry Applications, 50 (2), 1448-1458.