LMI Approach for Sliding Mode Control and Analysis of DC-DC Converters

Circuits’ and in particular DC/DC converters’ switching behavior is analyzed in this paper using the equivalent control modeling of the dynamic systems’ sliding mode regime. As a representative example and also being one of the most complex circuits among DC/DC converters, the Ćuk converter is chosen. It is shown how the converter’s behavior in the steady state regime can be studied and analyzed by the linear matrix inequalities based stability conditions for linear dynamic systems with nonlinear sector bounded perturbations. The maximization of the nonlinear sector bound provides a limit for applying the linear ripple approximation in the converter operation analysis. Furthermore, our approach is validated by providing simulation results for two different switching surfaces of practical interest.


INTRODUCTION
In this work, we study a sliding mode regime as a nonlinear (due to switching) system modeling behavior [1][2] of DC/DC converters using the concept of equivalent control as introduced and developed by Utkin (see, for example, [1,3]).In particular, we consider the Ćuk converter as a representative, one of the most complex, and important example of DC/DC converters [12].We propose an approach in which a nonlinear dynamic model resulting from applying so called equivalent control [1], is linearized around its equilibria and represented as the sum of its linearized portion and the remainder which is nonlinear.This representation is known to be well suited for a Lyapunov stability analysis based on Linear Matrix Inequalities (LMI) [4].
LMI represent a special class of convex optimization formulations [5] which thus inherit all the benefits of convex programming algorithms such as uniqueness of solutions, convergence to a solution if it exists, and a definite test for feasibility of problems where the answer depends on whether or not convex algorithms converge.In particular, the LMI formula- tion enables an application of convex programming in computing a quadratic Lyapunov function which proves stability of the overall nonlinear system as well as switched systems [6].
(a) (b) Figure 1 -Ćuk converter containing (a) unidirectional switch S and a diode D; (b) bidirectional switches S1 and S2.One of the key ingredients in this formulation is the sector bounding of the additive nonlinearity which is thus treated as a perturbation [7].Furthermore we obtain a relationship between the simulation results and the domain in which the nonlinearity is sector bounded with the sector being maximized using LMI TEHNIKA -ELEKTROTEHNIKA 65 (2016) 5 formulation.This relationship enables us to estimate a size of the hysteresis used to design a switching sequence for which the converter operates in a proper steady state regime.
Most of the research on control of the Ćuk converter has been focusing on controlling two state variables, as in [8] where it is related to the sliding mode control under an assumption that 2 1

C C 
(for the capacitors C1 and C2 denoted as in Fig. 1) which results in an overall system reduction.
This reduction was performed in some particular cases by exactly one reactive element elimination [9].However, there are some other works concerning sliding mode control of the Ćuk converter which incorporated a full-order system model as reported in [10].
Publication [10] describes sliding mode controls of the Ćuk converter built with bidirectional switches, which do not produce any discontinuous regimes.LMI approach has been already used for a design of the sliding-mode controlled buck converter [11].
In this paper we show how the sliding mode control of the Ćuk converter in its all continuous and discontinuous regimes can be analyzed using an LMI based stability analysis for linear systems with an additive nonlinearity.Stability conditions in the case of each switching surface are formulated using LMI and the additive nonlinearity (treated as a perturbation) bound is computed.The sector bounded nonlinearity is shown to be in a direct connection with the linear ripple approximation [12].
The paper is organized as follows.In Section 2, the sliding mode analysis and how to compute equivalent control for a class of dynamic systems which model DC/DC circuits' switching behavior are presented.Dynamic properties of the Ćuk converter and its modeling are described and developed in Section 3. In Section 4, a switching surface analysis for two different surfaces of interest as well as corresponding simulation results are provided.Finally, some concluding remarks are summarized in Section 5.

SLIDING MODE ANALYSIS, STABILITY AND CONTROL
For a switching system such as DC/DC converter [13], the concept of the sliding mode control is materialized through an ON-OFF operation of a controllable switch.Considering that for any converter there is only one switching signal, there are two subsystems in the continuous conduction mode which can be rewritten in a compact matrix form as where x denotes the state vector consisting of n state variables and {0,1} u  is the scalar indicator showing whether the controllable switch is conducting or not.Matrices A, B, C and D are of appropriate dimensions determined by dimensions of x and u.Essentially the system (1) represents a set of n linear differential equations with a switching variable.For the control of DC/DC converters, the switching law is given by 1, switch to 1 when ( ) 0, switch to 0 when ( ) , where we assume that the sliding surface is of a linear form ( ) x M x x with the coefficients being entries of a row vector ,  is a positive number and a constant column vector  x is an equilibrium point and steady-state vector.
Sliding mode control technique is applied in order to analyze the converter's behavior on the switching surface using the concept of equivalent control [1].
The system's motion in the sliding mode is restricted to the switching surface 0 if the surface is attractive.In the steady-state is also 0 ) (  x S  , [1], and thus, the equivalent control can be obtained from the vector equation is nonsingular, which will be satisfied in the cases of our interest.After substituting eq u in Eq. (1) we obtain ( ) .
An equilibrium and steady-state vector  x can now be computed as a constant solution to Eq. ( 4).Furthermore, we represent the system as the sum of the linear part (computed using Taylor's expansion) and the exact nonlinear remainder computed as ( ) ( ) where x  is given in the Eq. ( 4).
The sliding surface equation 0 enables an elimination of one state variable, in general.By introducing the change of variables , where  x is an equilibrium, and denoting as z the vector of variables i y which are not eliminated, the reduced system dynamics can be written as ( 1) ( 1) ( 1) 1 ( ) where matrix  A and nonlinearity  h are obtained after one state variable is eliminated using the sliding surface equation.
System stability on the desired sliding surface is determined by finding a Lyapunov function candidate of the quadratic type ( ) where P is a constant positive definite matrix.The next step is to compute as follows: ) The stability analysis is performed using LMI [4] which is an efficient and almost the only way to compute Lyapunov functions for systems whose dynamics are represented by a linear and a sector bounded nonlinear part.The LMI problem is formulated using the S-procedure [14] which maximizes the bound on the nonlinear part.This approach due to being based on convex programming will always provide a solution if the problem is feasible and the convergence in the case when the solution exists is fast.Now, the LMI problem to examine systems stability can be formulated as follows [7]: . 0 A solution would provide matrix , for some positive value τ (see [7] for more details), which maximizes the sector bound , (6) where and the nonlinear term/remainder satisfies the following sector bound 2 .

T T T     h h z H Hz (7)
The nonlinear remainder's sector bound shape is determined by a matrix H and a positive parameter 0   .The following nonsingular transformation [7] ,  x Tx  (8) with matrix T being formed from the eigenvectors of the matrix  A provides larger values for α, that is, a better sector bound.Therefore, the LMI formulation in the transformed space ,  H HT  (9) for the preselected matrix H ~results in sector bound value denoted as  ~, which is related to α as 1 .
The maximization of the sector bound provides a limit for applying the linear ripple approximation, that is, a limit for considering state variables time constants to be negligible.

MODELING BEHAVIORS OF THE ĆUK CONVERTER
The Ćuk converter depicted in Fig. 1 is a complex fourth order DC/DC converter [12] which for those reasons is chosen to illustrate our approach.In particular, Figure 1a shows a Ćuk converter which applies switches realized by an unidirectional switch and a diode, while Figure 1b shows a synchronous Ćuk converter employing current and voltage bidirectional switches.For the synchronous converter bidirectional switches are controlled with state(S2)=¬state(S1), where ¬ denotes negation.

A. Sliding domain and operating modes
Depending on the Ćuk converter's switches realization, a few operating modes can be determined and analyzed.Both realizations of the Ćuk converter depicted in Fig. 1 operate in two continuous modes, that is, the first one being when switch S is on and a diode isn't (in Figure 1b switch S1 is on and S2 is off) and the second one being when switch S is off and diode D is conducting (switch S1 is off and S2 is on where again we refer to Figure 1b).
In the continuous conduction mode the state space equations are given by , , where 1  u is when S is on and D is off and 0  u when S is off and diode D is on.Additionally, switch S1 from Fig. 1b is controlled with signal u and switch TEHNIKA -ELEKTROTEHNIKA 65 (2016) 5 S2 with u .Eq. ( 15) can be rewritten in the compact matrix form suitable for applying sliding mode control as in Eq. ( 1), with matrices defined as follows: , with the prescribed syntax steady-state values being A synchronous converter depicted in Figure 1b can operate only in the continuous modes.Additionally, the Ćuk converter shown in Figure 1a operates in two discontinuous modes, discontinuous inductor current mode (DICM) and discontinuous capacitor voltage mode (DCVM).
DICM occurs when the switch S is turned off and the sum of the inductor currents goes to zero, that is, 0 , which turns off the diode.State space equations in DICM are as follows: DCVM occurs in the state when swith S is on and the diode is off and capacitor voltage 1 C v reaches zero.That creates conditions for the diode to turn on and converter enters DCVM resulting in the following state space equations: , , 0, .
In order to apply the sliding mode control technique to the Ćuk converter, inverting hysteresis regulator with thresholds   is used.
Regulator produces controlling signal u as defined in Eq. ( 2) with the sliding surface determined using the following equation: , m m  given with expression We primarily consider a Ćuk converter containing a switch and a diode which can operate in all continuous and discontinuous modes.For such a converter, state variable 3 x is positive and 4 x is negative, in the steady state, respectively.Thus, it is assumed

B. Ripple approximation
The characteristic values of the DC/DC converters' are adjusted so that the time constants of the capacitors and the inductors are much longer than the switching period.Thus, the capacitors' voltages and inductors' currents steady-state values are considered when the ripple is computed [12].This means that, by using the linear ripple approximation, inductors' currents of the Ćuk converter either linearly increase or decrease during a switching period.When the switch S is on, for eq S u T time interval, then the inductors' cur- rents increase as while when the diode is on during the rest of the period, .8 Using the steady state equations and the ripple of the state variables given in Eqs. ( 21), ( 22) and ( 23) one can determine the hysteresis value ∆.

SWITCHING SURFACE ANALYSIS
In this section we are going to provide analysis for the Ćuk converters' behavior on the switching surfaces as representative ones, yet the procedure is general and thus can be applied to any other switching surface in the same way.One of the surfaces of the importance is , , , .
In order to achieve 0 3  x and 0 4  x , the second solution is chosen.The first solution is negative and thus physically unacceptable.
The system's stability on the sliding surface can be verified through the introduced approach of linearization and by finding matrix P as in Eq. ( 9) to establish Lyapunov stability.
On the sliding surface During the switching subinterval when the switch S is on and the diode is off, the inductor current 1 L i , presenting controlling variable, linearly changes, according to the linear ripple approximation: . 2 Using Eq. ( 25), the switching period S T can be computed and then used in the ripple calculation for other state variables as given in Eqs. ( 21), ( 22) and (23). In (a) Calculated and measured values are .Furthermore, the diagrams in Fig. 2a show a high level of linear behavior and fit well the linear ripple approximation analysis.In Fig. 2b, the nonlinearity in TEHNIKA -ELEKTROTEHNIKA 65 (2016) 5 the ripple becomes evident especially in the waveform for 1 L i .It shows that the time constants caused by real parts of eigenvalues are comparable to the switching period.
The system stability is verified using the LMI approach and computing the nonlinearity bound α.The .By choosing matrix H to be a 3 by 3 matrix with all entries equal to zero except the one at position (2,2) being equal to 1, and using the LMI convex program as in Eq. ( 9) we obtain the maximized sector size.In particular, one can show that the nonlinear term satisfies , where 2 T is the second column of the inverse of a transformation matrix T as in Eq. ( 14).
The nonlinearity perturbation bound is computed as  In Figure 3, diagrams , as depicted in Figure 2b.It can be also observed that the left side of the diagram in Figure 3c shows steeper increasing of the error while decreasing 3 y .In this case the error forces the system to approach the other infeasible equilibrium (the first solution in Eq. ( 24)).
Considering obtained sector bound α in Eq. ( 11), the limit for applying linear ripple approximation can be determined as crossing point of nonlinearity * .It can be seen that is a limit for considering linear ripple approximation in this example and from Eq. ( 21), the following can be derived: which provides m 17   as a limit.In Figure 3d this is depicted as a limit ripple confirming the LMI procedure computation.Clearly the calculated error for all simulated cases shows when and if the linear ripple approximation can be used.

B. Switching surface
The sliding surface allows hysteresis window current mode control [12] in which the average of a weighted sum of inductors' currents is kept at a desired constant value.The equivalent control produces equilibria 2 . Of our interest is the second solution because it provides a positive value for 3 x and a negative value for 4 x .This choice results in For the case of the switching surface We can compute S T , which is then used to compute inductors currents' and capacitors voltages' ripple as described by Eqs. ( 21), ( 22) and (23) based on In order to demonstrate a bound for the linear ripple approximation particular, the values for switching surface    , as depicted in Figure 4c.We can also observe maximal ∆ for which the influence of the system eigenvalues is negligible comparing to the switching period and estimate . By using the following expression: we compute m 90   . We can confirm that this is indeed the boundary value as depicted in Figure 5d The resulting phase diagram for this sliding surface is given in Figure 5.At a start up the system's trajectory follows 2 0 x  and 1  u , before it gets to the switching surface 0 . Then there is a small overshoot when 0  u until a steady state is reached.After that the system's motion stays on 0 ).In the case of the bidirectional realization there is no discontinuous regime, so when 5 m is very large the return to the switching surface occurs as depicted by green trajectories in Figure 5, that is,  are provided.One may also observe that DCVM reduces significantly the inductor currents' ripple and the capacitor C1's voltage.In addition, the phase diagrams shown in Fig. 7 confirm the previous analysis and agree with the phase plot from Figure 5.

CONCLUSION
In this paper we applied an equivalent control model of a sliding (or switching) surface to characterize the steady-state analysis of DC/DC converters.This is done by linearizing nonlinear sliding mode dynamics and representing it by a linear part and a sector bounded nonlinear remainder.Furthermore, using the linear matrix inequalities stability approach we estimated the size and the shape of the sector which the nonlinear remainder satisfies.The nonlinear sector bound is then used to determine the limit for applying linear ripple approximation in the converter operation analysis.This approach is demonstrated of the Ćuk converter's example for two switching surfaces of the practical importance.
switching surface.
used for controling output voltage disturbance as well as the input current [15], but it won't be analyzes in this paper.In terms of simulations, we used PLECS [16and we do not include units in  to simplify the notation yet it is obvious they match the corresponding variables, such as volts for voltages. .The equivalent control can be computed for is constant and thus the equation for1  x  can be eliminated.The stability analysis is then performed in

Figure 2 ,
steady state waveforms for different values of the regulator hysteresis bound

Figure 3 - 3 h is plotted in red and the sector bound 3 y
Figure 3 -The nonlinear term  3 h is plotted in red and for ∆ are shown.It can be seen that the error  model when the switch S is off and the diode D is on.The case when m the following steady-state values.

x 2 ˆ7
Using the LMI convex program we obtain the sector size parameter value as 3

Figure 4 - 3 h is plotted in red and the sector bound 3 y 4 .
Figure 4 -The nonlinear term  3 h is plotted in red and m is large enough so that DCVM occurs, after reaching the sliding surface, then voltage 1 C v drops to zero, so the system's motion in DCVM returns to the switching surface via yellow trajectories as depicted in Figure6(with 1  u

Figure 5 -
Figure 5 -Phase diagram x2(x1) in the case when S(x)=m1x1+m2x2-m5 In order to see the difference between the case of unidirectional switch S and a diode and bidirectional switches, in Figure 6 time diagrams when 4 ) ( 2 1   x x S xare provided.One may also observe that DCVM reduces significantly the inductor currents' ripple and the capacitor C1's voltage.In addition, the phase diagrams shown in Fig.7confirm the previous analysis and agree with the phase plot from Figure5.

Figure 6 -
Figure 6 -State variables versus time in the case when 1 2 ( ) 4 S x x    x when the switches are realized as (a) an unidrectional switch and a diode; (b) bidirectional