observer-based control for dc–dc converters
TRANSCRIPT
Arab J Sci Eng (2014) 39:4089–4102DOI 10.1007/s13369-014-1045-3
RESEARCH ARTICLE - ELECTRICAL ENGINEERING
Observer-Based Control for DC–DC ConvertersPractical Switching Control
Djekidel Kamri · Cherif Larbes
Received: 6 August 2012 / Accepted: 31 January 2013 / Published online: 9 April 2014© King Fahd University of Petroleum and Minerals 2014
Abstract The paper proposes a computational techniquefor the synthesis of a suitable observer-based control forDC–DC switching converters. The work will be carried outdirectly on the Piece-Wise Affine convincing representationobtained from the Bonds Graph modeling method. The aimwas to regulate by switching the converter outputs to an aver-age (not common) equilibrium reference. Based on the Lya-punov theory, a systematic state feedback control is derivedfrom a tractable Bilinear Matrix Inequality formulation of theproblem. The design control searches for a single Lyapunov-like function that satisfies practical quadratic stabilizationproperties in an appropriate continuous state space partitionand provides a way to drive the system states into a quantifi-able small ball around the non-equilibrium desired reference.To complete the method, a simple state estimation procedureis introduced to avoid state measurements, the technique isbased on Luenberger-like observer structure. The approachcan be implemented as an embedded software system to gen-erate several continuous supply levels. Satisfactory simula-tion results are obtained for several examples, the illustra-tive theoretical demonstrations for some of these switchingdevices are reported.
Keywords DC–DC converters · BMI · Hybridobservability · Hybrid systems · Practical stabilization ·PWA systems · Switching control · Switching systems
D. Kamri (B)Department of Electronic, University Ammar Telidji,Laghouat, Algeriae-mail: [email protected]; [email protected]
C. LarbesDepartment of Electronic, National Polytechnic School,Algiers, Algeriae-mail: [email protected]; [email protected]
1 Introduction
The DC–DC converters are power electronic devices thathave received great interest during the last decades. Thisinterest results from their use in realizations of machinesspeed variation and the various electronic device supplies.Their conception and control techniques do not stop develop-ing and improving in performances. Nowadays, these tech-niques of conversion are well developed and a wide vari-ety of reliable products is proposed for demanding applica-tions. Due to their behavior complexities (switching effect,multi equilibrium and state constraints), the control prob-lems associated with such devices still pose theoretical chal-lenges for academic researchers [1–3]. For these circuits,
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we often call for the averaging models or the discreditingtechniques to make the control operation more tractable [4].These techniques involve considerable approximations andproduce results that are less accurate. Whereas the function-ing nature of these circuits is well suited for hybrid modelingwhere the evolution of all state variables is captured withinswitching periods. In this context, they can be described byswitched Piece-Wise Affine (PWA) systems, which bene-fit from a large available analysis and control tools [5–8].Switched PWA systems constitute a well-known class ofhybrid systems; they are a widely used multi-model repre-sentation that offers a good modeling framework for dynam-ical engineering systems that involve nonlinear phenomena.These systems have received increasing interest over the lastyears and some related control problems remain unsolvedeven for simple practical examples.
Several control approaches have been applied for DC–DCconverters, there are at least four most recent ideas proposedby the mains groups working on these circuits [2]. Basedon the discrete PWA model, Morari’s group used the predic-tive control for these circuits [1]. An extension of sampleddata H∞ control with pulse width-modulated systems wasintroduced by Jönsson’s group [3]. The approach proposedby Rantzer group is based on the relaxed dynamic program-ming. While Buisson’s group developed at Supélec a stabiliz-ing Lyapunov-based control [9] using a port-control Hamil-tonian formulation of the continuous-time plant dynamic.The reports in [2,3] contain a comparative study of theseapproaches.
In this paper, we are interested in the investigation of state-dependent switching stabilization problem using estimatedstates. This is the most elusive problem in the switched sys-tems literature [10–12]. Based on the obtained PWA sys-tem of DC–DC converter, a complete practical stabiliza-tion method with state estimation technique is formulatedas Bilinear Matrix Inequality (BMI) that can be efficientlysolved by Matlab’s LMI toolbox after a grid-up operation ofthe scalars variables. Among the small number of papers thatconsider this class of PWA systems without (or not common)equilibrium are the works in [13] and [14]. Recently, manyother theoretical papers investigate the switched and PWAsystems, necessary and sufficient conditions for quadraticstabilization are obtained for bimodal systems only [6,10,12].
Most results of these approaches are based on the ideaof state feedback where a complete state of the system isrequired to achieve the control operation. In practice, how-ever, the whole system state is often unavailable from mea-surements. The state estimation problem represents anotherarea of interest for hybrid systems that has been the sub-ject of intensive investigations in the last decade. A surveyof observability analysis and observer design for PWA andswitched systems is beyond our scope. We will present a
simple state estimation technique that is applicable for ourconverters. Precise definitions of observability and results forsome classes of PWA and switched systems can be found in[5,15–17]. Among the first, concrete propositions for hybridasymptotic observer design are the works in [18,19]. In [18],an asymptotic observer design for switched linear systemsis formulated as LMI when the active mode is known. Anextension of the method to the more general and difficultcase where the active mode is not known is presented in[19]. Another state estimation approach that is applicableto the more general PWA system is considered in [20]. Itis based on moving horizon estimation; the proposed effi-cient observability test requires computational effort (mixed-integer quadratic programming and multi-parametric pro-gramming). The output feedback controller design in [13,21]for controlled PWA systems invoked obligatory partial stateestimation; however, the observer design cannot be directlydistinguished from the control methodology in these works.
Since the results were mainly motivated by practical digi-tal implementations, most of these approaches are first devel-oped for discrete-time hybrid systems. In our case of DC–DCconverters, the obtained models are PWA with continuousdynamic where the active mode is determined by the con-trol approach. Since there is neither jump nor reset of statesfor this kind of systems [9], the design of Luenberger-likeobserver reduces to the computation of observer gains thatensure errors convergence. The obtained estimation errorsare also governed by a switched system; therefore, it is notsufficient to stabilize each error mode separately. As in thefirst control part, we will use a common Lyapunov-like func-tion to formulate the state estimation as BMI, which completethe proposed systematic control approach.
Paper organization: Sect. 2 represents a background forthe paper. It starts by defining the derived DC–DC convertermodel and the problem formulation. Section 3 contains mainresults of the paper; it exhibits practical quadratic stabiliza-tion as BMI of the PWA system described in an augmenteddimensional state space. In Sect. 4, we introduce a simplecomputational state estimation technique to avoid measure-ment. Section 5 illustrates the theoretical results on practicalexamples.
2 Problem Statement
Different energy-based approaches were used to derive mod-els for these circuits, such as circuits theory, Euler–Lagrange,Bond Graph [22] and the Hamiltonian approach [9]. Undersome assumptions (dissipative elements are linear, physi-cals switches are ideals, storage components are linear andindependents), all these approaches lead to the same follow-ing valid model (most results of this section are taken from[9,23]):
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y = (J (ρ) − R(ρ))x + g(ρ)u (1)
where ρ is a Boolean vector of mode, it describes the on/offconfiguration of switches. u ∈ Rm represents the source ofenergy which is constant in this case, y ∈ Rn is the statevector with dimension equal to the number of storage ele-ments. The state variables are the energy ones (fluxes link-age in the inductors, charges in the capacitors). x ∈ Rn iscalled the co-state vector, co-state variables are the corre-sponding co-energy variables (currents, voltages). In the casewhere the storage components are linear, the relation betweenthese two vectors is given by: x = Fy where F is a positivedefinite symmetric matrix. In simple cases, F is a diagonalmatrix whose elements are the inverse values of inductancesor capacitances [9,22].
g(ρ) is the matrix of input power, R(ρ) > 0 representsthe energy dissipative part of the circuit and J (ρ) = −J (ρ)t
corresponds to power continuous interconnections in the cir-cuit.
For this class of systems, it was shown [9] that the threematrices J (ρ), R(ρ) and g(ρ) can be put into an affine rela-tion. Replacing these quantities in (1), the next affine modeldynamics were obtained:
x = a(ρ)x + b(ρ) with ρ ∈ {0, 1}k (2)
For a circuit with k pairs of physical switches, we will have aswitched system with 2k operating modes that can be imposedby the configuration of all switches. Furthermore, we canassign to each mode a region of state space where it will beactivated. How and when switch between these various con-figurations is the classical question addressed by the synthesisoperation of switched systems. For DC–DC converters, thisquestion is investigated such that the objective is to regulatethe output to a certain reference value, which represents theequilibrium of the average model. The set of average equi-librium points was called the set of admissible references in[9], and they are calculated by equalizing the Eq. (1) to zero.
0 = (J (ρ0) − R(ρ0))x + g(ρ0)u (3)
where in this case, ρ0 is the vector mode formed by the dutiescycle components, hence ρ0i are included between 0 and 1.However, to determine the set of admissible reference pointswhen the abstract representation (4) of model (2) is used inhybrid formulation, we will use the convex hull definition (5)that is more suitable for LMI implementation.
The obtained switched affine model (2) can be investigatedby the mean of the following abstract general PWA model :
x = ai x + bi , with x ∈ �i and i ∈ Im . (4)
where ai ∈ Rn×n and bi ∈ Rn , Im = {1, 2, . . . , m} withm the number of discrete states or modes and �i are theircorresponding regions of activation. The time variable hasbeen omitted for clarity.
Here the objective is the synthesis of a state-dependentswitching law (where the system mode to be activated issolely determined by x) and state space partition defined byregions �i such that the full state space is covered.
For PWA systems, the desired reference point xa has toverify the average model equilibrium given by the followingconvex combination [14,23]:
m∑
i=1
αi (ai xa + bi ) = 0. (5)
with 0 ≤ αi ≤ 1 and∑m
i=1 αi = 1.
Since this average model equilibrium does not mean nec-essarily a common equilibrium of subsystems, there is nochance to approach these reference points without forcingand fast switching. However, when some subsystems havestable equilibria xei = −a−1
i bi , the control must preventthe system to stabilize at these points when xa �= xei . Onemust know that fast switching is not desirable; it is assumedthrough the paper that within a finite time interval, only afinite number of switches may occur. We will see in the lastsection the most used practical way to avoid fast switching.
In practice, we commonly search for a switching law thathas the ability to bring the system trajectory from any startinginitial condition to a quantifiable small neighboring regioncentered on the desired reference xa and maintain it aroundthis neighboring. Let the ball B(xa , εα) define a neighboringregion centered on xa .
Definition [23] The system (4) is globally asymptoticallypractically stabilizable by switching at a point xa ∈ Rn , iffor every εp ≥ εp min, there exists εα with 0 < εα ≤ εp anda switching law that brings the system trajectory from anystarting point x0 ∈ Rn to B(xa, εα) and maintains it insideB(xa, εp) for all future time.
Precisely, our objective is to find this state-dependentswitching law (with the help of state space partition) andthe scalar εα . We will show (19) that for PWA system, the εp
must satisfy εp ≥ εp min.Without loss of generality, we will consider the prob-
lem at the origin providing that the affine term is equal to:Bi = (ai xa + bi ). In order to make easier the use of the LMIapproach, we opt for the following change of variable:
z =[
x1
]. (6)
Then our system (4) will have the linear form (7) which willbe called augmented system:
z = Ai z, with Ai
[ai Bi
0 0
]; Ai ∈ R(n+1)×(n+1)
and z ∈ Rn+1. (7)
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Then the problem reduces to the stabilization of the system(7) at point z∗ = (0 · · · 0 1)t .
3 Quadratic Stabilization and LMI Formulation
Note that we are not interested in the whole space Rn+1, sinceour augmented system (7) is defined only on the followingstate space subset:
X = {z ∈ Rn+1/zn+1 = 1}. (8)
The objective is to find a systematic way leading to a commonscalar function that checks Lyapunov stabilizing conditionsas BMI for system (7) in the domain X .
Since the average equilibrium is only a mathematicalnotion, the physical energy of a system described by PWAmodel can never be nil at an average (not common) equilib-rium. Whereas in Lyapunov theory, the choice of the energyfunction for such systems may be crucial in control and sta-bilization. It follows that to stabilize these systems, the moreappropriate energy functions are those, which do not van-ish on these fictitious equilibrium points. However, when wewant to approach these points, we expect to reach their neigh-boring with minimum energy to meet best results and reduceswitching. Moreover, we must be able to maintain the sys-tem states at this neighboring. With the help of state spacepartitioning, we will present a method that drives asymptot-ically the system trajectory to the neighboring of an averageequilibrium reference and maintains it inside. The proposedmethod represents a more direct and less computational ver-sion of the approach in [23].
For positive symmetric matrix p ∈ Rn×n and positivescalar εp, let the following function V (z) define a candidateLyapunov function for system (7) and �i the quadratic regionwhere the mode i will be activated.
V (z) = zt Pz and �i ={
z ∈ Rn+1|zt Qi z ≥ 0}
. (9)
where P =[
p αp
αtp εp
], Qi =
[qi αqi
αtqi βqi
], iε Im with P, Qi
∈R(n+1)×(n+1) and p, qi ∈ Rn×n .
Observe that V (z∗) = εp �= 0.
Let D ⊂ Rn be a domain defined as follow:
D = {x ∈ Rn|xt px < εp
}. (10a)
This domain has an extension in the domain X defined by:
D ={
z ∈ X |zt[
p 00 0
]z < εp
}. (10b)
Let the time derivative of V (z) along the trajectory of the ithsubsystem of (7) be defined by:
Vi (z) = zt [At
i P + P Ai]
z. (11)
3.1 Switching Control Strategy
In our approach, each subsystem i of the augmented system(7) will be associated with a quadratic region �i where itwill be activated. This region is designed such that the timederivative Vi (z) of its associated subsystem is negative intothat region. We have opted for the stabilizing switching con-trol strategy that is based on the selection of the subsystemthat has the highest decrease of Lyapunov function Vi (z), inother words, at each switching time, the subsystem that hasthe minimum time derivative Vi (z) will be activated:
σ(z) = argminiε Im{Vi (z)}. (12)
where σ(z) = i is the subscript of the mode to be activated.This stabilizing switching control strategy will be called max-imum descent switching control strategy.
In general, without use of hysteresis or dwell time (seeSect. 3.2), sliding motions can arise when passing from oneregion to another if the corresponding vector fields both pointtowards their common boundary. The solution of this behav-ior is more complicated in the case of synthesis, since we donot know a priori which regions will intersect with each other.The reason is that the regions are not known in advance butare a result of the synthesis procedure. However, for bimodalsystems, it is easy to include explicitly in the BMI a conditionthat excludes this phenomenon. In interest of simplicity, wedo not address explicitly the question; see the discussion in[11] for more details.
Theorem 1 If there exist symmetric matrices qi , a symmetricpositive definite matrix p, vectors αpεRn, αqiεRn, positivescalars γ , δi , θi , εp and scalars βqi such that for iε Im, thefollowing BMI
(1) εp < ε0.
(2)
[p αp
αtp εp
]> 0.
(3)
[at
i p + pai + δi qi + γ In ati αp + pBi + δiαqi
Bti p + αt
pai + δiαtqi 2Bt
i αp + δiβqi
]< 0.
(4)m∑1
θi
[qi αqi
αtqi βqi
]≥ 0.
has a solution for fixed small value ε0, then using the maxi-mum descent control strategy (12), all trajectories of system(7) converge to the domain:
D0 ={
z ∈ X |zt[
p 00 0
]z ≤ (αt
p p−1αp+εp)
2
}. Hence, the sys-
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tem is practically asymptotically stabilizable at point z∗ε D0
from any initial condition z0εX .
Proof of Theorem 1 Notice that z∗ ∈ D. To ensure the prac-tical quadratic stabilizability of the augmented system (7) inD, we have to check if there exist a scalar function V (z),positive scalar γ and regions Xi such that:
(1) V (z) > 0, for all z ∈ {X − D}.(2) Vi (z) ≤ −γ zt z, for all z ∈ Xi , i = 1, m.
(3) ∪ Xi = (X − D).
(13)
The third condition cannot be tackled as LMI in {X − D} andsince (X − D) ⊂ Rn+1, we will use the whole augmentedstate space Rn+1 and the results will be valid in (X − D).
In fact, when V (z) > 0 in Rn+1, then it is positive in(X − D). On the other side, if Vi (z) < 0 for all z ∈ �i
with ∪�i = Rn+1 then Vi (z) < 0 for all z ∈ Xi whereXi = (X − D) ∩ �i with:
∪Xi = ∪((X − D) ∩ �i ) = (X − D) ∩ (∪�i )
= (X − D) ∩ Rn+1 = (X − D).
The third condition is satisfied if ∪�i = Rn+1 which ischecked if there exists θi > 0 such that
∑m1 θi Qi ≥ 0.
Remark 1 In reality, ∪Xi = (X − D) is satisfied if (X − D)
⊆ (∪�i ) even when the regions �i do not cover the com-pletely augmented space Rn+1 but this detail cannot beexploited as LMI. This means that the requirement ∪�i =Rn+1 or its sufficient condition
∑m1 θi Qi ≥ 0 is a strong
condition for our objective.
However, out of this statement, to ensure the practicalquadratic stabilizability we have to check if there exist ascalar function V(z), positive scalar γ and regions �i suchthat:
(1) V (z) > 0, for all z ∈ Rn+1.
(2) Vi (z) ≤ −γ zt z, for z ∈ �i , i = 1, m. (14)
(3) ∪ �i = Rn+1.
Using the S-procedure [24], we obtain the following tractableBMI:
If there exist symmetric matrices Qi , a positive symmetricmatrix P and positive scalars γ , δi , θi such that for i = 1, m:
(1) P > 0.
(2) Ati P + P Ai + +δi Qi + γ I ≤ 0 (15)
(3)
m∑
1
θi Qi ≥ 0.
where I =[
In 0n
0tn 0
], 0n =
⎛
⎜⎝0...
0
⎞
⎟⎠ and P =[
p αp
αtp εp
].
Then using the maximum descent switching control strat-egy (12), all trajectories of system (7) converge to the domainD and the system is practically asymptotically stabilizableat point z∗ ∈ D from any initial condition z0 ∈ X .
In order to obtain inequalities that are easier to use inMatlab’s LMI toolbox. We replace non-strict inequalities bystrict ones. One must know [24], when there is solution tostrict inequalities, there will be solution to non-strict ones.As a result, our BMI becomes:
(1)
[p αp
αtp εp
]> 0.
(2)
[at
i p + pai + δi qi + γ In ati αp + pBi + δiαqi
Bti p + αt
pai + δiαtqi 2Bt
i αp + δiβqi
]< 0.
(3)
m∑
1
θi
[qi αqi
αtqi βqi
]≥ 0, θi > 0. (16)
The first condition is implemented with minimization of εp
or a specified small higher bound ε0 on εp. But the thirdcondition (by definition) requires non-strict inequality and allmatrices Qi must be available for the checking. Thus we cancheck this condition for each resolution without including itin the LMI. A marginal feasibility is widely sufficient in thiscase. However, there will be no guaranty to obtain solutionthat checks the condition. Really, this condition is very strongfor our objective (see Remark 1). Nevertheless, when it issatisfied then ∪Xi = (X − D) which permits to fulfill ourobjective. This makes subsequent covering test (necessary)and sufficient to overcome the difficulty. Necessary in thesense, when this condition is included in the LMI, it is notpossible to find a strictly feasible solution.
In addition, we note that minimum of V(z) in X is at z∗∗ =(x∗∗1
)with x∗∗ = −p−1αp, it is given by:
Vmin(z) = V (z∗∗) = εp − αtp p−1αp. (17)
Note that z∗∗ is different from z∗ but necessarily very nearwhen εp is small. From (first condition) V (z) > 0 for allzεRn+1, the Schur lemma [24] shows that εp > αt p−1α.Then the smallest domain centered on z∗ that contains z∗∗ is
(see Fig. 1) included in D, it is defined in X by: zt[
p 00 0
]z ≤
z∗∗t[
p 00 0
]z∗∗. If we denote by Dα this domain then we
have:
Dα ={
z ∈ X |zt[
p 00 0
]z ≤ αt
p p−1αp
}, Dα ⊂ D. (18)
This domain is also an extension of the domain Dα ⊂ Rn
defined by:
Dα ={
x ∈ Rn|xt px < αtp p−1αp
}.
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4094 Arab J Sci Eng (2014) 39:4089–4102
V(z)=V(z*)=εp
Fig. 1 Simple construction in two dimensions
It follows from the above development that:
εα = εp min = αtp p−1αp. (19)
Remark 2 Observe that this method is direct and requiresless computation effort than the proposed approach in [23].However, the obtained results in [23] are more accurate since
the approach ended up with εp min = αtp p−1αp
4 .
Since V(z) > 0 for all z ∈ Rn+1 and Vi (z) < 0 for z ∈ �i
with ∪�i = Rn+1, then Vi (z) is negative for all z ∈ Xi ,i ∈ Im with Xi = X ∩ �i and ∪Xi = X , which leads to theconclusion that we have also V(z) > 0 and Vi (z) < 0 in D.This means that V(z) converges inevitably to Vmin(z). Sincethis minimum is at z∗∗ which is on the closure of Dα , it maybe constructive to stop switching before; for example whenV(z) reaches a chosen domain D0 such that Dα ⊂ D0 ⊂ D.
This can achieved by:
D0 ={
z ∈ X |zt[
p 00 0
]z ≤ (αt
p p−1αp + εp)
2
}(20)
When Dα or the chosen D0 is reached, the correspondingmaximum errors on states (distance from z∗) can be easilycalculated from the above equation once p and αp are com-puted.
In conclusion, we can affirm that if the two first condi-tions in (16) are checked for all z and we have a positivesubsequently covering test, then all trajectories of system (7)converge to the domain D. In light of the previous discussion,D is replaced by the smaller domain Dα or the chosen domainD0 with Dα ⊂ D0 ⊂ D. This end the proof of Theorem 1.
3.2 Practical Switching Control Strategy
Using the switching control strategy in (12), as stated before,we obtain a wide practical satisfaction, if we stop switch-
ing once the trajectory reaches the domain Dα (or the cho-sen D0) and restart when the trajectory tries to get out ofdomain D. This strategy limits switching frequency; how-ever, when εp is very small, there will be no significant dif-ference between these two domains and fast switching mayhappen. Moreover, due to the used switching control strat-egy, this phenomenon may appear far away from the desiredpoint. In this case, we call for the usually used simple tech-nique based on state-dependent hysteresis. It consists on theuse of a given negative higher bound on the Lyapunov deriv-ative: max(Vi (z)) = −εzt z with ε >0 and switch off whenLyapunov derivative Vi (z) of the active mode exceeds thisbound. This technique is combined with the practical sta-bilization to exclude this phenomenon. It is reasonable inthe synthesis operation with common Lyapunov function toallow regions to overlap with each other. If regions interiorsare pairwise disjoint, then it is clear that the use of hysteresisin our switching strategy will obligatorily lead to a selectionof a subsystem with increasing energy function. Similarly inour practical stabilization, when restart switching (if domainD is reached with a certain active mode), one must selectamong other subsystems that have an increasing energy func-tion in the region of the active mode. This is not necessarilythe case in overlapping regions, which permit a great flexi-bility in the choice (according to switching strategy) betweendifferent subsystems. Therefore, nothing has been done (inthe LMI) imposing regions to be disjoints. However, in ourpractical stabilization method even with overlapping regions,when we stop switching in the smallest domain Dα (or D0)the energy function of the active mode will increase (for ashort time) allowing the trajectory to reach the domain D.This will happen only when we approach the desired aver-age equilibrium. In fact, for all approaches, this is always thesituation when regulating around a not common equilibrium.
4 State Estimation
This section is devoted to state estimation problem whenthey are not all accessible for measure. As known, the moregeneral question of observability as well as controllabilityof switched systems may be gained or lost by switching [5].Furthermore, several works showed that there is no way toderive simple results about observability for switched sys-tems from properties of their individual components unlessthe subsystems present some specific properties that may beexploited. In this paper, we will not deal with the complextheories of observability and their various definitions [15–17], we will expose only our idea for a simple computationalestimation technique that will be used for DC–DC convertersviewed as PWA systems.
Generally, two main difficulties are encountered in theobservability of switched systems:
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• Estimated states jump.• Mode location.
The first one must provide a way that properly updates theestimated states when the observer mode change occurs. Thisis necessary to ensure convergence of the estimation error.The second difficulty is related to the active discrete moderecognition; except few concrete suggestions [15,19], thislast question is still in its early theoretical stage.
However, in our case the active discrete mode is deter-mined by the control approach. Therefore, the observerdesign will deal with known active mode [18]. In fact, whenthe same control signal is used to select both observer andcircuit modes, then at each time the correct active mode isavailable without any time delay. Furthermore, since thereare neither jumps nor reset in system state variables, therewill be no jumps and no resets in observer-estimated states,so it remains to ensure error convergence. An expansion ofthe Luenberger observer to switched systems may be doneas follow:
The continuous-time dynamic of Luenberger-like observeris given as follow:{ ˙x = ai x + bi + Li (s − s).
s = Ci x .(21)
Where xεRn , is the estimated state vector and Li are theobserver gains.
If we associate with each individual error a space region �i
such that these regions cover the space Rn . Then the dynamicof the estimation error x = x − x is defined by:
˙x = x − ˙x = (ai − Li Ci )x, for x ∈ �i (22)
with �i = {x ∈ Rn|x t qi x ≥ 0}.Notice that the obtained estimation errors are also gov-
erned by a switched system in (22); consequently, it is notsufficient to stabilize each error mode separately. As in thefirst control part, we will use a common Lyapunov-like func-tion to formulate the state estimation as BMI, which completethe proposed systematic control approach.
Using the following single-candidate Lyapunov function:
v(x) = x t px, with p = pt > 0. (23)
For x ∈ �i , the time derivative along the trajectory of theith error mode is given by:
˙vi (x) = x t (ati p + pai − Ct
i Lti p − pLi Ci )x . (24)
The accuracy of the estimated states in (21) is confirmed bythe convergence of their corresponding errors that is ensuredby the stability of the switched system in (22).
The following theorem resumes the Lyapunov stabilityconditions for our switched system in (22).
Theorem 2 If there exist symmetric matrices qi , a symmetricpositive definite matrix p, vectors Wi and positive scalars γ ,μi ,θi such that for iε Im, the following BMI:
(1) p > 0.
(2) ati p + pai + μi qi + γ In − Ct
i W ti − Wi Ci < 0.
(3)m∑1
θi qi ≥ 0.
has a feasible solution, then all errors of system (22) convergeasymptotically to zero and the observer gains are given byLi = p−1Wi ; hence, the estimated states (21) represent agood estimation of our system states that can be used insteadin the feedback to stabilize the converter output by switching.
Remark 3 We do not need to specify the switching strategysince the switching, i.e., the active mode is imposed by thecontrol approach. Here we mean a stability analysis and nota synthesis operation as in the control approach.
Proof of Theorem 2 Since the obtained estimation errors (22)are also governed by a switched system, we are looking forthe stability of the global error of the switched system (22).Using v(x) as a candidate common Lyapunov-like functionfrom the Lyapunov stability theory we state that if:
(1) v(x) > 0 for x ∈ Rn
(2) ˙vi (x) ≤ −γ x t x for x ∈ �i . (25)
(3) ∪ �i = Rn
Then the system (22) is stable at the origin x =⎛
⎜⎝0...
0
⎞
⎟⎠, i.e.,
the estimation errors in (22) converge asymptotically to zero.
• No problem with the first condition, it requires only thatit exists a matrix p = pt > 0 for v(x) = x t px .
• The second condition requires that:
x t (ati p+ pai −Ct
i Lti p− pLi Ci )x ≤−γ x t x for x ∈ �i .
Using the S-procedure to remove region constraints, it canbe rewritten as:
ati p+ pai +μi qi +γ In −Ct
i Lti p− pLi Ci ≤ O for x ∈ Rn
Because of the product Lti p and pLi , this is a nonlinear
matrix inequality NMI that cannot be tackled. In order toovercome this problem, let us make the following change ofvariable:
Wi = pLi (26)
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4096 Arab J Sci Eng (2014) 39:4089–4102
then the second condition becomes bilinear and equivalentto:
ati p + pai + μi qi + γ In ≤ Ct
i W ti + Wi Ci , for γ, μi > 0.
(27)
• The third condition is a simple covering property ofregions �i that is satisfied if there exists positive scalarsθi such that
∑m1 θi qi ≥ 0 when the regions �i are given
by �i = {x ∈ Rn|x t qi x ≥ 0}.
Replacing the non-strict inequality by a strict one in (27),then our conditions in (25) can be rewritten as:
(1) p > 0
(2) ati p+ pai+μi qi+γ In < Ct
i W ti +Wi Ci , for γ, μi > 0
(3)
m∑
1
θi qi ≥ 0 for θi > 0 (28)
When including these BMI in the BMI control, and if thereis a solution that leads to the unknowns p, qi and Wi , hencethe estimated errors in (22) converge to zero and the observergains are back calculated from (26):
Li = p−1Wi . (29)
This end the proof of Theorem 2.
Here, again the used sufficient condition, i.e., existence ofθi > 0 such that
∑m1 θi qi ≥ 0 associated with the covering
property ∪�i = Rn may cause problem for the observersynthesis.
The results of the considered DC–DC converter examples(bimodal PWA system) are direct and the simulation confirmsthe rapid convergence of the estimation errors of this simpleobserver. However, for multi-cellular converters (more thantwo modes) the rate of errors convergence is very slow andthe method does not apply without major change such as theuse of dwell time; the problem is still under investigation.
5 Simulation Results
Generally, for DC–DC converter models, only input supplyand resistive load variations are considered. These changesare useful to test the ability of any control approach. Thenoisy measure of the output may also be considered to showstate estimator quality.
Note that all admissible reference points are reached witherrors less than 2 % after a slightly prolonged rising time.Therefore, the selected reference points for simulation arearbitrarily chosen. The set of admissible points verifying theconditions in (5) may be obtained by resolving these condi-tions separately from the design control. However, since we
have opted for a complete systematic methodology, we havedirectly introduced these conditions in the LMI control aver-aging some approximations for LMI implementation. For theobserver, we have considered the current iL as a measuredoutput, however similar results are obtained when the outputvoltage is measured.
For the LMI optimization results, the obtained matricesare not unique; the results depend on the choice of ε0 andessentially on the grid-up method of unknown scalars in theobtained BMI.
5.1 Analysis of the Benchmark Converters
The benchmark examples (Figs. 2, 3) are theoretical schemesof the treated converters in [1–3,9] where the selected para-meters values still allow to capture the conventional func-tioning of these circuits. Nevertheless, some related analysisaspects of the converter parameters and their operating modesare briefly discussed in the following paragraph.
The two treated examples are simplified versions of theBoost and Buck–Boost converters with ideal components.Where the continuous source E has a negligible internal resis-tance, no energy is lost in the inductor L or in the capacitor C,the diode D has an ideal characteristic with no voltage dropin conducting mode and switch exactly at zero voltage level.
As a rule for the design of DC/DC converter, from a desiredoutput voltage and the nature of the energy source, a range ofswitching frequency, a nominal input voltage and load valuesare suggested. In practice, these nominal values may deviate.For example, 20 % line variation is expected or the nominalload may deviate to no-load or full load.
L
Fig. 2 Example 1: simplified electric circuit of the Boost converter
Fig. 3 Example 2: simplified electric circuit of the Buck–Boost con-verter
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Arab J Sci Eng (2014) 39:4089–4102 4097
In general, these converters feature three different modesof operation, where each mode has an associated linear contin-uous-time dynamic. Furthermore, constraints are presentwhich result from the converters topology. In particular, themanipulated variable (duty cycle) is bounded between zeroand one, and in the discontinuous current mode a state (induc-tor current) is constrained to be nonnegative. Additional con-straints may be imposed as safety measures, such as currentlimiting or soft starting.
However, in practice we commonly try to avoid magneticsaturation (high inductor current ripple) and load regulationproblems (a load-dependent converter ratio) such that thethird mode corresponding to discontinuous conduction modeDCM is discarded. Therefore, only two operating modesare considered for the target converters. Indeed, this canbe achieved by selecting appropriates values for L and C toforce the converter operating in continuous conduction modeCCM. The LC stage constitutes a low-pass filter whose cor-ner frequency fc = 1/2π LC is chosen to be sufficiently lessthan the switching frequency, so that the filter essentiallypasses only the dc component. To the extent that the induc-tor and capacitor are ideal, the filter removes the switchingharmonics without dissipation of power. Thus, the converterproduces a dc output voltage whose magnitude is control-lable via the duty cycle, using circuit elements that (ideally)do not dissipate power.
A simple mathematical calculation [25] permits to findthe critical value for the inductor to determine a boundarybetween the two conduction modes:
L ≥ Lcritical for (CCM).
L < Lcritical for (DCM).
When referring to Figs. 2 and 3, and for Ts = 1/ fs withfs the switching frequency, Pin the input power, E and V are,respectively, the input and output voltages, the critical valuefor the inductor is given by [25]:
Lcritical = (E)2Ts
2PinV
(1 − E
V
)for Boost converter
and Lcritical = (V )(E)Ts
2Pin
(E
V + E
)
for Buck–Boost converter
Here the design requires the two converters to operate inCCM under all conditions, hence the condition L ≥ Lcritical
should be satisfied. From the above formulas, it can be seenthat for the two converters, the highest Lcritical values occurwhen P in is in its minimum value and the input voltage E atits maximum value, which is considered to be the two worstcases for the converters to operate in CCM.
Moreover, in our practical method the switching frequencycannot be predetermined. A selection of an estimated averagevalue for fs (see Fig. 11) that may be high leads to a small
value for Ts which permits a large range to choose L andprivilege the CCM for these converters, i.e., one can select asmall value for L (which reduce the converter cost) and stilloperating in CCM. Once a value of L is selected, a C valuecan be deduced.
On the other hand, it is well known that for Boost andBuck–Boost converters, the efficiency is high at low dutycycles, but decreases rapidly to zero when the duty cycleapproaches 1 for non ideal inductor. However, maximumefficiency is generally obtained for relatively (to the load)very small internal resistance of the inductor. This analysiscannot be performed for the assumption of ideal convertercomponents; moreover, we have no direct access to the dutycycle in our control approach.
5.2 Example 1: Boost Converter [2]
J (ρ) =(
0 −ρ
ρ 0
), R(ρ) =
(0 00 1/R
), g(ρ) =
(10
), F =
(1/L 0
0 1/C
), E = 0.75 V, R = 1 �, L = 1 H, C = 1 F;
x =(
iLV
). Selected reference: V0 = 1 V, iL0 = (4/3) A.
The control objective is achieved with good accuracy fromany initial conditions as seen in Figs. 4 and 5. The same per-formances are obtained when the control is based on theobserver output. It must be pointed out that in absence ofout regulating loop, the load disturbances and input voltageirregularity should be limited since there is no perfect reject;however, the corresponding errors are still small for moderatedisturbances. Figure 5 shows the good quality of our estima-tor, where the rapid convergence of the estimation error isobserved. The ability of our observer-based control to dealwith the case of very high power measure noise is establishedin Figs. 6 and 7 which confirm the interest of the proposedapproach as a practical method.
5.3 Example 2: Buck–Boost Converter [9]
J (ρ) =(
0 ρ
−ρ 0
), R(ρ) =
(0 00 1/R
), g(ρ) =
(1 − ρ
0
),
F =(
1/L 00 1/C
), E = 1 V, R = 1 �, L = 1 H, C = 1 F;
x =(
iLV
).
Selected reference point: V0 = −1 V, iL0 = 2 A.Similar performances to those obtained in example 1 are
confirmed when the control approach and state estimationtechnique are applied to the Buck–Boost converter of exam-ple 2. As in the previous example, Fig. 8 presents control per-formances to regulate converter output at point (iL0, V0) fromany initial condition. Figure 9 shows state estimation quality
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4098 Arab J Sci Eng (2014) 39:4089–4102
−3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
5
IL (A)
V (
V)
Fig. 4 Example 1: output voltage vs. the inductor current for differentinitial conditions
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
1.2
1.4
t in sec
(a)
Cur
rent
IL (
A)
IL
estimated IL
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
t in sec
(b)
Vol
tage
V (
V)
V
estimated V
Fig. 5 Example 1: inductor current and its estimate (a), output voltageand its estimate (b); with system initial conditions different from theobserver ones
for this example whereas Fig. 10 confirms the applicabilityof the approach when the control is based on the estimatedstates.
0 5 10 15 20 25 300
0.5
1
1.5
2
t in sec
(a)
Noi
sy O
bser
ver
Out
put I
L (A
)
0 5 10 15 20 25 30-0.2
0
0.2
0.4
0.6
0.8
1
1.2
t in sec
(b)
Noi
sy O
bser
ver
Out
put V
(V
)
Fig. 6 Example 1: observer output with high measure noise, inductorcurrent (a) and output voltage (b)
5.4 Discussion and Comparison
Before proceeding the comparison, it must be pointed outthat our control approach can be classified as fixed switchingfrequency method. Variable switching frequency may be con-sidered as a disadvantage for switched mode DC–DC con-verters especially from electromagnetic interference (EMI)design problem [3]. As mentioned above for our approach, wedo not have direct access to the duty cycle and the switchingfrequency can neither be predetermined nor be subsequentlycalculated. However, according to the simulation results (seeFig. 11), in steady states, the converters operate at a quasi-constant duty cycle and an average switching frequency maybe selected. Hence, this undesirable effect is greatly reduced.
The obtained performances are stated for ideal circuits andcannot be guaranteed when the approach is applied to realconverters. For this reason and in order to make a constructivecomparison between our method and the developed meth-ods in [2,3], we have to pick the same non ideal benchmarkconverter and perform the same performances tests recom-mended in the HYCON meeting report Da51 [3]. This report
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Arab J Sci Eng (2014) 39:4089–4102 4099
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
1.2
1.4
t in sec
(a)C
onve
rter
Out
put I
L (A
)
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
t in sec
(b)
Con
vert
er O
utpu
t V (
V)
Fig. 7 Example 1: system output for control based on noisy observeroutput of Fig. 6, inductor current (a) and output voltage (b)
−6 −4 −2 0 2 4 6 8−8
−6
−4
−2
0
2
4
6
8
IL (A)
V (
V)
Fig. 8 Example 2: output voltage vs. the inductor current for differentinitial conditions
focused on the capability of hybrid systems techniques forhigh-performance design of power electronic devices.
Remind that the HYCON meeting participants groups areamong the more European experienced ones on switching
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
t in sec
(a)
Cur
rent
IL (
A)
Current IL
estimated IL
0 5 10 15 20 25 30-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
t in sec
(b)
Vol
tage
V (
V)
Voltage V
estimated V
Fig. 9 Example 2: inductor current and its estimate (a), output voltageand its estimate (b); with system initial conditions different from theobserver ones
mode converters and have a worldwide reputation, the affil-iation of these groups are cited in Table 1.
Each of these research groups proposed more than onecontrol method for switching mode DC–DC converters. TheCRAN approach used a continuous-time switched systemframework. Their method associates optimal control tech-niques to the sensitivity function analysis to tune the parame-ters of the controller. In collaboration with CNRS (NationalCenter for Scientific Research, Paris, France), they proposedalso a predictive and sliding mode methods for this bench-mark. The ETH group focused on two classes of discrete-time hybrid models: Mixed Logical Dynamic (MLD) andPWA systems. They employ a control action obtained byminimizing an objective function over a finite horizon sub-ject to the constraints of the MLD or PWA model and thephysical constraints on the manipulated variables. In orderto ensure problem feasibility, they opted for pre-solving off-line the optimization problem for the whole state space usingmulti-parametric programming that results in an equivalentstate feedback control law parameterized over the whole statespace. The starting point for KTH researchers is a class ofpulse width-modulated systems where the dynamics peri-odically switches between two affine vector fields in a given
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4100 Arab J Sci Eng (2014) 39:4089–4102
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
t in sec
(a)C
onve
rter
Out
put I
L (A
)
0 5 10 15 20 25 30-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
t in sec
(b)
Con
vert
er O
utpu
t V (
V)
Fig. 10 Example 2: system output for control based on highly noisyobserver output, inductor current (a) and output voltage (b)
order. Buck converters of standard topologies with fixedswitching frequency belong to this system class. They con-sider a closed loop systems as primary design criteria to sus-tain large changes in operating point due to load and/or sourcevariations. For this purpose, they derived sampled data mod-els for counterparts of the standard LQ and H∞ criteria.However, the resulting design criteria from their sampled datamodels are highly nonlinear and some approximations seemto be necessary during the design process to derive the synthe-sis algorithms corresponding to these two criteria. The LTHmethodology considers the periodically time-varying linearsystem obtained by linearization. For larger disturbances andtransients, it is not enough to consider linearized models. Forthis purpose, switching linear systems and piece-wise affinemodels are used. A linear time-periodic optimal control prob-lem is applied and relaxed dynamic programming is used tooptimize switch sequences for the pulse width modulation.They also address the observer problem to estimate systemstates that are not available for measurement. Instead, Sup-elec focused rather on Boost and Buck–Boost Converters.
Based on their certified concluding remarks and our exam-ination, the different proposed approaches in the report [3]have showed good performances that are strongly related.Most of them are based on sampled and discrete-time modelwhere some level of control complexity and off-line compu-
15 20 25 300.7
0.72
0.74
0.76
0.78
0.8
t in sec
(a)
Dut
y C
ycle
15 20 25 300
0.2
0.4
0.6
0.8
1
t in sec
(b)
Dut
y C
ycle
Fig. 11 Examples 1 and 2: duty cycle for the Boost (a) and the Buck–Boost (b) converters for start-up from zero initial conditions
Fig. 12 Example 3: non ideal electric circuit of the Buck converter
tation are required for their real applications. However, theirreported results provide a very good survey on the variousmethods that may be applied for the control of DC–DC con-verters. As a result, the global same performances displayedin this report may be considered as a certified orientation foracademic research. Due to space limitation, we restrict thecomparison to the non ideal step-down (Buck) converter inFig. 12.
The PWA model of the Buck converter is given by:
x = Ax + bi , i = 1, 2
with A =[
− 1xL
(rL + RrcR+rc
) − 1xL
RR+rc
1xc
RR+rc
− 1xc
1R+rc
],
bi =(
ρE/xL
0
), x =
(iLVc
).
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Arab J Sci Eng (2014) 39:4089–4102 4101
Table 1 Affiliations of participant groups in the HYCON meeting
University Affiliation Group leader
ETH Swiss FederalInstitute ofTechnology,Zurich, SwissFederal
M. Morari
KTH Royal Institute ofTechnology,Stockholm,Sweden
U. Jönsson
LTH Lund Institute ofTechnology, Lund,Sweden
A. Rantzer and R. Johansson
CRAN Research Center forAutomatic, Nancy,France
Pierre Riedinger
SUPELEC Laboratory ofHybrid Systems,Supelec, Rennes,France
Jean Buisson
The circuit parameters in per-unit system (normalized unit)are E = 1.8 p.u; xc = 70/2π p.u; xL = 3/2π p.u; rc =0.005 p.u; rL = 0.05 p.u; R = 1 p.u and the desired refer-ence V0 = 1 p.u; iL0 = 1 p.u.
Here we have V0 = RrcR+rc
iL + RR+rc
Vc and ρ is a Boolean(0 or 1) variable that denote the on/off state of the switch.
The obtained simulation results for this benchmark con-verter are assembled in Fig. 13 where accepted performancesare observed without start-up current limit process. In fact,this figure shows that the weakness of our approach lay inthe slightly observed high overshoot and prolonged timeresponse. However, these performances criteria can be greatlyimproved when set-up maximum limit on the inductor cur-rent is expected. On the other hand, from the various per-formed robustness tests (results not reported), we notice thatour control approach presents an excellent robustness againstsupply line variation whereas this is not the case for load fluc-tuation. This last problem may be resolved using outer reg-ulating loop to handle load regulation problems. The Table2 contains performances criteria obtained by the proposedapproach and those generally displayed in the HYCON meet-ing report [3]. The comparison table shows that our resultsare not very distinct from these professional results; how-ever, the proposed approach is a preliminary one and someadditional improvements (as mentioned above) are requiredbefore its implementation.
6 Conclusion
Systematic state feedback controls with simple state estima-tor have been derived for DC–DC converters’ ideal circuits.
0 10 20 30 400
1
2
3
4
5
t in sec
(a)
Indu
ctor
cur
rent
IL (
p.u)
0 10 20 30 400
0.2
0.4
0.6
0.8
1
t in sec
(b)
Out
put V
olta
ge V
0 (p
.u)
20 20.02 20.04 20.06 20.08 20.1
0
0.2
0.4
0.6
0.8
1
t in sec
(c)
Dut
y C
ycle
Fig. 13 Example 3: inductor current iL (a), output voltage V0 b andduty cycle (c) for the Buck start-up from zero initial conditions
The extensive simulation results are very promising for bothcontrol and observer. An advantage of the control part is itsapplicability to switched PWA systems with multiple dynam-ics (more than two modes). However, still deeper investiga-tion may be done to complete the method especially on theset neighboring points other than the average equilibriums.At some neighboring points, the system may be practicallystabilized with certainly much less accuracy. As mentionedbefore, the noticed weakness, i.e., time response and loadregulation will be easily improved on specifying maximumlimit on inductor current during start-up and outer regulatingloop. Possible additional constraints on duty cycle may also
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Table 2 Comparison table
Methods Performances criteria
Switchingfrequency atsteady state
Duty cycle atsteady state
With start-uplimit currentprocess atiLmax = 3 p.u
Inductor currentovershoot(
iLmaxiL0
)Output voltagerising time (s)
Robustnessagainst linefluctuation
Robustnessagainst loadfluctuation
HYCON Da51presented methods
Constant Constant at 0.58 Yes 3–3.5 7–10 Good Good
Our proposed method Quasi-constant Quasi-constant at 0.58 No ∼= 5 14 Very good Poor
be considered as an important criteria for practical imple-mentation. On the other hand, it is meaningful to try extend-ing the observability part to the more general PWA systemswith multiple dynamics to cover multi-cellular converters;this is an important point for the general hybrid observer thatis still under investigation. Another important mathemati-cal point must be revealed, we state that an advanced workis needed to inspect necessary conditions for the coveringproperty, this will resolve many associated problems. Noticethe advantage when working with BMI formulation, we caneasily deal with additional constraints (if any) to improveperformances and facilitate real implementation. In particu-lar, conditions excluding the occurrence of sliding motions(as in [11]) may be included. Finally, we expect to prove thistheory by experimental results in a future work.
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