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Practical stabilization for piecewise-affine systems: A BMI approach
Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
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Practical stabilization for piecewise-affine systems: A BMI approach D. Kamri a,∗ , R. Bourdais a , J. Buisson a , C. Larbes b a
Automatic and control group, Supélec-IETR, Rennes Campus, France
b
Department of Electronic, National Polytechnic School, Algiers, Algeria
article
info
Article history: Received 16 May 2010 Accepted 9 January 2012 Keywords: Hybrid systems Lyapunov theory LMI PWA systems Practical switching stabilization
abstract We propose in this paper, a systematic switching practical stabilization method for PWA switched systems around an average equilibrium. For these systems, the main difficulty comes from the fact that to end in BMI formulation, it is necessary to represent the system in an augmented state space but a restricted one. However, the derived stabilizing conditions are not tractable as BMI in the restricted domain. We will present a method that overcomes this difficulty and drives asymptotically system states into a ball centered on the desired non-equilibrium reference. The efficiency of this practical stabilization method is showed by the ball smallness and the good robustness against disturbances. The design control searches for a single Lyapunov-like function and an appropriate continuous state space partition to satisfy stabilizing properties. Therefore, the method constitutes a simple systematic state feedback computation; it may be useful for on-line applications. As a direct application, satisfactory simulation results are obtained for two illustrative examples, a Buck–Boost converter and a multilevel one. Due to their functioning nature, these devices constitute good examples of switched systems. They are electrical circuits controlled by switches to produce regulated outputs despite the load disturbances and power supply irregularities. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In most real-world practical applications, nonlinearity is the rule rather than the exception. To get rid of the complexity of nonlinear models, one often resorts to a widely used modeling strategy that represents the system behavior by a set of simple (linear) models and a switching rule that orchestrates between these operating regimes to obtain a hybrid representation of the system. Generally, the submodels (electrical circuits, biological systems etc.) have an affine relation or may be approximated by affine functions; hence they are called piecewise affine (PWA) switched systems [1,2]. In the above definition, one must distinguish between the two situations of analysis and synthesis. In the former case, the state partition is a result of the dynamics modeling effect; thus it is previously specified for the analysis operation. While our concern is the latter case where the aim, is to find this partition and the state dependent switching law such that the switching between possibly unstable subsystems results in a stable system. The PWA switched systems are a known class of hybrid systems. They are multi-models representation that offers a good modeling framework for complex dynamical systems that involve nonlinear phenomena. They have received increasing interest over the last years and still pose challenging control problems even for simple practical examples [3–6]. Generally, the proposed Lyapunov based approaches provide only sufficient conditions with no guaranties that a Lyapunov function can be found; however if one is found, the result is unambiguous. In this context, several constructive propositions have emerged in the literature for the analysis and the design
∗
Corresponding address: Department of Electric Engineering, University of Laghouat, BP 37G, 03000 Laghouat, Algeria. Tel.: +213 698276530. E-mail addresses: [email protected], [email protected] (D. Kamri), [email protected] (R. Bourdais), [email protected] (J. Buisson), [email protected] (C. Larbes). 1751-570X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2012.01.001
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of switched systems [7–12]. Even though PWA systems benefit enormously from the proposed approaches, few researches dealt explicitly with the PWA systems. The more specialized papers on PWA switched systems started with the reported methods in [5,13,14] where the analysis problems are formulated as LMI. In these approaches, some attempts to extend their results to the synthesis operation were formulated; however, the proposed feedback controls are still suited for systems with common equilibrium only. An integral of performance index is optimized for control law in [15], where lower and upper bounds are computed which increase the computational speed in the proposed approach. For PWA systems with multiple equilibriums, [16] addressed the dynamic output feedback synthesis question for controlled systems. Using the discretetime PWA models, many others computed state feedback control have been suggested [17,18]. However, in this paper we are interested in investigating the state-dependent switching stabilization problem for PWA systems. Our main motivation for this direction is the control of power electronic switching circuits where the affine term in the PWA model is essentially due to the input voltage. Apart from [16], the resumed work in [19] or the specific approaches developed by the main group working on electronic circuits [18]; most proposed approaches do not deal with PWA systems without (or not common) equilibrium. In this context, the available results are the elegant necessary and sufficient conditions for quadratic stabilization of pair LTI systems [7,20,21]. However, the generalization to more than two subsystems is not well established and the question remains open. Recently, many others papers investigated the switched and PWA systems without significant novelty such that the survey papers [9,18,22] remain a complete and detailed description of the problems arising in this area. The outline of this paper is as follow: Section 2 presents the model to be used and problem statement. Section 3 contains the proposed approach within its application to quadratic stabilization. Section 4 illustrates the theoretical results on practical examples. Notation. It will be assumed throughout the paper, that the numerical zero may specify a vector or a matrix of zero with appropriate dimension that can be understood from the context. 2. Problem statement Several situations of nonlinear approximation, saturating, abrupt changes in systems components as well as classical linearization around different operating points lead to the following general PWA model: x˙ (ξ ) = aσ x(ξ ) + bσ ,
ξ ∈ R+ ,
(1)
where x(ξ ) ∈ R is the state and σ is a piecewise constant function of time ξ , called switching signal. In our case, we are interested by the state-dependent switching where the system mode to be activated is determined exclusively by x. This leads to the following switching function σ (x) : Rn → Im = {1, . . . , m} with m number of mode or subsystems in (1), we say that the switched system is in the ith operating regime when σ (x) = i. In all future use of PWA model, the subscript σ is replaced by the subscript i and the time variable ξ is omitted for clarity. Generally, in practice we search for a switching law that has the ability to bring the system trajectory from any starting initial condition x0 to a quantifiable small neighboring region B(xa , εα ) centered on the desired reference xa and maintain it around this neighboring. n
Definition. The system (1) is globally asymptotically practically stabilizable by switching at a point xa ∈ Rn , if for every scalar εp ≥ εpmin , there exist εα with 0 < εα ≤ εp and a switching law that brings the system trajectory from any starting point x0 ∈ Rn to B(xa , εα ) and maintain it inside B(xa , εp ) for all future time. Precisely, our objective is to find this state dependent switching law and the scalar εα . We will show (17) that for PWA system, the εp must satisfy εp ≥ εpmin . As known for PWA systems, the desired reference point xa must belong to the set of average equilibrium points that verify the following convex combination: m i =1
αi (ai xa + bi ) = 0 with 0 ≤ αi ≤ 1 and
m
αi = 1.
(2)
i =1
Since the subsystems do not necessarily share any equilibrium, there is no chance to approach these reference points without forcing and fast switching. One must know that fast switching is not desirable; it is assumed through the paper that within a finite time interval, only a finite number of switches may occur. We will see in the last section the most used practical way to avoid fast switching. When state space partition is explicitly used in the synthesis, we associate with each mode a space region Ωi where it will be activated. The regions Ωi must cover the whole state space. In this case, the synthesis objective will be to determine these regions and the state dependent switching law such that the switching between possibly unstable subsystems results in a stable system.
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Without loss of generality, we will consider the problem at the origin providing that the affine term is equal to: Bi = ai xa + bi . In order to make easier the use of the LMI approach, we opt for the following change of variable:
z=
x . 1
(3)
The obtained linear form (4) will be called: augmented system.
z˙ = Ai z
with Ai =
ai 0
Bi 0
and i ∈ I m .
(4)
Hence, the problem reduces to the stabilization of the system (4) at point: 0
.. . z∗ = .
(5)
0 1
3. Quadratic stabilization and LMI formulation Note that for our augmented system (4), the synthesis of a switching law and its corresponding state partition concern only the defined following state space subset:
X = z ∈ Rn+1 |zn+1 = 1 .
(6)
Our main objective is to develop a systematic synthesis technique that leads to a common scalar function verifying Lyapunov stabilizing conditions as tractable BMI in a partition of the domain X. However, as will be seen later, this will require a partition of the whole state space Rn+1 . Let Σi be quadratic regions of the state space Rn+1 , defined by: with Qi ∈ R(n+1)×(n+1) and i ∈ I m .
Σi = z ∈ Rn+1 |z t Qi z ≥ 0
(7)
Observe that the number of regions to be designed is equal to the number of subsystems in our system (4). Consider the following candidate Lyapunov function for our system (4): V1 (z ) = z t P1 z
with P1 =
αp
2p
α
t p
0
and p = pt > 0
(8)
where z ∈ Rn+1 , p ∈ Rn×n and αp ∈ Rn . Let us define a domain D ⊂ Rn as follows: D = x ∈ Rn |xt px < εp .
(9a)
This domain has an extension in the domain X, defined by: ⌣
D =
z ∈ X|z t
p 0
0 z < εp . 0
(9b)
Since the candidate Lyapunov function in (8) is not positive in the whole domain X, we will try to prove its positivity in ⌣ ⌣ a sub domain X − D and content our self to Practical stability. However, even in this sub domain X − D we cannot directly prove the strict positivity of this function; the following matrix P2 is used as an intermediate to meet our objective.
P2 =
p
α
t p
αp , εp
εp > 0.
(10)
Let V˙ 1i (z ) define the time derivative of V1 (z ) along the trajectory of the ith subsystem in system (4) that is given by: ∂ V (z ) V˙ 1i (z ) = ∂1z ddzξ with ddzξ = Ai z and ξ ∈ R+ , This lead to the following expression for V˙ 1i (z ): V˙ 1i (z ) = z t Ati P 1 + P1 Ai z .
(11)
Control strategy. In our approach, each subsystem i of the augmented system (4) will be associated with a quadratic region Σi where it will be activated. This region is designed such that the time derivative V˙ 1i (z ) of its associated subsystem is negative into
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that region. One used stabilizing switching control strategy is based on the selection of the subsystem that has the highest decrease of Lyapunov function V1 (z ):
σ (z ) = argmini∈Im V˙ 1i (z )
(12)
where σ (z ) = i, is the subscript of the mode to be activated. We refer to this stabilizing control switching strategy by maximum descent control switching strategy. Lemma. If there exist a symmetric positive matrix p ∈ Rn×n , symmetric matrices Qi ∈ R(n+1)×(n+1) , vector αp ∈ Rn and positive scalar εp such that: P2 > 0 and V˙ 1i (z ) is negative when z ∈ Σi and i ∈ Im with ∪Σi = Rn+1 then using the maximum descent control switching ⌣ strategy, all trajectories of system (4) converge to the domain D . Hence, the system is practically asymptotically stabilizable at ⌣ ∗ point z ∈ D from any initial condition z0 ∈ X. ⌣
Proof. Observe that V1 (z ∗ ) = 0 and z ∗ ∈ D . From (9a) we have xt px ≥ εp for all x ∈ {Rn − D}. This will be written in the domain X as follow: z
t
p 0
Then z t
p 0
0 0 z ≥ zt 0 0
0 0
z − zt
0 0
0
0
εp
εp z
⌣
for all z ∈ X − D .
z
(13)
⌣ ≥ 0 for all z ∈ X − D . Well, if P2 > 0 (this can be checked as LMI) i.e. z t P2 z > 0 for
z ∈ Rn+1 , and since X ⊂ Rn+1 then we have for all z ∈ X: z
t
p 0
0 0 z + zt αpt 0
αp
0
z+z
When we add a positive quantity (z t ⌣
t
p 0
0 0
0 0
0
εp z − zt
z > 0.
0 0
0
εp z) (positive for z
⌣
∈ {X − D }) to the left term, the strict inequality
holds for z ∈ X − D then we obtain: V1 ( z ) = z
t
2p
αpt
αp
0
z>O
⌣
for all z ∈ X − D .
(14)
On the other side, we have: V˙ 1i (z ) = z t
2(ati p + pai ) 2Bti p + αpt ai
ati αp + 2pBi z 2Bti αp
(15)
where V˙ 1i (z ∗ ) ̸= 0 for Bi αp ̸= 0 (the system cannot be stabilized at a not common equilibrium). If V˙ 1i (z ) < 0 for z ∈ Σi and i ∈ Im with ∪Σi = Rn+1 (this can be checked as LMI) then we have: V˙ 1i (z ) < 0 for all z ∈ Xi ⌣ with Xi = X − D ∩ Σi and
⌣ ⌣ ⌣ ∪Xi = ∪ X − D ∩ Σi = X − D ∩ (∪Σ i ) = X − D . The results are resumed below: ⌣
• V1 (z ∗ ) = 0 and z ∗ ∈ D . ⌣ • V1 (z ) > 0 for all z ∈ X − D . • V˙ 1i (z ) < 0 for all z ∈ Xi and i ∈ Im . ⌣ • ∪Xi = X − D . ⌣
This means that, using the maximum descent control switching strategy, V1 (z ) converges to D that contains z ∗ and the system (4) is practically asymptotically stabilizable at this point. This ends the proof. Important note. ⌣ ⌣ Really, ∪Xi = X − D is satisfied if X − D ⊆ (∪Σi ) even when the regions Σi do not cover the whole augmented space Rn+1 but this fact cannot be exploited. In fact, the only available mathematical m condition (that can be exploited as LMI) for the covering propriety, is the existence of positive scalars θi such that 1 θi Qi ≥ 0 which suffices to ensure the covering of the whole space Rn+1 (∪Σi = Rn+1 ). This means that the requirement of ∪Σi = Rn+1 or its sufficient condition m 1 θi Qi ≥ 0 in (20) is a strong condition for our objective.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
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Fig. 1. Simple construction in two dimensions.
Note of application. Notice the guaranteed practical stabilizability means that we cannot stabilize at the desired point. We can only bring the ⌣ system trajectory to the small domain D and maintain it inside. This is sufficient since the desired point is just an average equilibrium. However for best results, a small value of εp is required. Although it seems that we can choose εp arbitrary small from p > 0, we are limited to minimum εp that ensures P2 > 0 and V˙ 1i (z ) < 0. Really, if we know resolve LMI in X then εp may be extremely small. The resort to resolve the LMI in the whole state space overvalues the effective εp and affects slightly the obtained matrices. This can be considered as the price to pay when working in the whole space. Fortunately, we have other information to exploit. The results can be made more accurate by specifying smaller domain of convergence ⌣ inside D (see Fig. 1). Indeed, since V˙ 1i (z ) < 0 for z ∈ Σi with ∪Σi = Rn+1 then it is negative for all z ∈ Xi and i ∈ Im ⌣ ⌣ ˙ where Xi= X ∩ Σi and ∪Xi = X. This means that V1i (z ) is also negative in D . Furthermore, V1 (z ) has its minimum in D at z ∗∗ =
x∗∗ 1
−p−1 αp
with x∗∗ =
2
. Therefore, V1 (z ) converge inevitably to its minimum. Note that the point z ∗∗ is different
⌣ ∗ ∗∗ from z ∗ but necessarily very near when εp is small. Let D α0 be the smallest domain centered on z and contains z , it is 0 t p ∗∗t p 0 ∗∗ defined in X by z 0 0 z ≤ z z . This leads to: 0 0
⌣
D α0 =
z ∈ X|z
t
p 0
αpt p−1 αp 0 z≤ 0 4
.
(16a)
This domain is an extension of the domain Dα 0 defined in Rn by xt px ≤ x∗∗t px∗∗ which leads to:
Dα 0 =
n
t
x ∈ R |x px ≤
αpt p−1 αp
.
4
(16b) ⌣
⌣
Since P2 > 0, then from Schur complement [23], we have: αpt p−1 αp < εp thus D α 0 ⊂ D or equivalently Dα 0 ⊂ D. It follows from the above development that:
εα = εpmin =
αpt p−1 αp 4
.
(17) ⌣
Since the minimum of V1 (z ) is at z ∗∗ which is on the closure of D α 0 , it may be useful to stop switching before. For example ⌣ when V1 (z ) reaches the domain D α 1 (that can also be specified as an extension of a domain Dα 1 of Rn ) defined by: ⌣
D α1 =
z ∈ X|z t
p 0
0 z ≤ αpt p−1 αp . 0
(18)
⌣
When D α 1 is reached, the maximum errors on states (distance from z ∗ ) can be easily calculated from (18) once p and αp are computed. Remark 1. A particular case worth being clarified, when we impose αp = 0, this will lead to the following results:
• V1 (z ∗ ) = 0.
• V1 ( z ) = z t
2p 0
0 0
⌣
z > 0 for all z ∈ X − D since εp > 0 and z t
p 0
0 0
z ≥ zt
0 0
0
⌣
εp z in {X − D }.
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D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
• From Eq. (15), we observe that the low right corner element in V˙ 1i (z ) is zero, therefore to check V˙ 1i (z ) < 0 for z ∈ Σi as LMI, all ((n + 1), (n + 1)) elements in Qi must be necessarily negative. This will automatically alter the covering property otherwise asymptotic stability should have been obtained since the minimum (in X) of V1 (z ) is at z ∗ and V1 (z ∗ ) = 0, moreover V˙ 1i (z ∗ ) = 0 for all i ∈ Im . As application of the previous lemma, V1 (z ) has been used to derive stabilizing conditions for the augmented system (4). Note when minimization of εp is not formulated, then a user given small higher bound ε0 may be fixed. The following theorem states the obtained results. Theorem. If there exist symmetric matrices qi , a symmetric positive matrix p, vectors αp ∈ Rn , αqi ∈ Rn , positive scalars γ , δi , θi , εp and scalars βqi such that for i ∈ Im , the following BMI: (1) εp < ε0 . (2) (3) (4)
p
αpt 2
αp εp
> 0.
ati p + pai + δi qi + γ In t 2Bti p + αpt ai + δi αqi
m 1
ati αp + 2pBi + δi αqi 2Bti αp + δi βqi
θi
α qi βqi
qi
αqi t
< 0.
≥ 0.
has a solution for fixed small ε0 , then using the maximum descent control switching strategy, all trajectories of system (4) converge to the domain: ⌣
p 0
D α 1 = z ∈ X|z t
0 0
⌣ z ≤ αpt p−1 αp . Hence, the system is practically asymptotically stabilizable at point z ∗ ∈ D α 1 from
any initial condition z0 ∈ X.
Proof. We search for system (4), regions Σi and a Lyapunov function V1 (z ) as defined in (7) and (8), so we have: V1 (z ) = z t P1 z
and
Σi = z ∈ Rn+1 |z t Qi z ≥ 0 .
(19)
With the following matrices structures:
P1 =
αp
2p
α
t p
0
and Qi =
qi
α
t qi
αqi ; βqi
P1 , Qi ∈ R(n+1)×(n+1) .
Note that V1 (z ∗ ) = 0. To ensure the quadratic practical asymptotic stabilization of the augmented system (4), we have to check: If there exist a scalar function V1 (z ), positive scalar γ and regions Xi such that: ⌣
(1) V1 (z ) > 0 for all z ∈ {X − D }. (2) V˙ 1i (z ) ≤ −γ z t z for all z ∈ Xi , i = 1, m. ⌣ (3) X1 ∪ X2 · · · ∪ Xm = X − D .
(20)
⌣
Since these conditions do not provide a BMI form in {X − D }, with the help of our lemma we replace the first condition by n+1 P2 > 0 and resolve the second that is checked if mcondition in Σi instead of Xi . The third condition is satisfied if ∪Σi = R there exist θi > 0 such that 1 θi Qi ≥ 0 (see the Important Note). Using the S-procedure [5,23], we obtain the solvable following LMI (tractable BMI): If there exist symmetric matrices Qi , a positive symmetric matrices P1 , P2 and positive scalars γ , δi , θi such that:
(1) P2 > 0. (2) Ati P 1 + P1 Ai + δi Qi + γ ˇI ≤ 0, m (3) θi Qi ≥ 0
i = 1, m.
(21)
1
where ˇI =
In 0
0 0
p
αp
and P2 = α t ε . p p Then using the maximum descent control switching strategy, the augmented system (4) is practically asymptotically stabilizable at point z ∗ . It is necessary to develop the obtained conditions to derive inequalities that are easier to use in Matlab’s LMI toolbox. In particular, we replace non-strict inequalities by strict ones. One must know [23], when there is solution to strict inequalities, there will be solution to non-strict ones, whereas this logic is not correct for the absence of a solution.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
865
(1) the first condition is given by:
p
α
t p
αp > 0. εp
(22)
This condition is implemented with minimization of εp or a specified small higher bound ε0 . (2) the second condition is equivalent to: ati Bti
0 0
2p
α
t p
αp 0
+
2p
α
t p
αp
0
ai 0
q Bi + δi ti αqi 0
αqi I +γ n βqi 0
0 < 0 ∀i = 1, m. 0
Rearranged as: 2(ati p + pai ) + δi qi + γ In t 2Bti p + αpt ai + δi αqi
ati αp + 2pBi + δi αqi 2Bti αp + δi βqi
< 0.
(23)
(3) the third condition is given by:
m q θi ti αqi 1
αqi ≥ 0, βqi
θi > 0.
(24)
Since this condition require non-strict inequality and all matrices Qi must be available for the checking. Then we can check this condition for each resolution without include it in the LMI. In this case, a marginal feasibility is widely sufficient. However, there will be no guaranty to obtain solution that checks the condition. Really, this condition is very strong for ⌣ our objective. However, when it is satisfied then ∪Xi = (X − D ) which is exactly what we need. This subsequent covering test is (necessary) and sufficient to overcome this difficulty. Necessary in the sense, when this condition is included in the LMI, it is not possible to find a strictly feasible solution. Finally, our claim, if the conditions (22) and (23) are checked for all z and we have a positive subsequently covering test, ⌣ ⌣ then all trajectories of system (4) converge to the domain D . In light of the previous note of application, D is replaced by a ⌣ smaller domain D α 1 . These are the results of the theorem that states practical stability. This ends the proof. Remark 2. Note that from a given discrete state, it may be not possible to visit all others discrete states. So that, switching can be only between neighboring regions. Two regions Σi and Σj are neighbors if their intersection is not empty i.e. clΣi ∩ clΣj ̸= ∅. Let Si,j = clΣi ∩ clΣj represents the switching surface when passing from mode i to mode j and cl stands for closure of region. For all i ∈ Im Lyapunov derivatives V˙ 1i (z ) become nil when system states reach these boundaries; this will make maximum descent control switching strategy more suitable. In general, without use of dwell time or hysteresis (see control strategy), there can be an issue for solutions on the common boundaries. The so-called sliding motions can arise if the vector fields corresponding to two adjacent regions both point toward their common boundary. The classical independent solution (according to Filippov convex definition) of this behavior is more complicated in our case of synthesis, since we do not know a priori which regions will intersect each other. The reason is that the regions are not known in advance but a result of the synthesis procedure. However, it may be useful to include explicitly (in the LMI) a condition that excludes this phenomenon. In interest of simplicity, we do not address explicitly the question; see the discussion in [11] for more details. Computation note It is clear that not all the obtained conditions are in LMI form, but a tractable BMI since the reduced number of parameters makes easier the grid up of unknown scalars. For fixed values of these parameters, the verification of the remaining unknown variables becomes an LMI problem, which can be efficiently solved by Matlab’s LMI toolbox. Practical switching control strategy The used switching control strategy is based on the selection of the subsystem that has the highest decrease of Lyapunov function defined in (12). ⌣ As stated before, we obtain a wide practical satisfaction, if we stop switching once trajectory reaches the domain D α 1 ⌣ and restart when trajectory tries to get out the domain D . This strategy limit switching frequency when we approach the reference, however when εp is very small, there will be no significant difference between these two domains and sliding mode may occur. Moreover, due to the maximum descent control switching strategy, this phenomenon may appear far away from the desired point such that its occurrence becomes inevitable. As a preventive method, usually we call for the simple technique that is the hysteresis-based state-dependent switching strategy. It consists on the use of a given negative higher bound on the Lyapunov derivative: max(V˙ 1i (z )) = −ε z t z with ε > 0 and switch off when Lyapunov derivative V˙ 1i (z ) of the active mode exceeds this bound. This technique is combined with the practical stabilization to exclude sliding motion. It is reasonable in the synthesis operation with common Lyapunov function to allow regions to overlap each other, especially, when a possible suitable state space partition is made before. Since, if regions interiors are pairwise disjoints, it is clear that the use of hysteresis in our switching strategy will obligatory lead to a selection of subsystem with increasing ⌣ energy function. Similarly in practical stabilization, when restart switching (if domain D is reached with a certain active mode), one must select among others subsystems that have an increasing energy function in the region of the active
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Fig. 2. Simplified electric circuit of the Buck–Boost converter.
mode. This is not necessary the case in overlapping regions, which permit a great flexibility in the choice (according to switching strategy) between different subsystems since the uniqueness of switching sequence is not crucial for our approach. Therefore, nothing has been done (in the LMI) imposing regions to be disjoints. However, in our practical stabilization ⌣ method even with overlapping regions, when we stop switching in the smallest domain D α 1 the energy function of the ⌣ active mode will increase (for a short time) allowing the trajectory to reach the domain D . This will happen only when we approach the desired average equilibrium. In fact, for all approach, this is always the situation when regulating around a not equilibrium reference. The min projection switching control strategy (proposed by Stefan Pettersson) based on the selection of subsystem that has a vector field pointing toward the average equilibrium, may be efficiently used in our practical stabilization method; it leads also to good performances in simulation. Applications As application, DC–DC converters have been selected from [24]. The used PWA model can be obtained by different energetic modeling approaches. The derived model from the Bond Graph method has the following form: x˙ = a(ρ)x + b(ρ), where ρ is a Boolean vector of mode describing the on/off configuration of switches. Its dimension equals the number of pair physical switches. x ∈ Rn is the co-state vector, co-state variables are the corresponding co-energy variables (current, voltages) and n is the number of storage elements [24]. According to the number of physical switches, each on/off configuration of switches will lead to one mode. The first example in Fig. 2 is a multi-models of two 21 functioning modes. Whereas the multi-cellular converter depicted in Fig. 5 functions with (8 = 23 ) operating regimes represented by 8 submodels. For these circuits, only input supply and resistive load variations are considered. These changes may be useful to test the ability of our control approach. Note that the neighboring of any admissible reference point (points verifying the average model equilibrium) can be reached such that all admissible points are attained with errors less than 2%. So, the selected points for simulation are arbitrarily chosen. The set of admissible points verifying the conditions in (2) may be obtained by resolving these conditions separately from the design control. However, since we have opted for a complete systematic methodology, we have directly introduced these conditions in the LMI control with some approximations for LMI implementation. To alleviate the paper, the figures of robustness test are omitted. The computation for the first example is direct, but when the number of subsystems is high some know how to deal with BMI is required. The LMI optimization results have been reported for the second example (with eight subsystems) but the obtained matrices are not unique. These results depend on the choice of ε0 and essentially on the grid up method of unknown scalars in the obtained BMI.
Example 1 (Buck–Boost Converter [24]). a(ρ) =
ρ
0
−
ρ C
−
, b(ρ) =
L 1
E 1 − ρ L
0
RC
;x =
iL V
. E = 1 V , R = 1 , L = 1 H
and C = 1 F. Selected reference: V0 = −1 V, iL0 = 2 A. For this simple example with two modes, several quadratic stabilization methods have been efficiently used. However, our control approach is very adequate see Figs. 3–4. Note in absence of out regulating loop, the load disturbances and input supply irregularity must be limited to a reasonable level. However, the corresponding errors are still smalls for low disturbances. Example 2 (Multi-Cellular Converter [24]). R
−L ρ ρ 1− 2 a(ρ) = C1 ρ2− ρ3 C2
iL x=
V1 V2
,ρ =
ρ 1
ρ2 . R ρ3
ρ2− ρ1
ρ3− ρ2
L
L
0
0
0
0
,
ρ1 E
b(ρ) = L ; 0 0
= 20 , E = 90 V, L = 0.075 H, and C 1 = C 2 = 0.001 F.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
867
a
b
Fig. 3. Inductor current iL in (a) and voltages V in (b) for zero initial conditions.
Fig. 4. Voltage V vs. the inductor current iL for different initials conditions.
Selected reference: iL0 = 2 A, V10 = E /3, V20 = 2E /3. For the selected reference, the LMI optimization corresponding to ε0 = 5 · 10−3 and δi equals, lead to εp = 3 · 10−3 and the following matrices (the ±0.0000 mean a relatively very small number): 53.0482 −0.0000 P2 = −0.0000 −0.2845
−0.0000 42.8618 −0.0000 0.0000
−0.0000 −0.0000 42.8618 0.0000
−0.2845 0.0000 ; 0.0000 0.0030
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D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
Fig. 5. Simplified electric circuit of the multi-cellular converter.
106.0964 −0.0000 P1 = −0.0000 −0.2845
−0.0000 85.7235 −0.0000 0.0000
8.3913 +3 0.0020 Q1 = 10 0.0020 8.4168
0.0020 0.0020 0.0020 0.0120
−0.0000 −0.0000 85.7235 0.0000 0.0020 0.0020 −0.0129 0.0120
−0.2845 0.0000 0.0000 0 8.4168 0.0120 ; 0.0120 0.0450
0.9893 +5 −0.0012 Q2 = 10 3.3668 0.0001
−0.0012 −0.0024 −0.0012 0.0001
3.3668 −0.0012 0.0014 0.0001
0.0001 0.0001 0.0001 0.0073
1.7198 +5 2.6981 Q3 = 10 −2.7001 0.0001
2.6981 −0.0060 0.0029 0.0001
−2.7001 0.0029 −0.0060 −5.5136
0.0001 0.0001 ; −5.5136 0.0077
1.4184 +5 3.4640 Q4 = 10 −0.0012 0.4664
3.4640 −0.0043 −0.0012 6.9650
−0.0012 −0.0012 −0.0024 −0.0000
0.4664 6.9650 −0.0000 0.0073
0.9893 +5 −3.3691 Q5 = 10 −0.0012 −0.9198
−3.3691 0.0014 −0.0012 −6.9132
−0.0012 −0.0012 −0.0024 −0.0000
−0.9198 −6.9132 ; −0.0000 0.0073
0.4998 +5 −2.5839 Q6 = 10 2.5819 −2.9201
−2.5839 0.0022 −0.0053 −5.4786
2.5819 −0.0053 0.0022 5.4786
−2.9201 −5.4786 5.4786 0.0078
1.4184 +5 −0.0012 Q7 = 10 −3.4664 0.4664
−0.0012 −0.0024 −0.0012 −0.0000
−3.4664 −0.0012 −0.0043 −6.9650
0.4664 −0.0000 ; −6.9650 0.0073
0.8071 +4 −0.0006 Q8 = 10 −0.0006 −1.0115
−0.0006 −0.0012 −0.0006 0.0000
−0.0006 −0.0006 −0.0012 0.0000
−1.0115 0.0000 . 0.0000 0.0049
These regions are conic and may be overlapping, it‘s also noted that only marginal feasibility is guaranteed for the covering property (see important note). The corresponding simulation results are shown below. The simulation results shown in Figs. 6–7 prove that our control approach is a very interesting systematic method that is more suitable for multi-model systems like multi-cellular converter. The neighboring of all tested admissible points is reached with high accuracy. Note for this example, the set (infinite) of admissible points is characterized by a maximum
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
869
a
b
c
Fig. 6. Inductor current iL in (a), voltages V1 in (b) and V2 in (c) for zero initial conditions.
current iLmax = E /R = 4.5 A and an arbitrary choice of voltages V1 and V2 . For small current values of the selected reference, high variations of the load and power supply do not affect the control performances. However when the current value of the selected reference is high, it becomes increasingly sensitive to the load variations whereas the voltages remain undisturbed for all these perturbations. 4. Conclusion A simple practical switching stabilization methodology has been shown for PWA systems without equilibrium. The approach provides a systematic way to search for sufficient stabilizing conditions; therefore it has an on-line applicability. It overcomes the control problems encountered when dealing with these systems in an augmented state space. For these systems, the need of systematic approaches allows seeing the present method as a contribution not negligible. However, still deeper investigation may be done to complete the method especially, on the set neighboring points other than the average equilibriums. At some neighboring points, the system may be practically stabilized with certainly much less accuracy. When the analysis of sliding motions is complicated or may alter the systematic calculation, the use of hysteresis suffices in practice. Generally, there is an advantage when working with LMI formulation, we can easily deal with others constraints (if any) to improve performances and facilitate real implementation. In particular, conditions related to the occurrence of sliding
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Fig. 7. Current iL vs. V1 and V2 for different initial conditions.
motions may be included. For the application of the method, the obtained simulation results are very promising for all treated examples. References [1] S. Azuma, J. Imura, T. Sugie, Lebesgue piecewise affine approximation of nonlinear systems, Nonlinear Analysis: Hybrid Systems 4 (2010) 92–102. [2] E.D. Sontag, Nonlinear regulation: the piecewise linear approach, IEEE Transactions on Automatic Control 26 (2) (1981) 346–358. [3] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, IEEE Transactions on Automatic Control 45 (2000) 1864–1876. [4] M. Johansson, Piecewise Linear Control Systems: A Computational Approach, in: M. Thoma, M. Morari (Eds.), Ser. Lecture Notes in Control and Information Sciences, vol. 284, Springer-Verlag, 2003. [5] S. Pettersson, Analysis and design of hybrids systems, Ph.D. Dissertation, Chalmers University, Goteborg Sweden, 1999. [6] L. Rodrigues, S. Boyd, Piecewise affine slab systems using convex optimization, Systems and Control Letters 54 (2005) 835–853. [7] B. De Schutter, W.P.M.H. Heemels, Modeling and Control of Hybrid Systems, Lecture Notes of the DISC Course October 2006, Delft Center of Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. http://www.dcsc.tudelft.nl. [8] J. Hespanha, Stabilization through hybrid control, in: UNESCO Encyclopaedia of Life Support Systems, 2004. www.ece.ucsb.edu/~hespanha/.../6-4328-7-hy-stab.pdf. [9] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19 (1999) 59–70. [10] J. Geromel, P. Colaneri, Stability and stabilization of continuous time switched System, SIAM Journal on Control and Optimization 45 (5) (2006) 1915–1930. [11] S. Pettersson, Synthesis of switched linear systems, in: Proceedings of the 42nd IEEE Conference on Decision Control, 2003, pp. 5283–5288. [12] R. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88 (7) (2000) 1069–1082. [13] A. Hassibi, S. Boyd, Quadratic stabilization and control of piecewise-linear systems, in: Proceedings of the American Control Conference, 1998, pp. 3659–3664. [14] M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998) 555–559. [15] A. Rantzer, M. Johansson, Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45 (2000) 629–637. [16] L. Rodrigues, A. Hassibi, P. Jonathan How, Output feedback controller synthesis for piecewise-affine systems with multiple equilibria, in: Proceedings of the American Control Conference, 2000, pp. 1784–1789. [17] S. Zhendong, S.S. Ge, Analysis and synthesis of switched linear control systems, Automatica 41 (2005) 181–195. [18] S. Mariéthoz, S. Almér, M. Bâja, A.G. Beccuti, D. Patino, A. Wernrud, J. Buisson, H. Cormerais, T. Geyer, H. Fujioka, U.T. Jönsson, C.-Y. Kao, M. Morari, G. Papafotiou, A. Rantzer, P. Riedinger, Comparison of hybrid control techniques for Buck and Boost DC–DC converters, IEEE Transaction on Control Systems Technology 18 (5) (2010) 1126–1145. [19] X. Xu, G. Zhai, Results on practical asymptotic stabilizability of switched affine systems, in: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006, pp. 15–21. [20] P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, in: Proceedings of the American Control Conference, 2004, pp. 3890–3895. [21] R. Shorten, K. Narendra, Necessary and sufficient conditions for the existence of a CQLF for a finite number of stable LTI systems, International Journal of Adaptive Control and Signal Processing 16 (10) (2002) 709–728. [22] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems, a survey of recent results, IEEE Transactions on Automatic Control 54 (2) (2009) 308–322. [23] S. Boyd, L. El Ghaoui, E. Feron, H. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, in: SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1994. [24] J. Buisson, H. Cormerais, P.Y. Richard, On the stabilisation of switching electrical power converters, in: Hybrid Systems: Computation and Control, in: M. Morari, L. Thiele (Eds.), Ser. LNCS, vol. 3414, Springer, 2005, pp. 184–197.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Practical stabilization for piecewise-affine systems: A BMI approach D. Kamri a,∗ , R. Bourdais a , J. Buisson a , C. Larbes b a
Automatic and control group, Supélec-IETR, Rennes Campus, France
b
Department of Electronic, National Polytechnic School, Algiers, Algeria
article
info
Article history: Received 16 May 2010 Accepted 9 January 2012 Keywords: Hybrid systems Lyapunov theory LMI PWA systems Practical switching stabilization
abstract We propose in this paper, a systematic switching practical stabilization method for PWA switched systems around an average equilibrium. For these systems, the main difficulty comes from the fact that to end in BMI formulation, it is necessary to represent the system in an augmented state space but a restricted one. However, the derived stabilizing conditions are not tractable as BMI in the restricted domain. We will present a method that overcomes this difficulty and drives asymptotically system states into a ball centered on the desired non-equilibrium reference. The efficiency of this practical stabilization method is showed by the ball smallness and the good robustness against disturbances. The design control searches for a single Lyapunov-like function and an appropriate continuous state space partition to satisfy stabilizing properties. Therefore, the method constitutes a simple systematic state feedback computation; it may be useful for on-line applications. As a direct application, satisfactory simulation results are obtained for two illustrative examples, a Buck–Boost converter and a multilevel one. Due to their functioning nature, these devices constitute good examples of switched systems. They are electrical circuits controlled by switches to produce regulated outputs despite the load disturbances and power supply irregularities. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In most real-world practical applications, nonlinearity is the rule rather than the exception. To get rid of the complexity of nonlinear models, one often resorts to a widely used modeling strategy that represents the system behavior by a set of simple (linear) models and a switching rule that orchestrates between these operating regimes to obtain a hybrid representation of the system. Generally, the submodels (electrical circuits, biological systems etc.) have an affine relation or may be approximated by affine functions; hence they are called piecewise affine (PWA) switched systems [1,2]. In the above definition, one must distinguish between the two situations of analysis and synthesis. In the former case, the state partition is a result of the dynamics modeling effect; thus it is previously specified for the analysis operation. While our concern is the latter case where the aim, is to find this partition and the state dependent switching law such that the switching between possibly unstable subsystems results in a stable system. The PWA switched systems are a known class of hybrid systems. They are multi-models representation that offers a good modeling framework for complex dynamical systems that involve nonlinear phenomena. They have received increasing interest over the last years and still pose challenging control problems even for simple practical examples [3–6]. Generally, the proposed Lyapunov based approaches provide only sufficient conditions with no guaranties that a Lyapunov function can be found; however if one is found, the result is unambiguous. In this context, several constructive propositions have emerged in the literature for the analysis and the design
∗
Corresponding address: Department of Electric Engineering, University of Laghouat, BP 37G, 03000 Laghouat, Algeria. Tel.: +213 698276530. E-mail addresses: [email protected], [email protected] (D. Kamri), [email protected] (R. Bourdais), [email protected] (J. Buisson), [email protected] (C. Larbes). 1751-570X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2012.01.001
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of switched systems [7–12]. Even though PWA systems benefit enormously from the proposed approaches, few researches dealt explicitly with the PWA systems. The more specialized papers on PWA switched systems started with the reported methods in [5,13,14] where the analysis problems are formulated as LMI. In these approaches, some attempts to extend their results to the synthesis operation were formulated; however, the proposed feedback controls are still suited for systems with common equilibrium only. An integral of performance index is optimized for control law in [15], where lower and upper bounds are computed which increase the computational speed in the proposed approach. For PWA systems with multiple equilibriums, [16] addressed the dynamic output feedback synthesis question for controlled systems. Using the discretetime PWA models, many others computed state feedback control have been suggested [17,18]. However, in this paper we are interested in investigating the state-dependent switching stabilization problem for PWA systems. Our main motivation for this direction is the control of power electronic switching circuits where the affine term in the PWA model is essentially due to the input voltage. Apart from [16], the resumed work in [19] or the specific approaches developed by the main group working on electronic circuits [18]; most proposed approaches do not deal with PWA systems without (or not common) equilibrium. In this context, the available results are the elegant necessary and sufficient conditions for quadratic stabilization of pair LTI systems [7,20,21]. However, the generalization to more than two subsystems is not well established and the question remains open. Recently, many others papers investigated the switched and PWA systems without significant novelty such that the survey papers [9,18,22] remain a complete and detailed description of the problems arising in this area. The outline of this paper is as follow: Section 2 presents the model to be used and problem statement. Section 3 contains the proposed approach within its application to quadratic stabilization. Section 4 illustrates the theoretical results on practical examples. Notation. It will be assumed throughout the paper, that the numerical zero may specify a vector or a matrix of zero with appropriate dimension that can be understood from the context. 2. Problem statement Several situations of nonlinear approximation, saturating, abrupt changes in systems components as well as classical linearization around different operating points lead to the following general PWA model: x˙ (ξ ) = aσ x(ξ ) + bσ ,
ξ ∈ R+ ,
(1)
where x(ξ ) ∈ R is the state and σ is a piecewise constant function of time ξ , called switching signal. In our case, we are interested by the state-dependent switching where the system mode to be activated is determined exclusively by x. This leads to the following switching function σ (x) : Rn → Im = {1, . . . , m} with m number of mode or subsystems in (1), we say that the switched system is in the ith operating regime when σ (x) = i. In all future use of PWA model, the subscript σ is replaced by the subscript i and the time variable ξ is omitted for clarity. Generally, in practice we search for a switching law that has the ability to bring the system trajectory from any starting initial condition x0 to a quantifiable small neighboring region B(xa , εα ) centered on the desired reference xa and maintain it around this neighboring. n
Definition. The system (1) is globally asymptotically practically stabilizable by switching at a point xa ∈ Rn , if for every scalar εp ≥ εpmin , there exist εα with 0 < εα ≤ εp and a switching law that brings the system trajectory from any starting point x0 ∈ Rn to B(xa , εα ) and maintain it inside B(xa , εp ) for all future time. Precisely, our objective is to find this state dependent switching law and the scalar εα . We will show (17) that for PWA system, the εp must satisfy εp ≥ εpmin . As known for PWA systems, the desired reference point xa must belong to the set of average equilibrium points that verify the following convex combination: m i =1
αi (ai xa + bi ) = 0 with 0 ≤ αi ≤ 1 and
m
αi = 1.
(2)
i =1
Since the subsystems do not necessarily share any equilibrium, there is no chance to approach these reference points without forcing and fast switching. One must know that fast switching is not desirable; it is assumed through the paper that within a finite time interval, only a finite number of switches may occur. We will see in the last section the most used practical way to avoid fast switching. When state space partition is explicitly used in the synthesis, we associate with each mode a space region Ωi where it will be activated. The regions Ωi must cover the whole state space. In this case, the synthesis objective will be to determine these regions and the state dependent switching law such that the switching between possibly unstable subsystems results in a stable system.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
861
Without loss of generality, we will consider the problem at the origin providing that the affine term is equal to: Bi = ai xa + bi . In order to make easier the use of the LMI approach, we opt for the following change of variable:
z=
x . 1
(3)
The obtained linear form (4) will be called: augmented system.
z˙ = Ai z
with Ai =
ai 0
Bi 0
and i ∈ I m .
(4)
Hence, the problem reduces to the stabilization of the system (4) at point: 0
.. . z∗ = .
(5)
0 1
3. Quadratic stabilization and LMI formulation Note that for our augmented system (4), the synthesis of a switching law and its corresponding state partition concern only the defined following state space subset:
X = z ∈ Rn+1 |zn+1 = 1 .
(6)
Our main objective is to develop a systematic synthesis technique that leads to a common scalar function verifying Lyapunov stabilizing conditions as tractable BMI in a partition of the domain X. However, as will be seen later, this will require a partition of the whole state space Rn+1 . Let Σi be quadratic regions of the state space Rn+1 , defined by: with Qi ∈ R(n+1)×(n+1) and i ∈ I m .
Σi = z ∈ Rn+1 |z t Qi z ≥ 0
(7)
Observe that the number of regions to be designed is equal to the number of subsystems in our system (4). Consider the following candidate Lyapunov function for our system (4): V1 (z ) = z t P1 z
with P1 =
αp
2p
α
t p
0
and p = pt > 0
(8)
where z ∈ Rn+1 , p ∈ Rn×n and αp ∈ Rn . Let us define a domain D ⊂ Rn as follows: D = x ∈ Rn |xt px < εp .
(9a)
This domain has an extension in the domain X, defined by: ⌣
D =
z ∈ X|z t
p 0
0 z < εp . 0
(9b)
Since the candidate Lyapunov function in (8) is not positive in the whole domain X, we will try to prove its positivity in ⌣ ⌣ a sub domain X − D and content our self to Practical stability. However, even in this sub domain X − D we cannot directly prove the strict positivity of this function; the following matrix P2 is used as an intermediate to meet our objective.
P2 =
p
α
t p
αp , εp
εp > 0.
(10)
Let V˙ 1i (z ) define the time derivative of V1 (z ) along the trajectory of the ith subsystem in system (4) that is given by: ∂ V (z ) V˙ 1i (z ) = ∂1z ddzξ with ddzξ = Ai z and ξ ∈ R+ , This lead to the following expression for V˙ 1i (z ): V˙ 1i (z ) = z t Ati P 1 + P1 Ai z .
(11)
Control strategy. In our approach, each subsystem i of the augmented system (4) will be associated with a quadratic region Σi where it will be activated. This region is designed such that the time derivative V˙ 1i (z ) of its associated subsystem is negative into
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that region. One used stabilizing switching control strategy is based on the selection of the subsystem that has the highest decrease of Lyapunov function V1 (z ):
σ (z ) = argmini∈Im V˙ 1i (z )
(12)
where σ (z ) = i, is the subscript of the mode to be activated. We refer to this stabilizing control switching strategy by maximum descent control switching strategy. Lemma. If there exist a symmetric positive matrix p ∈ Rn×n , symmetric matrices Qi ∈ R(n+1)×(n+1) , vector αp ∈ Rn and positive scalar εp such that: P2 > 0 and V˙ 1i (z ) is negative when z ∈ Σi and i ∈ Im with ∪Σi = Rn+1 then using the maximum descent control switching ⌣ strategy, all trajectories of system (4) converge to the domain D . Hence, the system is practically asymptotically stabilizable at ⌣ ∗ point z ∈ D from any initial condition z0 ∈ X. ⌣
Proof. Observe that V1 (z ∗ ) = 0 and z ∗ ∈ D . From (9a) we have xt px ≥ εp for all x ∈ {Rn − D}. This will be written in the domain X as follow: z
t
p 0
Then z t
p 0
0 0 z ≥ zt 0 0
0 0
z − zt
0 0
0
0
εp
εp z
⌣
for all z ∈ X − D .
z
(13)
⌣ ≥ 0 for all z ∈ X − D . Well, if P2 > 0 (this can be checked as LMI) i.e. z t P2 z > 0 for
z ∈ Rn+1 , and since X ⊂ Rn+1 then we have for all z ∈ X: z
t
p 0
0 0 z + zt αpt 0
αp
0
z+z
When we add a positive quantity (z t ⌣
t
p 0
0 0
0 0
0
εp z − zt
z > 0.
0 0
0
εp z) (positive for z
⌣
∈ {X − D }) to the left term, the strict inequality
holds for z ∈ X − D then we obtain: V1 ( z ) = z
t
2p
αpt
αp
0
z>O
⌣
for all z ∈ X − D .
(14)
On the other side, we have: V˙ 1i (z ) = z t
2(ati p + pai ) 2Bti p + αpt ai
ati αp + 2pBi z 2Bti αp
(15)
where V˙ 1i (z ∗ ) ̸= 0 for Bi αp ̸= 0 (the system cannot be stabilized at a not common equilibrium). If V˙ 1i (z ) < 0 for z ∈ Σi and i ∈ Im with ∪Σi = Rn+1 (this can be checked as LMI) then we have: V˙ 1i (z ) < 0 for all z ∈ Xi ⌣ with Xi = X − D ∩ Σi and
⌣ ⌣ ⌣ ∪Xi = ∪ X − D ∩ Σi = X − D ∩ (∪Σ i ) = X − D . The results are resumed below: ⌣
• V1 (z ∗ ) = 0 and z ∗ ∈ D . ⌣ • V1 (z ) > 0 for all z ∈ X − D . • V˙ 1i (z ) < 0 for all z ∈ Xi and i ∈ Im . ⌣ • ∪Xi = X − D . ⌣
This means that, using the maximum descent control switching strategy, V1 (z ) converges to D that contains z ∗ and the system (4) is practically asymptotically stabilizable at this point. This ends the proof. Important note. ⌣ ⌣ Really, ∪Xi = X − D is satisfied if X − D ⊆ (∪Σi ) even when the regions Σi do not cover the whole augmented space Rn+1 but this fact cannot be exploited. In fact, the only available mathematical m condition (that can be exploited as LMI) for the covering propriety, is the existence of positive scalars θi such that 1 θi Qi ≥ 0 which suffices to ensure the covering of the whole space Rn+1 (∪Σi = Rn+1 ). This means that the requirement of ∪Σi = Rn+1 or its sufficient condition m 1 θi Qi ≥ 0 in (20) is a strong condition for our objective.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
863
Fig. 1. Simple construction in two dimensions.
Note of application. Notice the guaranteed practical stabilizability means that we cannot stabilize at the desired point. We can only bring the ⌣ system trajectory to the small domain D and maintain it inside. This is sufficient since the desired point is just an average equilibrium. However for best results, a small value of εp is required. Although it seems that we can choose εp arbitrary small from p > 0, we are limited to minimum εp that ensures P2 > 0 and V˙ 1i (z ) < 0. Really, if we know resolve LMI in X then εp may be extremely small. The resort to resolve the LMI in the whole state space overvalues the effective εp and affects slightly the obtained matrices. This can be considered as the price to pay when working in the whole space. Fortunately, we have other information to exploit. The results can be made more accurate by specifying smaller domain of convergence ⌣ inside D (see Fig. 1). Indeed, since V˙ 1i (z ) < 0 for z ∈ Σi with ∪Σi = Rn+1 then it is negative for all z ∈ Xi and i ∈ Im ⌣ ⌣ ˙ where Xi= X ∩ Σi and ∪Xi = X. This means that V1i (z ) is also negative in D . Furthermore, V1 (z ) has its minimum in D at z ∗∗ =
x∗∗ 1
−p−1 αp
with x∗∗ =
2
. Therefore, V1 (z ) converge inevitably to its minimum. Note that the point z ∗∗ is different
⌣ ∗ ∗∗ from z ∗ but necessarily very near when εp is small. Let D α0 be the smallest domain centered on z and contains z , it is 0 t p ∗∗t p 0 ∗∗ defined in X by z 0 0 z ≤ z z . This leads to: 0 0
⌣
D α0 =
z ∈ X|z
t
p 0
αpt p−1 αp 0 z≤ 0 4
.
(16a)
This domain is an extension of the domain Dα 0 defined in Rn by xt px ≤ x∗∗t px∗∗ which leads to:
Dα 0 =
n
t
x ∈ R |x px ≤
αpt p−1 αp
.
4
(16b) ⌣
⌣
Since P2 > 0, then from Schur complement [23], we have: αpt p−1 αp < εp thus D α 0 ⊂ D or equivalently Dα 0 ⊂ D. It follows from the above development that:
εα = εpmin =
αpt p−1 αp 4
.
(17) ⌣
Since the minimum of V1 (z ) is at z ∗∗ which is on the closure of D α 0 , it may be useful to stop switching before. For example ⌣ when V1 (z ) reaches the domain D α 1 (that can also be specified as an extension of a domain Dα 1 of Rn ) defined by: ⌣
D α1 =
z ∈ X|z t
p 0
0 z ≤ αpt p−1 αp . 0
(18)
⌣
When D α 1 is reached, the maximum errors on states (distance from z ∗ ) can be easily calculated from (18) once p and αp are computed. Remark 1. A particular case worth being clarified, when we impose αp = 0, this will lead to the following results:
• V1 (z ∗ ) = 0.
• V1 ( z ) = z t
2p 0
0 0
⌣
z > 0 for all z ∈ X − D since εp > 0 and z t
p 0
0 0
z ≥ zt
0 0
0
⌣
εp z in {X − D }.
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• From Eq. (15), we observe that the low right corner element in V˙ 1i (z ) is zero, therefore to check V˙ 1i (z ) < 0 for z ∈ Σi as LMI, all ((n + 1), (n + 1)) elements in Qi must be necessarily negative. This will automatically alter the covering property otherwise asymptotic stability should have been obtained since the minimum (in X) of V1 (z ) is at z ∗ and V1 (z ∗ ) = 0, moreover V˙ 1i (z ∗ ) = 0 for all i ∈ Im . As application of the previous lemma, V1 (z ) has been used to derive stabilizing conditions for the augmented system (4). Note when minimization of εp is not formulated, then a user given small higher bound ε0 may be fixed. The following theorem states the obtained results. Theorem. If there exist symmetric matrices qi , a symmetric positive matrix p, vectors αp ∈ Rn , αqi ∈ Rn , positive scalars γ , δi , θi , εp and scalars βqi such that for i ∈ Im , the following BMI: (1) εp < ε0 . (2) (3) (4)
p
αpt 2
αp εp
> 0.
ati p + pai + δi qi + γ In t 2Bti p + αpt ai + δi αqi
m 1
ati αp + 2pBi + δi αqi 2Bti αp + δi βqi
θi
α qi βqi
qi
αqi t
< 0.
≥ 0.
has a solution for fixed small ε0 , then using the maximum descent control switching strategy, all trajectories of system (4) converge to the domain: ⌣
p 0
D α 1 = z ∈ X|z t
0 0
⌣ z ≤ αpt p−1 αp . Hence, the system is practically asymptotically stabilizable at point z ∗ ∈ D α 1 from
any initial condition z0 ∈ X.
Proof. We search for system (4), regions Σi and a Lyapunov function V1 (z ) as defined in (7) and (8), so we have: V1 (z ) = z t P1 z
and
Σi = z ∈ Rn+1 |z t Qi z ≥ 0 .
(19)
With the following matrices structures:
P1 =
αp
2p
α
t p
0
and Qi =
qi
α
t qi
αqi ; βqi
P1 , Qi ∈ R(n+1)×(n+1) .
Note that V1 (z ∗ ) = 0. To ensure the quadratic practical asymptotic stabilization of the augmented system (4), we have to check: If there exist a scalar function V1 (z ), positive scalar γ and regions Xi such that: ⌣
(1) V1 (z ) > 0 for all z ∈ {X − D }. (2) V˙ 1i (z ) ≤ −γ z t z for all z ∈ Xi , i = 1, m. ⌣ (3) X1 ∪ X2 · · · ∪ Xm = X − D .
(20)
⌣
Since these conditions do not provide a BMI form in {X − D }, with the help of our lemma we replace the first condition by n+1 P2 > 0 and resolve the second that is checked if mcondition in Σi instead of Xi . The third condition is satisfied if ∪Σi = R there exist θi > 0 such that 1 θi Qi ≥ 0 (see the Important Note). Using the S-procedure [5,23], we obtain the solvable following LMI (tractable BMI): If there exist symmetric matrices Qi , a positive symmetric matrices P1 , P2 and positive scalars γ , δi , θi such that:
(1) P2 > 0. (2) Ati P 1 + P1 Ai + δi Qi + γ ˇI ≤ 0, m (3) θi Qi ≥ 0
i = 1, m.
(21)
1
where ˇI =
In 0
0 0
p
αp
and P2 = α t ε . p p Then using the maximum descent control switching strategy, the augmented system (4) is practically asymptotically stabilizable at point z ∗ . It is necessary to develop the obtained conditions to derive inequalities that are easier to use in Matlab’s LMI toolbox. In particular, we replace non-strict inequalities by strict ones. One must know [23], when there is solution to strict inequalities, there will be solution to non-strict ones, whereas this logic is not correct for the absence of a solution.
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(1) the first condition is given by:
p
α
t p
αp > 0. εp
(22)
This condition is implemented with minimization of εp or a specified small higher bound ε0 . (2) the second condition is equivalent to: ati Bti
0 0
2p
α
t p
αp 0
+
2p
α
t p
αp
0
ai 0
q Bi + δi ti αqi 0
αqi I +γ n βqi 0
0 < 0 ∀i = 1, m. 0
Rearranged as: 2(ati p + pai ) + δi qi + γ In t 2Bti p + αpt ai + δi αqi
ati αp + 2pBi + δi αqi 2Bti αp + δi βqi
< 0.
(23)
(3) the third condition is given by:
m q θi ti αqi 1
αqi ≥ 0, βqi
θi > 0.
(24)
Since this condition require non-strict inequality and all matrices Qi must be available for the checking. Then we can check this condition for each resolution without include it in the LMI. In this case, a marginal feasibility is widely sufficient. However, there will be no guaranty to obtain solution that checks the condition. Really, this condition is very strong for ⌣ our objective. However, when it is satisfied then ∪Xi = (X − D ) which is exactly what we need. This subsequent covering test is (necessary) and sufficient to overcome this difficulty. Necessary in the sense, when this condition is included in the LMI, it is not possible to find a strictly feasible solution. Finally, our claim, if the conditions (22) and (23) are checked for all z and we have a positive subsequently covering test, ⌣ ⌣ then all trajectories of system (4) converge to the domain D . In light of the previous note of application, D is replaced by a ⌣ smaller domain D α 1 . These are the results of the theorem that states practical stability. This ends the proof. Remark 2. Note that from a given discrete state, it may be not possible to visit all others discrete states. So that, switching can be only between neighboring regions. Two regions Σi and Σj are neighbors if their intersection is not empty i.e. clΣi ∩ clΣj ̸= ∅. Let Si,j = clΣi ∩ clΣj represents the switching surface when passing from mode i to mode j and cl stands for closure of region. For all i ∈ Im Lyapunov derivatives V˙ 1i (z ) become nil when system states reach these boundaries; this will make maximum descent control switching strategy more suitable. In general, without use of dwell time or hysteresis (see control strategy), there can be an issue for solutions on the common boundaries. The so-called sliding motions can arise if the vector fields corresponding to two adjacent regions both point toward their common boundary. The classical independent solution (according to Filippov convex definition) of this behavior is more complicated in our case of synthesis, since we do not know a priori which regions will intersect each other. The reason is that the regions are not known in advance but a result of the synthesis procedure. However, it may be useful to include explicitly (in the LMI) a condition that excludes this phenomenon. In interest of simplicity, we do not address explicitly the question; see the discussion in [11] for more details. Computation note It is clear that not all the obtained conditions are in LMI form, but a tractable BMI since the reduced number of parameters makes easier the grid up of unknown scalars. For fixed values of these parameters, the verification of the remaining unknown variables becomes an LMI problem, which can be efficiently solved by Matlab’s LMI toolbox. Practical switching control strategy The used switching control strategy is based on the selection of the subsystem that has the highest decrease of Lyapunov function defined in (12). ⌣ As stated before, we obtain a wide practical satisfaction, if we stop switching once trajectory reaches the domain D α 1 ⌣ and restart when trajectory tries to get out the domain D . This strategy limit switching frequency when we approach the reference, however when εp is very small, there will be no significant difference between these two domains and sliding mode may occur. Moreover, due to the maximum descent control switching strategy, this phenomenon may appear far away from the desired point such that its occurrence becomes inevitable. As a preventive method, usually we call for the simple technique that is the hysteresis-based state-dependent switching strategy. It consists on the use of a given negative higher bound on the Lyapunov derivative: max(V˙ 1i (z )) = −ε z t z with ε > 0 and switch off when Lyapunov derivative V˙ 1i (z ) of the active mode exceeds this bound. This technique is combined with the practical stabilization to exclude sliding motion. It is reasonable in the synthesis operation with common Lyapunov function to allow regions to overlap each other, especially, when a possible suitable state space partition is made before. Since, if regions interiors are pairwise disjoints, it is clear that the use of hysteresis in our switching strategy will obligatory lead to a selection of subsystem with increasing ⌣ energy function. Similarly in practical stabilization, when restart switching (if domain D is reached with a certain active mode), one must select among others subsystems that have an increasing energy function in the region of the active
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D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
Fig. 2. Simplified electric circuit of the Buck–Boost converter.
mode. This is not necessary the case in overlapping regions, which permit a great flexibility in the choice (according to switching strategy) between different subsystems since the uniqueness of switching sequence is not crucial for our approach. Therefore, nothing has been done (in the LMI) imposing regions to be disjoints. However, in our practical stabilization ⌣ method even with overlapping regions, when we stop switching in the smallest domain D α 1 the energy function of the ⌣ active mode will increase (for a short time) allowing the trajectory to reach the domain D . This will happen only when we approach the desired average equilibrium. In fact, for all approach, this is always the situation when regulating around a not equilibrium reference. The min projection switching control strategy (proposed by Stefan Pettersson) based on the selection of subsystem that has a vector field pointing toward the average equilibrium, may be efficiently used in our practical stabilization method; it leads also to good performances in simulation. Applications As application, DC–DC converters have been selected from [24]. The used PWA model can be obtained by different energetic modeling approaches. The derived model from the Bond Graph method has the following form: x˙ = a(ρ)x + b(ρ), where ρ is a Boolean vector of mode describing the on/off configuration of switches. Its dimension equals the number of pair physical switches. x ∈ Rn is the co-state vector, co-state variables are the corresponding co-energy variables (current, voltages) and n is the number of storage elements [24]. According to the number of physical switches, each on/off configuration of switches will lead to one mode. The first example in Fig. 2 is a multi-models of two 21 functioning modes. Whereas the multi-cellular converter depicted in Fig. 5 functions with (8 = 23 ) operating regimes represented by 8 submodels. For these circuits, only input supply and resistive load variations are considered. These changes may be useful to test the ability of our control approach. Note that the neighboring of any admissible reference point (points verifying the average model equilibrium) can be reached such that all admissible points are attained with errors less than 2%. So, the selected points for simulation are arbitrarily chosen. The set of admissible points verifying the conditions in (2) may be obtained by resolving these conditions separately from the design control. However, since we have opted for a complete systematic methodology, we have directly introduced these conditions in the LMI control with some approximations for LMI implementation. To alleviate the paper, the figures of robustness test are omitted. The computation for the first example is direct, but when the number of subsystems is high some know how to deal with BMI is required. The LMI optimization results have been reported for the second example (with eight subsystems) but the obtained matrices are not unique. These results depend on the choice of ε0 and essentially on the grid up method of unknown scalars in the obtained BMI.
Example 1 (Buck–Boost Converter [24]). a(ρ) =
ρ
0
−
ρ C
−
, b(ρ) =
L 1
E 1 − ρ L
0
RC
;x =
iL V
. E = 1 V , R = 1 , L = 1 H
and C = 1 F. Selected reference: V0 = −1 V, iL0 = 2 A. For this simple example with two modes, several quadratic stabilization methods have been efficiently used. However, our control approach is very adequate see Figs. 3–4. Note in absence of out regulating loop, the load disturbances and input supply irregularity must be limited to a reasonable level. However, the corresponding errors are still smalls for low disturbances. Example 2 (Multi-Cellular Converter [24]). R
−L ρ ρ 1− 2 a(ρ) = C1 ρ2− ρ3 C2
iL x=
V1 V2
,ρ =
ρ 1
ρ2 . R ρ3
ρ2− ρ1
ρ3− ρ2
L
L
0
0
0
0
,
ρ1 E
b(ρ) = L ; 0 0
= 20 , E = 90 V, L = 0.075 H, and C 1 = C 2 = 0.001 F.
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
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a
b
Fig. 3. Inductor current iL in (a) and voltages V in (b) for zero initial conditions.
Fig. 4. Voltage V vs. the inductor current iL for different initials conditions.
Selected reference: iL0 = 2 A, V10 = E /3, V20 = 2E /3. For the selected reference, the LMI optimization corresponding to ε0 = 5 · 10−3 and δi equals, lead to εp = 3 · 10−3 and the following matrices (the ±0.0000 mean a relatively very small number): 53.0482 −0.0000 P2 = −0.0000 −0.2845
−0.0000 42.8618 −0.0000 0.0000
−0.0000 −0.0000 42.8618 0.0000
−0.2845 0.0000 ; 0.0000 0.0030
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D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
Fig. 5. Simplified electric circuit of the multi-cellular converter.
106.0964 −0.0000 P1 = −0.0000 −0.2845
−0.0000 85.7235 −0.0000 0.0000
8.3913 +3 0.0020 Q1 = 10 0.0020 8.4168
0.0020 0.0020 0.0020 0.0120
−0.0000 −0.0000 85.7235 0.0000 0.0020 0.0020 −0.0129 0.0120
−0.2845 0.0000 0.0000 0 8.4168 0.0120 ; 0.0120 0.0450
0.9893 +5 −0.0012 Q2 = 10 3.3668 0.0001
−0.0012 −0.0024 −0.0012 0.0001
3.3668 −0.0012 0.0014 0.0001
0.0001 0.0001 0.0001 0.0073
1.7198 +5 2.6981 Q3 = 10 −2.7001 0.0001
2.6981 −0.0060 0.0029 0.0001
−2.7001 0.0029 −0.0060 −5.5136
0.0001 0.0001 ; −5.5136 0.0077
1.4184 +5 3.4640 Q4 = 10 −0.0012 0.4664
3.4640 −0.0043 −0.0012 6.9650
−0.0012 −0.0012 −0.0024 −0.0000
0.4664 6.9650 −0.0000 0.0073
0.9893 +5 −3.3691 Q5 = 10 −0.0012 −0.9198
−3.3691 0.0014 −0.0012 −6.9132
−0.0012 −0.0012 −0.0024 −0.0000
−0.9198 −6.9132 ; −0.0000 0.0073
0.4998 +5 −2.5839 Q6 = 10 2.5819 −2.9201
−2.5839 0.0022 −0.0053 −5.4786
2.5819 −0.0053 0.0022 5.4786
−2.9201 −5.4786 5.4786 0.0078
1.4184 +5 −0.0012 Q7 = 10 −3.4664 0.4664
−0.0012 −0.0024 −0.0012 −0.0000
−3.4664 −0.0012 −0.0043 −6.9650
0.4664 −0.0000 ; −6.9650 0.0073
0.8071 +4 −0.0006 Q8 = 10 −0.0006 −1.0115
−0.0006 −0.0012 −0.0006 0.0000
−0.0006 −0.0006 −0.0012 0.0000
−1.0115 0.0000 . 0.0000 0.0049
These regions are conic and may be overlapping, it‘s also noted that only marginal feasibility is guaranteed for the covering property (see important note). The corresponding simulation results are shown below. The simulation results shown in Figs. 6–7 prove that our control approach is a very interesting systematic method that is more suitable for multi-model systems like multi-cellular converter. The neighboring of all tested admissible points is reached with high accuracy. Note for this example, the set (infinite) of admissible points is characterized by a maximum
D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
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a
b
c
Fig. 6. Inductor current iL in (a), voltages V1 in (b) and V2 in (c) for zero initial conditions.
current iLmax = E /R = 4.5 A and an arbitrary choice of voltages V1 and V2 . For small current values of the selected reference, high variations of the load and power supply do not affect the control performances. However when the current value of the selected reference is high, it becomes increasingly sensitive to the load variations whereas the voltages remain undisturbed for all these perturbations. 4. Conclusion A simple practical switching stabilization methodology has been shown for PWA systems without equilibrium. The approach provides a systematic way to search for sufficient stabilizing conditions; therefore it has an on-line applicability. It overcomes the control problems encountered when dealing with these systems in an augmented state space. For these systems, the need of systematic approaches allows seeing the present method as a contribution not negligible. However, still deeper investigation may be done to complete the method especially, on the set neighboring points other than the average equilibriums. At some neighboring points, the system may be practically stabilized with certainly much less accuracy. When the analysis of sliding motions is complicated or may alter the systematic calculation, the use of hysteresis suffices in practice. Generally, there is an advantage when working with LMI formulation, we can easily deal with others constraints (if any) to improve performances and facilitate real implementation. In particular, conditions related to the occurrence of sliding
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D. Kamri et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 859–870
Fig. 7. Current iL vs. V1 and V2 for different initial conditions.
motions may be included. For the application of the method, the obtained simulation results are very promising for all treated examples. References [1] S. Azuma, J. Imura, T. Sugie, Lebesgue piecewise affine approximation of nonlinear systems, Nonlinear Analysis: Hybrid Systems 4 (2010) 92–102. [2] E.D. Sontag, Nonlinear regulation: the piecewise linear approach, IEEE Transactions on Automatic Control 26 (2) (1981) 346–358. [3] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, IEEE Transactions on Automatic Control 45 (2000) 1864–1876. [4] M. Johansson, Piecewise Linear Control Systems: A Computational Approach, in: M. Thoma, M. Morari (Eds.), Ser. Lecture Notes in Control and Information Sciences, vol. 284, Springer-Verlag, 2003. [5] S. Pettersson, Analysis and design of hybrids systems, Ph.D. Dissertation, Chalmers University, Goteborg Sweden, 1999. [6] L. Rodrigues, S. Boyd, Piecewise affine slab systems using convex optimization, Systems and Control Letters 54 (2005) 835–853. [7] B. De Schutter, W.P.M.H. Heemels, Modeling and Control of Hybrid Systems, Lecture Notes of the DISC Course October 2006, Delft Center of Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. http://www.dcsc.tudelft.nl. [8] J. Hespanha, Stabilization through hybrid control, in: UNESCO Encyclopaedia of Life Support Systems, 2004. www.ece.ucsb.edu/~hespanha/.../6-4328-7-hy-stab.pdf. [9] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19 (1999) 59–70. [10] J. Geromel, P. Colaneri, Stability and stabilization of continuous time switched System, SIAM Journal on Control and Optimization 45 (5) (2006) 1915–1930. [11] S. Pettersson, Synthesis of switched linear systems, in: Proceedings of the 42nd IEEE Conference on Decision Control, 2003, pp. 5283–5288. [12] R. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88 (7) (2000) 1069–1082. [13] A. Hassibi, S. Boyd, Quadratic stabilization and control of piecewise-linear systems, in: Proceedings of the American Control Conference, 1998, pp. 3659–3664. [14] M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998) 555–559. [15] A. Rantzer, M. Johansson, Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45 (2000) 629–637. [16] L. Rodrigues, A. Hassibi, P. Jonathan How, Output feedback controller synthesis for piecewise-affine systems with multiple equilibria, in: Proceedings of the American Control Conference, 2000, pp. 1784–1789. [17] S. Zhendong, S.S. Ge, Analysis and synthesis of switched linear control systems, Automatica 41 (2005) 181–195. [18] S. Mariéthoz, S. Almér, M. Bâja, A.G. Beccuti, D. Patino, A. Wernrud, J. Buisson, H. Cormerais, T. Geyer, H. Fujioka, U.T. Jönsson, C.-Y. Kao, M. Morari, G. Papafotiou, A. Rantzer, P. Riedinger, Comparison of hybrid control techniques for Buck and Boost DC–DC converters, IEEE Transaction on Control Systems Technology 18 (5) (2010) 1126–1145. [19] X. Xu, G. Zhai, Results on practical asymptotic stabilizability of switched affine systems, in: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006, pp. 15–21. [20] P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, in: Proceedings of the American Control Conference, 2004, pp. 3890–3895. [21] R. Shorten, K. Narendra, Necessary and sufficient conditions for the existence of a CQLF for a finite number of stable LTI systems, International Journal of Adaptive Control and Signal Processing 16 (10) (2002) 709–728. [22] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems, a survey of recent results, IEEE Transactions on Automatic Control 54 (2) (2009) 308–322. [23] S. Boyd, L. El Ghaoui, E. Feron, H. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, in: SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1994. [24] J. Buisson, H. Cormerais, P.Y. Richard, On the stabilisation of switching electrical power converters, in: Hybrid Systems: Computation and Control, in: M. Morari, L. Thiele (Eds.), Ser. LNCS, vol. 3414, Springer, 2005, pp. 184–197.