Input to state set stability for pulse width modulated control systems with disturbances

Transcription

1 Input to state set stability for pulse width modulated control systems with disturbances A.R.eel and L. Moreau and D. Nešić Abstract New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. he results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. he nonsmooth nature of PWM systems is accommodated by working with upper semicontinuous set-valued maps, locally Lipschitz inflations of these maps, and locally Lipschitz parameterizations of locally Lipschitz set-valued maps. Keywords: stability theory, averaging theory, set-valued systems theory, pulse width modulation, power electronics, power converters. Introduction Pulse width modulated (PWM) control is a common paradigm in systems where the values of the actuators are discrete, like {, }. he idea is to maintain a constant frequency of switching between the values and and to use measurements from the plant to determine the duty ratio, i.e., the ratio of time spent at to the time spent at. his kind of control strategy can be useful for systems controlled by on-off valves and/or on-off switches. PWM control techniques are used extensively in state-of-the-art AC-DC converters as well as DC-DC converters, 8, 9, ] which find wide application in industry. he main idea used in PWM control, and many other power electronic systems 7], is to model the switching actuator by its average behavior. While the intuition behind this idea is sound, classical averaging theory is not suitable for these discontinuous systems. However, Lehman and Bass 9] have shown how averaging theory can be extended to address switching power electronic systems when the actuators don t switch infinitely often on a finite time interval, i.e., they don t chatter. For switching power electronic systems where exogenous disturbances are considered and chattering is allowed, perhaps caused by the exogenous disturbances, there has not been a suitable Supported in part by the AFOSR under grant F and the NSF under grant ECS Electrical and Computer Engineering Department, University of California, Santa Barbara, CA, 936, USA. Postdoctoral Fellow of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.-Vlaanderen). his paper presents research results of the Belgian Programme on Inter-university Poles of Attraction, initiated by the Belgian State, Prime Minister s Office for Science, echnology and Culture. he scientific responsibility rests with the author. SYSeMS-EESA, Ghent University, echnologiepark 94, 95 Zwijnaarde, Belgium. Supported by the Australian Research Council under the Discovery ARC grants scheme. Department of Electrical and Electronic Engineering, he University of Melbourne, Parkville, 35, Victoria, Australia.

2 version of averaging stability theory until recently 4]. (Earlier work in this direction can be found in ].) In this paper, we will apply the generalized two time scale stability theory of 4] to study input-to-state stability ] in PWM control systems. he paper is organized as follows: In Section we present mathematical preliminaries. PWM control is introduced in Section 3 and an asymptotic stability result for PWM systems without disturbances is stated in heorem. Section 4 contains the proof of heorem. heorem in Section 5 gives conditions for ISS of PWM systems with disturbances. he proof of heorem is sketched in Section 6. Preliminaries A function α : R R belongs to class-k if it is continuous, zero at zero, strictly increasing and unbounded. A function β : R R R is said to belong to class-kl if it is nondecreasing in it first argument, nonincreasing in its second argument and lim s + β(s, t) = lim t β(s, t) =. Given an open set H R n and a compact set A H, a continuous function ω : H R is said to be positive definite with respect to A if ω(x) = x A. It is said to be proper with respect to H if for each sequence x i approaching the boundary of H or approaching infinity, we have ω(x i ) as i. he open and closed unit balls are denoted B and B, respectively. Given a set F R n, the closed convex hull of the set (that is, the smallest closed convex set containing F) is denoted as cof. For a function f : R m R n R r, we define f(y, B) := { w : w = f(y, x), x B }. () A set-valued map F : R n (subsets of R n ) is upper semicontinuous if for each x and ε > there exists δ > such that F (x + δb) F (x) + εb () where x+δb denotes the ball of radius δ around the point x. A set-valued map F L : R n (subsets of R n ) is said to be locally Lipschitz if, for each x there exists a neighborhood U of x and L > such that x, x U = F L (x ) F L (x ) + L x x B. (3) If a set-valued map is locally Lipschitz then it is upper semicontinuous. For the notion of a measurable set-valued map, we refer the reader to, Section 8.]. We will use the integral of a set-valued map which corresponds to the set of integrals of integrable selections from the set-valued map. See, Section 8.6]. For a discontinuous differential equation ẋ = f(x, t) (4) where f is locally bounded and measurable, the generalized Krasovksii, respectively Filippov, solutions of (4) correspond to the solutions of the differential inclusion ẋ F (x, t) := δ> cof(x + δb, t) (5)

3 respectively ẋ F (x, t) := δ> meas(n)= cof((x + δb\n), t). (6) (he set-valued maps on the right-hand sides are upper semicontinuous in x. See, e.g., 4, p. 85].) he generalized solution concepts of Filippov and Krasovskii agree for the systems that we consider in the sequel. For a general comparison, see 6]. 3 Pulse width modulated control In pulse width modulated control systems, the closed-loop system can typically be written in the form ] ẋ = ε f(x) + g i (x)u(h i (x) p i (t)) (7) where ε is a small positive parameter, u : R, ] is the unit step function (u() = ), the functions h i : R n, ], f and g i are continuous, and the functions p i : R, ] are measurable, bounded and periodic with period one. Often they have the form p i (t) = t mod, as in Figure. (Actually, the period of p i is typically parameterized by ε and the system (7) represents the original system in a transformed time scale.) t Figure : A typical (triangle) switching signal p i (t) in PWM control systems. he behavior of the closed-loop system hinges on the nondecreasing, possibly discontinuous functions v σ i (v) := u(v p i (t))dt = meas {t, ] : v p i (t)} (8) which takes values in, ], and its corresponding upper semicontinuous set-valued map v S i (v) := co σ i (v + µb). (9) µ> At points v where σ i ( ) is continuous, we have S i (v) = {σ i (v)}. he upper semicontinuity of S i is standard, since S i is constructed in the same way that a Filippov or Krasovskii differential inclusion is constructed. If p i (t) then σ i (v) = u(v) = sgn(v) + ] and the closed-loop system (7) can be related to sliding mode control. If, on the other hand, p i (t) = t mod, as in Figure, we have S i (v) = {v} for all v, ]. For example, on the interval, ], u(.8 (t mod )) spends.8 seconds at the value and. seconds at the value, and the integral over the period, ] is equal to.8. his situation is illustrated in Figure. 3

4 u(.8 p i (t)) t Figure : Duty ratio for v =.8 using p i (t) = t mod. he main result in pulse width modulated control systems of the form (7) is the following: Since ε > is small, the state x changes slowly compared to p i and so the effect of p i on (7) can be averaged, as in (8), and the analysis reduced to the analysis of the system which becomes simply ẋ f(x) + ẋ = f(x) + when p i (t) = t mod, using that h i takes values in, ]. he following formal statement is enabled by the results in 4]: g i (x)s i (h i (x)) () g i (x)h i (x) () heorem Suppose the functions f, g i, h i are continuous and that for () the compact set A is asymptotically stable with basin of attraction H. Under these conditions, the set H is open and for each continuous function ω : H R that is positive definite with respect to A and proper with respect to H, there exists β KL, and, for each δ > and compact K H, there exists ε > such that ε (, ε ], x(t ) K = the (generalized Krasovskii/Filippov) solutions of (7) exist for all t t and satisfy ω(x(t)) β ( ω(x(t )), ε(t t ) ) + δ, t t. () 4 Proof of main result 4. he function β As we have asserted above, the set valued map x f(x) + g i (x)s i (h i (x)) =: F (x) (3) 4

5 is upper semicontinuous, and for each x, is nonempty, compact and convex. According to 3, Proposition 3], the set H is open and for each ω : H R that is continuous, positive definite with respect to A and proper with respect to H, there exists β KL such that all solutions of the system () that start in H satisfy ω(x(t)) β (ω(x()), t) t. (4) he general averaging results in 4] give conditions under which the bound (4), which holds for the averaged system, also holds with an arbitrarily small offset for the actual system. o use the explicit sufficient conditions given in 4], F ( ) should be locally Lipschitz rather than just upper semicontinuous. he next computations are aimed at obtaining a bound like (4) for a system ẋ F L (x) where F (x) F L (x) and F L is locally Lipschitz. According to the combination of 3, Proposition, heorem 3, and heorem ], there exist α, α K and a smooth function V : H R such that, for all x H, α (ω(x)) V (x) α (ω(x)) (5) and Define max V (x), w V (x). (6) w F (x) ( ) β(s, t) := α α (s)e t. (7) 4. he value ε 4.. Embedding () in a Lipschitz system Let δ > and the compact set K H be given. Using that ω is proper with respect to H and perhaps by enlarging K and thereby considering a larger set of initial conditions, we can assume without loss of generality that, with the definition we have c := max ω(x), (8) x K {x : ω(x) δ} K, & δ α α (c). (9) Using (7), it follows that δ/ β(c, ). Following the proof of 3, Lemma 8], which relies on the continuity of x V (x), there exist two strictly positive real numbers ρ and ρ such that, with the definitions F (x) := cof (x + ρ B) + ρ B () and we have α ( δ ) F (x) := cof (x + ρ B) + ρ B, () V (x) α (c) = max V (x), w V (x). () w F (x) 5

6 Using 3, Lemma 8], there exists a set-valued map F L : R n (subsets of R n ) with nonempty compact, convex values that is locally Lipschitz and satisfies F (x) F L (x) F (x) x. (3) Combining () and (3) we have ( ) δ α V (x) α (c) = max V (x). w F L (x) (4) It follows, like in, p. 44], that the solutions of ẋ εf L (x) (5) starting in K satisfy { ω(x(t)) max β(ω(x()), εt), δ } t. (6) 4.. Embedding (7) in a continuous system with average contained in F L We define U(s) := µ> co u (s + µb), i.e., U(s) = s > s <, ] s = (7) and note that the Krasovskii/Filippov solutions of (7) correspond to the solutions of ] ẋ ε f(x) + g i (x)u(h i (x) p i (t)) =: ε F (x, t). (8) We define X := {x : ω(x) β(c, )}. (9) It follows from the properties of ω that X is a compact subset of H, and thus there exists ν > be such that C := X + νb H. (3) Claim here exist M > and a set-valued map (x, t) F c (x, t) such that. (x, t) C R = F (x, t) Fc (x, t) MB ; (3). Fc (x, t) is nonempty, compact and convex for each (x, t) C R; 3. the mapping t F c (x, t) is measurable for each x C; 6

7 4. for each ε > there exists δ > such that 5. there exists > such that Proof of claim: We define x, x C, x x δ = Fc (x, t) F c (x, t) + εb ; (3) (x, t ) C R, = M := max x C Recall the value ρ used in (). We define and U c (s) = ( F c (x, t) := f(x) + f(x) + F c (x, t + t )dt F L (x). (33) ) g i (x). (34) s < ρ, s ] + s ρ ] ρ, ] s, s, ρ ] ρ s > ρ (35) g i (x)u c (h i (x) p i (t)). (36) It is clear that items and 3 of the claim hold. Since ( U(s) U c (s) U s + ρ ) B s (37) it follows from (8) and (34) that (3) holds. Moreover, since the set-valued mapping U c is globally Lipschitz with Lipschitz constant /ρ and the functions f, g i and h i are continuous, and thus uniformly continuous on C, it follows that item 4 holds. It remains to establish item 5. Since F c (x, t + t )dt = f(x) + g i (x) U c (h i (x) p i (t + t ))dt (38) and (3), () and (3) hold, it is enough to show that there exists > such that for all t R, v R and, U c (v p i (t + t ))dt S i (v + ρ B) + ρ B. (39) We claim that this condition holds with = ρ. o see this we first observe, using the second inclusion in (37), the definition of σ i ( ) in (8), and the definition of S i in (9) that ( U c (v p i (t))dt U v p i (t) + ρ ) B dt ( σ i v 3 ) 4 ρ, σ i (v + 3 )] 4 ρ (4) S i (v + ρ B). 7

8 Next, we break the integral on the left-hand side of (39) into integration over the intervals, ],, ],..., n, n], n, ] where n is the largest integer not larger than, and we denote the value of the jth integral by w j for j =,..., n +. Since U c (s) U ( s + ρ B ), ] for all s, it follows that w j for all j =,..., n +. Also, using (4) and that p i ( ) is periodic with period one, w j S i (v + ρ B) for all j =,..., n. Since S i (v + ρ B) is convex by definition, the convex combination n n j= w i S i (v + ρ B). So, using ρ, the integral on the left-hand side of (39) can be written as n+ n w i = n w i + ( n ) j= j= j= n w i + w n+ S i (v + ρ B) + ( ( n) + ) B S i (v + ρ B) + B (4) S i (v + ρ B) + ρ B, i.e., (39) holds Working with enlarged systems We now have arrived at the situation where we have that:. the trajectories of (7) that remain in C = X + νb are subsumed by the trajectories of the system ẋ ε F c (x, t) εmb (4) where F c (, t) is continuous uniformly in t, and F c (x, ) is measurable.. the trajectories of where F L is locally Lipschitz, satisfy x avg () K = ω(x avg (t)) max ẋ avg εf L (x avg ), (43) { β(ω(x avg ()), εt), In particular, the trajectories of (43) don t leave the compact set X ; 3. on C and for sufficiently large, the set-valued maps F c and F L are related by } δ. (44) F c (x, t + t )dt F L (x). (45) hese conditions lead to the following statement: Proposition here exists > and for each there exists ε > such that for each ε (, ε ], each x K, each t R and each solution x( ) of (4) satisfying x(t ) = x there exists a solution x avg ( ) of (43) satisfying x(t ) = x and ω(x(t)) ω(x avg (t)) + δ t t, t + ] (46) ε 8

9 and x(t) K t t + ε, t + ] ε. (47) We defer the proof of Proposition. First we will show how it gets used to prove the bound () for the trajectories of (4) starting in K, and hence enables the proof of heorem. Claim For sufficiently small ε, the bound () holds for the trajectories of (4) starting in K. Proof. Let > come from Proposition, and recall the definition of c in (8). Let be such that β(c, τ) δ ) τ,. (48) Let ε > come from Proposition for this. Let ε (, ε ]. Now we have, for all t t, t +/ε], ω(x(t)) ω(x avg (t)) + δ { max β(ω(x ), ε(t t )), Using (48) it also follows that for t t + /(ε), t + /ε], } δ + δ (49). ω(x(t)) δ (5) and x(t + /(ε)) K. he above argument can now be applied repeatedly to obtain ω(x(t)) δ t t + /(ε), ). (5) he conclusion then follows by combining (49) and (5). Remark 4. his proof technique is illustrated graphically in Figure 3. he solid line, in both plots, is the graph of ω(x(t)) (on a time scale with ε = and time indicated relative to t ). Over the initial seconds, ω(x(t)) is compared to ω(x avg (t)), indicated by the dashed graph in the upper plot, where x avg ( ) is a solution of the averaged system that is close to x( ). After information about ω(x(t)) over the first seconds is derived, a new close solution of the average system is introduced and considered for seconds starting / seconds after the original initial time. his solution is indicated by the dashed curve in the lower plot. It gives information about ω(x(t)) between and 3/. his process is continued iteratively, as suggested by the vertical dashed line at the bottom of the figure, to arrive at the conclusion about ω(x(t)) for all time. 4.3 Proof of Proposition We make use of the following result, which is a combination of, heorem 9.6., heorem 9.7.]. Lemma If F : R n R (subsets of R n ) has compact, convex values and is such that F (, t) is continuous (respectively, locally Lipschitz) uniformly in t and F (x, ) is measurable then there exists a function f : R n R R n R n such that. f(x, t, B) = F (x, t); 9

10 ω(x(t)) - solid ω(x avg (t)) - dashed δ δ/ 3 δ δ/ 3 Figure 3: Graphical illustration of proof technique for Claim.. f(, t, d) is continuous (respectively, locally Lipschitz) uniformly in d B and t; 3. f(x,, d) is measurable; 4. f(x, t, ) is continuous. Via what is known as Filippov s lemma (see 3, Exercise 3.7.] or, heorem 8..]), it can be shown that the solutions of ẋ F (x, t) are the same as the solutions of ẋ = f(x, t, d(t)), where d( ) ranges over the set of measurable functions taking values in B. Lemma (Filippov (special case)) Assume f : R n R R n R n has the properties in the conclusion of Lemma. Let x( ) be absolutely continuous and satisfy, for almost all t, ẋ(t) f(x(t), t, B). (5) hen there exists a measurable function d : R B such that, for almost all t, ẋ(t) = f(x(t), t, d(t)). (53) Remark 4. In the notation of 3, Exercise 3.7.], ϕ : R R n R n is given by ϕ(t, u) = ẋ(t) f(x(t), t, u) (with ẋ(t) defined arbitrarily in f(x(t), t, B) for times (a set, if nonempty, of measure zero) where it is not otherwise defined), and Γ = B. Using the above results, we now know that the enlarged problem is the same as the problem where we have that:

11 . the trajectories of (7) that remain in C = X + νb are subsumed by the trajectories of ẋ = εf c (x, t, d) εmb (54) where d : R B is measurable, f c (, t, d) is continuous, uniformly in t and d, f c (x,, d) is measurable, f c (x, t, ) is continuous;. the trajectories of ẋ avg = εf l (x avg, e) (55) where e : R B is measurable, f l is locally Lipschitz, satisfy (44); in particular, the trajectories of (55) don t leave the compact set X ; 3. on C and for sufficiently large, f c and f l are related by: for each measurable d : R B, and t, there exists e B such that f c (x, t + t, d(t))dt = f l (x, e). (56) In this situation, the result of Proposition follows from (the proof of) 4, Proposition 8]. 5 Input-to-state stability results It is clear from the development of the proof that it should be straightforward to handle the situation where certain types of disturbances d appear explicitly in the differential equation (7). For example, we could consider ] ẋ = ε f(x, d f, d s ) + g i (x, d s )u(h i (x, d s ) p i (t)) (57) where d f : R V f represents an arbitrary, measurable function and d s : R V s is slowly varying. We will denote these classes of functions D f and D s,ε respectively. More precisely (cf. 4, Assumptions and ]): Assumption he sets V s and V f are compact. Assumption he set D s,ε is such that. it is shift invariant, i.e., if t d s (t) belongs to D s,ε then t d s (t + t ) belongs to D s,ε for all t.. For each > and ρ > there exists ε > such that, for all ε (, ε ], For simplicity, we will assume: d s D s,ε = d s (t) d s () ρ t, ]. (58) Assumption 3 he functions f, g i and h i are locally Lipschitz and p i (t) = t mod.

12 In this case, the natural averaged system to consider is ] ẋ = ε f(x, d f, d s ) + g i (x, d s )sat (h i (x, d s )). (59) Here we enforce that h i takes values in, ] by passing h i through the function sat defined to be the nondecreasing function that is linear on, ] and with range, ]. In fact, the actual average system that we will consider is ] ẋ = ε f(x, d f, d s ) + g i (x, d s )sat (h i (x, d s ) + e) =: εf avg (x, d f, d s, e) (6) where e is measurable and takes values in ρ 3 B where ρ 3 > is arbitrarily small. We will use E to denote this class of functions. We note that f avg is locally Lipschitz. We assume: Assumption 4 Given continuous, positive semidefinite functions ω o and ω i and the function β KL such that. ω o is proper with respect to H;. the trajectories of the system (6) satisfy (compare with (6) or (44)) ω o (x(t)) max {β(ω o (x()), εt), ω i (d f, d s ) } (6) for all initial conditions in an open set H and all measurable (d f, d s, e) D f D s,ε E. Remark 5. We note that ω i (, ) may be strictly positive to account for the fact that e does not appear explicitly in the bound (6). E.g., we may have ω i (d f, d s ) = ω i (d f, d s ) + γ(ρ 3 ) (6) where γ( ) and ω i (, ) are positive definite, and ρ 3 represents the worst case infinity norm for e E. We can then make the following statement: heorem Under the Assumptions -4, for each δ > and compact set K H there exists ε > such that ε (, ε ], x(t ) K, (d f, d s ) D f D s,ε = (63) the (generalized Krasovskii/Filippov) solutions of (57) exist for all t t and satisfy ω o (x(t)) max {β(ω o (x(t )), ε(t t )), ω i (d f, d s ) } + δ. (64)

13 6 Proof of heorem We follow the proof of heorem, however ω (which now is ω o ) and β are already given and the averaged system is already locally Lipschitz. herefore, after taking the definition of c from (8), we pick up the proof at Section 4... We define U(s) as in (7) and note that the Krasovskii/Filippov solutions of (57) correspond to the solutions of (cf. (8)) ] ẋ ε f(x, d f, d s ) + g i (x, d s )U(h i (x, d s ) p i (t)) =: ε F (x, t, d f, d s ) (65) We define (cf. (9)) { { }} X := x : ω o (x) max β(c, ), max ω i (d f, d s ) (d f,d s) V f V s. (66) Like before, it follows from the properties of ω that X is a compact subset of H, and thus there exists ν > be such that (cf. (3)) C := X + νb H. (67) We define Φ to be the set of measurable functions taking values in B. hen we have the following claim, which parallels Claim : Claim 3 here exist M > and a map (x, t, d f, d s, φ) f c (x, t, d f, d s, φ) such that. (x, t, d f, d s ) C R V f V s = F (x, t, df, d s ) f c (x, t, d f, d s, B) MB (68). the mapping (x, d f, d s, φ) f c (x, t, d f, d s, φ) is continuous for each t; 3. the mapping t f c (x, t, d f, d s, φ) is measurable for each (x, d f, d s, φ) C V f V s B; 4. for each ε >, there exists δ > such that (x, x, d f, d s, φ) C C V f V s B, x x δ = (69) f c (x, t, d f, d s, φ) f c (x, t, d f, d s, φ) ε ; 5. for each ρ > there exists > such that for each (x, t, d s, d f, φ) C R V s D f Φ, there exists e E such that fc (x, t + t, d f (t), d s, φ(t)) f avg (x, d f (t), d s, e(t))] dt ρ. (7) 3

14 Proof. he proof of this claim is like the proof of Claim with the extra step that we explicitly parameterize the set-valued map U c (s) defined in (35) using the function u c (s, φ) = ( ) ] s sat + + { ρ max s }, φ. (7) ρ We note that s u c (s, φ) is globally Lipschitz uniformly in φ B, and ( U(s) u c (s, B) U s + ρ ) B. (7) We define and f c (x, t, d f, d s, φ) := f(x, d f, d s ) + M = max (x,d f,d f ) C V f V s ( g i (x, d s )u c (h i (x, d s ) p i (t), φ) (73) f(x, d f, d s ) + ) g i (x, d s ). (74) he first four conditions of the claim follow immediately from Assumption 3 and the properties of u c. Finally, the last condition of the claim is established by following the calculations (38)-(4) at the end of the proof of Claim, using the fact that u c (s, B) = U c (s). At this point we have the analogy of Proposition, which follows from (the proof of) 4, Proposition 8], and then the analogy of Claim to complete the proof of heorem. 7 Conclusion Pulse width modulation is an important control technique used extensively in state-of-the-art AC- DC converters as well as DC-DC converters such as the boost and buck converters. he control design intuition behind pulse width modulation is often based on an averaging approach, but a rigorous justification of averaging techniques as a design tool is highly nontrivial because of the discontinuous nature of the resulting control schemes. he present paper provides a mathematically sound framework for the (input-to-state) stability analysis of pulse width modulated control systems via averaging techniques. Crucial ingredients in our analysis are a recent generalization of two time scale stability results for systems with disturbances 4] and methods from nonsmooth systems theory. References ] J.P. Aubin and H. Frankowska. Set-valued analysis, Birkhauser, Boston, 99. ] R.M. Bass and P.. Krein. Large-signal design alternatives for switching power converter control. In IEEE Power Electronics Specialists Conference Record, June 99, pp ] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski. Nonsmooth analysis and control theory, Springer,

15 4] A.F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, ] R.A. Freeman and P. Kokotovic. Robust nonlinear control design: state-space and Lyapunov techniques, Birkhauser, Boston, ] O. Hajek. Discontinuous differential equations, I. Journal of Differential Equations, vol. 3, 49 7 (979). 7] P..Krein, J.Bentsman, R.M.Bass and B.L.Lesieutre. On the use of averaging for the analysis of power electronic systems. ransactions on Power Electronics, vol. 5, No., April 99, pp ] B. Lehman, R.M. Bass. Switching frequency dependent averaged models for PWM DC-DC converters. IEEE ransactions on Power Electronics, vol., no., January 996, pp ] B. Lehman, R.M. Bass. Extensions of averaging theory for power electronic systems. IEEE ransactions on Power Electronics, vol., no. 4, July 996, pp ] D. Nešić and A.R.eel, Input-to-state stability for nonlinear time-varying systems via averaging. Math. Contr. Sign. Sys., vol. 4,, pp ] E.D. Sontag. Smooth stabilization implies coprime factorization. IEEE ransactions on Automatic Control, vol. 34, no. 4, April 989, pp ] D.G. aylor. Pulse width modulated control of electromechanical systems. IEEE ransactions on Automatic Control, vol. 37, no. 4, pp , April 99. 3] A.R. eel and L. Praly. A smooth Lyapunov function from a class KL estimate involving two positive semidefinite functions. ESAIM: Control, Optimisation, and Calculus of Variations, June, vol. 5, ] A.R. eel, L. Moreau and D. Nešić. A unified framework for input-to-state stability in systems with two time scales. IEEE ransactions on Automatic Control, 3, to appear. 5