On Stability of Nonlinear Slowly Time-Varying and Switched Systems

Transcription

1 2018 IEEE Conference on Decision nd Control CDC) Mimi Bech, FL, USA, Dec , 2018 On Stbility of Nonliner Slowly Time-Vrying nd Switched Systems Xiobin Go, Dniel Liberzon, nd Tmer Bşr Abstrct In this pper, we pply totl vrition pproch to bridge the stbility criteri for nonliner time-vrying systems nd nonliner switched systems. In prticulr, we derive set of stbility conditions pplying to both nonliner time-vrying systems nd nonliner switched systems. We show tht the derived stbility conditions, when pplied to nonliner time-vrying systems nd nonliner switched systems, recover the existing stbility results in the literture. We lso show tht the derived stbility conditions cn be pplied to qulittively recover unified stbility criterion for slowly time-vrying liner systems nd switched liner systems proposed in our recent work. I. INTRODUCTION The stbility criteri for slowly time-vrying systems nd switched systems re fundmentl, nd they hve drwn extensive reserch interest over the pst decdes. For slowly time-vrying liner or nonliner) system, group of stbility results in [1] [6] stte tht the system is globlly symptoticlly stble if the system is stble t ech frozen time 1, nd the system prmeters vry slowly enough. Similrly, group of stbility results in [7] [10] stte tht switched liner or nonliner) system is globlly symptoticlly stble if ech subsystem is stble, nd the system switches slowly enough mong its subsystems. Although the two groups of results re similr nd switched system cn be viewed s time-vrying system with piecewise constnt system prmeters, the stbility conditions for switched systems cnnot be directly recovered from the stbility conditions for time-vrying systems. The gp blocking this trnsition is cused by the common pproch to chrcterize the vrition of system prmeters in time-vrying system. To be more specific, the common pproch for exmple, conditions involving the integrl of the norm of system prmeters time derivtives) requires tht the systems prmeters re differentible or Lipschitz continuous) in time, nd thus it cnnot be directly pplied to the piecewise constnt cse. In our recent work [11], [12], set of stbility conditions ws derived for slowly time-vrying liner systems. Differently from the prior works, the vrition of system prmeters is chrcterized by their totl vrition over the time intervl. The concept of totl vrition cn be pplied to both differentible functions nd piecewise constnt functions. Benefitting from this property, it ws shown tht This reserch ws supported in prt by the Office of Nvl Reserch ONR) MURI grnt N , in prt by NSF grnt CMMI , nd in prt by AFOSR grnt FA All uthors re with the Coordinted Science Lbortory, University of Illinois t Urbn-Chmpign, Urbn, IL 61801; emils: {xgo16, liberzon, bsr1}@illinois.edu. 1 A time-vrying system is stble t frozen time mens the system with prmeters fixed t tht frozen time is stble. the stbility conditions derived in [11], [12] cn be pplied to recover both the stbility conditions derived in [1] [3], for slowly time-vrying liner systems, nd the stbility conditions derived in [8] [10], for switched liner systems 2. The work in [11], [12] brings up wreness of the fct tht the studies of stbility conditions for slowly time-vrying systems nd switched systems should not be disjoint, nd it provides n pproch to trnslte the results from one field to the other. Thus, it motivtes us to pply the totl vrition pproch to further bridge the gp between stbility conditions for nonliner time-vrying systems nd switched systems. For nonliner time-vrying systems, group of stbility results hve been derived in [] [6]. The stbility results in [] [6] impose slightly different conditions on system dynmics, yet they ll require tht the vrition of system prmeters, chrcterized by the time integrl of the norm of the time derivtive of the prmeters, should be upper bounded by n ffine function of the length of the time intervl. More results on this line of reserch cn be found in [13] nd the references therein. For nonliner switched systems, the stbility criteri proposed in [8] [10] cn lso be pplied to the nonliner cse. We re interested in unifying the bove two groups of results. In this pper, we first recll the concept of totl vrition Section II). Then, we derive set of stbility conditions for nonliner time-vrying systems min results, Section III), which generlize the stbility results in [] to incorporte piecewise differentible system prmeters. Next, we pply the derived stbility conditions to nonliner switched systems nd show tht they mtch the stbility conditions in [10] for the nonliner cse Section IV). Then, we pply the min results to liner time-vrying systems. We show tht with one minor dditionl ssumption, one cn recover qulittively) the unified stbility criteri for slowly time-vrying nd switched liner systems proposed in [11], [12] Section V). Finlly, we drw concluding remrks nd discuss future directions for reserch Section VI). II. PRELIMINARIES: TOTAL VARIATION The concept of totl vrition for rel-vlued nd vectorvlued functions hs been well documented in the literture [1]. This concept hs been generlized to the mtrix cse in the recent work [12]. We recll the definition of totl vrition here. We denote by P := {t i i = 0,..., k} prtition of intervl [, b], where = t 0 < < < t k = b. Let P be the set of ll prtitions of [, b]. Given vectorvlued function u ) : R R n, its totl vrition over [, b] 2 The results derived in [8] [10] re for generl switched systems; they were pplied to the liner cse in [11], [12] /18/$ IEEE 658

2 is defined s du := sup P P k ut i ) ut i 1 ). The norm in the expression bove is the Eucliden norm. In prticulr, if u ) stisfies some regulrity conditions Condition 1), its totl vrition over n intervl hs n explicit expression Lemm 1). We use ut ) to denote the left limit of u ) t t, nd we introduce the following set of conditions. Condition 1 [12], Assumption 2): A vector-vlued function u ) stisfies i) u ) is continuous from the right everywhere on [, b) nd hs left limits everywhere on, b]. ii) u ) hs finite number of discontinuities over, b), denoted by {d 1, d 2,..., d m }, nd := d 0 < d 1 < < d m < d m+1 =: b. iii) u ) is continuously differentible on d i, d i+1 ), nd u ) is Riemnn integrble on [d i, d i+1 ], for ll i {0, 1,..., m}. iv) u ) is Riemnn integrble on [d i, d i+1 ], for ll i {0, 1,..., m}. Lemm 1 [12], Lemm 1): Given vector-vlued function u ) stisfying Condition 1, its totl vrition over [, b] hs the following expression: du = m i=0 di+1 d i ut) dt + m+1 udi ) ud i ). In [12], Condition 1 nd Lemm 1 were stted for mtrixvlued function, yet vector-vlued function cn be viewed s specil cse of mtrix-vlued function. III. STABILITY OF NONLINEAR SLOWLY TIME-VARYING SYSTEMS In this section, we derive set of stbility conditions for nonliner time-vrying systems, which generlize the existing stbility results in [, Section 9.6]. Consider nonliner time-vrying system ẋ = fx, ut)), 1) where x R n is the system stte nd ut) Γ R m is the time-dependent system prmeter. Assume tht fx, u) is loclly Lipschitz over R n Γ. Furthermore, given ny u Γ, the eqution fx, u) = 0 is ssumed to hve solution x = 0. Then, system 1) hs n equilibrium point t x = 0, which is invrint over u Γ. Remrk 1: As strting point, we consider in this pper the time-vrying system with fixed equilibrium point. However, it is more generl to ssume tht the system eqution with time-vrying system prmeters dmits n equilibrium point tht vries in time on mnifold. This is not pursued in this pper. Theorem 1: The nonliner time-vrying system 1) is globlly exponentilly stble 3 if the following conditions re stisfied. i) The set of ll possible system prmeters, Γ, is compct nd convex. ii) There exist cndidte Lypunov function V x, u), continuously differentible in x nd u, nd positive constnts c 1, c 2, c 3, c such tht for ll x R n nd u Γ, c 1 x 2 V x, u) c 2 x 2, 2) V x fx, u) c 3 x 2, 3) V u c x 2. ) iii) There exist positive constnts µ nd α, with µ < c 1 c 3 /c 2 c, such tht for ny bounded time intervl [, t 2 ], u ) stisfies Condition 1, nd its totl vrition stisfies du µt 2 ) + α. 5) Remrk 2: The inequlity 3) implies tht for ll u Γ, the eqution fx, u) = 0 hs unique solution, i.e., x = 0. Remrk 3: The exct choice of the system prmeter ut), in prticulr its dimension, ffects the totl vrition t2 du, nd thus it ffects condition iii) in Theorem 1. For exmple, one could dd virtul dimension to ut), such tht the totl vrition becomes lrger nd condition iii) becomes more restrictive. Pursuing this issue further is out of the scope of this pper. Before proving Theorem 1, we would like to clrify the novelty of this theorem. The set of stbility conditions proposed in [, Section 9.6] requires item ii) in Theorem 1, yet it replces item iii) in Theorem 1 by the stronger condition tht u ) is continuously differentible, nd ut) dt µt 2 ) + α, where µ < c 1 c 3 /c 2 c nd α > 0. By introducing the concept of totl vrition, Theorem 1 generlizes the result in [, Section 9.6] to the cse where u ) is piecewise differentible with discontinuities t the non-differentible points. This generliztion enbles us to further derive set of stbility conditions for switched systems, which will be discussed lter in Section IV. One my find the ssumption described by ) not s stndrd s those described by 2) nd 3). The following lemm Lemm 2) from [] justifies tht if the system stisfies some generl regulrity conditions, nd if the system with fixed prmeters is globlly exponentilly stble, then there exists cndidte Lypunov function stisfying 2)-) 3 The definition of globl exponentil stbility cn be found in [, Definition.5]. 659

3 in Theorem 1. The proof of Lemm 2 pplies the converse Lypunov theorem, for exponentil stbility. Lemm 2 [], Lemm 9.8): Consider the system 1) nd suppose tht the Jcobin mtrices fx, u)/ x nd fx, u)/ u stisfy fx, u) x L 1, fx, u) u L 2 x, x R n, u Γ for some L 1, L 2 0. Furthermore, ssume there exist positive constnts k nd γ such tht the trjectories of the system, with constnt ut) u Γ, stisfy xt) k x0) e γt, x0) R n, t 0. Then, there exists function V x, u) defined over R n Γ stisfying 2)-). We re now in position to prove Theorem 1. Proof: Proof of Theorem 1) Let n rbitrry time intervl [, t 2 ] be given. Without loss of generlity, we ssume tht u ) hs only one discontinuity over [, t 2 ], denoted by t d. By Condition 1, u ) is differentible nd u ) is integrble over the sub-intervls, t d ) nd t d, t 2 ). Furthermore, u ) is right continuous nd hs left limit t t d. Denote by V t) the Lypunov function evluted long xt) nd ut), nmely, V t) = V xt), ut)). Since u ) hs discontinuity t t d, V ) lso hs discontinuity t t d. Consider the intervl [, t d ). By 2) ) in Theorem 1, we hve V = V V fx, u) + x u u c 3 x 2 + c x 2 u c 3 + c ) u V. c 2 c 1 By the Gronwll-Bellmn inequlity, we obtin td V t d ) V ) exp c 3 + c ) u dt c 2 c 1 = V ) exp c 3 t d ) + c td ) u dt. c 2 c 1 6) We pply similr rgument on the intervl [t d, t 2 ], nd we obtin V t 2 ) V t d ) exp c 3 t 2 t d ) + c ) u dt. 7) c 2 c 1 t d Consider ny x 1, x 2 R n nd u 1, u 2 Γ. Since Γ is convex set, nd V x, u) is differentible in x nd u, by the Men Vlue Theorem, there exists λ [0, 1] such tht V x 2, u 2 ) V x 1, u 1 ) = V x λx λ)x 2, λu λ)u 2 ) x 2 x 1 ) + V u λx λ)x 2, λu λ)u 2 ) u 2 u 1 ). Denote by z 1 = [x 1, u 1 ] vector whose elements re x 1 nd u 1. Similrly, let z 2 = [x 2, u 2 ]. Then, the Men Vlue Theorem sttes tht there exists λ [0, 1] such tht V z 2 ) V z 1 ) = V λz λ)z 2 ) z 2 z 1 ), where V = [ V x, V ] T u. Tking x 1 = x 2 = x, we hve By ), V x, u 2 ) V x, u 1 ) = V u x, λu λ)u 2 ) u 2 u 1 ). V x, u 2 ) V x, u 1 ) c x 2 u 2 u 1. 8) Tking gy) = expy 1) y, y R, nd then we hve g y) = expy 1) 1, g y) = expy 1). It cn be checked tht g y) = 0 if nd only if y = 1. In ddition, g y) > 0, y R. Hence, gy) ttins globl minimum t y = 1, nd g1) = 0. This implies tht Hence, V x, u 2 ) V x, u 1 ) y expy 1), y R. ) V x, u2 ) exp V x, u 1 ) 1 ) V x, u2 ) V x, u 1 ) = exp. V x, u 1 ) Plugging 2) nd 8) into the inequlity bove, we further hve V x, u 2 ) c x 2 ) u 2 u 1 exp V x, u 1 ) c 1 x 2 ) 9) c = exp u 2 u 1 c 1 for ll u 1, u 2 Γ, x R n. In prticulr, tking x = xt d ), u 1 = ut d ) nd u 2 = ut d ), we obtin ) V t d ) V t d ) exp c ut d ) ut d c ). 10) 1 Note tht ut d ) Γ since Γ is compct nd thus closed) set. Combining 5)-7) nd 10) together, we obtin V t 2 ) V ) exp c 3 t 2 ) + c ) du c 2 c 1 V ) exp c 3 c µ ) t 2 ) + c ) α. c 2 c 1 c 1 Since µ < c 1 c 3 /c 2 c, V ) decys exponentilly fst with rte c 3 /c 2 c /c 1 )µ. By 2), we conclude tht x ) lso decys exponentilly fst. IV. STABILITY OF SWITCHED NONLINEAR SYSTEMS In this section, we pply the min result obtined in Section III to derive set of stbility conditions for switched systems, nd we show tht the derived results mtch the existing stbility conditions in [10]. Let set of systems ẋ = fx, u p ), p P 660

4 be given, where x R n is the system stte, u p is the system dependent prmeter, nd P is the index set, which is finite or countbly infinite. The system prmeter u p is picked from set Γ R m, which could be n uncountble set. Assume tht f, ) is loclly Lipchitz over R n Γ. In ddition, ssume tht for ny u Γ, the eqution fx, u) = 0 hs solution t x = 0. Now consider switched system ẋ = f σt) x), 11) where σ ) : [0, ) P is the switching signl nd f σt) x) := fx, u σt) ). By convention, it is ssumed tht σ ) is piecewise constnt nd is continuous from right everywhere. We introduce Nσ pq t, t + T ), p, q P s the number of switches from subsystem p nd subsystem q over the time intervl [t, t+t ]. By Theorem 1, we hve the following corollry. Corollry 1: The switched system 11) is globlly exponentilly stble if the following conditions hold. i) Γ is compct nd convex set. ii) There exists Lypunov function V x, u), continuously differentible over R n Γ, which stisfies 2)-) in Theorem 1 with constnts c 1, c 2, c 3, c. iii) There exist positive constnts µ nd α, with µ < c 1 c 3 /c 2 c, such tht Nσ pq t, t+t ) u p u q µt +α, t, T 0. 12) Proof: By the piecewise constnt property of σ ), it cn be redily seen tht u σ ) is lso piecewise constnt. Hence, the totl vrition of u σ ) over [t, t + T ] cn be expressed by t+t t du σ = N pq σ t, t + T ) u p u q. Then 12) is equivlent to 5), nd thus globl exponentil stbility cn be estblished by Theorem 1. By [10, Theorem 5], the switched system 11) is globlly exponentilly stble if the following conditions re stisfied 5 : for ech p P, there exists Lypunov function V p x), continuously differentible in x, stisfying I) There exist positive constnts c 1 nd c 2 such tht c 1 x V p x) c 2 x, p P, x R n. II) There exists positive constnt γ such tht V p x fx, u p) γv p x), p P, x R n. III) For ech pir of p, q P, there exists positive constnt ν pq such tht V p x) ν pq V q x), x R n. 5 This result ws recovered in [12, Theorem 5] for the purpose of compring it with stbility conditions for switched liner system. Here, we compre it with Corollry 1 which dels with nonliner switched systems. IV) There exists positive constnt η such tht Nσ pq t, t + T ) ln ν pq < γt + η, t, T 0. We now compre the bove stbility result with Corollry 1. Assume tht items i) nd ii) in Corollry 1 hold. By 2), condition I) in [10, Theorem 5] is stisfied with the Lypunov functions By 2) nd 3), we hve V p x) := V x, u p ), p P. V p x fx, u p) c 3 x 2 c 3 c 2 V p x), p P, x R n. Therefore, condition II) is stisfied with γ = c 3 /c 2. By 9) in the proof of Theorem 1, we hve ) V p x) V q x) exp c u p u q, p, q P, x R n. c 1 Hence, condition III) is stisfied with ν pq = expc /c 1 u p u q ). By plugging in the bove expressions for γ nd ν pq, condition IV) cn be expressed by Nσ pq t, t + T ) c u p u q < c 3 T + η, c 1 c 2 which is equivlent to Nσ pq t, t + T ) u p u q < c 1c 3 T + c 1 η. c 2 c c The bove inequlity is further equivlent to Nσ pq t, t + T ) u p u q µt + α, with µ < c 1c 3 c 2 c, α = c 1 c η. Hence, condition IV) is exctly the sme s item iii) in Corollry 1. In view of this, we conclude tht Corollry 1 recovers the result [10, Theorem 5] under the sme ssumptions on the Lypunov function, i.e., items i) nd ii). V. STABILITY OF SLOWLY TIME-VARYING LINEAR SYSTEMS In this section, we pply Theorem 1 to slowly time-vrying liner systems. We show tht Theorem 1, when pplied to the liner cse, qulittively recovers the stbility criteri proposed in our erlier works [11], [12]. To present nd prove this sttement, we need to introduce some nottions nd definitions. We use nd F to denote the stndrd induced 2-norm nd the Frobenius norm, respectively, for mtrices. Given mtrix-vlued function A ) : R R n n, its totl vrition over [, b] is defined s da := sup p P k At i ) At i 1 ), 661

5 where p is prtition of [, b], nd P is the set of ll prtitions of [, b]. Similrly, the totl vrition of A ) over [, b], under the Frobenius norm, is defined s da F := sup p P k At i ) At i 1 ) F. A well-known property of mtrix norms see [15, eqn ]) sttes tht given ny mtrix A R n n, Then, we hve A A F. 13) da da F. 1) We define mpping U ) : R n n R n2 such tht for ny A R n n, UA ) := [ 11,..., 1n, 21,..., 2n,..., n1,..., nn] T, where n A = n1... nn It is cler tht U ) is bijection, nd we denote by U 1 ) the inverse mpping of U ). Furthermore, by the definitions of the Eucliden norm for vectors nd the Frobenius norm for mtrices, we hve UA n n ) = 2 ij = A F. 15) j=1 Given mtrix-vlued function A ), we denote by u A ) := UA )) 16) the composition between U ) nd A ). By 15), we hve du A = da F. To pply Theorem 1 to slowly time-vrying liner systems, we need to rely on the following lemm. The lemm sttes tht when system 1) is liner in x, Lypunov function stisfying 2) ) in Theorem 1 exists, nd the constnts c 1, c 2, c 3, c dmit explicit expressions. Lemm 3 [], Lemm 9.9): Consider system 1) with system eqution liner in x, tht is, ẋ = fx, ut)) = Āut))x, 17) where x R n, ut) = [u 1 t),..., u m t)] T Γ R m, nd Ā ) : R m R n n. Assume tht the following conditions hold. i) Āu) is continuously differentible in u. Furthermore, there exist positive constnts b i, i = 1,..., m, such tht Āu) bi, u Γ, i = 1,..., m. u i ii) There exists positive constnt L such tht iii) Āu) L, u Γ. Āu) is Hurwitz for ll u Γ, nd there exist positive constnts c nd λ such tht eāu)s ce λs, s 0, u Γ. Let V x, u) = x T P u)x be the cndidte Lypunov function, where P u) is the solution to the Lypunov eqution P u)āu) + ĀT u)p u) = I. Then, V x, u) stisfies 2) ) in Theorem 1 for ll x R nd u Γ, with c 1 = 1 2L, c 2 = c2 2λ, c 3 = 1, c = c m 2λ 2 b 2 i. Combining Lemm 3 nd Theorem 1, we hve the following corollry. Corollry 2: Consider liner time-vrying system ẋ = At)x, 18) where x R n nd At) A R n n. The system is globlly exponentilly stble if the following conditions re stisfied. i) A is compct nd convex set. ii) A is Hurwitz for ll A A, nd there exist positive constnts c nd λ such tht e A s ce λs A A, s 0. iii) For ny time intervl [, t 2 ], u A ) stisfies Condition 1, nd its totl vrition stisfies du A µt 2 ) + α, where u A ) is defined in 16), α > 0, µ < β 1 2nβ2 3, β 1 = 1 2L, β 2 = c2 2λ, 19) nd L := mx A A A. Proof: The system eqution 18) of the liner timevrying system cn be written s ẋ = At)x = U 1 U At) )) x = U 1 u A t))x, which mtches 17) by letting Ā ) = U 1 ), ut) = u A t), nd Γ = UA) R n2. It cn be redily seen tht Ā/ u i, i = 1,..., n 2, is sprse mtrix with one element being one nd ll other elements being zero. Hence, we hve Āu) Āu) F = 1 u R n 2, i = 1,..., n 2, u i u i where the inequlity holds due to 13). Then, condition i) in Lemm 3 is stisfied. By the compctness of A, it cn be checked tht condition ii) in Lemm 3 is stisfied. Furthermore, condition iii) in Lemm 3 is stisfied due to condition ii) in Corollry 2. Therefore, we cn pply Lemm 3 nd 662

6 obtin cndidte Lypunov function stisfying 2) ), with constnts c 1 = 1 2L, c 2 = c2 2λ, c 3 = 1, c = c n2 2λ 2 1 = nc 2λ 2. 20) To pply Theorem 1, we need to further justify tht the conditions i) nd iii) in its sttement hold. Since Γ = UA), the compctness nd convexity of A implies 6 those of Γ, nd thus condition i) in Theorem 1 is stisfied. Furthermore, it cn be computed tht β 1 2nβ 3 2 = 2λ3 nlc 6 = 1 2L λ3 nc 6 = c 1c 3 c 2 c, where β 1 nd β 2 re s in 19) nd c 1 c re chrcterized by 20). Hence, condition iii) in Corollry 2 is equivlent to condition iii) in Theorem 1. Combining the rguments bove, we conclude tht the stbility conditions in Theorem 1 re stisfied, nd thus the liner time-vrying system is globlly exponentilly stble. The result in [12, Theorem 3] sttes tht the liner time-vrying system is globlly exponentilly stble if the following conditions re stisfied. i) A is compct set. ii) A is Hurwitz for ll A A, nd there exist positive constnts c nd λ such tht e A s ce λs A A, s 0. iii) For ny time intervl [, t 2 ], the totl vrition of A ) stisfies da µt 2 ) + α, where α > 0, µ < β 1 2β2 3, β 1 = 1 2L, β 2 = c2 2λ. Compring this result with Corollry 2, we observe tht the convexity ssumption on A ws not mde in [12, Theorem 3]. Furthermore, du A = da F da, where the inequlity holds due to 1), nd β 1 2nβ 3 2 β 1 2β2 3. Hence, with minor dditionl ssumption on the convexity of A, Corollry 2 recovers [11, Theorem 3] qulittively but not quntittively. 6 The convexity of Γ is esy to check, yet the proof of compctness requires n ppliction of the equivlence of norms, between the induced 2-norm nd the Frobenius norm. VI. CONCLUSIONS AND FUTURE WORK In this pper, we hve derived set of stbility conditions unifying the stbility results for nonliner time-vrying systems nd nonliner switched systems. By pplying the derived stbility conditions to liner time-vrying systems, we hve lso recovered qulittively the unified stbility criteri for slowly time-vrying liner systems nd switched liner systems, which were derived in n erlier work inititing this line of reserch. Bsed on the results of this pper, there re two interesting directions for future study: first, it is ssumed in this pper tht the time-vrying system is stble t ech frozen time, nd the switched system contins only stble subsystems. However, stbility of slowly time-vrying systems with stble nd unstble system dynmics t different frozen times, nd stbility of switched systems with both stble nd unstble subsystems, hve been well studied in the literture. It is worth trying to bridge the two groups of results vi the totl vrition pproch. Second, in this work, it is ssumed tht the systems with different prmeters dmit the sme equilibrium point. It is worth generlizing the stbility results derived in this pper to the cse where the time-vrying system with time-vrying prmeters dmits time-vrying equilibrium point on mnifold. REFERENCES [1] C. A. Desoer. Slowly vrying system ẋ = Ax. IEEE Trnsctions on Automtic Control, 16):339 30, [2] A. Ilchmnn, D. H. Owens, nd D. Prätzel-Wolters. Sufficient conditions for stbility of liner time-vrying systems. Systems Control Letters, 92): , [3] P. A. Ionnou nd J. Sun. Robust Adptive Control. Prentice-Hll, [] H. K. Khlil. Nonliner Systems, Third Edition. Prentice-Hll, [5] D. A. Lwrence nd W. J. Rugh. On stbility theorem for nonliner systems with slowly vrying inputs. IEEE Trnsctions on Automtic Control, 357):860 86, [6] H. K. Khlil nd P. V. Kokotovic. On stbility properties of nonliner systems with slowly vrying inputs. IEEE Trnsctions on Automtic Control, 362):229, [7] A. S. Morse. Supervisory control of fmilies of liner set-point controllers, pr: exct mtching. IEEE Trnsctions on Automtic Control, 11): , [8] J. P. Hespnh nd A. S. Morse. Stbility of switched systems with verge dwell-time. In Proceedings of the 38th IEEE Conference on Decision nd Control, pges , [9] X. Zho, L. Zhng, P. Shi, nd M. Liu. Stbility nd stbiliztion of switched liner systems with mode-dependent verge dwell time. IEEE Trnsctions on Automtic Control, 577): , [10] A. Kundu nd D. Chtterjee. Stbilizing switching signls for switched systems. IEEE Trnsctions on Automtic Control, 603): , [11] X. Go, D. Liberzon, J. Liu, nd T. Bşr. Connections between stbility conditions for slowly time-vrying nd switched liner systems. In Proceedings of the 5th IEEE Conference on Decision nd Control, pges , [12] X. Go, D. Liberzon, J. Liu, nd T. Bşr. Unified stbility criteri for slowly time-vrying nd switched liner systems. Automtic, 96: , [13] Y. U. Choi, H. Shim, nd J. H. Seo. On stbility property of nonliner systems with periodic inputs hving slowly vrying verge. IFAC Proceedings Volumes, 381): , [1] C. G. Hu nd C. C. Yng. Vector-Vlued Functions nd Their Applictions, volume 3. Springer Science & Business Medi, [15] G. H. Golub nd C. F. Vn Lon. Mtrix Computtions, volume 3. JHU Press,