On external semi-global stochastic stabilization of linear systems with input saturation

Transcription

1 1 On exernal semi-global sochasic sabilizaion of linear sysems wih inpu sauraion Anon A. Soorvogel 1,AliSaberi 2 and Siep Weiland 3 Absrac This paper inroduces he noion of exernal semi-global sochasic sabilizaion for linear plans wih sauraing acuaors, driven by a sochasic exernal disurbance, and having random Gaussiandisribued iniial condiions. The aim of his sabilizaion is o conrol such plans by a possibly nonlinear saic sae feedback law ha achieves global asympoic sabiliy in he absence of disurbances, while guaraneeing a bounded variance of he sae vecor for all ime in he presence of disurbances and Gaussian disribued iniial condiions. We repor complee resuls for boh coninuous- and discree-ime open-loop criically sable plans. I. INTRODUCTION Inernal and exernal sabilizaion of linear plans wih acuaors subjec o sauraion has been he subjec of inense renewed ineres among he conrol research communiy for he pas wo decades. The number of recen books and special issues of conrol journals devoed o his subjec maer evidence his inense research focus, see for insance [1], [11], [4], [15] and he references herein. I is now considered a classical fac ha boh coninuous- and discree-ime linear plans wih sauraing acuaors can be globally inernally sabilized if and only if all of he open-loop poles are locaed in he closed lef half plane in he coninuous-ime case), and are in inside 1 Deparmen of Mahemaics, and Compuing Science, Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Neherlands. Deparmen of Elecrical Engineering, Mahemaics and Compuer Science, Delf Univ. of Technology, P.O. Box 5031, 2600 GA Delf, The Neherlands. A.A.Soorvogel@ue.nl. 2 School of Elecrical Engineering and Compuer Science, Washingon Sae Universiy, Pullman, WA , U.S.A. saberi@eecs.wsu.edu. 3 Deparmen of Elecrical Engineering, Eindhoven Universiy of Technology, P.O. Box 513, 5600 MB Eindhoven, The Neherlands, s.weiland@ue.nl.

2 2 and/or on he uni circle in he discree-ime case). These condiions on open-loop plans can be equivalenly saed as a requiremen ha he plan be asympoically null conrollable wih bounded conrol ANCBC). I is also a classical fac ha, in general, global inernal sabilizaion of ANCBC plans requires nonlinear feedback laws. See for insance. [10], [2], [16]. By weakening he noion of global inernal sabilizaion o a semiglobal framework, Saberi e al. showed ha ANCBC plans wih sauraing acuaors can be inernally sabilized in he semiglobal sense using only linear feedback laws. Such a relaxaion from a global o semiglobal framework is, from an engineering sandpoin, boh sensible and aracive. A body of work exiss proposing design mehodologies low-gain, low-high gain, and variaions) for boh coninuousand discree-ime ANCBC plans wih sauraing aciaors. See for insance [3], [17], [6], [5] Wih regard o exernal sabilizaion, he picure is complicaed. Unlike linear plans, inernal sabiliy does no necessarily guaranee exernal sabiliy when sauraion is presen. Hence, sabilizaion mus be done simulaneously in boh he inernal and exernal sense. There is a body of work ha examines he sandard noion of L p sabiliy for ANCBC plans wih sauraing acuaors when he exernal inpu disurbance) is inpu-addiive and a he same ime requiring eiher global or semi global inernal sabiliy in he absence of disurbnce, see [7]. For he more general case in which he disurbance is no necessarily inpu-addiive, here are surprising resuls in he lieraure which poin o he complexiy of he noion of exernal sabiliy for ANCBC plans wih sauraing acuaors. Noable among hese is he resul from [13] of Soorvogel e.al., ha for a double inegraor wih sauraing acuaors, any linear inernally sabilizing conrol law can achieve L p sabiliy wihou finie gain for p [1, 2], bu no linear inernally sabilizing conrol law can achieve L p sabiliy for any p 2, ]. Moreover, considering he recenly developed noion of inpu-o-sae sabiliy ISS) as a framework for simuaneous exernal and inernal sabiliy, such a double-inegraor sysem canno achieve ISS sabiliy wih a linear feedback law, see [8]. All hese resuls poin oward he delicacy of he exernal sabiliy concep for linear plans wih sauraing acuaors. Moreover, his lieraure also poins ou ha, for susained disurbances, he induced L norm does no yield a suiable problem formulaion. I conains mahemaical funcions ha are no reasonable models for disurbances, and i is impossible o ge a good sable response if we use hese funcions as models for disurbance. In fac one would need o consider a class of sensible susained disurbances from an engineering poin of view. All hese consideraions lead us o believe ha a suiable noion of exernal

3 3 sabiliy for linear plans wih sauraing acuaors, and indeed for general nonlinear sysems in he presence of disurbances and nonzero iniial condiions, is ye o be developed. In his paper we look a he simulaneous exernal and inernal sabilizaion of ANCBC plans wih sauraing acuaors when he exernal inpu is a sochasic disurbance. Specifically, we consider a linear ime-invarian sysem subjec o inpu sauraion, sochasic exernal disurbances and random Gaussian disribued iniial condiions, independen of he exernal disurbances. The aim will be o conrol his sysem by a possibly nonlinear saic sae feedback law ha achieves global asympoic sabiliy in he absence of disurbances, while guaraneeing a bounded variance of he sae vecor for all ime. This problem seems a naural exension of he resuls in [9], [12]. In [14], a resul was claimed. In his paper we give a complee proof for he case ha he sysem is neurally sable, i.e., he eigenvalues of he sysem marix of he coninuous-ime discree-ime) sysem are in he closed lef half plane open uni disc) and he eigenvalues on he imaginary axis uni circle) have equal algebraic and geomeric mulipliciy. The paper is organized as follows. For boh discree- and coninuous-ime sysems, a formal problem formulaion and he main resuls are saed in Secion II. A formal proof of he discreeime case for neurally sable sysems is given in Secion III. A discussion on he implicaions of he main resuls, some exensions and conclusions are colleced in Secion V. II. PROBLEM FORMULATION AND MAIN RESULTS A. The discree-ime case In discree ime, we consider sysems of he form xk +1)=Axk)+Bσuk)) + Ewk) 1) where he sae x, he conrol u and he disurbance w are vecor-valued signals of dimension n, m and l, respecively. Here, k Z +, w is a whie noise sochasic process wih variance Q and mean 0, he iniial condiion x 0 of 1) is a Gaussian random vecor independen of wk) for all k 0. Moreover σ is he sandard sauraion funcion given by: 1 u< 1 σu) = u 1 u 1 1 u>1

4 4 An admissible feedback is a nonlinear feedback of he form uk) =fxk)) 2) where f : R n R m is a coninuous map wih f0) = 0. We herefore consider nonlinear saic sae feedbacks. We will be ineresed in he following problem. Problem II.1 Given he sysem 1), he semiglobal exernal sochasic sabilizaion problem is o find an admissible feedback 2) such ha he following properies hold: 1) in he absence of he exernal inpu w, he equilibrium poin x =0of he conrolled sysem 1)-2) is globally asympoically sable. 2) he variance Varxk)) of he sae of he conrolled sysem 1)-2) is bounded over k 0. The fac ha conrollers exis ha achieve global asympoic sabiliy in he absence of disurbances as described in condiion 1) is well-known. The main objecive of his paper is o look a he addiional requiremen on he variance. The following is he main resul of his paper for discree ime sysems: Theorem II.2 Consider he sysem 1) and suppose ha A, B) is sabilizable while A is neurally sable, i.e. he eigenvalues of A are in he closed uni disc and he eigenvalues on he uni circle have equal geomeric and algebraic mullipliciy. In he above case, here exiss a linear feedback which solves he semiglobal exernal sochasic sabilizaion problem as defined in Problem II.1. We claim ha he condiion above is only sufficien and ha we can weaken he condiion of A neurally sable o he requiremen ha all eigenvalues of A are in he closed uni disc. However, his does, in general, require a nonlinear feedback. Conjecure II.1 Consider he sysem 1). Then for any given variance Q of he whie noise sochasic process w here exiss a feedback 2) which solves he semiglobal exernal sochasic sabilizaion problem as defined in Problem II.1 if and only if A, B) is sabilizable while he eigenvalues of A are in he closed uni disc.

5 5 B. The coninuous-ime case In coninuous ime we consider he sochasic differenial equaion dx) = Ax)d + Bσu))d + Edw) 3) where he sae x, he conrol u and he disurbance w are vecor-valued signals of dimension n, m and l, respecively. Here w is a Wiener process a process of l independen Brownian moions) wih mean 0 and rae Q, hais, Var[W )] = Q and he iniial condiion x 0 of 3) is a Gaussian random vecor which is independen of w. Is soluion x is rigorously defined hrough Wiener inegrals and is a Gauss-Markov process. Like in he discree ime case, σ denoes he sandard sauraion funcion and admissible feedbacks are possibly nonlinear saic sae feedbacks of he form 2) where f is a Lipschiz-coninuous mapping wih f0) = 0. Problem II.3 Given he sysem 3), he semiglobal exernal sochasic sabilizaion problem is o find an admissible feedback 2) such ha he following properies hold: 1) in he absence of he exernal inpu w, he equilibrium poin x =0of he conrolled sysem 3)-2) is globally asympoically sable. 2) he variance Varx)) of he sae of he conrolled sysem 3)-2) is bounded over 0. Like in he discree ime, he fac ha conrollers exis ha achieve global asympoic sabiliy in he absence of disurbances as described in condiion 1) is well-known. The main objecive of his paper is o look a he addiional requiremen on he variance. The following is he main resul of his paper for coninuous ime sysems: Theorem II.4 Consider he sysem 3) and suppose ha A, B) is sabilizable while A is neurally sable, i.e. he eigenvalues of A are in he closed lef half plane and he eigenvalues on he imaginary axis have equal geomeric and algebraic mullipliciy. Then for any given rae Q of he Wiener process w, here exiss a linear feedback which solves he semiglobal exernal sochasic sabilizaion problem as defined in Problem II.3.

6 6 We claim ha he condiion above is only sufficien and ha we can weaken he condiion of A neurally sable o he requiremen ha all eigenvalues of A are in he closed lef half plane. However, his does, in general, require a nonlinear feedback. Conjecure II.2 Consider he sysem 3). Then for any given variance Q of he whie noise sochasic process w) here exiss a feedback 2) which solves he semiglobal exernal sochasic sabilizaion problem as defined in Problem II.3 if and only if A, B) is sabilizable while he eigenvalues of A are in he closed lef half plane. III. PROOFS FOR THE DISCRETE-TIME CASE We firs presen a lile lemma ha we need laer: Lemma III.1 For any a R m and b R m and n>1 we have: σa + b) 2 ) 1 1 2n σa) 2 8n b 2 where σ is he sandard sauraion funcion. Proof: I is easily verified ha: σa + b) σa) b where he inequaliy is componenwise and b R m is defined as he absolue value of each componen of b. This implies: Using hen yields he required resul. σa + b) 2 σa) 2 2 b T σa)+ b 2 b T σa) 4n b n σa) 2 Using a simple basis ransformaion we can assume wihou loss of generaliy ha: A = A 11 0, B = B 1, E = E 1 0 A 22 B 2 E 2 wih A 11 asympoically sable while A T 22 A 22 = I. This is guaraneed by he fac ha A is neurally sable. Nex we consider he subsysem: x 2 +1)=A 22 x 2 )+B 2 u)+e 2 w) 4)

7 7 If we find an admissible feedback of he form u) =fx 2 )) 5) which solves he semiglobal sochasic sabilizaion problem for he sysem 4) hen i is easily verified ha his conroller also solves he semiglobal sochasic sabilizaion problem for he original sysem 1). In oher words we can resric aenion o he sysem 4). We will use a conroller of he form: uk) = B2 T A 22x 2 k) and we will esablish ha for any given variance Q of he whie noise here exiss small enough such ha he resuling sysem has a bounded variance for he sae. The choice of is independen of he variance of he iniial condiion x 0. Proof: [Proof of Theorem II.2] To keep noaion simple, we assume he original sysem saisfies A T A = I and we consider a feedback of he form: uk) = B T Axk) 6) The general case of he heorem can hen easily be esablished using he reducion o he sysem 4) as described earlier. If he sysem is no affeced by noise, i is well-known ha he feedback 6) achieves asympoic sabiliy which is also easily verified by noing ha V x) =x T x is a suiable Lyapunov funcion. To sudy he effec of he noise, we firs noe ha due o he Markov propery of he sae, we have: E [xk + r) T xk + r) xk + s)] = E [E [xk + r) T xk + r) xk + )] xk + s)] for r s. Nex we consider: E [xk +1) T xk +1) xk)]

8 8 Using he feedback 6), he dynamics 1) and he fac ha A T A = I we obain: E [xk +1) T xk +1) xk)] = xk) T xk) 2xk) T A T BσB T Axk)) + σb T Axk)) T B T BσB T Axk)) + race EQE T xk) T xk) 2 σbt Axk)) T σb T Axk)) + σb T Axk)) T B T BσB T Axk)) + race EQE T xk) T xk) 1 σbt Axk)) 2 + race EQE T 7) provided is small enough such ha B T B<I. Nex, we consider: [ σbt E A xk +1)) ] 2 xk) for some ineger 1. Wege: [ σbt E A xk +1)) ] 2 xk) [ σbt = E A +1 xk) B T A BσB T Axk)) + B T A Ewk)) ] 2 xk) 1 1 2n ) σbt A +1 xk) 2 8n E [ B T A BσB T Axk)) +B T A Ewk)) 2 xk) ] 1 1 2n ) σbt A +1 xk)) 2 8n 2 B T A B 2 σb T Axk)) B T A 2 race EQE T where he firs inequaliy follows from Lemma III.1. If we choose such ha B T B<I[as used earlier in deriving 7)] and 8n B T A 2 1, 8n B T A B 2 1 for all, hen i is easy o see ha for all < and any posiive ineger we have: [ σbt E A xk +1)) ] 2 xk) 1 1 2n ) σbt A +1 xk)) 2 3/2 σb T Axk)) 2 3/2 race EQE T The following lemma presens a crucial inequaliy:

9 9 Lemma III.2 We have for small enough ha E [xk + s) T xk + s) xk)] xk) T xk) 1 s =1 1 ) σbt A xk)) 2 + s 1+ s ) race EQE T 8) 2n n Proof: We will esablish his inequaliy recursively. Noe ha we already esablished his inequaliy for s =1given 7). Nex, consider any s>1 and assume he above inequaliy is rue for s and any k). Then we obain: E [xk + s +1) T xk + s +1) xk)] = E [E [xk + s +1) T xk + s +1) xk +1)] xk)] [ E xk +1) T xk +1) 1 s 1 ) σbt A xk +1)) 2 2n =1 +s 1+ s ) ] race EQE T xk) n Using he previously obained inequaliy we have: [ s E 1 ) σbt A xk +1)) 2 xk)] 2n =1 1 1 ) [ s 1 ) σbt A +1 xk)) ] 2 s 3/2 σb T Axk)) 2n 2n =1 s 3/2 race EQE T s 1 +1 ) σbt A +1 xk)) 2 s 3/2 σb T Axk)) s 3/2 race EQE T 2n =1

10 10 We hen obain: E [xk + s +1) T xk + s +1) xk)] = E [E [xk + s +1) T xk + s +1) xk +1)] xk)] xk) T xk) 1 σbt Axk)) 2 + race EQE T 1 s 1 +1 ) σbt A +1 xk)) 2 2n =1 xk) T xk) 1 + s σb T Axk)) 2 + s race EQE T + s s+1 =1 provided < is such ha 1+ s ) race EQE T n 1 ) σbt A xk)) 2 +s +1) 1+ s +1 ) race EQE T 2n n 1 n, s3/2 1 2n. Nex we noe ha since A, B) is conrollable while A is inverible, he marix B T A B T A 2. B T A n is injecive which implies ha here exiss α he smalles singular value of his marix divided by n) such ha for any x here exiss a posiive ineger n such ha and hence B T A x α x 1 { } σbt A x) 2 min α 2 x 2, 1 Bu hen 8) yields: } E [xk + n) T xk + n) xk)] xk) T xk) {α 1 min 2 xk) 2, 1 +2nrace EQE T 2 We define M = 1 2 α 2

11 11 and yk) =max{ xk) T xk),m} The following lemma presens anoher crucial resul: Lemma III.3 We have: E [yk + n) yk)] yk) 9) provided is small enough such ha provided is small enough such ha he inequaliy of Lemma III.2 is valid for s =1,...,n and: 1 2 > 2n race EQET and α 2 < 1 Moreover, have: E[yk + pn)] E[yk)], p =1, 2,... 10) Proof: This follows from he fac ha if yk) =xk) T xk) and hence xk) T xk) M we E [xk + n) T xk + n) xk)] } yk) {α 1 min 2 xk) 2, 1 +2nrace EQE T 2 yk) 1 +2nrace EQET 2 yk) while for he case yk) =M and hence xk) T xk) M we have: E [xk + n) T xk + n) xk)] } xk) T xk) {α 1 min 2 xk) 2, 1 +2nrace EQE T α2 )xk) T xk)+2n race EQE T α2 )yk)+2n race EQE T yk) 1 2 α2 M +2n race EQE T yk) bu hen using ha yk) M): E [yk + n) xk)] yk)

12 12 and finally, E [yk + n) yk)] = E [E [yk + n) xk)] yk)] E [yk) yk)] = yk) Similarlywherewehaveobecarefulsinceyk) is no Markov), E [yk +2n) yk)] = E [E [yk +2n) xk + n),xk)] yk)] = E [E [E [yk +2n) xk + n),xk)] xk)] yk)] = E [E [E [yk +2n) xk + n)] xk)] yk)] E [E [yk + n) xk)] yk)] E [yk) yk)] = yk) Using a simple recursion, we obain 10). We have, using 7): E[xk +1) T xk +1) xk)] xk) T xk) + race EQE T which yields: E[xk) T xk)] Var[x0)] + k race EQE T for k =1,...,n 1 and hence: E[yk)] max { M,Var[x0)] + k race EQE T } for k =1,...,n 1. This implies ha he expecaion of y0),...,yn 1) is bounded. Using 10) we find ha he expecaion of yk) is bounded in k. Usinghayk) xk) T xk) his yields ha he variance of xk) is bounded in k. IV. PROOFS FOR THE CONTINUOUS-TIME CASE Using a simple basis ransformaion we can assume wihou loss of generaliy ha: A = A 11 0, B = B 1, E = E 1 0 A 22 B 2 E 2

13 13 wih A 11 asympoically sable while A 22 + A T 22 =0. This is guaraneed by he fac ha A is neurally sable. Nex we consider he subsysem: dx 2 ) =A 22 x 2 )d + B 2 σu))d + E 2 dw) 11) If we find an admissible feedback of he form u) =fx 2 )) 12) which solves he semiglobal exernal sochasic sabilizaion problem for he sysem 11) hen i is easily verified ha his conroller also solves he semiglobal sochasic sabilizaion problem for he original sysem 3). In oher words we can resric aenion o he sysem 11). We will use a conroller of he form: u) = B2x T 2 ) and we will esablish ha for any given rae Q of he Wiener process here exiss small enough such ha he resuling sysem has a bounded variance for he sae. The choice of is independen of he variance of he iniial condiion x 0. Proof: [Proof of Theorem II.4] To keep noaion simple, we assume he original sysem saisfies A + A T =0and we consider a feedback of he form: u) = B T x) 13) The general case of he heorem can hen easily be esablished using he reducion o he sysem 11) as described earlier. If he sysem is no affeced by noise, i is well-known ha he feedback 13) achieves asympoic sabiliy which is also easily verified by noing ha V x) =x T x is a suiable Lyapunov funcion. To sudy he effec of he noise, we firs noe ha: τ τ xτ) =e Aτ ) x)+ e Aτ s) Bσus))ds + e Aτ s) Edws) = e Aτ ) x)+v τ) We noe ha σu) is bounded by 1 and he second inegral has bounded variance for all τ such ha τ [, +1]. This implies ha here exiss a consan N such ha: Var[v τ)] <N 14)

14 14 for all τ such ha τ [, +1]. Nex, we noe ha σb T xτ)) 2 = σ B T e Aτ ) x)+b T v τ) ) σbt e Aτ ) x)) B 2 v τ) 2 15) where we applied Lemma III.1 wih n =1. Nex we noe ha: dx T x =2x T Bσu) + race E T E)d +2x T Edw using ha A + A T =0and Iô s lemma. This yields ha x +1) T x +1) x) T x) = +1 2xτ) T Bσuτ))dτ + race E T E xτ) T Edwτ) and hence: E[x +1) T x +1) x)] On he oher hand, we have: = x) T x)+e +1 2xτ) T Bσuτ))dτ x) + race E T E Combining he las wo equaions we obain: 2xτ) T Bσuτ)) = 2xτ) T Bσ B T xτ)) 2 σbt xτ)) 2 E[x +1) T x +1) x)] x) T x) + race E T E 2 E +1 σb T xτ)) 2 dτ x) Using 14), 15) and he fac ha x) and v τ) are independen for τ [, +1] we obain a crucial inequaliy: +1 E[x +1) T x +1) x)] x) T x)+v 1 σb T e Aτ ) x)) 2 dτ 16)

15 15 where V =16 B 2 N + race E T E. Lemma IV.1 Assume ha A, B) is conrollable. Then here exis suiably defined consans α and R such ha for all < we have ha: 1 +1 σbt e Aτ ) x)) 2 dτ min{α x) 2, R} Proof: If he sauraion does no ge acivaed and hence we ge: +1 σbt e Aτ ) x)) 2 dτ = 2 x) T Qx) where: Q = 1 e Aτ BB T e Aτ dτ Conrollabiliy of A, B) implies ha Q is inverible and hence we obain: 0 x)qx) α x) 2 where α is he smalles singular value of Q. Hence we obain: 1 +1 σbt e Aτ ) x)) 2 dτ α x) 2 17) On he oher hand, in case he sauraion does ge acivaed hen i is easily seen ha here mus exis an N independen of ) such ha: We obain: +1 We consider he inegral: x) > N σbt e Aτ ) x)) 2 dτ +1 σ B T e +1 σ B T e Aτ ) Nx) x) Aτ ) Nx) x) ) 2 dτ ) 2 dτ 18) In case sauraion does no arise in his inegral we immediaely find ha he erm is larger or equal o 2 α Nx) x) 2 αn 2

16 16 using he same echniques as for he case ha he original inegral did no conain a erm which sauraes. Bu hen we find 1 +1 σbt e Aτ ) x)) 2 dτ 1 αn 2 19) Finally we look a he case where he sauraion in he inegral 18) does ge acivaed. In ha case we noe ha here exiss τ such ha: ) σ B T Aτ ) Nx) 2 e 1 x) Bu he derivaive wih respec o τ of his erm is bounded by some consan γ independen of our choice for τ [, +1] and independen of our choice for x). Thisimpliesha This yields σ B T e Aτ ) Nx) σbt e Aτ ) x)) 2 dτ 1 x) +1 ) 2 dτ 1 γ σ B T e Aτ ) Nx) x) ) 2 dτ 1 γ Combining 17) wih 19) and 20), he resul of he lemma follows immediaely for R =min{n 2, 1 γ }. 20) The above lemma yields, wih he help of 16), ha: E[x +1) T x +1) x)] x) T x) min{α x) 2, R } + V We define: and M = R 2 α y) =max{ x) T x),m} The following lemma presens anoher crucial resul:

17 17 Lemma IV.2 We have: E [y +1) y)] y) 21) provided is small enough such ha: < R V and α < 1. Moreover, have: Proof: E[y + p)] E[y)], p =1, 2,... 22) This follows from he fac ha if y) =x) T x) and hence x) T x) M we E [x +1) T x +1) x)] { } y) min α x) 2, R + V y) R + V y) while for he case y) =M and hence x) T x) M we have: bu hen using ha y) M): and finally, E [x +1) T x +1) x)] { } x) T x) min α x) 2, R + V 1 α)x) T x)+v 1 α)y)+v y) αm + V y) E [y +1) x)] y) E [y +1) y)] = E [E [y +1) x)] y)] E [y) y)] = y)

18 18 The inequaliy 22) hen follows using he same argumens as in he proof of Lemma III.3. I is easy o esablish ha: E[y)] is bounded on he inerval [0, 1]. Combined wih 22) we find ha he variance of y) is bounded. Using ha y) x) T x) his yields ha he variance of x) is bounded in. V. DISCUSSION AND CONCLUSIONS This paper is he coninuaion of our earlier effors oward undersanding and suiably formulaing he noion of simulaneous exernal and inernal sabilizaion for linear plans wih sauraing acuaors. Here we have shown ha i is possible o conrol an open-loop criically sable linear plan in coninuous or discree ime, which is subjec o inpu sauraion, sochasic exernal disurbances and Gaussian-disribued iniial condiions, via a linear sae feedback law ha achieves global asympoic sabiliy of he closed-loop sysem in he absence of disurbances, while guaraneeing bounded variance of he sae vecor for all ime in he presence of disurbances. We have ermed his conrol problem he exernal semi-global sabilizaion problem. Furhermore, we have conjecured ha such a problem is also solvable for open-loop criically unsable linear plans, albei via a nonlinear sae feedback law. REFERENCES [1] D.S. BERNSTEIN AND A.N. MICHEL, Gues Eds., Special Issue on sauraing acuaors, In. J. Robus & Nonlinear Conrol, 55), 1995, pp [2] A.T. FULLER, In-he-large sabiliy of relay and sauraing conrol sysems wih linear conroller, In. J. Conr., 104), 1969, pp [3] P. HOU, A. SABERI, Z. LIN, AND P. SANNUTI, Simulaneously exernal and inernal sabilizaion for coninuous and discree-ime criically unsable sysems wih sauraing acuaors, Auomaica, 3412), 1998, pp [4] V. KAPILA AND G. GRIGORIADIS, Eds., Acuaor sauraion conrol, Marcel Dekker, [5] Z. LIN AND A. SABERI, Semi-global exponenial sabilizaion of linear discree-ime sysems subjec o inpu sauraion via linear feedbacks, Sys. & Conr. Leers, 24, 1995, pp [6] Z. LIN, A. SABERI, AND A.A. STOORVOGEL, Semi-global sabilizaion of linear discree-ime sysems subjec o inpu sauraion via linear feedback - an ARE-based approach, IEEE Trans. Au. Conr., 418), 1996, pp [7] A. SABERI, P. HOU, AND A.A. STOORVOGEL, On simulaneous global exernal and global inernal sabilizaion of criically unsable linear sysems wih sauraing acuaors, IEEE Trans. Au. Conr., 456), 2000, pp [8] G. SHI AND A. SABERI, On he inpu-o-sae sabiliy ISS) of a double inegraor wih sauraed linear conrol laws, in Proceedings of he American Conrol Conference, Anchorage, Alaska, 2002, pp

19 19 [9] G. SHI, A.SABERI, AND A.A. STOORVOGEL, OnL p l p) performance wih global inernal sabiliy for linear sysems wih acuaors subjec o ampliude and rae sauraion, in American Conrol Conference, Chicago, IL, 2000, pp [10] E.D. SONTAG AND H.J. SUSSMANN, Nonlinear oupu feedback design for linear sysems wih sauraing conrols, in Proc. 29h CDC, Honolulu, 1990, pp [11] A. SABERI AND A.A. STOORVOGEL, Gues Eds., Special Issue on conrol problems wih consrains, In. J. Robus & Nonlinear Conrol, 910), 1999, pp [12] A.A. STOORVOGEL,A.SABERI, AND G. SHI, On achieving L p l p) performance wih global inernal sabiliy for linear plans wih sauraing acuaors, in Proc. 38h CDC, Phoenix, AZ, 1999, pp [13] A.A. STOORVOGEL, G. SHI, AND A. SABERI, Exernal sabiliy of a double inegraor wih sauraed linear conrol laws, Dynamics of Coninuous Discree and Impulsive Sysems, Series B: Applicaions & Algorihms, 114-5), 2004, pp [14] A.A. STOORVOGEL,S.WEILAND, AND A. SABERI, On sabilizaion of linear sysems wih sochasic disurbances and inpu sauraion, in Proc. 43rd CDC, The Bahamas, 2004, pp [15] S. TARBOURIECH AND G. GARCIA, Eds., Conrol of uncerain sysems wih bounded inpus, vol. 227 of Lecure noes in conrol and informaion sciences, Springer Verlag, [16] A.R. TEEL, Global sabilizaion and resriced racking for muliple inegraors wih bounded conrols, Sys. & Conr. Leers, 183), 1992, pp [17] Y. YANG, Global sabilizaion of linear sysems wih bounded feedback, PhD hesis, Rugers Universiy, New Brunswick, 1993.