A Limiting Property of the Matrix Exponential
|
|
|
- Elisabeth Quinn
- 5 years ago
- Views:
Transcription
1 A Limiting Property of the Matrix Exponential Sebatian Trimpe Student Member IEEE and Raffaello D Andrea Fellow IEEE K(α u(t ẋ f(t ẋ (t xf(t x (t A11 A 12 Abtract A limiting property of the matrix exponential i proven: if the (11-block of a 2-by-2 block matrix become arbitrarily mall in a limiting proce the matrix exponential of that matrix tend to zero in the (11- (12- and (21-block The limiting proce i uch that either the log norm of the (11-block goe to negative infinity or for a certain polynomial dependency the matrix aociated with the larget power of the variable that tend to infinity i table The limiting property i ueful for implification of dynamic ytem that exhibit mode with ufficiently different time cale The obtained limit then implie the decoupling of the correponding dynamic Index Term Matrix exponential limiting property logarithmic norm time-cale eparation I INTRODUCTION The ubject of tudy in thi paper i the matrix exponential ( A11 K(α A exp 12 t t > (1 in the limit a K(α grow large for α in ome ene to be made precie later All matrice are complex and α i a real parameter For different clae of K(α we derive ufficient (and in one cae alo neceary condition on K(α uch that for all t > ( A11 K(α A 12 lim exp t e A22t (2 That i we are intereted in condition guaranteeing that the coupling block (12 and (21 vanih (in addition to the (11- block In addition to being an intereting matrix problem the reult can be applied to control ytem that exhibit ignificantly different time cale uch a ytem with high-gain feedback on ome tate For example conider the ytem ẋ f (t A 11 x f (t+a 12 x (t+u(t (3 ẋ (t A 21 x f (t+a 22 x (t (4 with tatic feedback on the tate x f (t (index f for fat and for low u(t K(αx f (t (5 The matrix function K(α then repreent the feedback gain parametrized by α The feedback ytem i depicted in Fig 1 A more general multi-loop feedback ytem with additional reference input i conidered in 1 The matrix exponential (1 i a fundamental matrix (ee eg 2 of the feedback ytem (3 (5 The limit (2 mean that the dynamic of x f (t and x (t are decoupled in the limit S Trimpe i with the Max Planck Intitute for Intelligent Sytem 7276 Tübingen Germany ( trimpe@tuebingenmpgde R D Andrea i with the Intitute for Dynamic Sytem and Control ETH Zurich 892 Zurich Switzerland ( rdandrea@ethzch Thi work wa upported by the Swi National Science Foundation (SNSF Fig 1 Linear ytem with feedback on the firt part of the tate vector the fat tate x f a K(α grow large In thi context we eek to determine what type of feedback yield a decoupling of the tate in feedback from the remaining one in the limit a the feedback gain become arbitrarily large Thi quetion i of interet for example when deigning multi-loop control ytem with high-gain inner loop ince a decoupling of the tate allow for a implified ytem decription and hence a implified control deign The matrix reult herein i applied in 1 to derive a time-cale eparation algorithm for a cacaded control ytem with high-gain inner feedback loop The algorithm yield a ytem decription that include the plant dynamic and the effect of the inner feedback loop The obtained repreentation i ueful for example for deigning an outer-loop controller Thi methodology i applied in the deign of a cacaded feedback control ytem for an inverted pendulum in 1 and for a balancing cube (a multi-body 3-D inverted pendulum in 3 Related to the problem tudied herein i the work by Campbell et al 4 5 The author conider the matrix exponential with it argument being a polynomial in 1/ε and derive condition for it convergence in the limit a ε + In 4 for example Campbell et al preent a neceary and ufficient condition for pointwie convergence of exp((a+b/εt t > (6 a ε + While they are intereted in general convergence to ome limit we eek condition that yield the particular limit (2; that i where the cro coupling block (12 and (21 vanih Before deriving the technical reult thi article continue with notation and preliminarie in Sec II In Sec III we etablih lemma and cite theorem that are required for the development of the main reult which follow in Sec IV The main reult are three different condition on K(α that guarantee (2: a ufficient condition that i baed on the log norm (defined in (8 of the next ection of K(α and that make no prior aumption on the function type of K(α (Theorem 3; a neceary and ufficient condition for the cae when K(α i affine (Theorem 4; and another ufficient condition for the cae when K(α ha an affine term and an additional polynomial term α r r 2 (Theorem 5 The latter two reult are baed on 4 5 while the firt one i etablihed independently of thoe Numerical example illutrating the applicability of the different theorem are given throughout in Sec IV The article conclude with remark in Sec V A preliminary verion of Theorem 3 wa firt publihed in 1 Accepted final verion Publihed in: IEEE Tranaction on Automatic Control vol 59 no 4 Apr 214 DOI: 1119/TAC c 214 IEEE Peronal ue of thi material i permitted Permiion from IEEE mut be obtained for all other ue in any current or future media including reprinting/republihing thi material for advertiing or promotional purpoe creating new collective work for reale or reditribution to erver or lit or reue of any copyrighted component of thi work in other work
2 II NOTATION AND PRELIMINARIES We ue R C and R + to denote real number complex number and nonnegative real number repectively For the derivation in the paper we work excluively with the vector 2-norm and it induced matrix norm; that i for x C n and A C pn ( n x x i 2 1/2 A max Ax (7 i1 x 1 For A C nn µ(a denote the log norm of A 6 8: µ(a : max{µ µ an eigenvalue of(a+a /2} (8 where A i the conjugate tranpoe of A We hall exploit the following propertie of µ(a 6 8: e At e µ(at (9 µ(a A (1 µ(a+b µ(a+ B (11 where AB C nn and t R + Let pec(a denote the pectrum of A C nn (the et of all eigenvalue of A ignoring algebraic multiplicity and let OLHP denote the open left half plane in C (ie OLHP : {x C : Rex < } 7 The matrix A i called table if pec(a OLHP; and it i called emitable if pec(a OLHP {} and if pec(a then i emiimple (ie it algebraic and geometric multiplicity are identical 7 The index of A denoted Index A i the mallet nonnegative integer j uch that ranka j ranka j+1 7 The Darzin invere of A i the unique matrix A D atifying AA D A D A A D AA D A D anda j+1 A D A j withj IndexA 7 9 For AB C nn define A;B : (I B D BA(I B D B 5 where I i the identity matrix The following two fact are ueful in later derivation; the proof are traightforward and therefore omitted Fact 1: Conider the matrix differential equation Ż(t AZ(t+BU(t t Z( Z (12 where Z : R + C np continuouly differentiable U : R + C mp continuou A C nn B C nm and Z C np The unique olution of (12 i Z(t e At Z + e A(t τ BU(τdτ (13 Fact 2: Let A : ab C nm be continuou Then b b A(t dt A(t dt (14 a a III LEMMAS Thi ection etablihe preliminary lemma and retate two theorem from 4 5 which are ued in Sec IV to prove the main reult of thi paper Conider the matrix ordinary differential equation (ODE: Ẋ(t (A 11 K(αX(t+A 12 Y(t X( X (15 Ẏ(t A 21 X(t+A 22 Y(t Y( Y (16 with X : R + C np and Y : R + C mp continuouly differentiable and complex matrice A 11 A 12 A 21 A 22 K(α X and Y of appropriate dimenion Notice that the matrix exponential (1 i a fundamental matrix of the ODE ytem given by (15 and (16 which i why the tudy of (15 (16 will be ueful in the later development By Fact 1 the unique olution to (15 and (16 (conidered individually are for all t X(t e (A11 K(αt X + Y(t e A22t Y + e (A11 K(α(t τ A 12 Y(τdτ (17 e A22(t τ A 21 X(τdτ (18 Lemma 2 and 3 below treat the olution (17 and (18 for different initial condition in the limit a the log norm of K(α tend to negative infinity To etablih thee two lemma the following Gronwall-type inequality i ued: Lemma 1 (adapted from 1 Theorem 19: Let v(t a(t b(t be real-valued nonnegative continuou function on J t t 1 Let κ(t be a real-valued nonnegative continuou function for t t t 1 and uppoe Then v(t a(t+b(t t κ(tv(d t J v(t ā(texp ( b(t κ(td t J t where ā(t : up τ tta(τ b(t : up τ ttb(τ and κ(t : up τ t κ(τ Lemma 2: Conider the olution (17 and (18 with the initial condition X I and Y If lim µ( K(α then for t > lim X(t and lim Y(t (19 Proof: Since lim µ( K(α there exit an α R uch that for all α α µ(a 11 K(α A 11 +µ( K(α < (2 µ(a 11 K(α A 22 < 1 (21 In the following we conider ufficiently large α uch that α α Subtituting (17 into (18 and uing the initial condition X I and Y yield Y(t + e A22(t τ A 21 e (A11 K(ατ dτ τ + e A22(t τ A 21 e (A11 K(α(τ A 12 Y(ddτ e A22(t τ A 21 e (A11 K(ατ dτ e A22(t τ A 21 e (A11 K(α(τ A 12 Y(dτ d (22 where the order of integration in the lat term wa interchanged Thi i valid by Fubini theorem 11 Prop 536
3 and the fact that the integrand i continuou and the integration region can be expreed in either of the two way: {(τ : τ t τ} or {(τ : t τ t} Uing (9 (1 Fact 2 and ubmultiplicativity of the induced matrix norm we obtain the inequality Y(t A 21 A 21 a(t+ where e A22(t τ e (A11 K(ατ dτ + A 21 A 12 e A22(t τ e (A11 K(α(τ Y( dτ d e A22 (t τ e µ(a11 K(ατ dτ + A 21 A 12 e A22 (t τ e µ(a11 K(α(τ Y( dτ d (23 a(t : A 21 κ(t Y( d (24 κ(t : A 21 A 12 e A22 (t τ e µ(a11 K(ατ dτ e A22 (t τ e µ(a11 K(α(τ dτ Applying Lemma 1 to (24 yield for all t ( Y(t ā(texp κ(td (25 where ā(t up τ t a(τ and κ(t up τ t κ(τ Next we derive bound for a(t ā(t and κ(t κ(t uing the propertie (2 (21 Firt a(t A 21 e A22 t e (µ(a11 K(α A22 τ dτ A 21 ( e A 22 t e µ(a11 K(αt ξ(α }{{} (1 by (2 A 21 ξ(α e A22 t M 1(t ξ(α where ξ(α : A 22 µ(a 11 K(α > 1 by (21 and M 1 (t : A 21 e A22 t i a continuou function in t Therefore A 21 ā(t up a(τ up τ t τ t ξ(α e A22 τ A 21 ξ(α e A22 t M 1(t ξ(α (26 Similarly we obtain a bound for κ(t With t κ(t A 21 A 12 e A22 t e µ(a11 K(α e (µ(a11 K(α A22 τ dτ A 21 A 12 ( e A 22 (t e µ(a11 K(α(t ξ(α }{{} (1 by (2 A 21 A 12 e A22 t M 2(t ξ(α ξ(α where M 2 (t : A 21 A 12 e A22 t i a continuou function in t Therefore κ(t up κ(τ up τ t τ t A 21 A 12 e A22 τ ξ(α A 21 A 12 e A22 t M 2(t ξ(α ξ(α (27 With (26 and (27 we can now bound (25 Y(t M ( 1(t ξ(α exp M 2 (t ξ(α d M ( 1(t ξ(α exp M2 (t ξ(α t M 1(t ξ(α etm2(t M(t ξ(α (28 where M(t : M 1 (te tm2(t i continuou Since lim ξ(α lim Y(t follow directly from (28 Furthermore with (17 and X I X(t e µ(a11 K(αt + A 12 e µ(a11 K(αt + A 12 ξ(α e } µ(a11 K(α(t τ {{} (1 Y(τ dτ M(τdτ (29 Therefore lim X(t for t > Lemma 3: Conider the olution (17 and (18 with the initial condition X and Y I If lim µ( K(α then for t > lim X(t and lim Y(t ea22t (3 Proof: The proof i eentially analogou to the proof of Lemma 2 Let α α uch that (2 and (21 hold Subtituting (18 into (17 and uing the initial condition X and Y I yield after interchange of integration in the econd term X(t + e (A11 K(α(t τ A 12 e A22τ dτ and therefore X(t A 12 e (A11 K(α(t τ A 12 e A22(τ A 21 X(dτ d e µ(a11 K(α(t τ e A22 τ dτ + A 12 A 21 e µ(a11 K(α(t τ e A22 (τ X( dτ d (31 Now conider the ubtitution τ t τ for the firt term in (31 and τ t+ τ for the inner integral of the econd term which yield X(t A 12 e A22 (t τ e µ(a11 K(ατ dτ + A 12 A 21
4 e A22 (t τ e µ(a11 K(α(τ X( dτ d (32 Comparing thi inequality to (23 we find that (32 i obtained from (23 by the ubtitution Y( X( A 12 A 21 and A 21 A 12 Therefore we can derive an upper bound on X(t the ame way a in the proof of Lemma 2 Correponding to (28 we get for all t X(t L(t ξ(α where the continuou function L(t i obtained from M(t by ubtituting A 12 A 21 and A 21 A 12 Thu lim X(t Furthermore with (18 and Y I Y(t e A22t A 21 A 21 ξ(α e A22 (t τ X(τ dτ e A22 (t τ L(τdτ Therefore lim Y(t e A22t and hence lim Y(t e A22t Lemma 2 and 3 are ued in the next ection to etablih one of the main reult of thi note (Theorem 3 The other two reult (Theorem 4 and 5 preented in the next ection are baed on 4 5 and in particular on: Theorem 1 (4 Theorem 1: Let AB C nn Then e (A+B/εt converge pointwie a ε + for t > if and only if B i emitable If B i emitable then lim e (A+B/εt e (I BB D At (I BB D (33 ε + Theorem 2 (5 Theorem 1: Suppoe IndexC 1 and C i emitable Then e (A+B/ε+C/εr t r > 1 (34 converge a ε + for an r 2 for all t > if and only if B;C i emitable Suppoe B;C i emitable If r > 2 then (34 converge to e A;C;B;Ct (I B;C D B;C(I C D C; (35 if r 2 then the limit of (34 i the ame a (35 except that a term i added into the exponential BC D B;C;B;Ct (36 IV MAIN RESULTS Thi ection etablihe condition on K(α that guarantee (2 In Sec IV-A a ufficient condition i preented that i baed on the log norm of K(α (Theorem 3 In Sec IV-B and IV-C we conider the cae when K(α ha a particular polynomial tructure; namely K(α K +αk 1 and (37 K(α K +αk 1 +α r K 2 r 2 (38 repectively For the affine cae (37 a neceary and ufficient condition i derived (Theorem 4; and for (38 we preent a ufficient condition (Theorem 5 Following each theorem we give numerical example in order to illutrate the applicability of the different reult If K(α repreent a feedback controller gain uch a in (5 equation (37 and (38 decribe explicit parametrization of the controller gain via the calar tuning parameter α If one eek to analyze a controller parametrization that i not explicitly given a a function of α the log norm condition can be ueful a hall be illutrated later in Example 1 The pecific functional dependencie conidered in (37 and (38 (affine and polynomial correpond to thoe that are alo tudied in 4 5 (therein a polynomial in 1/ε cf Theorem 1 and 2 A Condition Baed on the Log Norm of K(α A ufficient condition for (2 i the log norm of K(α becoming arbitrarily mall Thi reult i obtained by conidering the matrix ODE that i olved uniquely by the matrix exponential (1 and then applying Lemma 2 and 3 of Sec III Theorem 3: Let A A 11 A 12 C (n+m(n+m and let K : R C nn be a matrix function of the real parameter α If lim µ( K(α then (2 hold for all t > Proof: By Fact 1 the matrix exponential ( A11 K(α A X(t : exp 12 i the unique olution to the matrix ODE X(t t A11 K(α A 12 X(t t X( I (39 Note that X : R + C (n+m(n+m i continuouly differentiable By ubdividing X(t into block matrice of appropriate dimenion X11 (t X X(t 12 (t X 21 (t X 22 (t we can write (39 equivalently a X 11 (t A11 K(α A X 12 X11 (t 21 (t X 21 (t X 12 (t A11 K(α A X 12 X12 (t 22 (t A 22 X 22 (t A 21 X11 ( X 21 ( X12 ( X 22 ( I (4 I (41 Notice that (4 and (41 repreent the matrix ODE conidered in Lemma 2 and 3 repectively Uing thee two lemma we therefore conclude that for t > lim exp lim ( A11 K(α A 12 X11 (t X 12 (t X 21 (t X 22 (t t e A22t We illutrate Theorem 3 with the following example Example 1 (Dicretization and high-gain feedback: Conider the feedback control example from the introduction given by (3 (5 and aume a diagonal tructure for the feedback gain K(α with diagonal element k i (α Suppoe
5 we are intereted in a dicrete-time decription of the cloedloop ytem (3 (5 at a rate T > The dicretized ytem read xf (t+t x (t+t ( A11 K(α A exp 12 T xf (t (42 x (t Now aumek i (α α that i the individual controller gain are at leat a large a α Then lim µ( K(α lim max ( ki (α lim α i (43 and by Theorem 3 (42 become in the limit a x f (t+t (44 x (t+t e A22T x (t; (45 that i the low and fat dynamic are decoupled In 1 and 3 p 65 Theorem 3 i applied to control ytem that are more general than (3 (5 where in particular the controller (5 i modified to track a reference input changing at the rate T The obtained dicrete-time model of the control ytem i then ued to deign an outer-loop controller that command the reference input to the modified controller (5 which then act a the inner-loop controller of the cacaded control ytem Remark 1: Note that the function K(α i not given explicitly in Example 1 The etimate k i (α α with the diagonal tructure of K(α i enough to verify the condition of Theorem 3 In contrat the convergence reult in Sec IV-B and IV-C require an explicit decription of K(α Example 2: Conider (1 with A 1 α α 2 K(α α and t 1 Notice that K(α i table for all α > (both eigenvalue are α and that both eigenvalue go to negative infinity a α But lim µ( K(α lim max{ α α2 α 1 2 α2 } and the limit of (1 for α i (can be computed uing 7 Fact which i clearly different from (2 in the (12-block Remark 2: The preceding example how that it doe not uffice for (2 to hold that the eigenvalue of K(α tend to negative infinity B K(α Affine We tudy the limit of (1 with affine K(α a in (37 The following reult provide a neceary and ufficient condition for (2 It i obtained uing Theorem 1 Theorem 4: Let A A 11 A 12 C (m+n(m+n and let K(α K +αk 1 with K K 1 C nn and α R Then (2 hold for t > if and only if K 1 i table Proof: We firt prove ufficiency Let A11 K Ã : A 12 B : K1 (46 Since K 1 i table B i emitable and it follow from Theorem 1 (by ubtituting 1/ε with α that lim e(ã+α Bt lim e (Ã+ B/εt (I B B e D Ãt (I B B D ε + (47 Since K 1 i table it i invertible and B D K 1 1 Hence we have ( e (I B B D Ãt A11 K exp A 12 I ( exp t t I e A22t (48 where the lat equality follow from 7 Fact and i a placeholder left unpecified Therefore we get from (47 lim e(ã+α Bt e (I B B D Ãt (I B B D (49 I e A22t I e A22t (5 which complete the ufficiency part of the proof For the neceity proof aume (2 hold Firt notice that for the limit lim e (Ã+α Bt lim ε + e (Ã+ B/εt to exit it follow from Theorem 1 that B i emitable From the definition of B in (46 it can be een that thi implie that K 1 i emitable which further implie that pec( K 1 OLHP {} (51 From B being emitable and Theorem 1 it follow that (47 hold Hence the limit in (47 i equal to the limit in (2: e (I B B D Ãt (I B B D e A22t (52 Now let E11 E E 12 : e (I B B D Ãt (53 E 21 E 22 Uing B D K D 1 and (53 it follow from (52 (by conidering the firt block column that E11 (I K 1 K1 D (54 E 21 Since the matrix exponential i noningular 7 Prop 1128 E in (53 i noningular and E 11 E 21 ha full column rank Therefore (54 implie (I K 1 K1 D K 1 K1 D I From thi and the rank formula 7 Lemma 252 n rank(i rank(k 1 K1 D min{rank(k 1 rank(k1} D n it follow that K 1 ha full rank Thu alo K 1 ha full rank which implie / pec( K 1 7 Cor 266 Prop 552 Thi and (51 imply pec( K 1 OLHP ie K 1 i table Example 3: Conider 1 2 K(α αk 1 with K 1 1 Then K 1 i table and by Theorem 4 (2 hold for any A Remark 3: For K(α a in Example 3 we compute lim µ( K(α lim max{ 2α} Example 3 hence how that the condition in Theorem 3 i not a neceary condition
6 C K(α Polynomial We tudy the limit of (1 with K(α a in (38; that i compared to (37 K(α poee an additional power α r with r 2 A ufficient condition for convergence i derived uing Theorem 2 The condition i different from the ufficient condition in Theorem 3 (one doe not imply the other a hall be pointed out later Theorem 5: Let A A 11 A 12 C (m+n(m+n and let K(α K + αk 1 + α r K 2 with K K 1 K 2 C nn and αr R r 2 If K 2 i table then (2 hold for t > Proof: Let à B be a in (46 and let C : K2 Since K 2 i table C i emitable Furthermore Index C 1 ince rank(k2 2 rank( K 2 ( K 2 ha full rank Thu the aumption of Theorem 2 are atified With C D K 1 2 we get (I CD C I and B; C (I C D C B(I CD C which i emitable Therefore by Theorem 2 lim e (Ã+α B+α r Ct lim ε + e (Ã+ B/ε+ C/ε r t converge to the limit pecified by (35 and (36 where A B C are replaced by à B C We next compute the expreion (35 and (36 From B; C we get (I B; C D B; C I D I Furthermore Ã; C I A11 K A 12 I A 22 and hence Ã; C; B; C Ã; C (55 A 22 Uing thee reult expreion (35 yield the deired limit in (2 e Ã; C; B; Ct (I B; C D B; C(I C D C e A22t Since B C D K1 K B; C 2 1 K 1 I I expreion (36 i Example 4: Conider K(α αi +α 2 K 2 with K Then K 2 i table and by Theorem 5 (2 hold for any A Notice that the intability of K 1 I i irrelevant From lim µ( K(α lim max{α+ 2α2 5 α 3α2 5 } we ee that Theorem 3 i not helpful here Example 5: Conider K(α αk 1 +α r K 2 with 1 < r < and K 1 K 1 2 Theorem 5 i not helpful here ince r < 2 (neither i Theorem 2 But µ( K(α max{ α2α α r } a α ; hence (2 follow from Theorem 3 Remark 4: Example 4 and 5 how that there are problem with K(α K +αk 1 +α r K 2 which are covered by Theorem 3 but not by Theorem 5; and vice vera In general both theorem provide ufficient condition for different problem clae V CONCLUDING REMARKS The three theorem preented in thi technical note guarantee the limiting property (2 of the matrix exponential (1; eentially a large enough K(α in the (11-block force all but the (22-block of the matrix exponential to tend to zero in the limit Theorem 3 tate a ufficient condition for (2 baed on the log norm of K(α and Theorem 4 and 5 provide ufficient condition for K(α having a particular polynomial form For the affine cae (Theorem 4 the obtained condition i alo neceary Theorem 4 and 5 herein are obtained uing the reult by Campbell et al 4 5 Theorem 3 however i obtained independently of thoe reult It method of proof i baed on the matrix differential equation that i olved uniquely by the matrix exponential (1 and on bounding it olution uing a Gronwall-type inequality In contrat Campbell et al make ue of Cauchy integral formula to prove their reult in 4 for example The numerical example herein were choen to highlight pecific mathematical propertie of the reult The limiting property of the matrix exponential ha been applied to practical example in 1 (an inverted pendulum and 3 (a balancing cube Therein Theorem 3 i ued to derive a time-cale eparation algorithm that compute a dicrete-time model repreenting the plant dynamic under high-gain feedback on ome of the plant tate Thi model i then ued to deign a tabilizing outer-loop controller REFERENCES 1 S Trimpe and R D Andrea A limiting property of the matrix exponential with application to multi-loop control in Proc of the Joint 48th IEEE Conference on Deciion and Control and 28th Chinee Control Conference Shanghai PR China Dec 29 pp P Hartman Ordinary Differential Equation 2nd ed Philadelphia: Society for Indutrial and Applied Mathematic 22 3 S Trimpe and R D Andrea The Balancing Cube: A dynamic cuplture a tet bed for ditributed etimation and control IEEE Control Sytem Magazine vol 32 pp Dec S L Campbell and N J Roe Singular perturbation of autonomou linear ytem SIAM Journal on Mathematical Analyi vol 1 no 3 pp S L Campbell Singular perturbation of autonomou linear ytem II Journal of Differential Equation vol 29 no 3 pp C Moler and C van Loan Nineteen dubiou way to compute the exponential of a matrix twenty-five year later SIAM Review vol 45 no 1 pp D S Berntein Matrix mathematic: theory fact and formula 2nd ed Princeton New Jerey: Princeton Univerity Pre 29 8 I Higuera and B García-Celayeta Logarithmic norm for matrix pencil SIAM Journal on Matrix analyi and Application vol 2 no 3 pp S L Campbell and C D Meyer Generalized invere of linear tranformation Pitman Pub London 1979 ch The Darzin invere pp D Bainov and P Simeonov Integral inequalitie and application er Mathematic and it application Eat European Serie Dordrecht The Netherland: Kluwer Academic Publiher 1992 vol S Ghorpade and B Limaye A coure in multivariable calculu and analyi er Undergraduate text in mathematic Springer New York 21
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3
Bogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281
72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition
The Hassenpflug Matrix Tensor Notation
The Haenpflug Matrix Tenor Notation D.N.J. El Dept of Mech Mechatron Eng Univ of Stellenboch, South Africa e-mail: dnjel@un.ac.za 2009/09/01 Abtract Thi i a ample document to illutrate the typeetting of
IEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation
IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between
A Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems
A Contraint Propagation Algorithm for Determining the Stability Margin of Linear Parameter Circuit and Sytem Lubomir Kolev and Simona Filipova-Petrakieva Abtract The paper addree the tability margin aement
Computers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order
Computer and Mathematic with Application 64 (2012) 2262 2274 Content lit available at SciVere ScienceDirect Computer and Mathematic with Application journal homepage: wwweleviercom/locate/camwa Sharp algebraic
Chapter 4. The Laplace Transform Method
Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination
One Class of Splitting Iterative Schemes
One Cla of Splitting Iterative Scheme v Ciegi and V. Pakalnytė Vilniu Gedimina Technical Univerity Saulėtekio al. 11, 2054, Vilniu, Lithuania rc@fm.vtu.lt Abtract. Thi paper deal with the tability analyi
March 18, 2014 Academic Year 2013/14
POLITONG - SHANGHAI BASIC AUTOMATIC CONTROL Exam grade March 8, 4 Academic Year 3/4 NAME (Pinyin/Italian)... STUDENT ID Ue only thee page (including the back) for anwer. Do not ue additional heet. Ue of
Copyright 1967, by the author(s). All rights reserved.
Copyright 1967, by the author(). All right reerved. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or
Research Article Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations
International Scholarly Reearch Network ISRN Mathematical Analyi Volume 20, Article ID 85203, 9 page doi:0.502/20/85203 Reearch Article Exitence for Nonocillatory Solution of Higher-Order Nonlinear Differential
ONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES
ONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE, AND KOTA SAITO In thi online
EE Control Systems LECTURE 14
Updated: Tueday, March 3, 999 EE 434 - Control Sytem LECTURE 4 Copyright FL Lewi 999 All right reerved ROOT LOCUS DESIGN TECHNIQUE Suppoe the cloed-loop tranfer function depend on a deign parameter k We
An Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem
An Inequality for Nonnegative Matrice and the Invere Eigenvalue Problem Robert Ream Program in Mathematical Science The Univerity of Texa at Dalla Box 83688, Richardon, Texa 7583-688 Abtract We preent
Unbounded solutions of second order discrete BVPs on infinite intervals
Available online at www.tjna.com J. Nonlinear Sci. Appl. 9 206), 357 369 Reearch Article Unbounded olution of econd order dicrete BVP on infinite interval Hairong Lian a,, Jingwu Li a, Ravi P Agarwal b
Convex Hulls of Curves Sam Burton
Convex Hull of Curve Sam Burton 1 Introduction Thi paper will primarily be concerned with determining the face of convex hull of curve of the form C = {(t, t a, t b ) t [ 1, 1]}, a < b N in R 3. We hall
TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL
GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National
Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004
18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem
Math 273 Solutions to Review Problems for Exam 1
Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c
Nonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamic in High Energy Accelerator Part 6: Canonical Perturbation Theory Nonlinear Single-Particle Dynamic in High Energy Accelerator Thi coure conit of eight lecture: 1. Introduction
Multi-dimensional Fuzzy Euler Approximation
Mathematica Aeterna, Vol 7, 2017, no 2, 163-176 Multi-dimenional Fuzzy Euler Approximation Yangyang Hao College of Mathematic and Information Science Hebei Univerity, Baoding 071002, China hdhyywa@163com
Linear System Fundamentals
Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept
696 Fu Jing-Li et al Vol. 12 form in generalized coordinate Q ffiq dt = 0 ( = 1; ;n): (3) For nonholonomic ytem, ffiq are not independent of
Vol 12 No 7, July 2003 cfl 2003 Chin. Phy. Soc. 1009-1963/2003/12(07)/0695-05 Chinee Phyic and IOP Publihing Ltd Lie ymmetrie and conerved quantitie of controllable nonholonomic dynamical ytem Fu Jing-Li(ΛΠ±)
MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:
MATEMATIK Datum: 20-08-25 Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: 073-07926. Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what
Local Fractional Laplace s Transform Based Local Fractional Calculus
From the SelectedWork of Xiao-Jun Yang 2 Local Fractional Laplace Tranform Baed Local Fractional Calculu Yang Xiaojun Available at: http://workbeprecom/yang_iaojun/8/ Local Fractional Laplace Tranform
Control Systems Analysis and Design by the Root-Locus Method
6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If
SOME RESULTS ON INFINITE POWER TOWERS
NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite
Minimal state space realization of MIMO systems in the max algebra
KULeuven Department of Electrical Engineering (ESAT) SISTA Technical report 94-54 Minimal tate pace realization of MIMO ytem in the max algebra B De Schutter and B De Moor If you want to cite thi report
Gain and Phase Margins Based Delay Dependent Stability Analysis of Two- Area LFC System with Communication Delays
Gain and Phae Margin Baed Delay Dependent Stability Analyi of Two- Area LFC Sytem with Communication Delay Şahin Sönmez and Saffet Ayaun Department of Electrical Engineering, Niğde Ömer Halidemir Univerity,
Problem Set 8 Solutions
Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem
Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)
Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable.
ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang
Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang
Chapter 13. Root Locus Introduction
Chapter 13 Root Locu 13.1 Introduction In the previou chapter we had a glimpe of controller deign iue through ome imple example. Obviouly when we have higher order ytem, uch imple deign technique will
Social Studies 201 Notes for November 14, 2003
1 Social Studie 201 Note for November 14, 2003 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
Laplace Transformation
Univerity of Technology Electromechanical Department Energy Branch Advance Mathematic Laplace Tranformation nd Cla Lecture 6 Page of 7 Laplace Tranformation Definition Suppoe that f(t) i a piecewie continuou
CHEAP CONTROL PERFORMANCE LIMITATIONS OF INPUT CONSTRAINED LINEAR SYSTEMS
Copyright 22 IFAC 5th Triennial World Congre, Barcelona, Spain CHEAP CONTROL PERFORMANCE LIMITATIONS OF INPUT CONSTRAINED LINEAR SYSTEMS Tritan Pérez Graham C. Goodwin Maria M. Serón Department of Electrical
Stochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions
Stochatic Optimization with Inequality Contraint Uing Simultaneou Perturbation and Penalty Function I-Jeng Wang* and Jame C. Spall** The John Hopkin Univerity Applied Phyic Laboratory 11100 John Hopkin
CHAPTER 4 DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL
98 CHAPTER DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL INTRODUCTION The deign of ytem uing tate pace model for the deign i called a modern control deign and it i
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat Thi Online Appendix contain the proof of our reult for the undicounted limit dicued in Section 2 of the paper,
EXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS. Otto J. Roesch, Hubert Roth, Asif Iqbal
EXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS Otto J. Roech, Hubert Roth, Aif Iqbal Intitute of Automatic Control Engineering Univerity Siegen, Germany {otto.roech,
Digital Control System
Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital
Explicit formulae for J pectral factor for well-poed linear ytem Ruth F. Curtain Amol J. Saane Department of Mathematic Department of Mathematic Univerity of Groningen Univerity of Twente P.O. Box 800
WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD
WELL-POSEDNESS OF A ONE-DIMENSIONAL PLASMA MODEL WITH A HYPERBOLIC FIELD JENNIFER RAE ANDERSON 1. Introduction A plama i a partially or completely ionized ga. Nearly all (approximately 99.9%) of the matter
into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get
Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}
Approximate Analytical Solution for Quadratic Riccati Differential Equation
Iranian J. of Numerical Analyi and Optimization Vol 3, No. 2, 2013), pp 21-31 Approximate Analytical Solution for Quadratic Riccati Differential Equation H. Aminikhah Abtract In thi paper, we introduce
CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. VIII Decoupling Control - M. Fikar
DECOUPLING CONTROL M. Fikar Department of Proce Control, Faculty of Chemical and Food Technology, Slovak Univerity of Technology in Bratilava, Radlinkého 9, SK-812 37 Bratilava, Slovakia Keyword: Decoupling:
Singular perturbation theory
Singular perturbation theory Marc R. Rouel June 21, 2004 1 Introduction When we apply the teady-tate approximation (SSA) in chemical kinetic, we typically argue that ome of the intermediate are highly
Efficient Methods of Doppler Processing for Coexisting Land and Weather Clutter
Efficient Method of Doppler Proceing for Coexiting Land and Weather Clutter Ça gatay Candan and A Özgür Yılmaz Middle Eat Technical Univerity METU) Ankara, Turkey ccandan@metuedutr, aoyilmaz@metuedutr
Evolutionary Algorithms Based Fixed Order Robust Controller Design and Robustness Performance Analysis
Proceeding of 01 4th International Conference on Machine Learning and Computing IPCSIT vol. 5 (01) (01) IACSIT Pre, Singapore Evolutionary Algorithm Baed Fixed Order Robut Controller Deign and Robutne
Introduction to Laplace Transform Techniques in Circuit Analysis
Unit 6 Introduction to Laplace Tranform Technique in Circuit Analyi In thi unit we conider the application of Laplace Tranform to circuit analyi. A relevant dicuion of the one-ided Laplace tranform i found
Midterm Test Nov 10, 2010 Student Number:
Mathematic 265 Section: 03 Verion A Full Name: Midterm Tet Nov 0, 200 Student Number: Intruction: There are 6 page in thi tet (including thi cover page).. Caution: There may (or may not) be more than one
Manprit Kaur and Arun Kumar
CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar manpreet2410@gmail.com, arun04@rediffmail.com Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA. Abtract:
Social Studies 201 Notes for March 18, 2005
1 Social Studie 201 Note for March 18, 2005 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS. Volker Ziegler Technische Universität Graz, Austria
GLASNIK MATEMATIČKI Vol. 1(61)(006), 9 30 ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS Volker Ziegler Techniche Univerität Graz, Autria Abtract. We conider the parameterized Thue
Pacific Journal of Mathematics
Pacific Journal of Mathematic OSCILLAION AND NONOSCILLAION OF FORCED SECOND ORDER DYNAMIC EQUAIONS MARIN BOHNER AND CHRISOPHER C. ISDELL Volume 230 No. March 2007 PACIFIC JOURNAL OF MAHEMAICS Vol. 230,
On mild solutions of a semilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem in Banach spaces
MAEMAIA, 16, Volume 3, Number, 133 14 c Penerbit UM Pre. All right reerved On mild olution of a emilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem in Banach pace
List coloring hypergraphs
Lit coloring hypergraph Penny Haxell Jacque Vertraete Department of Combinatoric and Optimization Univerity of Waterloo Waterloo, Ontario, Canada pehaxell@uwaterloo.ca Department of Mathematic Univerity
A Simplified Methodology for the Synthesis of Adaptive Flight Control Systems
A Simplified Methodology for the Synthei of Adaptive Flight Control Sytem J.ROUSHANIAN, F.NADJAFI Department of Mechanical Engineering KNT Univerity of Technology 3Mirdamad St. Tehran IRAN Abtract- A implified
STOCHASTIC DIFFERENTIAL GAMES:THE LINEAR QUADRATIC ZERO SUM CASE
Sankhyā : he Indian Journal of Statitic 1995, Volume 57, Serie A, Pt. 1, pp.161 165 SOCHASIC DIFFERENIAL GAMES:HE LINEAR QUADRAIC ZERO SUM CASE By R. ARDANUY Univeridad de Salamanca SUMMARY. hi paper conider
New bounds for Morse clusters
New bound for More cluter Tamá Vinkó Advanced Concept Team, European Space Agency, ESTEC Keplerlaan 1, 2201 AZ Noordwijk, The Netherland Tama.Vinko@ea.int and Arnold Neumaier Fakultät für Mathematik, Univerität
On Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces
On Uniform Exponential Trichotomy of Evolution Operator in Banach Space Mihail Megan, Codruta Stoica To cite thi verion: Mihail Megan, Codruta Stoica. On Uniform Exponential Trichotomy of Evolution Operator
Control of Delayed Integrating Processes Using Two Feedback Controllers R MS Approach
Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) Control of Delayed Integrating Procee Uing Two Feedback Controller R MS Approach LIBOR
OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE
LTV System Modelling
Helinki Univerit of Technolog S-72.333 Potgraduate Coure in Radiocommunication Fall 2000 LTV Stem Modelling Heikki Lorentz Sonera Entrum O heikki.lorentz@onera.fi Januar 23 rd 200 Content. Introduction
Multicolor Sunflowers
Multicolor Sunflower Dhruv Mubayi Lujia Wang October 19, 2017 Abtract A unflower i a collection of ditinct et uch that the interection of any two of them i the ame a the common interection C of all of
Beta Burr XII OR Five Parameter Beta Lomax Distribution: Remarks and Characterizations
Marquette Univerity e-publication@marquette Mathematic, Statitic and Computer Science Faculty Reearch and Publication Mathematic, Statitic and Computer Science, Department of 6-1-2014 Beta Burr XII OR
ROBUST CONTROLLER DESIGN WITH HARD INPUT CONSTRAINTS. TIME DOMAIN APPROACH. Received August 2015; revised November 2015
International Journal of Innovative Computing, Information and Control ICIC International c 2016 ISSN 1349-4198 Volume 12, Number 1, February 2016 pp. 161 170 ROBUST CONTROLLER DESIGN WITH HARD INPUT CONSTRAINTS.
A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES. Sanghyun Cho
A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES Sanghyun Cho Abtract. We prove a implified verion of the Nah-Moer implicit function theorem in weighted Banach pace. We relax the
Analysis and Design of a Third Order Phase-Lock Loop
Analyi Deign of a Third Order Phae-Lock Loop DANIEL Y. ABRAMOVITCH Ford Aeropace Corporation 3939 Fabian Way, MS: X- Palo Alto, CA 94303 Abtract Typical implementation of a phae-lock loop (PLL) are econd
arxiv: v2 [math.nt] 30 Apr 2015
![arxiv: v2 [math.nt] 30 Apr 2015](/thumbs/74/69750607.jpg "arxiv: v2 [math.nt] 30 Apr 2015")
A THEOREM FOR DISTINCT ZEROS OF L-FUNCTIONS École Normale Supérieure arxiv:54.6556v [math.nt] 3 Apr 5 943 Cachan November 9, 7 Abtract In thi paper, we etablih a imple criterion for two L-function L and
THE SPLITTING SUBSPACE CONJECTURE
THE SPLITTING SUBSPAE ONJETURE ERI HEN AND DENNIS TSENG Abtract We anwer a uetion by Niederreiter concerning the enumeration of a cla of ubpace of finite dimenional vector pace over finite field by proving
THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY STABILIZING CONTROLLERS. Tatsuya Hoshikawa, Kou Yamada and Yuko Tatsumi
International Journal of Innovative Computing, Information Control ICIC International c 206 ISSN 349-498 Volume 2, Number 2, April 206 pp. 357 370 THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY
Preemptive scheduling on a small number of hierarchical machines
Available online at www.ciencedirect.com Information and Computation 06 (008) 60 619 www.elevier.com/locate/ic Preemptive cheduling on a mall number of hierarchical machine György Dóa a, Leah Eptein b,
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS VOLKER ZIEGLER Abtract We conider the parameterized Thue equation X X 3 Y (ab + (a + bx Y abxy 3 + a b Y = ±1, where a, b 1 Z uch that
A Short Note on Hysteresis and Odd Harmonics
1 A Short Note on yterei and Odd armonic Peter R aoput Senior Reearch Scientit ubocope Pipeline Engineering outon, X 7751 USA pmaoput@varco.com June 19, Abtract hi hort note deal with the exitence of odd
SECTION x2 x > 0, t > 0, (8.19a)
SECTION 8.5 433 8.5 Application of aplace Tranform to Partial Differential Equation In Section 8.2 and 8.3 we illutrated the effective ue of aplace tranform in olving ordinary differential equation. The
Simple Observer Based Synchronization of Lorenz System with Parametric Uncertainty
IOSR Journal of Electrical and Electronic Engineering (IOSR-JEEE) ISSN: 78-676Volume, Iue 6 (Nov. - Dec. 0), PP 4-0 Simple Oberver Baed Synchronization of Lorenz Sytem with Parametric Uncertainty Manih
Lecture 4 Topic 3: General linear models (GLMs), the fundamentals of the analysis of variance (ANOVA), and completely randomized designs (CRDs)
Lecture 4 Topic 3: General linear model (GLM), the fundamental of the analyi of variance (ANOVA), and completely randomized deign (CRD) The general linear model One population: An obervation i explained
Codes Correcting Two Deletions
1 Code Correcting Two Deletion Ryan Gabry and Frederic Sala Spawar Sytem Center Univerity of California, Lo Angele ryan.gabry@navy.mil fredala@ucla.edu Abtract In thi work, we invetigate the problem of
MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005
MA 66 FINAL EXAM INSTRUCTIONS May, 5 NAME INSTRUCTOR. You mut ue a # pencil on the mark ene heet anwer heet.. If the cover of your quetion booklet i GREEN, write in the TEST/QUIZ NUMBER boxe and blacken
Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is
EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt -
Lecture 10 Filtering: Applied Concepts
Lecture Filtering: Applied Concept In the previou two lecture, you have learned about finite-impule-repone (FIR) and infinite-impule-repone (IIR) filter. In thee lecture, we introduced the concept of filtering
February 5, :53 WSPC/INSTRUCTION FILE Mild solution for quasilinear pde
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde Infinite Dimenional Analyi, Quantum Probability and Related Topic c World Scientific Publihing Company STOCHASTIC QUASI-LINEAR
FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS
FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 Abtract. Thi paper concern the filtering problem for a cla of tochatic nonlinear ytem
Assessment of Performance for Single Loop Control Systems
Aement of Performance for Single Loop Control Sytem Hiao-Ping Huang and Jyh-Cheng Jeng Department of Chemical Engineering National Taiwan Univerity Taipei 1617, Taiwan Abtract Aement of performance in
Jump condition at the boundary between a porous catalyst and a homogeneous fluid
From the SelectedWork of Francico J. Valde-Parada 2005 Jump condition at the boundary between a porou catalyt and a homogeneou fluid Francico J. Valde-Parada J. Alberto Ochoa-Tapia Available at: http://work.bepre.com/francico_j_valde_parada/12/
Stability. ME 344/144L Prof. R.G. Longoria Dynamic Systems and Controls/Lab. Department of Mechanical Engineering The University of Texas at Austin
Stability The tability of a ytem refer to it ability or tendency to eek a condition of tatic equilibrium after it ha been diturbed. If given a mall perturbation from the equilibrium, it i table if it return.
Research Article A New Kind of Weak Solution of Non-Newtonian Fluid Equation
Hindawi Function Space Volume 2017, Article ID 7916730, 8 page http://doi.org/10.1155/2017/7916730 Reearch Article A New Kind of Weak Solution of Non-Newtonian Fluid Equation Huahui Zhan 1 and Bifen Xu
Hyperbolic Partial Differential Equations
Hyperbolic Partial Differential Equation Evolution equation aociated with irreverible phyical procee like diffuion heat conduction lead to parabolic partial differential equation. When the equation i a
REPRESENTATION OF ALGEBRAIC STRUCTURES BY BOOLEAN FUNCTIONS. Logic and Applications 2015 (LAP 2015) September 21-25, 2015, Dubrovnik, Croatia
REPRESENTATION OF ALGEBRAIC STRUCTURES BY BOOLEAN FUNCTIONS SMILE MARKOVSKI Faculty of Computer Science and Engineering, S Ciryl and Methodiu Univerity in Skopje, MACEDONIA mile.markovki@gmail.com Logic
Electronic Theses and Dissertations
Eat Tenneee State Univerity Digital Common @ Eat Tenneee State Univerity Electronic Thee and Diertation Student Work 5-208 Vector Partition Jennifer French Eat Tenneee State Univerity Follow thi and additional
CDMA Signature Sequences with Low Peak-to-Average-Power Ratio via Alternating Projection
CDMA Signature Sequence with Low Peak-to-Average-Power Ratio via Alternating Projection Joel A Tropp Int for Comp Engr and Sci (ICES) The Univerity of Texa at Autin 1 Univerity Station C0200 Autin, TX
STOCHASTIC EVOLUTION EQUATIONS WITH RANDOM GENERATORS. By Jorge A. León 1 and David Nualart 2 CINVESTAV-IPN and Universitat de Barcelona
The Annal of Probability 1998, Vol. 6, No. 1, 149 186 STOCASTIC EVOLUTION EQUATIONS WIT RANDOM GENERATORS By Jorge A. León 1 and David Nualart CINVESTAV-IPN and Univeritat de Barcelona We prove the exitence
FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS
FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS Nguyen Thanh Lan Department of Mathematic Wetern Kentucky Univerity Email: lan.nguyen@wku.edu ABSTRACT: We ue Fourier erie to find a neceary
A BATCH-ARRIVAL QUEUE WITH MULTIPLE SERVERS AND FUZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH
Mathematical and Computational Application Vol. 11 No. pp. 181-191 006. Aociation for Scientific Reearch A BATCH-ARRIVA QEE WITH MTIPE SERVERS AND FZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH Jau-Chuan
MATH 251 Examination II April 6, 2015 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 6, 2015 FORM A Name: Student Number: Section: Thi exam ha 12 quetion for a total of 100 point. In order to obtain full credit for partial credit problem, all work mut be hown.
OBSERVER DESIGN FOR DISCRETE-TIME LINEAR SWITCHING SYSTEMS 1
OBSERVER DESIGN FOR DISCRETE-TIME LINEAR SWITCHING SYSTEMS 1 E. De Santi, M.D. Di Benedetto Department of Electrical Engineering and Computer Science, Univerity of L Aquila. Email: (deanti,dibenede@ing.univaq.it
The Laplace Transform , Haynes Miller and Jeremy Orloff
The Laplace Tranform 8.3, Hayne Miller and Jeremy Orloff Laplace tranform baic: introduction An operator take a function a input and output another function. A tranform doe the ame thing with the added
Stochastic Perishable Inventory Control in a Service Facility System Maintaining Inventory for Service: Semi Markov Decision Problem
Stochatic Perihable Inventory Control in a Service Facility Sytem Maintaining Inventory for Service: Semi Markov Deciion Problem R.Mugeh 1,S.Krihnakumar 2, and C.Elango 3 1 mugehrengawamy@gmail.com 2 krihmathew@gmail.com