Enhancement of Robust Tracking Performance via Switching Supervisory Adaptive Control

Transcription

1 nhancement of Robut Tracking Performance via Switching Suerviory Adative Control Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 O. Namaki-Shouhtari* and A. Khaki Sedigh* Abtract: When the roce i highly uncertain, even linear minimum hae ytem mut acrifice deirable feedback control benefit to avoid an exceive cot of feedback, while reerving the robut tability. In thi aer, the control tructure of uerviory baed witching Quantitative Feedback Theory (QFT) control i rooed to control highly uncertain lant. According to thi trategy, the uncertainty region i uitably divided into maller region. It i aumed that a QFT controller-refilter exit for robut tability and robut erformance of the individual uncertain et. The rooed control architecture i made u by thee local controller, which commute among themelve in accordance with the deciion of a high level deciion maker called the uervior. The uervior make the deciion by comaring the candidate local model behavior with the one of the lant and elect the controller correonding to the bet fitted model. A hyterei witching logic i ued to low down witching for tability reaon. Beide, each controller i deigned to be table in the whole uncertainty domain, and a accurate in command tracking a deired in it uncertainty ubet to reerve the robut tability from any failure in the witching. Keyword: Switching Suerviory Adative Control, Robut Control, Quantitative Feedback Theory. Introduction According to Quantitative Feedback Theory, the feedback control i only jutified to reduce the cloed loo enitivity to any kind of uncertainty, in the lant modeling or in the unmeaurable diturbance []. Thu, the amount of feedback i directly roortional to the amount of uncertainty and to the enitivity reduction required. Beide, a unique controller cannot erform a well for a wide uncertainty lant et a for a maller domain of uncertainty. QFT i a owerful deign methodology that rovide a tranarent trade-off between different often conflicting deign ecification. It ugget a controller with minimum cot of feedback that atifie the et of erformance ecification in ite of the lant uncertainty [], [3]. Let conider the loo tranfer function L(jω) = G(jω) P(jω). Uing ufficiently large feedback (i.e. L >> P ) the effect of the lant uncertainty and of the diturbance could be diminihed. On the Iranian Journal of lectrical & lectronic ngineering,. Paer firt received May and in revied form 7 Nov.. * The Author are with the Deartment of Sytem and Control, Faculty of lectrical and Comuter ngineering, K. N. Tooi Univerity of Technology, P.O. Box , Tehran 43749, Iran. -mail: onamaki@ieee.org and edigh@kntu.ac.ir. other hand, any ractical L(jω) mut go to zero a ω, but robut tability imlie that L decreae relatively lowly with ω, [4]. Therefore, there i an intermediate frequency range where the exceive loo bandwidth due to feedback benefit amlifie enor noie or diturbance at the lant inut. Thi effect, which i called cot of feedback, reult in the ueful ignal comonent of the inut command cannot a into due to the aturation of element of G and P. For a highly uncertain roce, in which arametric uncertaintie are very large, finding a ingle controller that can deal with the entire range of arameter variation may be imoible or imly a oor erformance. Therefore, the uncertainty reduction remain a the only olution. An infinite uncertainty diviion tranlate into claical adative control cheme where a articular controller i reonible for a unique lant identified in the uncertain domain. In order to overcome the limitation of adative control and to deign a atifactory control ytem in the reence of large modeling uncertaintie, noie, and diturbance, a hierarchical control tructure can be ued. The control tructure conit of a bank of candidate controller uervied by a logic-baed witching [5]. 8 Iranian Journal of lectrical & lectronic ngineering, Vol. 8, No., March

2 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 In each fixed, redetermined region of uncertainty, the local controller can achieve the deired erformance. Switching i made between the local controller to uort all range of uncertaintie. The overall control architecture conit of a bank of controller (multicontroller), and a uervior. The uervior i alo comoed of a bank of model (multi-etimator), a monitoring ignal generator, and a witching logic. At each time intant, a high level deciion maker, the o called uervior, determine which controller hould be laced in the feedback loo. In other word, when the etimation of the lant i changed, a new controller may be elected, which i imilar to the idea of adative control. But unlike the traditional adative control trategie, thi adatation take lace in a dicrete fahion. A a reult, the overall cloed loo ytem can be viewed a a hybrid ytem. One of the main advantage of the uerviory control i it modularity [5]. Deigning multi-etimator, multi-controller, and witching logic can be done mutually indeendent. Thi feature enable the deigner, to ue off-the-helf robut control law. Baed on thi idea, in [6], a multi-model adative PID controller i develoed and evaluated in a imulation tudy for a nonlinear H neutralization roce. A methodology that blend robut non-adative mixed μ-ynthei deign and tochatic hyothei-teting concet i introduced in [7]. In [8], linear limitation of linear robut controller overcame by combining witching and QFT. Combining robut deign and witching, the deigned controller otimize the time reone of the ytem by fat adatation of the controller arameter during the tranient reone baed on the error amlitude [8]. Thi aer how the feedback tracking control limitation due to the uncertainty ize in LTI minimum hae lant. Thee are dicued from the QFT bound, comuted with the formula develoed in [9]. The goal i to divide the uncertainty. Then, everal QFT controller are emloyed to attain robut tability and erformance requirement deite uncertaintie and diturbance. They alo imrove the feedback benefit and minimize the cot of feedback in their uncertainty ubet. A uerviory architecture orchetrate controller election. Thi election i baed on the value of monitoring ignal. To reerve robut tability deite of failure in the witching, each controller will be deigned robutly table for the full uncertainty range and a accurate in et-oint tracking a deired in it uncertainty ubet. Thi roblem i alo tackled in []. Thi aer i organized a follow: Section examine fundamental of QFT with the nature and everity of the QFT bound. It alo mention the difficultie in loo haing, revealing the limitation in the tracking feedback erformance. In Section 3, witching uerviory control i reviewed. Our rooed deign tructure i decribed in Section 4. Simulation reult and concluion are made in Section 5, and 6, reectively. Quantitative Feedback Theory The general QFT feedback tructure i hown in Fig.. The block to be deigned by QFT in Fig. are the controller and the re-filter. The QFT deign, erformed in the frequency domain, follow very cloely claical deign uing Bode lot. The model for the oen-loo dynamic can either be fixed or include uncertainty. If the roblem require that the ecification be met with the uncertain dynamic, it i called a robut erformance roblem. That i, the erformance ecification mut be atified for all oible cae admitted by the ecific uncertainty model []. Several erformance ecification could be laced on any ingle loo cloed loo relation. Some of the tyical ecification conidered in QFT are a follow: gain and hae margin P G F + W claical -DOF QFT tracking roblem + Wa F b W Fig.. The two-degree of freedom tructure of QFT [3]. Namaki-Shouhtari & Khaki Sedigh: nhancement of Robut Tracking Performance 9

3 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 rejection of diturbance at lant outut F + W rejection of lant inut diturbance P F W4 + where W i denote the maximum tolerance model on the magnitude of certain tranfer function from ome inut to ome outut in the frequency domain which i correond to an ecification laced on the magnitude of the tranfer function. Then, to deign the controller G(jω), QFT tranlate thee frequency domain ecification of an uncertain feedback ytem into bound on the Nichol Chart that the nominal loo tranmiion L o (jω) mut meet. Therefore, the bound arrangement i the key oint in the controller ynthei roce. The neceary trade-off among conflicting cloed loo requirement are dicued in term of QFT bound in [9]. The loo-haing L o =GP o (P o i the nominal lant) erformed on the bound arrangement uch that the QFT bound are atified which reult in the demanded erformance ecification. Robut tability bound make L decreaing comaratively lowly with ω. Thu, there i an intermediate range where L(jω) << but L/P(jω) > (to remove the effect of the lant ignorance). That mean a dangerou amlification of the noie N at Y and U. Satifyingly mall outut deviation can be achieved with available intrumentation. However, large L/P eak roduce unavoidable large eak in control inut U [], which i the main rice aid for feedback, alo referred in QFT a the cot of feedback []. A large cot may aturate element of G and P, in uch a way that the ueful ignal comonent due to inut command cannot get through. 3 3 Switching Suerviory Control In uerviory control of uncertain ytem, the baic idea i to dicretize the uncertainty et into a finite number of nominal value and then emloy a family of controller, one for each nominal value. Switching among the controller i orchetrated by a uervior in uch a way that cloed-loo tability i guaranteed. The benefit gained by thi aroach include (i) imlicity and modularity in deign: controller deign amount to controller deign for known linear time-invariant ytem for which variou comutationally efficient controller deign tool are available; (ii) ability to handle larger clae of ytem than i oible with other aroache (ee [5] for more dicuion). We quickly review here the uerviory control framework for adatively controlling lant with large modeling uncertainty (ee Fig. ); for detail, ee e.g. ([3], Chater 6), [4] or [5] and the reference therein. Conider an uncertain linear lant M arameterized by a arameter where * denote the true but unknown arameter: x = A x+ B u M : () y = C x, where x R n_x i the tate, u R n_u i the inut, and y R n_y i the outut. The arameter * R n_ belong to a known finite et P: = {,, m }, where m i the cardinality of P. It i aumed that (A, B ) i tabilizable and (A, C ) i detectable for every P. The uervior comrie a multi-etimator, monitoring ignal, and a witching logic. The etimator-baed uerviory control deign for the ytem () i briefly outlined below: Fig.. The uerviory control framework [3]. 3 Iranian Journal of lectrical & lectronic ngineering, Vol. 8, No., March

4 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 Multi-etimator: A multi-etimator i a collection of model, one for each fixed arameter P. The multi-etimator take in the inut u and roduce a bank of outut y, P. The multi-etimator hould have the following matching roerty: there i ˆ P uch that y λe ( t t ) () t c e y ( t y ( ) ˆ t) y e ˆ ) ( t () for all t t, for all u, and for ome c e and λ e >. One uch multi-etimator for () i the following dynamic x = A x + B u + L ( y y = C x, y), for all P, and the roerty () i atified with ˆ = *. The matrix L in (3) i uch that the eigenvalue of A + L C have negative real art for each P. (ince (3) with = * and () imly that (d/dt)(x * x) = (A +L C ) (x * x) and y=c * x). Multi-controller: A family of candidate controller {G } i deigned uch that the cloed loo ytem meet the deired robut tability and erformance ecification for every P. Then the family of controller i (3) u, P. (4) Monitoring ignal: Monitoring ignal μ, P are norm of the outut etimation error, y y. Here, the monitoring ignal are generated a t λ ( t ) μ : = ε + e γ y ( ) y ( ) for ome γ, ε, λ >. The number γ, ε, and λ are deign arameter and need to atify < λ < λ (6) for ome contant λ related to the eigenvalue of the cloed-loo ytem (for detail on λ, ee [3]). Switching logic: A witching logic roduce a witching ignal that nominate the active controller at each time intant. In thi aer, we ue the cale indeendent hyterei witching logic [5]: if q P uch that arg min μq ( t) σ (): t = q ( h) μq ( t) μ P + ( t) σ ( t ) σ ( t ) ele, d (5) (7) where h > i called a hyterei contant and i a deign arameter that revent exceive witching. Both in theory and in ractice, it i imortant that exceive witching be avoided. The ue of a hyterei term conveniently atifie thi requirement. The control ignal alied to the lant i u(t) = u σ (t). 4 Switching Suerviory QFT Control Combining witching and QFT wa firt introduced in [8] to rioritize ome ecification over other according to the tate of the ytem at each time. Switching i ued to elect the aroriate controller, which i determined baed on the error amlitude. Two controller are ued: the fat, more table and imrecie controller i ued when the outut i too far from the reference, or equivalently when the error amlitude i large. When the error i mall, a controller, which reduce the bandwidth, i ued to avoid the effect of noie, and meanwhile to increae the low frequency gain in order to minimize the jitter and the tracking error. In that method, both of the controller are uoed to have the ame ole, o that grahical tability criteria can be utilized. Thi contraint limit the tye of controller and therefore the erformance of the ytem. So, a more general aroach i introduced to overcome thi limitation. 4. Problem Formulation In the cae of highly uncertain lant, two ditinct cae can occur: A ingle QFT controller exit for the entire uncertainty range. However, the cloed loo erformance may not be imroved further a deired. A ingle QFT controller doe not exit to enure cloed loo robut tability and erformance. In both cae, the QFT trategy need imrovement to fulfill the ractical deign requirement and to meet the challenge of the control of difficult highly uncertain lant. 4. Cla of Admiible Plant The goal i to deign a control ytem, which can track a redetermined et oint in cae of lant uncertainty and diturbance. The lant i aumed to be modeled by a tabilizable and detectable SISO linear ytem with control inut u and meaured outut y, erturbed by a bounded diturbance inut d. It i alo aumed that the lant tranfer function belong to a known cla of admiible tranfer function of the form: M: = M P where i a arameter taking value in ome index et. M i alo a family of tranfer function centered around ome known nominal roce model tranfer function ν [6]. Allowable unmodeled dynamic around the nominal roce model tranfer function ν could be ecified a: Namaki-Shouhtari & Khaki Sedigh: nhancement of Robut Tracking Performance 3

5 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 { v δ δ δ, λ ε δ, λ ε} M : = ( + ) + :,, m a m a where ε > and λ are two arbitrary number. Here,.,λ denote e λ t -weighted H norm of a tranfer function: v, λ := u Re[] v(- λ) [6]. Throughout the aer, we will take P to be a comact ubet of a finite-dimenional normed linear vector ace. By thi definition, the entire region of lant uncertainty i artitioned into a et of maller region. ach maller region i reented by a arameter value, and M i a model of the lant in that mall region. Conidering all ermiible uncertaintie and diturbance in each maller region, a controller i deigned to erform robut tability and robut et oint tracking ecification, via QFT. 4.3 Multi-timator and Multi-Controller A tate-hared multi-etimator of the form x = A x + L y + B u, y = C x, e = y y, (8) where P, and A an aymtotically table matrix, i utilized here. Thi tye of tructure i quite common in adative control ( [7]). Note that even if P i an infinite et, the above dynamical ytem i finite-dimenional. In thi cae the multi-etimator formally ha an infinite number of outut; however they can all be comuted from x. Here we ue tate-haring not only to generate the etimation error e, but alo in the monitoring ignal generator for μ. For ractical reaon the bank of local controller can be efficiently imlemented (multirealized) by mean of a tate-hared arameter deendent feedback ytem. The rovided method can imlement bumle tranfer, which i an effective way to imrove oor tranient reone of witched ytem [7], [8]. 5 Simulation Reult In thi ection, a ractical examle i ued to illutrate the rooed deign method. Conider a DC field-controlled motor, whoe nominal arameter are given in Table. Auming that the angular velocity Ω i the lant outut to be controlled, and the field voltage V f i the maniulated variable, the tranfer function correond to the DC motor i ( [9]): Ω K m R f K M ( ) = = =, (9) V f J + b J + b where the electrical time contant L f /R f i ignored comared to the mechanical time contant J/b (ee Table ). Due to temerature variation and ome other effect, the motor arameter may differ from the nominal value. Therefore, a % variation in b and a 5% deviation in K from their nominal value are conidered in thi examle. Moreover, the larger difference for thi real alication relate to the rotor inertia J. It value i exected to change between J =. a the nominal value which i correond to no load and J =. a the full load condition, reectively. The cloed-loo deired ecification for eed control are: (): Robut tability in term of a margin ecification (L i the loo gain) L.3, ω > + L Thi would indicate a gain margin and hae margin of 4.9dB and 45deg reectively. (): Robut tracking ecification enforced to ω h = rad/ec ( + 3) Tr ( + 3) ( + 4) ( + ) ( + 7) P G where T r = F i the tranfer function from + reference inut to the outut. (3): A low enough high frequency loo gain to reduce the cot of feedback, e.g. L(jω hf ) < - db, ω hf rad/ec. The main te involved in the deign of the QFT controller, uch a temlate generation, loo haing, re-filter deign, maniulation of tolerance bound within the available freedom, temlate ize conideration and election of nominal tranfer function all require much exerience and exertie. In [] the variou te of quantitative deign are reformulated and reented in term of aroriate cot function and reective algebraic contraint. The reulting nonlinear contrained otimization roblem can eaily be olved uing the Genetic Algorithm. Here, the QFT deign are adoted from [9]. The QFT controller and re-filter, which i deigned for the whole uncertainty range, are the following: Table Nominal Parameter Value for a DC Field-Controlled Motor Symbol Parameter Nominal value J No Load Rotor Inertia (kg m ). b Friction (N m ). K m Motor Contant (N m/a).5 R f Field Reitance (Ω) L f Field Inductance (H) <<. 3 Iranian Journal of lectrical & lectronic ngineering, Vol. 8, No., March

6 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th G ( ) = 37.7., F ( ) = ( + ) Note that the controller hould have a ole at the origin to avoid teady tate tracking error. In thi cae, a ingle QFT deign could not cover the whole uncertainty range. It i imoible to find a QFT controller-refilter that meet exlicit bound on tability and tracking ecification in () and (), and that cut off imultaneouly the imlicit cot of feedback in (3). Thi erformance limitation wa argued in Section. Fig. 3 imulate the time domain erformance of the aforementioned QFT deign for the lant with J=Jmax, b=bmax, K=Kmin. Thi i the lant with more difficultie to track the inut ignal ince it rereent the bigget load and friction for the mallet motor contant; and it need the maximum control effort value; thi lant i alo the nominal lant taken in the QFT deign. Neverthele, the wort aturation effect (higher cot of feedback) occur in the lant with the bigget L, that i, the lant which correond to J=Jmin, b=bmin and K=Kmax. Fig. 3(a) how an accetable tracking erformance for J=Jmax, b=bmax and K=Kmin. However, the enor noie i highly amlified at the control inut a Fig. 3(b). The ituation deteriorate for J=Jmin, b=bmin and K=Kmax (Fig. 4(b)). In ractice, thi huge cot of feedback aturate the armature core, oiling the exected erformance in Fig. 3(a) a hown by Fig. 4(a). The cot of feedback can be reduced by relaxing the tracking ecification. In other word, the cot of feedback reduction i not for free. It alo imlie oiling the control tracking erformance. The olution lanned in thi aer i the diviion of the uncertainty. Thu, witching QFT deign i imlemented to overcome thi difficulty. An identifier-baed etimator Σ for thi ytem may be contructed of the following form a in [7]. x y A = = c x x A b + y + u b () where (A, b ) a table arameter-indeendent, controllable air, and A b + c A,, c b () i a tabilizable realization of M(). A tate-hared imlementation of thi multi-etimator i then ued in the uervior. It i oible to contruct a multi-etimator Σ for thi examle by icking any one-dimenional controllable air (A, b ) with A table, and then defining c o that it rereent the lant uncertainty. For imlicity, (A, b ) i choen to be in control canonical form and that A characteritic olynomial ω i + ω, where ω i a deign arameter. Under thee condition c aear to be the following vector: b K c : = ω J J The erformance ignal μ, P are then contructed uing the idea of tate-haring in a imilar way a in [3], [7]. rotor eed (rad/) field voltage (V) (a) (b) Fig. 3. Simulation reult for the lant J=Jmax, b=bmax and K=Kmin in the full uncertainty et (Single QFT deign) (a) outut tracking erformance and (b) control effort. rotor eed (rad/) field voltage (V) (a) (b) Fig. 4. Simulation reult for the lant J=Jmin, b=bmin and K=Kmax in the full uncertainty et (Single QFT deign) (a) outut tracking erformance and (b) control effort. Namaki-Shouhtari & Khaki Sedigh: nhancement of Robut Tracking Performance 33

7 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 The J uncertainty contribute trongly in both magnitude and hae uncertainty in the temlate at mid frequencie. In other word, J i the mot ignificant uncertain arameter in the arameter ace. In the following, we divide the uncertain ytem into maller ubytem baed on thi uncertain arameter. So, to deign the multile-model baed witching architecture the rotor inertia uncertainty i divided into two maller art, J = [..5] and J = [.5.]. Now, for each uncertainty region, a QFT deign i emloyed to achieve robut tability and erformance deite the correonding uncertaintie. The reulting controller are a follow: G ) = + 6.., G ( + ) 6.7 ( ( ) = ( + ) 8.6 The refilter for thee new deign are the ame a the reviou one ued in the firt QFT deign. ventually, a uerviory architecture determine the active controller, which hould be laced in the feedback loo. Thi election i baed on the value of monitoring ignal. A chematic diagram of the overall multile-model baed witching algorithm i deicted in Fig. 5. Further detail on the imlementation of Adative Control via Switching and Suerviory can be find in [3] (chater 6) and [4] (chater 5). To roceed with imulation, aume that the arameter of the real (unknown) lant are J=Jmax, b=bmax, K=Kmin. In addition, the hyterei contant h in the witching law (7) i et to h =.. Suoe that the econd candidate controller (G ) i connected into the loo initially, that i, σ() =. Fig. 6 deict the cloedloo inut outut trajectorie, which how atifactory cloed-loo erformance. The witching ignal identifie the right controller via one witch. Furthermore, we et the firt controller initially, i.e., rotor eed (rad/) field voltage (V) (a) (b) Fig. 6. Suerviory Switching QFT control imulation for for the lant J=Jmax, b=bmax and K=Kmin (a) outut tracking erformance and (b) control effort. rotor eed (rad/) field voltage (V) (a) (b) 6 4 Fig. 5. The Suerviory Baed Switching QFT control architecture [] Fig. 7. Suerviory Switching QFT control imulation for the lant J=Jmin, b=bmin and K=Kmax (a) outut tracking erformance and (b) control effort. σ() =, and carry out the imulation again for lant which correond to J=Jmin, b=bmin and K=Kmax. Simulation reult are given in Fig. 7. The right controller i identified by the uervior and the cloed loo erformance i atifactory. Fig. 6 and 7 verify the time erformance of the witching QFT deign. Comare them to Fig. 3(a) and Fig. 4(a), and note that G and G imrove the tracking erformance yielded by G (ingle QFT deign for the full uncertainty). Beide, G and G reduce coniderably the cot of feedback. 34 Iranian Journal of lectrical & lectronic ngineering, Vol. 8, No., March

8 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 6 Concluion Thi aer dealt with the feedback erformance limitation for highly uncertain lant. A large uncertainty may imly a oor tracking erformance to avoid an exceive cot of feedback, which would aturate the actuator, and guaranteeing an accetable robut tability. The olution rooed wa to uitably divide the uncertainty region. Different QFT controller were emloyed to maximize the feedback benefit and minimize the cot of feedback in their uncertainty ubet. In addition, they were robutly table in the full uncertainty domain. The alication of QFT in witching multile model baed adative control wa reented. The control tructure rooed conit of a bank of candidate tate-hared comenator and a uervior. ach of the candidate controller i deigned in order to achieve the demanded erformance in a region of the lant uncertainty. The uervior conit of a tate-hared multi-etimator, a erformance ignal generator and a hyterei witching logic cheme. The uervior chooe the active controller correonding to the local model which bet fit the lant data. Simulation reult on a ractical examle how the erformance of thi algorithm for controlling highly uncertain lant. Reference [] Horowitz I.M., Synthei of Feedback Sytem. Academic Pre: New York, 963. [] Horowitz I., Survey of quantitative feedback theory (QFT), International Journal of Control; Vol. 53, No.,. 55-9, 99. Alo in International Journal of Robut and Nonlinear Control; Vol., ,. [3] Houi C. H., Ramuen S. J. and Garcia-Sanz M., Quantitative Feedback Theory: Fundamental and Alication (nd edition). A CRC Pre Book, Taylor & Franci: Florida, 6. [4] Bode H.W., Network Analyi and Feedback Amlifier Deign. Van Notrand: New York, USA, 945. [5] Heanha J. P., Liberzon D. and More A. S., Overcoming the limitation of adative control by mean of logic-baed witching, Sytem and Control Lett., Vol. 49, No., , 3. [6] Böling J. M., Seborg D.. and Heanha J. P., Multi-model adative control of a imulated H neutralization roce, Control ngineering Practice, Vol. 5, , 7. [7] Fekri S., Athan M. and Pacoal A., Iue, rogre and new reult in robut adative control, Int. J. Adat. Control Signal Proce.; Vol., , 6. [8] Garcia-Sanz M. and lo J., Beyond the linear limitation by combining Switching & QFT. Alication to Wind Turbine Pitch Control Sytem. Secial Iue: Wind Turbine: New Challenge and Advanced Control Solution. International Journal of Robut and Nonlinear Control, Vol. 9, No.,. 4-58, 9. [9] Gil-Martinez M. and Garcia-Sanz M., Simultaneou meeting of robut control ecification in QFT, International Journal of Robut and Nonlinear Control, Vol. 3, , 3. [] Namaki-Shouhtari O. and Khaki Sedigh A. Deign of Suerviory Baed Switching QFT Controller for Imroved Cloed Loo Performance in Proceeding of the 8th Iranian Conference on lectrical ngineering, IC, Ifahan, Iran, May, [] Borgheani C., Chait Y., and Yaniv O., Matlab Quantitative Feedback Theory Toolbox v., Teraoft, Inc., 3. [] Horowitz I. M. and Sidi M., Synthei of feedback ytem with large lant ignorance for recribed time-domain tolerance, Int. J. Control, Vol. 6, No., , 97. [3] Liberzon D., Switching in Sytem and Control. Birkhäuer, Boton, 3. [4] Sun Z., and Ge S. S., Stability Theory of Switched Dynamical Sytem, Sringer-Verlag London,. [5] Heanha J. P., Liberzon D., and More A. S., Bound on the number of witching with cale indeendent hyterei: Alication to uerviory control, In Proc. of the 39th Conf. on Deciion and Contr., Vol. 4, , Dec.. [6] Heanha J. P., Tutorial on uerviory control. Lecture note for the workho control uing logic and witching for the 4th I CDC, Orlando, FL,. [7] More A. S., Suerviory control of familie of linear et-oint controller-art : exact matching, I Tran. Automat. Control, Vol. 4, No., , 996. [8] More A. S., Control uing logic-baed witching. In A. Iidori, editor, Trend in Control: an uroean Perective, age Sringer-Verlag, London, 995. [9] Gil-Martinez M. and Garcia-Sanz M., Robut tracking erformance enhancement through uncertainty diviion, In Proc. of the 7th uroean Control Conference, CC'3, Cambridge, United Kingdom, Setember 3. [] Khaki-Sedigh A. and Luca C., Otimal deign of robut quantitative feedback controller uing random otimization technique, International Journal of Sytem Science; Vol. 3, No. 8,. 43-5,. Namaki-Shouhtari & Khaki Sedigh: nhancement of Robut Tracking Performance 35

9 Downloaded from ijeee.iut.ac.ir at 5:43 IRST on Saturday January 5th 9 Omid Namaki-Shouhtari received hi B.Sc. and M.Sc. degree in lectrical ngineering (control ytem) from Univerity of Tehran, Tehran, Iran, in and 5, reectively. He received hi Ph.D. in lectrical ngineering (control ytem) in from K.N. Tooi Univerity of Technology, Tehran, Iran. Hi reearch interet include witching uerviory control, advanced roce control, robut adative control and nonlinear control ytem. Ali Khaki Sedigh i currently a rofeor of control ytem with the Faculty of lectrical and Comuter ngineering, K.N. Tooi Univerity of Technology, Tehran, Iran. He obtained an honor degree in mathematic from Univerity of Newcatle uon Tyne in 983, mater in Control Sytem from UMIST in 985, and Ph.D. in Control Sytem from Univerity of Salford in 988, all in UK. He i the author and co-author of about eventy journal aer and ha ublihed ten book in the area of control ytem. Hi main reearch interet are adative and robut multivariable control ytem, chao control, hybrid ytem, and the hitory of control. 36 Iranian Journal of lectrical & lectronic ngineering, Vol. 8, No., March