Design of Robust Reference Input Tracker via Delayed Feedback Control Method

Transcription

1 Design of Robust Reference Inut Tracker via Delayed Feedback Control Method Zahed Dastan, Mahsan Tavakoli-Kakhki * Faculty of Electrical Engineering, KN Toosi University of Technology, Tehran, Iran * Eail: atavakoli@kntuacir Abstract: In this aer using the delayed feedback control ethod, a new robust tracking syste is designed The roosed syste utilizes all the features of the conventional state feedback with integral action and it is able to reject higher order disturbances coared to the conventional ethod In addition, the roosed tracking syste tracks the ra-shae reference inut signal as well, which it is not ossible through the conventional state feedback Since the roosed syste is easy to ileent, it can be used in ractical alications effectively Moreover, the siulated annealing algorith is used to address the issues in designing the roosed control syste araeters Finally, by an exale, the roosed reference inut tracker is siulated and the results are resented Keywords: Delayed feedback, Integral control, Robust tracking Introduction In recent years, the effects of tie delays on different systes behaviors have been widely taken into consideration This is due to the existence of delays in any tye of systes, including biological [], econoic [, 3], engineering [4] and social systes [5] A ublic view in the field of control syste engineering is that delays are always harful and they ay cause instability or decrease the stability argins It ight be asked that whether it is ossible to use delays in order to stabilize or irove the systes erforance or not An attet to resond this uestion is tie delay control or TDC This ethod at first was roosed to overcoe uncertainties in syste dynaics [6] TDC is used in a variety of contexts [7-] Furtherore, tie delays can be used as tie delay filters [3-6] and tie delay observers [7, 8] Another secial for of utilizing delays in the syste control ethods is tie-delayed feedback control or tiedelay auto synchronization The ain idea of using delayed feedback was roosed by Pyragas in the field of stabilizing unstable oscillations in chaotic systes [9] In this ethod, the control force is created by the difference between the current and the delayed value of the syste states, eg, x ( t x ( t τ Delayed feedback has been used in different alications such as engineering [0], cheistry [, ] and edicine [3] This tye of feedback has several advantages Firstly, it can be ileented easily Secondly, this ethod is non-invasive It eans that when the desired state is achieved, the control inut vanishes In addition, delayed feedback has been used as an aroxiation of derivative feedback and hase synchronizer [4, 5] Detailed inforation about the delayed feedback control ethod can be found in [6] Delayed feedback can stabilize fixed oints and steady states as well as eriodic orbits [7] As a result, its alications are beyond stabilizing eriodic orbits Delays would be exloited to irove the syste erforance as a tracker For instance, in [8] the tracking error has been considered as a erforance criterion and it has been shown that the existence of delay in the feedback loo decreases the steady state tracking error in the certain tyes of systes However, to the best knowledge of the author, utilizing delayed feedback features in tracking the reference inut has not been widely discussed In this aer, by using delayed feedback ethod in the structure of the conventional integral control [9], a control loo is designed in order to stabilize an unstable syste and track ste/ra reference inut Most of the roosed ethods in the context of the disturbance rejection benefit fro observers in order to eliinate the disturbance effects on the control syste outut [30-38] The roosed reference inut tracker in this aer utilizes delayed feedback control features by which it is ossible to reject certain tye of disturbances without eloying observers In addition, using non-invasive feature of the delayed feedback, an extra ter is alied to the integral state which causes rearkable benefits to the tracking syste The roosed delayed feedback control ethod brings several advantages to the syste However, by eerging delay ters in the syste euations, it is laborious to find the aroriate values of the controller araeters in order to stabilize the unstable syste Several ethods have been roosed in the literature to stabilize systes that involve delays in their euations These ethods would be classified to freuency doain tests such as atrix encil [39] or tie doain techniues that utilize Lyaunov s direct ethod [40] Detailed discussions about these ethods can be found in [4] A thorough search of the relevant literature revealed that designing the roosed delayed feedback controller through these ethods would be a cubersoe task For instance, in [4] a delay deendent LMI ethod has been roosed to exaine the stability of a tie delayed syste Although this ethod deterines a delay interval for which the syste can be stabilized, any attets to find the roosed delayed feedback controller araeters by such a ethod arises a nonlinear atrix ineuality which obviously reuires a lot of effort to solve On the other hand, the freuency doain techniues in ost cases set a bound on the delay values for which the syste is stable But these techniues norally do not reresent a direct rocedure to deterine the controller araeters [4]

2 In this aer, the delayed feedback control structure is used to introduce a new for of the reference inut tracking syste This new ethod which is based on the structure of the conventional state feedback with integral action tracking syste enhances the conventional tracking ethod robustness and tracking roerties Moreover, due to the aforeentioned difficulties for designing the roosed delayed feedback controller, utilizing siulated annealing algorith, we have roosed a new ethod for deterining the delayed feedback controller araeters This aer is organized as follows In section, the conventional tracking syste based on state feedback with integral action is resented The roosed tracking syste is designed in section 3 and the siulated annealing algorith is introduced in order to deterine the roosed syste araeters Robustness and tracking characteristics of the roosed syste is investigated in section 4 and 5 resectively Section 6 illustrates a nuerical exale of the roosed syste and the aer ends with conclusions drawn in Section 7 Preliinaries State feedback with integral action is an ordinary reference tracking aroach in the state sace odel This tracking aroach eliinates the effects of the ste-tye disturbances on the syste outut and it is alost robust against the changes of the syste araeters This tracking ethod has been fully described in [9] Consider the state sace odel of a controllable syste as follows where n x R, xɺ ( t = Ax ( t Bu( t, y = Cx ( t A n n R, B n r R, C n R and ( y R The integral state ( t R is considered such that ( holds ɺ ( t = y ( t r ( t = Cx ( t r ( t, ( where r ( t stands for the reference inut Considering the integral state ( t, the state sace odel of the augented syste is obtained as xɺ ( t A 0 x ( t B 0 u ( t r, ( t = C 0 ( t 0 I ɺ x ( t y ( t = [ C 0 ] ( t In conventional ethod, it is shown that for a controllable x ( t syste there is a state feedback u( t = [ K K ] ( t such that the closed loo syste becoes stable and for roerly chosen values of the coefficients K and K, the integral sate ( t goes to a steady state In other words, the following relation holds li ɺ ( t = li ( y ( t r = 0, (4 t t (3 which results in li y ( t = r (5 t Such a control syste rejects the effect of the ste-tye disturbances on the syste outut The block diagra of an integral control syste is deicted in Fig Reference tracking by delayed feedback controller In this section, the tracking syste with the delayed feedback control structure is designed Using the noninvasive feature of the delayed feedback, the integral state ( t is odified and finally an algorith is roosed to deterine the roosed delayed feedback control structure araeters 3 Stabilizing augented syste In this subsection, instead of using the conventional state feedback for stabilizing the augented syste described in section, a new for of the delayed feedback controller is used to stabilize the augented syste To this end, the integral state ( t is considered for which relation ( holds and it is roosed that the control r signal u( t R is alied as follows u ( t = K ( x ( t i K ( t, i τ i = 0 i (6 r n r where the coefficients K R, K R and the delay value τ > 0 are chosen such that the delayed closed loo augented syste becoes stable Substituting (6 in (3 the state sace odel of the delayed closed loo augented syste is obtained as (7 xɺ ( t A BK BK x ( t ( t = C 0 ( t ɺ BK 0 i x ( t iτ 0 ( r, 0 0 i = i ( t iτ I x ( t y ( t = [ C 0] ( t Choosing u ( t as (6, which is a new for of delayed feedback, does not change the non-invasive feature of the delayed feedback structure since as x ( t reaches to the steady Fig Block diagra of the conventional state feedback with integral action (7

3 Fig Block diagra of the roosed delayed feedback tracker with the odified for of the integral state state the value of i ( x ( t iτ i = 0 i goes to zero Moreover, this new for of the delayed feedback structure leads to roerly disturbance rejection by the closed loo syste This issue will be discussed later 3 Modified for of the integral state As it was entioned, delayed feedback is a noninvasive control ethod Using this feature, in this aer it is roosed to add an extra ter to ɺ ( t This extra ter brings noticeable interests to the closed loo syste which are elaborated in the following sections The roosed ɺ ( t is as follows ɺ ( ( ( ( (, (8 t = Cx t r K t t τ Where the coefficient K R can be chosen roerly by the designer Modifying ɺ ( t as (8, does not change the steady state value of the syste outut Because when the syste outut reaches to its steady state, ( t will be eual to ( t τ and the extra ter in (8 will be eliinated The advantages of adding this new ter to ɺ ( t are higher order disturbance rejection and tracking the ra shae reference inuts by the closed loo syste Eventually, by considering the integral state ( t in (8, the state sace odel of the odified closed loo syste is obtained as xɺ ( t A BK BK x ( t ( t = C K ( t ɺ BK 0 i x ( t iτ ( 0 0 i = i ( t iτ 0 0 x ( t τ 0 r, 0 K ( t τ I x ( t y ( t = [ C 0] ( t (9 Accordingly, the araeters ( K, K, K, τ, τ are designed such that (9 is stabled and at the sae tie the reference tracking objective is fulfilled The roosed delayed feedback tracker with the odified for of the integral state is deicted in Fig 33 Designing rocedure for the controller araeters ( K, K, K, τ, τ In this subsection, it will be discussed that how the araeters of the roosed delayed feedback control structure are deterined in order to stabilize the closed loo syste and track the reference inut The syste euation (9 can be rewritten as follows i Gɺ ( t = A0G ( t A ( G ( t iτ i = i A G ( t τ Tr ( t, where x ( t A BK BK BK 0 G ( t, A0, A ( t C K A =, T = 0 K I (0 The characteristic euation of the syste euation (0 is given by i siτ det si A0 A ( e Ae = 0 i = i ( i By defining E (, i (, the characteristic i = i euation ( can be rewritten in the following coact for si s ( si A0 AE i e τ Ae τ det (, = 0 ( The riary objective is to deterine the araeters of the delayed feedback structure such that all the roots of the characteristic euation ( lie in the left half lane As it was 3

4 entioned before, usually deterining the araeters of the delayed feedback control structure through the available ethods for the stability of delayed systes reuires a lot of effort However, in [43] a ethod has been roosed called continuous ole laceent which is based on the sensitivity of rightost roots with resect to the controller araeters This ethod oves rightost roots of the characteristic euation to the left half lane by alying sall erturbations to the controller araeters The sall changes of the controller araeters are obtained fro the following euation = Λ, (3 d K S T where K = [ k k ] n indicates the sall dislaceent of the controller araeters, λi is the n λi characteristic root, S = si, j R, si, j = k is the sensitivity atrix, S is the Moore Penrose inverse of S T d d d and Λ = λ λ is sall dislaceent of the rightost roots This algorith decreases the real art of the rightost roots until two roots coincide and the sensitivity atrix becoes large In essence, in [43] there are no evidences to indicate that while this algorith breaks down, the rightost roots are in the left half lane In order to tackle this liitation, it is straightforward to use a nuerical algorith which is able to avoid getting stuck in local inia Siulated annealing (SA algorith introduced by Kirkatrick [44], is a robabilistic techniue to find a global otiu of a function This algorith was firstly insired by statistical echanics and has been alied to several otiization robles such as traveling salesan roble [45] Since we are not searching for a recise global iniu of the rightost eigenvalue of (, and in ost ractical alications, decreasing rightost root below a threshold fulfills our objectives, this algorith would be a suitable otion SA algorith has been widely discussed and used for variety of alications [46-50] Accordingly, assutions and atheatical details of the algorith will not be discussed herein, and we directly utilize this ethod for designing delayed feedback controller araeters In order to aly SA to our roble, an energy function E ( W ust be defined which shows the energy of the syste at the state W In this case, E ( W indicates the rightost root of the characteristic euation ( and the state W is the set of the controller araeters or W ( K, K, K, τ, τ Another iortant eleent in SA is the teerature T which should be initialized aroriately for different otiization robles Finally, the P robability of acceting a state change is deterined by the Boltzann distribution of the energy difference of the two states as follows j By these definitions, we introduce the following SA algorith for deterining the delayed feedback controller araeters W in order to stabilize (9 The SA Algorith for deterining the roosed delayed feedback controller araeters Ste Initializing the controller araeters W by rando values Ste Initializing T with a large value Ste 3 Reeat: a Reeat: I Alying rando erturbation to the controller araeters as W = W W II Couting E = E ( W W E ( W : if E < 0, kee the new set of the controller araeters W else accet the new set of the controller araeters with the robability: E T P = e until the nuber of the acceted changes is less than a secific threshold level b Set T = T T until T is close enough to zero or E ( W reaches a desired value which can be deterined by the designer In this algorith, E ( W or the rightost root of ( can be calculated effectively using the Matlab ackage DDE- BIFTOOL [5] Notice that in each iteration, W ust be chosen according to the hysical constraints on the controller araeter values, ie delays ust be ket ositive during the algorith Moreover, in [5], it has been shown that if T is decreased such that T k T 0 (, k =,, ln( k (5 where k indicates the iteration nuber and T0 is a sufficiently large initial teerature, the SA algorith will find the global otiu with robability value one Disturbance Rejection In this section, the robustness of the roosed control aroach is discussed in the resence of the disturbance signals d ( t and d ( t shown Fig 4 Rejection of disturbance signal d ( t In this subsection, the effect of the disturbance signal d ( t on the syste states is studied To this end, the disturbance signal d ( t is added to the control signal u ( t in (6 and afterwards by considering ( 0 r t = the closed loo syste euations are obtained as below E T P = e (4 4

5 xɺ ( t A BK BK x ( t ( t = C K ( t ɺ BK 0 i x ( t iτ ( 0 0 i = i ( t iτ 0 0 x ( t τ 0 K ( t τ BK 0 i d ( t iτ ( 0 0 i = 0 i 0, x ( t y ( t = [ C 0] ( t (6 According to (8, the Lalace transfor of ( t, ie Q ( s in ters of the Lalace transfor of x ( t, ie X ( s is described as ( ( ( ( Q s = si K e CX s (7 In addition, based on the syste euations (6 the following euality holds { ɺ } = { L x ( t L Ax ( t BK ( t BK Substituting Q ( s following euality i = 0 i ( x ( t iτ i i BK ( d( t iτ i = 0 i (8 fro (7 into (8, eventuates to the s X ( s = si A BK ( e τ BK ( si K ( e C BK ( e D ( s (9 where D ( s is the Lalace transfor of d ( t The euality (9 can be also rewritten in the following for ( = ( X s ss I sa sbk e e BK ( I K ( C BK ( e D ( s s Now, suose D ( s follows (0 affecting all the easured states as D( s = = V n, s ( s n where V = Substituting D ( s in ( into (0 and n alying the final value theore results in ( s 0 s 0 li sx ( s = li s s I sa sbk ( e e BK( I K ( C BK ( e V s s ( e Noticing that li = τ 0, relation ( is silified s 0 s to li ( τ ( s 0 s 0 BK I K C BK e V = s ( ( τ ( τ li det( I K BK adj I K C BK ( e V s (3 In (3, adj( stands for the adjoint of the atrix Fro (3, it can be inferred that the steady state value of x ( t in the resence of ( d t = t V euals zero if we have the following conditions det( = 0, (4 I K τ According to (4, in order to reject the disturbance signal ( d t = t V, the atrix K τ ust have soe eigenvalues in It is worthwhile to state that if (4 is not fulfilled, the closed loo syste is able to reject the disturbance signal ( d t = t V Benefiting fro the control signal u ( t in (6 the roosed control structure is able to reject all the disturbance signal, Z d t t V ( = for Moreover, using the integral state ( t as (8 ay irove the disturbance rejection roerty of the closed loo syste by one ore order 4 Rejection of disturbance signal d ( t In this subsection, it is investigated that for an aroriate choice of the atrix K τ, the closed loo syste rejects the ste/ra tye of the disturbance signal d ( t Considering the disturbance signal d ( t and r ( t = 0 the closed loo syste euation is obtained as follows 5

6 xɺ ( t A BK BK x ( t ( t = C K ( t ɺ BK 0 x ( t iτ i ( 0 0 i = i ( t iτ 0 0 x ( t τ I d ( t, 0 K ( t τ 0 x ( t y ( t = [ C 0] ( t (5 According to (5, the Lalace transfor of x ( t is given by X ( s = si A BK ( e (6 BK ( si K ( e C D ( s Siilar to (0, X ( s can be rewritten in the following for According to (9 and considering Q ( s as the Lalace transfor of ( t the following euality is resulted (9 X ( s = si A BK ( e BK Q ( s Fro (8 and (9 it is concluded that Q ( s = si C si A BK ( e BK K ( e R ( s (30 Moreover fro (8 and according to Fig, the Lalace transfor of the tracking error e( t = y ( t r ( t is given by E ( s = si K ( e Q ( s (3 Substituting Q ( s fro (30 into (3 results in ( = ( X s ss I sa sbk e e (7 BK ( I K ( C D ( s s E ( s = si K ( e [ si K ( e C si A BK ( e BK R( s (3 Suose the Lalace transfor of d ( t as D ( s = V n s Now, alying the finite value theore results in li sx ( s = s 0 ( ( I K τ BK ( I K τ C V det( adj (8 Fro (8 it is induced that the steady state value of x ( t euals zero ie the ste/ra tye disturbance d ( t is rejected if the euality (4 is fulfilled Fro the detailed discussions in this section, it is concluded that the roosed control ethod in this aer is robust against the disturbance signal d ( t of any order showing in Fig Moreover, it is robust against the ste/ra tye disturbance signal d ( t in Fig As a result, the roosed control syste iroves the disturbance rejection characteristic of the conventional state feedback by alying sile and ileentionally convenient delayed feedback control ethod As it has been discussed in literatures, an aroach to reject a ra tye disturbance by a conventional tracking control ethod is to use a derivative feedback [53] But such an aroach causes a large control signal which is not adissible in ractical alications Tracking ste and ra tye reference inuts In this section, it is shown that satisfying a secific condition the roosed control syste in this aer recisely tracks the ra tye reference inuts as well as the ste tye inuts Suose the Lalace transfor of the reference inut r ( t as R ( s = M where M k = s Alying the final value theore yields to the following euality e li se ( s = li s I K ( s 0 s 0 ( ( ( si C si A BK e BK K ( e M k = s ( ( ( li I K τ C A BK M s 0 k s s (33 Fro (33, it is inferred that for k = ie the ra reference inut, the steady state value of the tracking error euals zero if the euality I K τ = 0 is satisfied Also, fro (33 it is obvious that tracking the reference inut is definitely achieved for k = ie for the ste tye reference inut Nuerical Exale In this section, a nuerical exale is rovided to show the efficiency of the roosed control aroach in reference tracking and disturbance rejection by the closed loo syste Consider the state sace atrices of the controllable syste ( are given as follows A =, B =, C = [ 5 ], 5 6

7 a b Fig 3 Designing the delayed feedback controller araeters with SA algorith (a Delayed feedback controller araeters K = [ k k ], K = k and τ with resect to the iteration nuber of the SA algorith for = and τ = 0 (b The real art of the rightost root of the characteristic euation ( with resect to the iteration nuber of the SA algorith for = and τ = Disturbance on x Disturbance on x Real Part of the Rightost Root Iteration Nuber 0 Reference Inut Conventional State Feedback Proosed Delayed Feedback d (t y(t a b u(t c Fig 4 The corresonding signals of the roosed delayed feedback with = and τ = 0 and the conventional state feedback with ste reference inut (a Disturbance signal d ( t iosed on the syste states x ( t and x ( t (b Coarison of the closed loo syste oututs by alying the conventional state feedback and the roosed delayed feedback ethod in the resence of the disturbance signal d ( t in Fig 4a (c Control signal u( t corresonding to Fig 4b (d Tracking error signals corresonding to Fig 4b d where the atrix A has two eigenvalues 05 and 5 in the right half lane Considering u ( t in (6 for = and the integral state as (8 for τ = 0, the SA algorith is used to find the aroriate =, K = k and τ in (6 Fig 3a indicates the delayed feedback controller araeters with resect to the iteration nuber of the SA algorith Furtherore, Fig 3b shows the real art of the rightost root of the characteristic euation ( with values of the controller araeters K [ k k ] resect to the iteration nuber of the SA algorith As it is seen in these figures, the SA algorith stos when the real art of the rightost root becoes less than a redefined threshold which is considered to be eual - in this exale According to the obtained results the values of these K = 0339, K = 0553, araeters are chosen as [ ] and τ = 04 The disturbance signal d ( t on the syste sates is shown in Fig 4a As this figure deicts the disturbance signal 7

8 a b Fig 5 Designing the delayed feedback controller araeters with SA algorith (a Delayed feedback controller araeters K = [ k k ], K = k,τ and τ with resect to the iteration nuber of the SA algorith for = and τ 0 (b The rightost root of the characteristic euation ( with resect to the iteration nuber of the SA algorith for = and τ 0 a Fig 6 Disturbance signals iosed on the closed loo syste in Fig (a d ( t arabolic disturbance signal iosed on x ( t and x ( t states (b d ( t disturbance signal iosed on x ( t and x ( t states b y(t d (t a b c Fig 7 The corresonding signals of the roosed delayed feedback with = and τ 0 and outut of the conventional state feedback with ste reference inut (a Closed loo syste outut with the roosed delayed feedback in the resence of the siultaneous disturbance signals d ( t in Fig 6a and d ( t in Fig 6b (b Control signal u( t corresonding to Fig 7a (c Tracking error signal corresonding to Fig 7a (d Closed loo syste outut with the conventional state feedback in the resence of the siultaneous disturbance signals d ( t in Fig 6a and d ( t in Fig 6b d 8

9 y(t u(t a b c Fig 8 The corresonding signals of the roosed delayed feedback with = and τ 0 and outut of the conventional state feedback with a cobination of ra and ste reference inut (a Closed loo syste outut with the roosed delayed feedback by alying ste and ra reference inuts and in the resence of the siultaneous disturbance signals d ( t in Fig 6a and d ( t in Fig 6b (b Control signal u( t corresonding to Fig 8a (c Tracking error signal corresonding to Fig 8a (d Closed loo syste outut with the conventional state feedback by alying ste and ra reference inuts and in the resence of the siultaneous disturbance signals d ( t in Fig 6a and d ( t in Fig 6b d d ( t on the syste sate x ( t is alied as a cobination of ste and ra signals Oututs of the closed loo syste by alying the conventional state feedback ethod and the roosed delayed feedback ethod in the resence of the disturbance signal d ( t are shown in Fig 4b Moreover, according to the Fig 4c, the control signal u( t of the roosed delayed feedback reains bounded during the tracking rocess Also, Fig 4d indicates the error signals corresonding to Fig 4b As it is seen in these figures, the conventional state feedback ethod cannot reject the disturbance signal d ( t, while the roosed delayed feedback ethod oits the effect of the disturbance signal d ( t in the closed loo syste outut coletely It is worth entioning that to ake a fair coarison, three roots of the characteristic euation by alying the conventional state feedback ethod has been located in the sae lace of the three doinant roots of the characteristic euation by alying the roosed delayed feedback ethod These laces are 36 ± i, 679 In the following, the delayed feedback signal u ( t in (6 is considered with = and τ 0 Moreover, K = τ is considered Siilar to the revious design, / the SA algorith is used to obtain the other control araeters K = [ k k ], K = k and τ in (6 and τ in (8 According to the obtained result in Fig 5a the value of K = , these araeters are chosen as [ ] K = 4346, τ = 034 and τ = 044 Fig 5b also shows the real art of the rightost root of the characteristic euation ( with resect to the iteration nuber of the SA algorith Fig 6a and Fig 6b deicts the disturbance signals d ( t and d ( t on the syste states As it is seen, the disturbance signal d ( t is a arabolic signal and the disturbance signal d ( t is a cobination of ste and ra signals Outut of the closed loo syste by alying the roosed delayed feedback ethod in the resence of the siultaneous disturbance signals d ( t and d ( t is reresented in Fig 7a In addition, Fig 7b ilies that under these circustances, the control signal u( t of the roosed delayed feedback still reains bounded Also, the tracking error of the roosed control syste in this condition is deicted in Fig 7c As it is seen, the roosed syste is able to track the reference inut signal and reject both of the disturbances in the steady state without any error Moreover, as it is obvious fro Fig 7d, the conventional state feedback ethod cannot handle such a cobination of disturbance signals Therefore, other corresonding signals have not been coared in this case Finally, Fig 8a shows the reference inut tracking of the designed control syste by alying a cobination of ste and ra reference inut signals and also iosing siultaneous disturbance signals d ( t and d ( t Likewise, it is seen fro Fig 8b that the control signal u( t is bounded Also, the tracking error under these circustances is resented in Fig 8c As what was roved in Section 4, this 9

10 sohisticated cobination of reference inut and disturbance signals can be also effectively handled by the design control syste based on the delayed feedback ethod Whereas, Fig 8 shows that under these circustances, the conventional state feedback is coletely unable to track the reference inut signal In suary, the coarison between the roosed delayed feedback and the conventional state feedback tracking ethods is given in Table Conclusion In this study, based on the conventional integral control tracking ethod and delayed feedback control schee, a new for of a set oint tracker has been roosed The roosed syste stabilizes conventional integral control tracking syste using delayed feedback and also benefits fro all of the state feedback ethod features such as ste disturbance rejection and robustness to the changes of the syste araeters However, due to the delayed feedback characteristics, the roosed ethod is able to reject higher order disturbances Besides, based on the non-invasive feature of the delayed feedback, a new ter is added to the integral state which increases the closed loo syste disturbance rejection roerty by one order Moreover, by satisfying a sile condition, this new ter will cause the closed loo syste to track ste/ra reference inut signals The roosed syste ossesses a convenient ileentation feature which akes it a suitable choice for ost ractical alications Table Coarison between the roosed delayed feedback and the conventional state feedback tracking ethods Order One Two Higher d ( t rejection by the roosed delayed Yes Yes Yes feedback ethod d ( t rejection by the conventional state Yes No No feedback ethod d ( t rejection by the roosed delayed Yes Yes No feedback ethod d ( t rejection by the conventional state Yes No No feedback ethod Reference inut tracking by the roosed delayed Yes Yes No feedback ethod Reference inut tracking of the conventional state feedback Yes No No References [] MacDonald, N, 'Biological Delay Systes: Linear Stability Theory', (Cabridge University Press, 989, edn [] Chukwu, EN, 'On the Controllability of Nonlinear Econoic Systes with Delay: The Italian Exale', Alied Matheatics and Coutation, 998, 95, (, [3] Zhou, L and Li, Y, 'A Dynaic Is-L Business Cycle Model with Two Tie Delays in Caital Accuulation Euation', Journal of Coutational and Alied Matheatics, 009, 8, (, 8-87 [4] Allen-Prince, M, Thoas, C, and Yi, S, 'Analysis of the Effects of Counication Delays for Consensus of Networked Multi-Agent Systes', International Journal of Control, Autoation and Systes, 07, 5, (5, [5] Peña, C and Moreno, E, 'Delay at Bus Stos of Transilenio Transort Syste According to Paraeters Measured in Situ Case Study Bogotá-Colobia', Procedia - Social and Behavioral Sciences, 04, 60, -9 [6] Youcef-Toui, K and Ito, O, 'A Tie Delay Controller for Systes with Unknown Dynaics', in, 988 Aerican Control Conference, (988 [7] Jung, JH, Chang, PH, and Stefanov, D, 'Discretisation Method and Stability Criteria for Non-Linear Systes under Discrete-Tie Tie Delay Control', IET Control Theory & Alications, 0, 5, (, [8] Cui, P, Zhang, D, Yang, S, et al, 'Friction Coensation Based on Tie-Delay Control and Internal Model Control for a Gibal Syste in Magnetically Susended Cg', IEEE Transactions on Industrial Electronics, 07, 64, (5, [9] Tavakoli, SD, Kadkhodaei, G, Hazeh, M, et al, 'Non- Linear Tie-Delay Controller for Dc/Dc Power Converters in Alication of Electric Vehicles', IET Electrical Systes in Transortation, 07, 7, (4, [0] Roy, S, Kar, IN, Lee, J, et al, 'Adative-Robust Tie- Delay Control for a Class of Uncertain Euler Lagrange Systes', IEEE Transactions on Industrial Electronics, 07, 64, (9, [] Jin, M, Kang, SH, Chang, PH, et al, 'Robust Control of Robot Maniulators Using Inclusive and Enhanced Tie Delay Control', IEEE/ASME Transactions on Mechatronics, 07,, (5, 4-5 [] Cho, GR, Chang, PH, Park, SH, et al, 'Robust Tracking under Nonlinear Friction Using Tie-Delay Control with Internal Model', IEEE Transactions on Control Systes Technology, 009, 7, (6, [3] Jing, M and Zhongyi, C, 'Residual Vibration Suression of Flexible Sacecraft Via Otial Arbitrary Tie-Delay Filter Cobined with Adative Controller', in, 0 IEEE International Conference on Couter Science and Autoation Engineering, (0 [4] Liang, C and Xie, J, 'Pd Tye of Tie Delay Filter Design in the Feedback Control Syste to Reduce High Freuency Noise', in, Proceedings of the 4th World Congress on Intelligent Control and Autoation (Cat No0EX57, (00 [5] Chunyan, L and Jianying, X, 'Tie Delay Filter Design for the Non-Miniu Phase Systes', in, Proceedings of the 000 IEEE International Conference on Control Alications Conference Proceedings (Cat No00CH376, (000 [6] Liang, C, Zhong, Q, and Xie, J, 'Tie Delay Filter Techniue in Tdo Feedback Control for Uncertain Flexible Systes', in, Proceedings of the 4th World Congress on 0

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