Uncertainty Reduction Through Active Disturbance Rejection

Transcription

1 8 Amerian Cntrl Cnferene Westin Seattle Htel, Seattle, Washingtn, USA June -3, 8 FrAI. Unertainty Redutin Thrugh Ative Disturbane Rejetin Jeffrey Csank and Zhiqiang Ga Department f Eletrial and Cmputer Engineering Cleveland State University Cleveland, Ohi 4445 Abstrat The theme f mdern ntrl is hw t get arund the unknwns, i.e. mdel unertainties and disturbanes, s that they d nt degrade what is valued: stability and perfrmane. That is, the unknwns are aepted as part f the system. Anther ptin perhaps, prpsed here, is t first make a frntal attak n the unknwns, t redue their effets and then, nly then, invke the existing well-established methdlgy t deal with the remnants. In partiular, it is shwn that the amunt f unertainties an be redued by way f ative disturbane rejetin, implemented in an inner lp t prdue a well-behaved plant, whih is then regulated by anther ntrller in the uter lp. What's new here is a tw degree f freedm design t deal with the unknwns: they are first atively estimated and rejeted; then the remaining unertainty, mstly in high frequeny, is dealt with by, say, an H ntrller. The result is a hybrid H -Ative Disturbane Rejetin Cntrl (H-ADRC) strategy. A mtin ntrl senari is used t illustrate hw the new apprah uld benefit prblem-slving in the real wrld. I. INTRODUCTION The thery and pratie f ntrl system design have lng had a symbiti relatinship. The frmer prvides the insight and understanding; the latter the utility. The mathematis f feedbak were disvered in lassial ntrl, whereas in mdern ntrl new ntrllers frm mathematial mdels, r idealizatins f physial plants, are synthesized. The utility f suh design hinges upn, f urse, the disrepany between the real and mdeled dynamis, als knwn as mdel unertainty. This has been the fus f mdern ntrl thery fr the last several deades. Mdel unertainty has been haraterized in literature as strutured and unstrutured, refleting the nature and degree f unertainty. Strutured unertainty desribes the unknwns in the parameters f an therwise expliitly given mathematial mdel, ften referred t as parametri unertainty. Unstrutured unertainty, n the ther hand, pints t unknwns beynd thse in the parameters in whih ase the dynamis, suh as thse at high frequeny r thse t mpliated t desribe, itself is negleted [][][3]. In the mdern ntrl paradigm, the unertainty, regardless f its type, is separated frm the nminal plant mdel and desribed in ne f three general frms: ) Multipliative unertainty: G G + W, () ( ) p I ) Inverse multipliative unertainty: G p G, () + W 3) Additive unertainty: ( ) I G G + W, (3) p I where G p represents the unertain plant, G is the nminal plant mdel, is the unertainty whse magnitude is bunded by ±, and W I is the unertain frequeny weight whih sales the unertainty. The subsript I represents unertainty at the input, but fr Single Input Single Output (SISO) system, where this disussin is limited t fr the sake f simpliity, the unertainty at the input is equivalent t the unertainty at the utput. u w u P K Figure. -P-K Struture y z v In the mdern ntrl paradigm, the nminal plant mdel, unertainty, and lsed lp ntrller are rganized int the -P-K struture as shwn in Figure where: P is knwn as the general plant and written as: y u z P w (4) v u with inputs: u as unertainty, w as external inputs, and u as plant input. The utputs f P are: y, the input t the unertainty blk abve, z, the desired press variable t be minimized, and v, the traking errr whih is the input t the ntrller. The prblem bemes that f finding a ntrller K suh that the perfrmane speifiatins are met and, mre imprtantly, the lsed-lp system remains stable fr all pssible unertainties given in equatins () t (3). A typial slutin is H design where the rbustness is attained based n the small-gain therem, with the premise that the unertainty is small [8]. µ-synthesis ffers anther slutin with a given unertainty weight funtin, but the resulting ntrller is ften f a high rder and may be diffiult t implement [3][8]. Bth methds are learly limited in the amunt f unertainties they an handle, whih then pses the questin f whether the amunt f unertainty an be redued first befre rbust ntrl is applied. The ative disturbane rejetin paradigm [9][3][4] prvides an alternative t mdel-based design. Its entral /8/$5. 8 AACC. 3689

2 premise is that ertain unknwns in physial systems, inluding bth dynamis and disturbanes, an be estimated frm the input-utput data and mpensated fr in real time, thus transfrming a highly unertain system int a wellbehaved ne. T this end, several disturbane bserver tehniques have been prpsed, inluding the unknwn input bserver (UIO) [6][7], perturbatin bserver (PB) [5], the disturbane bserver (DB) [4], and the extended state bserver (ESO) [9]-[5] as a few examples. The differenes in these disturbane bservers an be seen in terms f ) the amunt f mdeling infrmatin required; ) representatin: transfer funtin r state spae; 3) bserver gains: linear r nnlinear. Of all the bservers, the ESO appears t require the least amunt f mdeling infrmatin; is implemented in the state spae frm, whih ffers better numerial prperties; and an emply bth linear r nnlinear gains fr maximum perfrmane benefits. It is fr these reasns that the Ative Disturbane Rejetin Cntrl (ADRC) emplys the ESO as its re, althugh ther disturbane bservers an als be viewed as speial ases f ADRC. The bjetive f this researh is t mbine the ative disturbane rejetin ideas with the mdern rbust ntrl methdlgy t frm a pwerful ne-tw punh in making a ntrl system truly rbust. Instead f passively ping with unertainties as nstraints in design, a pr-ative stane is taken in first attempting t redue the amunt f unertainty thrugh ADRC and then applying the rbust ntrl paradigm t deal with the remnants. The paper is rganized as fllws. The main idea f unertainty redutin is presented in Setin II, fllwed by the design fr the annial plant in setin III. Setin IV mpares the traditinal ADRC ntrller against the hybrid H -Ative Disturbane Rejetin Cntrl (H-ADRC) ntrller, fllwed by a rbust stability analysis in Setin V. Finally, nluding remarks are inluded in setin VI. II. UNCERTAINTY REDUCTION Any real physial plant ntains unertainties, inluding bth the external disturbane and unknwn dynamis. T deal with the latter, the main apprah in ntrl thery nsists f three steps: ) determine the mathematial mdel as aurately as pssible, leaving the smallest amunt f mdeling unertainty as pssible; ) determine the bund f mdel unertainty, mstly in frequeny dmain; 3) use the unertainty bund as a design nstraint t find a slutin that is a mprmise between rbust stability and perfrmane. Parallel t this apprah, ADRC asks a different questin: an the ttal unertainty, inluding bth types mentined abve, be redued first, leaving the feedbak ntrl lp t deal with a system that is rather ertain and deterministi? Perhaps withut realizing it, disturbane bservers are different answers t this questin. Althugh mst were designed t estimate and anel external disturbanes, these disturbane bservers, as shwn belw, all have the additinal benefit f reduing mdel unertainty. If pssible, fr the time being, ignre the differenes and nentrate n the mmnalities amng these disturbane bservers, whih an be redued t the frm f Figure, using the transfer funtin metaphr. Here Q is a nise filter, P f G - Q, and G p is the perturbed plant whih may be in any f the frms f equatins () t (3). The riginal intent f the disturbane bserver design is t estimate the external disturbane, d, and anel it suh that the new plant frm u t y is disturbane free. Suh harateristis have been well established in pratie and analysis. What is f interest here is the effet suh a disturbane bserver has n the unertain dynamis. In partiular, in the absene f d, is the mdel unertainty redued in Figure? That is, if the plant G p is f the frm f equatin (), (), r (3), is there less unertainty in the transfer funtin frm u t y? Frm Figure : Gp Gyu (5) Q + G P r, if P f is replaed by G - Q: G pg G yu Q( G G) + G whih was first shwn in the analysis f the DB [4]. u - - Q Figure. Equivalent Blk Diagram f Varius Disturbane Observers Nw, nsider a plant with multipliative unertainty as written in equatin (), substituting () int (6) results in the transfer funtin f: ( Q) WI G yu G + (7) + QWI and as Q apprahes unity, assuming that nise is negligible, (7) redues t: G yu G (8) whih demnstrates that, under ideal nditins, the mdel unertainty is mpletely remved by the disturbane bserver! Of urse it is unrealisti t believe that suh feat an be pulled ff in pratie, as previus researhers have demnstrated; it is nnetheless an imprtant disvery, the nnetin between the external disturbane remval and the mdel unertainty redutin. And this is nt limited t multipliative unertainty. Cnsider a system with inverse multipliative unertainty, equatin (), and an ative disturbane rejetin tehnique designed arund the plant as shwn in Figure. The transfer funtin f the inner plant bemes: G G yu (9) + ( Q) WI and when Q, (9) redues t: G yu G () The same may be shwn fr additive unertainty where the plant in (3) is substituted int (6) whih results in: p d P f p f G p y (6) 369

3 G( G + WI ) G yu () G + QWI and when Q, () redues t: G yu G () These results demnstrate that the disturbane bservers have the effet f reduing the amunt f unertainty in a plant, fring it t behave like the nminal transfer funtin G. It is here that ADRC takes ne mre bld step: making G a asaded integral plant f rder t the real plant, regardless f its dynamis. That is, in the ADRC framewrk, bth the external disturbane and internal dynamis are estimated and aneled, leaving the feedbak ntrl lp t deal nly with a simple asaded integral plant. A general nnlinear, timevarying send rder plant will be used as an illustratin. A. Unertainty Redutin via Ative Disturbane Rejetin Fr the purpse f illuminating the idea f ative disturbane rejetin and evaluate its ptential in unertainty redutin, a send rder plant with unity gain is seleted here: & y f (&, y y, t, d) + u (3) where f( ), generally unknwn, represents the nnlinear, time-varying dynamis, and the effet f external disturbane, d. A unity gain is hsen fr the sake f simpliity. A nventinal apprah wuld start with mdeling, i.e. btaining the apprximate mathematial expressin f f( ), upn whih the ntrl design wuld fllw. The key idea f ADRC is t target f( ) as a general disturbane t be estimated and rejeted (aneled) and, if suessful, redue the prblem t the ntrl f a duble integral plant. That is, if f frm equatin (3) an be fairly estimated as fˆ, the ntrl law: u fˆ + u (4) redues the plant in (3) t: & y u (5) thus transfrming a nnlinear, unknwn, and time-varying plant t a well behaved, easy t ntrl ne. The suess f this ative disturbane rejetin apprah t ntrl design hinges upn the timely and aurate estimatin f f. T this end, the extended state bserver (ESO) is intrdued. If (3) is written in state spae frm and augment the state vetr with f as an extra, r extended state, then: x& x x& x3 + u (6) x& f& 3 and the state bserver f whih, the ESO, an be nstruted as: z& Az + Bu + L ( y yˆ) (7) yˆ Cz where: A, B T (8) l C, L l l3 and L is the gain vetr t be seleted. Fr the ease f tuning, it was suggested [] that the bserver be parameterized by the bserver bandwidth, ω, suh that its harateristi equatin is: ( λ ω ) 3 3 ( s) λ + lλ + lλ + l3 λ + (9) Nte that the ESO an be nverted frm state spae frm t transfer funtin frm in the frm f Figure with: 3 ω Q () P f ( s + ω ) 3 3 ω s () ( s + ω ) 3 as shwn in []. A detailed mparisn f the ESO and ther bservers, suh as DB and UIO, is beynd the spe f this paper. It suffies t say that the ESO ffers distint advantages in ) numerial effiieny in implementatin; ) ease f tuning thrugh parameterizatin; 3) requiring the least amunt mdel infrmatin. T demnstrate the effetiveness f the ESO in unertainty redutin, nsider a send rder plant with a nminal transfer funtin f: G s( s + 3) () fr whih the unknwn dynamis is haraterized by the weight, adpted frm [8], in the frm f: τs + r Wud (3) τ s + r where r is the mdeling errr in steady state, r is an unertainty salar at high frequeny, and τ - is the frequeny at whih the system is mpletely unknwn. Fr this example, assume that there is % mdeling errr at steady state, r, the frequeny at whih the system is unknwn is.hz, r τ -.π, and r is hsen randmly as r 5. The perturbed plant is f the frm: G G + W, (4) ( ) p ud The magnitude plt f the perturbed plant is depited in Figure 3. The amunt f unertainty redutin by the ESO is shwn in Figure 4. Bde plts f the transfer funtin frm u t y in Figure are shwn fr different bserver bandwidths, ω. Clearly, the quality f unertainty redutin is diretly rrelated t the bandwidth: the higher the ω, the lser the mpensated plant is t the ideal duble integral plant. Frm Figure 4 it is nluded that the plant frm u t y is redued t a pure duble integratr with very small errr up t the frequeny f.ω. That is, the ntrl design prblem is redued t dealing with a pure duble integral plant at r belw the frequeny f.ω. 369

4 Magnitude (db) Bde Diagram Frequeny (rad/se) Figure 3. Magnitude Plt f the Perturbed Plant Bde Diagram design is applied t the mpensated plant f (5) in the hpe f enhaning the rbustness f ADRC against the unertainties that uld nt mpletely be estimated and rejeted by the ESO. In the H frmulatin, this mpensated plant is rewritten as: & Ax + B w B u (8) x + z C x + Dw + D y C x + Dw + D u (9) u (3) where fr this system: Magnitude (db) - - /s w w w w Frequeny (rad/se) Figure 4. Magnitude Plt f the Cmpensated Plant III. CONTROL DESIGN FOR A CONANICAL PLANT Nw turn t the prblem f designing a frnt end ntrller fr the mpensated plant frm u t y. Depending n hw high an ω is pratially attainable, the rbustness prblem may r may nt have t be dealt with. When ω an be made suffiiently higher than the ntrl lp bandwidth, a PD ntrller fr the duble integral plant will suffie. When this is nt the ase, t deal with the unertainty abve the.ω frequeny is where, perhaps, an pprtunity exists t take advantage f the vast prgress made in rbust ntrl ver the last several deades. A. Parameterized Prprtinal Derivative Cntrller Fr the ideal duble integral plant, a parameterized PD ntrller [] is prpsed with ne tuning parameter, the ntrller bandwidth ω. The design gal is t make the lsed lp transfer funtin, frm the referene r t utput y: y r ω (5) ( s + ω ) With the ESO prviding the estimated states, the PD ntrl law fr the duble integral plant f (5): u k ( r z k z (6) with the gains f: p ) + k p d d ω k ω (7) This is a mmn ADRC ntrller nfiguratin [], [3], and [5] that is simple and effetive, allwing intuitive tuning n the fly based n the well-knwn fat that an inrease in the ntrl bandwidth results in a mre aggressive lsed lp system. With the bserver bandwidth set as a multiple f the ntrller bandwidth, the entire system is tuned by adjusting the ntrl bandwidth. Suh simpliity is very attrative t pratitiners. T ensure stability rbustness at high frequeny, it may be required t turn t a mre advaned ntrl design methdlgy. B. H -ADRC Cntrl T pe with mdel unertainties, H design is a predminant slutin in the literature. In this setin this A C C B [ ] D [ ] D [ ] [ ] D [ ] D [ ] B (3) Using γ-iteratin, the ptimal H slutin may be determined. Mre realistially, a subptimal H ntrller may be btained by speifying a desired γ and a tlerane, r auray. Fr this system, the H design slutin was determined iteratively using the Matlab Tlbx in the fllwing manner: ) an initial γ was seleted; ) Matlab was used t determine whether a rrespnding ntrller K exists; 3) if it did, then γ was redued and step is repeated fr the new γ. This press ntinued until the γ was redued t the pint where a ntrller did nt exist. At this pint, the lwest γ rrespnds t the ptimal ntrller: x & Ax + Be (3) u Cx + De (33) where: A B.966 (34) 6 4 C [ ] D [ ] where e is the traking errr between the referene, r, and the estimated psitin frm the ESO, z, r the atual psitin, y, depending n whih signal is fed bak t the ntrller, and u is the input t the mpensated plant shwn in Figure (4). This hybrid H and ADRC design is dented as H-ADRC. Nte that in H-ADRC ntrller there is nly ne tuning parameter, the bserver bandwidth. Fr the feedbak ntrl lp, H-ADRC an nt be tuned, but nly redesigned fr different design speifiatins. This innveniene an be alleviated smewhat by the fat that sine the H ntrller is always designed fr the duble integral plant in this ase, it an be designed ff-line, arding t varius requirements, and stred in a lk up table t be seleted by the users. Of urse, tuning n the fly, as in the PD slutin, is nt an ptin. IV. SIMULATION AND HARDWARE VERIFICATION In this setin the simple plant mdel will be simulated in Matlab t demnstrate the similarity f H-ADRC t ADRC. Bth ntrllers have been implemented n a simple mtin ntrl testbed t mpare the tw ntrllers. A. Simulatin Cmparisn Fr mparisn purpses, bth the ADRC and H-ADRC ntrllers are applied in simulatin t the plant f equatin 369

5 (). Bth are tuned fr apprximately the same settling time. The ADRC ntrller has ntrller gains f ω 65,.5 Psitin Respnse (rev) Psitin Errr (rev) Cntrl Signal (v) H-ADRC ADRC Time (s) Figure 5. Cmparisn Between ADRC and H-ADRC.5 Psitin Respnse(rev) Psitin Errr (rev) Cntrl Signal (v) H-ADRC ADRC Time (sends) Figure 6. Hardware Results Between ADRC and H-ADRC ω 35, while the H-ADRC ntrller uses the same ω 35. The ADRC and H-ADRC system respnses are shwn in Figure 5, whih shws that eah ntrller prdues a respnse with apprximately the same settling time, nise level in the ntrl signal, and apprximately same psitin errr. In ther wrds, perfrmane wise, the tw ntrllers are rughly the same. B. Hardware Results T verify the simulatin results, the tw ntrllers are further tested in hardware, implemented in and tested n the Eduatin Cntrls Prduts (ECP) Mdel with a sampling rate f khz. The ECP Mdel is a trsinal system that ntains tw mtrs and three plates, ne that attahes t the drive mtr, ne that attahes t the disturbane mtr whih injets a disturbane t the system, and ne wheel that is the lad itself. The system is apprximately a 3: trque inreaser, and ntains weights that may be added t the system t hange the amunt f inertia in the system. The system has a linear time invariant mdel f: & y.4y& + 3. u (35) In this ase, the estimate f b is hsen t be bˆ 4 fr bth ADRC and H-ADRC. The H-ADRC ntrller has an bserver bandwidth f 75 while the ADRC ntrller has a ntrller bandwidth f 5 and bserver bandwidth f. Figure 6 shws the differene between H-ADRC and ADRC ntrllers. In Figure 6, the H-ADRC and ADRC ntrllers are shwn t have similar system respnses, apprximately the same settling time and steady state errr. The differene between the tw lsed lp systems is the amunt f nise that is present in the ntrl signal. Furthermre, if the bserver bandwidth f H-ADRC is inreased t be the same as that in ADRC, the tw systems will have apprximately the same amunt f nise in the ntrl signal, refleting the same bservatin in simulatin. That is, perfrmane wise, the tw ntrllers are very similar. V. ROBUST STABILITY AND TRADE-OFFS The real benefit f the H design ver the PD design in ADRC turns ut t be in rbustness. In bth ases, the plant is redued t a nminal pure duble integratr with an unertain weight that is a funtin f the bserver bandwidth: ( ) s +..ω WI (36) ( ) s +.ω (5) That is, the target f ntrl design is apprximately a pure duble integral plant in the frequeny range frm DC up t the frequeny f.ω, beynd whih there is signifiant dynami unertainty. This is f urse the result f the unertainty redutin shwn in setin II. This partiular type f unertainty is knwn as multipliative unertainty and the perturbed plant is written as shwn in equatin (). The general rbust stability nditin is: σ ( M ( jω) ) <, ω (37) where M is the N y u transfer funtin frm the N- struture. An alternative rbust stability nditin fr a SISO system with multipliative unertainty: TW (38) I where T is the mplimentary sensitivity funtin [8]. The inverse f T W I is deemed as the rbust stability bund, whih desribes the tlerane in the amunt unertainty while system stability is still assured. Therefre the rbust stability bund must be greater than t guarantee rbust stability. Fr the ADRC and H-ADRC ntrllers tested in simulatin abve, the rbust stability bund is fund t be.4 fr the frmer and 7. fr the latter. This learly shws that, given the same perfrmane, the rbustness f H- ADRC is superir than that f ADRC. Fr ADRC t meet the rbust stability nditin, the ntrller bandwidth needs t be detuned, leading t a less desired perfrmane. On the ther hand, there is, f urse, a st and trade-ff assiated with H-ADRC. In Figure 5, the bserver bandwidth fr bth ADRC and H-ADRC are equal, and the system respnse, psitin errr, and ntrl signal are apprximately the same. Hwever, in the hardware test, Figure 6, the H-ADRC bserver bandwidth is smaller than the ADRC bserver bandwidth, resulting in a ntrl signal that is less nisy, but apprximately the same settling time and psitin errr. This differene shws that by dereasing the bserver bandwidth the nise in the ntrl signal is dereased. The real differene between these tw ntrllers lies in the frnt-end ntrller. The parameterized PD design allws fr the ability t easily hange the ntrller bandwidth and quikly adjust aggressiveness f the system t suit the peratinal needs. This an be very advantageus, fr example, in serv systems where the rssver frequeny 3693

6 shuld be maintained as high as pssible but must be limited t avid exitatin f mehanial resnane. The disadvantage f this design, as shwn abve, is the lak f rbust stability fr large unertainties in the system at higher frequenies. The H design, n the ther hand, guarantees minimizatin f the wrst ase errr, resulting in imprved stability rbustness. Perfrmane wise, it is similar t the riginal ADRC with a PD ntrller, as shwn in bth simulatin and hardware tests. The main disadvantage f this ntrller is its rigidity, r the lak f flexibility t be tuned fr different peratin nditins, and the inability t hange the aggressiveness f the system by adjusting the ntrller bandwidth. One pssible remedy is t design many H ntrllers fr the duble integral plant ff line and put them in a lk up table t be swithed in and ut, arding the hange in needs. This is f urse far mre mplex than tuning the PD ntrller using a single parameter, ω. VI. CONCLUSION In this paper it is demnstrated fr the first time that the unertainty stemming frm bth the external disturbane and the unknwn internal dynamis, whih is the subjet f intense researh effrts in the last few deades, an be greatly redued thrugh ative disturbane rejetin. Ardingly, it is demnstrated that ntrl f unertain system an be arried ut in tw steps: ) reduing the unertain plant, via ative disturbane rejetin, t a lass f asaded integral plants; and ) design the frnt end ntrller fr these mpensated plants. This paper shws quantitatively hw muh unertainty redutin an be ahieved, whih, nt surprisingly, is prprtinal t the bandwidth f the disturbane bserver. Furthermre, ne the unertain plant is redued t a asaded integral ne, bth PD and H design an be applied t ntrl it. Thrugh a mparisn f the tw ntrllers in bth simulatin and hardware tests, it is nluded that they are similar in perfrmane but drastially different in rbustness and ease f tuning. In partiular, the H design ahieves better rbustness at the st f ease f tuning. Further researh is needed t ndut a mprehensive study n hw t make the ntrller bth rbust and easy t tune. [5] SangJ Kwn and Wan Kyun Chung, Rbust Perfrmane f the Multilp Perturbatin Cmpensatr, ASME Transatins n Mehatrnis, Vl.7 N., June. [6] C.D. Jhnsn, Ammdatin f External Disturbanes in Linear Regulatr and Servmehanism Prblems, IEEE Transatins n Autmati Cntrl, Vl. AC-6 N. 6, Deember 97. [7] Jseph A. Prfeta, William G. Vgt, and Marlin H. Mikle, Disturbane Estimatin and Cmpensatin in Linear Systems, IEEE Transatins n Aerspae and Eletrni Systems, Vl. 6 N., Marh 99. [8] Sigurd Skgestad and Ian Pstlethwaite, Multivariable Feedbak Cntrl: Analysis and Design, Jhn Wiley and Sns, Ltd, Send Editin, 5. [9] J. Han A Class f Extended State Observers fr Unertain Systems, Cntrl and Deisin, Vl. N., 995. (in Chinese) [] Zhiqiang Ga, Saling and Bandwidth- Parameterizatin Based Cntrller Tuning, Amerian Cntrl Cnferene, pp , June 3. [] Qing Zheng, Linga Q. Ga, and Zhiqiang Ga, On Estimatin f Plant Dynamis and Disturbane frm Input-Output Data in Real Time, IEEE Multinferene n Systems and Cntrl, Otber -3, 7. [] Gang Tian and Zhiqiang Ga, Frequeny Respnse Analysis f Ative Disturbane Rejetin Based Cntrl System, IEEE Multi-nferene n Systems and Cntrl, Otber -3, 7. [3] Zhiqiang Ga and Yi Huang, Jingqin Han, An Alternative Paradigm fr Cntrl System Design, IEEE Cnferene n Deisin and Cntrl Cnferene. [4] Zhiqiang Ga, Ative Disturbane Rejetin Cntrl, A Paradigm Shift in Cntrl System Design, Preeding f the 6 Amerian Cntrl Cnferene. [5] Yi Hu, Zhiqiang Ga, Fangjun Jiang, and Brian T Bulter, Ative Disturbane Rejetin Cntrl fr Web Tensin Regulatin, IEEE Cnferene n Deisin and Cntrl,. REFERENCES [] Lenard Lublin, Simn Grtt, and Mihael Athans, H and H Cntrl, The Cntrl Handbk, CRC Press, pp 65-66, 996. [] Jhn C. Dyle, Brue A. Franis, and Allen R. Tannenbaum, Feedbak Cntrl Thery, Mamillan Publishing Cmpany, New Yrk, NY, 99. [3] Kemin Zhu, Jhn C. Dyle, and Keith Glver, Rbust and Optimal Cntrl, Prentie-Hall In, Upper Saddle River, NJ, 996. [4] Erwin Shrijver and Jhannes Van Dijk, Disturbane Observers fr Rigid Mehanial Systems: Equivalene, Stability, and Design, ASME, Jurnal f Dynami Systems and Cntrl, Vl. 4, Deember. 3694