Controller Design of Nonlinear TITO Systems with Uncertain Delays via Neural Networks and Error Entropy Minimization

Transcription

1 Controer Desgn of Nonnear TITO Systes wth Uncertan Deays va Neura Networs Error Entroy Mnzaton J. H. Zhang A. P. Wang* H. Wang** Deartent of Autoaton North Chna Eectrc Power Unversty Bejng 6 P. R. Chna (e-a: zhangwu@ ubc.bta.net.cn). *Insttute of Couter Scences Anhu Unversty Anhu P. R. Chna ** Contro Systes Centre The Unversty of Manchester PO Box 88 Manchester M6 QD UK (e-a: hong.wang@anchester.ac.u) Abstract: In ths aer a nove contro agorth for nonnear two nut two outut (TITO) systes wth ro nut outut deays s resented. Due to the stochastc characterstcs nduced by uncertan te deays TITO feedbac contro systes are st nto a genera fraewor where the controers are desgned based uon nzng the entroes of tracng errors. The controers that have been eented by BP neura networs are obtaned wthout decoung. The convergence n the ean square sense s anayzed. Suaton resuts show the effectveness of the roosed aroaches.. INTRODUCTION Nonnear dynac utvarabe systes wth ro nut outut te deays exst wdey n ract ndustra contro rocesses. Due to the exstence of nonnearty utvarabe ro deays t s very dffcut to desgn controers for such coted cosed-oo systes. U t now tte research has been erfored to nvestgate ths nd of systes. Fuzzy contro ethod was deveoed n (Zhang et a. 7) where Taag-Sugeno (T-S) fuzzy ode wth a te-deay ter was used to dea wth the stabzaton robes of nonnear syste wth te deays. In ths context a guaranteed cost controer was desgned va a state feedbac the stabty condton was roosed by near atrx nequaty (LMI) technques. Suaton resuts of two nut snge outut nonnear systes wth certan deays were gven. In ths aer a stochastc contro ethod based on the tracng error entroy nzaton s roosed. Beuse of the stochastc characterstcs nduced by uncertan deays the controers are desgned under a genera fraewor. Foowng the recent deveoents on the odeng contro va nzng error entroy (Wang ; Wang Zhang ; Wang ; Guo Wang 5a; Guo Wang 5b; Yue Wang 3; Erdogus Prce a; Erdogus Prce b) a feedbac contro strategy s roosed by neuro-pid controers wthout decoung. The nnovaton of ths schee n be suarzed as foows: ) the controers are desgned drecty utzng a new erforance ndex constructed by entroes of tracng errors; ) the stabty condton of the feedbac contro syste s gven. Secton 3 deveos the contro agorth; Secton 4 resents a suaton exae to ustrate the effcency feasbty of the roosed aroach. The ast secton concudes ths aer.. PROBLEM FORMULATION Consder a nonnear TITO syste wth uncertan nut outut deays shown n Fg. n whch a nonnear TITO ant are controed by two neura controers. In ths fgure τ sc τ sc are ro deays fro the sensor to the controer τ τ are ro deays fro the controer to the actuator. Moreover r r are the setonts ε ε are the cosed oo tracng errors. Fg.. A nonnear TITO syste wth uncertan deays In ths aer the nonnear ARMAX ant n be descrbed as The reer of ths aer s organzed as foows: Secton foruates the robe of desgnng controer for nonnear TITO systes wth ro nut/outut deays.

2 y = f( y y... y n y y... y n u u... u u u... u ) y = f ( y y... y n y y... y n u u... u u u... u ) As such the tracng errors are defned by ε = r y =. Beuse these uncertan deays ay not obey Gaussan dstrbutons t s necessary to consder hgher order statstcs rather than the ean varance of tracng error when desgnng feedbac controers. Recent aers have addressed ths ssue both n contro terature (Wang ; Wang Zhang ; Wang ; Guo Wang 5a; Guo Wang 5b; Yue Wang 3) n the sgna rocessng achne earnng terature (Erdogus Prce a; Erdogus Prce b). It has been onted out that n certan ses nzng the error entroy s equvaent to nzng the dstance between the robabty dstrbutons of the desred syste oututs (Erdogus Prce a). The controers n be soved by nzng the entroes of the two tracng errors n fgure as foows: J ( Δ u ) = H( ε ) + H( ε ) + H( ε ε ) n () where Δ u s the ncreenta outut of the controer H( ε ) H ( ε ) are the entroy of the tracng error ε ε at nstant resectvey. Moreover H ( ε ε ) n () denotes the jont entroy of the two tracng errors. The objectve of the erforance ndex s to ae the shae of the robabty densty functon (df) of each tracng error to be as narrow as ossbe. Moreover the controer n each oo needs an ntegra oerator n order to ensure that the steady state tracng error aroaches to zero. Hence n ths aer neuro-pid controers are utzed n each contro oo. Reny extended the concet of entroy gven by Shannon the defned Reny s α entroy s gven by H ( ) og α α ε = γε ( ε) de α The jont entroy of two ro varabes s cuated fro Hα ( x y) = γ xy ( x y) dxdy. α α Reny quadratc entroy ( α = ) n be cuated n a nonaraetrc ract effcent way when the robabty densty functon (PDF) s estated by Parzen wndow wth Gaussan ernes. Therefore Reny quadratc entroy n conjuncton wth Parzen wndow wth Gaussan erne s utzed here to read () H ( ε ) og V( ε ) = (4) ε κξ σ ξ (5) N V( ) = ( ( )) e d N = where V ( ε ) s naed as nforaton otenta n (Erdogus Prce b) whch n be cuated fro the saes { e e... e N } usng Gaussan erne n the foowng way. V e e (6) ( ε ) = κ( j σ ) N = j= Snce Reny s quadratc entroy s a onotonc functon of the nforaton otenta V ( ε ) we n equvaenty axze the nforaton otenta V ( ε ) nstead of nzng Reny s entroy. As such we resent the foowng nove erforance ndex shown n () so as to desgn the requred controer J ( Δ u ) = n V( ε ) + V( ε ) + V( ε ε ).e. J u V ε V ε V ε ε ( Δ ) = ( ( ) + ( ) + ( )) n. The jont nforaton otenta V ( ε ε ) n aso be cuated fro the saes of two tracng errors. 3. DESIGN OF NEURAL CONTROLLERS The neura controer n each cosed oo s served as an adatve PID controer ts agorth s structured n the foowng (7) u = u +Δ u (8) Δ u = ( ε ε ) + ε + ( ε ε + ε ) (9) d In ths aer the BP neura networs are utzed to eent the PID controer. Tang the neura controer as an exae the contro agorth n be obtaned. For such a neura networ the nut ayer s defned as O () q = x( q) q =... Q () where the tracng error ε s a rary node n the nut ayer the other nodes n the nut ayer ay ncude the tracng error ε the set-ont r the transtted outut of ant y. The nut outut of the hdden ayer are denoted by Q () () () q q q= net = w O () O = f( net ) =... P () () () The nut outut of the outut ayer are gven by

3 P () = net = w O O = g( net ) = 3 (4) O ( ) = (5) O ( ) = (6) O 3 ( ) = (7) d () where w q w are weghts traned n BP neura networs the actvaton functons have been defned that ex e x ex f( x) = gx () =. ex + e x ex+ e x In ths aer the adatve crteron of the BP neura networs based controer s to nze the su of Reny s entroes defned n () (or to axze the su of nforaton otentas) due to the reason stated before. Usng the steeest descent aroach the tranng agorth n be obtaned. At nstant the statst nforaton of the tracng error ε ε s estated by saes { e e... e } N { } e e... e resectvey. The jont nforaton otenta s N cuated fro V e e e e ( ε ε ) = κ( σ ) κ( σ ) j j N = j=. For the outut ayer the tranng of the weghts are foruated as foows J( Δu) Δ w = η [ V( ε ) + V( ε ) + V( ε ε )] = η (8) y = y u O ( ) ( ) net = u O net w =... N Sary t n be obtaned. [ V ( ε )] [ V ( ε ε )] The Jacoban nforaton y y n the above u u foruaton n be reaced by y sgn y sgn u u resectvey. In addton they n be cuated by usng ode redcton agorth of the ant under foowng assuton. y y u ( τ ) = u u( τ ) u y u( ) u( ) u( τ )... = u( τ ) ( ) ( ) ( u u u τ + ) y u ( τ ) = () For a hdden ayer the tranng of the reevant neura networ weghts n be foruated to gve [ V( ε ) V( ε ) V( ε ε )] Δ wq =η + + q () For the three ters n the rght-h-sde of equaton () the foowng cuatons n be ade η η are the re-secfed earnng factors. where ths stage t n be shown that [ V ( ε )] = ( e e ) j N σ κ = j= ( j e e σ )[ ] j where t n be further seen that At (9) [ V ( ε )] = ( e e ). j N σ q κ where t n be shown that y = q = j= ( j e e σ )[ ] j q q q y u O = u O net net O net O net () () () () q =... N ()

4 [ V ( ε )] [ V ( ε ε )] q q n be obtaned e equaton (). Moreover q n be obtaned e equaton. Obvousy another neura controer n be desgned n a sar way usng the sae erforance ndex shown as (7). The agorth n be suarzed as foows: Ste : Perfor Monte Caro test at nstant obtan the entroes. A set of stochastc te deays are osed on the syste for cuatng the saes of tracng errors { e e... e ( )} { e e... e ( )}. The N N entroes of the tracng errors H( ε ) H ( ε ) H( ε ε ) n be obtaned by Ste : Tran the two neura networs by udatng the weghts of BP neura networs usng equatons (8)-. Ste 3: Foruate the contro nut u usng (8) (9) after obtanng va the BP neura networs. d Ste 4: Ay contro nut u to actuate the ant set + go bac to ste. The use of varabe ste-sze gradent agorth or addng oentu ter n enabe roved tranng of the BP neura networs. In addton the erne sze (the wndow wdth of Parzen estator) n be set exerentay after a renary anayss of the dynac range of the tracng error. Once the contro agorths are obtaned t s ortant to further anayze the stabty of cosed oo systes. Theore: If the araeter satsfes < η [ Γ ( t) + ε ] where ε s an arbtrary sa ostve constant Γ ( t) = V( ε( t)) + V( ε( t)) + V( ε( t) ε( t)) then the contro agorth gven by (8)- w be convergent. Proof: Seect a Lyaunov functon as foows π = + ( Δut ( )) Γ () t + ε then t n be obtaned that π Γ( t) u( t) u( t) = + Δut ( ) t [ Γ ( t) + ε ] u( t) t t Γ( t) u( t) = ( + Δut ( )) [ Γ ( t) + ε ] u( t) t (4) Aroxatey we have the foowng Δπ Γ( t)) Δu( t) = ( + Δut ( )) Δt [ Γ ( t) + ε ] u( t) Δt (5) Snce the neura controer s desgned by nzng the entroes (.e. axzng the nforaton otenta) of the tracng errors wth steeest descent aroach the above aroach to desgn controer s n fact of the foowng for () t Δ ut () = η Γ (6) ut () Fro (5) (6) t n be foruated that Γ( t) Γ( t) Γ( t) Δ π = [ + η ] η [ Γ () t + ε ] u() t u() t u() t Γ( t) = η( + η)( ) t ε u t [ Γ ( ) + ] ( ) () t E( π ) E η Γ ( ) ( ) ut () [ () t ε ] η Δ = + Γ + (7) (8) To ensure that the agorth s convergent (8) shoud be non-ncreasng (.e. the rght-h sde of (8) shoud be non-ostve. Snce η > Γ() t ( ) hod ut () η [ ( t) ε ] + shoud Γ + be non-ostve so as to guarantee that (7) s not greater than zero. At ths stage we n obtan the condton for convergence as foows: < η (9) [ Γ ( t) + ε ] The roosed neura controer s very se. The desgn of the controer s straghtforward based on nzng entroy of the tracng error or axzng the nforaton otenta of the tracng error. 4. SIMULATIONS In order to show the abty of the roosed contro agorth for nonnear TITO systes wth nut outut uncertan deays et us consder the foowng feedbac contro syste whch conssts of the ant reresented as foows y = (.8 y( ) + ( + y ( )) u( ) +. u( 3)) y = (.9 y( ) + ( + y ( )).3 u( 3) + u( ))

5 In ths suaton the PDFs of these stochastc deays τ τ τ τ are gven n order to roduce stochastc sc sc tracng errors. These PDFs obey the foowng β - dstrbuton ( ) x α βα + λ+ = ( x) λdx λα. Therefore t n be seen that γ ( x) τ γ ( x) τ γ ( x) τ sc [3 βα ( + λ+ )] x (3 x) x (3) otherwse α+ λ+ α λ = [4 βα ( + λ+ )] x (4 x) x (4) otherwse α+ λ+ α λ = [4 βα ( + λ+ )] x (4 x) x (4) otherwse α3+ λ3+ α3 λ3 3 3 = (33) [3 α4+ λ4+ βα ( λ4 + )] xα (3 x) λ x (3) γ ( x) = τ sc otherwse (34) where α = λ = 4 α = λ = α 3 = 3 λ 3 = 5 α 4 = 3 λ =. In ths aton the set-onts are set to r = r = 4 the sang erod T =. For the suaton the nodes n nut ayer of each neura controer conssts of two tracng errors the nuber of nodes n hdden ayer of the neura controers s 5 the earnng factor η =. the erne sze used to estate the entroy s exerentay set at σ =.5 the saes for neura controers are traned wth a segent of N = at each nstant. The suaton resuts are shown n Fgs. -7. of PDF of two tracng errors becoe narrower aong wth the ncreasng te. Fgs ndte that the contro syste has a sa uncertanty n ts cosed oo oeraton. error error te(s) te(s) Fg. 3 The tracng errors d te(s) te(s) te(s).75.7 Fg. 4 Paraeters of controer Ve te(s) te(s) te(s) Fg. The nforaton otentas of tracng errors d.98 Fg. shows the ncreasng the nforaton otentas of tracng errors whch ustrates the entroes of tracng errors decease wth te. The tracng errors are shown n Fg. 3. The varances of PID araeters tuned by neura controer are ustrated n Fg. 4 Fg. 5. In Fg. 6 Fg. 7 both the range of tracng error the PDF of tracng error are gven where t n be seen that the shae te(s) Fg. 5 Paraeters of controer

6 5 Ths wor s suorted by Natona Natura Scence Foundaton of Chna under grants (No ) Bejng Natura Scence Foundaton under grant (No. 47). These are gratefuy acnowedged. 5 PDF REFERENCES Suaton te Fg. 6 PDF of the tracng error PDF Suaton te Fg. 7 PDF of the tracng error Error range.3.. Error range Zhang H. Yang D. Cha T. (7). Guaranteed cost networed contro for T-S fuzzy systes wth te deays IEEE Trans. Syst. Man. Cybern. C Wang H. (). Bounded Dynac Stochastc Systes: Modeng Contro Srnger-Verag London Ltd. Wang H. Zhang J. H. (). Bounded stochastc dstrbuton contro for seudo ARMAX systes IEEE Transactons on Autoatc Contro Wang H. (). Mnu entroy contro of non-gaussan dynac stochastc systes IEEE Transactons on Autoatc Contro Guo L. Wang H. (5). PID controer desgn for outut PDFs of stochastc systes usng near atrx nequates IEEE Trans. Sys. Man Cyb Guo L. Wang H. (5). Generazed dscrete-te PI contro of outut PDFs usng square root B-sne exanson. Autoat Yue H. Wang H.. Mnu entroy contro of cosed-oo tracng errors for dynac stochastc systes IEEE Trans. on Autoatc Contro Erdogus D. Prce J.C. (a). An error entroy nzaton agorth for suervsed tranng of nonnear adatve systes IEEE Transactons on Sgna Processng Erdogus D. Prce J.C. (b) Generazed nforaton otenta crteron for adatve syste tranng IEEE Transacton on Neura Networs CONCLUSIONS An aroach to contro nonnear TITO systes wth uncertan nut/outut te deays s resented based on nzng tota entroes whch conssts of entroy of each tracng error ther jont entroy. For sae of effcent coutaton easy reazaton Reny quadratc entroy n conjuncton wth Parzen wndow wth Gaussan erne s eoyed to construct a nove erforance ndex so as to obtan contro agorth. The controers are rred out by BP neura networs. The an resuts have foowng feature: ) the aroach n be easy extended to generay nonnear stochastc utvarabe contro systes wth uncertan nut/outut deays; ) the contro agorth n dea wth stochastc utvarabe systes wthout decoung; 3) the erforance ndex s estabshed based on entroes of each tracng error besdes the jont entroy of tracng errors the controed syste n aroach zero steady errors beuse controers ncude ntegra strategy; 4) the convergent condton of the roosed contro agorth s gven. Fnay a suaton exae s gven to show the effectveness of the roosed agorth. ACKNOWLEDGEMENTS