Full-state feedback LQR with integral gain for control of induction heating of steel billet

Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 A ailable online 4 June 2024 2215-0986/© 2024 THE AUTHORS. Published by Else ie BV on behal o Ka abuk Uni e si y. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). Full Leng h A icle Full-s a e eedback LQR wi h in eg al gain o con ol o induc ion hea ing o s eel bille Sohaibullah Za ghoon a , Samuel Emebu b , c , e , * , Radek Ma uˇ sů b , Cy il Bela ý a , Luk´ aˇ s Ba alský a , S anisla ˇ Du iˇ s a , Sabi Husnain , Cla a Mendoza Ma inez d a Ins i u e o Au oma ion, In o ma iza ion, and Measu emen , Facul y o Mechanical Enginee ing, Slo ak Uni e si y o Technology, N´ am. slobody 17, 812 31 B a isla a, Slo akia b Depa men o Au oma ion and Con ol Enginee ing, Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, Nad S ´ anˇ emi 4511, 760 05 Zlín, Czech Republic c Depa men o Chemical Enginee ing, Facul y o Enginee ing, Uni e si y o Benin, PO Box 1154, Benin Ci y, Nige ia d Depa men o Ene gy, Lappeen an a-Lah i Uni e si y o Technology LUT, Yliopis onka u 34, FI-53850 Lappeen an a, Finland e Depa men o Sepa a ion Science, Lappeen an a-Lah i Uni e si y o Technology LUT, Yliopis onka u 34, FI-53850 Lappeen an a, Finland Depa men o Mechanical, Mecha onics and Manu ac u ing Enginee ing, Uni e si y o Enginee ing and Technology Laho e, Laho e, Punjab, Pakis an ARTICLE INFO Keywo ds: Induc ion hea ing Skin e ec s Ca bon s eel bille Semi-nume ical me hod Full-s a e eedback Linea Quad a ic Regula o (LQR) ABSTRACT Induc ion hea ing is widely used in indus ial u naces due o i s apid esponse and ene gy e iciency. Compu e - aided modelling and simula ion a e necessa y o he design and op imisa ion o hese u naces. This a icle ocuses on modelling and simula ing induc ion hea ing o a one-dimensional spa ial pa ial di e en ial model o a ec angula ca bon s eel bille . The simula ion conside ed empe a u e dynamics, including he skin e ec and hea loss ia adia ion. The pa ial di e en ial model was con e ed in o a s a e-space model o designing a ull- s a e eedback Linea Quad a ic Regula o (LQR) wi h in eg al ac ion. This con olle was e ec i ely applied o egula e he bille ’s co e empe a u e om 1000 ◦C o 1200 ◦C, unde di e en inpu , R and s a e, Q, weighing ma ices o he LQR, as well as in he p esence o dis u bances. 1. In oduc ion Induc ion hea ing is a widely used hea ing echnique in a ious in- dus ies due o i s en i onmen al bene i s. T adi ional hea ing me hods, such as bu ne hea ing u naces, consume mo e ene gy when compa ed o induc ion hea ing equipmen . As a esul , inco po a ing induc ion hea ing sys ems in ho - o ging plan s helps educe ene gy consump ion, as well as ai pollu ion [1]. This me hod e icien ly gene a es hea di ec ly wi hin he molecula le el o me allic induc i e ma e ials such as aluminium, b ass, coppe , s eel, and semiconduc ing ma e ials like silicon ca bide [2], h ough elec omagne ic induc ion. The e o e, in- duc ion hea ea men has become a globally ecognised echnique o epai ing, welding bo h mic oscopic componen s and la ge pa s, o - e ing economic as well as p ac ical ad an ages [3]. In p ac ice, he design o induc ion hea ing u naces o a ious ap- plica ions o en in ol es a leng hy and expensi e ial-and-e o p ocess. Measu ing and e alua ing key ac o s such as cu en equency, em- pe a u e, and hea dis ibu ion can pose challenges. And compu e modelling is a sui able app oach o such e alua ion, as well as he design o induc ion hea ing u naces [4]. The modelling o he induc ion hea ing p ocess is based on he Maxwell equa ions, which cons i u e a se o Pa ial Di e en ial Equa ions (PDEs) as such a nonlinea model ha can be di icul o sol e. Howe e , he solu ions can be ob ained o simpli ied using modelling echniques such as analy ical, nume ical, and semi-analy ical app oaches [5–8]. De eloping and simula ing models o induc ion hea ing u naces a e essen ial o op imising and con olling he p ocess o achie e he desi ed empe a u e o he hea ed bille . This p ima ily aims o p e en haza ds and ensu e e icien ene gy u ilisa ion [9]. The induc ion hea ing p ocess can be con olled using ei he nonlinea o linea con olle s. Linea con olle s a e commonly employed in indus ial con ol sys ems since many nonlinea p ocesses can be accu a ely desc ibed by linea models unde speci ic nominal condi ions h ough app oxima ion me hods like ans e unc ions and s a e-space models [10]. Addi ionally, linea con olle s ha e a well- es ablished ounda ion, and iden i ying linea models based on p ocess da a is ela i ely s aigh o wa d. They p o ide a good app oxima ion in he icini y o he designed ope a ing condi ions, o e as esponse imes, and a e cos -e ec i e o implemen [11]. * Co esponding au ho . E-mail add ess: [email p o ec ed] (S. Emebu). Con en s lis s a ailable a ScienceDi ec Enginee ing Science and Technology, an In e na ional Jou nal jou nal homepage: www.else ie .com/loca e/jes ch h ps://doi.o g/10.1016/j.jes ch.2024.101721 Recei ed 20 Decembe 2023; Recei ed in e ised o m 20 Ap il 2024; Accep ed 21 May 2024 Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 2 Al hough he impo ance o modelling, simula ion, and con ol has been discussed, i is wo h no ing ha mos li e a u e s udies ha e p i- ma ily ocused on he modelling and simula ion o induc ion hea ing u naces. These s udies ha e o en compa ed he esul s ob ained om nume ical models, pa icula ly he Fini e Elemen Me hod (FEM), wi h expe imen al da a, and ha e gene ally ound good ag eemen [8,12–17]. Howe e , Demido ich e al. [18] no only conside ed FEM bu also explo ed and compa ed o he nume ical me hods such as ini e- di e ence me hods (FDM), bounda y elemen me hod (BEM), in ini e elemen me hod (IEM), as well as hei combina ions, wi hou he use o expe imen al da a. Fu he mo e, some epo s ha e compa ed analy ical models wi h expe imen al da a [5,15]. Fo ins ance, Li e al. [5] de eloped an analy ical model o p edic he empe a u e p o ile in a plana mo ing induc ion hea ing p ocess. The s udy demons a ed ha he empe a u e p o ile and cu en in ensi y calcula ed by he analy - ical model we e simila o he esul s ob ained om he nume ical model. Impo an ly, he analy ical model exhibi ed highe compu a- ional e iciency compa ed o he nume ical model. The ad an age o educed expe imen al and compu a ional complexi y o e ed by he analy ical model p o ides suppo o s udies such as Pa ida e al.’s[6] epo on he combina ion o analy ical and nume ical me hods, as well as A ei ioau ena e al.’s[7] semi-analy ical me hod o modelling he induc ion hea ing p ocess. Based on he discussion hus a , he e a e limi ed epo s on he con ol o he induc ion hea ing p ocess using he de eloped non-linea PDEs models. Cambe e al. [19] employed he dis ibu ed pa ame e heo y o he con ol o a con inuous s eel bille induc ion hea e sys em as lumped-inpu s (i.e., al e na ing cu en ) and dis ibu ed-ou pu s ( empe a u e p o ile o bille ). Goodwin e al. [20] modelled he in- duc ion hea ing p ocess ia a non-linea PDE, con olled he p ocess using nonlinea model p edic i e con ol (MPC), as well as compa ed expe imen al and simula ed esul s. Fu he mo e, Roe ze e al. [21] also modelled he induc ion hea ing p ocess ia a non-linea PDE, con olled and compa ed he p ocess wi h di e en ypes o con olle s (i.e., P opo ional in eg al (PI), and linea -quad a ic Gaussian (LQG) con olle s). Howe e , as highligh ed by Goodwin e al. [20], he inco po a ion o models o accoun o skin e ec phenomena, and he empe a u e dependency o pa ame e s, a e s ill limi ed in li e a u e. Pa icula ly he con ol o ma e ial wi h he inco po a ion o skin e ec [22] and he empe a u e dependency o ma e ial p ope ies a he han cons an alues (such as densi y, emissi i y, elec ical conduc i i y, he mal conduc i i y, and speci ic hea capaci y) [23], can esul in a e y complex nonlinea sys em ha is di icul o sol e as well as con- ol. In p ac ice, his kind o model can be compu a ionally demanding, slow, and expensi e o con ol using nonlinea con olle s [24]. The e- o e, o educe cos and inc ease he con olle esponse ime, a ises he need o explo e linea con olle s such as he LQR, o con ol he in- duc ion hea ing p ocess using simpli ied app oxima ions o non-linea PDEs models, wi h bo h conside a ion o skin e ec , and empe a u e dependency o ma e ial p ope ies. Al hough a emp s ha e been made o co e his esea ch gas [25,26], howe e esul s we e s ill inadequa e. Fo ins ance, Asadzadeh e al. [26] applied an analy ical app oxima- ion, speci ically he in e se model o simula e he hea ing p ocess wi h empe a u e dependency o he ma e ial p ope ies and a emp ed o conside skin e ec . Howe e , he in e se model nega ed he spa ial dis ibu ion o empe a u e along he ma e ial, hence being unable o e ec i ely po ay he skin e ec on he ma e ial. Fu he mo e, Kapus a e al. [25] applied a nume ical model o he simula ion o induc ion hea ing o s eel bille and con ol ia dis ibu ed-inpu and dis ibu ed- pa ame e -ou pu sys ems. Al hough Kapus a e al. [25] conside ed he empe a u e dependency o he ma e ial on he model, howe e , he model may no ha e adequa ely conside ed skin e ec on he spa ial empe a u e dis ibu ion wi hin he dep h o he ma e ial. The e o e, i is his esea ch gap ha his wo k a emp s o co e . Hence he no el y o his wo k en ails he con ol ia linea con olle o acili a e he apid esponse o he highly nonlinea model o induc ion hea ing o s eel bille wi h adequa e conside a ion o skin e ec , along wi h empe a u e dependency o i s ma e ial p ope ies. Speci ically, he LQR is a eliable linea con olle ha can be applied, because o i s abili y o deli e op imal as well as apid con ol esponse o he sys em pe he designe ’s speci ica ions (e.g. ac ua o s cha ac e is ics o limi a ions) [27]. LQR is howe e mo e sui able o sys ems wi h limi ed s eady-s a e e o on applica ion o he LQR eedback gain ma ix o he con olled sys em [28]. The e o e, o sys ems wi h high s eady-s a e e o he LQR is usually inco po a ed wi h in eg al gain, p opo ional in eg al gain, p opo ional de i a i e gain, e c [27,29,30]. This addi ional gain o he LQR gain ma ix se es o elimina e he high s eady-s a e e o , as well as inc ease he con olle esponse. Howe e , his may esul in an un- desi able o e shoo on he con olle esponse [31]. The e o e, o limi he e ec o con olle o e shoo , i is necessa y o op imise he LQR weighing ma ices [32]. The induc ion hea ing o s eel bille is cha ac- e ised by high s eady-s a e e o , hence in his s udy, he LQR wi h an in eg al gain is a po en ial linea con olle ha can be applied. Which o he bes o he au ho s’knowledge, ha e no been epo ed in li e a u e. The e o e, his s udy aims o model, simula e, and con ol he in- duc ion hea ing p ocess o a ec angula s eel bille . This will be ach- ie ed h ough he ollowing objec i es: Sou ce da a on he p ope ies o s eel as a unc ion o empe a u e; De elop a model and simula e he induc ion hea ing p ocess using a semi-nume ical me hod, inco po- a ing skin e ec phenomena and he p ope ies o s eel as a unc ion o empe a u e; Linea ise he model using he Jacobian me hod; Apply a linea con ol scheme (i.e., LQR wi h in eg al ac ion in conjunc ion wi h op imised weighing ma ices) o egula e he co e empe a u e o he s eel bille . By accomplishing hese objec i es, he s udy in ends o p o ide a comp ehensi e unde s anding o he induc ion hea ing p o- cess and p opose an e ec i e con ol s a egy o main aining he desi ed empe a u e in he s eel bille . Fu he mo e, he inno a i e con ibu ion o his s udy would en ail a p edic i e es ima ion and con ol o he nonlinea spa ial empe a u e a ia ion along he dep h o s eel bille because o skin e ec , as well as he non-uni o m ene gy o cu en dis ibu ion, especially ega ding powe losses associa ed wi h his phenomenon. 2. Me hodologies 2.1. P ocess desc ip ion and model de elopmen The Fig. 1a illus a es an induc ion hea ing u nace, in which he coil ca ies high- equency al e na ing cu en (AC) and gene a es ime- a ying elec omagne ic ields. These a ying ields, in u n, induce eddy cu en s in he bille , leading o hea gene a ion h ough he Joule e ec . The go e ning model o desc ibe he induc ion hea ing can be deduced om Ampe e’s law, Equa ions (1) a subse o Maxwell’s law [33] in ela ion o Equa ions (2), (3), and (4) o magne ic lux densi y, B (N.A -1 m −1 ), elec ic lux densi y, D (C.m −2 ), and conduc ion cu en densi y, J(A.m −2 ) espec i ely. ∇ × H=J+ ∂ D ∂ (1) D= ε E (2) B= μ H (3) J= σ E (4) In sol ing o elec ic ield in ensi y, E (N.C -1 ), and magne ic ield in- ensi y, H (A.m −1 ), i is mo e con enien o in oduce he magne ic ec o po en ial, A(N.A -1 ), and elec ic scala po en ial, φ(V), as gi en by Equa ion (5) –(6). The pa ame e s ε , μ ,and σ a e espec i ely emissi i y, magne ic pe meabili y (H.m −1 ), and elec ical conduc i i y (S.m −1 ) o he bille . S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 3 B= ∇ × A(5) E= ∇φ+ ∂ A ∂ (6) Subs i u ing Equa ions (2), (3), and (4) in o (1) yields Equa ion (7d). No ing ha ∂ε / ∂ =0, since ε is a cons an , and ( σ E+ ε∂ E/ ∂ ) ≈ σ E. 1 μ ∇B= σ E+ ∂ ( ε E) ∂ (7a) 1 μ ∇B= σ E+ ε∂ E ∂ +E ∂ε ∂ (7b) 1 μ ∇B= σ E+ ε∂ E ∂ (7c) 1 μ ∇B= σ E (7d) Fu he mo e, subs i u ing Equa ions (5) and (6) in o (7d), yields Equa- ion (8b) upon ea anging. No ing ha J = − σ ∇φ. 1 μ ∇2A= σ (∇φ+ ∂ A ∂ )= σ ∇φ+ σ∂ A ∂ (8a) 1 μ ∇2A= − J+ σ∂ A ∂ (8b) Assuming hea ing o he s eel bille is implemen ed by al e na ing cu - en , ypically a sinusoidal ha monic, so ha he magne ic ec o po- en ial, A, and cu en densi y, J, can be desc ibed as A=Aoej ω and J=Jsej ω espec i ely. No ing ha Aois he maximum magne ic ec o po en ial and Jsis he maximum cu en densi y on bille su aces [8]. The e o e, Equa ion (8b) can be exp essed in e ms o he sinusoidal ha monic, Equa ion (9c). In his equa ion, j implies an imagina y s a e, ω =2 π , is he angula equency and, is he ime o obse a ion. In sol ing his Equa ion (9c), he Di ichle (Ao=0) and Neumann bounda y condi ions a e espec i ely used in he co e and ou e su aces o he bille . 1 μ ∇2Aoej ω = − Jsej ω + σ∂ (Aoej ω ) ∂ (9a) 1 μ ∇2Aoej ω = − Jsej ω + σ Aoj ω ej ω (9b) 1 μ ∇2Ao= − Js+j ωσ Ao(9c) The Equa ion (9c) can be modi ied o inco po a e he skin e ec , i.e., he endency o he sou ce cu en o be g ea es a he su ace. This implies ha a highe cu en is induced a he su ace o he bille bu dec eases owa ds i s co e as desc ibed by Equa ion (10) [4,7] and Fig. 2. In Equa ion (10),dis he dis ance o he co e o he bille o i s su ace, δis he skin dep h, Equa ion (11) [4,7], i.e., he dep h below he su ace o he bille a which he cu en densi y has allen by he in e se exponen o he maximum sou ce cu en a he bille ou e su ace, Jso. In Equa ion (11), (Hz) is he equency o he sou ce cu en , and μ is he ela i e magne ic pe meabili y o he bille o acuum. Js=Jsoe−d/δ(10) δ= 1/( πμσ ) √≅503  1/( σμ ) √(11) The gene ic magne ic ec o po en ial on he bille su ace, Ao, can be deduced om Equa ion (9c) –(11), and subsequen ly he sou ce hea Fig. 1. Illus a ion o induc ion hea ing o a ec angula s eel bille . Fig. 2. Illus a ion o he skin e ec o a ec angula bille . S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 4 lux, q, can be calcula ed as gi en by Equa ion (12) in e ms o he conduc ion cu en densi y, J, which can also be exp essed in e ms o Aoas gi en by Equa ion (13) [8]. In his equa ion, Rsigni ies he eal s a e, and (AoAo)*indica es a p oduc o complex conjuga e. J= σ E= − j ωσ Ao(12) q=[R(J)]2 σ =1 2 ω 2 σ (AoAo)*(13) The hea di usion equa ion o induc ion hea ing on he bille can be exp essed by Equa ion (14). In he equa ion, ρ (Kg.m −3 ) is he densi y, Cp(J. Kg −1 . K −1 ) is he speci ic hea capaci y, and k (W.m −1 . k −1 ) is he he mal conduc i i y o he bille . ρ Cp ∂ T ∂ =k∇2T+q (14) The Equa ions (9c) and (14) a e sol ed using he o dina y di e en ial equa ion sol e in MATLAB (speci ically, ode15s) in collabo a ion wi h Equa ions (10), (11), and (13). In sol ing hese equa ions he spa ial empe a u e dis ibu ion o pa ial di e en ial, ∇, was app oxima ed using he Cen al scheme o he Fini e Di e ence Me hod (FDM). This esul s in N h numbe o equa ions ha ep esen s each space poin , i = 1, 2, 3…, Nxas shown in Fig. 1b. Fu he mo e, in Equa ion (14) he Neumann bounda y condi ion is applied o he co e, and o he ou e su ace o he bille , adia ion hea loss, i.e., Equa ion (15), is applied. The Equa ion (15) ela es he en i onmen o su ounding ai empe - a u e, TE, o he ou e empe a u e o he bille , T, ia he S e an- Bol zmann cons an , σ *, he mal conduc i i y, and emissi i y. k∇T= σ * ε (T4−T4 E)(15) These bounda y condi ions applied a e based on he ope a ing condi ion o indus ial induc ion hea ing u naces as well as popula ly epo ed in li e a u e, o which bo h con ec ion and adia i e hea losses a e conside ed [34,35]. Howe e , conside ing ha he adia i e hea loss is ≫con ec i e hea loss[36], only adia i e hea loss was conside ed in his wo k. 3. P ocess con ol heo y To e ec i ely con ol he sys em, i is necessa y o ini ially in es i- ga e he sys em p ope ies such as he sys em’s dynamics, s abili y, and pe o mance (i.e., he di e ence in s eady-s a e e o ). P io obse a- ions o induc ion hea ing sys ems om li e a u e [7,15] indica e ha he sys em is gene ally s able bu may exhibi signi ican s eady-s a e e o . To mi iga e his s eady-s a e e o , a ull-s a e eedback Linea Quad a ic Regula o (LQR) can be employed. In gene al, he LQR con ol aims o con ol linea sys em s a e a iables using op imal con ol cos unc ion in ini e ho izon o in ini e ho izon. I aims o deduce gain ma ices ha minimises a cos unc ion o ensu e adequa e sys em s a- bili y and con ol pe o mance. The e o e, he choice o weighing ma ices (i.e., s a e weighing and inpu weighing ma ices) is c ucial o adequa ely design he LQR [37]. The ull-s a e eedback LQR me hodology is based on he concep o a cos unc ion, deno ed as, j, whe ein he objec i e is o de e mine an op imal inpu o ac ua o e o , u, ha changes he s a e ec o , x, h ough minimisa ion o Equa ion (16) by adjus ing he s a e weighing ma ix, Q. The inpu weighing ma ix, R, is u ilised o penalise he ac ua o e o and achie e he desi ed con ol pe o mance. j=∫(xTQx +uTRu)d (16) Q and R a e diagonal ma ices, which mus be posi i e-de ini e o ensu e ha when mul iplied by he s a e, x(i.e., empe a u e), and inpu , u(i.e., sou ce hea lux), he column ec o s yield xTQx ≥0 and uTRu >0. The dimension Q and R ma ices a e de ined by he numbe o s a es, s, and inpu s, w, as gi en by Equa ion (17). Q=⎡ ⎢ ⎢ ⎣ Q1,10⋯0 0 Q2,2⋯0 0 0 ⋱ ⋮ 0 0 ⋯Qs,s ⎤ ⎥ ⎥ ⎦ &R=⎡ ⎢ ⎢ ⎣ R1,10⋯0 0 R2,2⋯0 0 0 ⋱ ⋮ 0 0 ⋯Rw,w ⎤ ⎥ ⎥ ⎦(17) Typically, he LQR using he in ini e ho izon cos unc ion, Equa ion (16), can be s uc u ed as gi en by Equa ion (18). This ollows he adi ional ze o- e minal quad a ic cos unc ion, applied in conjunc ion wi h dec e ised s a e-space model, Equa ion (22) –(23) as cons ain s. min uj=1 2∑ N k=1 (x(k )*−x(k ))TQ(x(k )*−x(k ))+u(k )TRu(k )(18) Subjec o: x(k +1) = Ax(k ) + Bu(k ) y(k ) = Cx(k ) In Equa ion (18), he ime scala o he p ocess model is di ided in o disc e e poin s, k ∈ [1 N ], and x(k)*is he e e ence alue o ou pu s. The solu ion o Equa ion (18) is equi alen o sol ing he Ricca i Fig. 3. Illus a ion o a ull-s a e eedback sys em wi h in eg al ac ion. Table 1 Simula ion pa ame e s u ilised o he induc i e hea ing u nace. Pa ame e , symbol Value, uni Du a ion o simula ion, 5000 s Thickness o bille ,ϑ0.12 m Numbe o disc e ised spa ial poin s,Nx5 Numbe o disc e ised ime poin s,N 100 Di e en ial dis ance be ween disc e ised spa ial poin s, Δxϑ/2Nx S e an-Bol zmann cons an , σ *5.6704 ×10 8 Cu ie empe a u e o s eel,Tc750 ℃ Pe meabili y ac o ,n 0.5 Magne ic pe meabili y o acuum, μ o4 π ×10 -7 H.m −1 Rela i e pe meabili y o ca bon s eel a oom empe a u e, μ o 10H.m −1 F equency o al e na ing cu en , [17] 100 Hz Maximum cu en densi y on bille su ace,Jso 50 ×10 5 A.m −2 Su ounding ai empe a u e,TE35 ℃ Ini ial empe a u e o bille ,Tin 1000 ℃ Ini ial maximum magne ic ec o po en ial,Ao0 N.A -1 S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 5 equa ion [38]. In sol ing he Ricca i equa ion, o deduce he inpu , u, Equa ion (21), he so-called “s a e ansi ion ma ix”, P, mus be deduced ia Equa ion (19), o es ima e he con olle gain, K, Equa ion (20). P=ATP+PA−PBR−1BTP+Q (19) K=R−1(BTP)(20) u= − R−1(BTPx)(21) The ull-s a e eedback LQR wi h in eg al ac ion wi hin he eed o wa d pa hway connec ing he e o compa a o and he sys em is depic ed in Fig. 3. In summa y o he diag am shown in Fig. 3, is he desi ed se - poin , u is he p edic ed con olle inpu , A, B,and C a e ma ices ep- esen ing he p ocess model, Kxis he LQR eedback gain ma ix and Kiis Fig. 4. P ope ies o ca bon s eel as a unc ion o empe a u e. S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 6 he in eg al gain. The addi ion o in eg al ac ion o he con ol sys em allows he con olle o ack he se poin (ins ead o elying on he classical null s abilisa ion) and, as such ensu e a be e pe u ba ion obus ness [39]. The aim is o s abilise he con olled sys em as well as in eg a e he s eady-s a e e o . Howe e , he in eg al ac ion can lead o con olle o e shoo s, he e o e o esol e his issue[40], adequa e measu es mus be aken o op imise he LQR h ough i s weigh ing ma ices, i.e. Q and R. Adequa e design o uning o hese wo ma ices a e he mos impo an pa ame e s o he LQR which can be edious and challenging[41]. To apply he p ocess model o he LQR, i s s a e and ou pu dynamics a e usually exp essed as s a e-space models, Equa ion (22) and (23). The s a e-space model de ined by ma ices, A, B,and C can be deduced om he linea isa ion o he dynamic models, Equa ion (9c) and (14). No e ha C is a ma ix o ze os and ones ha de ines he ou pu s om he s a es. ˙ x=Ax +Bu (22) y=Cx (23) ˙ I= −y= −Cx (24) The esul ing e o om he sys em, ˙ I, can be deduced om he di e - ence be ween he desi ed se poin , , and ou pu , y, o he sys em. Equa ions (22) and (24), can be used o design a new s a e-space model wi h in eg al ac ion as gi en by he s a e-space ma ices, Equa ion (25), Fig. 5. Illus a ion o empe a u e dynamics o he disc e ised bille su aces. Table 2 E alua ion me ic o he compa ison o de eloped model and da a o ca bon s eel p ope ies. Ca bon s eel p ope ies R 2 Densi y, ρ 0.9985 Speci ic hea capaci y,Cp0.9997 The mal conduc i i y, k 0.9998 Emissi i y, ε 1.0000 Elec ical conduc i i y, σ 0.9950 Fig. 6. Illus a ion o he esul ing magne ic ec o po en ial om he cu en densi y on bille su aces. S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 7 which can be u he simpli ied in o Equa ion (26). [˙ x ˙ I]=[A 0 −C 0][x I]+[B 0]u+[0 1] (25) ˙ x=AIxI+BIu+B (26) The con olle inpu , u, o he ull-s a e eedback LQR can be exp essed as gi en in Equa ion (27). Subs i u ing Equa ion (27) in o (26) yields Equa ion (28). The gain ma ices o he con olle a e gi en as K = [Kx⋮Ki]. Typically Kxis a single- ow ma ix whose column size is he same as he numbe o s a e a iables, x, i.e. he i e empe a u es o each disc e ised poin , Kx=[Kx1, Kx2, Kx3, Kx4, Kx5] and Kia single alue. The e o e, K =[Kx1, Kx2, Kx3, Kx4, Kx5, Ki]. u= − KxI(27) ˙ x= (AI−KBI)xI+B (28) In he a o emen ioned equa ions, he gain ma ix, K, can be de e mined h ough he u ilisa ion o he “lq ” unc ion in MATLAB R2021b by sol ing he Ricca i equa ion, Equa ion (19), and he con ol sys em is implemen ed wi h SIMULINK R2021b in he same design amewo k gi en by Fig. 3. 4. Resul s and discussion 4.1. Simula ion o induc ion hea ing To simula e he induc ion hea ing p ocess, Equa ions (9c) and (14) a e conside ed as one-dimensional spa ial di e en ial, ∇, along he x- axis, and disc e ised in o i e segmen s (i.e., Nx=5), Fig. 1b. The cen al di e ence app oxima ions a e used o de i e i e equa ions o ep esen Equa ions (9c) and (14) each. These equa ions a e hen sol ed simul- aneously wi h Equa ions (10), (11), and (13). The simula ion esul s, ob ained using he pa ame e s speci ied in Table 1, and Equa ions (29) – (34), a e p esen ed in Figs. 4 –5. The Equa ion (29) –(34) shows he empe a u e dependence (i.e., T in kel ins, K) o he densi y, ρ [42,43], speci ic hea capaci y, Cp [42,44–48], he mal conduc i i y, k [42–48], emissi i y, ε [49], elec- ical conduc i i y, σ [50], and ela i e magne ic pe meabili y, μ [34,51,52], o ca bon s eel. The Equa ions (29) –(33) we e cu e- i ed as a unc ion o empe a u e, using da a om he espec i e highligh ed li e a u e. Howe e , Equa ion (34) is as epo ed in li e a u e. The esul ing compa ison o he de eloped models and da a is shown in Fig. 4, wi h hei espec i e R-squa ed (R 2 ) e alua ion me ics gi en in Table 2, and he esul showed adequa e i (R 2 >0.95). ε =⎧ ⎨ ⎩ T≤663→0.28 653 ≤T≤793→ T≥793→0.69 0.6999exp(−(T−848.7278 203.6169 )2)(32) μ = μ μ o =⎧ ⎪ ⎨ ⎪ ⎩ T≤1023→1+ ( μ o −1)(1−(T−273 TC(◦C))n) T≥1023→1 (34) In Fig. 5a, upon he applica ion o sou ce cu en o he bille su ace, Js, he ou e mos su ace (i.e., Nx5) ini ially expe iences a dec ease in empe a u e due o adia ion hea loss, Equa ion (15) o he su ounding ai ilm. This hea loss occu s because o he sha p empe a u e con as ρ (kg.m3)=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 273 ≤T≤1673K→7849exp(−(T−146.6 4858 )2)+367.2exp(−(T−1914 468 )2) 1673 ≤T≤1803K→12420exp( − 0.0003T) 1803 ≤T≤1973K→7120exp(1.635E−8T) (29) Cp(J.kg−1K−1)=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 273 ≤T≤993K→4780exp(−(T−996.9 13.39 )2)+710.2exp(−(T−2046 2627 )2) 993 ≤T≤1973K→4412exp(−(T−990.6 42.22 )2)+603.7exp(−(T−21320 356500 )2)(30) k(W.m−1K−1)=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 273 ≤T≤1673K→12.13exp(−(T−349.3 384.4)2)+3.292E+15exp(−(T−641800 113100 )2) 1673 ≤T≤1773K→23.32exp( − 0.0002T) + 1.706E−13exp(0.0197T) 1773 ≤T≤1973K→259.7exp(7.473E−7T) (31) σ (S.m−1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 273 ≤T≤1773K→5364015.153exp(−(T−4977.4071 4746.0594 )2)+239071.5586exp(−(T−781.0628 317.3960 )2) 1773 ≤T≤1803K→1.4860exp(0.0074T) 1803 ≤T≤1973K→2.2269T2−8614.6218T+9202966.681 (33) S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 8 be ween he bille su ace (1000 ℃) and he su ounding ai empe a- u e (35 ℃). This hea loss akes place e en hough he skin e ec he- o em p edic s ha Jsshould be maximum a Nx5as such should expe ience an ins an aneous inc ease in empe a u e (i.e., in he absence o hea loss). The e o e, due o his coun e ac ing e ec , a e his ini ial hea loss, he su ace begins o s eadily inc ease in empe a u e. In con as o he obse a ion a he ou e mos su ace, he inne su aces (i.e., Nx4 o Nx1) expe ience an ins an aneous inc ease in empe a u e. Du ing he ini ial phase o hea ing (≤205 s), he middle su ace, Nx3, expe iences he highes hea ing a e, ollowed by Nx2and Nx1. This is because he skin e ec has a g ea e in luence on he Nx3su ace compa ed o adia ion hea loss. On he o he hand, he Nx4su ace expe iences he lowes hea ing a e due o i s p oximi y o he ou e su ace, esul ing in mo e adia ion hea loss o he su ounding su ace, Nx5. Howe e , as he hea ing p ocess con inues (>205 s), he e is mo e hea accumula ion owa d he co e, Nx1, o he bille due o negligible adia ion hea loss o he su oundings. This con inues e en hough, acco ding o he skin e ec heo em, Jsdec eases owa ds he co e, Nx1. Consequen ly, as he hea ing p og esses, when all su aces each hei s eady-s a e empe a u es a app oxima ely 4400 s, he co e empe a- u e inally eaches he highes empe a u e. The p eceding su ace empe a u es ollow he o de : Nx1(≅1259 ◦C) >Nx2(≅1258 ◦C) > Nx3(≅1246 ◦C) >Nx4(≅1217 ◦C) >Nx5(≅1164 ◦C). Conside ing ha he spa ial disc e isa ion o s eel bille was linea , i.e., Δx=ϑ/2Nx. I would ha e been expec ed ha he spa ial dis ibu ion o empe a u e along he dep h o he s eel bille should be co espondingly linea . The esul s, (Tx1−Tx2≅1◦C, Tx2−Tx3≅12 ◦C, Tx3−Tx4≅29 ◦C, Tx4−Tx5≅53 ◦C) and Fig. 5a, howe e shows o he wise. This highly nonlinea spa ial empe a u e dis ibu ion can be a ibu ed o he skin e ec , as well as he nonlinea bounda y condi ions (i.e., he Di ichle bounda y condi ion ia adia i e hea loss on Nx5, and he Neumann bounda y on Nx1). Fu he mo e, Fig. 5b shows a su ace plo illus a ing he dynamic changes in empe a u e ac oss he disc e ised su aces as ime p o- g esses. I isually ep esen s how he empe a u e e ol es h oughou he bille . In addi ion, Fig. 6 showcases he esul ing magne ic ec o po en ial, A, on he bille su aces. As an icipa ed, A, dec eases owa ds he co e o he bille , ollowing he in luence o he skin e ec on Js. This demons a es he dis ibu ion and beha io o he magne ic ield wi hin he induc ion hea ing p ocess. The esul s o he simula ion o his model hus a con o m o hose epo ed in li e a u e. Luozzo e al.[34] o ca bon s eel ube, Hansson and Fisk [35] o s ainless s eel ube, Jelicic [53] and Baldan [54] o solid s eel bille , all epo ed simila esul s ha e i y he esul o his wo k. Conside ing ha bo h adia ion and con ec ion hea losses on he su ace we e conside ed, he inne su ace also expe ienced highe empe a u es han he ou e su ace, as ound in his s udy. Howe e , in his wo k, he empe a u e di e ence was highe compa ed o hose epo ed in li e a u e. The di e ence can be a ibu ed o he ob ious di e ence in ini ial condi ion (i.e. 1000 ◦C in his wo k as compa ed o abou 20 o 35 ◦C o o he wo k), emissi i y, as well as he assump ion o con ec i e hea ans e conside ed. Fu he mo e Jankowski e al. [55] al hough conside ed a cylind ical s eel bille , he end o he esul also showed simila i y wi h ha o his wo k, e en hough he spa ial Fig. 8. LQR empe a u e con ol o bille co e su ace based on changes in inpu weighing ma ix. Fig. 7. LQR empe a u e con ol o bille co e su ace based on changes in s a e weighing ma ix. S. Za ghoon e al. Enginee ing Science and Technology, an In e na ional Jou nal 55 (2024) 101721 9 empe a u e dis ibu ion wasn’ conside ed in hei wo k. Janu ien˙ e and Maˇ zeika [56] howe e epo ed highe empe a u es on he bille su - ace han a he co e o a solid cylind ical ca bon s eel bille , con a y o he esul o his wo k. This disc epancy is a esul o negligible hea loss assumed a he su ace, as such due o he p e ailing skin e ec , he esul is expec ed o such an assump ion. 5. P ocess con ol o induc ion hea ing To con ol he induc ion hea ing p ocess, he s a e-space model desc ibed by Equa ion (22) –(28) is u ilised. To deduce he A and B ma ices equi ed o his pu pose, he dynamic models, Equa ions (9c) and (14) in conjunc ion wi h Equa ions (10), (11), and (13) a e line- a ised using Taylo ’s se ies o Jacobian linea isa ion heo em, as ex ensi ely desc ibed in li e a u e [11]. The alue o Jso gi en in Table 1 was used as he s eady-s a e alue o he maximum cu en densi y, Jso, o he linea isa ion p ocess, along wi h he p elimina y s eady-s a e empe a u e alues o su aces Nx1 o Nx5i.e., T1=1259 ◦C,T2= 1258 ◦C, T3=1246 ◦C, T4=1217 ◦C andT5=1164 ◦C. These empe a u e alues a e used o he p elimina y calcula ion o ca bon s eel p ope ies as highligh ed in Equa ion (29) –(34). The esul ing ma ices ob ained om he linea isa ion p ocess a e gi en by Equa ion (35), wi h he co esponding ou pu ma ix, C = [ 10000], and po en ially he dis u bance o di ec ansmission ma ix, D =0. The ypical applica ion o he ma ices A, B, and C o he s a e-space model, Equa ion (22) – (23), is gi en in Equa ion (30b). To e i y he ea lie assump ion o sys em s abili y, he open-loop s abili y o he induc ion hea ing sys em was e alua ed based on he eigen alues, λ, ob ained om he de e minan o he ma ix A and i s iden i y ma ix, I, i.e, |A−λI|= 0. The sys em s abili y is con i med i all eigen alues ha e nega i e eal pa s, i.e., λ<0. The esul ing eigen- alues (−0.0017, −0.0308, −0.0954, −0.1850, −0.1599) a i m he sys em is s able. Wi h he ma ices A and B de e mined, and s abili y con i med, he LQR con ol sys em can be designed. The esul ing AI, and BIma ices a e gi en by Equa ion (38), wi h he p e iously speci ied ma ix, B =[ 0 1 ]T. The LQR, i espec i e o he speci ic alues assigned o he Q and R pa ame e s, se es as design elemen s o penalise bo h he s a e a iables and he con ol signals [57,58]. In his s udy, he Q and R ma ices we e es ed unde wo case s udies. In case- s udy-1, R was kep cons an while Q was a ied, and in case-s udy-2, Q was kep cons an while R was assigned di e en alues. The objec i e o hese case s udies was o e alua e he sys em’s ou pu when a emp ing o each a se poin o 1200 ℃a he co e o he bille . In case-s udy-1, he ollowing alues o Q and R ma ices we e espec i ely applied, i.e., Q1=diag (1,1,1,1,1,100), Q2=diag (1,1,1,1,1,150), Q3=diag (1,1,1,1,1,200), Q4=diag (1,1,1,1,1,300), and Q5=diag (1,1,1,1,1,350), Q6=diag (1,1,1,1,1,600) as well as R = 2, i.e., he R ma ix is kep cons an . Fig. 7a illus a es he esponse o he sys em upon he implemen a ion o a ull-s a e eedback LQR, wi h he esul ing gain ma ices gi en by Equa ion (39). The obse ed beha iou indica es ha as he alue o Q inc eases, he sys em app oaches he se poin wi h minimal o e shoo and an imp essi e se ling ime bo h be o e and a e he occu ence o dis u bance. Gene ally, in a con ol sys em, a as e con olle esponse o se ling ime o each he se poin is desi able, while o e shoo is conside ed undesi able. Among he es ed alues o Q, Q3demons a es he bes pe o mance as i esul s in minimal o e shoo and an adequa e se ling ime. Inc easing he alue o Q gene ally leads o as e se ling ime bu a he expense o highe o e shoo ing due o he ini ial o e eac ion o he con olle [59], which may ha e esul ed in o ampli ied by he in eg al ac ion added o he LQR. This can be obse ed in he esponse o Q6(i.e., he highes Q alue), which achie es he as es se ling ime bu wi h he highes o e shoo , while Q1(i.e., he lowes Q alue) exhibi s he opposi e beha iou . O e all, he con olle p o es o be e ec i e e en in he p esence o dis u bance a 4000 s, as he sys em quickly e u ns o i s se poin . The esul ing maximum cu en densi y, Jso, o achie ing empe a u e con ol o he bille co e su ace is depic ed in Fig. 7b. A=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ −0.0908 0.0908 0.0000 0.0000 0.0000 0.0454 −0.0908 0.0454 0.0000 0.0000 0.0000 0.0455 −0.0909 0.0455 0.0000 0.0000 0.0000 0.0456 −0.0911 0.0456 0.0000 0.0000 0.0000 0.0915 −0.1091 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ &B =⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 5.1363E −8 1.7614E −7 3.2884E −7 4.6366E −7 5.3544E −7 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (35) ˙ x=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ dT1/d dT2/d dT3/d dT4/d dT5/d ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ =⎡ ⎢ ⎢ ⎢ ⎢ ⎣ −0.0908 0.0908 0.0000 0.0000 0.0000 0.0454 −0.0908 0.0454 0.0000 0.0000 0.0000 0.0455 −0.0909 0.0455 0.0000 0.0000 0.0000 0.0456 −0.0911 0.0456 0.0000 0.0000 0.0000 0.0915 −0.1091 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⏟⏞⏞⏟ A ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ T1 T2 T3 T4 T5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⏟⏞⏞⏟ x +⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 5.1363E −8 1.7614E −7 3.2884E −7 4.6366E −7 5.3544E −7 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⏟⏞⏞⏟ B q ⏟⏞⏞⏟ u (36) Fig. 9. LQR empe a u e con ol o bille co e su ace based on speci ic s a e and inpu weighing ma ix. S. Za ghoon e al.