338 IEEE/CAA JOURAL OF AUTOMATICA SIICA VOL. 3 O. 3 JULY 26 An Approach to Deign MIMO FO Controller for Untable onlinear Plant Arturo Roja-Moreno Senior Member IEEE Abtract Thi paper develop an approach to control untable nonlinear multi-input multi-output (MIMO) quare plant uing MIMO fractional order (FO) controller. The controller deign ue the linear time invariant (LTI) tate pace repreentation of the nonlinear model of the plant and the diagonal cloedloop tranfer matrix (TM) function to enure decoupling between input. Each element of the obtained MIMO controller could be either a tranfer function (TF) or a gain. A TF i aociated in turn with it correponding FO TF. For example a D (Derivative) TF i related to a FO TF of the form D δ δ =. Two application were performed to validate the developed approach via experimentation: control of the angular poition of a manipulator and control of the car and arm poition of a tranlational manipulator. Index Term Fractional calculu modeling of nonlinear ytem control of manipulator multivariable decoupling multivariable nonlinear ytem. I. ITRODUCTIO CURRETLY reearch for the deign application and tuning rule of SISO (Single-input-ingle-output) FO PID controller uch a P I λ D δ i notably growing becaue controller deigned via fractional calculu improve the control performance and robutne over conventional IO (Integer order) PID controller due mainly to the preence of two more tuning parameter 3 : fractional number λ and δ. However few reult about FO control of MIMO plant have been publihed. In 4 the two interacting conical tank proce a twoinput two-output table plant wa controlled by a multiloop FO PID configuration employing two FO PID controller that were tuned uing the cuckoo algorithm. Reference 5 deal with the tuning of FO PID controller employing a genetic algorithm. Thoe controller were applied to a MIMO proce. A MIMO FO PID controller wa deigned in 6 to control table MIMO time-delay plant. The reulting controller ha a diagonal form with each diagonal element being a FO PI controller whoe parameter were tuned uing CMAES (Covariance matrix adaptation evolution trategy). In 7 a diagonal-form MIMO FO PI controller wa deigned to control table time-delay ytem. The deign procedure i baed on a teady tate decoupling of the MIMO ytem. A MIMO IO (Integer order) PID controller wa deigned uing LMI Manucript received September 6 25; accepted February 26. Recommended by Aociate Editor YangQuan Chen. Citation: Arturo Roja-Moreno. An approach to deign MIMO FO controller for untable nonlinear plant. IEEE/CAA Journal of Automatica Sinica 26 3(3): 338 344 Arturo Roja-Moreno i with the Department of Electrical Engineering Univeridad de Ingenieria y Tecnologia (www.utec.edu.pe) Lima 563 Peru (email: aroja@utec.edu.pe). (Linear matrix inequality) to control a MIMO FO plant in 8. Deign approache developed in 4 6 and 7 were teted via imulation. Thi paper develop an approach to control table and untable nonlinear MIMO quare plant uing MIMO FO controller. The deign procedure employ the LTI tate pace repreentation of the nonlinear model of the the plant. Fig. depict the linear feedback control ytem where the TM function G p () of the plant i computed from it pace tate decription. A elected diagonal cloed-loop TM function denoted a G T () enure decoupling between input. For deign purpoe each element of thi TM function ha the form of a firt order TF with unity gain. The tep repone of thi TF contitute the deired output repone of the variable under control. Knowing G p () and G T () the MIMO controller G c () actually the tructure of the FO MIMO controller G cf O () hown in Fig. 2 can be eaily computed. The element of G c () could be either tranfer function or gain. Replacing each TF of G c () with it correponding FO TF G c () become G cf O (). For intance the FO form of the Laplace variable i δ while the FO form of i λ where δ and λ are fractional number between and. The validity of the developed deign approach wa verified via experimentation uing the FO nonlinear control ytem of Fig. 2. Two application were performed for uch a purpoe: control of the angular poition of a manipulator and control of the car and arm poition of a tranlational manipulator. Fig.. Fig. 2. Block diagram of the linear feedback control ytem. Block diagram of the nonlinear FO feedback control ytem. The MIMO FO controller deigned in thi work i novel due to it abilitie to control not only MIMO table plant but alo nonlinear MIMO untable one. On the other hand the developed approach wa verified not only via imulation but alo by mean of two real-time application. Thi paper i organized a follow. Section II deal with the deign of the MIMO FO controller for MIMO nonlinear pla-
ROJAS-MOREO: A APPROACH TO DESIG MIMO FO COTROLLERS FOR USTABLE OLIEAR PLATS 339 nt. The firt and econd application are decribed in Section III and IV repectively while in Section V ome concluion derived from thi work are preented and dicued. II. DESIG OF THE MIMO FO COTROLLER A MIMO nonlinear plant can be decribed by the following teady tate repreentation Ẋ = f(x U) () where f i the function vector that decribe the ytem dynamic. X and U are the tate and control vector repectively. All vector and function are of known order. The correponding LTI tate-model can be obtained by linearization of () about a nominal trajectory. That i ẋx = Ax + Bu y = Cx (2) where A B and C are the tate control and output matrice repectively and y i the output vector. The TM function of the linear plant (2) i obtained from G p () = C(I A) B + D. (3) In (3) I i the identity matrix and all vector and matrice are of known order. Fig. depict the block diagram of the MIMO LTI control ytem where G c () i the MIMO controller G() i the open-loop matrix function G() = G p ()G c () (4) G T () i the cloed-loop matrix function G T () = G() + I G() (5) and r e u and y repreent reference error control and output vector repectively. Conider the following diagonal cloed-loop matrix function to enure complete decoupling between different m input From (5) G T () = G T... G T mm. (6) G() = G T ()I G T (). (7) Since G T i diagonal I G T and I G T are alo diagonal matrice. Therefore matrix G take on the diagonal form G() = G T G T... G T mm G T mm. (8) From Fig. y() = G T ()r() where r() i the reference vector. The ytem error e(t) i given by e() = r() y() = I G T ()r(). (9) The neceary condition to make e(t) = in (9) i Introducing condition () in (5) reult lim G T () = I. () I + G() = G(). () Thi requirement mean that each element of the diagonal matrix G mut contain at leat one integrator. Uing (4) in (5) we obtain the MIMO controller G c () depicted in Fig.. That i G c () = G p () G T ()I G T (). (2) Element of G c () can be either gain or TF of the form K c ; K ( + z i i) ; K i= d; G x () = K (3) Π ( + p j) j= where Kc K i K d and K are real gain and z i and p j are pole and zero of G x (). ote in (3) that G x () i the general form of a TF. To formulate the FO MIMO controller denoted a G cf O term Ki and K d of (3) are written a K i λ ; K d δ (4) where δ and λ are poitive fractional number. It i worth mentioning that intenive reearch i performed in finding the FO counterpart of the TF G x () of (3). For example a particular cae of G x () i the following lead or lag compenator K ( + /ω b) ( + /ω h ). (5) The correponding FO counterpart of (5) i written a 9 ( ) r + /ωk K K Π + /ω h k= ( M Π + /ω k + /ω h where < ω b < ω h K > and ω k and ω k are corner frequencie that are computed recurively. Fig. 2 depict the block diagram of the FO feedback control ytem. For real-time implementation it i required to have the dicrete form of the controller G cf O. The dicretization method by Muir recurion etablihe δ ( ) δ 2 A n (z δ) T A n (z δ) (6) where T i the ample time. In (6) polynomial A n (z δ) and A n (z δ) can be computed in recurive form from A n (z δ) = A n (z δ) c n z n A n (z δ) A (z δ) = { δ/n if n i odd c n = if n i even. ) (7)
34 IEEE/CAA JOURAL OF AUTOMATICA SIICA VOL. 3 O. 3 JULY 26 For n = 3 (7) take on the form M (qq )q q + P (qq q )q + d (qq ) = u A3 (z δ) = 3 δz 3 + A3 (z δ) = 3 δz 3 + 2 2 δz + 3δ z 2 2 + δz +. 3δ z (8) The developed deign procedure will be validated experimentally with two application decribed below. M = d= III. F IRST A PPLICATIO Bae poition q and arm poition q2 of a manipulator of 2DOF (2 degree of freedom) will be controlled uing a MIMO FO controller. Fig. 3 how the experimental etup. The bae and the arm of the manipulator are driven by two DC ervomotor having reduction mechanim and quadrature encoder to ene angular poition. A I crio-973 (Compact reconfigurable input/output) device wa employed to embed the MIMO FO controller. Module I 9263 and I 94 were ued to acquire angular poition and generate the control ignal repectively. Such ignal were amplified uing two PWM (Pule width modulation) Galil motion control amplifier. M d2 M22 P = q q= q2 (9) P2 P22 u u= u2 P P2 J + Jeq + 2Ma in2 q2 (J2 + Jeq ) M22 = µ n2 Km Kb P = Beq + Bq + P2 = (4Ma q in q2 co q2 ) P2 = (2Ma q in q2 co q2 ) µ n2 Km Kb P22 = Beq + Bq2 + d2 = (in q2 ). M = In (9) M = M T i the diagonal inertia matrix matrice P and d contain Corioli and centripetal force and gravitational torque repectively. u repreent the control vector. Table I decribe the valued parameter. Uing in (9) the approximation: in2 q2 q22 q in q2 co q2 q q2 in q2 q2 we obtain M = Fig. 3. M P = P P 2 P 2 P 22 d= d2 q2 (2) The experimental etup of the manipulator of 2DOF. TABLE I VALUED PARAMETERS OF THE MAIPULATOR Symbol Ma J J2 Jeq Bq Bq2 Beq D n KA Km Kb g Decription M. I. (Moment of inertia) M. I. M. I. Equivalent M. I. F.C. (Friction contant) F. C. Equivalent F. C. Torque Gear ratio Armature reitance Amplifier gain Servomotor contant Back EMF contant Gravitational contant Value.57.394.767.89.2.2 3.2287.648 2.5 3.5 2.5.42.565 9.8 Unit kg m2 kg m2 kg m2 kg m2 m /rad m /rad m /rad m Ω m/a V /rad m/2 The following dynamic model of the manipulator wa obtained uing Lagrange equation (J + Jeq ) = M22 = P P 2 = P 2 = P 22 = P22 =. M = P d2 Defining a tate variable: x = q x2 = q2 x3 = q and x4 = q 2 the linear model (2) can be tranformed into x = Ax + Bu x Bu A= a42 a44 a33 C= y = Cx Cx B = M (2)
ROJAS-MOREO: A APPROACH TO DESIG MIMO FO COTROLLERS FOR USTABLE OLIEAR PLATS 34 a 33 = P M a 42 = d 2 a 44 = P 22 b 3 = M b 42 = (22) It i not difficult to demontrate that output of model given by (9) and (2) have untable tep repone. In order to atify condition () matrix G T () i elected a G T () = +T (23) +T 2 where T and T 2 are the choen time contant of the controlled variable q and q 2. Subtituting (23) into (7) we obtain T G() =. (24) T 2 Oberve that G() in (24) meet requirement (). ote that G T () matrice having diagonal element a M Π (T i + ) i= Π (T j + ) j= fulfill condition () but not requirement (). Therefore uch matrice can not be employed to calculate the controller G c () uing (2). On the other hand tep repone of firt order tranfer function a ued in (23) contitute a good meaure of deign pecification to be met by the controlled output becaue thoe how no overhoot null teady tate error and time contant that are about one quarter of the ettling time of the output under control. G p () and G c () matrice are obtained uing equation (3) and (2) repectively. The controller G c () ha the form G c () = Kc + K d K c22 + K d22 + Ki22. (25) Parameter in (25) are function of thoe of (22). The correponding FO controller i expreed a G cf O () = Kc + K d δ K c22 + K d22 δ + Ki22 λ From Fig. 2 the FO control force i decribed by. (26) In (28) all a i b j and h j are known contant for intance a 6 = δλ a = b 6 = 9 K dδλ( 2 T )δ 9 K cδλ h 6 = 9 K d22δλ( 2 T )δ 9 K c22δλ 9 K i22δλ( 2 T ) λ where T i the ampling time elected a m in thi work. The FO control ytem wa imulated in Mathcript with time contant T and T 2 and FO parameter δ and λ et to.5.5.5 and.9 repectively. The other gain were taken from (25): K c =.7 K d =.866 K c22 = 26.74 K d22 = 4.66 K i22 = 3.2786. For the experimentation phae K c wa et to 5. Fig. 4 and 5 depict the experimental reult. Fig. 4. Controlled bae poition q (t) of the manipulator with repect to tep wie reference. u() = G cf O ()e(). (27) Subtituting (6) with n = 3 into (26) and uing the hift property z n u i (z) = u i (k n) and z n e i (z) = e i (k n) i = 2 where k i the dicrete time we obtain the following difference equation for the control force u (k) = Σ 6 a i u (k i) + Σ 6 b j e (k j) a i= j= u 2 (k) = Σ 6 a i u 2 (k i) + Σ 6 h j e 2 (k j). (28) a i= j= Fig. 5. Controlled arm poition q 2(t) of the manipulator with repect to tep wie reference.
342 IEEE/CAA JOURAL OF AUTOMATICA SIICA VOL. 3 O. 3 JULY 26 IV. SECOD APPLICATIO The linear poition q of the car and the angular poition q 2 of the arm of a tranlational manipulator of 2DOF hown in Fig. 6 will be controlled with a MIMO FO controller. The manipulator poee two DC ervomotor with reduction mechanim and quadrature encoder to ene angular poition. One ervomotor i attached to the axi of one of the two pulley. Thoe pulley carry a cable to tranmit the force to tranlate the car which i mounted on rail. The other ervomotor i mounted on the car to drive the arm. Thi application ue the ame experimental etup a the firt one. TABLE II VALUED PARAMETERS OF THE MAIPULATOR Symbol Decription Value Unit r p Pulley radio.5 m K A Amplifier gain 2.5 K x Contant.9858 /A K b Emf contant.565 V /rad R a Armature reitance 5.3 Ω K m Motor contant.42 m/a J x Ma 45.65 kg J t Moment of inertia.325 kg m 2 J eq2 Moment of inertia.26 kg m 2 B T Friction contant.8 m/rad/ B F Friction contant 2.8 kg/ B x Contant 2 39.5 kg//m 2 B eq2 Friction contant 3.2287 kg/ m Ma 2.2 kg m 2 Work.695 kg m g Gravitational contant 9.8 m/ 2 n Gear ratio 2.5 On uing the approximation co θ in θ θ and θ in θ in (29) we obtain Fig. 6. The tranlational manipulator of 2DOF. The dynamic model of the manipulator wa alo obtained employing Lagrange equation. The reulting nonlinear model take on the form q = q M = r = q 2 θ M(q) q + P (q q) q + d(q) = u (29) u u = d = u 2 d 2 M M 2 P P P = 2 M 2 P 22 M = J x + m M 2 = m 2 co θ K x K A K x K A m 2 M 2 = co θ = J eq2 + J t r p K x K A2 r p K x K A2 P = B x + B F K x K A P 2 = m 2 K x K A θ in θ P 22 = n 2 K mk b + B eq2 + B T r p K x K A2 d 2 = gm 2 r p K x K A2 in θ. Oberve that matrix M in (29) i neither diagonal nor ymmetric which make the manipulator more challenging to be controlled. Recall that matrix M in (2) wa diagonal. Table II decribe the valued parameter of the tranlational manipulator. M = M M 2 = m 2 m 2 M 2 = K x K A r p K x K A2 = P = P P 2 = P 22 = P 22 d 2 = gm 2 r p K x K A2. (3) Defining a tate variable: x = r x 2 = θ x 3 = ṙ and x 4 = θ the nonlinear model of (29) uing (3) can be tranformed into the following tate equation A = ẋx = Ax + Bu a 32 a 33 a 34 a 42 a 43 a 44 C = B = y = Cx (3) b 3 b 32 b 4 b 42 a 32 = M 2d 2 den a 33 = P den a 34 = M 2P 22 den a 42 = M d 2 den a 43 = M 2P den a 44 = M P 22 den b 3 = den b 32 = M 2 den b 4 = M 2 den b 42 = M den den = M M 2 M 2. (32) It i not difficult to prove that output of model given by (29) and (3) poe untable tep repone. According to () G T () matrix i choen a in (23). G p () G c () and G()
ROJAS-MOREO: A APPROACH TO DESIG MIMO FO COTROLLERS FOR USTABLE OLIEAR PLATS 343 matrice are obtained uing (3) (2) and (4) repectively. G() reult in a matrix that ha the ame form a (24). The G c () matrix i calculated uing relation (2) G c () = K c + K d K c2 + K d2 K c2 + Ki2 K c22 + Ki22 + K d2 + K d22. (33) Parameter of (33) are function of parameter decribed in (32). The correponding FO controller i expreed a G cf O () K c + K d δ = K c2 + K d2 δ K c2 + Ki2 λ K c22 + Ki22 λ + K d2 δ + K d22 δ. (34) Fig. 7. Controlled car poition q (t) of the manipulator with repect to tep wie reference. Compare (33) with (25) and (34) with (26). Replacing (6) with n = 3 in (34) and then employing the hift property we obtain the following difference equation for the vector control (27) u (k) = Σ 6 a i u (k i) + Σ 6 b j e (k j) a i= j= + Σ 6 c j e 2 (k j) j= u 2 (k) = Σ 6 a i u 2 (k i) + Σ 6 g j e 2 (k j) a i= j= + Σ 6 h j e 2 (k j). (35) j= In (35) k i the dicrete time and all a i b j and h j are known contant a in (27). T the ampling time wa elected to be m. The FO control ytem for thi manipulator wa imulated in Mathcript with time contant T and T 2 and FO parameter δ and λ et to.3.5.6 and.75 repectively. The other gain were taken from (3): K c =22 K d = K c2 = K i2 = ; K d2 =.28 K c2 = K d2 =.93 K c22 = 2 K d22 = K i22 = 5. For the experimentation phae K c wa et to 25. Fig. 7 and 8 depict the experimental reult. V. COCLUSIOS In light of the reult in Section III and IV the main goal of thi work ha been achieved: experimental verification of the deign approach to control nonlinear MIMO procee uing MIMO FO controller. The nonlinear model of the plant i neceary to obtain the linear model required to deign the tructure of the FO controller and to tet via imulation the deigned FO controller. In the imulation phae controller parameter were tuned uing the trial and error method. Such valued parameter were ued with few modification for the experimentation phae. Fig. 8. Controlled arm poition q 2(t) of the manipulator with repect to tep wie reference. Intenive work ha been done in tuning rule development for FO SISO (Single-input ingle-output) controller. It i till under reearch to extend the reult for FO MIMO controller. o tuning method for controller of the form given by (26) and (34) have been reported. Moreover depending on the application the tructure of a FO MIMO controller can change (compare (26) with (34)) making the development of proper tuning rule difficult. For uch reaon thi work employed the trial and error method. MIMO FO controller deigned in 4 6 and 7 were tuned uing different method becaue uch controller have diagonal form with each diagonal element being a FO SISO controller. Thi work i novel becaue unlike other the deigned FO MIMO controller can be applied not only to table MIMO plant but alo to nonlinear untable MIMO plant. Thi approach wa teted not only via imulation but alo via experimentation. The propoed deign procedure can alo be applied to MIMO time-delay plant. In order to obtain a LTI tate pace
344 IEEE/CAA JOURAL OF AUTOMATICA SIICA VOL. 3 O. 3 JULY 26 decription of the plant TF containing time-delay need to be replaced by equivalent TF. For example 2 + e 2 ( + ) 4 It i neceary to perform more reearch related to the deign of a FO MIMO controller when the tructure of the controller matrice (25) and (33) for example ha term of the form M Π ( + z i) i= G x () = K Π ( + p j) j= REFERECES Karad S Chatterji S Suryawanhi P. Performance analyi of fractional order PID controller with the conventional PID controller for bioreactor control. International Journal of Scientific & Engineering Reearch 22 3(6): 6 2 Singh S Koti A. Comparative tudy of integer order PI-PD controller and fractional order PI-PD controller of a DC motor for peed and poition control. International Journal of Electrical and Electronic Engineering & Telecommunication 25 4(2): 22 26 3 Chen Y Petrǎš I Xue D. Fractional order control - a tutorial. In: Proceeding of the 29 American Control Conference. St. Loui MO USA: IEEE 29. 397 4 4 Banu U S Lakhmanaprabu S K. Multiloop fractional order PID controller tuned uing cuckoo algorithm for two interacting conical tank proce. World Academy of Science Engineering and Technology International Journal of Mechanical and Mechatronic Engineering 25 2(): 742 5 Moradi M. A genetic-multivariable fractional order PID control to multiinput multi-output procee. Journal of Proce Control 24 24(4): 336 343 6 Sivananaithaperumal S Bakar S. Deign of multivariable fractional order PID controller uing covariance matrix adaptation evolution trategy. Archive of Control Science 24 24(2): 235 25 7 Murean C I Dulf E H Ionecu C M. Multivariable fractional order PI controller for time delay procee. In: Proceeding of the 22 International Conference on Engineering and Applied Science. Colombo Sri Lanka 22. 8 Song X Chen Y Q Tejado I Vinagre B M. Multivariable fractional order PID controller deign via LMI approach. In: Proceeding of the 8th IFAC World Congre. Milano Italy: IFAC 2. 396 3965 9 Malti R Melchior P Lanue P Outaloup A. Toward an object oriented CROE toolbox for fractional differential ytem. In: Proceeding of the 8th IFAC World Congre. Milano Italy: IFAC 2. 83 835 Vinagre B M Chen Y Q Petrǎš I. Two direct Tutin dicretization method for fractional-order differentiator/integrator. Journal of the Franklin Intitute 23 34(5): 349 362 Michalowki T. Application of MATLAB in Science and Engineering. Rijeka Croatia: InTech 2. Arturo Roja-Moreno graduated from Univeridad acional de Ingenieria (UI) Peru in 973. He received the M. Sc. degree from UI in 979 and the Ph. D. degree from Utah State Univerity USA in 995. He i currently a profeor at the Electrical Engineering Department of Univeridad de Ingenieria y Tecnologia (UTEC) Peru. Hi reearch interet include nonlinear modelling proce control engineering and motion control epecially the control of robotic manipulator and drone.