Smith Compensator Using Modified IMC for Unstable Plant with Time Delay

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1 2 International Conerence on Inormation an Electronics Engineering IPCSIT vol.6 (2) (2) IACSIT Press, Singapore Smith Compensator Using Moiie IMC or Unstable Plant with Time Delay Hiroki Shibasaki, Manato Ono, Naohiro Ban an Kazusa Matsumoto, Kazuhiro Sasaki, Junichi Eno an Yoshihisa Ishia Department o Science an Technology Meiji University Kanagawa, JAPAN Abstract. In this paper, a esign o smith compensator using moiie IMC or an unstable plant with time elay is propose. An unstable plant with time elay is controlle by the metho o a preicte-state eeback. However, when we introuce an input-sie isturbance in the system, it has large inluence response or the isturbance to control an unstable plant with time elay. In aition, we introuce the isturbance compensator to overcome the problem in the preicte-state eeback. We applie the preictestate eeback with isturbance compensator. Furthermore, we introuce this metho to the moiie smith preictor. In these simulation stuies, it is shown that the propose metho has a superior perormance in an input sie isturbance o the plant response. The nominal system can reuce large inluence or isturbance. Furthermore, the robust system was conirme high robustness. Keywors: Unstable Plant, Time elay, Internal Moel Control, Smith compensator. Introuction Smith Preictor [] an Internal Moel Control (IMC) [2] are eective methos to control a plant with time elay. However, these methos cause a steay state error by an input sie isturbance or the plants with an integrator. Astrom et al. [3] propose new Smith preictor with a superior perormance. However, this metho has the problem that parameter ajustments are complicate. Authors [4] propose the iscrete moiie IMC. This metho oes not cause a steay-state error by an input-sie isturbance or plants with an integrator. However, Smith Preictor an Internal Moel Control o not control an unstable plant with time-elay. In recent year, there are many methos [propose against the problem. We introuce some papers. De Paor et al. [5] [6] propose a moiie Smith preictor which has a constraint on ratio o time elay to the time constant. Watanabe et al. [7] metho is base on the output preiction o a plant. Furukawa et al. [8] propose a control strategy base on a preicte-state eeback technique with an observer. K.K. Tan s metho [9] is base on Generalize Preictive Control approach. Basilio el-muro-cuellar et al. [] use an observer-base preictor with partitions o time elay to stabilize an unstable plant. Above all, authors [] propose the moiie smith preictor which can control an unstable plant with time elay an eliminates a steay-state error cause by an input-sie isturbance. In this paper, we propose the preicte-state eeback system with the isturbance compensator to eliminate large inluence or the input-sie isturbance. Furthermore, we introuce the preicte-state eeback system with isturbance compensator to the moiie smith preictor []. Our propose metho has high robustness against an input-sie isturbance. 2. Plant Preictor Hiroki Shibasaki. Tel.: 8() aress: meiji.shibasaki@gmail.com. 8

2 We state the esign metho o a plant preictor. State space equations which are a controllable an an observable controlle plant with time elay express () an (2). ( k ) = ( k) u( k ) x Ax b () ( k) y k = cx (2) Then we put k k in (). We get 2 sample preicte state o the plant x ( k 2). ( k 2) = ( k ) u( k ) 2 = Ax( k) Ab u( k ) bu( k ) x Ax b (3) Repeating the same operation, we get a preicte state o the plant ( k ) x. ( k ) = ( k) u( k ) u( k) x A x A b b i ( k) u( k i ) = A x A b (4) Then, a preicte output o the plant y ( k ) is ( ) = cx ( ) y k k (5) The above preicte state equations are not observe irectly, then, an observer which estimates states o the plant is use. ( k ) = ( k) u( k ) L{ y( k) y( k) } xˆ Axˆ b ˆ (6) ( ) = ˆ ( ) ŷ k cx k (7) Where, L is observer gain. However, the above observer has time elay in the input u (k-). 3. Preicte-State Feeback with Disturbance Compensator We propose the plant preictor to a state eeback control with a isturbance compensator M to eliminate large inluence or an input- sie isturbance in this system or a plant with time elay. Fig. is a block iagram o the preicte-state eeback system with the isturbance compensator. A plant preictor is connecte by the observer in this section. For this reason, (4) is written by (8). r( k ) ( k) u k y( k) Plant i ( k ) = ( k) u( k i ) xˆ A xˆ A b (8) k PI Controller rk uk yk zk G c Plant F Plant Preictor M Observer ˆx ( k ) yˆ ( k) y k F Plant Preictor M Observer ˆx( k) ŷk c Smith Compensator K DC yk ˆ yk Fig. Preicte-State Feeback System with Disturbance Compensator Fig.2 Block iagram o the propose metho 9

3 The preicte-state eeback system with isturbance compensator is written by (), (6), an (8). Note that (), (6), an (8) are change (9) (), an () by z- transormation. A manipulate valuable U (z) is written (2) ( z ) ( z) z U( z) = I A X b (9) ( z ) ( z) z U( z) ( z) I A Lc X ˆ b LcX = () ˆ = ( )( ) z Xˆ z A X z I A z zi A b U z () { } = ˆ ˆ From (9), (), an (2), we express in matrix orm. U z Fz X z M Y z Y z (2) ( z) zia bz X z z ˆ Lc I A Lc b X z = M M ( z )( z ) Uz c FA c F IA IA b (3) The characteristic equation is given by zi A bz φ ( z) = et Lc zi A Lc bz c FA c F I A I A b = et M M ( z )( z ) ( zi A Lc) et ( zi A bf) (4) I the characteristic equation oes not equal by an observer gain L an a Feeback gain F, the system is stable. When we input a reerence input R (z), a manipulate valuable U (z) is written (5). { } ˆ ˆ U z = Fz X z M Y z Y z R z (5) From (9), (), an (5), the target value response Y (z)/r (z) is zi A bz Yz = [ c ] Lc zi A Lc bz Rz M M ( z )( z ) c FA c F I A I A b (6) (6) is change into a small matrix an we solve the equation. An we get the transer unction with a eeback gain. Y z R z = G z z z = c I AbF b (7) 2

4 When we input a isturbance D (z), a plant is written (8). An (8) is change (9) by z- transormation. ( k ) = ( k) { u( k ) D( k) } x Ax b (8) z z z U( z) = z D( z) From (), (2), an (9), the target value response Y (z)/r (z) is I A X b b (9) zi A bz bz Y( z) [ ] = c Lc zi A Lc bz Dz M M ( z )( z ) c FA c F I A I A b (2) We solve (2) by same operation in such way (7) Y z ( z )( z ) ( )( ) F I A I A b = G c FA I A Lc b D z M z z (2) In case o an unstable plant, (2) is written by Y z G = z M z z D z i F A i b c FA I A Lc b (22) To eliminate the inluence o isturbance, we solve the ollowing the equation. An we get the isturbance compensator M in (24). F A b c FA I A Lc b (23) i i lim z M z z z = i FA I A Lc b F A b M = c I A Lc b ( ) (24) In this metho, the isturbance is eliminate by the isturbance compensator M. Furthermore, we introuce the preicte-state eeback system with isturbance compensator to the moiie smith preictor propose by authors []. Fig.2 is a block iagram o the propose metho. 4. Simulation Stuy In this section, we show the simulation result o propose metho. Fig.3 is shown a response o nominal system an Fig.4 is shown a response o robust system. A unit step input is introuce at time t=[sec]. We iscrete the system by the zero-orer hol metho at sampling time T s =[msec]. Ls = 2 G s e 2 e s 2.2s Ls (25) The weight o the cost unction are Q/R=.2. The observer gain is L = [ ] T an the eeback gain is F= [ ]. The isturbance compensator is M= The compensate gain is K DC =.862. The PI controller parameters are K P =., an K I =.288. An input-sie isturbance is introuce D(s) = -./s at time t=5[sec]. We a 2% error in time elay to show the robustness o the propose metho. 2

5 Output Output Step Input Preicte-State Feeback with M.2 Moiie Smith Preictor Propose Metho Time[sec] 5. Conclusion In this paper, we have propose smith compensator using moiie IMC or an unstable plant with time elay. The irst o all, we have propose the preicte-state eeback system with the isturbance compensator to eliminate the large inluence or the input-sie isturbance an it is introuce to the moiie smith preictor. In this simulation stuy, our propose metho has high robustness against an input-sie isturbance. 6. Reerences Fig.3 Nominal System [] O.J. Smith, A Controller to Overcome Dea Time, ISA Journal, vol.6, pp , Feb., 959 [2] M. Morari an E. Zairiou, Robust Process Control, Prentice Hall, New Jersey, Step Input.2 Preicte-State Feeback with M Moiie Smith Preictor Propose Metho Time[sec] Fig.4 Robust System (2% error in time elay) [3] K.J. Astrom, C.C. Hang an B.C. Lim, A Smith preictor or controlling a process with an integrator an long ea time, IEEE Trans. On Automatic Control, vol.39, pp , Feb., 994. [4] I. Yoichi, M. Yuichi an I. Yoshihisa, Discrete Moiie Internal Moel Control or plants with an integrator an time elay, IEICE Trans. On Funamentals o Electronics, Communication an Computer Sciences, vol.j9-a, pp , Aug., 27. (in Japanese) [5] Annraoi M.De Paor, A Moiie Smith Preictor an Controller or Unstable Processes with Time Delay, Int. J.Control,Vol. 4,No.4,985,pp [6] Annraoi M.De Paor an Ruth P. K. Egan, Extension an Partial Optimization o a Moiie Smith Preictor an Controller or Unstable Process with Time Delay, Int. J.Control,Vol. 5(4),pp35-326,989. [7] K. Watanabe an M. Ito, A Process-moel Control or Linear Systems with Delay, IEEE Trans. Automatic Control, vol. AC-26, no. 6, Dec., 98. [8] T.Furukawa an E. Shimemura, Preictive Control or Systems with Time Delay, Int.J.control,Vol. 37,No.2,983,pp [9] K. K. tan, T.H.Lee, an F. M. Leu, Optimal Smith-preictor Design Base on a GPC Approch, In.Eng.Chem.Res. 4, pp , 22. [] B. el-muro-cuellar, M. Velasco-Villa, O. Jiménes-Ramírez, G. Fernánez-Anaya, an J. Álvarez-Ramírez, Observer-Base Smith Preiction Scheme or Unstable Plus Time Delay Processes, In. Eng. Chem. Res., 46(4), pp , 27. [] O.Manato, S. Hiroki, M. Kazusa, S. Kazuhiro an I. Yoshihisa, Discrete Moiie Smith Preictor Base on Optimal Control Metho or a Plant with an Integrator, 2 IEEE International Conerence on Systems, Man an Cybernetics (SMC 2),pp ,Octorber,2 22