A practical control method for multivariable systems

A PRACTICAL CONTROL METHOD FOR MULTIVARIABLE SYSTEMS Vasile Cîrtoaje, Sanda Frâncu, Alina Băieşu Control Engineering and Computers Department, ” Petroleum-Gas” University of Ploiesti Bd. Bucuresti, 39, Ploiesti, 2000, Romania E-mail:[email protected] Abstract: The main purpose of the paper is to present a robust practical method for experimentally decoupling, compensating and control of two input-two output process. The decoupler channels are first order lag plus dead time elements, which satisfy the following requirements: the direct channels have unit gain and two channels have dead time equal to zero. The decoupler can be simplified in addition taking two appropriate channels with the lag time constant equal to zero. After decoupling, each output of the decoupled process is controlled by a special method, which consists in monotonic compensating and standard IMC control of the both direct channels of the decoupled process. The results obtained by simulation validate the proposed control procedure. Key words: decoupling controller, decoupler, monotonic compensation, standard IMC algorithm. 1. INTRODUCTION Many industrial processes are multivariable, exhibiting input-output cross-coupling which cannot be neglected because it provides difficulties in process control. Multivariable process control using monovariable controllers cannot yield satisfactory performance, because of mutual interactions between monovariable loops. The multivariable controller use makes possible partial or total elimination (only in the theoretical case) of these self-disturbing interactions. Usually, a multivariable controller with n inputs and n outputs consists of a block with n monovariable controllers (possibly of PID type) and a process decoupling block. Using a decoupling controller, the system tuning problem is reduced to the independent tuning of the monovariable controllers of every control loop. Most of the controller synthesis methods for multivariable control systems are based on knowing the process model as precise as possible. The relation between the process, decoupler and decoupled process transfer matrixes has the form: 0  G11 G12   D11 D12  (G11)d .   = G G  D D   0 (G22 ) d  21 22 21 22 From decoupling equations (1) G11D12 + G12 D22 = 0 ,  G21D11 + G22 D21 = 0 (2) we obtain the decoupled transfer functions:  (G ) = (1− f )D11 11 d G11  ,  (G ) = (1− f )D22  22 d G22 (3) where f= G12G21 G11G22 (4) is the process coupling factor. Because there are two decoupling equations and the decoupler has four transfer functions, two of the decoupler transfer functions, usually the diagonal transfer functions D11 and D22 , can be arbitrary chosen. The non-diagonal transfer functions of the decoupler are then given by the expressions: D = −G12 D22  12 G11  D21 = − G21D11 G22  (5) Usually, to have a simple decoupler structure, the diagonal transfer functions D11 and D22 are chosen equal to 1. If the transfer functions D12 and D21 are improper (not realizable), then D11 and D22 are chosen as follows D11 = 1/(τ1s +1) , D22 = 1/(τ 2s +1) , (6) where the time constants τ1 and τ 2 are 5…10 times less than the process dominant time constant. In this case  D = −G12  12 G11(τ1s +1) .  − G21 D =  21 G22 (τ 2 s +1) (7) Usually, process modeling and identification tasks are time-consuming and demand specific knowledge in control theory and an advanced practical experience. Moreover, the decoupling controller structure is dependent of the process model structure. In the case of strong cross-interaction, even if the process has a monotonous and finite step response on the direct channels, the direct channels of the decoupled process can be non-monotonic (of non-minimal phase, with large overshoot or of oscillating type) or even unstable. For such decoupled process, the monovariable controllers design is not a simple problem and the control performance may not be acceptable. The proposed control method eliminates or reduces these disadvantages. In the proposed structure of a multivariable control system (fig. 1), F1 and F2 are serial filters for decoupled process compensating, and C1 and C2 are standard IMC controllers. k12 = −K p12 / K p11 , k21 = −K p21 / K p22 , T11 = Tt 22 T T T , T12 = t12 , T21 = t 21 , T22 = t11 . 4 4 4 4 (9) From the third decoupler property and the dead time compensation relations τ11 +τ p21 = τ 21 +τ p22 , τ 22 +τ p12 = τ12 +τ p11 , (10) we get the decoupler dead-times such as: τ p22 ≤ τ p21 ⇒ τ11 = 0 , τ 21 = τ p21 −τ p22 ; τ p22 >τ p21 ⇒ τ 21 = 0 , τ11 =τ p22 −τ p21 ; τ p11 ≤ τ p12 ⇒ τ 22 = 0 , τ12 = τ p12 −τ p11 ; τ p11 >τ p12 ⇒ τ12 = 0 , τ 22 =τ p11 −τ p12 . (11) From here, it follows that depending on the process dead times, the decoupler structure can be in four ways: with zero dead time at 1-1 or 2-1 channel, and at 1-2 or 2-2 channel. The decoupler has eight parameters: two gains, four lag time constants and two dead times. The decoupler designed in such a way can be experimentally refined by suitably adjusting the two lag time constants T12 and T21 . If the steady state decoupling is perfect, the Fig. 1. Proposed structure for multivariable control system refining operation yields ∞ The proposed decoupling solution is based on the following idea: two parallel-opposite channels with the same gain, the same dead time and very close transient time accomplish a satisfactory compensation for a process having all the input-output channels of proportional type (with finite step response). The transient time is the whole response time to input step change minus the dead time. Each process channel is thus described by 3 parameters, which can be easily determined by experimental way: gain (K p )ij , dead time (τ p )ij and transient time (Tt )ij , i.e.  (K p11, τ p11, Tt11) (K p12, τ p12, Tt12 )  Process :  . (K p21,τ p21, Tt 21) (K p22, τ p22, Tt 22 ) The proposed decoupler fulfils the following conditions: 1) each channel is at most first order lag plus dead time element; 2) the direct input-output channels have gain equal to 1; 3) at least two channels are dead time equal to zero. Taking into account the first two decoupling conditions, the decoupler structure is in the form with k12e−τ12 s  T12 s +1  , e−τ 22 s  T22 s +1  0 1 1 0 2 2 (12) where y1 and y2 are the step input responses of decoupled process crossing channels. As an example, for the multivariable process proposed by Ho, et al. (1996)  12.8e−s 16.7s +1 G ( s) =   6.6e−7s 10.9s +1 −18.9e−3s  21s +1  , −19.4e−3s  14.4s +1  (13) the unadjusted transfer matrix of the first type decoupler is 2. FIRST TYPE DECOUPLER  e−τ11s  T s +1 D =  11 −τ 21s  k21e  T21s +1 ∞ ∫ [ y (t) − y (0)]dt ≅ 0 , ∫ [ y (t) − y (0)]dt ≅ 0 , (8) 1.48e−2s   1   D(s) =  14.4s−+41s 21s +1  . 1   0.34e  10.9s +1 16.7s +1  (14) 3. SECOND TYPE DECOUPLER The general structure of the second type decoupler has the form (8), but it fulfils in addition the condition: 4) at least two channels are lag time constant Tij equal to zero. Consequently, the decoupler has the structure simpler than that of the first type decoupler. Moreover, the decoupled process is faster. Taking into account the fourth decoupler property and also the relations 4T11 + Tt 21 = 4T21 + Tt 22 , 4T22 + Tt12 = 4T12 + Tt11 , (15) which approximately express the equality of mutual compensation channel transient times, it follows the decoupler lag time constants: Tt 22 ≤ Tt 21 ⇒ T11 = 0 , T21 = (Tt 21 −Tt 22 ) / 4 ; Tt 22 >Tt 21 ⇒ T21 = 0 , T11 = (Tt 22 −Tt 21) / 4 ; Tt11 ≤ Tt12 ⇒ T22 = 0 , T12 = (Tt12 − Tt11) / 4 ; Tt11 >Tt12 ⇒ T12 = 0 , T22 = (Tt11 −Tt12 ) / 4 . (16) From here it follows that depending on the process transient times, the decoupler structure can be in four ways: with zero lag constant time on 1-1 or 2-1 channel, and on 1-2 or 2-2 channel. The decoupler gains and dead times follow from (9) and (12), like the first decoupler. Therefore, depending on the process dead times, the decoupler structure can be also in four ways. Each of the 16 possible structures can be experimentally refined by suitably adjusting the non-zero time constants. The decoupler has only six parameters: two gains, two lag time constants and two dead times. For example, for the multivariable process (13), the unadjusted transfer matrix of the second type decoupler is 1.48e−2s   1 D(s) =  3.5s +1 4.3s +1  .   − 4s 1  0.34e Compensating both direct channels of the decoupled process has the aim to improve its dynamics, so that the input step response of each channel to be monotonic and as fast as possible (Cîrtoaje, 2002). The most usual method of monovariable process compensating is to connect a lead-lag filter in front of the process. The leadlag transfer function has the form (18) For robustness reason, in the case of a stable and monotonic process, we recommend a filter time constant with the value T f = Tt /10 , Each direct channel of the compensated process is controlled by the standard IMC method (fig. 2). The controller R is a serial connection between a proportional element with the gain K and a positive feedback loop, which is designed by means of the compensated process model (Gm )c (s) . In the forward path of the controller loop there is a proportional element having the gain equal to the inverse of the model gain. Fig. 2. Proposed IMC variant Since the input step response of the compensated process is monotonic, we may consider its model in the form (Gm )c (s) = K me−τ m s , (Tm s +1)2 Tm = (Tt )c / 6 , (17) 4. DECOUPLED PROCESS COMPENSATION K f T f s +1 . T f s +1 5. DECOUPLED PROCESS CONTROL (20) with The second decoupler form (17) is much simpler and faster that the first form (14). G f (s) = sufficiently large so that the compensated process to become monotonic. (19) where Tt is the transient time. The filter gain K f must be chosen as large as possible, but respecting two condition: a) K ≤ 5 , for robustness reason; b) the compensated process to remain also monotonic. In the case of a stable but non-monotonic process, we recommend that K f to be equal to zero and T f to be (21) where (Tt )c is the transient time of the compensated process. Hence, the controller R has the continuous transfer function 1 H R (s) = K ⋅ , K m 1− e−τ m s /(Tm s +1)2 (22) or the discrete transfer function 1− 2 pz −1 + p 2 z −2 H R ( z) ≅ K ⋅ , K m 1− 2 pz −1 + p 2 z − 2 − (1− p)2 z −lm −1 (23) where p = e−T /Tm and lm = Tm / T ( T - sample time). The gain K has the standard value 1. Increasing/decreasing K , the control output becomes stronger/weaker. According to (23), the controller equation in the time domain is as follows ck = 2 pck −1 + p 2ck − 2 − (1− p)2 ck −lm −1 + + K [(ek − e0 ) − 2 p(ek −1 − e0 ) + p 2 (ek − 2 − e0 )]. Km (24) 6. APPLICATION Consider the multivariable process without dead time proposed by Menani and Koivo (1996), but completed here with dead times on all channels:  0.5e−0.5s  0.1s +1 1 G(s) = 2  (0.1s +1)(0.2s +1)  −0.3s e   − e−0.8s  . 2.4e−0.6s  0.5s +1  (25) From the process response to step input change (fig. 3 and fig. 4), we get the process parameters: K p11 = 0.5 , K p12 = −1 , K p21 =1 , K p22 = 2.4 ; τ p11 = 0.6 , τ p12 = 0.8 , τ p21 = 0.3 , τ p22 = 0.7 ; (26) Tt11 =1.4 , Tt 21 =1.36 , Tt12 =1.36 , Tt 22 = 3 . Fig. 6. Decoupled process response to u2 =1(t) In order to compensate the direct channels of the decoupled process, we chose the filter time constants T f 1 = Tt1 /10 ≅ 0.4 , T f 2 = Tt 2 /10 ≅ 0.4 . By using the filter gains K f 1 = 2 and K f 2 =1.5 , the unit step responses of the both direct input-output channels of the decoupled and compensated process remain monotonous, but faster (fig. 7 and fig. 8). From these responses, we get the parameters of the direct channels of compensated process: Fig. 3. Process response to u1 =1(t) (K p1)c = 0.92 , (Tt1)c =1.9 , (τ p1)c =1 , (K p2 )c = 4.4 , (Tt 2 )c = 2.4 , (τ p2 )c = 0.7 . According to (20) and (21), we build the suitable models (H m1)c (s) = −0.7s 0.92e−s , (H m2 )c (s) = 4.4e 2 . 2 (0.317s +1) (0.4s +1) The control result for K =1 (obtained in MatlabSimulink) is presented in fig. 9 and fig. 10. Fig. 4. Process response to u2 =1(t) A. First type decoupler From (8)… (11) and (26), we obtain the decoupler 2e−0.2s   e−0.4s  0.75s +1 0.35s +1 D1 =  . 1   − 0.417  0.35s +1 0.35s +1 (27) The decoupling performance is shown in fig. 5 and fig. 6. Notice that it is not necessary to adjust the time constant T12 or T21 . Fig. 5. Decoupled process response to u1 =1(t) Fig. 7. Compensated process response to c1 =1(t) Fig. 8. Compensated process response to c2 =1(t) Fig. 12. Decoupled process response to u2 =1(t) Fig. 9. Controlled process response to r1 =1(t) we get the compensated process responses from fig. 13 and fig. 14, which are also monotonous. These responses yield (K p1)c = 0.92 , (Tt1)c =1.64 , (τ p1)c = 0.9 , (K p2 )c = 4.4 , (Tt 2 )c =1.6 , (τ p2 )c = 0.54 , and Fig. 10. Controlled process response to r2 =1(t) B. Second type decoupler From (8), (9), (11), (16) and (26), we obtain the decoupler  e−0.4s − 0.2s   0.4s +1 2e  D2 =  . 1  − 0.417  0.01s +1 −0.9s −0.54s , (H m2 )c (s) = 4.4e . (H m1)c (s) = 0.92e 2 (0.273s +1) (0.267s +1) 2 The control performance (for K =1 ) is little better than the performance obtained by means of first type decoupler (fig. 15 and fig. 16). (28) The decoupling performance does not require adjusting the time constants T11 = 0.4 or T22 = 0.01 (fig. 11 and fig. 12). Fig. 13. Compensated process response to c1 =1(t) Fig. 11. Decoupled process response to u1 =1(t) By using T f 1 = Tt1 / 10 ≅ 0.3 , T f 2 = Tt 2 /10 ≅ 0.3 , K f 1 =1.8 , K f 2 =2 , Fig. 14. Compensated process response to c2 =1(t) Fig. 15. Controlled process response to r1 =1(t) We propose two practical decoupler type. The first decoupler has eight parameters: two gains, four lag time constants and two dead times, while the second decoupler has six parameters: two gains, two time constants and two dead times. Moreover, the second decoupler is simpler and faster than the first. To accomplish decoupled process control, we used Cirtoaje-IMC method, based on decoupled process compensation and use of the standard variant of internal model control for the decoupled process. The simulation results have proven that the proposed control procedure is simple, practical and robust. REFERENCES Fig. 16. Controlled process response to r2 =1(t) 7. CONCLUSIONS This paper presents a practical method of experimentally decoupling, compensating and control for two input-two output process. 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