ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΧΗΜΙΚΩΝ ΜΗΧΑΝΙΚΩΝ Εργαστήριο Β Χημικής Μηχανικής Τ.Θ. 472 54124 Θεσσαλονίκη PROCESS DYNAMICS & CONTROL 2014 Final Review Series of Solved Problems Problem 1: Consider the following feedbak ontrol system of Figure 1. Figure 1: Feedbak ontrol-loop. The proess and disturbane transfer funtions are assumed to be equal. KΚ p G(s) p G(s) d (τ s1)(τ s1)(τ s1)(τ s1) 1 2 3 4 τ1 1 min ; τ 2 2 min ; τ 3 4 min ; τ 4 7 min ; Kp 1 i) Calulate the Ziegler-Nihols settings (K, τ I and τ D ) of a PID ontroller based on the ontinuous yling method using the harateristi equation of the losed-loop system. ii) Compare the alulated values of K u and P u (see question (i) ) with the results of Figure 2. iii) Calulate an approximate FOPTD model of the proess using the experimental results of Figure 3. iv) Use the identified FOPTD model to determine the Cohen-Coon and ITAE settings (K, τ I and τ D ) for a PID ontroller. Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Figure 2: Experimental alulation of the ultimate gain, K u = 6.35 and of the ultimate period, P u. Figure 3: Open-loop response to a unit step hange in the input signal (Proess reation urve) Solution of Problem 1: i) The harateristi equation of the losed-loop system under P-ontrol is: Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Κ 1G OL(s) 0 1 1 0 (τ1s1)(τ 2s1)(τ 3s1)(τ 4s1) (s1)(2s1)(4s1)(7s1) 56s 4 106s 363s 2 14s (1 K ) 0 To alulate K u and ω, we substitute s = jω in the above harateristi equation. Thus, we have: 3 j 56ω 4 106ω 3 63ω 2 14 jω (1 K ) 0 Rearranging the above equation in the omplex form (Re + jim) we obtain: Κ (1 K 4 2 3 5 6 ω 63ω ) j( 106ω 14ω ) 0 This omplex equation is satisfied if both the real and imaginary parts are equal to zero. Real part (Re): (1K 56ω 4 63ω 2) 0 (a) Imaginary part (Im): ( 106ω314ω)jjω(106ω 2 14) 0 (b) From Eq. (b), we get: ω 2 14 106 and thus, ω 0.3634. Substituting the value of ω into Eq. (a), we alulate the ultimate value of K u. K u 6.3439 1 K 4 2 u 56 0.3634 63 0.3634 0 Aordingly, P u 2π 23.14 P u 17.28 ω 0.3634 From the above results we an alulate the following Z-N tuning settings of a PID ontroller: PID: K = 0.6K u = 3.8063 ; τ I = P u /2 = 8.64 and τ D = P u /8 = 2.16 ii) From Figure 2, we an verify the above alulated values of the ultimate gain and ultimate period are: K u 6.35 6.3439 and Pu 17.4 17.28 Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
P u Figure 2: Experimental alulation of the ultimate gain, K u = 6.35 and of the ultimate period, P u. ii) From the results of Fig. 3 we an obtain the following parameter values for a FOPTD model. τ θ Figure 3: Open-loop response to a unit step hange in the input signal (Proess Reation Curve) K 1 ; τ 16.9 ; θ 3.85 Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Therefore, the FOPTD model of the proess will be: (1) e 3.85s G(s) (16.9s 1) iv) From the attahed two Tables 1 and 2, we an alulate the Cohen-Coon and ITAE settings (K, τ I and τ D ) of a PID ontroller. Cohen-Coon settings 1 τ 3θ 16τ K 6.1028 K θ 12τ ; I 4θ τ D 1.3443 11 2(θ τ) θ[32 6(θ τ)] τ 8.6667 13 8(θ τ) ; ITAE settings Set-point Settings KK 0.85 0.965(3.85 16.9) 3.3930 K 3.3930 1 K 3.3930 ττ 0.796 0.1465(3.85 16.9) 0.7626 τ 16.9 0.7626 I I I τ 0.929 D τ 0.308(3.85 16.9) 0.0779 τ D 0.0779 16.9 τ 22.1603 τ D 1.3171 Load Settings KK 0.947 1.357(3.85 16.9) 5.5075 K 5.5075 1 K 5.5075 ττ 0.738 I 0.842(3.85 16.9) 2.5085 τ I 16.9 2.5085 τ I 6.7370 τ 0.995 D τ 0.381(3.85 16.9) 0.0874 τ D 0.0874 16.9 τ D 1.4777 Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Table 1: Cohen-Coon Tuning Relations Controller K T i T d P-only ( 1/ )( / )[1 /3 ] K p PI ( 1/ )( / )[0.9 /12 ] K p [303( / )] 9 20( / ) PID (1/ K p 3 16 )( / )[ ] 12 [326( / )] 138( / ) 4 11 2( / ) Table 2: Controller Design Based on ITAE Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Problem 2: Consider the problem of ontrolling the level in tank 3 of the three tank proess shown in Figure 4. Figure 4: Three noninterating tanks in series with proportional feedbak ontrol To aomplish this, the inlet flowrate to the first tank, F o (t), is manipulated based on a signal from a proportonal feedbak ontroller. The proess transfer funtion for the three noninterating tanks in series is: G p (s) = K p / [(τ 1 s +1)( τ 2 s + 1)( τ 3 s + 1)] where K p = 6 and τ 1 = 2, τ 2 = 4, τ 3 = 6. The time onstants are given in minutes. The trnsfer funtions of the measurement element (level transmitter) and final ontrol element (ontrol valve) are equal to one, i.e. G m (s) = G v (s) = 1. i) Draw the blok diagram for the above ontrol system. ii) Identify the transfer funtion for eah element in the ontrol system and alulate the losed loop transfer funtion between the ontrolled variable Y(s) and the set-point value, Y sp (s). iii) Using the Routh array alulate the values of the ultimate gain, K u, and ultimate period of sustained osillations, P u. iv) Use the method of diret substitution s = jω and the harateristi equation, to verify the results of question (iii). Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
v) Calulate the Ziegler-Nihols settings of a PID ontroller, using the tuning rules of the following Table 3. Table 3: Ziegler-Nihols ontroller tuning parameters vi) For feedbak ontrolling tuning purposes, the approximate model most ommonly employed has a first-order-plus-time-delay (FOPTD) transfer funtion. Figure 5: Proess reation urve for the three-tank in series (time is in min) To estimate the parameters of the approximate model (FOPTD) for the three-tank system, the proess reation urve was applied using a unit step hange as a proess input. The results of the proess reation urve are given in Figure 5. Applying the graphial proedure, identify the parameters K p, θ, and τ of the approximate FOPTD proess model. vii) Use the identified FOPTD model to determine the ITAE settings (K, τ I and τ D ) of a PID ontroller for set point hanges (see Table 4). Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
Table 4: Controller Design Based on ITAE and FOPTD Model Solution of Problem 2: i) The blok diagram of the ontrol system is illustrated in the figure below: Kp G p (s) (τ s1)(τ s1)(τ s1) 1 2 3 ii) Identifiation of the transfer funtions for eah element: TF of measurement devie TF of ontrol valve TF of P-ontroller TF of proess G (s) K 1 m v Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr m G (s) K 1 v G (s) K
K p 6 G p(s) (τ s1)(τ s1)(τ s1) (2s1)(4s1)(6s1) 1 2 2 Derive CLTF: 6K Y(s) K mg (s)g v(s)g p(s) (2s 1)(4s 1)(6s 1) Y 6K sp(s) 1 G (s)g v(s)g p(s)g m(s) Y(s) 6K 1 (2s 1)(4s 1)(6s 1) Y 3 2 sp(s) 48s 44s 12s (1 6K ) iii) The Charateristi Equation is: The Routh Array for this third-order system is: 1 a a 1 n n2 48s 344s 2 12s (1 6K ) 0 n1 an3 1 2 1 2 48 12 2 a 2 44 1 6K 3 b b 3 528 48(1 6K ) 44 0 4 41 6K 0 To have a stable system, eah element of the first olumn in the Routh array must be positive, i.e. 528 48(1 6K ) 0 K 5 3 and 1 6K 0 K 1 6 44 We onlude that the system will be stable if, 0.167 K 1.667. Therefore, the ontroller s gain at the point of marginal stability, i.e. the ultimate gain, is equal to: K u 5 3 1.667. We an alulate the purely imaginary roots from the following equation: 2 2 5 44s (1 6K 2 u ) 0 44s (1 6 ) 44s 11 0 s j0.5, whih yields, ω 0.5 3 Hene, Pu 2πω 2π 0.5 4π Pu 12.56 (in min/yle). iv) By substituting s = jω into the above harateristi equation, we get: Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
48jω3 44ω 2 12 jω (1 6K ) 0 Rearranging the above equation in the omplex variable form: (Re + jim) we obtain: (16K 2 3 44ω ) j( 48ω 12ω) 0 This omplex equation is satisfied if and only if both the real and imaginary parts are equal to zero. Real part (Re): (16K 44ω 2) 0 (a) Imaginary part (Im): ( 48ω 312ω)jjω(48ω 2 12) 0 (b) ω 0 From Eq. (b), we get: ω(48ω 2 12) 0 (48ω 2 12) 0 and thus, ω 0 or 0.5. Substituting the values of ω into Eq. (a), we an alulate the K, i.e. For ω = 0 we obtain: 1 6K 0 K 1 6 For ω = 0.5 we obtain: 16K 440.52 0 K 5 3 whih onfirm the results of question (iii). v) The Z-N PID ontroller tuning parameters will be: K 0.6K u K 1 ; τ I Pu 2 τ I 6.28 min ; τ D Pu 8 τ D 1.57 min vi) From Figure 2 we an alulate the numerial values of the FOPTD model parameters: Kp 6 ; τ 15 min ; θ 3 Therefore, the FOPTD model of the proess will be: 6e 3s G(s) (15s 1) vii) Finally, from the Table 4 and the above model we get the following ITAE settings of a PID ontroller for set point hanges: K 0.632 KK 0.85 0.965(3 15) 3.790 K 3.790 6 Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr
ττ 0.796 0.1465(3 15) 0.7667 τ 15 0.7667 τ I 19.56 min I I τ D τ 0.929 D τ 0.308(3 15) 0.069 τ D 0.069 15 1.04 min Τηλέφωνο: (+30) 2310 996211 Fax: (+30) 2310 996198 e mail: ypress@peri.erth.gr (+30) 2310 996212 ypress@eng.auth.gr