Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics PID Controller design for multiple time delays system Asma Karoui, Rihem Farkh, Moufida Ksouri Laboratoire d’Analyse, Conception et Commande des Systèmes Université de Tunis El Manar Ecole Nationale d’Ingénieurs de Tunis BP 37, Le Belvedère 1002, Tunis, Tunisia
[email protected] [email protected] [email protected]Abstract— This paper presents an approach of stabilization and parameters or numerically in the case of the knowledge of the control of time invariant linear system of an arbitrary order that delay system frequency response [6], [12], [13]. include several time delays. In this work, the stability is ensured Since these methods have been developed mainly for the by PI, PD and PID controller. The method is analytical and case of a single system delay, the contribution of our work needs the knowledge of transfer function parameters of the plant. concerns the stabilization of system with several time delays It permits to find stability region by the determination of K p , by using the PI, PD and PID controllers. The proposed K i and K d gains. approach is based on the extension of the analytical method developed in [6], [12], [13]. Keywords— PID Controller, multiple delay systems, stability The considered feedback structure is depicted in Fig. 1 and region the related transfer functions of the process G ( s ) and the I. INTRODUCTION controller C ( s ) are given by: Time delay systems are often encountered in various N engineering systems such as electrical and communication G ( s) = ∑ Gi ( s )e −τ i s (1) i =1 network, chemical process, turbojet engine, nuclear reactor, hydraulic system; it is frequently a source of instability, Ki oscillation and poor performance in many dynamic systems. C (s) = K p + + Kd s (2) s Furthermore, delay makes system analysis and control design much more complex [7], [17]. where N is the number of delays, τ i is the time delay and The PID controller is widely applied in control engineering Gi is a continuous linear system of any order, K p , K i and applications for many industries. The choice of the PID controller parameters leads to obtain a closed loop stable K d are the PID parameters. system. Many researches have been applied the PID controller to different classes of dynamical systems [1]. Among these the +- C(s) G(s) particular class of time delay system has been investigated by means of several methods [8], [14], [15], [16], of which the Controller Process Nyquist criterion, a generalization of the Hermite-Biehler Theorem, and the root location method. The main objective to Figure 1: The closed-loop system with controller design the PID controller is to ensure closed loop stability. Indeed, by using the Hermit-Biehler theorem applicable to the quasi-polynomials [9], [10], [11], a characterization of all The region of stability is found in the ( K p , K i ) plane for values of the PI/PID stabilization gains for stable first order delay system is addressed. However, these results are not PI controller, ( K p , K d ) plane for PD controller and applicable to the second order delay system. In [2], [3], the stabilizing problem of PI/PID controller for second order ( K p , K i , K d ) space for PID controller. This approach has delay system is analysed and then used to obtain all PI and been applied to continuous time linear systems of any order, PID gains that stabilize an interval first and second order with multiple time delays. delay system [4], [5]. The design methods of PID controllers can be analytically The paper is organized as follows: In part II stabilization determined in the case of knowledge of the transfer function approach with PI controller is presented, similarly 254 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics stabilization approach with PD controller is developed in N Ki N section III. The case of PID controller is discussed in section K p ∑ Ri (ω ) + ω ∑ I i (ω ) = −1 i =1 i =1 IV. Finally, simulation results are given in section V. N N (8) K K p ∑ I i (ω ) − i ∑ Ri (ω ) = 0 i =1 ω i =1 II. STABILIZATION SEVERAL TIME DELAYS SYSTEM USING PI Similar, CONTROLLER N N ω K p ∑ Ri (ω ) + K i ∑ I i (ω ) = −ω The considered plant is a continuous linear time-invariant i =1 i =1 system of any order that contains several time delays. It is N N (9) ω K described by its transfer function given by (1). p∑ i I (ω ) − K i∑ i R (ω ) = 0 i =1 i =1 In this section, the stabilization of the plant is assured by The Kp and K i parameters are determined by solving the the PI controller designed as follows: following system to ensure closed loop stability Ki K p s + Ki N N K C (s) = K p + = (3) s s ω ∑ Ri (ω ) ∑ I i (ω ) p −ω The proposed method leads to an efficient calculation of the i =1 i =1 = (10) N N proportional and integral gains K p and K i achieving ω ∑ I i (ω ) −∑ Ri (ω ) K i 0 i =1 i =1 stability. Let’s note ∆(s) the closed loop characteristic polynomial of the process shown in Fig. 1. The obtained expressions of K p and K i are: In frequency domain, the characteristic polynomial is N defined by: ∑ Ri (ω ) K p (ω ) = − i =1 G ( jω ) 2 ∆( jω ) = 1 + C ( jω )G ( jω ) = R∆ (ω ) + jI ∆ (ω ) (4) (11) N ∑ Ii (ω ) Ki (ω ) = −ω the transfer function can be then written as: i =1 G ( jω ) 2 N G ( jω ) = ∑ Ri (ω ) + jI i (ω ) (5) where i =1 2 2 N N G ( jω ) = ∑ Ri (ω ) + ∑ I i (ω ) 2 N N (12) = ∑ Ri (ω ) + j ∑ I i (ω ) i =1 i =1 i =1 i =1 where R∆ and I ∆ are the real and the imaginary parts of the III. STABILIZATION SEVERAL TIME DELAYS SYSTEM USING PD characteristic polynomial, respectively. Ri and I i are the real CONTROLLER In this section, the same plant (1) is stabilized with a PD and the imaginary parts of the transfer function Gi ( jω ) . controller as shown in Fig. 1. The stability region is determined when ∆ ( jω ) is equal to The transfer function of this controller is given by: zero: C (s) = K p + K d s (13) N N Ki To obtain stability region in terms of proportional and ∆( jω ) = 1 + ( K p − j )(∑ Ri (ω ) + j ∑ Ii (ω )) = R∆ (ω ) + jI ∆ (ω ) = 0 (6) ω i =1 i =1 derivative gains K p and K d , the previous approach is According to equations (4) and (5), the following results are obtained: applied in the case of several time delay system with PD controller N Ki N ∆R (ω ) = 1 + K p∑ i R (ω ) + ∑i I (ω ) The closed loop characteristic polynomial ∆ ( s ) is written i =1 ω i =1 (7) as: N I (ω ) = K Ki N p ∑ I i (ω ) − ∑ Ri (ω ) N N ∆( jω ) = 1 + ( K p + jK d ω )(∑ Ri (ω ) + j ∑ I i (ω )) = R∆ (ω ) + jI ∆ (ω ) (14) ∆ i =1 ω i =1 i =1 i =1 Applying the real part and the imaginary part equal to zero by setting ∆ ( jω ) equal to zero: leads to the following equations: ∆( jω ) = R∆ (ω ) + jI ∆ (ω ) = 0 (15) where: 255 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics N N Real part and the imaginary part are setting to zero to R∆ (ω ) = 1 + K p ∑ Ri (ω ) − K d ω ∑ I i (ω ) obtain equation system of three unknown variable i =1 i =1 N N (16) N N K pω ∑ Ri (ω ) + Ki ∑ I i (ω ) = −ω + K d ω 2 ∑ I i (ω ) N I (ω ) = K p ∑ I i (ω ) + K d ω ∑ Ri (ω ) i =1 i =1 i =1 ∆ i =1 i =1 N N N (24) K ω I (ω ) − K p ∑ i∑ i ∑ the real part and the imaginary part equal to zero lead to the i R (ω ) = − K d ω 2 I i ( ω ) i =1 i =1 i =1 following equations: In the first step, the K d parameter is fixed. The ( K p , K i ) N N p∑ i K R ( ω ) − K d ∑ i ω I ( ω ) = −1 plane is then determined by solving the following system: i =1 i =1 (17) N K N N N N −ω + K ω 2 ∑ I i (ω ) p ∑ I i (ω ) + K d ω ∑ Ri (ω ) = 0 ω ∑ Ri (ω ) ∑ I i (ω ) p d K i =1 (25) i =1 i =1 i =1 i =1 = N N Equivalently the system can be written as: ω ∑ I i (ω ) −∑ Ri (ω ) K i − K ω 2 R (ω ) N N N K ∑ Ri (ω ) −ω ∑ I i (ω ) p −1 i =1 i =1 d ∑ i i =1 i =1 i =1 = N N (18) leading to the K p and K i expressions: ∑ I i (ω ) ω ∑ Ri (ω ) K d 0 i =1 i =1 N Solving (18), the results are as follows: ∑ Ri (ω ) K p (ω ) = − i =1 N G ( jω ) 2 ∑ Ri (ω ) (26) K p (ω ) = − i =1 N G ( jω ) 2 ∑ I i (ω ) (19) iK (ω ) = ω 2 K − ω i =1 G ( jω ) d 2 N ∑ I i (ω ) G ( jω ) is given by (12). 2 K d (ω ) = i =1 where ω G ( jω ) 2 In the second step, the K i parameter is now fixed. The G ( jω ) is given by (12). 2 where ( K p , K d ) plane is then determined by solving the following system: N K ω + Ki ∑ I i (ω ) IV. STABILIZATION SEVERAL TIME DELAYS SYSTEM USING PID N N ∑ i −ω ω ω ∑ ω 2 CONTROLLER R ( ) I ( ) i =1 i p (27) Considering the same system (1) shown in Fig. 1, we i =1 i =1 = N N N attempt to achieve stabilization with PID controller presented ω ∑ i =1 I i (ω ) ω 2 ∑ i =1 R i ( ω ) K K i ∑ Ri (ω ) d by: i =1 Ki K p s + Ki + K d s 2 K p is given by the same equations (26) and: C ( s) = K p + + Kd s = (20) N s The same approach is applied in the case of several time s Ki ∑ I (ω ) i K d (ω ) = + i =1 (28) delay system with PID controller. ω 2 ω G ( jω ) 2 The closed loop characteristic polynomial ∆(s) is written as: − K pω + j ( K i − K d ω 2 ) N N ∆( jω ) = 1 − ( )(∑ Ri (ω ) + j ∑ I i (ω )) = R∆ (ω ) + jI ∆ (ω ) V. SIMULATION RESULTS ω i =1 i =1 (21) The proposed approach is illustrated on a linear system defined by two parallel subsystems, having two different time The stabilization region is determined by setting ∆( jω ) to delays. The first subsystem is of order one and the second is of order three given by: zero: ∆( jω ) = R∆ (ω ) + jI ∆ (ω ) = 0 (22) G ( s ) = G 1 ( s ) e −τ1s + G 2 ( s ) e −τ 2 s (29) where: where N N N R∆ (ω ) = ω + K pω ∑ Ri (ω ) − K d ω ∑ I i (ω ) + K i ∑ I i (ω ) K 2 + K3s 2 K1 , i =1 i =1 i =1 G1 ( s) = G2 ( s ) = (30) N N N (23) 1 + a1s 1 + b1s + b2 s 2 + b3 s 3 I (ω ) = K ω I (ω ) + K ω 2 R (ω ) − K p ∑ i ∑ i ∑ Ri (ω ) ∆ i =1 d i =1 i i =1 256 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics K2 = 1 0.14 K1 = 0.5 K 3 = −0.5 0.12 We suppose that: and b1 = 1 a1 = 2 0.1 τ = 1.5 1 b2 = 3 b3 = 2 0.08 Stability region Ki τ 2 = 0.6 0.06 0.04 noting: 0.02 G( jω ) = G1 ( jω ) [ cos(τ1ω ) − j sin(τ 1ω )] + G2 ( jω ) [ cos(τ 2ω ) − j sin(τ 2ω )] 0 0 0.05 0.1 0.15 0.2 0.25 Kp (31) Figure 2: Stability region with PI Controller Developing (31), the results are as follows: 1 R1 (ω ) = 1 + A (ω ) 2 [ K1 cos(τ 1ω ) − K1 A 1 (ω )sin(τ 1ω ) ] In ( K p , K i ) plane, the curve described by values of Kp 1 (32) and K i given by equations (34) bounds a zone that represents 1 I (ω ) = − [ K1 sin(τ 1ω ) + K1 A1 (ω ) cos(τ1ω )] 1 1 + A1 (ω ) 2 stability region, which is displayed as shaded in Fig. 2. where Closed loop responses of this system are represented by the A1 (ω ) = a1ω following figures: and 1 7 R2 (ω ) = B (ω ) 2 + B (ω ) 2 [ X 1 (ω ) cos(τ 2 w) + X 2 (ω ) sin(τ 2 w)] 2 1 6 (33) 1 I (ω ) = [ X 2 (ω ) cos(τ 2 w) − X 2 (ω ) sin(τ 2 w)] 2 B2 (ω )2 + B1 (ω ) 2 5 where 4 B1 (ω ) = b1ω − b3ω 3 3 B2 (ω ) = 1 − b2ω 2 2 X 1 (ω ) = K 2 B2 (ω ) + ω K 3 B1 (ω ) 1 X (ω ) = ω K B (ω ) − K B (ω ) 2 3 2 2 1 0 R1 , I1 are the real and imaginary parts of G1 , respectively -1 0 100 200 300 400 500 600 700 800 900 1000 R2 , I 2 are the real and imaginary parts of G2 , respectively Figure 3: Closed loop step responses with PI Controller (Kp= 0.05, Ki=0.1: red response; Kp= 0.1, Ki=0.077: green response): stable case A. Stabilization with PI controller 2000 K p and K i parameters are given by: 1500 R1 (ω ) + R2 (ω ) K p (ω ) = − ( R (ω ) + R (ω )) 2 + ( I (ω ) + I (ω )) 2 1000 1 2 1 2 500 K (ω ) = −ω I 1 (ω ) + I 2 (ω ) (34) i ( R1 (ω ) + R2 (ω )) 2 + ( I1 (ω ) + I 2 (ω )) 2 0 -500 Substituting (32) and (33) into (34) leads to determine -1000 stability region in ( K p , K i ) plane shown in the following figure: -1500 -2000 0 100 200 300 400 500 600 700 800 900 1000 Figure 4: Closed loop step response with PI Controller (Kp= 0.2, Ki=0.04): unstable case 257 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics The parameters of the PI controller are altered according to Figure 5: Stability region with PD Controller their belonging to stability region. Starting with K p = 0.05, K i = 0.1, the closed loop 4 response system is described by the red response shown in 3.5 Fig. 3. 3 Corresponding response for these gains displays that the 2.5 system is stable in the shaded zone. 2 For K p = 0.1, K i = 0.077, these proportional and integral 1.5 gains define a point belonging to the border of the shaded 1 zone. Corresponding closed loop behavior system is shown by 0.5 the green response in Fig. 3. In the transient regime, the system presents oscillations and it is stabilized in the steady 0 state which represents the limit of the stability. -0.5 0 100 200 300 400 500 600 700 800 900 1000 Finally, K p and K i parameters are chosen so that they Figure 6: Closed loop step response with PD Controller (Kp= 0.2, Kd=0.5: define a point out of the region of stability. For K p = 0.2, red response; Kp= 0.41, Kd=0.5: green response): stable case 9 x 10 K i = 0.04, the closed loop response system is presented in 1.5 Fig. 4. The result shows that in the not shaded region, the 1 system becomes unstable. 0.5 B. Stabilization with PD controller The same approach is applied in the case of stabilization 0 with PD controller. The K p and K d expressions are given by: -0.5 R1 (ω ) + R2 (ω ) -1 K p (ω ) = − ( R (ω ) + R (ω )) 2 + ( I (ω ) + I (ω )) 2 1 2 1 2 (35) -1.5 K (ω ) = I1 (ω ) + I 2 (ω ) 0 100 200 300 400 500 600 700 800 900 1000 d ω (( R1 (ω ) + R2 (ω ))2 + ( I1 (ω ) + I 2 (ω ))2 ) Figure 7: Closed loop response step with PD Controller: (Kp= 0.5, Kd=0. 4): Substituting (32) and (33) into (35) leads to obtain K p and unstable case K d values. As seen previously, the stability region which is shaded in For K p = 0.2, K d =0.5, the closed loop system is stable. Fig. 5 is bounded by the curve defined in ( K p , K d ) plane. In the case of K p = 0.41, K d =0.5, the system is on the Closed loop responses of this system for different K p and limit of the stability in the closed loop. K d values are represented by the figures Fig. 6, Fig. 7. Finally, for K p = 0.5, K d =0.4, the system is unstable in the closed loop. The obtained results show the effectiveness of the proposed 3.5 approach. 3 2.5 C. Stabilization with PID controller 2 Considering the previous approach, two cases are presented. Kd Stability region In the first step, the stability region can be determined in 1.5 the ( K p , K i ) plane with fixed K d . The results are as follows 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 kp 258 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics R1 (ω ) + R2 (ω ) K p (ω ) = − ( R (ω ) + R (ω )) 2 + ( I (ω ) + I (ω )) 2 50 1 2 1 2 40 K (ω ) = ω 2 K − ω I 1 (ω ) + I 2 ω) ( (36) 30 ( R1 (ω ) + R2 (ω )) + ( I1 (ω ) + I 2 (ω ))2 i d 2 20 10 In the second step, the stability region can be determined in 0 the ( K p , K d ) plane with fixed K i . The results obtained are -10 as follow: -20 R1 (ω ) + R2 (ω ) -30 K p (ω ) = − ( R (ω ) + R (ω ))2 + ( I (ω ) + I (ω )) 2 -40 1 2 1 2 (37) -50 K K (ω ) = i + I 1 (ω ) + I 2 (ω ) 0 100 200 300 400 500 600 700 800 900 1000 d ω 2 ω (( R1 (ω ) + R2 (ω ))2 + ( I1 (ω ) + I 2 (ω ))2 ) Figure 10: Closed loop step response with PID Controller (Kp= 0.6, Ki=2.085, Substituting (32) and (33) into (36) and (37) leads to Kd=4.2): stable case obtain K p , K i and K d values. x 10 24 2 For each value of K i , a stability region is defined in the 1.5 ( K p , K d ) plane. A three dimensional curve is then obtained 1 by varying K i as shown in Fig. 8. 0.5 0 3 -0.5 2.5 2 -1 1.5 Stability region Ki -1.5 1 0.5 -2 0 100 200 300 400 500 600 700 800 900 1000 0 4 1.5 2 1 0.5 Figure 11: Closed loop step response with PID Controller (Kp= 0.5, Ki=0.05, 0 0 -2 -1 -0.5 Kd=0.1): unstable case Kd Kp Figure 8: Stability region with PID Controller For K p =1, K i = 2.085 and K d =4.2, the closed loop The figures Fig. 9, Fig. 10 and Fig. 11 show the closed loop system is stable. responses of the system for different chosen PID parameters. In the case of K p =0.6, K i = 2.085 and K d =4.2, the 6 system is on the limit of the stability in the closed loop. 5 Finally, for K p =0.5, K i = 0.05 and K d =0.1, the system is 4 unstable in the closed loop. 3 2 VI. CONCLUSIONS The main contribution of this paper concerns the 1 stabilization of continuous linear time invariant system of any 0 order and which presents several delays using a PID controller. The proposed approach is based on mathematical calculation -1 0 100 200 300 400 500 600 700 800 900 1000 of the proportional, derivative and integral gains by extracting the real and imaginary parts of the system transfer function. Figure 9: Closed loop step response with PID Controller (Kp= 1, Ki=2.085, This not complicated method leads to the determination of the Kd=4.2): stable case stability regions in ( K p , K i ) plane for PI controller, ( K p , K d ) plane for PD controller and ( K p , K i , K d ) space for PID controller. 259 Proceedings of the 2013 International Conference on Systems, Control, Signal Processing and Informatics The simulation results point out the correspondence [8] G. J. Silva, A. Datta and S. P. Bhattacharyya, PID controllers for time- delay systems. Boston: Birkhäuser, 2005. between the time domain responses and the obtained stability [9] G. J. Silva, A. Datta and S. P. Bhattacharyya, "Stabilization of time regions. 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