TUNING OF TWO-DEGREE-OF-FREEDOM PI/PID CONTROLLER FOR SECOND-ORDER UNSTABLE PROCESSES

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1 TUNING OF TWO-DEGREE-OF-FREEDOM PI/PID CONTROLLER FOR SECOND-ORDER UNSTABLE PROCESSES CRISTIANE G. TAROCO, HUMBERTO M. MAZZINI, LUCAS C. RIBEIRO Departamento de Engenharia Elétrica Universidade Federal de São João del-rei Praça Frei Orlando, 170 C.E.P.: São João del-rei, M.G. Abstract The proportional-integral-derivative (PID) controllers are still widely used in the process industries even though more advanced control techniques have been developed. The main reason is its relatively simple structure, which can be easily understood and implemented in practice. Over the years, there are many formulas derived to tune the PID controllers for stable processes. However, unstable systems are fundamentally and quantifiably, more difficult to control than stable ones. Despite these difficulties, research for unstable process control has been increasingly active. Many different approaches of controller design for unstable processes have been reported in the literature. However, they usually either show excessive overshoot and large settling time or have complicated formulas. Moreover, it is difficult to achieve all goal since control system design involves inherent conflicts and trade-offs. For example, a controller design that minimizes the effect of load disturbances tends to produce large overshoots for set-point changes. This paper presents a simple tuning method for twodegrees-of-freedom (-DOF) controllers for second-order unstable processes. The proposed method obtains explicit expression of PID tuning parameters using specification of the desired closed-loop transfer function for disturbance responses. Although the PID are designed for disturbance rejection, the set-point responses are usually satisfactory and can be independently tuned. Simulation results demonstrate that the proposed method provides good control performance in terms of disturbance rejection, set-point tracking, and robustness. This paper assumes that a second-order plus time-delay (SOPTD) model of the underlying plant is available for control design. Keywords Unstable Process, PID Control, Two-degree-of-freedom Controller 1 Introduction The proportional-integral-derivative (PID) controller remains the most popular controller for industrial control applications despite continual advances in control theory. This is not only due to the simple structure, which is conceptually easy to understand and makes manual tuning possible, but also due to the vast majority of applications because they perform well for a wide class of processes (Zhang, 006). The PID controller is used for wide range applications: process control, motor drives, magnetic and optic memories, automotive, flight control, etc. In process control, more than 95 percent of the control loops are of PID type; most loops are actually PI control (Astrom and Hagglund, 006). For stable systems, method of designing PID controller is well established. A system whose transfer function has at least one pole lie in the right half plane is known as unstable system. Open loop instability means the system will move away from the steady state even for the small perturbation of the system parameters or operating conditions. The some of examples are the polymerization furnaces, continuous stirred tank reactors (CSTRs), and the batch chemical reactor, which has a strong nonlinearity due to heat generation term in the energy balance (Leu et al., 006). Open-loop unstable processes are much more difficult to control than that of the stable processes. Desired closed-loop performance cannot be achieved with the conventional proportional-integralderivative (PID) controller for any adjustable parameters of the controller (Rao et al., 007). As for conventional PI/PID methods within the framework of a unity feedback control structure, improved tuning rules had been provided by (Poulin and Pomerleau, 1996), (Hwang and Hwang, 004), etc. However, the set-point response is usually accompanied with excessive overshoot and large settling time according to the aforementioned methods, owing to that the water-bed effect between the setpoint response and the load disturbance response becomes severe for unstable processes (Liu et al., 005). Recently, tuning of controllers for a time delay unstable process has been an active area of research in the literature: (DePaor and O Malley, 1989), (Venkatashankar and Chidambaram, 1994) and (Huang and Lin, 1995). More recently, (Ho and Xu, 1998) derived PID tuning formulas based on gain and phase margin specifications. In (Majhi and Atherton, 1999), an internal feedback loop is used across the delay-free part of the process model and the same amount of signal is sent to the original process to stabilize the unstable process. (Visioli, 001) proposed optimal PID parameters tuning in terms of IAE, ISTE and ITSE specifications via genetic algorithm. (Park et al., 1998) and (Majhi and Atherson, 000a) developed PID-PI and PI-PD strategies respectively, both using inner feedback loops. Besides, (Wang and Cai, 00) employed the PID-PI structure to design an equivalent two degree-of-freedom (-DOF) single loop PID Control scheme in terms of gain and phase margin specifications. In (Tan et al., 00), a modified IMC structure is proposed for unstable proc-

2 esses. The structure extends the standard IMC structure for stable processes to unstable processes and controllers do not have to be converted to conventional ones for implementation. (Keel et al., 00) have given a solution to the problem of stabilization of a digital control system using PID controllers. In (Lu et al., 005), four controllers are placed to separately tune the denominators and numerators of closed-loop transfer functions from the set-point and disturbance responses. It should be noted that most existing control methods restricted attention on unstable processes modeled in the form of a first-order time-delayed processes (FOPDT), which in fact, cannot represent a variety of industrial unstable processes well enough (see (Jung et al., 1999), (Majhi and Atherton, 000) and (Sree et al., 004)). In this work are proposed PI and PID settings for second-order plus dead time (SOPTD) Direct synthesis methods are usually based on specification of the desired closed loop transfer function for set-point changes. Consequently, the resulting controllers tend to perform well for set-point changes, but the disturbance response might not be satisfactory. For example, (Middleton and Graebe, 1999) have investigated the relationship between input disturbance responses and robustness. (Lee et al., 1998) extended the IMC design approach for two degree of freedom controllers to improve disturbance performance. An other approach was proposed by (Szita and Sanathanan, 1997). They specify the desired disturbance rejection characteristics in terms of a closed loop transfer function for disturbances. the resulting controller usually is not a PI or PID controller and might be of high order. In (Chen and Seborg, 00) and (Mazzini, 00) the authors proposed a direct synthesis design method to improve disturbance rejection for several process models. However, their work did not deal with unstable process. The research done by (Araki and Taguchi, 00) can be mentioned too. Recently, (Rao and Chidambaram, 006) have proposed a simple design of a PID controller with a lead-lag compensator based on a direct synthesis method for unstable second-order processes that have one unstable pole or two unstable poles. This paper presents a simple procedure for tuning two-degree-of-freedom PID controllers using direct synthesis approach for disturbance rejection for second-order unstable processes. The equations incorporate a single design parameter, the desired closed loop time constant. One controller (PI or PID) is designed for set-point changes. Another (P or PD) is used for disturbance rejection. The PI/PID settings obtained from the direct synthesis approach are based on specifying the closed loop transfer function for disturbance. The set-point responses can be independently tuned by the P/PD controllers. This work assumes that a second-order plus time-delay (SOPTD) model of the underlying plant is available for control design. The paper is organized as follows. Section addresses the two-degree-of-freedom control scheme, and section gives the controller structures. Section 4 deals with the controllers design. In section 5, simulation results are explained and finally conclusion are given in section 6. Two-degree-of-freedom control scheme The proposed method is based on the two-degreeof-freedom control structure shown in Fig. 1, where Gp(s) is the plant to be controlled, Gc (s) is the controller for disturbance rejection and Gc 1 (s) is the controller for set-point tracking. Fig. 1. Two-degree-of-freedom control structure In this system, R(s) is the set-point, D(s) the load disturbance and Y(s) the controlled process variable. The design objective is to determine the controller parameters in Gc (s) and Gc 1 (s) so that the system behaves well with respect to changes in the signals R(s) and D(s). The set-point response is given by: Y(s) Gp(s)Gc1(s ) (1) R(s) 1 + Gp(s) [ Gc1(s) + Gc(s) ] The load disturbance response is given by: Y(s) D(s) Gp(s) 1 + Gp(s) [ Gc1(s) + Gc(s) ] Controller structures () The process is assumed to be linear, time invariant, which is analytic with finite poles. The PID controller Gc 1 (s) is described by: 1 Gc 1(s) Kc ( 1 α ) + + (1 β )Tds Tis The PD controller Gc (s) is described by: () Gc (s) Kc( α + βt ds ) (4) The controller parameters are proportional gain Kc, integral time Ti, and derivative time Td. In the

3 following, α and β will be treated as adjustable parameters. The conventional PID controller is obtained by setting α β 0, the preceded-derivative PID by setting α 0 and β 1, and the I-PD by setting α β 1 (Araki and Taguchi, 00). In this paper, we use the preceded-derivative PID. Equation () tells that the load disturbance response depends on both Gc 1 (s) and Gc (s). On the other hand, by adjusting α 0 and β 1, we obtain: where Gc (s) is given by Gc (s) Gc 1( s) + Gc ( s), (5) 1 Gc (s) Kc Tds Tis (6) The equation (), that is the load disturbance response, can be rearranged for the controller: Gc (s) Where Y(s) A(s) D(s) 1 1 A(s) Gp(s) (7) Assume that a process model Gm(s) and a desired disturbance model Ad(s) are available. Then: Gc(s) 1 1 (8) Ad(s) Gm(s) 4 Controllers design Design starts with the selection of a reference model for load disturbances whose response meets the design specifications. Subsequently, Kc, Ti and Td parameters are determined so that the closed loop and the reference responses are closely matched. Assume that the process is described by a secondorder delayed unstable process (SODUP): -sl Kpe Gp(s) (9) (T1s -1)(Ts + 1) The simplest approach to approximate a time delay is to choose a first-order Taylor series expansion e -sl 1 Ls. Because the error occurs only in terms involving the third power of s and higher, this approach should serve as a reasonable approximation for small values of Ls (Seborg et al.,1989). The closed loop transfer function for disturbances in () becomes: Kp(1 - Ls) Y(s) Ts + 1) D(s) Kp(1 Ls) 1 1+ Kc(1 + + Td ) Ts + 1) Tis (10) If the desired closed loop transfer function for disturbances is specified as in: A d(s) Ti Kc e -sl s (s + 1) Where is the desired closed loop time constant. The Kc, Ti e Td parameters can be obtained: Kp Kc Ti (11) [ + L + (T + LT 1 LT ) + T L T L ] L + (T ( + L) + LT 1 LT ) + T + T 1L - T L - L + T L T ( T + L T 1) + T T 1 + T L + T L Td ( T + T 1L - T L - L ) Ti Now, assume that the process is described by: L (1) (1) (14) -sl Kpe Gp(s) (15) (T1s -1)(Ts 1) If a PID controller is used and approximating the time delay by e -sl 1 Ls, then the closed loop transfer function in () becomes: Kp(1 - Ls) Y(s) Ts 1) D(s) Kp(1 - Ls) 1 1+ Kc(1 + + Td ) Ts 1) Tis (16) If the desired closed loop transfer function for disturbance is specified as in (11), then the Kc, Ti e Td parameters can be obtained: K Kc T Ti [ T 1 T L L ( LT 1 + LT T ) T 1L T L ] P c + LT c ( + L) + ( LT 1 + LT T ) + T 1L ( T 1 + T ) L T L ( T 1 + T L) + T + T L + T ct L Td Ti( T 1L T L + L + T ) (17) + T L T L (18) (19) Selecting the design parameter the designer establishes the desired control system response

4 speed. As the value becomes lower, the system response becomes faster, but its robustness decreases. been observed that there is only a minor effect on the responses for a 10% estimating error in delay time. 5 Simulation results In this section, we demonstrate our designs in the proceeding section by two examples, one for each case. Example 1 Consider an unstable process as (Majhi and Atherton, 1999): -,5s e Gp(s) (10s -1)(s + 1) For performance comparison, we use the proposed method by (Lu et al., 005) and (Majhi and Atherton, 1999). For the proposed tuning rule, the value b0.80 was adjusted. is chosen with a value of PID parameters are obtained as: Kc6.0, Ti0.10 and Td.00. The parameters of the previous approaches are adopted from those given in the papers. With these controller settings, the methods are compared by giving a unit step change in the setpoint and a unit step change input in the load disturbance at t50 s. Figure shows the responses for perfect model parameters. Fig. Responses for a perturbation of +10% in process time delay for Example 1 Example Now, consider the process studied in (Tan et al., 00) : -0.s e Gp(s) (s -1)(s 1) Here, for performance comparison, we use the proposed method by (Rao and Chidambaram, 006) and (Tan et al., 001). In this example, for the proposed tuning rule, the value b1.0 was adjusted and is chosen with a value of PID parameters are obtained as: Kc0.7, Ti1.90 and Td8.7. Again, the parameters of the controllers of the others approaches are adopted from those given in the papers. Add a unit step change to the set-point input at t0 s and a step change of load disturbance to the input of the process at t15s. Simulation results are shown in figure 4. Fig. Set-point and disturbance responses for example 1. The figure illustrates that the proposed design and the controllers from Majhi give superior performance for set-point tracking when compared to the design method of Xiang Lu. Once again, both methods have better disturbance rejection performance. Now suppose that there exists 10% error for estimating the process time. The perturbed system responses are provided in figure. Majhi s method and the proposed approach provide a faster closed-loop response with a small overshoot and an improved disturbance rejection. It has Fig. 4 Set-point and disturbance responses for example. The Rao and Tan controllers give the fastest setpoint responses. However, the Rao method gives large overshoot. The load responses of the three schemes are acceptable.

5 The effects of 0% time delay deviation are showed in figure 5. Fig. 5 Responses for a perturbation of +0% in process time delay for Example It is seen from the figure 5 that the Rao method tends to be unstable and the Tan scheme becomes more oscillatory, whereas the proposed scheme changes little. 6 Conclusions A simple design of a PI/PID controller based on a direct synthesis method is proposed for unstable second-order processes. Ensuring good disturbance rejection is the primary requirement in many industrial control applications. The method presented in this paper allows direct specification of the disturbance response of the closed-loop system. Its procedure is simple and easy to implement. The parameters of the controllers are tuned by simple tuning formulas and the resulting responses for two examples indicate that the designs are relatively robust to delay variations. The simulations show that the proposed method furnishes a convenient and flexible design method that provides good performance in terms of disturbance rejection and set-point tracking. References Araki, M. and Taguchi, H. (00). Two-degree-offreedom PID controllers. International Journal of Control, Automation, and Systems, 1: 401:411. Astrom, K. J. and Hagglund, T. (006). Advanced PID Control, Instrument Society of America: Research Triangle Park, NC Chen, D. and Seborg, D. (00). PI/PID controller design based on direct synthesis and disturbance rejection, Ind. Eng. Chem. Res., 41: De Paor, A. M. and O Malley, M. (1989). Control of Ziegler-Nichols type for unstable processes with time delay, International Journal of Control, 49: Ho, W. K. and Xu, W. (1998). PID tuning for unstable processes based on gain and phasemargin specifications, IEE Proceedings on Control Theory and Applications, 145:9-96. Huang, C. T. and Lin, Y. S. (1995). Tuning of PID controller for open-loop unstable processes with time delay, Chem. Eng. Commun., 1:11-0. Hwang, C. and Hwang, J. H. (004). Satibilization of first-order-plus-dead-time unstable process using PID controllers, IEE Proceedings Part D, 151 (1): Jung, C. S., Song, H. K. and Hyun, J. C. (1999). A direct synthesis tuning method of unstable firstorder-plus-time-delay processes, Journal of Process Control, 9: Keel, L. H., Rego, J. I. and Bhattacharyya, S. P. (00). A new approach to digital PID controller design, IEEE Transactions on Automatic Control, 48 (4): Lee, Y., Lee, J. and Park S. (000). PID controller tuning for integrating and unstable processes with time delay, Chemical Engineering Science, 55: Lee, Y., Park, S., Lee, M. and Brosilow, C. (1998). PID controller tuning for desired closed loop responses for SI/SO Systems, AIChE Journal, 44: Leu, J., Tsay, S. and Hwang, C. (006). Optimal tuning of PID-deadtime controllers for integrating and unstable time-delayed processes, Journal of the Chinese Institute of Engineers, : 9-9. Liu, T., Zhang, W. and Gu, D. (005). Analytical design of two-degree-of-freedom control scheme for open-loop unstable processes with time delay, Journal of Process Control, 15: Lu, X., Yang, Y., Wang, Q. and Zheng, W. (005). A double two-degree-of-freedom control scheme for improved control of unstable delay processes, Journal of Process Control, 15: Majhi, S. and Atherton, D. P. (1999). Modified Smith predictor and controller for processes with time delay, IEE Proceedings-Control Theory Applications, 146 (5): Majhi, S. and Atherton, D. P. (000). Online tuning of controllers for an unstable FOPDT process, IEE Proceedings-Control Theory Applications, 147 (4): Mazzini, H. M. (00). Estudo sobre a compensação de processos integradores com atraso e a proposição de uma nova abordagem. Tese de doutorado. Universidade Federal de Uberlândia. Middleton, R.H. and Graebe, S.F. (1999). Slow stable open loop poles: to cancel or not to cancel", Automatica, 5: Park, J. H., Sung., S. W. and Lee, I. B. (1998). An enhanced PID control strategy for unstable processes, Automatica, 4: Poulin, E. and Pomerleau, A. (1996). PID tuning for integrating and unstable processes, IEE Proceedings on Control Theory and Applications, 14:

6 Rao, A. S. and Chidambaram, M. (006). Enhanced two-degrees-of-freedom control strategy for second-order unstable processes with time delay, Ind. Eng. Chem. Res., 45: Rao, A. S., Rao, V. S. R. and Chidambaram, M. (007). Simple analytical design of modified Smith predictor with improved performance for unstable first-order plus time delay (FOPTD) processes, Ind. Eng. Chem. Res., 46: Seborg, D., Edgar, T. F. and Mellichamp, D. A. (1989). Process Dynamics and Control, John Wiley and Sons. Shamsuzzoha, M. and Lee, M. (007). IMC-PID controller design for improved disturbance rejection of time-delayed processes, Ind. Eng. Chem. Res., 46: Sree, R. P., Srinivas, M. N. and Chidambaram, M. (004). A simple method of tuning PID controllers for stable and unstable FOPTD systems, Computers and Chemical Engineering, 8 (11): Szita, G. and Sanathanan, C. K. (1997). Robust design for disturbance rejection in time delay systems, Automatica, 4: Tan, W., Marquez, H. J. and Chen, T. (00). IMC design for unstable processes with time delays, Journal of Process Control, 1: 0-1. Venkatashankar, V. and Chidambaram, M. (1994). Design of P and PI controllers for unstable first order plus time delay systems, International Journal of Control, 60: Visioli, A. (001). Optimal tuning of PID controllers for integral and unstable processes, IEE Proc.Control Theory Appl., 148: Wang, Y. G. and Cai, W. J. (00). Advanced proportional-integral-derivative tuning for integrating and unstable processes with gain and phase margin specifications, Ind. Eng. Chem. Res., 41: Zhang, W. (006). Optimal design of the refined Ziegler-Nichols proportional-integral-derivative controller for stable and unstable process with time delay, Ind. Eng. Chem. Res., 45: Acknowledgements This work was supported by the FAPEMIG Fundação de Amparo à Pesquisa do Estado de Minas Gerais