Control of integral processes with dead time Part IV: various issues about PI controllers

Control of integral processes with dead time Part IV: various issues about PI controllers B. Wang, D. Rees and Q.-C. Zhong Abstract: Various issues about integral processes with dead time controlled by a PI controller are discussed. First, the region of the control parameters to guarantee the system stability is characterised. Then, the control parameters to achieve the given gain and/or phase margins (GPM) are determined. Furthermore, the constraint on achievable GPM is derived. These results are obtained on the basis of the normalised system that involves only two free parameters. 1 Introduction An integral process with dead time (IPDT) is a typical case of unstable processes. The classical Smith predictor (SP) is not applicable to IPDT because the integral mode will result in a steady-state error for a constant load disturbance. In the last two decades, many modified SP and other schemes have been presented to overcome this shortcoming and to further improve the system performances. Watanabe and Ito [1] proposed a process-model control to obtain the zero steady-state error and the desired transient response for the step disturbance. Åström et al. [2] designed a structure to decouple the disturbance response and the set-point response so that they can be designed separately. Normey-Rico and Camacho [3] introduced a filter into the scheme of [1] and achieved the same set-point and disturbance responses, while the robustness was taken into account explicitly. Zhong and Li [4] used a repetitive control scheme to improve the disturbancerejection ability and to make the system possessing PID control advantages. Lu et al. [5] proposed a double twodegree-of-freedom (2-DOF) control scheme to improve the performance of set-point and disturbance responses. A number of tuning methods for PID controllers have been proposed to control IPDT. Rivera et al. [6] applied the internal model control method to tune a PI controller so that the tuning formula is only related to the closed-loop time constant. Tyreus and Luyben [7] showed that the closed-loop time constant in [6] must be chosen carefully to avoid a very oscillatory response and then proposed an alternative approach by specifying a closed-loop damping coefficient (relating to the smallest closed-loop time constant). Luyben [8] extended the method in [7] to a PID controller and achieved a smaller closed-loop time constant for the same closed-loop damping coefficient. Poulin and Pomerleau [9] obtained the PI settings for the maximum peak resonance specification according to the ultimate cycle information of the process. Wang and Cluett [10] designed a PID controller in terms of the desired control signal trajectory. Visioli [11] optimised PID settings by minimising ISE, ISTE or ITSE with the genetic algorithm. Chidambaram and Sree [12] proposed a simple coefficientmatching method to obtain similar results to [11]. In the above-mentioned papers, it is still not clear how the control parameters affect the system stability. This motivated the research in this paper, which is Part IV of a series of papers devoted to the control of IPDT. In Part I [13], a disturbanceobserver-based control scheme, which can reject arbitrary disturbances, was proposed and the resulting disturbance responses are in fact sub-ideal. In Part II [14], the robust stability regions, the disturbance response specifications and the stability of the controller itself were analysed quantitatively. In Part III [15], a dead-beat disturbance response was designed to reduce the long recovery time of the disturbance response, while robust stability was still guaranteed. This paper characterises the stability region of the control parameters. In this region, the effect of the control parameters on the system relative stability is further described quantitatively by using stability margins. Consequently, the controller is designed according to the specified gain and/or phase margins (GPM). It is also found that there exists fundamental limitation on the achievable GPM (see [16] and the references therein for more details about fundamental performance limitations in control). These results are based on the normalised system which involves only two free parameters. A relevant paper to this theme is [17], where the complete set of stabilising PID parameters is determined for both open-loop stable and unstable firstorder plus dead time processes (FOPDT), by applying the Hermite Biehler theorem to quasi-polynomials. 2 Control scheme under consideration Consider the 2-DOF control scheme shown in Fig. 1a, where # IEE, 2006 IEE Proceedings online no. 20050065 doi:10.1049/ip-cta:20050065 Paper first received 3rd March and in revised form 19th July 2005 B. Wang and D. Rees are with the School of Electronics, University of Glamorgan, Pontypridd CF37 1DL, UK Q.-C. Zhong is with the Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, UK E-mail: zhongqc@ieee.org 302 GðsÞ ¼PðsÞe ts ¼ k s e ts is the process with the rational part P(s) ¼ k/s (k. 0) and the dead time t. 0. The pre-filter F(s), the first degree of freedom, is designed to obtain the desired set-point response. The feedback controller C(s), the second degree of freedom, is designed to reject disturbances and to achieve robustness, with an essential compromise having to be made between

3 Achievable performance of the system 3.1 Normalisation of the system The loop transfer function of the system is LðsÞ ¼CðsÞGðsÞ ¼k p 1 þ 1 k T i s s e ts ð1þ which involves four parameters. If the process parameters k and t are normalised into Fig. 1 Two-degree-of-freedom control structure for integral processes with dead time these two requirements. Here, C(s) is chosen as a PI controller CðsÞ ¼k p 1 þ 1 T i s where k p. 0 is the proportional gain and T i. 0 is the integral time constant. The transfer function from the set-point r to the output y is If F(s) is designed as CðsÞPðsÞe ts G r ðsþ ¼FðsÞ 1 þ CðsÞPðsÞe ts FðsÞ ¼ 1 þ CðsÞPðsÞe ts CðsÞPðsÞ then the desired set-point response is G r ðsþ ¼ 1 ls þ 1 e ts 1 ls þ 1 This is independent of the second degree of freedom C(s). Here, 1/(ls þ 1) (l. 0) is selected to make F(s) proper. Note that there is no unstable zero-pole cancellation between F(s) and the feedback loop because the closedloop system is designed to be stable. On the other hand, the transfer function from the disturbance d to the output y is G d ðsþ ¼ PðsÞe ts 1 þ CðsÞPðsÞe ts This is independent of the first degree of freedom F(s). Therefore the set-point response and the disturbance response are decoupled from each other. With the integral action in C(s), the steady-state error for the step disturbance is 0 because G d (s)! 0 when s! 0. An equivalent structure of Fig. 1a is shown in Fig. 1b for better understanding. It is also useful for implementation because F(s) in Fig. 1a is dependent on the parameters of C(s). The desired set-point response is separated out as the reference output y d. C(s) acts only when the error between the actual output y and the reference output y d is not equal to 0. If there are no disturbances or process uncertainties in the system, then y is always equal to y d and C(s) has no effect on the system, that is the system becomes open-loop. k p ¼ tkk p and T i ¼ T i ð2þ t then the loop transfer function is correspondingly normalised as LðsÞ ¼k p 1 þ 1 1 T i s s e s ð3þ This involves only two parameters. As a result, L (s) can be regarded as a system with the process and the controller GðsÞ ¼ 1 s e s CðsÞ ¼k p 1 þ 1 T i s where k p. 0 and T i. 0 are the normalised proportional gain and the normalised integral time constant, respectively. As a matter of fact, (3) can be obtained from (1) by substituting s with (1/t)s. This means that the Nyquist plots of L(s) and L (s) have the same form, but arrive at the same point with different frequencies. Therefore the design of C(s) for L(s) can be done via designing C (s) for L (s) and then recovering the parameters of C(s) from (2). As can be seen later, this has considerably simplified the system analysis and design. 3.2 Stability region Rewrite L (s) asl ( jv) ¼ Re(v) þ j Im(v), where k ReðvÞ ¼ p cos v k p sin v T i v 2 v k ImðvÞ ¼ p sin v k p cos v T i v 2 v When v! 0, there are 8 < 1 T i. 1 lim ReðvÞ ¼ 1; lim ImðvÞ ¼ 0 T i ¼ 1 v!0 v!0 : þ1 0, T i, 1 The Nyquist plots for these cases are shown in Fig. 2. In the case of 0, T i, 1orT i ¼ 1, the Nyquist curve starts above the real axis and encircles the point (21, 0). In the case of T i. 1, it is possible for the Nyquist curve not to encircle the point (21, 0). Therefore a necessary condition for the closed-loop system stability is T i. 1. This means that the integral time constant of the PI controller C(s) must be greater than the dead time of the process G(s). Furthermore, a necessary and sufficient condition for the stability is given below. Theorem 1: The closed-loop feedback system with the open-loop transfer function given in (3) is stable if and 303

Fig. 2 system Nyquist plot of the loop transfer function L (s) of the only if k p and T i satisfy T i v k 2 p p, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 i v2 p þ 1 where v p [ (0, p/2] is the solution of v p ¼ arctanð T i v p Þ Proof: The closed-loop feedback system is stable if and only if the Nyquist curve crosses the real axis from the right side of the point (21, 0), i.e., satisfying Re(v).21 when Im(v) ¼ 0, which gives k p cos v p T i v 2 p ð4þ ð5þ þ k p sin v p v p, 1 ð6þ sin v p T i v p cos v p ¼ 0 ð7þ where v p is called the phase crossover frequency. Because T i. 1, the minimum solution of v p in (7) lies in the interval (0, p/2], where p/2 is obtained when T i!þ1 (in this situation, the PI controller degenerates to a P controller). This means that the Nyquist curve crosses the real axis for the first time at a frequency not more than p/2. Thus, (7) can be converted into (5). Furthermore, simplifying (6) with (7) gives (4). A When the Nyquist curve crosses the point (21, 0), then T i v k 2 p p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 i v2 p þ 1 Fig. 3 system Region of control parameters k p and T i to stabilise the where v p is the phase crossover frequency defined in (5) and v g is the gain crossover frequency defined by jlð jv g Þj ¼ 1 ð10þ Conventionally, A m, which is larger than 1, is converted so that the gain margin has the unit db. Here it is kept dimensionless to simplify the expression and calculation. Before the Nyquist curve reaches the real axis for the first time, v p/2 and Re(v) is always negative, so f m is in the range of 0, f m, p/2. Substituting (3) into (8) and (9) gives T i v 2 p A m ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11þ k p T 2 i v2 p þ 1 f m ¼ arctanð T i v g Þ v g ð12þ where v p is given in (5) and v g, according to (10), is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 u k 4 v g ¼ k 2 u p 2 þ p þ 4 k 2 t t p ð13þ For a specified gain margin A m, the parameters k p and T i satisfying (11) and (5) can be solved numerically. The solutions for typical specifications of A m ¼ 2, 3, 4, 5, 6 are shown in Fig. 4 and are called gain-margin curves. When k p! 0, then T i! 1. This is the point (0, 1). When T i!þ1, then k p! k pa with k A p ¼ p 2A m T 2 i From this, together with (5), the relationship between k p and T i can be solved numerically, which is shown in Fig. 3 as the curve c. Note that when T i!þ1, then v p! p/2 and k p! p/2; when k p! 0, then v p! 0 and T i! 1. Obviously, the filled area in Fig. 3 corresponds to (4), which gives the stability region. 3.3 Achievable stability margins The well-known gain margin A m and phase margin f m are defined as 1 A m ¼ ð8þ jlð jv p Þj 304 f m ¼ arg½lð jv g ÞŠ þ p ð9þ Fig. 4 Relationship between stability margins A m and f m and control parameters k p and T i

This is the intersection point of the gain-margin curve and the horizontal axis. Similarly, for a specified phase margin f m, the parameters k p and T i satisfying (12) and (13) can be solved numerically as well. The solutions for typical specifications of f m ¼ p/6, p/4, p/3 are also shown in Fig. 4 and are called phase-margin curves. When T i!þ1, then k p! 0ork p! k pf with k f p ¼ p 2 f m This is the right-side endpoint of the phase-margin curve. The curve for A m ¼ 1orf m ¼ 0 is the curve c shown in Fig. 3. So far, the system for a specified gain margin or phase margin has been designed. When both margins are specified, the parameters of C (s) can be obtained from the intersection point of the relevant gain-margin curve and phase-margin curve in Fig. 4. However, no arbitrary A m and f m can be achieved simultaneously. There is a constraint on the achievable stability margins. Theorem 2: The achievable stability margins A m and f m satisfy the following constraint: f m p 2 1 1 A m Proof: For the arbitrary pair (A m, f m ), there exists either a unique pair (k p, T i) or no pair (k p, T i) to meet them, depending on whether or not the relevant gain-margin and phasemargin curves intersect in Fig. 4. They intersect with each other when k pa k pf, that is p 2A m p 2 f m where the ¼ is satisfied when T i!þ1. This gives the condition in the theorem. A The achievable stability margins are shown in the filled area of Fig. 5 (including the curve). When A m approaches þ1, f m approaches p/2. When the system has been designed for the specified stability margins A m and f m, the uncertainties of the process parameters are determined too. Assuming that there exists an uncertainty k D in the process gain k (k D þ k. 0), according to (2) and the definition of the gain margin, the system is robustly stable if stable when 1 t D t, f m v g 4 An illustrative example Consider the following widely-studied process [2 4]: GðsÞ ¼ 1 s e 5s The parameter l in F(s) is set as l ¼ 1. In addition, the GPM are chosen as A m ¼ 3 and f m ¼ p/4, as often used in the literature. It can be found that k p 0.5 and 1/T i 0.15 from Fig. 4 and then k p ¼ 0.1 and T i ¼ 33.3 from (2). Simulation results with a unit-step input r(t) and a step disturbance d(t) ¼ 20.1, acting at t ¼ 20 s, are shown in Figs. 6 and 7 for the nominal case and the case with uncertainty t D ¼ 1.4 in the delay, respectively. When A m ¼ 2.3 and f m ¼ p/5.6, the disturbance response is much faster with a smaller dynamic error. The sub-ideal response generated from the scheme in [4] with a ¼ 4, which can also be obtained from other control schemes, for example in [3, 13, 18], is shown in Figs. 6 and 7 for comparison. The sub-ideal response is the fastest but the robustness is the worst. This is the well-known trade-off 1, k D k, A m 1 Similarly, if there exists an uncertainty t D in the process dead time t (t D þ t 0), then the system is robustly Fig. 5 Achievable stability margins A m and f m Fig. 6 Nominal responses a Output y b Control action u 305

6 Acknowledgment This work was supported in part by the EPSRC under Grant No. EP/C005953/1. 7 References Fig. 7 Robust responses with t D ¼ 1.4 a Output y b Control action u between disturbance rejection and robustness. These two can be easily compromised for the sub-ideal response by tuning a single parameter, for example the a for the scheme in [4]. However, for the PI control case, it is not easy to find a simple rule to determine the GPM. From this point of view, it is easier to use a scheme that generates a sub-ideal disturbance response. 5 Conclusion This paper discusses the stability region of integral processes with dead time controlled by PI controllers and then designs the control parameters according to the specified GPM. The constraint on the achievable stability margins in this system is also obtained. 1 Watanabe, K., and Ito, M.: A process-model control for linear systems with delay, IEEE Trans. Autom. Contr., 1981, 26, (6), pp. 1261 1269 2 Åström, K.J., Hang, C.C., and Lim, B.C.: A new Smith predictor for controlling a process with an integrator and long dead-time, IEEE Trans. Autom. Contr., 1994, 39, (2), pp. 343 345 3 Normey-Rico, J.E., and Camacho, E.F.: Robust tuning of dead-time compensators for processes with an integrator and long dead-time, IEEE Trans. Autom. Contr., 1999, 44, (8), pp. 1597 1603 4 Zhong, Q.-C., and Li, H.X.: Two-degree-of-freedom PID-type controller incorporating the Smith principle for processes with deadtime, Ind. Eng. Chem. Res., 2002, 41, (10), pp. 2448 2454 5 Lu, X., Yang, Y.-W., Wang, Q.-G., and Zheng, W.-X.: A double twodegree-of-freedom control scheme for improved control of unstable delay processes, J. Process. Contr., 2005, 15, (5), pp. 605 614 6 Rivera, D.E., Morari, M., and Skogestad, S.: Internal model control 4: PID controller design, Ind. Eng. Chem. Process. Des. Dev., 1986, 25, pp. 252 265 7 Tyreus, B.D., and Luyben, W.L.: Tuning PI controllers for integrator/dead-time processes, Ind. Eng. Chem. Res., 1992, 31, pp. 2625 2628 8 Luyben, W.L.: Tuning proportional-integral-derivative controllers for integrator/deadtime processes, Ind. Eng. Chem. Res., 1996, 35, pp. 3480 3483 9 Poulin, E., and Pomerleau, A.: PI settings for integrating processes based on ultimate cycle information, IEEE Trans. Contr. Syst. Tech., 1999, 7, (4), pp. 509 511 10 Wang, L., and Cluett, W.R.: Tuning PID controllers for integrating processes, IEE Proc., Control Theory Appl., 1997, 144, (5), pp. 385 392 11 Visioli, A.: Optimal tuning of PID controllers for integral and unstable processes, IEE Proc., Control Theory Appl., 2001, 148, (2), pp. 180 184 12 Chidambaram, M., and Sree, R.P.: A simple method of tuning PID controllers for integrator/dead time processes, Comput. Chem. Eng., 2003, 27, pp. 211 215 13 Zhong, Q.-C., and Normey-Rico, J.E.: Control of integral processes with dead time. Part I: a disturbance observer-based 2DOF control scheme, IEE Proc., Control Theory Appl., 2002, 149, (4), pp. 285 290 14 Zhong, Q.-C., and Mirkin, L.: Control of integral processes with dead-time. Part II: quantitative analysis, IEE Proc., Control Theory Appl., 2002, 149, (4), pp. 291 296 15 Zhong, Q.-C.: Control of integral processes with dead time. Part III: deadbeat disturbance response, IEEE Trans. Autom. Contr., 2003, 48, (1), pp. 153 159 16 Su, W., Qiu, L., and Chen, J.: Fundamental performance limitations in tracking sinusoidal signals, IEEE Trans. Autom. Contr., 2003, 48, pp. 1371 1380 17 Silva, G.J., Datta, A., and Bhattacharyya, S.P.: New results on the synthesis of PID controllers, IEEE Trans. Autom. Contr., 2002, 47, (2), pp. 241 251 18 Zhang, W., and Sun, Y.: Modified Smith predictor for controlling integrator/time delay process, Ind. Eng. Chem. Res., 1996, 35, pp. 2769 2772 306