A Method for Designing Modified PID Controllers for Time-delay Plants and Their Application

Transcription

1 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application 53 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application Kou Yamada 1, Takaaki Hagiwara, and Yosuke Shimizu 3, Non-members ABSTRACT In this paper, we examine a method for designing modified proportional-integral-derivative (PID controllers for use in stable and/or minimum-phase time-dela plants. The PID controller structure is the one used most widel in industrial applications. Recentl, the parameterization of all stabilizing PID controllers has been considered. However, no method which guarantees the stabilit of a PID control sstem for an stable and/or minimum-phase time-dela plants, and in which the admissible sets of P-, I- and D-parameters are independent from each other, has been published. In this paper, we propose a method for designing modified PID controllers such that the modified PID controller makes the feedback control sstem for an stable and/or minimum-phase timedela plant stable and the admissible sets of P-, I- and D-parameters are independent from each other. Numerical examples and application in a heat flow experiment are shown to illustrate the effectiveness of the proposed method. Kewords: PID control, time-dela plant, stabilit, admissible set, heat flow experiment 1. INTRODUCTION The proportional-integral-derivative (PID controller is the most widel used controller structure in industrial applications [1 3]. Its structural simplicit and abilit to solve man practical control problems have contributed to this wide acceptance. Several papers on tuning methods for PID parameters have been published [4 14] but these methods do not guarantee the stabilit of a closed-loop sstem. Several methods of designing PID controllers to guarantee the stabilit of closed-loop sstems have been proposed [15 18]. However, it is difficult to tune PID parameters to meet control specifications using these methods. If the admissible sets of PID parameters that would guarantee the stabilit of a closed-loop Manuscript received on Jul, 007 ; revised on November 6, ,,3 The authors are with department of Mechanical Sstem Engineering, Gunma Universit Tenjincho, Kiru Japan, amada@me.gunma-u.ac.jp,m06m430@gs.eng.gunma-u.ac.jp and a3m040@ug.eng.gunmau.ac.jp sstem can be determined, we can easil design stabilizing PID controllers to meet control specifications. Obtaining admissible sets of PID parameters that guarantee the stabilit of a closed-loop sstem is a parameterization problem [3, 19, 0]. The parameterization of all stabilizing PID controllers is considered in [3, 19, 0]. However, the difficult remaining with these methods is that the admissible sets of P-, I- and D-parameters in [3, 19, 0] are related to each other. In other words, if the P-parameter is changed, then the admissible sets of I- and D-parameters also change. From a practical point of view, it is desirable for the admissible sets of P-, I- and D-parameters to be independent from each other. Yamada and Moki first approached this problem and proposed a method for designing modified PI controllers for an minimum-phase sstem such that the admissible sets of P- and I-parameters were independent from each other [1]. Yamada expanded the results in [1] and proposed a method for designing modified PID controllers for minimum-phase plants such that the admissible sets of P-, I- and D-parameters were independent from each other []. For stable plants, a method for designing modified PID controllers was considered in [3, 4]. However, no paper gives a method for designing modified PID controllers for an stable and/or minimum-phase time-dela plant such that the stabilit of the control sstem is guaranteed and the admissible sets of P-, I- and D-parameters are independent. In this paper, we expand the results in [1 4] and propose a method for designing modified PID controllers such that the controller makes the feedback control sstem stable for an stable and/or minimumphase time-dela plant and the admissible sets of P-, I- and D-parameters are independent. This paper is organized as follows: In Section., the problem considered in this paper is clarified. In Section 3., we describe the basic concepts of designing modified PID controllers such that the controller makes the feedback control sstem stable for an stable and/or minimum-phase time-dela plant and the admissible sets of P-, I- and D-parameters are independent. In Section 4. and Section 5., we propose methods of designing modified PID controllers for use in stable plants and minimum-phase plants, respectivel. In Section 6., a simple numerical example is shown. In Section 7., an application in a heat flow experiment is

2 54 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 Februar 008 shown to illustrate the effectiveness of the proposed method. Notation R The set of real numbers. R(s The set of real rational functions with s. RH The set of stable proper real rational functions. U The set of unimodular functions on RH. That is, U(s U implies both U(s RH and U 1 (s RH.. PROBLEM FORMULATION Consider the feedback control sstem in Fig. 1. Here, G(se st is the single-input/single-output time-dela plant, G(s R(s is assumed to be strictl proper and have no zero on the origin, T > 0 is the time dela, C(s is the controller, r R is the reference input, u R is the control input, R is the output, d R is disturbance and e R is the error. r + à e Fig.1: C(s d + + u G(se àst Feedback control sstem. When the controller C(s can be written as C(s = a P + a I s + a Ds, (1 the controller C(s is a PID controller [1 3, 19, 0], where a P R is the P-parameter, a I R is the I- parameter and a D R is the D-parameter. a P, a I and a D are set so that the feedback control sstem in Fig. 1 has desirable characteristics such as stead state control and transient control. For simplicit, we call C(s in (1 the conventional PID controller. The transfer function from r to in Fig. 1 is written as = G(s ( a P + a I s + a D s e st 1 + G(s ( a P + a I s + a D s r. ( e st It is obvious that when a P, a I and a D are set at random, the stabilit of the feedback control sstem in Fig. 1 is not guaranteed. If a stabilizing PID controller C(s in (1 for G(s exists, we can use the results in [3, 19, 0] to design a PID controller C(s in (1 to stabilize the feedback control sstem in Fig. 1. In other words, the admissible sets of a P, a I and a D able to stabilize the feedback control sstem in Fig. 1 can be obtained using results from [3, 19, 0]. According to these authors, the admissible sets of a P, a I and a D are related to each other i.e. if a P is changed then the admissible sets of a I and a D also change. However, from a practical point of view, it is desirable for the admissible sets of a P, a I and a D to be independent. Furthermore, the transfer function in ( generall has an infinite number of poles. When the transfer functions from r to and from d to have an infinite number of poles, it is difficult to obtain desirable input output, disturbance attenuation and other such control characteristics. The purpose of this paper is to propose a method for designing modified PID controllers C(s in Fig. 1, with the following characteristics. 1. The stabilit of the feedback control sstem in Fig. 1 is guaranteed..the transfer functions from r to and from d to in Fig. 1 have a finite number of poles. 3.The admissible sets of P-parameters a P, I- parameters a I and D-parameters a D that guarantee stabilit of the feedback control sstem in Fig. 1 are independent. 4.The roles of P-parameters a P, I-parameters a I and D-parameters a D in the modified PID controller are equivalent to those of the conventional PID controller in (1. 3. BASIC CONCEPTS In this section, we describe the basic concept of designing modified PID controllers C(s with the characteristics described in Section. In order to design modified PID controllers C(s that can be applied to an stable and/or minimumphase time-dela plants and ensure the transfer functions from r to and d to have a finite number of poles, we adopted the parameterization of all stabilizing modified Smith predictors for an stable and/or minimum-phase time-dela plant in [5]. The reason for not adopting the Smith predictor, which is an effective time-dela compensator for a time-dela plant, in [6] but adopting the parameterization of all stabilizing modified Smith predictors in [5] is that the Smith predictor in [6] cannot be applied to unstable time-dela plants but the parameterization of all stabilizing modified Smith predictors in [5] can be applied to an stable and/or minimum-phase time-dela plant. The basic concept of designing modified PID controllers C(s with the characteristics described in Section., using the results in [5], is summarized as follows. 1. When G(s is stable. According to [5], the parameterization of all stabilizing modified Smith predictors for stable timedela plants is written as C(s = Q(s, (3 st 1 Q(sG(se where Q(s RH is an function. Using the controller in (3, the transfer functions from r to and d to in Fig. 1 are written as = Q(sG(se st r (4

3 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application 55 and = (1 Q(sG(se st G(se st d, (5 respectivel. Assuming that Q(s RH, the transfer functions in (4 and (5 have a finite number of poles. In addition, it is obvious that the stabilit of the feedback control sstem in Fig. 1 is guaranteed. In order to design modified PID controllers with the characteristics described insection., the free parameter Q(s in (3 is set so that C(s in (3 has the same characteristics as the conventional PID controller C(s in (1. Thus, we now describe the role of the conventional PID controller C(s in (1, in order to clarif the conditions that the modified PID controller C(smust satisf. The P-parameters a P, I-parametersa I and D-parameters a D are determined according to and a P = lim s { s d ds ( } 1 s C(s, (6 a I = lim s 0 {sc(s} (7 d a D = lim {C(s}, (8 s ds respectivel, from (1, using C(s. Therefore, if the controller C(s in (3 holds (6, (7 and (8, the role of controller C(s in (3 is equivalent to that of the conventional PID controller C(s in (1. In other words, we can design a stabilizing modified PID controller such that the role of controller C(s in (3 is equivalent to that of the conventional PID controller C(s in (1.. When G(s is of minimum-phase. According to [5], the parameterization of all stabilizing modified Smith predictors for minimumphase time-dela plants is written as C(s = C f (s, (9 1 C f (sg(se st C f (s = Ḡu(s ( 1 + Q(s, (10 G u (s G u (s where Q(s RH is an function, Ḡ u (s U satisfies Ḡ u (s i = 1 G s (s i e s it (i = 1,..., n, (11 n is the number of unstable poles of G(s, s i (i = 1,..., n are the unstable poles of G(s, G s (s is a stable minimum-phase function of G(s, i.e. wheng(s is factorized as G(s = G u (sg s (s, (1 G u (s is the unstable biproper minimum-phase function and G s (s is the stable minimum-phase function. Using the controller in (9, the transfer functions from r to and d to are written as = Ḡu(s and = ( 1 + Q(s G u (s G s (se st r (13 { ( 1 Ḡu(s 1 + Q(s G u (s G s (se st } G(se st d, (14 respectivel. From Q(s RH, Ḡ u (s U, G s (s RH and 1/G u (s RH, the transfer functions in (13 and (14 have finite numbers of poles. In addition, it is obvious that the stabilit of the feedback control sstem in Fig. 1 is guaranteed. If the controller C(s in (9 holds (6, (7 and (8, the role of controller C(s in (3 is equivalent to that of the conventional PID controller C(s in (1. In other words, we can design a stabilizing modified PID controller such that the role of controller C(s in (9 is equivalent to that of the conventional PID controller C(s in (1. In Section 4. and Section 5., we use the ideas discussed above to describe a method for designing the modified PID controller C(s in (3 or (9 that works as a modified PID controller. In the following, we call C(s 1. the modified P controller if C(s in (3 or (9 satisfies (6,.the modified I controller if C(s in (3 or (9 satisfies (7, 3. the modified D controller if C(s in (3 or (9 satisfies (8, 4. the modified PI controller if C(s in (3 or (9 satisfies (6 and (7, 5.the modified PD controller if C(s in (3 or (9 satisfies (6 and (8, 6.the modified PID controller if C(s in (3 or (9 satisfies (6, (7 and (8. 4. MODIFIED PID CONTROLLER, WHEN G(S IS STABLE In this section, we describe a method for designing the modified PID controller C(s in (3 that works as a modified PID controller Modified P controller a modified P controller C(s in (3 that holds (6, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable time-dela plant G(se st. The modified P controller C(s satisfing (6 is written as (3, where Q(s = a P. (15

4 56 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 Februar 008 Since Q(s in (15 is included in RH, the controller C(s in (3 with (15 makes the feedback control sstem in Fig. 1 stable for an stable time-dela plant G(se st independent from a P. 4. Modified I controller a modified I controller C(s in (3 that holds (7, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable time-dela plant G(se st. The modified I controller C(s satisfing (7 is written as (3, where q 1 = τ 1 G(0 G (0 Q(s = q 0 + q 1 s + τ 1 s, (16 q 0 = G(0, (17 ( 1 T G(0 + d a I ds {G(s} (18 and τ i R > 0(i = 0, 1. For τ i > 0(i = 0, 1, Q(s in (15 is included in RH. This implies that the controller C(s in (3 with (16 makes the feedback control sstem in Fig. 1 stable for an stable timedela plant G(se st independent from a I Modified D controller a modified D controller C(s in (3 that holds (8, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable time-dela plant G(se st. The modified D controller C(s satisfing (8 is written as (3, where Q(s = a D s. (19 Since Q(s in (19 is improper, Q(s in (19 is not included in RH. In order for Q(s to be included in RH, (19 must be modified according to Q(s = a Ds 1 + τ D s, (0 where τ D R > 0. For τ D > 0 in (0, Q(s in (0 is included in RH. This implies that the controller C(s in (3 with (0 makes the feedback control sstem in Fig. 1 stable for an stable time-dela plant G(se st independent from a D Modified PI controller a modified PI controller C(s in (3 that holds (6 and (7, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable time-dela plant G(se st. The modified PI controller C(s satisfing (6 and (7 is written as (3, where q 1 = τ 1 G(0 G (0 Q(s = q 0 + q 1 s + q s + τ 1 s + τ s, (1 q 0 = G(0, ( { 1 T G(0 + d } a I ds (G(s, (3 q = τ a P (4 and τ i R > 0(i = 0, 1,. For τ i > 0(i = 0, 1,, Q(s in (1 is included in RH. This implies that the controller C(s in (3 with (1 makes the feedback control sstem in Fig. 1 stable for an stable time-dela plant G(se st independent from a P and a I Modified PD controller a modified PD controller C(s in (3 that holds (6 and (8, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable time-dela plant G(se st. The modified PD controller C(s satisfing (6 and (8 is written as (3, where Q(s = a P + a D s. (5 Since Q(s in (5 is improper, Q(s in (5 is not included in RH. In order for Q(s to be included in RH, (5 must be modified according to Q(s = a P + a Ds 1 + τ D s, (6 where τ D R > 0. For τ D > 0 in (6, Q(s in (6 is included in RH. This implies that the controller C(s in (3 with (6 makes the feedback control sstem in Fig. 1 stable for an stable time-dela plant G(se st independent from a P and a D.

5 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application Modified PID controller a modified PID controller C(s in (3 that holds (6, (7 and (8, makes the feedback control sstem in Fig. 1 stable and is applicable to an stable timedela plant G(se st. The modified PID controller C(s satisfing (6, (7 and (8 is written as (3, where q 1 = Q(s = q 0 + q 1 s + q s + τ 1 s + τ s + q 3s, (7 τ 1 q 0 = G(0, (8 G(0 q 3 τ { 0 1 G T G(0 + d } (0 a I ds (G(s, (9 q = τ a P, (30 q 3 = a D (31 and τ i R > 0(i = 0, 1,. Since Q(s in (7 is improper, Q(s in (7 is not included in RH. In order for Q(s to be included in RH, (7 must be modified according to Q(s = q 0 + q 1 s + q s + τ 1 s + τ s + q 3s 1 + τ D s, (3 where τ D R > 0. For τ i > 0(i = 0, 1, and τ D > 0 in (3, Q(s in (3 is included in RH. This implies that the controller C(s in (3 with (3 makes the feedback control sstem in Fig. 1 stable for an stable time-dela plant G(se st independent from a P, a I and a D. 5. MODIFIED PID CONTROLLER, WHEN G(S IS OF MINIMUM-PHASE In this section, we describe a method for designing the modified PID controller C(s in (9 that works as a modified PID controller Modified P controller a modified P controller C(s in (9 that holds (6, makes the feedback control sstem in Fig. 1 stable and is applicable to an minimum-phase time-dela plant G(se st. The modified P controller C(s satisfing (6 is written as (9, where Q(s = lim u (s s Ḡ u (s a P G u (s = const. (33 Since Q(s in (33 is included in RH, the controller C(s in (9 with (33 makes the feedback control sstem in Fig. 1 stable for an minimum-phase timedela plant G(se st independent from a P. 5. Modified I controller a modified I controller C(s in (9 that holds (7, makes the feedback control sstem in Fig. 1 stable and is applicable to an minimum-phase time-dela plant G(se st. The modified I controller C(s satisfing (7 is written as (9, where q 1 = Q(s = q 0 + q 1 s + τ 1 s, (34 ( Gu (0 q 0 = Ḡ u (0G s (0 G u(0, (35 ( d Ḡ u (0G s (0 ds (G u(s G u (0 d (Ḡu (s d ds ds (G(s Ḡ u(0g s (0 Ḡ u (0G { s(0 τ 1 + τ } 0 (τ 1 T G u (0 Ḡ u (0G s (0 a I Ḡ u (0G s(0 (36 and τ i R > 0(i = 0, 1. For τ i > 0(i = 0, 1, Q(s in (34 is included in RH. This implies that the controller C(s in (9 with (34 makes the feedback control sstem in Fig. 1 stable for an minimumphase time-dela plant G(se st independent from a I Modified D controller a modified D controller C(s in (9 that holds (8, makes the feedback control sstem in Fig. 1 stable and is applicable to an minimum-phase time-dela plant G(se st. The modified D controller C(s satisfing (8 is written as (9, where and q 0 = a D Q(s = q 0 s (37 lim s u (s. (38 Ḡ u (s Since Q(s in (37 is improper, Q(s in (37 is not included in RH. In order for Q(s to be included in RH, (37 must be modified according to Q(s = a Ds 1 + τ D s, (39

6 58 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 Februar 008 where τ D R > 0. For τ D > 0 in (39, Q(s in (39 is included in RH. This implies that the controller C(s in (9 with (39 makes the feedback control sstem in Fig. 1 stable for an minimum-phase time-dela plant G(se st independent from a D Modified PI controller a modified PI controller C(s in (9 that holds (6 and (7, makes the feedback control sstem in Fig. 1 stable and is applicable to an minimum-phase timedela plant G(se st. The modified PI controller C(s satisfing (6 and (7 is written as (9, where q 1 = Q(s = q 0 + q 1 s + q s + τ 1 s + τ s, (40 ( Gu (0 q 0 = Ḡ u (0G s (0 G u(0, (41 ( d Ḡ u (0G s (0 ds (G u(s G u (0 d (Ḡu (s d ds ds (G(s Ḡ u(0g s (0 Ḡ u (0G { s(0 τ 1 + τ } 0 (τ 1 T G u (0 Ḡ u (0G s (0 q = τ lim u (s s Ḡ u (s a P G u (s (43 and τ i R > 0(i = 0, 1,. For τ i > 0(i = 0, 1,, Q(s in (40 is included in RH. This implies that the controller C(s in (9 with (40 makes the feedback control sstem in Fig. 1 stable for an minimumphase time-dela plant G(se st independent from a P and a I. and [ G q 0 = lim u (s s Ḡ u (s a P G u (s + a Ds G u (s Ḡ u (s d }] ds (G u(s q 1 = a D lim s G u (s d (Ḡu (s ds Ḡ u (s (45 u (s. (46 Ḡ u (s Since Q(s in (44 is improper, Q(s in (44 is not included in RH. In order for Q(s to be included in RH, (44 must be modified according to Q(s = q 0 + q 1s 1 + τ D s, (47 where τ D R > 0. For τ D > 0 in (47, Q(s in (47 is included in RH. This implies that the controller C(s in (9 with (47 makes the feedback control sstem in Fig. 1 stable for an minimum-phase time-dela plant G(se st independent from a P and a D Modified PID controller a modified PID controller C(s in (9 that holds a I Ḡ u (0G s(0, (6, (7 and (8, makes the feedback control sstem in (4 Fig. 1 stable and is applicable to an minimum-phase time-dela plant G(se st. The modified PID controller C(s satisfing (6, (7 and (8 is written as (9, where Q(s = q 0 + q 1 s + q s + τ 1 s + τ s + q 3s, (48 ( Gu (0 q 0 = Ḡ u (0G s (0 G u(0, ( Modified PD controller a modified PD controller C(s in (9 that holds (6 and (8, makes the feedback control sstem in Fig. 1 stable and is applicable to an minimum-phase timedela plant G(se st. The modified PD controller C(s satisfing (6 and (8 is written as (9, where Q(s = q 0 + q 1 s, (44 q 1 = ( d Ḡ u (0G s (0 ds (G u(s G u (0 d (Ḡu (s d ds ds (G(s Ḡ u(0g s (0 Ḡ u (0G { s(0 (τ 1 + τ } 0 (τ 1 T G u (0 Ḡ u (0G s (0 a I Ḡ u (0G s(0 + a D lim s u (s Ḡ u (s, (50

7 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application 59 [ G q = lim u (s s Ḡ u (s a P G u (s G u (s d (Ḡu (s ds Ḡ u (s d }] ds (G u(s, (51 + a Ds G u (s Ḡ u (s and τ D = 1. Using the parameters mentioned above, the modified PID controller C(s is designed according to (9 and (53. The step response of the control sstem using the modified PID controller C(s is shown in Fig.. Figure shows that the modified PID controller C(s q 3 = a D lim s u (s Ḡ u (s ( and τ i R > 0(i = 0, 1,. Since Q(s in (48 is improper, Q(s in (48 is not included in RH. In order for Q(s to be included in RH, (48 must be modified according to Q(s = q 0 + q 1 s + q s + τ 1 s + τ s + q 3s 1 + τ D s, (53 where τ D R > 0. For τ i > 0(i = 0, 1, and τ D > 0 in (53, Q(s in (53 is included in RH. This implies that the controller C(s in (9 with (53 makes the feedback control sstem in Fig. 1 stable for an minimum-phase time-dela plant G(se st independent from a P, a I and a D. 6. NUMERICAL EXAMPLE In this section, we use a numerical example to show the effectiveness of the proposed method. Consider the problem of designing a modified PID controller C(s for an unstable minimum-phase timedela plant G(se st written as G(se st = where T = 0.15[sec] and G(s = 100 s 3 0.1s 0.7s 0. e 0.15s, ( s 3 0.1s 0.7s 0.. (55 Note that G(se st in (54 has no stabilizing conventional PID controller. Since G(s in (55 is unstable and of minimumphase, we design a modified PID controller using the method described in Section 5.6 a P, a I and a D are settled b a P = 55 a I = a D = (56 q 0, q 1, q and q 3 are determined b (49, (50, (51 and (5, respectivel, where = 1 τ 1 = τ = 1 ( Fig.: Step response of the control sstem using a modified PID controller. makes the feedback control sstem stable. On the other hand, using a conventional PID controller with (56, the step response of the control sstem is shown in Fig. 3. Figure 3 shows that the 1 x Fig.3: Step response of the control sstem using a conventional PID controller. conventional PID control sstem is unstable. This is obvious, since no stabilizing conventional PID controller for the plant G(se st in (54 exists. Contrar to this, the stabilit of the modified PID control sstem is guaranteed to be independent from a P, a I and a D. Therefore, we find that the modified PID controller can be applied to plants that have no stabilizing conventional PID controller.

8 60 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 Februar 008 When a P, a I and a D in the modified PID controller are varied, the step responses can be compared. First, the step responses for various a P, with a P = 30, a P = 40 and a P = 50, are shown in Fig. 4. Here, the solid line, dotted line and broken line a P = 30 a P = 40 a P = 50 dotted line and broken line show the step responses of the modified PID control sstem using a I = , a I = and a I = , respectivel. Figure 5 shows that as the value of a I increased, the overshoot became smaller and convergence became more rapid. Since these characteristics are equivalent to those of the conventional PID controller, the role of the I-parameter a I in the modified PID controller is equivalent to that in a conventional PID controller. Thirdl, the step responses for various a D, with a D = 1, a D = 50 and a D = 100, are shown in Fig. 6. Here, the solid line, dotted line and broken Fig.4: Step response using modified P controller with a P = 30, 40, 50. show the step responses of the modified PID control sstem using a P = 30, a P = 40 and a P = 50, respectivel. Figure 4 shows that as the value of a P increased, the overshoot became larger and the rise time became shorter. Since these characteristics are equivalent to those of the conventional PID controller, the role of the P-parameter a P in the modified PID controller is equivalent to that in a conventional PID controller. Secondl, the step responses for various a I, with a I = , a I = and a I = , are shown in Fig. 5. Here, the solid line, a I = 0:00001 a I = 0:00005 a I = 0: Fig.5: Step response using modified I controller with a I = , , à a D = 1 a D = 50 a D = 100 à t(s Fig.6: Step response using modified D controller with a D = 1, 50, 100. line show the step responses of the modified PID control sstem using a D = 1, a D = 50 and a D = 100, respectivel. Figure 6 shows that as the value of a D increased, the response was smoothed. Since this characteristic is equivalent to that of the conventional PID controller, the role of the D-parameter a D in the modified PID controller is equivalent to that in a conventional PID controller. Thus, we have shown that we can easil design a stabilizing modified PID controller for an unstable minimum-phase time-dela plant, with the same characteristics as a conventional PID controller, and guarantee the stabilit of the feedback control sstem. 7. APPLICATION IN A HEAT FLOW EX- PERIMENT In this section, we show an application of the proposed modified PID controller in a heat flow experiment, to illustrate its effectiveness. The heat flow apparatus is shown in Fig. 7. The heat flow apparatus comprises a duct equipped with a heater and blower at one end and three temperature sensors located along the duct, as shown in Fig. 7. V h and V b denote the voltage to the heater and blower, respectivel. S 1, S and S 3 are terminals for measurement of temperatures at Sensor 1, Sensor and Sensor 3. We denote temperature measurements at Sensor i as T i. The problem considered in this section is to design

9 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application 61 Blower Heater Sensor1 Sensor Sensor Vh Vb 570 S1 S S3 100 = T3[ î C] Fig.7: Apparatus used in the heat flow experiment. a modified PID controller to make T 3, the temperature at Sensor 3, remain stead at 30[deg]. When we set V b = 5[V ], we found that the transfer function from V h to T 3 could be written as T 3 = s e 1.5s V h. (58 T 3 and V h are considered the output and control input u, respectivel, in the control sstem in Fig. 1. From (58, G(se st in Fig. 1 can be written as G(se st = where T = 1.5[sec] and G(s = s e 1.5s, ( s. (60 The reference input r in Fig. 1 is set so that r(t = 30[deg]. Since G(s in (60 is stable, we designed a modified PID controller using the method described in Section 4. a P, a I and a D are set according to a P = a I = a D = (61 q 0, q 1, q and q 3 are determined b (8, (9, (30 and (31, respectivel, where = 1 τ 1 = τ = 1 (6 and τ D = 1. Using the parameters given above, the modified PID controller C(s was designed according to (3 and (3. The experimental step response of the control sstem using this modified PID controller C(s is shown in Fig. 8. Figure 8 shows that the output = T 3 followed the step reference input r = 30 with negligible stead-state error and high convergence speed, compared to open-loop response. When a P, a I and a D in the modified PID controller are varied, the step responses can be examined. First, the step responses for various a P, with Fig.8: Experimental step response using the modified PID controller. = T3[ î C] a P = 1 a P = a P = t(s Fig.9: Step response using modified P controller with a P = 1,, 3. a P = 1, a P = and a P = 3, are shown in Fig. 9. Here, the solid line, dotted line and broken line show the step response of the modified PID control sstem using a P = 1, a P = and a P = 3, respectivel. Figure 9 shows that as the value of a P increased, the overshoot became larger and the rise time became shorter. Since those characteristics are equivalent to those of the conventional PID controller, the role of the P-parameter a P in the modified PID controller is equivalent to that in the conventional PID controller. Secondl, the step responses for various a I, with a I = 0.1, a I = 0. and a I = 0.3, are shown in Fig. 10. Here, the solid line, dotted line and broken line show the step response of the modified PID control sstem using a I = 0.1, a I = 0. and a I = 0.3, respectivel. Figure 10 shows that as the value of a I increased, convergence became more rapid. Since this characteristic is equivalent to that of the conventional PID controller, the role of the I-parameter a I in the modified PID controller is equivalent to that in the conventional PID controller. Thirdl, the step

10 6 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.6, NO.1 Februar 008 = T3[ î C] a I = 0:1 a I = 0: a I = 0: t(s Fig.10: Step response using modified I controller with a I = 0.1, 0., 0.3. responses for various a D, with a D = 1, a D = and a D = 3, are shown in Fig. 11. Here, the solid line, = T3[ î C] a D = 1 a D = a D = Fig.11: Step response using modified D controller with a D = 1,, 3 dotted line and broken line show the step response of the modified PID control sstem using a D = 1, a D = and a D = 3, respectivel. Figure 11 shows that as the value of a D increased, the response was smoothed. Since this characteristic is equivalent to that of the conventional PID controller, the role of the D-parameter a D in the modified PID controller is equivalent to that in the conventional PID controller. On the basis of these results, we found that the proposed modified PID controller provided effective control in this heat flow experiment. 8. CONCLUSIONS In this paper, we proposed a method for designing modified PID controllers such that the modified PID controller makes the feedback control sstem for an stable and/or minimum-phase time-dela plants asmptoticall stable and the admissible sets of P-, I- and D-parameters are independent from each other. A numerical example and application in a heat flow experiment were also shown to illustrate the effectiveness of the proposed method. References [1] N. Suda, PID Control, Asakura Shoten, Toko, 199 (in Japanese. [] K. Astrom and T. Hagglund, PID controllers: Theor design, and tuning, Instrument Societ of America, North Carolina, [3] A. Datta, M. T. Ho and S. P. Bhattachara, Structure and Snthesis of PID Controllers, Springer-Verlag, London, 000. [4] J.G. Zieglae and N.B. Nicholes, Optimum settings for automatic controllers, Trans. ASME, 64, pp , 194. [5] P. Hazebroek and B.L. van der Warden, The Optimal Adjustment of Regulators, Trans. ASME, 7, pp , [6] P. Hazebroek and B.L. van der Warden, Theoretical Considerations on the Optimal Adjustment of Regulators, Trans. ASME, 7, pp , [7] W.A. Wolf, Controller Setting for Optimum Control, Trans. ASME, 73, pp , [8] K.L. Chien, J.A. Hrones and J.B. Reswick, On the Automatic Control of Generalized Passive Sstems, Trans. ASME, 74, pp , 195. [9] G.H. Cohen and G.A. Coon, Theoretical Consideration of Retarded Control, Trans. ASME, 75, pp , [10] A.M. Lopez, J.A. Miller, C.L. Smith and P.W. Murrill, Tuning Controllers with Error-Integral Criteria, Instrumentation Technolog, 14, pp.5 6, [11] J.A. Miller, A.M. Lopez, C.L. Smith and P.W. Murrill, A Comparison of Controller Tuning Techniques, Control Engineering, 14, pp.7 75, [1] T. Kitamori, A Method of Control Sstem Design Based upon Partial Knowledge About Controlled process, Transactions of the Societ of Instrument and Control Engineers, 15-4, pp , 1979 (in Japanese. [13] T. Kitamori, Design Method for PID Control Sstems, Journal of the Societ of the Instrument and Control Engineers, 19-4, pp , 1980 (in Japanese. [14] P. Cominos and N. Munro, PID Controllers: Recent Tuning Methods and Design to Specification, IEE Proceedings, 149, pp.46 53, 00. [15] F. Zheng, Q.G. Wang and T.H. Lee, On the Design of Multivariable PID Controllers via LMI Approach, Automatica, 38-3, pp , 00. [16] C. Lin, Q.G. Wang and T.H. Lee, An Improvement on Multivariable PID Controller Design via Iterative LMI Approach, Automatica, 40-3, pp , 004.

11 A Method for Designing Modified PID Controllers for Time-dela Plants and Their Application 63 [17] N. Viorel, M. Constantin, A. Dorel and C. Emil, Aspects of Pole Placement Technique in Smmetrical Optimum Method for PID Controller Design, Preprints of the 16th IFAC World Congress DVD-ROM, 005. [18] K. Tamura and K. Shimizu, Eigenvalue Assignment Method b PID Control for MIMO Sstem, Transactions of The Institute of Sstems, Control and Information Engineers, 19-5, pp.193 0, 006 (in Japanese. [19] J. Yang, Parameter Plane Control Design for a Two-tank Chemical Reactor Sstem, Journal of the Franklin Institute, Vol.331B, pp.61 76, [0] M. T. Ho, A. Datta and S. P. Bhattachara, A linear programming characterization of all stabilizing PID controllers, Proceedings of the American Control Conference 1997, pp , [1] K. Yamada and T. Moki, A design method for PI control for minimum phase sstems, Intelligent Engineering Sstems Through Artificial Neural Networks, Vol.13, pp , 003. [] K. Yamada, Modified PID controllers for minimum phase sstems and their practical application, Proceedings of The 005 Electrical Engineering/Electronics, Computer, Telecommunication, and Information Technolog (ECTI International Conference, Vol.II, 005. [3] K. Yamada, N. Matsushima and T. Hagiwara, A design method for modified PID controllers for stable plants, Proceedings of the 006 Electrical Engineering/Electronics, Computer, Telecommunication, and Information Technolog (ECTI International Conference, Vol.I of II, pp.13 16, Ubon Ratchathani, Thailand, 006. [4] K. Yamada, N. Matsushima and T. Hagiwara, A design method for modified PID controllers for stable plants, ECTI Transactions on Electrical Eng., Electronics, and Communications, Vol.5-1, pp.31 40, 007. [5] K. Yamada and N. Matsushima, A design method for Smith predictors for minimumphase time-dela plants, ECTI Transactions on Computer and Information Technolog, Vol.-, pp , 005. [6] O.J.M. Smith, A controller to overcome deadtime, ISA Journal, Vol. 6, pp. 8 33, [7] M. Morari and E. Zafiriou, Robust process control, PTR Prentice Hall, New Jerse, [8] M. Vidasagar, Control sstem snthesis A factorization approach, MIT Press, London, [9] H. Kimura, Is the Model a Good Controller? Perspectives on Brain Motor Control, Proceedings of CDC, 000. Kou Yamada was born in Akita, Japan, in He received B.S. and M.S. degrees from Yamagata Universit, Yamagata, Japan, in 1987 and 1989, respectivel, and the Dr. Eng. degree from Osaka Universit, Osaka, Japan in From 1991 to 000, he was with the Department of Electrical and Information Engineering, Yamagata Universit, Yamagata, Japan, as a research associate. Since 000, he has been an associate professor in the Department of Mechanical Sstem Engineering, Gunma Universit, Gunma, Japan. His research interests include robust control, repetitive control, process control, and control theor for inverse sstems and infinite-dimensional sstems. Dr. Yamada received the 005 Yokoama Award in Science and Technolog and the 005 Electrical Engineering/Electronics, Computer, Telecommunication, and Information Technolog International Conference (ECTI-CON005 Best Paper Award. Takaaki Hagiwara was born in Gunma, Japan, in 198. He received a B.S. degree in Mechanical Sstems Engineering from Gunma Universit, Gunma, Japan, in 006. He is a current M.S. candidate in Mechanical Sstems Engineering at Gunma Universit. His research interests include process control and PID control. Yosuke Shimizu was born in Gunma, Japan, in He received a B.S. degree in Mechanical Sstems Engineering from Gunma Universit, Gunma, Japan, in 007. His research interests include process control and PID control.