A Design Method for Modified PID Controllers for Stable Plants And Their Application

A Design Method for Modified PID Controllers for Stable Plants And Their Application 31 A Design Method for Modified PID Controllers for Stable Plants And Their Application Kou Yamada 1, Nobuaki Matsushima 2, and Takaaki Hagiwara 3, Non-members ABSTRACT In this paper, we examine a design method for modified PID (Proportional-Integral-Derivative) controllers for stable plants. PID controller structure is the most widel used one in industrial applications. Recentl the parametrization of all stabilizing PID controller has been considered. However no method has been published to guarantee the stabilit of PID control sstem for an stable plants and the admissible sets of P-parameter, I-parameter and D-parameter to guarantee the stabilit of PID control sstem are independent from each other. In this paper, we propose a design method for modified PID controllers such that the modified PID controller make the closed-loop sstem for an stable plants stable and the admissible sets of P-parameter, I-parameter and D-parameter are independent from each other. Numerical examples and an application to a heat flow experiment are shown to illustrate the effectiveness of the proposed method. Kewords: PID control, stable plant, stabilit, admissible set, heat flow experiment 1. INTRODUCTION PID (Proportional-Integral-Derivative) controller is most widel used controller structure in industrial applications [1 3]. Its structural simplicit and sufficient abilit of solving man practical control problems have contributed to this wide acceptance. Several papers on tuning methods for PID parameters have been considered [4 14]. However the method in [4 14] do not guarantee the stabilit of closed-loop sstem. The reference in [15 18] propose design methods of PID controllers to guarantee the stabilit of closed-loop sstem. However, using the method in [15 18], it is difficult to tune PID parameters to meet control specifications. If admissible sets of PID parameters to guarantee the stabilit of closed-loop sstem are obtained, we can easil design stabilizing PID controllers to meet control specifications. Manuscript received on June 3, 26 ; revised on August 24, 26. 1,2,3 The authors are with Department of Mechanical Sstem Engineering, Gunma Universit 1-5-1 Tenjincho, Kiru 376-8515, Japan (amada@me.gunma-u.ac.jp) The problem to obtain admissible sets of PID parameters to guarantee the stabilit of closed-loop sstem is known as a parametrization problem [3, 19, 2]. If there exists a stabilizing PID controller, the parametrization of all stabilizing PID controller is considered in [3, 19, 2]. However the method in [3, 19, 2] remains a difficult. The admissible sets of P-parameter, I-parameter and D-parameter in [3, 19, 2] are related each other. That is, if P-parameter is changed, then the admissible sets of I-parameter and D-parameter change. From practical point of view, it is desirable that the admissible sets of P- parameter, I-parameter and D-parameter are independent from each other. Yamada and Moki initiall tackle this problem and propose a design method for modified PI controllers for an minimum phase sstems such that the admissible sets of P-parameter and I-parameter are independent from each other [21]. Yamada expand the result in [21] and propose a design method for modified PID controllers for minimum phase plant such that the admissible sets of P-parameter, I-parameter and D-parameter are independent from each other [22]. However the method in [21, 22] cannot appl for an stable non-minimumphase plants. Since PID controllers are applied for man stable plants, the problem to design stabilizing PID controller such that the admissible sets of P-parameter, I-parameter and D-parameter are independent from each other is important. In this paper, we expand the result in [21, 22] and propose a design method for modified PID controllers such that modified PID controller makes the closed-loop sstem stable for an stable plants and the admissible sets of P-parameter, I-parameter and D-parameter to guarantee the stabilit of closed-loop sstem are independent from each other. The basic idea of modified PID controller is ver simple. In order to appl an stable non-minimum-phase plants, the parametrization of all stabilizing controllers for stable plants in [23, 24] is used. The parametrization of all stabilizing controllers for stable non-minimumphase plants includes the model of plant and free parameter. There exists the idea of the model driven control [25], which is to improve the control performance using the model of the plant. Using the fusion of the idea of model driven control in [25] and the parametrization of all stabilizing controllers for stable non-minimum phase plants, if the free-parameter

32 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 Februar 27 in the parametrization includes PID parameter in adequate form, the role of controller will be equivalent to PID controller. We present a design method for free-parameter to make the role of modified PID controller be equal to that of PID controller. Then designed modified PID controller, of which the roles are equivalent to that of PID controller, stabilizes stable plant independent of P-parameter, I-parameter and D-parameter. Numerical examples and an application to a heat flow experiment are 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. 2. PROBLEM FORMULATION Consider the closed-loop sstem written b { = G(s)u, (1) u = C(s)(r ) where G(s) R is the single-input/single-output strictl proper plant written b b n 1 s n 1 + b n 2 s n 2 + + G(s) = a n s n + a n 1 s n 1 + a n 2 s n 2, + + a (2) where a i R(i =,..., n) and b i R(i =,..., n 1). G(s) is assumed to have no pole in the closed right half plane, to have no zero on the origin and to be coprime. C(s) R(s) is the controller, u R is the control input, R is the output and r R is the reference input. When the controller C(s) has the form written b C(s) = a P + a I s + a Ds, (3) then the controller C(s) is called the PID controller [1, 3, 19, 2], 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 settled so that the closed-loop sstem in (1) has desirable control characteristics such as stead state characteristic and transient characteristic. For eas explanation, we call C(s) in (3) the conventional PID controller. The transfer function from r to in (1) is written b = G(s) ( a P + a I s + a D s ) 1 + G(s) ( a P + a )r. (4) I s + a D s It is obvious that when a P, a I and a D are settled at random, the stabilit of the closed-loop sstem in (1) does not guaranteed. Using the result in [3, 19, 2], if a stabilizing PID controller C(s) in (3) for G(s) exists, we can design PID controller C(s) in (3) to stabilize the closed-loop sstem in (1). That is, the admissible sets of a P, a I and a D to stabilize the closed-loop sstem in (1) are obtained in [3, 19, 2]. According to [3, 19, 2], the admissible sets of a P, a I and a D are related each other. That is, if a P is changed, then the admissible sets of a I and a D change. From practical point of view, it is desirable that the admissible sets of a P, a I and a D are independent from each other. However the references in [3, 19, 2] do not give the solution to the problem to find admissible sets of a P, a I and a D which are independent from each other. The purpose of this paper is to propose a modified PID controllers C(s) to make the closed-loop sstem in (1) stable for an stable plant G(s) such that the admissible sets of P-parameter a P, I-parameter a I and D-parameter a D to guarantee the stabilit of closed-loop sstem are independent from each other. The modified PID controller is obtained using the fusion of the parametrization of all stabilizing controllers C(s) for stable plants [23, 24] and the model driven control [25]. In the next section, we describe the detail of basic idea for design method for modified PID controllers C(s). 3. THE BASIC IDEA In this section, we describe the basic idea to design for modified PID controllers C(s) to make the closedloop sstem in (1) stable for an stable plant G(s) such that the admissible sets of P-parameter a P, I- parameter a I and D-parameter a D to guarantee the stabilit of closed-loop sstem are independent from each other. In order to design modified PID controllers C(s) that can be applied to an stable plants, we adopt the parametrization of all stabilizing controllers for stable plants. According to [23, 24], the parametrization of all proper internall stabilizing controllers C(s) for stable plants G(s) is written b C(s) = Q(s) 1 Q(s)G(s), (5) where Q(s) RH is an function. On the parametrization of all stabilizing controllers C(s) in (5) for G(s), the controller C(s) in (5) includes plant model G(s). There exists the idea of the model driven control [25], which is to improve the control performance using the model of the plant. Using the idea of the model driven control and the parametrization of all stabilizing controllers C(s) for G(s), we propose a design method for modified PID controllers C(s) to make the closed-loop sstem in (1) stable and to be able to appl to an stable plant G(s). In order to design the modified PID controllers C(s) using the idea of model driven control, the free parameter Q(s) in (5) is settled for C(s) in (5) to have the same characteristics to conventional PID controller C(s) in (3). Therefore, next, we describe the role of conventional PID controller C(s) in (3) in order to clarif the condition that the modified PID controller C(s) must be satisfied. From (3), using C(s), the P-parameter

A Design Method for Modified PID Controllers for Stable Plants And Their Application 33 a P, the I-parameter a I and the D-parameter a D are decided b and { a P = lim s 2 d ( )} 1 s ds s C(s), (6) a I = lim s {sc(s)} (7) d a D = lim {C(s)}, (8) s ds respectivel. Therefore, if the controller C(s) in (5) holds (6), (7) and (8), the role of controller C(s) in (5) is equivalent to the conventional PID controller C(s) in (3). That is, we can design stabilizing modified PID controllers such that the role of controller C(s) in (5) is equivalent to the conventional PID controller C(s) in (3). Next, we describe a design method for the free parameter Q(s) in (5) to make the controller C(s) in (5) works as a modified PID controller. In the following, we call C(s) 1. the modified P controller if C(s) in (5) satisfies (6), 2. the modified I controller if C(s) in (5) satisfies (7), 3. the modified D controller if C(s) in (5) satisfies (8), 4. the modified PI controller if C(s) in (5) satisfies (6) and (7), 5. the modified PD controller if C(s) in (5) satisfies (6) and (8) 6. and the modified PID controller if C(s) in (5) satisfies (6), (7) and (8). 4. MODIFIED PID CONTROLLER In this section, we describe a design method for the free parameter Q(s) in (5) to make the controller C(s) in (5) works as a modified PID controller. 4. 1 Modified P controller In this subsection, we mention a design method for modified P controller C(s) that holds (6), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified P controller C(s) satisfing (6) is written b (5), where Q(s) = a P. (9) Since Q(s) in (9) is included in RH, the controller C(s) in (5) with (9) make the closed-loop sstem in (1) stable for an stable plant G(s) independent of a P. 4. 2 Modified I controller In this subsection, we mention a design method for modified I controller C(s) that holds (7), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified I controller C(s) satisfing (7) is written b (5), where q 1 = Q(s) = q + q 1 s τ + τ 1 s, (1) q = τ 1 G() τ = a τ 1 a2 τ b 2 τ G() = a τ, (11) G 2 + d ) () a I ds {G(s)} s= + a b 1 a 1 a I a 2 ), (12) τ R > and τ 1 R >. From τ R > and τ 1 R >, Q(s) in (9) is included in RH. This implies that the controller C(s) in (5) with (1) make the closed-loop sstem in (1) stable for an stable plant G(s) independent of a I. 4. 3 Modified D controller In this subsection, we mention a design method for modified D controller C(s) that holds (8), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified D controller C(s) satisfing (8) is written b (5), where Q(s) = a D s. (13) Since Q(s) in (13) is improper, Q(s) in (13) is not included in RH. In order for Q(s) to be included in RH, (13) is modified as Q(s) = a Ds 1 + τ D s, (14) where τ D R >. From τ D R > in (14), Q(s) in (14) is included in RH. This implies that the controller C(s) in (5) with (14) make the closedloop sstem in (1) stable for an stable plant G(s) independent of a D. 4. 4 Modified PI controller In this subsection, we mention a design method for modified PI controller C(s) that holds (6) and (7), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified PI controller C(s) satisfing (6) and (7) is written b (5), where Q(s) = q + q 1 s + q 2 s 2 τ + τ 1 s + τ 2 s 2, (15)

34 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 Februar 27 q 1 = q = τ 1 G() τ = a τ 1 a2 τ b 2 τ G() = a τ, (16) G 2 + d ) () a I ds {G(s)} s= + a b 1 a 1 a I a 2 ), (17) q 2 = τ 2 a P, (18) τ R >, τ 1 R > and τ 2 R >. From τ R >, τ 1 R > and τ 2 R >, Q(s) in (15) is included in RH. This implies that the controller C(s) in (5) with (15) make the closed-loop sstem in (1) stable for an stable plant G(s) independent of a P and a I. 4. 5 Modified PD controller In this subsection, we mention a design method for modified PD controller C(s) that holds (6) and (8), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified PD controller C(s) satisfing (6) and (8) is written b (5), where Q(s) = a P + 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) is modified as Q(s) = a P + a Ds 1 + τ D s, (2) where τ D R >. From τ D R > in (2), Q(s) in (2) is included in RH. This implies that the controller C(s) in (5) with (2) make the closedloop sstem in (1) stable for an stable plant G(s) independent of a P and a D. 4. 6 Modified PID controller In this subsection, we mention a design method for modified PID controller C(s) that holds (6), (7) and (8), makes the closed-loop sstem in (1) stable and is able to appl to an stable plant G(s). The modified PID controller C(s) satisfing (6), (7) and (8) is written b (5), where Q(s) = q + q 1 s + q 2 s 2 τ + τ 1 s + τ 2 s 2 + q 3s, (21) where q, q 1, q 2 and q 3 are settled as follows: 1. when the relative degree of G(s) is greater than or equal to 3 q 1 = τ 1 G() q 3τ τ ( 1 G 2 + d ) () a I ds {G(s)} s= = a τ 1 q 3 τ a2 τ b 2 a I + a b 1 a 1 a 2 ) (23) q 2 = τ 2 a P (24) q 3 = a D (25) 2. when the relative degree of G(s) equals 2 q 1 = τ 1 q = τ G() = a τ (26) G() q 3τ τ ( 1 G 2 + d ) () a I ds {G(s)} s= = a τ 1 q 3 τ a2 τ b 2 a I + a b 1 a 1 a 2 ) (27) q 2 = τ 2 a P b n 2 a n τ 2 a 2 D (28) q 3 = a D (29) 3. when the relative degree of G(s) equals 1 q 1 = τ 1 q = τ G() = a τ (3) G() q 3τ τ ( 1 G 2 + d ) () a I ds {G(s)} s= = a τ 1 q 3 τ a2 τ b 2 a I + a b 1 a 1 a 2 ) (31) q 2 = a n b n 1 a D τ 2 a P b n 2 τ 2 a 2 D (32) a n a n q = τ G() = a τ (22) q 3 = a D 1 + b n 1 a n a D (33)

A Design Method for Modified PID Controllers for Stable Plants And Their Application 35 Since Q(s) in (21) is improper, Q(s) in (21) is not included in RH. In order for Q(s) to be included in RH, (21) is modified as Q(s) = q + q 1 s + q 2 s 2 τ + τ 1 s + τ 2 s 2 + a Ds 1 + τ D s, (34) where τ D R >. From τ D R > in (34), Q(s) in (34) is included in RH. This implies that the controller C(s) in (5) with (34) makes the closedloop sstem in (1) stable for an stable plant G(s) independent of a P, a I and a D. 5. NUMERICAL EXAMPLE In this section, we illustrate numerical examples to show the effectiveness of the proposed method. 5. 1 Numerical example for minimum phase plant Consider the problem to design a modified PID controller C(s) for a stable and minimum phase plant G(s) written b G(s) = s + 5 s 4 + 1s 3 + 35s 2 + 5s + 24. (35) a P, a I and a D are settled b a P = 1 a I = 1 a D =.1. (36) Since the relative degree of G(s) equals 3, q, q 1, q 2 and q 3 are determined b (22), (23), (24) and (25), respectivel, where τ = 1 τ 1 = 1 τ 2 = 1 (37) and τ D is selected b τ D =.1. Using abovementioned parameters, the modified PID controller C(s) is designed b (5) with (34). The step response of the control sstem using modified PID controller C(s) is shown in Fig. 1. Figure 1 shows that the modified PID controller C(s) makes the closed-loop sstem stable. On the other hand, using conventional PID controller with (36), the step response of the control sstem is shown in Fig. 2. Figure 2 shows that the conventional PID control sstem is unstable. The reason wh the conventional PID control sstem is unstable is the stabilit of the conventional PID control sstem depends on a P, a I and a D. Therefore, when a P, a I and a D are settled b (36), the conventional PID control sstem is unstable. Contrar to this, the stabilit of modified PID control sstem is guaranteed independence of a P, a I and a D. Next, when a P, a I and a D in the modified PID controller are varied, the comparison of step responses 1.4 1.2 1.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Fig.1: Step response of the control sstem using modified PID controller 1.5 x 18 1.5.5 1 1.5 2 2 4 6 8 1 12 14 16 18 2 Fig.2: Step response of the control sstem using conventional PID controller is examined. First, the comparison of step responses for various a P as a P =.1, a P = 1 and a P = 1 is shown in Fig. 3. Here, the solid line, the dotted line and the broken line show the step response of the modified control sstem using a P =.1, a P = 1 and a P = 1, respectivel. Figure 3 shows that as the value of a P increased, the overshoot is larger and the rise time is shorten. Since this characteristics is equivalent to the conventional PID controller, the role of P-parameter a P in the modified PID controller is equivalent to that of the conventional PID controller. Secondl, the comparison of step responses for various a I as a I =.1, a I = 1 and a I = 1 is shown in Fig. 4. Here the solid line, the dotted line and the broken line show the step response of the modified PID control sstem using a I =.1, a I = 1 and a I = 1, respectivel. Figure 4 shows that as the value of a I increased, the overshoot is smaller and the convergence speed is faster. Since this characteristics is equivalent to the conventional PID controller,

36 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 Februar 27 2.5 1 2.8 1.5.6 1.4.5.2.5 1 2 3 4 5 6 7 8 9 1 Fig.3: Step response using modified P controller with a P =.1, 1, 1.2 1 2 3 4 5 6 7 8 9 1 Fig.5: Step response using modified D controller with a D =.1, 1, 1 2 2 4 6 8 1 12 14 16 1 2 3 4 5 6 7 8 9 1 Fig.4: Step response using modified I controller with a I =.1, 1, 1 the role of I-parameter a I in the modified PID controller is equivalent to that of the conventional PID controller. Thirdl, the comparison of step responses for various a D as a D =.1, a D = 1 and a D = 1 is shown in Fig. 5. Here, the solid line, the dotted line and the broken line show the step response of the modified PID control sstem using a D =.1, a D = 1 and a D = 1, respectivel. Figure 5 shows that as the value of a D increased, the response is smoothl. Since this characteristics is equivalent to the conventional PID controller, the role of D-parameter a D in the modified PID controller is equivalent to that of the conventional PID controller. In this wa, it is shown that we can easil design a stabilizing modified PID controller for stable minimum phase plant, which has same characteristic to conventional PID controller, and guarantee the stabilit of the closed-loop sstem. 5. 2 Numerical example for non-minimum phase plant Consider the problem to design a modified PID controller C(s) for a stable and non-minimum phase plant G(s) written b G(s) = a P, a I and a D are settled b a P =.1 a I = 1 a D = 1 s 2 1 s 3 + 9s 2 + 26s + 24. (38). (39) Since the relative degree of G(s) equals 1, q, q 1, q 2 and q 3 are determined b (3), (31), (32) and (33), respectivel, where τ = 1 τ 1 = 1 τ 2 = 1 (4) and τ D is selected b τ D =.1. Using abovementioned parameter, the modified PID controller C(s) is designed b (5) with (34). The step response of the control sstem using modified PID controller C(s) is shown in Fig. 6. Figure 6 shows that the modified PID controller C(s) makes the closed-loop sstem stable. On the other hand, using conventional PID controller with (39), the step response of the control sstem is shown in Fig. 7. Figure 7 shows that the conventional PID control sstem is unstable. The reason wh the conventional PID control sstem is unstable is the stabilit of the conventional PID control sstem depends on a P, a I and a D. Therefore, when a P, a I and a D are settled b (36), the conventional PID control sstem is unstable. Contrar to this, the stabilit of modified PID control sstem is guaranteed independence of a P, a I and a D.

A Design Method for Modified PID Controllers for Stable Plants And Their Application 37 2.5.8 2 1.5 1.5.5 1.6.4.2.2.4 1.5 2 4 6 8 1 12 14 16 18 2 Fig.6: Step response of the control sstem using modified PID controller.6.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fig.8: Step response using modified P controller with a P =.1, 1, 1 2 x 15 2 4 3 2 1 6 8 1 12 14 16 2 4 6 8 1 12 14 16 18 2 Fig.7: Step response of the control sstem using conventional PID controller Next, when a P, a I and a D in the modified PID controller are varied, the comparison of the step responses is examined. First, the comparison of step responses for various a P as a P =.1, a P = 1 and a P = 1 is shown in Fig. 8. Here, the solid line, the dotted line and the broken line show the step response of the modified PID control sstem using a P =.1, a P = 1 and a P = 1, respectivel. Figure 8 shows that as the value of a P increased, the overshoot is larger and the rise time is shorten. Since this characteristics is equivalent to the conventional PID controller, the role of P-parameter a P in the modified PID controller is equivalent to that of the conventional PID controller. Secondl, the comparison of step responses for various a I as a I =.1, a I = 1 and a I = 1 is shown in Fig. 9. Here the solid line, the dotted line and the broken line show the step response of the modified PID control sstem using a I =.1, a I = 1 and a I = 1, respectivel. Figure 9 shows that as the value of a I increased, the overshoot is smaller and the convergence speed is faster. 1 2 3 4.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fig.9: Step response using modified I controller with a I =.1, 1, 1 Since this characteristics is equivalent to the conventional PID controller, the role of I-parameter a I in the modified PID controller is equivalent to that of the conventional PID controller. Thirdl, the comparison of step responses for various a D as a D =.1, a D = 1 and a D = 1 is shown in Fig. 1. Here, the solid line, the dotted line and the broken line show the step response of the modified PID control sstem using a D =.1, a D = 1 and a D = 1, respectivel. Figure 1 shows that as the value of a D increased, the response is smoothl. Since this characteristics is equivalent to the conventional PID controller, the role of D-parameter a D in the modified PID controller is equivalent to that of the conventional PID controller. In this wa, it is shown that we can easil design a stabilizing modified PID controller for stable nonminimum phase plant, which has same characteristic to conventional PID controller and guarantee the stabilit of the closed-loop sstem.

38 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 Februar 27.8.6.4.2.2.4.6.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Fig.1: Step response using modified D controller with a D =.1, 1, 1 Then, from (41), G(s) in (1) is written b G(s) = 6.4 1 + 15.7s. (42) The reference input r in (1) is settled b r(t) = 4[deg]. Then the problem is to design a stabilizing modified PID controller C(s) using the method described in Section 4.6. When a P, a I and a D in the modified PID controller are settled b a P = 2.49 a I =.112 a D =.1391. (43) Experimental step response is shown in Fig. 12. Figure 12 shows that the output follows the step 45 6. APPLICATION FOR A HEAT FLOW EX- PERIMENT In this section, we present an application of the proposed modified PID controller for a heat flow experiment. The illustrated heat flow experiment is shown in Fig. 11. The heat flow experiment consists of a [ C] 4 35 3 Blower Heater Sensor1 Sensor2 35 Sensor3 25 5 1 15 2 25 3 5 8 17 Fig.11: 37 Vh Vb 57 75 S1 S2 S3 16 Illustrated heat flow apparatus duct equipped with a heater and a blower at one end and three temperature sensors located along the duct as shown in Fig. 11. V h and V b denote the voltage to heater and that to blower, respectivel. S 1, S 2 and S 3 are terminals for measurements of temperatures at Sensor 1, Sensor 2 and Sensor 3. We denotes T i a measurements of temperatures at Sensor i. The problem considered in this section is to design a modified PID controller to make T 1, which is temperature at Sensor 1, 4[deg] steadil. When we settle V b = 5[V ], we find the transfer function from V h to T 1 which is temperatures at Sensor 1 is written b 1 T 1 = 6.4 1 + 15.7s V h. (41) T 1 and V h are considered as the output and the control input u in the control sstem in (1), respectivel. Fig.12: Experimental step response using the modified PID controller reference input r = 4 negligible stead state error with high convergence speed comparing to open-loop response. When the disturbance d(t) =.5 exists as = G(s)(u + d), (44) the experimental step response is shown in Fig. 13. Figure 13 shows that even if the disturbance d(t) =.5 exists, the output follows the step reference input r = 4 negligible stead state error. From above discussions, we find that the proposed modified PID controller is effective controlling heat flow experiment. 7. CONCLUSION In this paper, we proposed a design method for modified PID controllers such that modified PID controllers make the closed-loop sstem for an stable and non-minimum phase plants asmptoticall stable and the admissible sets of P-parameter, I-parameter and D-parameter are independent from each other. Numerical examples and an application to a heat flow experiment were shown to illustrate the effectiveness of the proposed method.

A Design Method for Modified PID Controllers for Stable Plants And Their Application 39 [ C] 45 4 35 3 25 5 1 15 2 25 Fig.13: Experimental step response using the modified PID controller, when d(t) =.5 exists References [1] N. Suda, PID Control, Asakura Shoten, Toko, 1992(in Japanese). [2] K. Astrom and T. Hagglund, PID controllers: Theor design, and tuning, Instrument Societ of America, North Carolina, 1995. [3] A. Datta, M. T. Ho and S. P. Bhattachara, Structure and Snthesis of PID Controllers, Springer-Velag, London, 2. [4] J.G. Zieglae and N.B. Nicholes, Optimum settings for automatic controllers, Trans. ASME, 64, pp.759 768, 1942. [5] P. Hazebroek and B.L. van der Warden, The Optimal Adjustment of Regulators, Trans. ASME, 72, pp.317 332, 195. [6] P. Hazebroek and B.L. van der Warden, Theoretical Considerations on the Optimal Adjustment of Regulators, Trans. ASME, 72, pp.39 315, 195. [7] W.A. Wolf, Controller Setting for Optimum Control, Trans. ASME, 73, pp.413 418, 1951. [8] K.L. Chien, J.A. Hrones and J.B. Reswick, On the Automatic Control of Generalized Passive Sstems, Trans. ASME, 74, pp.175 185, 1952. [9] G.H. Cohen and G.A. Coon, Theoretical Consideration of Retaeded Control, Trans. ASME, 75, pp.857 834, 1953. [1] A.M. Lopez, J.A. Miller, C.L. Smith and P.W. Murrill, Tuning Controllers with Error-Integral Criteria, Instrumentation Technolog, 14, pp.52 62, 1967. [11] J.A. Miller, A.M. Lopez, C.L. Smith and P.W. Murrill, A Comparison of Controller Tuning Techniques, Control Engineering, 14, pp.72 75, 1967. [12] 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.549 555, 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.382 391, 198(in Japanese). [14] P. Cominos and N. Munro, PID Controllers: Recent Tuning Methods and Design to Specification, IEE Proceedings, 149, pp.46 53, 22. [15] F. Zheng, Q.G. Wang and T.H. Lee, On the Design of Multivariable PID Controllers via LMI Approach, Automatica, 38-3, pp.517 526, 22. [16] C. Lin, Q.G. Wang and T.H. Lee, An Improvement on Multivariable PID Controller Design via Iterative LMI Approach, Automatica, 4-3, pp.519 525, 24. [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, 25. [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.193-22, 26(in Japanese). [19] J. Yang, Parameter Plane Control Design for a Two-tank Chemical Reactor Sstems, Journal of the Franklin Institute, Vol.331B, pp.61 76, 1994. [2] 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.3922 3928, 1997. [21] 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.571 576, 23. [22] K. Yamada, Modified PID controllers for minimum phase sstems and their practical application, Proceedings of The 25 Electrical Engineering/Electronics, Computer, Telecommunication, and Information Technolog (ECTI) International Conference, Vol.II, 25. [23] M. Morari and E. Zafiriou, Robust process control, PTR Prentice Hall, New Jerse, 1989. [24] M. Vidasagar, Control sstem snthesis A factorization approach, MIT Press, London, 1985. [25] H. Kimura, Is the Model a Good Controller? Perspectives on Brain Motor Control, Proceedings of CDC, 2.

4 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 Februar 27 Kou Yamada was born in Akita, Japan, in 1964. He received the 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 1997. From 1991 to 2, he was with the Department of Electrical and Information Engineering, Yamagata Universit, Yamagata, Japan, as a research associate. Since 2, 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 25 Yokoama Award in Science and Technolog and The 25 Electrical Engineering/Electronics, Computer, Telecommunication, and Information Technolog International Conference (ECTI-CON25) Best Paper Award. Nobuaki Matsushima was born in Gunma, Japan, in 1981. He received the B.S. and M.S. degrees in Mechanical Sstem Engineering from Gunma Universit, Gunma Japan, in 24 and 26, respectivel. His research interests include process control and control theor for infinite-dimensional sstems. Takaaki Hagiwara was born in Gunma, Japan, in 1982. He received the B.S. degree in Mechanical Sstem Engineering from Gunma Universit, Gunma Japan, in 26. He is currentl M.S. candidate in Mechanical Sstem Engineering at Gunma Universit. His research interests include process control and PID control.