A Design Method of Compensator to Minimize Model Error

Transcription

1 SICE Journal of Control, Measurement, and System Integration, Vol. 6, No. 4, pp , July 2013 A Design Method of Compensator to Minimize Model Error Hiroshi OKAJIMA, Hironori UMEI, Nobutomo MATSUNAGA, and Toru ASAI Abstract : The robust control design method has been studied in recent decades. A control system works well under the modeling errors and disturbances if controller design is based on the robust control method. However, it is well known that in control systems, generally, there exists a trade-off between control performance and robustness. To overcome the trade-off problem, this paper proposes an internal model type compensator structure that minimizes the modeling gap between the nominal model and actual plant dynamics. By using the proposed compensator, the dynamics of the compensated system closes to that of the nominal model. Then, a design method of the compensator parameters is also proposed for minimizing a set of plant dynamics. The proposed design method is reduced to the standard μ design control problem. If we use the proposed compensator for control systems instead of the plant itself, the output performance might be better despite plant uncertainty. Given that the proposed compensator can be used for the control of not only linear but also nonlinear plants, we can easily achieve robust control of nonlinear systems. The effectiveness of the proposed method is shown by numerical examples. Key Words : 2DOF control systems, disturbance elimination, internal model control, robust control. 1. Introduction Many control system design methods can successfully achieve the desired control performance by using accurate plant models [1] [5]. If an accurate plant model is obtained, many important design specifications can be met by using the existing control system design methods. For example, the classical specifications such as rise time and overshoot, and explicit constraints of inputs and states can be handled by the model based design [6] and the model predictive control (MPC) [4],[5], respectively. Moreover, the continuous deadbeat control (CDBC), i.e. the deat-beat control in the continuous time domain, can be achieved [7]. However, it is difficult to obtain an accurate plant model owing to variations of the plant parameters, observation noise, and so on. When there exists a modeling gap between a plant and its model, the designed controller for the model may not exhibit the expected behavior for the actual plant. In order to guarantee stability and control performance against plant uncertainty and disturbances, robust control design methods have been developed in recent decades [1] [3]. In general, the goal of robust control is to design a controller that achieves a desired performance level for a set of plant dynamics. Because the controller is designed to satisfy the desired performance for all the plant models, the response of the control system tends to be slow. Moreover, important characteristics such as rise time and overshoot cannot be handled directly by robust control settings. Therefore, the class of the system design, i.e. the class of not only plant model sets but also control specifications, are limited compared to the design methods Graduate School of Science and Technology, Kumamoto University, Kurokami, Kumamoto , Japan Graduate School of Engineering, Osaka University, 1-1 Yamadaoka, Suita , Japan okajima@cs.kumamoto-u.ac.jp (Received July 10, 2012) (Revised November 26, 2012) for nominal models. If the modeling gap between the nominal model and the actual plant can be made small by using an additional compensator, the desired response given by the design methods for the nominal model would be achieved even if the modeling gap exists. In this study, the authors propose a novel compensator structure so as to reduce the modeling gap between the actual plant and its nominal model. Specifically, the proposed compensator is designed to minimize the worst-case error between the nominal model and the model set describing the plant uncertainty. The error is evaluated based on the H norm and hence the design problem is reduced to a μ synthesis problem. The proposed compensator can be applied to nonlinear plants such as the Hammerstein model, vehicle control systems and so on. Because the proposed compensator reduces the modeling gap, the control system given by the nominal system design might work well even if the modeling gap exists. In other words, the proposed method enables us to apply the existing many (possibly nominal) model-based design methods for systems with plant uncertainty. The proposed compensator structure is related to the IMC (Internal Model Control) structure [8] [13]. IMCs mainly aim to improve control performance by feedbacking the difference between the actual plant and the nominal model, where there is a feedback loop around not only the actual plant but also the nominal model. On the other hand, also by feedbacking the difference between the actual plant and the nominal model, the proposed compensator aims to ensure that the compensated plant dynamics is close to the nominal model, where there is no feedback loop around the nominal model. As a result, the proposed method enables to deal with robust stability while maintaining the nominal transfer characteristics and hence the nominal performance. The aim of the proposed method is also related to the loop transfer recovery (LTR) concept [14],[15] in the observer design. LTR methods consider the difference between a given JCMSI 0004/13/ c 2012 SICE

2 268 SICE JCMSI, Vol. 6, No. 4, July 2013 state feedback control system and the output feedback control system with an observer, and attempt to recover the gain margin while maintaining the nominal performance level by appropriate observer design. On the other hand, the proposed method aims to maintain stability and the nominal performance by feedbacking the error between the nominal and the actual plant outputs. This paper is organized as follows. In Section 2, the motivation behind this research and the basic idea are described. In particular, the specialty of this paper is that the proposed compensator minimizes the modeling gap between the plant and the nominal model in terms of the input-output relation. The structure of the proposed compensator is presented in Section 3. The proposed structure compensates for the difference between the expected output of the nominal model and the output of the actual plant. Then, a design method of the compensator parameters is proposed for the linear time-invariant plant in Section 4. The compensator dynamics are designed based on the given set of plant model dynamics. The compensator design method for minimizing the model set is reduced to the standard μ design problems. In Section 5, design methods for the nonlinear plants are introduced based on the result obtained for the linear plants. The effect of our proposed method is discussed in Section 6. In particular, we compare our proposed method and previous studies. Notation is standard. We denote the Laplace transform of u(t) as û(s). S is the set of proper real rational functions analytic in the closed right half plane. 2. Basic Idea The basic idea of this research is presented in this section. For simplicity, the linear time-invariant systems are considered. In model-based control system design, a nominal model P n is derived based on the output response of an actual plant P and some physical conditions. Then, the controller for P is designed based on the simulation by using P n as shown in left-hand side of Fig. 1. The symbol y n in Fig. 1 is the expected output by the control system with P n. Then, the designed controller, using P n, is applied to the actual plant P as shown in the right-hand side of Fig. 1. If P n and P are similar in terms of the input-output dynamics, the control system with P works well and achieves the expected performance. In contrast, if there exists a modeling gap between P n and P, the control system does not work as well as expected. We denote the model error as Δ P and assume that the following equation holds: Δ P = P P n. (1) If the model error of the plant can be minimized, it is expected that response variation will be reduced. Therefore, we consider a compensator to minimize the model error. The form of the compensated system P is shown in Fig. 2. The compensator H minimizes the modeling gap between P n and P. The compensated system P is used in the control system instead of the plant itself, as shown in Fig. 3. If the model error Δ P (= P P n ) is minimized by the compensator, the output of a control system with P might be similar to that with the nominal model P n, and thus, the desired control performance is successfully achieved. Compensator H is designed to minimize the effect of the model error Δ P, and it enforces the input-output relation from u to y, close to P n. We list the design specifications of H as follows. C1: The model error Δ P is smaller than that of Δ P in terms of the input-output relation. C2: P = P n holds when P = P n holds. If P equals P n, P = P n should be satisfied by the compensator H. The objective of this research is to find the compensator H that satisfies the design specifications C1 and C2. 3. Proposed Compensator Structure We consider the design of the compensator H, shownin Fig. 2. To achieve C2, we propose a compensator structure, as shown in Fig. 4. The proposed compensator structure includes the model P n for measuring the modeling gap. In Fig. 4, the response difference between P and P n is used as a feedback signal and the model error is reduced by a compensator D. We denote D as a differential compensator. The compensator H is represented by the dotted box in Fig. 4. In case P is a SISO linear time-invariant plant, the transfer function from u to y can be written as follows: ŷ(s) û (s) = 1 + P n (s)d(s) P (s) = 1 + (P n (s) +Δ P (s))d(s) (P n(s) +Δ P (s)). Fig. 2 Compensated system. Fig. 3 Control system with Fig. 2. (2) Fig. 1 Standard control system design sequence. Fig. 4 Proposed compensator structure in Fig. 2.

3 SICE JCMSI, Vol. 6, No. 4, July It can seen that P (s) = P n (s) holds when Δ P (s) = 0 holds in Eq. (2) for any D(s). Therefore, C2 is automatically satisfied by the proposed compensator structure. Moreover, if high gain feedback D(s) can be selected, the effect of Δ P (s) becomes small. It is expected that the dynamics of P (s) can be brought close to that of P n (s) by appropriate tuning of D(s). The effectiveness of the proposed structure in Fig. 4 is illustrated by a numerical example. First, we assume the plant is given by P(s) = K 1 /(T 1 s + 1) with parameter variation. The ranges of parameters T 1 and K 1 are as follows: T 1 [0.8, 1.2], (3) K 1 [0.9, 1.1]. (4) Then, the nominal model is given by P n (s) = 1/(s + 1). The feedforward (FF) controller C = (s+1)/(0.2s+1) is considered to control P(s) in this example. Then, the expected output ŷ n (s) is given by the following equation: ŷ n (s) = P n (s)c(s)ˆr(s), (5) where ˆr(s) is the reference signal. The differential compensator D(s) is given as the following PI controller: D(s) = (10s + 3)/s. (6) The output with the proposed compensator is given by the following equation: ŷ(s) = P (s)c(s)ˆr(s). (7) The simulation results of the step response (ˆr(s) = 1/s) are shown in Figs. 5 and 6. The outputs of P(s) with the FF controller are represented by red solid lines in Fig. 5, nominal response y n (t) by dashed line, and the step response of P n (s) by dash-dotted line. The parameters T 1 and K 1 are selected randomly from within the given parameter range. The outputs of the compensated system P (s) with the FF controller are denoted by red solid lines in Fig. 6. The response variation in Fig. 6 is smaller than that in Fig. 5. All outputs in Fig. 6 are close to the nominal response as opposed to the case in Fig. 5. As a result, owing to the proposed compensator structure, P (s)andp n (s) become similar in terms of the input-output relation. Figures 7 and 8 are the bode diagrams of P(s)andP (s), respectively. We can find that variation of red lines is small in low frequency for the case of P (s). 4. Design of Differential Compensator for Linear Plants 4.1 Case That P n is Stable In this section, a plant is assumed to be given by a rational transfer function P(s), and design of the differential compensator D(s) is considered. Let P(s) be a stable plant and the following equation holds: ŷ(s) = P(s)û(s) (8) P(s) can be divided into the nominal model P n (s) and the model error Δ P (s) as follows: P(s) = P n (s) +Δ P (s). (9) Moreover, we assume that the model error Δ P (s) satisfies the following condition: Δ P (s) =Δ(s)W(s)P n (s), Δ(s) S Δ, (10) where W(s) is a weighting function, and S Δ is given as follows: S Δ = {Δ(s) : Δ(s) 1 Δ(s) S}. (11) The set of Δ P (s) forallδ(s) is defined as follows: set Δ P := {Δ(s)W(s)P n (s) : Δ(s) S Δ }. (12) By using Eq. (2), Δ P (s) can be written as the following equation: Δ P (s) = (P n (s) +Δ P (s))d(s) Δ P(s). (13) Δ P (s) is represented as the product of Δ P(s) and the sensitive function of the standard unity feedback systems. To satisfy the design specification C1, we consider the following evaluation function Γ s. Γ s = min sup 1 D(s) Δ(s) 1 + P n (s)d(s)(1 + W(s)Δ(s)). (14) We design D(s) that minimizes Γ s for the worst possible Δ(s). If Γ s can be minimized, it is expected that the dynamics of P (s) and P n (s) are close to each for any Δ(s). Especially, by considering Eqs. (12) and (13), the following relation holds in case Γ s < 1. Δ P (s) Γ sset Δ P set Δ P, Δ(s) S Δ. (15) Then, the set of P is smaller than the set of P in the sense of the H norm as shown in Fig. 9, and the design specification C1 is satisfied. The problem of minimizing the model error can therefore be reduced to the problem of minimizing Γ s. However, because the relative degree of P(s) is greater than or equal to 1 in most cases, the optimal D(s) is obtained as a non proper function. Therefore, we consider the following modified design problem. [Problem A] Find D(s) that minimizes the following evaluation function Γ: Γ=min sup D(s) Δ(s) W e (s)γ(s), (16) where W e (s) is an evaluation weighting function and γ(s) is given as follows: 1 γ(s) = 1 + P n (s)d(s)(1 + W(s)Δ(s)). (17) Here, Problem A is the minimization of the performance level Γ for an uncertain system including Δ(s). It is necessary to give the relative degree of W e (s) which is greater than or equal to1inordertodesignaproperd(s). For example, the low-pass filter W e (s) = 1/(ɛ 0 s + 1) is one of the weighting functions. When we solve Problem A, it is necessary to simultaneously consider the robust stability for P (s). In Fig. 10, the stability of the system is essentially equivalent to the stability of the feedback system with the loop transfer function L(s) = P(s)D(s) because P n (s) is assumed to be stable. As P(s) has a multiplicative error, the stability condition of Fig. 10 is given by the following formula based on the small gain theorem: W(s) P n (s)d(s) 1 + P n (s)d(s) < 1. (18)

4 270 SICE JCMSI, Vol. 6, No. 4, July 2013 Fig. 5 Outputs of FF control with P. Fig. 11 Bode diagram of D(s) by solving Problem 1. Fig. 6 Outputs of FF control with proposed compensator. Fig. 12 Gain diagram for Eq. (20). Fig. 7 Bode diagram of P with randomly selected perturbation. Fig. 13 W e (s)γ(s) with various Δ. Fig. 8 Bode diagram of P. Fig. 14 Outputs by using the solution of Problem 1. It is necessary to minimize the complementary sensitivity function T(s) = P n (s)d(s)/(1 + P n (s)d(s)) for robust stability. Fig. 9 Sets of P and P. Fig. 10 Alternative view of Fig. 4. [Problem 1] Find D(s) that minimizes the following evaluation function Γ: Γ=min sup W e (s)γ(s), (19) D(s) Δ(s) W(s) P n (s)d(s) 1 + P n (s)d(s) < 1. (20) where W e (s) is an evaluation weighting function. The design method of D(s) with C1 is reduced to a robust performance problem. According to the main loop theorem [1], Problem 1 can be solved by standard μ synthesis. Then the following statement holds for the designed D(s). W e (s)δ P (s) Γset Δ P, Δ(s) S Δ. (21) If we can obtain a small Γ by solving Problem 1, the effect of Δ P is small at low frequencies because W e is a low-pass filter.

5 Thus, the compensated plant dynamics might be similar to the nominal model dynamics. Note that D(s) = 0 is a solution to satisfy (20), and Problem 1 is feasible for any W(s). SICE JCMSI, Vol. 6, No. 4, July Design Example of D(s) The numerical example in Problem 1 is discussed in this section. First, the proposed structure is applied to a 1-order minimum phase system given by P(s) = K 1 /(T 1 s + 1) with T 1 [0.8, 1.2] and K 1 [0.9, 1.1], as is presented in Section 3. We give the same FF controller C(s) = (s + 1)/(0.2s + 1). The weighting functions for Problem 1 are selected as follows: 0.375s W(s) =, W e (s) = s + 1 5s + 1. (22) The differential compensator is designed using MATLAB μ toolbox. The bode diagram of the designed differential compensator D(s) is shown in Fig. 11 of the previous page. The value of the evaluation function Γ is obtained by using μ as follows: Γ <μ= (23) We can see that a small Γ is achieved by solving Problem 1 because only a small value of μ is obtained. Figures 12 and 13 show the gain diagram of the stability condition and W(s)γ(s), respectively. We can confirm that (20) is satisfied and W(s)γ(s) <μholds for a great number of randomly selected Δ (50 lines). The output responses are shown in Fig. 14. The red solid lines in Fig. 14 are responses of the system with randomly selected K 1 and T 1. The response variation in Fig. 14 is very small as opposed to the case in Fig. 6, and we cannot distinguish each response. Therefore, the effectiveness of the proposed design method of D(s) is demonstrated. 4.3 General Case for P n (s) We showthe designof D(s) for feedback control systems. By using feedback systems, we can manage not only stable plants but also unstable plants. We consider the case that the feedback controller is already given to achieve a good nominal performance. Then we can divide the design of the control performance and the compensation of the modeling gap. D(s) should be designed to minimize the gap between the dynamics of the nominal control system and that of the compensated plant. We can use information about the designed feedback controller. A controller C FB for the nominal plant is assumed to be given as shown in Fig. 15. The control system is internally stable and achieves good nominal performance. The following equation holds for the feedback system with the nominal plant. ŷ n (s) = P n(s)c FB (s) ˆr(s). (24) 1 + P n (s)c FB (s) In contrast, by using the proposed compensator, the output ŷ(s) is given by the following equation: P (s)c FB (s) ŷ(s) = 1 + P ˆr(s), (25) (s)c FB (s) P 1 + P n (s)d(s) (s) = 1 + (P n (s) +Δ P (s))d(s) (P n(s) +Δ P (s)). (26) By subtracting (24) from (25), we can obtain the following equation: Fig. 15 Feedback control system with P n. Fig. 16 Feedback control system with Fig. 4. Fig. 17 Alternative view of Fig. 16. ŷ ŷ n = γ (s)δ P (s). (27) ˆr γ (s) is determined by the following function: C FB (s) γ (s) = (1 +P n (s)d(s))(1 +P n (s)c FB (s)) 2 + ψ(s), (28) ψ(s) = (1 + P n (s)c FB (s)) (C FB (s) + D(s) + P n (s)c FB (s)d(s))δ P (s). In the same manner as that in Section 4.1, the following evaluation function is considered for the feedback control systems: Γ = min sup D(s) Δ(s) W e (s)γ (s), (29) where W e (s) is an evaluation weighting function. The internal stability of the system in Fig. 16 is essentially equivalent to the stability of a feedback system having loop transfer function L(s) = P n (s)d(s). The proof for this statement is given in Appendix A. Moreover, to satisfy the robust stability of the closed loop system, the stability condition is given by the following formula based on the small gain theorem with the block-diagram in Fig. 17: W(s) P n(s)(c FB (s) + D(s) + P n (s)c FB (s)d(s)) (1 + P n (s)c FB (s))(1 + P n (s)d(s)) < 1. (30) When we choose D(s) = 0, (30) becomes the standard stability condition for the loop transfer function L(s) = P(s)C FB (s), i.e., the system without the proposed compensator. The differential compensator D(s) can be regarded as the design degree of freedom for achieving its robust stability while maintaining the same nominal performance. Based on the above results, the design problem of D(s) in Fig. 16 can be defined as follows: [Problem 2] Find D(s) that minimizes the evaluation function Γ in (29). Conditions for D(s) are as follows. The closed loop system with loop transfer function L(s) = P n (s)d(s) is internally stable.

6 272 SICE JCMSI, Vol. 6, No. 4, July 2013 Eq. (30) holds. The stability conditions are slightly different from the case of the stable plant in Problem 2. If we can obtain a small Γ by solving Problem 2, it is expected that the output of the feedback system with the compensated plant is similar to that with the nominal plant. As a simple numerical example, the following γ (s) and the stability condition that corresponds to (30) are given when P n (s) = 1/(s 1) and C FB (s) = 2 are assumed: 2(s 1) 3 γ (s) = (s + 1) 2 (s 1 + D(s)) + ψ(s), (31) ψ(s) = (s + 1)(s 1)(2s 2 + (s + 1)D(s))Δ P (s), (32) 1) + (s + 1)D(s) W(s)2(s (s + 1)(s 1 + D(s)) < 1. (33) We can see from (31) (33) that the design problem is similar to the design problem in Section 4.1, and it can be solved numerically by standard μ synthesis. 5. Design of Proposed Compensator for Nonlinear Plants The nonlinear model is used as P n in Fig. 4. Then, it is expected that the modeling gap between the nonlinear plant and the nonlinear model can be minimized by using the proposed compensator structure. The design of D for nonlinear plants is shown in this section. First, a design method is presented for the case of standard input-affine nonlinear systems [16],[17]. Second, the case of Hammerstein-type nonlinear systems [18] [21] is presented. The nonlinear plants are denoted as follows: 1. Input-affine nonlinear system P 1 2. Nonlinear Hammerstein-type mathematical model P 2 The differential compensators for P 1 and P 2 are denoted as D 1 and D 2, respectively. 5.1 Design of D 1 for Input-Affine Nonlinear Systems P 1 We consider a plant P 1 and that the plant is modeled by an input-affine nonlinear system of the form: ẋ = f (x) + g(x)u, (34) y = h(x), (35) which has states x R n, input u R, and output y R. Its corresponding nonlinear plant model is denoted by P n1. The functions f (x), g(x), and h(x) are assumed to be continuously differentiable for a sufficient number of times. g(x) is assumed as non zero. As an example, vehicle models can be written in the form of P 1 [22] [28]. One design example of D 1 is presented as follow: First, an input-output approximate linearization is applied for the nonlinear plant P 1. We denote P n1 with approximated linearization as P n1a. Second, the design of D 1 for linear systems in Section 4 is solved for P n1a and the derived D 1 is applied to the nonlinear plant P Design of D 2 for Hammerstein Systems P 2 Hammerstein models are composed of a static nonlinear gain and linear dynamics. In some cases, they may be a good approximation of many nonlinear plants [18] [21]. Fig. 18 Proposed compensator structure for Hammerstein model. Fig. 19 Equivalent system of Fig. 18. Consider a plant P 2, given by the following form: v = I(u), P 2 : (36) ŷ = T (s)ˆv. u and y are the plant input and output, respectively. v is an intervening variable. The function I, which represents the static gain, is non-identically null and satisfies the following conditions: I(0) = 0, (37) I(v) <, v, (38) v = I 1 (I(v)) = I(I 1 (v)), (39) where I 1 is the inverse nonlinear gain. An example of I is I( ) = sin( ) if the domain of definition is ( π/2,π/2). For this case, I 1 ( ) = sin 1 ( ) holds. T (s) is the dynamic part of P 2 which is considered as a linear time-invariant system with a perturbation term. The nominal model is also given as a Hammerstein model P n2 as follows: v = I(u), P n2 : (40) y = T n (s)v. We assume that the difference between P 2 and P n2 is written by the following equation: T (s) T n (s) Δ(s)W(s)T n (s), Δ( jω) 1, ω, (41) where W is a weighting function. Then, the compensated system with the differential compensator D 2 is shown in Fig. 18. The compensated system is equivalent to the system shown in Fig. 19. The right-hand side of the system can be regarded as a linear time-invariant system. Therefore, we can apply the design method of Section 4 to obtain D 2 (s). 6. Discussion 6.1 Design for Non-Minimum Phase Plants In the case that P(s) is given as a non-minimum phase plant, we have to use its non-minimum phase model in our method.

7 SICE JCMSI, Vol. 6, No. 4, July When we want to minimize Γ s, it is required to give D(s) as a high gain filter. However, it is difficult to achieve high gain feedback when a non-minimum phase plant is given because we have to stabilize the closed-loop in Fig. 4. Therefore, it is difficult to design the good compensator D(s) for non-minimum phase plants compared to the case of minimum phase plants. 6.2 Constraints About Input Signals The control input signal is modified by the proposed method and the input-output relation is enforced to achieve the dynamics of P n (s). Therefore, variation of the input signal becomes large if ΔP is large. When input constraints take an important role in the control system, it is not easy to use our proposed method directly because input signals are modified by our proposed compensator. To consider the variation of inputs, we have to design D(s) more carefully and our method should be improved to achieve good input performance by Problem 1 with an input cost. 6.3 Comparison to the Traditional Robust Control If we only focus on the linear control system, the proposed structure with the feedforward controller and robust control for 2-dof structure is equivalent in the meaning of the design degree of freedom. However, not only control structure but also controller parameter design is important in the control systems design. In the case of the standard robust control design problem, its objective is to achieve a certain performance such as H 2 performance for all plants with a given perturbation. On the other hand, our objective is to minimize the modeling gap between the nominal plant and the compensated plant. The designer can use any model-based control methods for the plant with our proposed compensator. This is the merit of our proposed method compared to the previous robust control methods. Moreover, we can use not only a linear time invariant controller but also a non-linear controller and other controllers such as a continuous dead-beat controller with our proposed method. This is the advantage for the previous robust control methods because our method divides the robustness design and transient performance design. 6.4 Comparison to the Disturbance Observer Disturbance observer can eject the input disturbance by using the model of the actual plant. The proposed method is similar to the disturbance observer in the meaning that the model is used to achieve a desired input-output relation. However, the objective is different for the disturbance observer. The exact model is assumed to be given if we use the disturbance observer. On the other hand, our objective is to enforce the input-output relation close to the given nominal model. A merit of our structure is that an inverse model is not required to realize the control systems. In case of the disturbance observer, an inverse model of the plant is required to construct the system. However, it is difficult to make the inverse model for the plants such as non-linear systems. We need only the plant model itself to construct our proposed structure. 7. Conclusion This paper proposed a compensator structure for minimizing the error between the plant and the nominal model. The proposed compensator includes the nominal model, and the difference between y and y r is used for compensating the control input. For linear time-invariant plants, the compensator D is designed to minimize a model set of the plant. With the use of an appropriate D, the effect of the model error is minimized and the control output obtained with the proposed compensator closes to the ideal response designed with the nominal model. Moreover, the proposed structure can be used for nonlinear control systems. Simple design of two nonlinear systems is presented. From the results, we can divide the design of the nominal control performance and the compensation of the robustness easily. The proposed compensator has a broad range of applications. In future, the authors intend to propose a design method for MIMO systems. References [1] P. Gahinet: Explicit controller formulas for LMI-based H synthesis, Automatica, Vol. 32, No. 7, pp , [2] J.C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis: State-space solutions to standard H 2 and H control problems, IEEE Transactions on Automatic Control, Vol. 34, No. 8, pp , [3] K. Zhou, J.C. Doyle, and K. Glover: Robust and Optimal Control, Prentice Hall, [4] J.M. Maciejowski: Predictive Control with Constraints, Prentice Hall, [5] J.A. Rossiter: Model-Based Predictive Control: A Practical Approach, CRC Press, [6] S. Fujimoto and K. 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9 SICE JCMSI, Vol. 6, No. 4, July Hiroshi OKAJIMA (Member) He received his M.E. and Ph.D. degrees from Osaka University, Japan, in 2004 and 2007, respectively. He is presently an Assistant Professor of Kumamoto University, Japan. His research interests include tracking control, analysis of non-minimum phase systems and data quantization for networked systems. He is a member of ISCIE. Hironori UMEI (Student Member) He received his B.E. degree from Kumamoto University, Japan, in He is a second year master s degree student at Kumamoto University, Japan. His research interests include robust control, internal model control structure and continuous deadbeat control. Nobutomo MATSUNAGA (Member) He received M.D. and Ph.D. degrees from Kumamoto University, Japan, in 1987 and 1993, respectively. He joined OMRON Corp. in Since 2002, he has been with the Department of Computer Science and Electrical Engineering, Kumamoto University, where he is a Professor. His research interests include thermal process control, automotive control, and human-machine system design. He is a member of IEEJ, ASME and IEEE. Toru ASAI (Member) He received his B.E., M.E. and Ph.D. degrees from the Tokyo Institute of Technology, Japan, in 1991, 1993 and 1996, respectively. He has worked as a Research Fellow of JSPS between 1996 and In 1999, he joined the faculty of Osaka University, where he is currently an Associate Professor of the Department of Mechanical Engineering. His research interests include robust control and switching control. He is a member of ISCIE and IEEE.