PAPER Sensitivity of Time Response to Characteristic Ratios

Transcription

1 520 IEICE TRANS FUNDAMENTALS, VOLE89 A, NO2 FEBRUARY 2006 PAPER Sensitivity of Time Response to Characteristic Ratios Youngchol KIM a), Member, Keunsik KIM b), and Shunji MANABE, Nonmembers SUMMARY In recent works [], [4], it has been shown that the damping of a linear time invariant system relates to the so-called characteristic ratios (α k, k,, n ) which are defined by coefficients of the denominator of the transfer function However, the exact relations are not yet fully understood For the purpose of exploring the issue, this paper presents the analysis of time response sensitivity to the characteristic ratio change We begin with the sensitivity of output to the perturbations of coefficients of the system denominator and then the first order approximation of the α k perturbation effect is computed by an explicit transfer function The results are extended to all-pole systems in order to investigate which characteristic ratios act dominantly on step response The same analysis is also performed to a special class of systems whose denominator is composed of so called K-polynomial Finally, some illustrative examples are given key words: sensitivity, characteristic ratios Introduction In control system design, time response specifications, mainly overshoot and the speed of response, are the most popular ways of describing transient response In [], some classical studies regarding transient response are reviewed The essence of these studies by Naslin [2] and Kessler [3] is to characterize the transient response in terms of coefficients of the characteristic polynomial rather than by its roots by defining the characteristic ratios Naslin claimed that the characteristic ratios are closely related to the overshoot of the system step response It was shown that all-pole systems with characteristic polynomials that share the same characteristic ratios give remarkably similar step responses This led to the new design techniques called the characteristic ratio assignment (CRA) in [] and the coefficient diagram method (CDM) [4] Kitamori [5] has investigated a partial model matching method based on the so called Independency from successors: IFS, wherein he suggested a standard form of all pole type reference model that can be simply applied to the problem of obtaining good transient responses with PID controller He has also pointed out that the step responses are mostly affected by the coefficients of lower-order terms in transfer function but are little affected Manuscript received August 2, 2004 Manuscript revised June 27, 2005 Final manuscript received November 4, 2005 The author is with the School of Electrical and Computer Engineering, Chungbuk National University, Korea The author is with the School of Automobile Engineering, Ajou Motor College, Korea The author is at Fujisawa-shi, Japan a) yckim@cbuackr b) kskim@motorackr DOI: 0093/ietfec/e89 a2520 by those of higher-order terms The CDM and CRA methods require properly choosing the characteristic ratios in order to build a target model that meets the desired time response However, the analytical relationship between time response and α k is not yet known This paper presents some new results about the time response characteristics of α k by analyzing the sensitivity of step response to the α k perturbations To consider the sensitivity of output y(t,α k )ofa system having a parameter of interest, α k, we compute the effect of a perturbation α k on the nominal response by using Taylor s series expansion, y(t,α k + α k ) y(t,α k ) + y α k +, () α k for k, 2,, n The first order approximation of the α k perturbation effect is y k (t) y α k (2) α k This function can be generated from the system itself as shown by Franklin et al [6] and Perkins et al [8] The output sensitivity model in [8] is derived using superposition and several block diagram manipulations However, the process of obtaining it is somewhat complicated In the present paper, we provide a new method for directly generating the time response sensitivity relative to the individual characteristic ratio change This sensitivity represents how effectively each characteristic ratio relates to the time response In a later section, we will establish the fact that the overshoot of the step response of all-pole systems is largely affected by the characteristic ratios,α 2 and α 3 This means that the rest of the α k s have little effect on the step response but are crucial for maintaining stability In [], a method that obtains a special all-pole system with small or no overshoot has been proposed This characteristic polynomial is completely characterized by the principal characteristic ratio and the remaining characteristic ratios are fixed functions of We call the polynomial K polynomial here It was also shown that the stability of K polynomial is always preserved for any 2 This paper also deals with the sensitivity analysis of K polynomial to the perturbation Some examples are given for illustration 2 Definitions and Preliminary Results In this section, we develop some preliminary results of time Copyright c 2006 The Institute of Electronics, Information and Communication Engineers

2 KIM et al: SENSITIVITY OF TIME RESPONSE TO CHARACTERISTIC RATIOS 52 Fig Kessler s 2 loop structure response sensitivity to coefficient changes Consider a linear system whose transfer function is T(s) n(s) δ(s) b ms m + b m s m + + b 0 (3) a n s n + a n s n + + a 0 The characteristic ratios are defined as: a2,α 2 a2 2,,α n a2 n (4) a 0 a 2 a a 3 a n 2 a n and the generalized time constant is defined to be τ : a (5) a 0 Conversely, the coefficients a i of δ(s) may also be represented in terms of α i s and τ as follows: a a 0 τ (6) a 0 τ i a i, for i2,, n (7) α i α 2 i 2α3 i 3 αi 2 2 αi Although the analytical relationships between damping and characteristic ratio are not yet known, the dependency can be explained by using the Kessler s multiple loop structure [3] For the purpose of this discussion, let us consider the 2-loop system shown in Fig The transfer function of the overall system is T(s) + τ s [ + τ 2 s( + τ 3 s)] τ τ 2 τ 3 s 3 + τ τ 2 s 2 + τ s + According to Eq (4), the characteristic ratios of T(s) are τ, α 2 τ 2 (8) τ 2 τ 3 Now, we first consider the dynamics of the inner loop system Its transfer function and the characteristic ratio are as follows: T i (s) + τ 2 s( + τ 3 s) τ 2 τ 3 s 2 + τ 2 s +, α τ 2 α 2 (9) τ 3 Secondly, if we assume that with τ >τ 2 τ 3, T(s) can be approximated as follows: T(s) + τ s( + τ 2 s) τ τ 2 s 2 + τ s + :T 0(s) Then the characteristic ratio of T 0 (s) becomes αˆ τ (0) τ 2 Furthermore, it is easily seen that the damping ratio of an second-order system is identical to ζ 2 Since the allpole system of arbitrary order is also developed in the same manner, we can say from Eqs (9), (0) that the characteristic ratio of an all-pole system is closely related to the damping It was shown in [] that τ represents the speed of the response of a system with denominator δ(s) The speed of time response can be controlled by the generalized time constant independent on the characteristic ratios As mentioned in the introduction, our concern of interest is to find out how the step response of the system in Eq (3) changes as each characteristic ratio changes We now define the unnormalized function sensitivity to the perturbation of the j-th coefficient of δ(s) as USa T j : T(s) () /a j Let the unit step response of the unnormalized function sensitivity be ya s j (t) That is, Ya s j (s) USa T j (s) R(s) wherethe input R(s) /s As shown earlier, the effect of a perturbation a j about a nominal a j can be similarly evaluated by a Taylor s series: y(t, a j + a j ) y(t, a j ) + y a j +, (2) for j 0,, 2,, n y(t, a j ) : y a j can be determined by the following Lemma, which will be used for the proof of the main results in a later section Lemma : Consider a linear system in Eq (3) Then the unnormalized function sensitivity and the first order approximation of output response change to the perturbation of j-th coefficient of δ(s) are determined by (i) USa T j a js j T(s), for j 0,,, n (3) δ(s) (ii) y(t, a j ) a j ya s a j (t) (4) j Proof: It is straightforward to derive part (i) For the proof of part (ii), let us consider the Taylor s series of T(s) when a j perturbs T(s, a j + a j ) T(s, a j ) + T a j +, for j 0,, 2,, n The first order approximation of a j perturbation effect, T(s, a j ), becomes as follows: T(s, a j ): T a j a j USa T a j j Since y(t, a j ) is the step response of T(s, a j ), the proof is completed

3 522 We now introduce another type of sensitivity that has the relative form It is seen from Eqs (6) and (7) that the coefficients of δ(s) are nonlinear functions of the characteristic ratios We define the coefficient sensitivity as S a j α k : /a j α k (5) α k /α k a j α k The following relationship will be used in the derivation of the main result We state the Lemma without the proof Lemma 2: For the linear system in Eq (3), the coefficient sensitivity of a j with respect to the perturbation of α k is { S a j ( j k) ifk < j, for k, 2,, n, α k (6) 0, if k j, j 0,, 2,, n 3 Time Response Sensitivity to Characteristic Ratio Change In this section, we will present how the step response changes as the characteristic ratio changes Time response sensitivities are studied for three cases: () a general transfer function as in Eq (3), (2) all-pole systems of degree n,(3)a special class of all-pole system whose denominator shall be composed of K polynomial 3 Time Response Sensitivity: A General Case We first define the other unnormalized function sensitivity relative to characteristic ratios as follows: US T α k : T(s) α k /α k (7) Recall Eqs () and (2) Let the unit step response of US T α k be Y s α k (s) That is, Y s α k (s) US T α k R(s), (8) where R(s) /s yα s k (t) indicates the inverse Laplace transform of Yα s k (s) Then the following Theorem states that y(t) in Eq (2) can be computed by an explicit transfer function Theorem : Given a stable T(s) as in Eq (3), the unnormalized function sensitivity and the first order approximation of step response perturbation to the α k change are determined by (i) US T α k : jk+ ( j k)a j s j T(s) (9) δ(s) (ii) y k (t) α k yα s α k (t), for k,, n (20) k Proof: Since δ(s) is stable, all a j are non-zero and positive Therefore, T : T(s, a 0, a,, a n ) is continuously differentiable with respect to a j and the coefficients a j are also differentiable to α k because they are functions of α k s Using the so-called chain rule we have IEICE TRANS FUNDAMENTALS, VOLE89 A, NO2 FEBRUARY 2006 T α k T a k+ + T a k+2 (2) a k+ α k a k+2 α k + T + a k+3 + T a n a k+3 α k a n α k Eq (22) was derived using the fact that a j is the function of α k for only k ( j ) Now rewrite Eqs () and (5) as T USa T a j, j (22) a j S a j α α k α k k (23) From Eqs (7) and (22), (23), we have US T α k α k T α k jk+ US T a j S a j α k (24) Using Lemma and Lemma 2 in this sequence, Eq (24) becomes USα T k ( j k)a j s j n(s) jk+ ( j k)a j s j T(s) δ(s) jk+ Therefore, part (i) has been proven The proof of part (ii) is similar to the one of Lemma Thus, from Eq (7), it is easily seen that T k (s) : T(s,α k ) T α k α k USα T α k α k k Since y k (t) is identical to the step response of T k (s), the proof is completed Remark : (The effect of τ on the output sensitivity function): So far, we have only dealt with the time response sensitivity for characteristic ratios under the assumption that the generalized time constant is constant In [], it was shown that if two systems, whose generalized time constants are τ and τ 2 respectively, share the same characteristic ratios, then the step response is exactly time-scaled by factor β τ /τ 2 From this result, it is easy to derive that ( ) yτ s 2 (t) yτ s β t, (25) where yτ s k (t)fork, 2 indicate the output sensitivity functions of the systems having τ k while their characteristic ratios share the same values Example : (Time response sensitivities when and α 3 perturb): Consider the system in Eq (3) of which n(s) s 3 + 5s 2 + s + 5 δ(s) s s s s s 3

4 KIM et al: SENSITIVITY OF TIME RESPONSE TO CHARACTERISTIC RATIOS s 2 + 5s + 5 The generalized time constant is τ and the characteristic ratios of the nominal model are [,,α 6 ][2285, 777, 65, 65, 777, 2285] Let the step response of the nominal system be y o (t) Consider that α k for k, 3 are changed by ±0% individually while the rest of α k s are fixed at the same values as the nominal system The y p (t) denotes the step response of the corresponding perturbed model Then the first-order approximation y k (t) of step response perturbation corresponding to the deviation of α k, α k ±0α k is computed by using (9) and (20) Neglecting all the higher-order terms in (), the estimated value of y p (t) may be defined by y s α k (t) : y o (t) + y k (t) (26) Figure 2 and Fig 3 show y s and step responses for the cases where is changed by 0%, respectively The estimated output y s (t) is shown closely along with the true response y p (t) As predicted by part (ii) of Theorem, the parameter perturbation effect either increases or decreases according to the sign of y k (t) Figure 4 and Fig 5 show the cases where α 3 is changed by ±0% The results are almost the same However, it is noted that the profile of yα s 3 has the largest value at the second extremum, and the shape is quite different from that of yα s In order to look into the reason furthermore, rewriting (8) and (9), Yα s (a 2 s + 2a 3 s a 7 s 6 n(s) ) (27) Yα s 3 (a 4 s 3 + 2a 5 s 4 + 3a 6 s 5 + 4a 7 s 6 n(s) ) (28) Let the impulse response of H(s) : n(s) denote h(t) Taking the inverse Laplace transform to (27) and (28), we have yα s (t) a 2 h (t) + 2a 3 h (t) + + 6a 7 h (6), (29) yα s 3 (t) a 4 h (3) (t) + 2a 5 h (4) (t) + 3a 6 h (5) (t) (30) +4a 7 h (6), where h (n) indicates the n-th derivative of h(t) Comparing (29) with (3), it reveals evidently that yα s k (t) does not contain lower-order derivatives of h(t) than that of the k th As a result, the shapes of yα s k (t) shall be different according to the order k Also, it is remarkable that yα s k (t) depends on H(s) as well Fig 2 Step responses and output sensitivity function with (Example ) Fig 4 Step responses and output sensitivity function with α 3 (Example ) Fig 3 Step responses and output sensitivity function with 09 (Example ) Fig 5 Step responses and output sensitivity function with 09α 3 (Example )

5 524 IEICE TRANS FUNDAMENTALS, VOLE89 A, NO2 FEBRUARY Dominant Characteristic Ratios of All-Pole Systems Since the transient response is generally affected by zeros as well as poles, it is more appropriate to consider the all-pole systems in order to study the pure time response sensitivity of the characteristic ratio This section presents the issue As a result, we will establish the fact that the transient response of all-pole systems is dominantly affected by only the principal characteristic ratios,α 2 and α 3 Consider the stable all-pole system T A (s) a 0 δ(s) a 0 (3) a n s n + a n s n + + a 0 Then from Eq (9) in Theorem, the unnormalized function sensitivity for T A (s) isgivenby US T A α k : T A(s) α k /α k jk+ ( j k)a j s j T A (s) (32) δ(s) For the sake of convenience, we describe the output sensitivity functions Yα s k US T A α k R(s) when the degree of δ(s)is n 7 Yα s a 0(a 2 s + 2a 3 s 2 + 3a 4 s a 7 s 6 ) Yα s 2 a 0(a 3 s 2 + 2a 4 s 3 + 5a 7 s 6 ) (33) Yα s 5 a 0(a 6 s 5 + 2a 7 s 6 ) Yα s 6 a 0(a 7 s 6 ) Before going on to Eq (33), we introduce two important properties regarding the relationships between τ, α k and coefficients of δ(s) As mentioned in Remark (see also []), the change of τ in Eqs (6), (7), make the output sensitivity function in the time domain merely time-scaled In other words, the minimum and the maximum values of yτ s k remain the same regardless of the τ change Thus, we may set τ without loss of generality as long as we deal with the sensitivity problems of α k The other property comes from results developed in [9] and [4] Several sufficient conditions for stability in [9] are in terms of the coefficients of the characteristic polynomial Rewriting the conditions in terms of α k, a real polynomial is stable if one of the following is satisfied (i) αi α i+ > 4656 for i, 2,, n 2, (ii) α i 2374( α i + α i+ )fori, 2,, n 2 Furthermore, Lipatov and Sokolov discovered the fact that if α k 4forallk, 2,, n, then every root of δ(s) is distinctively located on the negative real axis Manabe [4] has investigated the problem of obtaining good transient response of control systems by means of the characteristic ratio and the generalized time constant (which he calls the stability index and the equivalent time constant) According to his observations, the all-pole system of any degree of which its characteristic ratios are 25,α i 2for i 2, 3,, n gives good damping Now, recalling that τ, and if we substitute the conditions of Lipatov and Sokolov above into Eq (7), it is obvious that higher degree coefficients of δ(s) should be much smaller than those of lower degree In other words, the following inequality holds: a 0 a > a 2 > a 3 > > a n, Applying this relationship to Eq (33) results in y s max > y s α 2 max > > y s α 6 max (34) where y s α k max indicates the maximum value of the impulse response of Y s α k According to (20) and (26), the α k that gives rise to the large y s α k (t) should dominantly act on step response However, it is difficult to quantitatively determine by what characteristic ratios its step response is significantly connected, because y s α k (t) depends upon δ(s) We have observed through many simulations that only three principal characteristic ratios,, α 2 and α 3, are the most dominant factors dictating the damping of transient response This means that the above result accords well the Kitamori s assertion which was cited in the introduction The following example explains this Example 2: Consider a stable all-pole system that is of Type I, n 7andτ The values of the characteristic ratios for the nominal model are chosen to be the same values as those in Example The coefficients of corresponding δ(s) are given as follows: [a 0, a,, a 7 ] [5, 5, 6565, 67, 024, , 0 3, ] Comparing the values of a 2, a 3, a 4 with a 6, a 7,itisverified immediately from Eq (33) that Eq (34) holds The problem of interest is to show by which values of α k for k, 2,, n the transient response is largely affected To do this, we make one characteristic ratio at a time either increased by 25 times or decreased by 08 times while the rest of α k s are fixed at nominal values Since every pole is real and negative if α k 4forallk, increasing the value more than 25 times is of no use The decreasing factor was determined near the marginal value for which the stability condition of Lipatov and Sokolov is not lost For all these cases, step responses and output sensitivity functions have been computed Figure 6 Fig 9 show the results for four cases The effects of α k for k 5 are vanishingly small and can be neglected In the figures, y 25 (t) andy 08 (t) indicate the step response of T A (s) for which only one α k is changed to 25α k and 08α k, respectively It is shown that,α 2 and α 3 have much greater influence than the rest

6 KIM et al: SENSITIVITY OF TIME RESPONSE TO CHARACTERISTIC RATIOS 525 Fig 6 Step responses and output sensitivity function when changes (Example 2) Fig 9 Step responses and output sensitivity function when α 4 changes (Example 2) remains for most stable T(s)in(3) 33 Time Response Sensitivity for A Special System with K Polynomial In the present section, we consider a special class of systems whose denominator shall be composed of K polynomial [] The K polynomial δ k (s) is defined as the polynomial whose coefficients are generated using Eqs (6),(7) and the following function: Fig 7 Step responses and output sensitivity function when α 2 changes (Example 2) Fig 8 Step responses and output sensitivity function when α 3 changes (Example 2) Remark 2: (The dominant characteristic ratios of system including zeros): The dominance of characteristic ratios on the step response has been analyzed for the all pole type transfer functions For the case where system T(s) contains zeros, higher-order characteristic ratios over, α 2 and α 3 may be connected to the time response This is the reason that as shown in (27) (3), y s α k (t) depends upon not only the denominator of T(s) but also the numerator However, it is our observations that the dominance of three, α 2 and α 3 (i) > 2 (35) (ii) α k sin ( ) ( ) kπ n + sin π n 2sin ( ), (36) kπ n for k 2,, n For instance, when n 6 and choose 2464, (37) results in [ α 2 α 3 α 4 α 5 ] [ ] Then it is easy to generate a δ k (s) using (6) and (7) In [], it was shown that such an all-pole system T A (s) a 0 δ k (s) guarantees the stability and its frequency magnitude function is monotonically decreasing Furthermore, it is important to note that the K polynomial is generated by only for a given τ Let sdefinet K (s) n(s) δ k (s) Now, we are going to examine the time response sensitivity when perturbs Note that T K (s) is a function of only when τ and a 0 are given Lemma 3: The coefficient sensitivity of T K (s) relative to the change is determined by S a j : /a j / j (k ) (37) k Proof: The proof is omitted here Let the unit step response of US T K be Yα ks (s) That is, Yα ks (s) US T K R(s), where R(s) /s y ks (t) indicates the inverse Laplace transform of Yα ks (s) Then, similarly to Section 3, we have the following result Theorem 2: Given a T K (s) as in Eq (3), the unnormalized

7 526 IEICE TRANS FUNDAMENTALS, VOLE89 A, NO2 FEBRUARY 2006 function sensitivity and the first order approximation of step response perturbation to the change are determined by (i) US T K : j a j s j δ k (s) T K(s) j (k ), (38) (ii) y(t, ) y ks α (t) (39) Proof: Since T K (s) is in the form of Eq (3), by Eq (3) we have US T K a j : T K(s) a js j n(s) /a j δ 2 k (s), for j, 2,, n (40) From Eq (22), T K k a j US T K a j (4) Since T K (s) T K (s, a 0, a,, a n ) is stable because K polynomial is always stable and its coefficients are continuously differentiable with respect to, by using the chain rule, we have T K T K a 2 + T K a T K a n (42) a 2 a 3 a n By Eqs (5), (6) and (4), the unnormalized function sensitivity of T K (s) is given as US T K : T k / j US T K a j S a j (43) Finally, by substituting Eq (40) and Lemma 3 into Eq (43), part (i) is proven The proof of part (ii) is trivial because it is derived in the same manner as Theorem Acknowledgment This work was supported by grant NoR from the Basic Research Program of the Korea Science & Engineering Foundation References [] YC Kim, LH Keel, and SP Bhattachayya, Transient response control via characteristic ratio assignment, IEEE Trans Autom Control, volac-48, no2, pp , Dec 2003 [2] P Naslin, Essentials of Optimal Control, Boston Technical Publishers, Cambridge, MA, 969 [3] C Kessler, Ein Beitrag zur Theorie mehrschleifiger Regelungen, Regelungstechnik, vol8, no8, pp26 266, 960 [4] S Manabe, Coefficient diagram method, Proc 4th IFAC Symposium on Automatic Control in Aerospace, pp99 20, Seoul, Korea, 998 [5] T Kitamori, A method of control system design based upon partial knoweldge about controlled process, Trans of Society of Instrument and Control Engineers in Japan, vol5, no4, pp , 979 [6] GF Franklin, JD Powel, and A Emani-Naeini, Feedback Control of Dynamic Systems, Prentice-Hall, Upper Saddle River, NJ, 2002 [7] LP Huelsman, Active and Passive Analog Filter Design: An Introduction, McGraw-Hill, 993 [8] WR Perkins, PV Kokotovic, T Bourret, and JL Schiano, Sensitivity function methods in control system education, Proc IFAC Symposium on Advances in Control Education, pp23 28, Massachusetts, USA, 99 [9] AV Lipatov and NI Sokolov, Some sufficient conditions for stability and instability of continuous linear stationary systems, Automation and Remote Coontrol, vol39, pp285 29, 979 [0] K S Kim, YC Kim, LH Keel, and SP Bhattacharyya, PID controller design with response specifications, Proc 2003 American Control Conference, pp , Denver, CO, Conclusions When we deal with the problem of time response control of a linear system, it has been mainly carried out in root space Recently, instead of pole and zero, a different method that uses the relationships between coefficients of characteristic polynomial and time response such as overshoot and response rate has started to attract attention The idea was initially provided by Naslin in the mid of 960s [2] The present paper presented some new results in this regard Through the analysis of time response sensitivities to the characteristic ratios, we have shown how the parameters relate to the step response In particular, if it is the all-pole system, the damping of step response is dominantly characterized by only,α 2 and α 3 If we recall that the speed of response for a given LTI system is completely characterized by the generalized time constant τ [], we can conclude after all that the dominant factors representing transient response behavior are,α 2, α 3 and τwehavealsoanalyzed the sensitivity of time response to a special class of system of which the denominator is composed of K polynomial [] These results can be useful for constructing a reference model by means of the characteristic ratio, as shown in CRA [], [0] and CDM [4] Youngchol Kim received his BS degree in Electrical Engineering from Korea Univ, Korea in 98, and his MS and PhD degrees in Electrical Engineering from Seoul National Univ, Korea in 983 and 987 respectively Since March 988, he has been with the School of Electrical &Computer Eng, Chungbuk National Univ, Korea, where he is currently a Professor He was a post-doctoral fellow at Texas A&M Univ, , and a visiting research fellow at the COE-ISM, Tennessee State Univ / Vanderbilt Univ, He served as an Associate Editor for the Korean Institute of Electrical Engineers, He is the director of Technical Society of Control and Instrument, KIEE since 2004 He is a member of IEEE, KIEE, IEEK, and ICASE His current research interests include robust control and low-order controller design in parameter space

8 KIM et al: SENSITIVITY OF TIME RESPONSE TO CHARACTERISTIC RATIOS 527 Keunsik Kim was born in Yesan, Korea, on May, 2, 963 He received his BS and MS degrees in Electronic Engineering from Hanyang University, Korea in 985 and 987, respectively, and his PhD degree in Electronic Engineering from Chungbuk National University, Korea in 2003 During , he worked in Agency for Defence Development (ADD), Korea In 994, he was employed by Korea Automotive Technology Institute as a research engineer In 996, he joined Ajou Motor College, where he is currently an associate professor His research interests include low-order controller design and robust control in parameter space Shunji Manabe received BE from Tokyo Univ in 952, MS from the Ohio State Univ in 954, and PhD in engineering from Tokyo Univ in 962 He has been engaged in system design of various control systems at Mitsubishi Electric Corporation, such as motor control, radar antenna control, computer control, and spacecraft control From 990 to 2000, he taught contol theory at Tokai University His current interest is the development of the Coefficient Diagram Method, a new simple and efficient control design approach He is a member of IEEE, AIAA, IEE-J, SICE, and IFAC aerospace technical committee He has served as an editor and board member at IEE-J and SICE on various occasions