Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles67 Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles Kou Yamaa, Takaaki Hagiwara, Iwanori Murakami 3, Shun Yamamoto 4, an Hieharu Yamamoto 5, Non-members ABSTRACT In the present paper, we consier a esign metho that provies low-sensitivity control with robust stability for multiple-input/multiple-output continuous time-invariant systems having an uncertain number of right half plane poles. First, the class of uncertainty consiere in the present paper is efine an the necessary an sufficient robust stability conition is presente for the system having this class of uncertainty uner the assumption that the number of close right half plane poles of the plant is equal to that of the nominal plant. The relationship between the plant an the nominal plant inclue in this class of uncertainty is clarifie. Using this relationship, we will show the necessary an sufficient robust stability conition for the system having an uncertain number of right half plane poles. Keywors: Robust Stability, The Relative Degree, Multiplicative Uncertainty, Uncertain Number of Right Half Plane Poles. INTRODUCTION In the present paper, we examine a esign metho that provies low-sensitivity control systems with robust stability for multiple-input/multiple-output continuous time-invariant systems having an uncertain number of right half plane poles. Several stuies have been conucte on the robust stabilization problem [ 7]. Doyle an Stein erive the basic solution for this problem [, ], an the necessary an sufficient conitions for the multiplicative uncertainty an aitive uncertainty were shown. Chen an Desoer erive the complete proof of the solution presente by Doyle an Stein [3]. Kishore an Pearson clarifie if a class of uncertainty is close set then the gap between the necessary robust stability conition an the sufficient one exist[4]. In aition, if a class of Manuscript receive on July 6, 009 ; revise on,.,,3,4,5 The authors are with Department of Mechanical System Engineering, Gunma University -5- Tenjincho, Kiryu 376-855 Japan, E-mail: yamaa@gunma-u.ac.jp, t088005@gunma-u.ac.jp, murakami@gunma-u.ac.jp, t098054@gunma-u.ac.jp an m07m5@gs.eng.gunmau.ac.jp uncertainty is open set then the necessary robust stability conition is equal to that of sufficient one [4]. In this way, the robust stability conition for the system with invariant number of right half plane poles completely clarifie. Kimura consiere the robust stabilizability problem for single-input/single-output systems [8]. Viyasagar an Kimura expane the finings of Kimura for multiple-input/multiple-output systems [9]. Accoring to various reports [ 3], in orer to maintain stability for a large uncertainty, the complementary sensitivity function must be small. Ensuring that the complementary sensitivity function is small reuces the performance of the control systems by isturbance attenuation, etc., because the value of the sensitivity function increases. Since sum of the sensitivity function an the complementary sensitivity function is equal to, obtaining either lowsensitivity or high robust stability characteristics is impossible. However, controllers that achieve lowsensitivity characteristics o not always make the system unstable. Maea et al. treate this problem as an infinite gain margin problem [0, ]. Nogami et al. clarifie the conition in which the hi-gain controller oes not make the system unstable an propose a esign metho [] by reucing the robust passivity problem. Doyle et al. consiere this lowsensitivity control problem from the another viewpoint: there exists a class of uncertainty that has lowsensitivity that makes the system robustly stable [3]. Therefore, from the uncertainty escribe by Doyle, Francis an Tannenbaum [3], we can construct lowsensitivity characteristics with robust stability. Thus, the uncertainty presente by Doyle, Francis an Tannenbaum is suitable for hi-performance robust control system esign. However, this uncertainty cannot be applie to a system having an uncertain number of close right half plane poles. There exist applications such that the number of right half plane poles changes. For example, the number of right half plane poles of a large flexible spacecraft changes when the configuration of the spacecraft is change [9]. The problem of obtaining the robust stability conition for the system having an uncertain number of the
68 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.8, NO. February 00 close right half plane poles is ifficult because the problem oes not reuce to the small gain theorem. In the present paper, we expan the uncertainty presente by Doyle, Francis an Tannenbaum [3, 4] to be applicable to the multiple-input/multipleoutput system having an uncertain number of open right half plane poles. First, the class of uncertainty to be consiere in the present paper is efine. If this class is assume that the number of right half plane poles of the nominal plant is equal to that of the plant, this class of uncertainty is the same to that efine by Doyle, Francis an Tannenbaum [3]. The necessary an sufficient robust stability conition for the system of this class of uncertainty is presente uner the assumption that the number of right half plane poles of the nominal plant is equal to that of the plant. Next, the conition uner which the set of the plant is inclue in the above-mentione class is clarifie. Using this relationship between the nominal plant an the plant, the necessary an sufficient robust stability conition is obtaine when the number of poles of the plant is not necessarily to be equal to that of the nominal plant. The robust stability conition escribe in the present paper is ientical, whether or not the number of poles of the nominal plant is equal to that of the plant. Generally, in previous stuies, the number of poles of the plant is assume to be equal to that of the nominal plant [ 8]. Verma, Helton an Jonckheere consiere the robust stabilizability problem [5], but they i not consier the class of uncertainty that is consiere in the present paper. The robust stability conition iscusse in the present paper is obtaine using a kin of phase information, that is, the relative egree of the plant an the nominal plant. Therefore, the robust stability conitions consiere in the present paper for the system having the same number of right half plane poles is ientical to that having an uncertain number of right half plane poles. Conversely, the robust stability conition use by Verma, Helton an Jonckheere for the system having the same number of right half plane poles is ifferent from that having an uncertain number of right half plane poles. Therefore, the finings presente by Verma, Helton an Jonckheere an those of the present paper are ifferent. Notation F u (P, Q) upper LFT, that is F u (P, Q) = P + [ P Q(I P] Q) P, where P P P = P P F l (P, Q) lower LFT, that is F l (P, Q) = P + P Q(I P Q) P R real part of C σ the maximum singular value of. PROBLEM FORMULATION Consier the control system as below. y(s) = G(s)u(s) u(s) = C(s) (r(s) y(s)) () Here, G(s) R p m (s) is the strictly proper multipleinput/multiple-output plant. C(s) R m p (s) is the controller, r(s) R p is the reference input, y(s) R m is the output an u(s) R m is the control input. The plant G(s) is assume to be stabilizable, etectable an p m. The state space escription of the plant G(s) is enote by A B G(s) = R p m (s). C 0 The nominal plant of the plant G(s) enotes Am B G m (s) = m R p m (s). C m 0 Here, the number of zeroes of the nominal plant G m (s) in the close right half plane is assume to equal that of the plant G(s). That is, the number of s 0 C in the close right half plane satisfying A s0 I B < n + p () C 0 is equal to that of s 0 C in the close right half plane satisfying Am s 0 I B m < n + p (3) C m 0 with counting multiplicity. (C m, A m, B m ) is assume to be stabilizable an etectable. Let the plant G(s) be enote using the nominal plant G m (s) an the multiplicative uncertainty by R R +e C R(s) [ A B C D ] the set of real numbers the set of real numbers with infinite the set of complex numbers the set of all real-rational transfer functions represents the state space escription of C(sI A) B + D H norm G(s) = (I + (s)) G m (s). (4) Without loss of generality, (s) is assume to be stabilizable an etectable an I + (s) is of normal full. The sensitivity function S(s) an the complementary sensitivity function T (s) of the control system () are enote by S(s) = (I + G m (s)c(s)) (5)
Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles69 an T (s) = I S(s) = G m (s)c(s) (I + G m (s)c(s)), (6) respectively. In the present paper, we consier the robust stability problem for the following class of plants. Definition : The set of plants is enote by Ω. The elementary of the set Ω satisfies following expressions. The number of zeroes of the plant G(s) in the close right half plane is equal to that of the nominal plant G m (s). The number of right half plane poles of the plant G(s) is not necessarily equal to that of the nominal plant G m (s). σ (I + (jω)) (jω) < W (jω) (ω R +e ), (7) where W (s) R(s) is a stable rational function. If (s) hols above expressions, we enote simply (s) Ω. As escribe in a later section, by aopting the set of uncertainty efine in Definition, the robust stability conition is irectly relate not to the complementary sensitivity function, but to the sensitivity function. In other wors, a low-sensitivity controller can guarantee robust stability. Before consiering the robust stability conition for the class of the plant set Ω, the robust stability conition for the class of the plants efine in Definition. Here, the number of right half plane poles of the plant G(s) is assume to be equal to that of the nominal plant G m (s). Definition : Ω is the set of plants. The elementary of the set Ω satisfies following expressions. The number of zeroes of the plant G(s) in the close right half plane is equal to that of the nominal plant G m (s). The plant G(s) has the same number of right half plane poles as that of the nominal plant G m (s). σ (I + (jω)) (jω) < W (jω) (ω R +e ), (8) where W (s) R(s) is a stable rational function. The robust stability conition for the set of plants Ω is summarize in the following theorem. Theorem : Assume that C(s) stabilizes the nominal plant G m (s). C(s) is a robust stabilizing controller for Ω if an only if hols. S(s)W (s) (9) an Proof: Let P (s) an (s) be W (s)i W (s)gm (s) P (s) = I G m (s) (0) (s) = (I + (s)) (s) W (s), () respectively. Proof is immeiately obtaine by applying Theorem 3.3 in the book written by Mcfarlane an Glover [5] to obtain F u (P (s), (s)) = (I + (s)) G m (s) () F l (P (s), C(s)) = (I + G m (s)c(s)) W (s) = S(s)W (s), (3) thereby completing the proof of Theorem. Theorem shows that if the plant G(s) can be place in the form of Definition, low-sensitivity can be achieve for the robust stability conition. 3. RELATIONSHIP BETWEEN THE NOM- INAL PLANT AND THE PLANT In this section, the relationship between the nominal plant G m (s) an the plant G(s) that satisfies Theorem is escribe. To maintain the internal stability conition, the system () must be well-pose. Therefore, the controller must be proper. Since (5) an the nominal plant G m (s) is assume to be the strictly proper, when the controller C(s) is proper, the sensitivity function has the property: lim (S(jω)) = I. (4) By σs(jω) is performe multiplication on both sies of (8), we have σ S(jω) (I + (jω)) (jω) σ S(jω) σ < σ S(jω) W (jω) (I + (jω)) (jω) = σ S(jω)W (jω). (5) In orer to satisfy (9), from (4) an (5), lim σ S(jω) (I + (jω)) (jω) < lim σ S(jω)W (jω) (6) is require. Hence lim σ (I + (jω)) (jω) < lim W (jω) (7)
70 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.8, NO. February 00 is require. From (7), we obtain the following theorem. Theorem : Necessary conition that there exist controllers to satisfy Theorem, that is necessary conition that (s) satisfies (7), is that I + (s) is biproper. That is, when I + (s) is enote by A B I + (s) =, (8) C D the necessary conition that there exist controllers to satisfy Theorem is D = p. (9) Proof: Proof is obtaine by showing that if the nominal plant G m (s) is not biproper, that is, (9) is not satisfie, then (7) is not satisfie. For simplicity, let (s) = I + (s), then (I + (s)) (s) = (s) ( (s) I ) = I (s). (0) If I + (s) is not biproper but proper, then (s) is not proper. This implies that I (s) is also improper. We have lim σ (I + (jω)) (jω) =. () Therefore (7) is not satisfie. Conversely, if I + (s) is improper, then (s) is not biproper but proper. This implies that I (s) is proper, lim ( (jω) ) < p an at least one of the eigen value of lim I (jω) is equal to. We have lim σ (I + (jω)) (jω). () This oes not satisfy (7), thereby completing the proof of Theorem. When I + (s) is biproper, following theorem is satisfie. Theorem 3: If I + (s) is biproper, then following expressions hol: the number of right half plane zeroes of I + (s) is sum of those of the plant G(s) an right half plane poles of the nominal plant G m (s). the number of right half plane poles of I + (s) is sum of those of the plant G(s) an the number of right half plane zeroes of the nominal plant G m (s). Proof of this theorem requires following lemmas. Lemma : Let Ḡ(s) = R n n. If [ A B C D ], where A Ḡ(s) = p, (3) then A si B = n + p. (4) C D A si B The matrix is calle the system matrix of Ḡ(s). C D Lemma : The zeroes of the system consists of the following four elements:. all transmission zeroes of the system. all uncontrollable an unobservable poles of the system 3. one or all uncontrollable an observable poles of the system 4. one or all controllable an unobservable poles of the system Theorem 3 is proven using above lemmas. Proof: Proof is to show. the number of right half plane zeroes of I + (s) is equal to sum of those of the plant G(s) an the number of right half plane poles of the nominal plant G m (s). the number of right half plane poles of I + (s) is equal to that of the plant G(s) an the number of right half plane zeroes of the nominal plant G m (s). For this, it is sufficient only to show. right half plane zeroes of I + (s) are consiste of those of the plant G(s) an right half plane poles of the nominal plant G m (s). That is, for easy explanation, when I + (s) is assume to have no poles in the close right half plane an some zeroes in the close right half plane, the zero of I + (s) in the close right half plane is either a zero of the plant G(s) or a pole of the nominal plant G m (s) will be proven.. right half plane poles of I + (s) are consiste of that of the plant G(s) an right half plane zeroes of the nominal plant G m (s). That is, for easy explanation, when I + (s) is assume to have no zeroes in the close right half plane an some poles in the close right half plane, the pole of I + (s) in the close right half plane is either a pole of the plant G(s) or a zero of the nominal plant G m (s) will be proven. At first it will be shown that right half plane zeroes of I + (s) are that of the plant G(s) or right half plane poles of the nominal plant G m (s). For easy explanation, I + (s) is assume to have no poles in the close right half plane an only some zeroes in the close right half plane. From Theorem, I + (s) is enote as A B I + (s) =, (5) C D where A R n n, B R n p, C R p n an D R p p is nonsingular. Therefore the state space
Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles7 escription of (I + (s))g m (s) is written by A B C m 0 (I + (s))g m (s) = 0 A m B m, (6) C D C m 0 Let s 0 a right half plane zero of I + (s). We have A s 0 I B < n C D + p. (7) From above equation, there exists ξ ξ 0 satisfying A ξ ξ s 0 I B = 0. (8) C D This equation implies s 0 is also a zero of (I + (s)) G m (s). From Lemma, s 0 is either a zero of the plant G(s), uncontrollable poles of (I + (s)) G m (s) or unobservable poles of (I + (s)) G m (s). When s 0 is not a right half plane zero of the plant G(s), s 0 is either a pole of I + (s) or a pole of the nominal plant G m (s). From the assumption that I+ (s) has no poles in the close right half plane, s 0 is a pole of the nominal plant G m (s). Therefore s 0 is either a zero of the plant G(s) or a pole of the nominal plant G m (s). From above iscussion, right half plane zeroes of I + (s) are equivalent to that of the plant G(s) or right half plane poles of the nominal plant G m (s) was shown. Next poles of I + (s) in the close right half plane are consiste of poles of the plant G(s) or zeroes of the nominal plant G m (s) in the close right half plane will be shown. For easy explanation, I + (s) is assume to have no zeroes in the close right half plane an some poles in the close right half plane. Since D is nonsingular, the state space escription of the nominal plant G m (s) is rewritten by G m (s) = (I + (s)) G(s) [ A B = D C B D ] A B C D C 0 A B D C B D C 0 = 0 A s 0 I B.(30) C D C 0 Let s 0 a right half plane pole of I + (s). From [ A B D C s 0 I B D ] C D I B A s = 0 I 0 0 I C D A s = 0 I 0 C D < n + p, (3) s 0 is also a zero of (I + (s)). There exists [ ξ ξ ] 0 satisfying [ A ξ ξ B D C s 0 I B D C D ] = 0. (3) From this equation an (30), for the system matrix of (I + (s)) G(s), we have ξt 0 ξ T A B D C s 0 I B D C 0 0 A s 0 I B C D C 0 T = 0. (33) From (6) an (8), for the system matrix of This implies that s (I + (s)) G m (s), we have 0 is also a zero of (I + (s)) G(s). ξ 0 ξ A Therefore form Lemma, s 0 is either a zero s 0 I B C m 0 of the nominal plant G m (s), uncontrollable poles 0 A m s 0 I B m = 0. (9) of (I + (s)) G(s) or unobservable poles of C D C m 0 (I + (s)) G(s). When s 0 is not a zero of the nominal plant G m (s), s 0 is either a pole of (I + (s)) or a pole of the plant G(s). From the assumption that I + (s) has no poles in the close right half plane, s 0 is a pole of the plant G(s). Therefore s 0 is either a pole of the plant G(s) or a zero of the nominal plant G m (s). From above iscussion, right half plane pole of I + (s) is that of the plant G(s) or right half plane zeroes of the nominal plant G m (s). We have thus complete the proof of this theorem. 4. ROBUST STABILITY CONDITION FOR THE PLANTS HAVING UNCERTAIN NUMBER OF RIGHT HALF PLANE POLES In this section, the robust stability problem in which the number of poles of the nominal plant G m (s) in the right half plane oes not equal that of the plant G(s) is consiere using the results in the previous section. That is, the robust stability conition for the set of plants Ω is consiere. The robust stability conition for the set of plants Ω, is summarize below. Theorem 4: Assume that C(s) is a stabilizing controller for the nominal plant G m (s). C(s) is a robust stabilizing controller for Ω if an only if S(s)W (s). (34) The proof of Theorem 4 requires following lemmas. Lemma 3: Suppose M RH an M > γ. Then there exists a ω 0 > 0 such that for any given ω [0, ω 0 ] there exists a (s) RH with (s) < /γ such that et(i M(s) (s)) has a zero on the imaginary axis [6]. Lemma 4: Let W (s) satisfy (7). The nominal plant G m (s) an the plant G(s) are assume to have p m number of poles in the close right half plane an p number of poles in the close right half plane, respectively. Then, the Nyquist plot of et(i + (s))
7 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.8, NO. February 00 encircles the origin (0, 0) p p m times in the counterclockwise irection. Proof: From the assumption that W (s) satisfies (7) an Theorem, I + (s) is biproper. From Theorem 3, the number of zeroes of I + (s) in the close right half plane is sum of those of the plant G(s) an poles of the nominal plant G m (s) in the close right half plane, an the number of poles of I + (s) in the close right half plane is sum of those of poles of the plant G(s) in the close right half plane an the number of zeroes of the nominal plant G m (s) in the close right half plane. From the assumption that the number of zeroes of the plant G(s) in the close right half plane is equal to that of the nominal plant G m (s) in the close right half plane, let p z the number of zeroes of the plant G(s) in the close right half plane, I + (s) has p m + p z of zeroes in the close right half plane an p + p z of poles in the close right half plane. Accoring to argument principle, the Nyquist plot of et(i + (s)) encircles the origin (0, 0) p + p z (p m + p z ) = p p m times in the counter-clockwise irection. Theorem 4 is proven using above lemmas. Proof: The characteristic matrix of the system () is given by I + G(s)C(s). If the Nyquist plot of et(i + G(s)C(s)) encircles the origin (0, 0) p + p c times in the counter-clockwise irection, then the system () is robustly stable. Here, p c means the number of poles of the controller C(s) in the close right half plane an p means the number of poles of the plant G(s) in the close right half plane. Determinant of characteristic polynomial is written as et (I + G(s)C(s)) = et I + (I + (s)) G m (s)c(s) = et [ I + (s)g m (s)c(s)(i + G m (s)c(s)) (I + G m (s)c(s))] = et [ (I + (s)) I (I + (s)) (s)s(s) (I + G m (s)c(s))] = et (I + (s)) et I (I + (s)) (s)s(s) et (I + G m (s)c(s)). (35) From the assumption that C(s) is a stabilizing controller of the nominal plant G m (s), the Nyquist plot of et(i + G m (s)c(s)) encircles the origin (0, 0) p m + p c times in the counter-clockwise irection, where p m is the number of poles of the nominal plant G m (s) in the close right half plane. Therefore, if the Nyquist plot of et (I + (s)) et ( I (I + (s)) (s)s(s) ) for all (s) Ω encircles the origin (0, 0) p p m times in the counter-clockwise irection, then C(s) is a robust stabilizing controller for the set Ω. From Lemma 4, the Nyquist plot of et(i + (s)) encircles the origin p p m times in the counter-clockwise irection. Therefore, the necessary an sufficient conition that C(s) is a robust stabilizing controller for the set Ω is that the Nyquist plot of et I (I + (s)) (s)s(s) oes not encircle the origin. Finally, the necessary an sufficient conition that the Nyquist plot of et I (I + (s)) (s)s(s) oes not encircle the origin must be proven. The conition is expresse in the same form as (34). It is obvious that if (34) is satisfie, then the Nyquist plot of et I (I + (s)) (s)s(s) oes not encircle the origin any time. From Lemma 3, if (34) oes not hol, then there exist (I + (s)) (s) RH with (I + (s)) (s)/w (s) < to let the Nyquist plot of et I (I + (s)) (s)s(s) across on the origin. Therefore it is prove that the necessary an sufficient conition that the Nyquist plot of et I (I + (s)) (s)s(s) oes not encircle the origin any times is equivalent to (34). From the above iscussion, Theorem 4 is proven. This theorem is very interesting because the robust stability conition is ientical whether or not the number of poles of the nominal plant in the close right half plane is equal to that of the plant. A similar robust stabilizability problem having an uncertain number of right half plane poles was consiere by Verma, Helton, an Jonckheere [5]. Since Verma, Helton, an Jonckheere [5] aopt aitive uncertainty an G(s) = G m (s) + (s) (36) (s) <, (37) the robust stability conition is not similar to that shown in Theorem 4. In aition, the robust stability conition for the system having an uncertain number of right half plane poles is not the same as that having an invariant number of right half plane poles.
Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles73 5. NUMERICAL EXAMPLE In this section, we show a numerical example to illustrate the effectiveness of the propose metho. Consier the problem to esign a robustly stabilizing controller for the set Ω, where G m (s) = an = s + 5 (s + ) (s + ) (s + ) (s + ) s + 4 (s + ) (s + ) (s + ) (s + ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (38) 4 3 0 0 3 0 0 W (s) = 0.77s + 7.5s + 87 s. (39) + 48s + 50 C(s) satisfying (34) is esigne using LMI control toolbox. Next, we confirm that the esigne controller C(s) satisfies (34). The maximum singular value plot of S(s) = (I + G m (s)c(s)) an the gain plot of /W (s) are shown in Fig.. Here, the soli Gain[B].8.6.4. 0.8 0.6 0.4 0. 0 0 0 0 0 0 ω[ra/sec] Fig.: Maximum singular value plot of S(s) an the gain plot of /W (s) line shows maximum singular value plot of S(s) = (I +G m (s)c(s)) an the otte line shows the gain plot of /W (s). Figure shows that the esigne controller C(s) satisfies (34). Let (s) be s + 5 0 (s) = (s + 3) (s ) s + (s + 3) (s ) (s ) 0 0 0 0 0 0 0 0 = 0 0 3 0 0 0 0 0 3 0, (40).5 0 0.5 0 0 0 0.5 0.5 0 0 0 that is, G(s) = = 3 s + 5 (s + ) (s ) (s + ) (s ) s + 4 (s + ) (s ) (s + ) (s ) 0 0 0 0 0 0 0 0 0.0000.0000.0000 0.6667.6667 0.6667 0 0 0 0 0 0 0.0000 0 0 0.6667 0 0. (4) From (38) an (4), the number of poles of the plant G(s) in the close right half plane is not equal to that of the nominal plant G m (s). The fact that (s) in (40) satisfies (7) is confirme by showing the maximum singular value plot of (I + (s)) (s) an the gain plot of W (s) as Fig.. Here, the soli line shows Gain[B] 0 0 0 0 0 30 40 50 0 0 0 0 0 0 ω[ra/sec] Fig.: Gain plot of W (s) an maximum singular value plot of (I + (s)) (s) the maximum singular value plot of (I + (s)) (s)
74 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.8, NO. February 00 an the otte line shows the gain plot of W (s). Figure shows that (s) in (40) satisfies (7). From (38) an (4), the number of zeroes in the close right half plane of the nominal plant G m (s) is equal to that of the plant G(s). In aition, the relative egree of the nominal plant G m (s) is equivalent to that of the plant G(s). Therefore, the plant G(s) in (4) is an element of the set Ω. Using the esigne controller C(s), the response of the output y (t) y(t) = (4) y (t) for the step reference input r(t) = is shown in Fig. y(t) 0.5 0.5.5 0 (43) 3 Here, the soli line shows the 0 4 6 8 0 t[sec] Fig.3: Response of the output y(t) for the step reference input r(t) response of y (t) an the otte line shows that of y (t). Figure 3 shows that the controller C(s) makes the control system in () stable, even if there exists uncertainty with uncertain number of poles in the close right half plane. 6. CONCLUSION In the present paper, the robust stability conition for multiple-input/multiple-output continuous time invariant systems having an uncertain number of right half plane poles was consiere. The robust stability conition presente in the present paper was ientical whether or not the number of poles of the nominal plant is equal to that of the plant in the close right half plane. Uner these conitions, a control system having low-sensitivity characteristics an robust stability can be constructe. When the robust control esign metho in the present paper applies to real plant, the relative egree of real plant is require. In some cases, exact relative egree of real plant cannot be foun. Even if exact relative egree of real plant can not be obtaine, using the iea of parallel fee forwar compensator propose by Iwai et al. [7], the metho of the present paper can be applie to the real plant. References [] J.C. Doyle an G. Stein. Multivariable feeback esign: concepts for a classical moern synthesis, IEEE Trans. on Automatic Cotrol, Vol. 6, pp.4 6, 98. [] J.C. Doyle, J.E. Wall an G. Stein. Performance an robustness analysis for structure uncertainty, Proc. st IEEE conf. CDC, pp. 69 636, 98. [3] M.J.Chen an C.A.Desoer. Necessary an Sufficient conition for robust stability of linear istribute feeback systems, International Journal of Control, Vol. 35, pp.55 67, 98. [4] A.P. Kishore an J.B. Pearson. Uniform stability an preformance in H, Proceeings of the 3st IEEE Conference on Decision an Control, pp. 99 996, 99. [5] M.S.Verma, J.W.Helton an E.A.Jonckheere. Robust stabilization of a family of plants with varying number of right half plane poles, Proc. 986 American Control Conference, pp.87 83, 986. [6] K. Glover an J.C.Doyle. State-space formulae for all stabilizing controllers that satisfy an H norm boun an relations to risk sensitivity, Systems & Control Letters, Vol., pp.67 7, 988. [7] J.C.Doyle, K.Glover, P.P.Khargonekar an B.A.Francis. State-space solution to stanar H an H control problems, IEEE Trans. Automatic Control, Vol. 34, pp. 83 847, 989. [8] H. Kimura. Robust stabilizability for a class of transfer functions, IEEE Transactions on Automatic Control, Vol. 9, pp. 788 793, 984. [9] M.Viyasagar an H.Kimura. Robust controllers for uncertain linear multivariable systems, Automatica, Vol., pp. 85 94, 986. [0] H. Maea, M. Viyasagar. Design of multivariable feeback systems with infinite gain margin an ecoupling, Systems & Control Letters, Vol. 6, pp. 7 30, 985. [] H. Maea, M. Viyasagar. Infinite gain margin problem in multivariable feeback systems, Automatica, Vol., pp. 3 33, 986. [] H. Nogami, H. Maea, M. Viyasagar, S. Koama. Design of high gain feeback system with robust stability, Transactions of the Society of Instrument an Control Engineers in Japan, Vol., pp. 04 00, 986. [3] J.C. Doyle, B. Francis, A. Tannenbaum. Feeback control theory, Macmillan Publishing, 99.
Achievement of low-sensitivity characteristics an robust stability conition for multi-variable systems having an uncertain number of right half plane poles75 [4] K. Yamaa. Robust stabilization for the plants with varying number of unstable poles an low sensitivity characteristics, Proc. 998 American Control Conference, pp. 050 054, 988. [5] D.C.McFarlane an K. Glover. Robust controller esign using normalize coprime factor plant escriptions, Springer-Verlag, 989. [6] K. Zhou, J.C. Doyle, K.Glover. Robust an optimal control, Prentice-Hall, 996. [7] Z. Iwai, I. Mizumoto, H. Ohtsuka. Robust an Simple aaptive Control System Design, International Journal of Aaptive Control an Signal Processing, Vol. 7, pp. 63 8, 993. Kou Yamaa was born in Akita, Japan, in 964. He receive B.S. an M.S. egrees from Yamagata University, Yamagata, Japan, in 987 an 989, respectively, an the Dr. Eng. egree from Osaka University, Osaka, Japan in 997. From 99 to 000, he was with the Department of Electrical an Information Engineering, Yamagata University, Yamagata, Japan, as a research associate. From 000 to 008, he was an associate professor in the Department of Mechanical System Engineering, Gunma University, Gunma, Japan. Since 008, he has been a professor in the Department of Mechanical System Engineering, Gunma University, Gunma, Japan. His research interests inclue robust control, repetitive control, process control an control theory for inverse systems an infinite-imensional systems. Dr. Yamaa receive the 005 Yokoyama Awar in Science an Technology, the 005 Electrical Engineering/Electronics, Computer, Telecommunication, an Information Technology International Conference (ECTI- CON005) Best Paper Awar, the Japanese Ergonomics Society Encouragement Awar for Acaemic Paper in 007, the 008 Electrical Engineering/Electronics, Computer, Telecommunication, an Information Technology International Conference (ECTI-CON008) Best Paper Awar, the 4th International Symposium on Applie Electromagnetics an Mechanics Best Poster Presentation Awar in 009 an the Best Paper Awar from the Japan Society of Applie Electromagnetics an Mechanics in 009. Iwanori Murakami was born in Hokkaio, Japan, in 968. He receive the B.S., M.S an Dr. Eng. egrees from Gunma University, Gunma, Japan in 99, 994 an 997, respectively. Since 997, he has been an assistant professor in the Department of Mechanical System Engineering, Gunma University, Gunma, Japan. His research interests inclue applie electronics, magnetics, mechanics an robotics. Dr. Murakami receive Fourth International Conference on Innovative Computing, Information an Control Best Paper Awar in 009. Shun Yamamoto was born in Gunma, Japan, in 986. He receive a B.S. egree in Mechanical System Engineering from Gunma University, Gunma, Japan, in 009. He is currently M.S. caniate in Mechanical System Engineering at Gunma University. His research interests inclue robust control an process control. Mr. Yamamoto receive Fourth International Conference on Innovative Computing, Information an Control Best Paper Awar in 009. Hieharu Yamamoto was born in Saitama, Japan, in 985. He receive a B.S. egree in Mechanical System Engineering from Gunma University, Gunma, Japan, in 007. He is currently M.S. caniate in Mechanical System Engineering at Gunma University. His research interests inclue process control. Mr. Yamamoto receive the 008 Electrical Engineering/Electronics, Computer, Telecommunication, an Information Technology International Conference (ECTI-CON008) Best Paper Awar. Takaaki Hagiwara was born in Gunma, Japan, in 98. He receive the B.S. an M.S. egrees in Mechanical System Engineering from Gunma University, Gunma Japan, in 006 an 008, respectively. He is currently a octor caniate in Mechanical System Engineering at Gunma University. His research interests inclue process control an PID control. Mr. Hagiwara receive the 008 Electrical Engineering/Electronics, Computer, Telecommunication, an Information Technology International Conference (ECTI-CON008) Best Paper Awar an the 4th International Symposium on Applie Electromagnetics an Mechanics Best Poster Presentation Awar in 009.