THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY STABILIZING CONTROLLERS. Tatsuya Hoshikawa, Kou Yamada and Yuko Tatsumi

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1 International Journal of Innovative Computing, Information Control ICIC International c 206 ISSN Volume 2, Number 2, April 206 pp THE PARAMETERIZATION OF ALL TWO-DEGREES-OF-FREEDOM SEMISTRONGLY STABILIZING CONTROLLERS Tatuya Hohikawa, Kou Yamada Yuko Tatumi Department of Mechanical Sytem Engineering Gunma Univerity -5- Tenjincho, Kiryu , Japan { t ; yamada; t }@gunma-u.ac.jp Received Augut 205; revied December 205 Abtract. A emitrongly tabilizing controller i a tabilizing controller that ha one pole at the origin other pole in the open left-half plane. Uing emitrongly tabilizing controller, we can overcome the problem of trong tabilization, that i, if an uncertainty in the plant or a tep diturbance occur, the output of the control ytem cannot follow the tep reference input without teady tate error. Accordingly, Hohikawa et al. propoed parameterization of all emitrongly tabilizable plant of all emitrongly tabilizing controller. In thi parameterization, we cannot pecify the input-output characteritic the feedback characteritic eparately. One way to achieve thi i to ue a two-degreeof-freedom control ytem. However, the parameterization of all two-degree-of-freedom emitrongly tabilizing controller ha not been examined. The purpoe of thi paper i to propoe uch a parameterization for emitrongly tabilizable plant. Keyword: Strong tabilization, Robut ervo, Semitrong tabilization, Two-degreeof-freedom control ytem, Parameterization. Introduction. In the parameterization problem, all tabilizing controller for a plant [, 2, 3, 4, 5, 6, 7, 8, 9, 0] all plant that can be tabilized [] are ought. Becaue thi parameterization can uccefully earch for all proper tabilizing controller, it i ued a a tool for many control problem. In practical control problem, both the tability of the cloed-loop ytem that of the tabilizing controller are important. In certain cae [2], the intability of tabilizing controller caue poor overall ytem enitivity to variation in plant parameter. On the other h, from [3], even if a plant i tabilizable, the plant i not necearily trongly tabilizable. In addition, the achievable control characteritic i retricted in comparion with the cae uing untable controller. It i thu deirable to chooe either a table controller or an untable one by the required control pecification. Since nontrongly tabilizable plant exit, two neceary ufficient condition that a plant i trongly tabilizable have been clarified. One wa clarified by Youla et al. i called the parity interlacing property, which i a condition on the placement of pole zero of trongly tabilizable plant [4, 3]. They alo propoed a method to find trongly tabilizing controller uing Nevanlinna-Pick interpolation [4, 3]. Thi reult wa developed further in everal paper about the deign method for trongly tabilizing controller [4, 5, 6, 7]. The other condition wa clarified by Hohikawa et al. i the parameterization of all trongly tabilizable plant, which how that trongly tabilizable plant have a common feedback tructure [9]. They alo propoed the parameterization of all trongly tabilizing controller, thu enabling the ytematic deign of trongly tabilizing controller. The trong tabilization problem ha thu been tudied extenively. 357

2 358 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI With trongly tabilizing controller, when there i an uncertainty in the plant or a tep diturbance, the output of the control ytem cannot follow the tep reference input without teady tate error. The reaon i that trongly tabilizing controller cannot have a pole at the origin. If the control require high tracking performance, tabilizing controller require an integrator. Therefore, it i neceary to examine controller deign that have a pole at the origin other pole in the open left-half plane. We call uch controller emitrongly tabilizing controller [9]. Becaue plant that are untabilizable by trongly tabilizing controller exit [3], it i expected that plant that cannot be tabilized by a emitrongly tabilizing controller alo exit. Accordingly, Hohikawa et al. clarified the parameterization of all emitrongly tabilizable plant [9]. In addition, Hohikawa et al. propoed the parameterization of all emitrongly tabilizing controller [20]. However, with their parameterization [20], we cannot pecify the input-output characteritic the feedback characteritic, that i, a diturbance attenuation characteritic robut tability, eparately. When we pecify one characteritic, other characteritic are alo decided. From the practical viewpoint, it i deirable to pecify the input-output characteritic the feedback characteritic eparately. One way to achieve thi i to ue a two-degree-of-freedom control ytem. In addition, becaue a two-degree-of-freedom control ytem can have no overhoot for the reference input, more accurate control can be expected. In thi paper, we propoe the parameterization of all two-degree-of-freedom emitrongly tabilizing controller for emitrongly tabilizable plant, in which the output of the control ytem can follow the tep reference input without teady tate error even if an uncertainty in the plant or a tep diturbance exit. Thi paper i organized a follow. In Section 2, we propoe the concept of a two-degreeof-freedom emitrongly tabilizing controller formulate the problem conidered in thi tudy. In Section 3, we propoe the parameterization of all two-degree-of-freedom emitrongly tabilizing controller for emitrongly tabilizable plant. In Section 4, we preent a deign method for the emitrongly tabilizing controller preented in Section 3. In Section 5, a numerical example i illutrated to how the effectivene of the propoed method. Section 6 give concluding remark. Notation R The et of real number. R() The et of real rational function with. RH U The et of table proper real rational function. The et of unimodular function on RH. That i, U() U implie both U() RH U () RH. 2. Two-Degree-of-Freedom Semitrongly Stabilizing Controller Problem Formulation. Conider the two-degree-of-freedom control ytem hown in Figure, which can pecify the input-output characteritic the feedback characteritic eparately. Here, G() R() i the plant, C() i the two-degree-of-freedom controller: u() i the control input: u() = C() C() = [ C () C 2 () ], () [ r() y() ] = [ C () C 2 () ][ r() y() ], (2) r() i the reference input, d () d 2 () are diturbance, y() i the output. In the following, we call C () R() the feed-forward controller C 2 () R() the feedback

3 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS 359 d () d 2 () r() C() u() G() + y() Figure. Two-degree-of-freedom control ytem controller. From the definition of internal tability [4], when all tranfer function V i () (i =,...,6): [ ] [ ] u() V () V = 2 () V 3 () r() d y() V 4 () V 5 () V 6 () () (3) d 2 () are table, the two-degree-of-freedom control ytem in Figure i table. Semitrongly tabilizing controller were defined in [9] a follow. Definition 2.. (emitrongly tabilizing controller) [9] We call the controller C() a emitrongly tabilizing controller if the tabilizing controller ha only one pole at the origin other pole in the open left-half plane. That i, if C() i: C() = + α Q(), (4) then we call C() a emitrongly tabilizing controller, where α R i any poitive real number Q() RH i any function atifying Q(0) 0. According to Definition 2., the difference between trongly tabilizing controller emitrongly tabilizing controller i whether or not the controller have only one pole at the origin. That i, the characteritic of emitrongly tabilizing controller i to make the output of the control ytem follow the tep reference input without teady tate error in the preence of an uncertainty in the plant or a tep diturbance. In addition, a plant tabilizable by a emitrongly tabilizing controller, o-called emitrongly tabilizable plant, i alo defined in [9] a follow. Definition 2.2. (emitrongly tabilizable plant) [9] We call the plant G() a emitrongly tabilizable plant if G() i tabilizable by a emitrongly tabilizing controller C() in (4). According to [9], the parameterization of all emitrongly tabilizable plant i defined by: where β R i given by: G() = Q 3 () RH i given by: β + Q 2 () ( + α) ( + Q 3 () Q ()Q 2 ()), (5) β = α Q (0), (6) Q 3 () = α βq (), (7)

4 360 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI Q () RH Q 2 () RH are any function atifying Q (0) 0. That i, the plant G() in Figure i decribed by the form of (5). In addition, Hohikawa et al. gave the parameterization of all emitrongly tabilizing controller for emitrongly tabilizable plant in (5) [20]. However, the parameterization of all emitrongly tabilizing controller [20] wa only conidered for one-degree-of-freedom control ytem. Thi mean that all characteritic are pecified for a one-degree-of-freedom emitrongly tabilizing controller. From the practical point of view, it i deirable to pecify the input-output characteritic the feedback characteritic eparately. One way to do thi i to ue a two-degree-of-freedom control ytem in Figure. From thi viewpoint, we conider a two-degree-of-freedom emitrongly tabilizing controller that make the output of the control ytem follow a tep reference input without teady tate error, under the exitence of an uncertainty in the plant or a tep diturbance. The concept of a two-degree-of-freedom emitrongly tabilizing controller i propoed a follow. Definition 2.3. (two-degree-of-freedom emitrongly tabilizing controller) We call the controller C() in () a two-degree-of-freedom emitrongly tabilizing controller if the following expreion hold true.. The feed-forward controller C () in () ha only one pole at the origin. That i, the feed-forward controller C () i defined by: C () = + γ Q f (), (8) where γ R i any poitive real number Q f () RH i any function atifying Q f (0) The feedback controller C 2 () in () work a a emitrongly tabilizing controller. That i, the feedback controller C 2 () i defined in the form of (4). 3. The two-degree-of-freedom control ytem in Figure i table. That i, all tranfer function V i () (i =,...,6) in (3) are table. From Definition 2.3, the feed-forward controller C () alo ha a pole at the origin. Thi mean the tranfer function V 4 () in (3) from the reference input r() to the output y() in Figure cannot have a zero at the origin with the origin pole of the feedback controller C 2 (), which i to enure that the output cannot have a teady tate error for the tep reference input. The problem conidered in thi paper i to obtain the parameterization of all twodegree-of-freedom emitrongly tabilizing controller C() defined in Definition The Parameterization of All Two-Degree-of-Freedom Semitrongly Stabilizing Controller. In thi ection, we propoe the parameterization of all two-degreeof-freedom emitrongly tabilizing controller C() for emitrongly tabilizable plant G() in the form of (5). Thi parameterization i ummarized in the following theorem. Theorem 3.. The parameterization of all two-degree-of-freedom emitrongly tabilizing controller C() for emitrongly tabilizable plant G() in the form of (5) i: C () = + γ Q c () Q u () (9)

5 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS 36 C 2 () = + α Q () + + α + Q c2 () + α Q 2() ) Q c2 (). (0) Here, α R γ R are any poitive real number, Q c () RH i any function, Q c2 () RH i given by: Q u () Q c2 () = β + α +, () + α Q 2() Q u () U i any function that make Q c2 () in () proper atifie: ( i ) ( Q u()) m i = 0 ( i =,..., n), (2) =i i (i =,...,n) are untable zero of β+q 2 (), the multiplicitie of i (i =,...,n) are denoted by m i (i =,..., n). Proof: Firt, the neceity i hown. That i, we how that if the controller C () C 2 () make the control ytem in Figure table, that i all tranfer function V i () (i =,...,6) in (3) are table, then C () C 2 () are defined by (9) (0), repectively. The tranfer function V i () (i =,..., 6) in (3) are: V () = C () + C 2 ()G(), (3) V 2 () = C 2()G() + C 2 ()G(), (4) C 2 () V 3 () = + C 2 ()G(), (5) V 4 () = V 5 () = V 6 () = C ()G() + C 2 ()G(), (6) G() + C 2 ()G(), (7) + C 2 ()G(). (8) From the aumption that all tranfer function in (3) to (8) are table, C 2 () make G() table. From [4], the parameterization of all tabilizing feedback controller i: C 2 () = X() + D() Q(), (9) Y () N() Q() where N() D() are coprime factor of G() on RH atifying: X() RH Y () RH are any function atifying: G() = N() D(), (20) N()X() + D()Y () =, (2)

6 362 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI Q() RH i any function. Therefore, we mut conider the condition to make C 2 () in (9) work a a emitrongly tabilizing controller. Since the emitrongly tabilizable plant G() i defined by the form of (5), when G() in (5) i factorized by (20): N() = β + α + + α Q 2() (22) D() = + Q 3 () Q ()Q 2 (). (23) From (22) (23), a pair of X() Y () atifying (2) are defined by: X() = Q () (24) Y () = Subtituting (22), (23), (24) (25) for (9), we have: + α. (25) C 2 () = Q () + ( + Q 3 () Q ()Q 2 ()) ( Q() β + α + α + ) + α Q 2() Q() = + α Q () + ( + Q 3 () Q ()Q 2 ()) Q() ( β + α + ). (26) + α + α Q 2() Q() From the aumption that C 2 () ha one pole at the origin, Q() become: Q() = + α Q c2(), (27) where Q c2 () RH i any function. Subtituting (7) (27) for (26), we have: ( Q () + + α βq ) () Q ()Q 2 () + α Q c2() C 2 () = ( β + α + α + + ). (28) α Q 2() + α Q c2() By imple manipulation, we have: C 2 () = + α Q () + Q c2 () + α + + ) α Q 2() Q c2 (). (29) We have therefore hown that C 2 () i defined by (0). The remaining problem i to confirm that: C 2 () = Q () + + α + Q c2 () + α Q 2() ) Q c2 () i table. Since Q () RH Q c2 () RH, the condition that (30) i table if only if Q c2 () in (30) reult in: + α + ) + α Q 2() Q c2 () U. (3) (30)

7 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS 363 That i, uing Q u () U, + α + ) + α Q 2() Q c2 () = Q u (). (32) Thi equation correpond to (). Since i (i =,..., n) denote untable zero of β + Q 2 () the multiplicitie of i (i =,...,n) are denoted by m i (i =,...,n), ( β ( i ) m i + α + ) + α Q 2() Q c2 () = 0 ( i =,...,n) (33) = i hold true. From (32) (33), (2) i atified. The fact that Q c2 () in () i included in RH i confirmed a follow: From Q u () U, if Q c2 () in () i untable, then untable pole of Q c2 () are equal to untable zero i (i =,..., n) of β + Q 2 (). Since Q u () atifie (2), untable zero i (i =,..., n) of β + Q 2 () are not equal to untable pole of Q c2 (). Therefore, (β/( + α) + Q 2 ()/( + α))q c2 () i table. That i, when we elect Q u () to make Q c2 () proper, Q c2 () in () i included in RH. In addition, uing Q u (), C 2 () in (30) i rewritten: C 2 () = Q () + Q c2() Q u (). (34) Since Q () RH, Q c2 () RH Q u () U, C2 () RH. In thi way, the fact that C 2 () work a a emitrongly tabilizing controller in (4) i hown. Next, we how that the feed-forward controller C () i decribed by (9). Uing C 2 () in (0), the tranfer function in (3) (6) are: V () = C ()Q u () V 4 () = repectively. From (6), { Q () + α + )} + α Q 2() (35) + α C ()Q u () + α + ) + α Q 2(), (36) Q () + α + 2()) + α =0 Q = 0 (37) in (35) hold true. For V () in (35) V 4 () in (36) to be table, C () RH or C () can have only one pole at the origin ha other pole in the open left-half plane. Therefore, C () in (8) work a a tabilizing controller. To pecify the input-output characteritic the feedback characteritic eparately, C () in (8) become: C () = + γ Q c () Q u (), (38) where, Q c () RH i any function. In thi way, when the plant G() take the form of (5), then C () C 2 () take the form of (9) (0). Thu, the neceity ha been hown. Next, the ufficiency i hown. That i, we how that if C () C 2 () are decribed by (9) (0), C () C 2 () make the control ytem in Figure table. Uing C () in (9) C 2 () in (0), tranfer function V i () (i =,...,6) are written: V () = + γ Q c () { Q () + α + + α Q 2() )}, (39)

8 364 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI V 3 () = + α V 2 () = Q u () { Q () (Q ()Q u () + Q c2 ()) + α + { Q () + α Q 2() + α + V 4 () = + γ + α Q c() + α + ) + α Q 2() V 5 () = + α Q u() + α + + α Q 2() )}, (40) )} + α Q 2(), (4), (42) ), (43) { V 6 () = Q u () Q () + α + )} + α Q 2(). (44) Since Q () RH, Q 2 () RH, Q c () RH, Q u () U, α i a poitive real number, (40), (42), (43) (44) are all table. In addition, ince (37) hold true, (39) (42) have no pole at the origin. From thi becaue Q c2 () RH, (39) (42) are alo table. Thu, the ufficiency ha been hown. We have thu proved Theorem 3.. Next, we explain the control characteritic of the control ytem in Figure uing the parameterization of all two-degree-of-freedom emitrongly tabilizing controller in (9) (0). Firt, the input-output characteritic i hown. The tranfer function from the reference input r() to the output y() i: y() r() = + γ + α Q c() + α + ) + α Q 2(). (45) For the output y() to follow the tep reference input r() without teady tate error: γ β α α Q c(0) = (46) mut be atified. From (6), (46) i rewritten: Therefore, we elect Q c () atifying: γ Q c (0) =. (47) α Q (0) Q c (0) = α γ Q (0). (48) Next, the diturbance attenuation characteritic, which i one of the feedback characteritic, i hown. The tranfer function from the diturbance d () to the output y() from the diturbance d 2 () to the output y() of the control ytem in Figure are: y() d () = + α Q u() + α + + α Q 2() ) (49) { ( y() β d 2 () = Q u() Q () + α + )} + α Q 2(), (50) repectively. Equation (49) how that the tep diturbance d () = / i attenuated effectively. In addition, ince (37) hold true, the tep diturbance d 2 () = / i alo attenuated effectively.

9 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS 365 Furthermore, we find that the input-output characteritic i pecified by Q c () in (9), the diturbance attenuation characteritic i pecified by Q u () in (2). That i, the propoed two-degree-of-freedom emitrongly tabilizing controller can pecify the input-output characteritic the diturbance attenuation characteritic eparately. 4. Deign Method for Q u (). In thi ection, we preent a deign method for Q u () U that atifie (2) make Q c2 () proper.. β/( + α) + Q 2 ()/( + α) i factorized: β + α + + α Q 2() = Q i ()Q o (), (5) where Q i () RH i the inner function atifying Q i (0) = Q o () RH i the outer function. 2. Uing Q o (), Q() RH i deigned: where Q() = q() Q o (), (52) q() = k (τ + ) ǫ, (53) τ R i an arbitrary poitive number, ǫ i an arbitrary poitive integer to make Q() proper, k R i a real number atifying 0 < k <. 3. Uing Q(), Q u () U i deigned: Q u () = + α + ) + α Q 2() Q(). (54) Next, we how that Q u () in (54) atifie (2) make Q c2 () proper. Firt, we how that Q u () in (54) atifie (2). Subtituting (52) for (54), Q u () in (54) i rewritten: Q u () = Q i ()q(). (55) Since i (i =,...,n) are untable zero of β + Q 2 (), m i (i =,..., n) denote multiplicitie of i (i =,...,n), Q i () i an inner function of β/( + α) + Q 2 ()/( + α): ( i ) m i Q i() = 0 ( i =,...,n) (56) = i hold true. From thi equation (53): ( i ) m i Q i()q() = 0 ( i =,..., n) (57) = i are alo atified. From (55) (57), Q u () in (54) atifie (2). Next, we how that Q u () in (54) make Q c2 () proper. Subtituting (54) for (), Q c2 () i rewritten: Q c2 () = Q(). (58) Since Q() RH, Q c2 () i proper. Therefore, Q u () in (54) make Q c2 () proper.

10 366 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI 5. Numerical Example. We provide a numerical example to compare repone of a one-degree-of-freedom control ytem [20] a two-degree-of-freedom control ytem to how the effectivene of the propoed method. The plant conidered in [20] i the angular velocity control of the two-inertia ytem in Figure 2. Here, τ M i the torque of the motor, J M i the moment of inertia of the motor, D M i the coefficient of friction of the motor, J L i the moment of inertia of the load, D L i the coefficient of friction of the load, K i the torional pring contant, ω L i the angular velocity of the load. In [20], J M = , D M = , J L = , D L =.8 0 3, K = 0.4. For thi plant, we deign a two-degree-offreedom emitrongly tabilizing controller, contrat the repone of the one-degreeof-freedom control ytem in [20] our two-degree-of-freedom control ytem. ü M J M K J L! L D M D L The plant in Figure 2 i decribed by: G() = Figure 2. Two-inertia ytem ( + 0.7) ( ). (59) In [20], the plant G() in (59) wa rewritten in the form of (5). Here, α =, β = : Q () = , (60) ( ) Q 2 () = ( )( ), (6) Q 3 () = 0. (62) Firt, we deign the feedback controller C 2 () in (0). To how that the feedback characteritic of the two-degree-of-freedom control ytem can be equal to that of the onedegree-of-freedom control ytem, we et C 2 () equal to C() in [20]. That i, Q i () Q o () in (5) are: Q i () = (63) Q o () = ( + ) ( )( ), (64) repectively. In addition, τ, ǫ, k R in (53) are et to τ = 0.02, ǫ = 3, k = 0.99, repectively. Uing thee parameter, Q u () in (54) C 2 () in (0) are given by: Q u () = ( ) ( ) ( + 50) 3 (65) C 2 () =.36 ( )( ) ( ) ( ), (66)

11 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS 367 repectively. Next, we deign the feed-forward controller C () in (9). Since the tranfer function from the reference input r() to the output y() i decribed by V 4 () in (42), Q c () in (9) i deigned a: Q c () = + α + γ (τ c + ) ǫ c β + α +, (67) + α Q 2() where τ c R i an arbitrary poitive number ǫ c i an arbitrary poitive integer to make Q c () proper. When γ, τ c, ǫ c are et to γ =, τ c = 0.02, ǫ c = 3, Q c () C () in (9) are given by: Q c () =.38 ( )( ) ( + )( + 50) 3 (68) C () =.38 ( )( ) ( ) ( ), (69) repectively. Uing the deigned C () in (69) C 2 () in (66), the repone of the output y(t) for tep diturbance d 2 (t) = of the one-degree-of-freedom control ytem uing C 2 () two-degree-of-freedom control ytem in Figure are hown in Figure 3 Figure 4, repectively. The olid line how the repone of the output y(t) the broken line how that of the tep diturbance d 2 (t) =. Figure 3 Figure 4 how that the tep diturbance d 2 (t) = i attenuated effectively. In addition, we find that the repone of the two-degree-of-freedom control ytem i the ame a that of the one-degree-of-freedom control ytem. On the other h, the repone of the output y(t) for the tep reference input r(t) = of the one-degree-of-freedom control ytem uing C 2 () the two-degree-of-freedom control ytem in Figure are hown in Figure 5 Figure 6, repectively. The olid line how the repone of the output y(t) the broken line how that of the tep reference input r(t) =. Figure 5 Figure 6 how that thee control ytem are table the output y(t) follow the tep reference input r(t) = without teady tate error. In d 2 (t), y(t) d 2 (t) y(t) t[ec] Figure 3. Repone of y(t) with the one-degree-of-freedom control ytem for d 2 (t) =

12 368 T. HOSHIKAWA, K. YAMADA AND Y. TATSUMI d 2 (t), y(t) d 2 (t) y(t) t[ec] Figure 4. Repone of y(t) with the two-degree-of-freedom control ytem for d 2 (t) = r(t), y(t) r(t) y(t) t[ec] Figure 5. Repone of y(t) for the one-degree-of-freedom control ytem for r(t) = r(t), y(t) r(t) y(t) t[ec] Figure 6. Repone of y(t) for the two-degree-of-freedom control ytem for r(t) =

13 PARAMETERIZATION OF 2DOF SEMISTRONGLY STABILIZING CONTROLLERS r(t), y(t) r(t) y(t) t[ec] Figure 7. Enlarged view from 0[ec] to 2[ec] of Figure r(t), y(t) r(t) y(t) t[ec] Figure 8. Enlarged view from 0[ec] to 2[ec] of Figure 6 addition, to compare the repone of Figure 5 Figure 6, enlarged view from 0[ec] to 2[ec] are hown in Figure 7 Figure 8. Figure 7 Figure 8 how that the repone of the two-degree-of-freedom control ytem ha no overhoot the ettling time of the two-degree-of-freedom control ytem i horter than that of the one-degree-of-freedom control ytem. We ee that with the propoed two-degree-of-freedom emitrongly tabilizing controller C(), the diturbance attenuation characteritic of the two-degree-of-freedom control ytem can be the ame a that of the one-degree-of-freedom control ytem the input-output characteritic of the two-degree-of-freedom control ytem can be different from that of the one-degree-of-freedom control ytem. That i, with the propoed controller, we can realize more accurate control for the reference input. 6. Concluion. In thi paper, we have propoed the parameterization of all two-degreeof-freedom emitrongly tabilizing controller for emitrongly tabilizable plant. We then preented a deign method for Q u () U that atifie (2) make Q c2 ()

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